TCER Working Paper Series

Estimating Flexible Functional Forms using Macroeconomic Data
Chihiro Shimizu
Erwin Diewert
Koji Nomura
April 2024
Working Paper E-206
https://www.tcer.or.jp/wp/pdf/e206.pdf

TOKYO CENTER FOR ECONOMIC RESEARCH
1-7-10-703 Iidabashi, Chiyoda-ku, Tokyo 102-0072, Japan©2024 by Chihiro Shimizu, Erwin Diewert and Koji Nomura.
All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without
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Abstract
The paper estimates a flexible functional form for a joint cost function using US
aggregate data for the years 1970-2022. There are four outputs (consumption,
government, investment and exports) and six inputs (imports, labour, machinery andequipment services, structure services, other capital services and land services).
Curvature conditions on the joint cost function are imposed without destroying theflexibility of the functional form. Various elasticities of supply and demand are
estimated along with technical progress bias terms for each input. When using
aggregate time series data based on the System of National Accounts, the paper showsthat it is probably better to estimate a joint cost function rather than a gross
output function or a GDP function. The paper also shows that assuming that an
aggregate production function has constant elasticities of substitution is not
appropriate for the US. Finally, the importance of including land as an aggregate
input is stressed.

Chihiro Shimizu
TCER
andHitotsubashi University
School of Social Data Science
2-1 Naka,Kunitachi,Tokyo,186-8601
c.shimizu@r.hit-u.ac.jp
Koji Nomura
Keio University
Keio Economic Observatory
Minato, Tokyo, 108-8345
nomura@sanken.keio.ac.jp

Erwin Diewert
University of British Columbia
Vancouver School of Economics
Vancouver, B.C., Canada, V6T 1L4.
erwin.diewert@ubc.ca

Estimating Flexible Functional Forms using
Macroeconomic Data
W. Erwin Diewert,∗ Koji Nomura† and Chihiro Shimizu.‡§
April 18, 2024.
Abstract
The paper estimates a flexible functional form for a joint cost function using US aggregate data for the years 1970-2022. There are four outputs (consumption, government,
investment and exports) and six inputs (imports, labour, machinery and equipment
services, structure services, other capital services and land services). Curvature conditions on the joint cost function are imposed without destroying the flexibility of thefunctional form. Various elasticities of supply and demand are estimated along withtechnical progress bias terms for each input. When using aggregate time series data
based on the System of National Accounts, the paper shows that it is probably betterto estimate a joint cost function rather than a gross output function or a GDP function.
The paper also shows that assuming that an aggregate production function has constant elasticities of substitution is not appropriate for the US. Finally, the importanceof including land as an aggregate input is stressed.
Keywords
Production theory, duality theory, production functions, joint cost functions, gross output functions, GDP functions, Shephard’s Lemma, Hotelling’s Lemma, Samuelson’s
Lemma, flexible functional forms, estimation of technical progress, user costs, modelingmonopolistic behavior, land as a factor of production.
JEL Classification Numbers
C01, C02, C32, C43, C51, C82, D24, D42, E01, E23, F11, O47∗ University of British Columbia and University of New South Wales: Email: erwin.diewert@ubc.ca† Keio University; Email: nomura@sanken.keio.ac.jp‡ Hitotsubashi University; Email: c.shimizu@r.hit-u.ac.jp§ The authors thank Ulrich Kohli and Alan Woodland for helpful comments.

1

1 Introduction
It is important to be able to estimate production functions for a large number of policy
purposes. If we attempt to estimate a twice continuously differentiable production function byregressing an output of a production unit on other outputs and inputs used by the production
unit over a time period, we run into multicollinearity problems if the number of other outputs
and inputs is large. If we want the estimated production function to be able to provide asecond order Taylor series approximation to the true production function, we require a large
number of parameters and we soon run into degrees of freedom problems. A partial solution tothese problems is to assume that the production unit is either minimizing the cost of producing
a vector of outputs (which leads to a joint cost function) or maximizing profits subject to one
or more inputs being fixed (which leads to a gross output function if all inputs are “fixed”)
or leads to a GDP function (if all primary inputs are “fixed”). The assumption of optimizing
behavior on the part of the production unit leads to many additional estimating equations andthus helps to overcome the degrees of freedom problem. Moreover, the additional estimating
equations can be obtained by simply differentiating the joint cost function or profit function
which is defined by the form of optimization. Thus the search for functional forms for joint
cost and gross output functions that can provide second order approximations to arbitrary
differentiable joint cost and gross output functions began about 50 years ago.*1
These dual representations of technology sets are important for policy purposes since theylead to estimates of technical change and measures of the biases in technical progress. They
also lead to estimates for elasticities of output supply and for input demand. These elasticities
are useful for a wide range of purposes. Index number methods are available for measuring
Total Factor Productivity and technical progress, as are nonparametric methods, but thesemethods cannot estimate elasticities or biases in technical change.
The early literature on finding flexible functional forms for joint cost functions or gross output
functions found that the estimated functional forms did not satisfy the curvature conditions
that optimizing behavior imposes on the dual representations of technology. For example, ajoint cost function, C(y, w) where y is a vector of outputs that the production unit produces
and w is a vector of input prices that it faces, must be a concave function in input prices
w. Similarly, a gross output function, G(p, x) where p is vector of output prices that the
production unit faces and x is a vector of input quantities, must be convex in output prices p.
In this paper, we will adapt the Normalized Quadratic functional form used by Diewert andWales (1987)[23] (1992)[25] to construct functions C(y, w) and G(p, x) that are flexible andsatisfy curvature conditions.*2 In section 2 below, we will define these functions and derivealternative sets of estimating equations that can be used to estimate technologies.
In section 3, we restrict our attention to the estimation of aggregate production functions
and technologies using macroeconomic data that are available from the national accounts ofa country. We show that using aggregate data, it is better (in terms of fitting the data) touse a Normalized Quadratic Joint Cost Function rather than using a Normalized Quadratic
Gross Output Function. Appendix B provides a proof of the flexibility of the NQ Joint Cost
Function.
*1 This approach to estimating technologies (and preferences) was developed by researchers at the Universityof California at Berkeley in the late 1960s and early 1970s; see McFadden (1966)[47] (1978)[48], Diewert(1971)[9] (1973)[10] (1974a)[11] (1974b)[12] and Christensen, Jorgenson and Lau (1973)[8].
*2 In 1989, Kohli was the first person to adapt the Normalized Quadratic functional form to the GDP
function. His paper was later published as Kohli (1993a)[41]. See also Kohli (1991)[40] for another early
application of the NQ functional form.

2Section 4 estimates a Normalized Quadratic Joint Cost Function for the US using aggregate
data for the years 1970-2022. We have the usual C + G + I + X macro aggregates as our 4
outputs and we constructed data for 6 inputs: M (imports), L (quality adjusted labour), and4 types of capital services, KM&E , KS , KO and KL (Machinery and Equipment, Structures,
Other Capital (mainly R&D and Inventories) and Land). The data are described more fully inAppendix A. We show that land is an important input*3 and that the aggregate US production
function is very far from being a Cobb-Douglas or CES function. We also show that just using
linear trends in each estimating equation to approximate the effects of technological change isa poor approximation.
Section 5 lists our econometric estimates for technical progress and for biases in technicalchange. Our technical progress estimates can be compared with index number methods
for computing Total Factor Productivity growth. The Diewert and Morrison (1986)[22] andKohli (1990)[39] (2003)[43] TFP growth accounting estimates for TFP are listed at the endof Appendix A. The Diewert and Fox (2018)[19] Nonparametric estimates of US TFP are
explained and listed in Appendix C and compared with the econometric estimates for technical progress.*4 Section 5 also lists the average price elasticities of input demand for our 6
aggregate inputs and the average elasticities of inverse output supply over 1970-2022.
For some purposes, it is useful to switch to estimating a GDP function or a Gross Output
function or a Variable Profit function where only a few inputs are fixed and all outputs andvariable inputs are free to vary. It turns out to be possible to get elasticity estimates for thesealternative functions using our estimated joint cost function. We show how this can be donein Appendix D.
Section 6 lists some possible extensions (and possible problems) with our models and Section
7 concludes.

2 Alternative Representations for Production Possibilities Sets
Consider a production unit that can produce M outputs using N inputs over a period of time,
which we take to be a year. Let x ≡ [x1 , . . . , xN ] ≥ 0N be a nonnegative vector of annual
inputs and let y ≡ [y1 , . . . , yM ] ≥ 0M be a nonnegative vector of annual outputs. Define theproduction possibilities set S to be the set of output vectors y that could be produced by
the technology given the availability of the vector of inputs x. Thus (y, x) ∈ S if the output
vector y and be produced by the input vector x. We assume that S is a nonempty, convex,
closed cone*5 that exhibits free disposal of inputs and outputs.
We consider alternative ways for estimating the technology set S using time series data.
Suppose we have information on outputs produced and inputs used for year t, [y t , xt ] ≡t, xt1 , . . . , xtN ] for t = 1, . . . , T . We could attempt to estimate a production function[y1t , . . . , yM
f which gives the maximum amount of output 1 y1t that could be produced conditional on
having available the vector of year t inputs xt and conditional on producing the amounts*3 For the importance of including land in production functions, see Kumhof, Tideman, Hudson and Good-

hart (2021)[45] and Muellbauer (2024)[50].
*4 The advantage of the nonparametric approach is that it distinguishes between technical progress andinefficiency. Of particular interest is the amount of inefficiency that occurred in the US economy during
the oil shocks of the 1970s and the Covid shocks of 2020-2022.
*5 Thus we are assuming that the technology exhibits constant returns to scale. The reason for this strong
assumption is that it proves to be too difficult to distinguish technical progress from returns to scaleusing time series data for a production unit. Our goal in this paper is to estimate aggregate technologies
using time series data.

3tof other outputs:
y2t , y3t , . . . , yMty1t = f (y2t , y3t , . . . , yM
, xt1 , . . . , xtN );t = 1, . . . , T.

(1)

If f is twice continuously differentiable with respect to its arguments, then we would like f to
be a flexible functional form; i.e., we would like f to be able to provide a second order Taylor
series approximation to an arbitrary twice continuously differentiable production function thatsatisfies the appropriate regularity conditions.*6 In the present context where we have assumed
constant returns to scale in production, a candidate function to be a flexible functional form
would require at least (M + N )(M + N − 1)/2 free parameters. If M + N = 10, a flexible
functional form for a production function would have to have at least 45 parameters and so
we would require over 45 years of data on the outputs and inputs produced by the productionunit.
In order to gain more degrees of freedom to enable the econometric estimation of a technology,
it is necessary to assume some form of optimizing behavior on the part of the producer. We
consider some alternative assumptions below.
Suppose the production unit faces the strictly positive output price vector p ≡ [p1 , . . . , pM ]and has the strictly positive input quantity vector x ≡ [x1 , . . . , xN ] at its disposal. Lety ≡ [y1 , . . . , yM ] be a nonnegative output quantity vector and let S be the production unit’s
production possibilities set that satisfies the above regularity conditions. The gross output
function, G(p, x) for this production unit is defined as follows:*7
G(p, x) ≡ maxy {p · y : (y, x) ∈ S}.*8(2)

Thus G(p, x) is the maximum revenue the production unit can generate if it faces output prices
p and uses the input vector x to produce the revenue maximizing output y(p, x) which solves
the constrained optimization problem (2). If intermediate inputs are included in the vector y(indexed by negative signs), then G(p, x) is a value added function or at the national level,
it is a GDP function. The properties of this function were studied by Samuelson (1953)[54],McFadden (1966)[47] (1978)[48], Diewert (1973)[10] (1974a)[11] (1974b)[12] (2022)[17] andothers. We assume constant returns to scale in production so that G(p, x) is linearly homogenous in p for fixed x and is linearly homogeneous in x for fixed p. If G(p, x) is differentiable
at a point (p, x), Hotelling’s Lemma (1932; 594)[28] implies that the vector of output supply
functions regarded as functions of p and x, y(p, x), can be obtained by differentiating G(p, x)
with respect to the components of p:y(p, x) = ∇p G(p, x).

(3)

Suppose further that the production unit faces the strictly positive input price vector w ≡[w1 , . . . , wN ]. Samuelson’s Lemma (1953; 10)[57]*9 implies that the producer’s system of
inverse input demand functions regarded as functions of p and x, w(p, x), can be obtainedby differentiating G(p, x) with respect to the components of x:w(p, x) = ∇x G(p, x).
*6 See Diewert (1974a;113)[11] on the concept of a flexible functional form.
*7 Alternative names for this function are the national product function Samuelson (1953;(4)10)[54], thegross profit function Gorman (1968)[27], the conditional profit function McFadden (1966)[47] (1978)[48],the variable profit function Diewert (1973)[10], the revenue function McFadden (1966)[47], Diewert(1974b)[12] and the GDP function Samuelson (1953)[54], Kohli (1978)[33] (1982)[36] (1991)[40]. These
functions differ from the gross output function by taking some variable inputs (like imports or labour
services) out of the list of inputs and treating them as negative outputs.
∑MT*8 Notation: p · y ≡m=1 pm ym ≡ p y is the inner product between the vectors p and y.
9* See also Diewert (1974a; 140)[11] (2022)[17].

4Instead of conditioning on output prices p and input quantities x, we can condition on output
quantities y and input prices w. Define the production unit’s joint cost function, C(y, w), as
the minimum cost of producing a given output vector y:C(y, w) ≡ minx {w · x : (y, x) ∈ S}.

(5)

Under our strong regularity conditions on the set S, it can be shown that C(y, w) is linearlyhomogeneous in the components of y holding w constant and is linearly homogeneous in thecomponents of w holding y constant. It is also a convex function in the components of yholding w fixed and a concave function in the components of w holding y fixed.*10
If C(y, w) is differentiable at a point y, w, then differentiating C(y, w) with respect to thecomponents of y will generate the vector of marginal costs. If the producer takes output
prices as being fixed and there are competitive markets, then this vector of marginal costs
will be equal to the vector of selling prices p. Thus the production unit’s system of inverse
output supply functions, p(y, w), can be obtained by differentiating C(y, w) with respect tothe components of y:p(y, w) = ∇y C(y, w).
(6)
Shephard’s Lemma (1953; 11)[57] (1970)[58] implies that the production unit’s system of input
demand functions regarded as functions of y and w, x(y, w), can be obtained by differentiating
C(y, w) with respect to the components of w:x(y, w) = ∇w C(y, w).

(7)

Thus we have two alternative methods for estimating a representation of the technology set
S: the first representation assumes a functional form for the gross output function, G(p, x),
and uses equations (3) and (4) as estimating equations and the second representation assumes
a functional form for the joint cost function, C(y, w), and uses equations (6) and (7) as
estimating equations. Note that both representations lead to (M + N )T estimating equations
if we have data on prices and quantities for T periods. This greatly increases degrees of
freedom compared to the degrees of freedom that are available when estimating a production
function.
In the following section, we ask whether we can choose between (3) plus (4) or (6) plus (7)when working with macroeconomic data.

3 Should We Estimate Gross Output Functions or Joint Cost
Functions?
Consider the following specialization of the general joint cost function C(y, w) defined by (5):C(y, w) ≡

∑M

m=1 c

m

(w)ym .

(8)

The function cm (w) is the unit cost function that is dual to the single output constant returns
to scale production function for sector m, ym = f m (xm ) for m = 1, . . . , M where xm is thevector of inputs used by sector m. This is the small open country framework considered by*10 See McFadden (1966)[47] (1978)[48] and Diewert (1974a)[11] (2022)[17] for references to the literature.

The first flexible functional form for a joint cost function that was estimated empirically was the translog
joint cost function introduced by Burgess (1974)[6]. The problem with this functional form is the difficultyin imposing curvature conditions on it without destroying its flexibility.

5Samuelson (1953)[54] in his seminal paper. It is sometimes called the Nonjoint Production
Model: each sector of the economy produces only a single product (or a single group of
products that does not overlap with the products produced in other sectors) and it does not
use the products of other sectors as intermediate inputs. Note that the aggregate input vector∑M
x is equal to m=1 xm and the mth output price is equal to pm = cm (w) for m = 1, . . . , M . Ifeach unit cost function is differentiable, then by applying Shephard’s Lemma to each sectoral
cost function, we can deduce that xm = ∇w cm (w)ym for m = 1, . . . , M . Making use of theseequalities, differentiate both sides of (8) with respect to the components of w. We obtain:∇w C(y, w) =

∑M

∑M
m
m
m=1 ∇w c (w)ym =m=1 x ≡ x.

(9)

Now further specialize the unit cost functions cm (w) to be linear functions of w:
cm (w) ≡∑N

n=1 wn dnm ;m = 1, . . . , M(10)where the dnm are N M constants. This means that the sectoral production functions are
Leontief (no substitution) production functions. Define the N by M matrix of the dnm as
D ≡ [dnm ]. Substitute definitions (10) into (8) and we obtain the following expression for theoverall joint cost function:
C(y, w) ≡

∑M

m=1

∑N

n=1 wn dnm ym = w · Dy.

(11)

Thus the joint cost function for this very special case of Leontief sectoral production functions
turns out to be a bilinear form in the vectors of input prices w and of output quantities y.
Equations (6) and (7) for this special case turn out to be the following estimating equations
for t = 1, . . . , T :pt = D T w t ;t

x = Dy

t

(12)
(13)

where pt , wt , y t and xt are the period t vectors of observed prices and quantities and DT is
the transpose of the matrix D. The estimating equations defined by (12) and (13) are linear
in the unknown M N parameters but of course, there are cross equation equality restrictions.
However, this model can readily be estimated using standard nonlinear regression packages.
Consider the following specializations of the general gross output function G(p, x) defined by
(2):
∑M

m
m=1 g (x)pm ;∑N
G(p, x) ≡ n=1 hn (p)xn ;∑M ∑N
G(p, x) ≡ m=1 n=1 xn enm pm = x · Ep

G(p, x) ≡

(14)
(15)
(16)

where E ≡ [enm ] is an N by M matrix of constants. The production model defined by (14)implies that output ym is equal to the function g m (x) of aggregate input x for each output
m = 1, . . . , M . In the context where outputs are produced by sectoral production functions,this model is not very sensible if M > 1: the output of sector m is produced by the sector mvector of inputs, xm ; not by the aggregate input vector x.
The production model defined by (15) implies that each unit of aggregate input n, xn , produces
the vector of outputs ∇p hn (p) independently of all other inputs. This is also not a very sensible
assumption if N > 1. Hence the bilinear model defined by (16), which is a special case of the6models defined by (14) and (15), is also not a sensible model of production in the sectoral
production function context.*11
Using (3) and (4), the estimating equations for the bilinear Gross Output function defined by(16) are as follows for t = 1, . . . , T :y t = ET xt ;ttw = Ep .

(17)
(18)

When we are estimating joint cost functions, it is convenient to start by estimating the bilinear
function defined by (11), C(y, w) = w ·Dy. When we are estimating Gross Output Functions,
it proves to be convenient to start by estimating the bilinear function defined by (16), G(p, x) =x · Ep. The analysis above suggests that the Bilinear Joint Cost Function option will fit thedata better than the Bilinear Gross Output Function option when we are using aggregate
national accounts data. The bilinear starting functional form defined by (11) will probably
provide a much better global approximation to the national technology if we are using national
macroeconomic data than will be provided by the starting bilinear functional form for a gross
output function defined by (16).
We will use US data for the years 1970-2022 to investigate which bilinear functional form does
better at describing the data: the Gross Output Bilinear Function or the Joint Cost Bilinear
Function.
Our data are based on the Augmented Productivity Database (APDB) for the US constructedby the Asian Productivity Organization and Keio University.*12 The data on outputs for the
US economy are based on the recent historical revision of the US national accounts made by
the Bureau of Economic Analysis (2023)[5]. The four outputs are the usual C, G, I and X
macroeconomic aggregates and the six inputs are Imports, Labour, Machinery and Equipment,
Structures, Other Capital (mainly R&D and Inventories) and Land (M, L, KM&E , KS , KO andKL ); see Appendix A for the details of the data construction. Denote the 4 dimensional output
price and quantity vectors for year t by pt and y t and the 6 dimensional input price andquantity vectors for year t by wt and xt for t = 1970, . . . , 2022.
We first estimated the parameters in the 10 equations that allow us to the Bilinear Gross
Output Function. The first 6 estimating equations were equations (18) and the last 4 equations
were equations (17). Although each set of equations is linear in the unknown parameters, there
are cross equation restrictions on the parameters which need to be taken into account. We used
the Nonlinear Estimation econometric package in Shazam*13 to do the estimation. The R2(between observed and predicted) were as follows for the 10 equations: 0.9159 0.9845 0.3454
0.9045 0.8703 0.5250 0.9981 0.4708 0.9675 0.9662. The equation with the lowest R2 (0.3454)
was the third equation in (18) which regressed the user cost of Machinery and Equipment on
the 4 prices of the outputs (the prices of C, G, I and X). The equation with the highest R2(0.9981) was the first equation in (17) which regressed the quantity of consumption on the 6aggregate quantities of inputs.
*11 There is an extensive literature on alternative specifications of joint and nonjoint production; see Samuel-son (1966)[56] and Kohli (1981a)[34] (1983a)[37] (1991; 42-46)[40] (1993b)[42] (2005)[44]. Our purpose is
not to test for the validity of alternative specifications of joint production: we simply wish to determine
whether a joint cost function or a gross output function will fit the data better, given that a bilinear
functional form is the initial functional form. Our paper does not answer this question definitively; i.e.,
which specification of technology is “best”.
*12 See the Asian Productivity Organization (2022)[1] for a description of the data base. The detailed database is not available to the public. However, the US data used in this study is listed in Appendix A.
*13 See White (2004)[59].

7We also estimated the 10 equations that estimate the parameters in the Bilinear Joint Cost
Function. The first 6 estimating equations were equations (13) and the last 4 equations were
equations (12). The R2 were as follows for the 10 estimating equations: 0.9908 0.9777 0.9740
0.9707 0.9506 0.9809 0.9895 0.9980 0.9930 0.9906. The equation with the lowest R2 (0.9506)
was the fifth equation in equations (13) which regressed the quantity of input 5 (the quantity
of Other Capital Services) on the quantities of the 4 outputs. The equation with the highest
R2 (0.9980) regressed the price of Government output on the 6 input prices.
It is clear that the Bilinear Joint Cost function fits the US data much better than the Bilinear
Gross Output function. These two bilinear functional forms are building blocks for the Normalized Quadratic Joint Cost and Gross Output functional forms which are flexible functional
forms as we shall see. Thus it is likely that we will end up with a functional form that fits
the data better if we estimate the Normalized Quadratic Joint Cost Function rather than theNormalized Quadratic Gross Output Function (because the bilinear special case for the joint
cost function provides a global fit which is much better than the fit for the bilinear special
case of the Normalized Quadratic Gross Output Function). In the following sections, we will
estimate several variants of the Normalized Quadratic Joint Cost Function.

4 Estimating the Normalized Quadratic Joint Cost Function for the
US
In the previous section, we defined pt and y t as the year t price and quantity vectors for our
4 gross outputs (C + G + I + X) and wt and xt were defined as the year t price and quantity
vectors for our 6 inputs (Imports, Labour, M&E services, Structure services, Other Capital
services (mainly R&D and Inventories) and Land). We normalized the prices and quantities of1970
the 4 outputs so that ym= 1 for m = 1, 2, 3, 4. We also normalized the prices and quantitiesof the 6 inputs so that wn1970 = 1 for n = 1, . . . , 6. The Normalized Quadratic Joint Cost
Function requires a vector of fixed nonnegative weights for input prices, α ≡ [α 1 , α 2 , α 3 ,∑2022
α 4 , α 5 , α 6 ]. Define x∗ ≡ t=1970 xt = [x∗1 , x∗2 , x∗3 , x∗4 , x∗5 , x∗6 ]. Thus x∗ is simply the sample
wide sum of the observed input quantity vectors. Define the sum of the x∗n as x∗sum . Finallydefine the components of the α vector as αn ≡ x∗n /x∗sum for n = 1, . . . , 6. Since all input
prices wnt equal 1 when t equals 1, we have α · w1970 = 1. Thus α · wt as t varies is a fixedbase index of input prices that starts out at the level 1 in 1970.
The Normalized Quadratic Joint Cost Function also requires a ∑vector of fixed nonnegative2022∗
weights for output quantities, β ≡ [β1 , β2 , β3 , β4 ]. Define p ≡ t=1970 pt = [p∗1 , p∗2 , p∗3 , p∗4 ].
Thus p∗ is simply the sample wide sum of the observed output price vectors. Define thesum of the p∗m as p∗sum . Finally define the components of the β vector as βm ≡ p∗m /p∗sum fort
m = 1, . . . , 4. Since all output quantities ym
equal 1 when t equals 1, we have β · y 1970 = 1.
Thus β · y t as t varies is a fixed base index of output quantities that starts out at the level 1
in 1970.
In order to condense the notation, we now let time t = 0, 1, 2, . . . , 52 instead oft = 1970, 1971, . . . , 2022. The Linear Time Trends Normalized Quadratic Joint Cost Function
at time period t, C(y t , wt , t), is defined as follows:C(y t , wt , t) ≡ (1/2)(wt · Awt )(α · wt )−1 (β · y t ) + (1/2)(y t · By t )(α · wt )(β · y t )−1+ wt · Dy t + (b · wt )(β · y t )t(19)

where A is a symmetric positive semidefinite matrix that satisfies the restrictions Aw0 = 06 ,B is a symmetric negative semidefinite matrix that satisfies the restrictions By 0 = 04 and8b ≡ [b1 , . . . , b6 ] is a vector of parameters which allow for biased technical change. Using thealgebra in Diewert and Wales (1987)[23], it can be shown that these conditions on the A and B
matrices are sufficient to imply the global convexity of C(y, w, t) in the components of y andthe global concavity of C(y, w, t) in the components of w. The year t estimating equations(7) and (6) become for t = 0, 1, 2, . . . , 52:
xt = Dy t + Awt (α · wt )−1 (β · y t ) + (1/2)(y t · By t )(β · y t )−1 α− (1/2)(wt · Awt )(α · wt )−2 (β · y t )α + b(β · y t )t;p = D w + By (α · w )(β · y )
t

T

t

t

t

t

t

t −2

− (1/2)(y · By )(β · y )

t −1

(20)

+ (1/2)(w · Aw )(α · w )tt(α · w )β + β(b · w )t.
t

t

t −1

β
(21)

The unknown parameters in the above equations are the components of the 6 by 4 matrix D,
the components of the 6 by 6 and 4 by 4 symmetric matrices A and B and the 6 components ofthe vector of technical progress parameters b which is 51 parameters in all, taking into accountthe symmetry restrictions on the A and B matrices. There are 53 times 10 observations so
we have 530 degrees of freedom. Note that all parameters appear in a linear fashion but
there are cross equations constraints on the parameters which means that we have to use
nonlinear regression techniques to estimate the unknown parameters. Note that when t = 0,
the technical progress terms involving the b vector on the right hand sides of (20) and (21)vanish.
It proved to be too difficult for the nonlinear estimation command in Shazam to estimate
the model defined by equations (20) and (21). Nonlinear estimation requires good starting
values for the parameters in the nonlinear regression. In the previous section, we essentially
set A, B and b equal to zero matrices and estimated the components of the D matrix. In thissection, we used the estimates for the components of the D matrix as starting coefficients forthe estimation of (20) and (21) with A and B set equal to zero matrices. The starting values
for the components of the b vector were zeros. The log likelihood for the new model thatallowed for technical progress increased by 238.85 log likelihood points for adding 6 technicalprogress parameters. Thus it is extremely important to allow for technical progress. The R2
for the 10 equations were as follows:
0.9891 0.9576 0.9868 0.9952 0.9967 0.8291 0.9980 0.9964 0.9705 0.9466.
We turn our attention to the estimation of the components of the input substitution matrix
A and the (inverse) output substitution matrix B. We need to ensure that our estimated A
matrix is a negative semidefinite symmetric matrix that satisfies Aw0 = A16 = 06 where 16
is a vector of ones of dimension 6.
The imposition of symmetry and negative semidefiniteness on A can be accomplished using atechnique due to Wiley, Schmidt and Bramble (1973)[60]: simply replace the matrix A by
A ≡ −UUT(22)

where U is a 6 by 6 lower triangular matrix ; i.e., uij = 0 if i < j.
The restrictions Aw0 = A16 = 06 on A can be imposed if we impose the following restrictions
on U:
U T 1 6 = 06 .
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The imposition of symmetry and positive semidefiniteness on B can be accomplished in asimilar fashion: set B equal to:
B ≡ VVT(24)
9where V is a 4 by 4 lower triangular matrix ; i.e., vij = 0 if i < j.
The restrictions By 0 = B14 = 04 on B can be imposed if we impose the following restrictions
on V:
VT 14 = 04 .
(25)
The restrictions (23) and (24) imply that the maximum rank for the A and B matrices is 5
and 3 respectively. Once the matrices A and B in the estimating equations (20) and (21) are
replaced by −UUT and VVT respectively, the resulting estimating equations are no longer
linear in the unknown parameters and a nonlinear regression package must be used. For moredetails on how to implement this method for imposing the restrictions on the substitution
matrices, see Diewert and Wales (1987)[23] (1988)[24] (1992)[25].
Setting A ≡ −UUT and B ≡ VVT can lead to difficulties in estimation due to the addition ofa large number of new parameters at the same time. A more fool proof method of proceeding
is to introduce the columns of U and V into the regression one column∑6 at a time. Thus weT
T
set A ≡ −uu where u ≡ [u1 , u2 , . . . , u6 ] and u1 is set equal to − n=2 un . Similarly, we∑4
set B = vv T where v T ≡ [v1 , v2 , v3 , v4 ] and v1 is set equal to − m=2 vm . We used the final
coefficients in the previous regression as starting values in this new regression along with thestarting values for the un and vm equal to 0. Thus this new nonlinear regression estimates
rank 1 substitution matrices.*14 The log likelihood for the new model increased by 76.088 log
likelihood points for adding 5 input and 3 output substitution parameters. The R2 for the 10
equations were as follows:
0.9954 0.9852 0.9866 0.9925 0.9984 0.9214 0.9717 0.9971 0.9852 0.9624.
Note that the R2 for equation 6 (the demand for land services) was 0.8291 in the previous nonlinear regression model but has now increased to 0.9214 when we allow for some substitution
between the 6 inputs.
Looking at the components of the output substitution matrix, it turned out that the components of the estimated v vector were as follows (v2∗ , v3∗ , v4∗ ) = (−0.0114, −0.1454, 0.0108)
and v1∗ = −(v2∗ + v3∗ + v4∗ ) = 0.1460. If the products produced by the 4 sectors in our model(the C, G, I and X sectors) were unique to each sector (i.e., if the Nonjoint Production Modeldefined by (8) above held), then the estimated v vector should be a zero vector. Since theestimated (v2∗ , v3∗ , v4∗ ) were significantly different from zero, it is likely that there is some joint
production in the US economy.*15
The process of adding an additional column to the input and output substitution matrices U
and V (and then running a new nonlinear regression using the final coefficients in the previous
regression as starting values in the new regression) continued until a rank 5 input substitution
matrix and a rank 3 output substitution matrix were estimated. The resulting model is themodel described by equations (20) - (25). The log likelihood for this 51 parameter model was
27.867 points higher than the final log likelihood for the model that had rank 1 input andoutput substitution matrices. We have added 13 new parameters to the previous rank 1 model.
However, 4 of the new substitution parameters converged to 0: the data only supported theestimation of a rank 4 input substitution matrix (maximum rank is 5) and a rank 1 output
substitution matrix (maximum rank is 3).
We need to develop formulae for various elasticities of substitution once we have estimates forthe components of A, B, D and b. Define the vector of input demand functions x(y t , wt , t) as
the right hand side of equations (20). The 6 by 6 matrix of derivatives of the input demand*14 There are now 38 parameters in our model.
*15 The standard errors for (v ∗ , v ∗ , v ∗ ) were (0.0229, 0.0187, 0.0052).
2 3 4

10

functions with respect to the components of the input price vector w evaluated at the year t
values for y and x, ∇w x(y t , wt , t), can be obtained by differentiating the right hand side of(20) with respect to the components of w:∇w x(y t , wt , t) = A(α · wt )−1 (β · y t ) − (Awt )α T (α · wt )−2 (β · y t )− α (Awt )T (α · wt )−2 (β · y t ) + α α T (wt · Awt )(α · wt )−3 (β · y t ).

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Define the year t elasticity of the demand for input n with respect to a change in the price of
input j, Extnj , as follows for n = 1, . . . , 6 and j = 1, . . . , 6:
Extnj ≡ ∂ ln xn (y t , wt , t)/∂ ln wj ≡ [∂xn (y t , wt , t)/∂wj ][wjt /xtn ](27)

where xtn ≡ xn (y t , wt , t) is the fitted demand for input n in year t defined by the nth component on the right hand side of equation (20) for year t and ∂xn (y t , wt , t)/∂wj is defined inequations (26).
Recall that when t = 0, w0 and A satisfy the equation Aw0 = 06 . Thus when t = 0, the righthand side of (26) when t = 0 collapses to ∇w x(y 0 , w0 , 0) = A(α · w0 )−1 (β · y 0 ). As timemarches on, the last 3 terms on the right hand side of (26) will tend to be fairly close to zeroso the elasticities defined by (27) will be approximately equal to the following expressions forn = 1, . . . , 6 and j = 1, . . . , 6:
Extnj ≈ anj (wjt /α · wt )/(xtn /β · y t )(28)

where anj is the component of the estimated A matrix in row n and column j. If there aredivergent trends in either the input prices wt or in the output quantities y t over time, thenit can be seen that the input substitution elasticities will have trends over time.
The problem of trending elasticities also arises with respect to output substitution elasticities.
Define the vector of (inverse) output supply functions p(y t , wt , t) as the right hand side ofequations (21). The 4 by 4 matrix of derivatives of the output supply functions with respect
to the components of the output quantity vector y evaluated at the year t values for y and x,∇y p(y t , wt , t), can be obtained by differentiating the right hand side of (21) with respect tothe components of y:∇y p(y t , wt , t) = B(α · wt )(β · y t )−1 − (By t )β T (α · wt )(β · y t )−2− β(By t )T (α · wt )(β · y t )−2 + ββ T (y t · By t )(α · wt )(β · y t )−3 .

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Define the year t elasticity of the (inverse) supply for output m with respect to a change inthe price of output k, Eptmk , as follows for m = 1, . . . , 4 and k = 1, . . . , 4:
Eptmk ≡ ∂ ln pm (y t , wt , t)/∂ ln yk ≡ [∂pm (y t , wt , t)/∂yk ][ykt /ptm ](30)

where ptm ≡ pm (y t , wt , t) is the fitted price for output m in year t defined by the mth component on the right hand side of equation (21) for year t and ∂pm (y t , wt , t)/∂yk is defined inequations (29).
Recall that when t = 0, y 0 and B satisfy the equation By 0 = 04 . Thus when t = 0, the righthand side of (29) when t = 0 collapses to ∇y p(y 0 , w0 , 0) = B(α · wt )(β · y t )−1 . The last
3 terms on the right hand side of (29) will tend to be fairly close to zero so the elasticities
defined by (30) will be approximately equal to the following expressions for m = 1, . . . , 4 andk = 1, . . . , 4:
Eptmk ≈ bmk (ykt /β · y t )/(ptm /α · wt )(31)
11

where bmk is the component of the estimated B matrix in row m and column k. If there aredivergent trends in either the input prices wt or in the output quantities y t over time, thenit can be seen that the output substitution elasticities will have trends over time.
Diewert and Lawrence (2002; 149-151)[21] noticed the above trending elasticities problem inthe context of estimating a Normalized Quadratic GDP function and they suggested a methodfor dealing with this problem: let the components of the A and B (or U and V) matrices
have linear time trends over the sample period. We implemented their method for our US
data. Thus the matrices A and B in the estimating equations (20) and (21) become functionsof time t, A(t) and B(t).*16 This new model has a total of 72 parameters. The log likelihoodfor this model increased by 78.021 points for adding 21 new parameters. The new R2 were as
follows:
0.9904 0.9914 0.9829 0.9875 0.9980 0.9383 0.9988 0.9982 0.9925 0.9745.
The lowest R2 was for the demand for land services equation which was 0.9383; the R2 for theremaining equations were all above 0.97. Ten out of the 42 substitution matrix parametersconverged to 0. Thus the ranks of the beginning and end of period input substitution matrices
were equal to 3 (maximum possible rank is 5) and the initial inverse output supply substitution
matrix had rank 1 and the final output substitution matrix had rank 2 (maximum possible
rank is 3).
The use of linear time trends to model technical progress will lead to smooth estimates ofoverall technical progress. But we know from index number estimates of technical progressthat the rate varies substantially from year to year and the trends also vary substantially fromdecade to decade.*17 Thus we will follow the example of Diewert and Wales (1992)[25] by
replacing the linear trends in equations 1-6 by piece-wise linear spline functions. In order todetermine the break points (or knots) for the spline functions, we looked at the plots of theregression residuals. These residual plots showed systematic trends that changed abruptly
from time to time. We used the observations where breaks in the residual trends occurredin equations 1-6 as our break points for the piece-wise linear functions that we use to modeltechnical progress. The break points occurred at the following observations where we now
number the years 1970-2022 as 1-53:
Equation 1: 8 18 23 37 51
Equation 2: 8 13 17 22 36 51
Equation 3: 10 17 26 37 40 51
Equation 4: 13 27 31 34 38 50
Equation 5: 5 17 22 31 35 51
Equation 6: 19 22 31 37.
The terms tbn for n = 1, . . . , 6 appear in the estimating equations (20) and (21). We need toreplace these linear functions of time t by piece-wise linear functions of time where the line
segments are linked together to form∑
continuous functions of time. Thus the linear function of
6time, tb1 , is replaced by the function i=1 b1i f1i (t) where the b1i are constants to be estimatedand the functions f1i (t) of time t are defined below:
f11 (t) ≡ t for t = 0, 1, . . . , 7; f11 (t) ≡ 7 for t = 8, 9, . . . , 52;
f12 (t) ≡ 0 for t = 0, 1, . . . , 7; f12 (t) ≡ t − 7 for t = 8, 9, . . . , 17;f12 (t) ≡ f12 (17) for t = 18, 19, . . . , 52;*16 Thus A(t) ≡ [1 − (t/52)]U0 U0T + [t/52]U1 U1T where U0 and U1 are lower triangular matrices thatsatisfy U0T 16 = 06 and U1T 16 = 06 and B(t) ≡ [1 − (t/52)]V0 V0T + [t/52]V1 V1T where V0 and V1
are lower triangular matrices that satisfy V0T 14 = 04 and V1T 14 = 04 .
*17 See Appendix C.

12

f13 (t) ≡ 0 for t = 0, 1, . . . , 17; f13 (t) ≡ t − 17 for t = 18, . . . , 22;
f13 (t) ≡ f13 (22) for t = 23, . . . , 52;
f14 (t) ≡ 0 for t = 0, 1, . . . , 22; f14 (t) ≡ t − 22 for t = 23, . . . , 36;
f14 (t) ≡ f14 (36) for t = 37, . . . , 52;
f15 (t) ≡ 0 for t = 0, 1, . . . , 36; f15 (t) ≡ t − 36 for t = 37, . . . , 50;
f15 (t) ≡ f15 (50) for t = 51, 52;
f16 (t) ≡ 0 for t = 0, 1, . . . , 50; f16 (t) ≡ t − 50 for t = 51, 52.
Note how the break points∑for equation 1, 8 18 23 37 51 appear in the above definitions for6
the f1i (t). Note also that i=1 f1i (t) = t for t = 0, 1, . . . , 52.
∑7
The linear function of time tb2 is replaced by the function i=1 b2i f2i (t) where the b2i areconstants to be estimated and the functions f2i (t) of time t are defined below using the breakpoints for equation 2:
f21 (t) ≡ t for t = 0, 1, . . . , 7; f21 (t) ≡ 7 for t = 8, 9, . . . , 52;
f22 (t) ≡ 0 for t = 0, 1, . . . , 7; f22 (t) ≡ t − 7 for t = 8, 9, . . . , 12;
f22 (t) ≡ f22 (12) for t = 13, . . . , 52;
f23 (t) ≡ 0 for t = 0, 1, . . . , 12; f23 (t) ≡ t − 12 for t = 13, . . . , 16;
f23 (t) ≡ f23 (16) for t = 17, . . . , 52;
f24 (t) ≡ 0 for t = 0, 1, . . . , 16; f24 (t) ≡ t − 16 for t = 17, . . . , 21;f24 (t) ≡ f24 (21) for t = 22, . . . , 52;
f25 (t) ≡ 0 for t = 0, 1, . . . , 21; f25 (t) ≡ t − 21 for t = 22, . . . , 35;
f25 (t) ≡ f25 (35) for t = 36, . . . , 52
f26 (t) ≡ 0 for t = 0, 1, . . . , 35; f26 (t) ≡ t − 35 for t = 36, . . . , 50;
f26 (t) ≡ f26 (36) for t = 51, 52;
f27 (t) ≡ 0 for t = 0, 1, . . . , 50; f27 (t) ≡ t − 50 for t = 51, 52.
The spline functions f3i (t), f4i (t) and f5i (t) for i = 1, . . . , 7 and f6i (t) for i = 1, . . . , 5 were
defined in a similar fashion using the break points∑ listed above. Thus replace each tbn termin equations (20) by the linear spline function i bni fni (t) for n = 1, . . . , 6. This adds 33
additional parameters to the previous model for a total of 105 parameters. In equations(21), the scalar term (b · wt )t appears on the right hand side. It needs to be replaced with the∑6∑∑∑following terms: w1t i=1 b1i f1i (t)+w2t 7i=1 b2i f2i (t)+w3t 7i=1 b3i f3i (t)+w4t 7i=1 b4i f4i (t)+∑7∑5w5t i=1 b5i f5i (t) + w6t i=1 b6i f6i (t).
We used the final coefficients in the previous nonlinear regression as starting values in the
new regression with 105 unknown parameters. The starting coefficient for each bni was set
equal to the final coefficient for bn in the previous regression for n = 1, . . . , 6. Somewhat
surprisingly, Shazam had no difficulty in converging to new estimates; the nonlinear option
took only 2.8 seconds with 700 iterations. The log likelihood increased by 273.11 points foradding 33 technical progress parameters. The R2 for the 10 equations were as follows:
0.9991 0.9978 0.9962 0.9984 0.9987 0.9901 0.9996 0.9994 0.9981 0.9791.
Our final Joint Cost Function model fits the data very well indeed. However, there are a large
number of parameters in our final model. In the following section, we will see if the elasticities
and the estimates for technical progress that the model generates are “reasonable”.*18*18 Standard errors for our estimated coefficients and our Shazam codes are available on demand.

13

5 Estimates for Elasticities and Biases in Technical Progress
In the previous section, we derived formula (27) (which drew on (26)) for the year t elasticity of demand for input n with respect to a change in the price of input j, Extnj ≡[∂xn (y t , wt , t)/∂wj ][wjt /xtn ]. This formula was derived using linear time trends to describe
technical progress but the same formula is also valid for year t using our spline model for technical progress provided that we replace the constant matrix A by the weighted average matrix
for year t, A(t). Thus we use our estimated coefficients to calculate these input elasticities of
input demand for each year. The resulting averages of these elasticities are listed in Table 1
below. The main diagonal elasticities for each year are listed in Appendix D.
Table 1 Average Price Elasticities of Input Demand over the Period 1970-2022
Exnj
x1
M
x2
L
x3 KM&E
x4 KS
x5 KO
x6 KL

w1
w2
w3
w4
w5
w6
M
L
KM&E
KS
KO
KL
-0.4352 0.4203 -0.2333 0.0513 0.1564 0.0404
0.0736 -0.2872 0.1614 0.0584 -0.0233 0.0172-0.2541 1.0037 -0.8031 -0.1320 0.2257 -0.0403
0.0366 0.1889 -0.0683 -0.0997 -0.0298 -0.0277
0.3426 -0.2464 0.4017 -0.0912 -0.3389 -0.0679
0.0667 0.1337 -0.0458 -0.0679 -0.0486 -0.0381

Note: Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

The main diagonal elements of the above matrix are negative, which is consistent with theconcavity in input prices property of a joint cost function. A pair of inputs n and j aresubstitutes if ∂xn (y, w, t)/∂wj = ∂xj (y, w, t)/∂wn is positive and are complements if thesepartial derivatives are negative. From Table 1, we see that there are 8 pairs of substitute inputs
and 7 pairs of complementary inputs on average. This means that it is unlikely that CobbDouglas or CES cost functions could provide a satisfactory approximation to the US aggregate
technology (because all inputs are substitutes using these functional forms). Another point tonote that follows from an examination of the above Table 1 is that most of the elasticities are
small in magnitude.
In the previous section, we derived a formula (30) (which drew on (29)) for the year t elasticity
of marginal cost for output m with respect to a change in the quantity of output k, Eptmk ≡[∂pm (y t , wt , t)/∂yk ][ykt /ptm ]. This formula was derived using linear time trends to describe
technical progress but the same formula is also valid using our spline model for technicalprogress, provided that for year t elasticities, we replace the constant matrix B in (29) by
the weighted average matrix B(t). Thus we use our estimated coefficients to calculate thesemarginal cost elasticities, which can also be interpreted as (inverse) output supply elasticities.
The resulting averages of these elasticities are listed in Table 2 below. The main diagonalelasticities for each year are listed in Appendix A.*19
The main diagonal elasticities are all positive and this follows from the fact that we have
imposed convexity in y on our estimated joint cost function. However, the elasticities in Table
2 are all quite small. This indicates that the US aggregate joint cost function is close to being*19 We did not attempt to compute standard errors for these elasticities.

The standard errors are likely to
be large so our tables of elasticities are only very rough approximations to the “true” elasticities.

14

Table 2 Average Elasticities of Inverse Output Supply over the Period 1970-2022
Epmk
p1 C
p2 G
p3 I
p4 X

y1
y2
y3
y4
C
G
I
X
0.0787 -0.0445 -0.0474 0.0131-0.1683 0.1195 0.0628 -0.0140-0.1279 0.0449 0.1225 -0.0396
0.0914 -0.0312 -0.0892 0.0914

Note: Consumption (C), Government (G), Investment (I), Export (X).

equal to the sum of 4 sectoral cost functions that produce non overlapping products; i.e., theassumption of no joint production almost holds.
In order to define our next set of elasticities, we need to list the functional form for the joint
cost function in year t:
C(y t , wt , t) ≡ (1/2)(wt · A(t)wt )(α · wt )−1 (β · y t ) + (1/2)(y t · B(t)y t )(α · wt )(β · y t )−1
[ ∑
∑7
∑
6
+ wt · Dy t + (β · y t ) w1t i=1 b1i f1i (t) + w2t i=1 b2i f2i (t) + w3t 7i=1 b3i f3i (t)
]
∑7
∑7
∑
+w4t i=1 b4i f4i (t) + w5t i=1 b5i f5i (t) + w6t 5i=1 b6i f6i (t)(32)

where the year t matrices A(t) and B(t) and the piece-wise linear in t spline functions fni were
defined in the previous section. The above definition for C(y t , wt , t) along with Shephard’s
Lemma, xn (y t , wt , t) = ∂C(y t , wt , t)/∂wn , are used to define the year t elasticity of input
demand n with respect to an increase in output m, Extnm , as follows for n = 1, . . . , 6, m =1, . . . , 4 and t = 0, 1, . . . , 52:tt/xtn ].
Extnm ≡ [∂xn (y t , wt , t)/∂ym ][ym/xtn ] = [∂ 2 C(y t , wt , t)/∂wn ∂ym ][ym(33)

The averages of these elasticities are listed in Table 3 below.
Table 3 Average Elasticities of Input Demand with Respect to Output Quantities
Exnm
x1
M
x2
L
x3 KM&E
x4 KS
x5 KO
x6 KL

y1
y2
y3
y4
C
G
I
X
0.0864 0.0600 0.4385 0.4152
0.6551 0.1717 0.1771 -0.0038
0.6367 -0.4275 0.5149 0.2760
0.4815 0.2818 0.0655 0.1712
0.8214 0.5049 -0.0554 -0.2709
0.5110 0.2280 0.1532 0.1079

Note: Consumption (C), Government (G), Investment (I), Export (X),
Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

On average, an increase in consumption y1 leads to substantial increases in the demand for all
inputs except imports where the elasticity was only 0.0864. It is somewhat puzzling that anincrease in government output y2 should lead to a substantial drop in the demand for input
3, Machinery and Equipment.*20*20 In retrospect, it may have been wiser to impose nonnegativity constraints on the components of the15

Definition (32) for the year t cost function, C(y t , wt , t), along with the price equals marginal
cost equations, pm (y t , wt , t) = ∂C(y t , wt , t)/∂ym , can be used to define the year t elasticity
of marginal cost m with respect to an increase in the price of input n, Eptmn , as follows form = 1, . . . , 4, n = 1, . . . , 6 and t = 0, 1, . . . , 52:
Eptmn ≡ [∂pm (y t , wt , t)/∂wn ][wnt /ptm ] = [∂ 2 C(y t , wt , t)/∂ym ∂wn ][wnt /ptm ].

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The averages of these elasticities over the years 1970-2022 are listed in Table 4 below.
Table 4 Average Elasticities of Output Prices with Respect to Input Prices
Epmn
p1 C
p2 G
p3 I
p4 X

w1
w2
w3
w4
w5
w6
M
L
KM&E
KS
KO
KL
0.0222 0.6119 0.0986 0.1328 0.2931 -0.1554
0.0476 0.6350 -0.2583 0.3060 0.0750 0.0596
0.2269 0.4618 0.2183 0.0492 0.1653 0.1043
0.5175 -0.003 0.2711 0.2931 -0.0056 0.0494

Note: Consumption (C), Government (G), Investment (I), Export (X),
Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

For the most part, an increase in an input price flows through to output prices and increases
the price of each of our four outputs. On average, a one percent increase in the price of labour,
w2 , substantially increases the prices of consumption, government and investment, p1 , p2 andp3 , but has a tiny negative effect on the price of exports. There are puzzles that an increasein the input price of Machinery and Equipment services w3 leads to a substantial drop in theprice of government output p2 and that an increase in the user cost of land w6 leads to a
decline in the price of consumption p1 .
We turn now to the determination of measures of technical progress and the biases in technicalprogress. The use of a joint cost function to measure technical progress can be traced back to
Salter (1960)[52] at least. The most comprehensive measure of technical change going fromyear t to year t + 1 using a joint cost function that depends on time, say C t (y, w) where y andw are arbitrary reference output quantity and input price vectors, is C t (y, w)/C t+1 (y, w).*21
Holding constant output levels y and input prices w, technical progress should reduce thecost of producing y with input prices held constant so that C t+1 (y, w) < C t (y, w) and hencetechnical progress is greater than 1. We will not use this measure of technical progress but wewill use a related measure which has the advantage of giving us a decomposition of technicalprogress into a sum of explanatory factors.
We define year t cost saving technical progress κt as the derivative of period t cost with respect
to time t divided by year t fitted cost:κt ≡ ∂ ln C(y t , wt , t)/∂t = [∂C(y t , wt , t)/∂t]/C(y t , wt , t).

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The year t fitted input n demand function is xn (y t , wt , t) = ∂C(y t , wt , t)/∂wn for n = 1, . . . , 6.
Define the year t input n cost saving bias, κtn , for n = 1, . . . , 6 as follows:κtn ≡ wnt [∂xn (y t , wt , t)/∂t]/C(y t , wt , t) = wnt [∂ 2 C(y t , wt , t)/∂wn ∂t]/C(y t , wt , t).

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matrix D in our model. The imposition of these nonnegativity constraints would lead to nonnegative
elasticities for Extnm and Eptmn for t = 0.
*21 This measure is due to Balk (1998; 58)[2]. For specializations and applications of this definition, see
Balk (2001)[3] (2003)[4] and Diewert (2014)[16].

16Thus κtn is the derivative of the year t cost of input n with respect to time, wnt ∂xn (y t , wt , t)/∂t,divided by total year t fitted input cost, C(y t , wt , t). Using the estimated cost function definedby (32), we can calculate κt and the bias terms κtn .*22 Euler’s Theorem on functions and thelinear homogeneity of our estimated C(y, w, t) in w implies that the following decompositionof κt into explanatory input bias terms will hold for each year t:∑6κt = n=1 κtn .
(37)

Table 5 below lists the technical progress estimates κt and the corresponding bias components.

From Table 5, we see that on average, technical progress reduced economy wide costs by
0.58 percentage points per year. For the years 1971-2022, the average cost reduction was
0.57 percentage points per year. In Appendix A, we show that the corresponding annual
average Total Factor Productivity (TFP) Growth estimates using the Jorgenson and Griliches(1967)[31], Diewert and Morrison (1986)[22] and Kohli (1990)[39] (2003)[43] index number
methodology was 0.59 percentage points per year. Using the Diewert and Fox (2018)[19]
nonparametric methodology for measuring TFP growth explained in Appendix C led to an
estimate of 0.59 percentage points per year over the period 1971-2022. Thus our econometricmethod for measuring technical progress is consistent with alternative methods for measuring
technical progress and TFP growth, at least over longer periods. However, the econometricmethod smooths the year to year fluctuations technical progress and so it does not capture
the annual fluctuations in technical progress that are captured by the index number andnonparametric methods, as shown in Figure 1.*23

Figure 1 Comparison of Technical Progress Estimates based on Different Approaches

Since κt2 , κt3 , κt4 and κt6 were negative on average, technical progress was on average Laboursaving, Machinery and Equipment saving, Structure saving and Land saving. Thus technical*22 The spline functions fni (t) that were defined in the previous section are not always differentiable withrespect to time t. However, the one sided right hand side directional derivatives exist and are straightforward to calculate. These one sided derivatives were used to calculate the various derivatives of theestimated cost function with respect to time.
*23 Note that TFPGt and NTFPt are virtually identical in Figure 1.

17

Table 5 Cost Saving Technical Progress and Input Bias Components for the US
Year t
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

κt
-0.0096-0.0097-0.0098-0.0097-0.0094-0.0079-0.0080-0.0082
0.00340.0032-0.0007-0.0009-0.0009-0.0117-0.0118-0.0118-0.01200.0006-0.0034-0.0027-0.0027-0.0024-0.0123-0.0075-0.0074-0.0072-0.0072-0.0080-0.0081-0.0080-0.0076-0.0062-0.0064-0.0065-0.0076-0.0074-0.0037-0.0026-0.0027-0.0026-0.0039-0.0037-0.0036-0.0036-0.0034-0.0034-0.0035-0.0037-0.0036-0.0037-0.0030-0.0119-0.0117-0.0058κt1
0.00080.00080.00080.00080.00070.00080.00080.00080.00130.00120.00110.00110.00100.00110.00110.00110.00110.0011-0.0031-0.0030-0.0031-0.0030-0.00290.00190.00200.00200.00200.00190.00180.00180.00180.00170.00170.00170.00170.00170.0016-0.0011-0.0012-0.0011-0.0010-0.0011-0.0010-0.0010-0.0009-0.0008-0.0008-0.0008-0.0008-0.0008-0.0008
0.00650.00670.0005κt2-0.0090-0.0091-0.0093-0.0095-0.0100-0.0101-0.0099-0.00990.00140.00130.00110.00120.0012-0.0078-0.0077-0.0079-0.00820.00410.00400.00390.00400.0040-0.0044-0.0044-0.0043-0.0043-0.0044-0.0045-0.0047-0.0049-0.0052-0.0054-0.0055-0.0055-0.0056-0.0056-0.0015-0.0016-0.0017-0.0017-0.0017-0.0018-0.0017-0.0017-0.0018-0.0017-0.0018-0.0018-0.0018-0.0018-0.0019-0.0167-0.0165-0.0040

κt3-0.0010-0.0009-0.0006-0.00020.00010.0000-0.0001-0.0001
0.00000.0001-0.0036-0.0035-0.0033-0.0035-0.0036-0.0035-0.0034-0.0049-0.0045-0.0043-0.0040-0.0038-0.0038-0.0039-0.0041-0.0042-0.0041-0.0040-0.0037-0.0034-0.0031-0.0028-0.0027-0.0025-0.0023-0.0022-0.00190.00100.00090.0010-0.0003-0.0003-0.0003-0.0003-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0001-0.0004-0.0005-0.001818

κt4-0.0001-0.0001-0.0001-0.0001
0.00000.00000.00000.00000.00000.00000.00010.00010.0001-0.0016-0.0018-0.0019-0.0020-0.0019-0.0019-0.0019-0.0017-0.0016-0.0016-0.0018-0.0020-0.0021-0.0023-0.0033-0.0034-0.0035-0.0035-0.0016-0.0016-0.0016-0.0021-0.0020-0.0017-0.0015-0.0012-0.0014-0.0017-0.0019-0.0021-0.0023-0.0024-0.0025-0.0025-0.0026-0.0025-0.0025-0.0017-0.0018-0.0018-0.0016κt5
0.00070.00060.00050.00050.00060.00220.00220.00220.00210.00200.00190.00180.00170.00180.00190.00190.00190.00370.00360.00350.00320.00310.00260.00270.00290.00300.00310.00310.00300.00300.00310.00250.00240.00230.00210.00260.00240.00210.00200.00210.00210.00220.00230.00240.00250.00240.00240.00230.00230.00220.00210.00140.00140.0022

κt6-0.0010-0.0011-0.0012-0.0011-0.0009-0.0009-0.0010-0.0011-0.0013-0.0014-0.0014-0.0016-0.0016-0.0018-0.0017-0.0015-0.0014-0.0014-0.0016-0.0009-0.0010-0.0011-0.0021-0.0021-0.0019-0.0016-0.0014-0.0012-0.0011-0.0010-0.0008-0.0005-0.0007-0.0009-0.0014-0.0019-0.0026-0.0015-0.0015-0.0014-0.0012-0.0010-0.0008-0.0007-0.0005-0.0005-0.0006-0.0006-0.0007-0.0008-0.0006-0.0007-0.0009-0.0012

progress reduced the demand for these inputs over time. By far the biggest contribution to
overall cost reduction due to technical progress was made by Labour. Since on average, κt1
and κt5 were positive, technical progress was Import augmenting and Other Capital (R&D andInventories) augmenting.
Table 5 shows a limitation of our econometric specification. We used piece-wise linear splines
to model technical progress and so as time t moves through a break point, the derivatives
with respect to time of the spline functions fni (t) change in a discontinuous manner and thisleads to discontinuous changes in the κtn . Thus κt2 changes in a discontinuous manner as the
spline functions f2i (t) move through break points. The discontinuous movements in κt2 largely
determine the discontinuous movements in κt .
For additional information on the index number and nonparametric methods for measuring
technical progress, see Appendices A and C.
For some purposes, it is useful to switch to estimating a GDP function or a Gross Output
function or a Variable Profit function where only a few inputs are fixed and all outputs andvariable inputs are free to vary. However, as we indicated in Section 3, better global fits can beobtained by estimating joint cost functions if we are using macroeconomic data. It turns out
to be possible to get elasticity estimates for these alternative functions using our estimatedjoint cost function. We show how this can be done in Appendix D.

6 Extensions and Areas for Future Research
It would be of interest to estimate models that used more disaggregated data.*24 As is explained in Appendix A, the Asian Productivity Data Base has data on more than 20 typesof capital for the US for the years 1970-2022. This data base also has information on hours
worked by workers classified by: education (2 classes), sex (2 classes), age (5 classes) and typeof worker (2, self employed and employee) or 40 types of labour in all.
However, as the number N of inputs and outputs increases, the number of parameters required
for a basic flexible functional form is N (N − 1)/2 and the number of degrees of freedom for asample of T periods is only N T so the curse of dimensionality becomes a problem. Diewert andWales (1988)[24] addressed this problem by limiting the number of columns that are allowed
to enter into the U and V lower triangular matrices defined in Section 4. The same idea could
be applied to the D matrix defined in Section 4; i.e., set D equal to a limited sum of rank one
matrices.*25
The choice of the fixed vectors α and β also needs to be examined: are there better ways ofchoosing α and β ?
Are there more efficient ways for specifying technical progress? Should we use quadratic
or cubic splines in place of the linear splines that we used? Should we impose nonnegativity
restrictions on the components of the D matrix? Answers to these questions require additional
research.
*24 Of course, it would be preferable to estimate the technology sets for each sector.

This option runs
into two problems: (i) typically the price and quantity data on capital stocks and labour input are notavailable by sector and (ii) when estimating sectoral Joint Cost functions or Gross Output functions
using sectoral data, we require data on the price and quantity of intermediate inputs, which is also notreadily available. When we work with national data, intermediate input flows between sectors cancel
out, except for imports.
*25 Alternatively, we could regard a small set of inputs as fixed and treat the remaining inputs as negative
outputs and estimate a variable profit function. This would fix the size of the A matrix at N by N
where N is small and the D matrix would have size N by M where M is large. The semi-flexible idea
could be applied to the B matrix; i.e., the rank of B would be limited.

19

Finally, an extension of the competitive model that we used is possible. Recall equations(6) which equated output prices ptm to the marginal cost ∂C(y t , wt )/∂ym for m = 1, . . . , M .
These equations can be replaced by the following equations for year t and m = 1, . . . , 4:
ptm = (1 − µtm )−1 ∂C(y t , wt )/∂ym(38)

where µtm is the year t ad valorem markup for output m.*26 If we assume that the markup
is constant over time, then this new specification for equations (6) would add an additional
4 markup parameters µm to be estimated.*27 However, the underlying data for this moregeneral model would have to be adjusted; i.e., we set the cost of capital equal to a rate of
return that causes cost to equal the value of output. For the markup model, we would needto specify an exogenous cost of capital. This is a tricky business: if the cost of capital is set
to a high level, markups could become negative. If the cost of capital is set too low, then the
resulting markups will be too large.
There are many other problems with our empirical application of flexible functional form
theory:• The General Government and Owner Occupied Housing Sectors should be treated separately from the Business Sector.
• There are missing assets in our model such as Natural Resource Deposits. Environmental Assets and Monetary Holdings.
• An exogenous cost of capital probably should be used in place of our endogenously
determined rt .
• There is a general problem with how to define expected asset inflation rates and how
to treat negative user costs when land and natural resource stocks are included in thelist of assets.
A host of additional measurement problems are discussed in the excellent survey paper on
productivity measurement by Martin and Riley (2024)[49].

7 Conclusion
Here are some of the important points that emerge from our paper:• It is important to include land as a factor of production in macroeconomic models of
the economy.
• It may be better to estimate joint cost functions rather than gross output functions
when working with macroeconomic data.
• The Normalized Quadratic Joint Cost function can be used to model aggregate production for an economy but the estimation is quite complex. The advantage of thisflexible functional form is that it can impose curvature conditions globally and it canbe used to model technical progress in a way that captures longer terms trends in the
more variable index number and nonparametric estimates of TFP growth.
• Elasticities of input substitution are far from being constant for aggregate production
functions once we disaggregate capital services. For our 6 input model, we found that 8
pairs of inputs were substitutes and 7 pairs were complements. Macroeconomic models
based on Cobb-Douglas or CES production functions are not satisfactory descriptionsof reality.
*26 See Diewert and Fox (2008; 176-177)[18] for an example of how this model could be implemented.
*27 More elaborate models for the markups could be estimated. For example, we could specify linear timetrends for the markups or we could specify more flexible spline models.

20• Sample wide linear trends used to model technical progress were not satisfactory. Linear
spline functions were used to model technical progress with some success.
• A next step in using the methodology outlined in this paper is to include markups in
the model. Instead of setting output price equal to marginal cost, set output price equal
to a markup plus marginal cost. However, this extension requires exogenous estimates
for the economy wide cost of capital and it is difficult to obtain a definitive estimate forthis important variable.

21

Appendix A: Data Construction
In this paper, we use the 1970-2022 US annual data on the prices and quantities of themacroeconomic aggregates C + G + I + X − M and the data for labour and capital stocks
developed by the joint project of the Asian Productivity Organization and Keio University
led by Koji Nomura. The Augmented Productivity Database (APDB) for the US was produced on November 15, 2023.*28 The APDB output data are consistent with the recent data
published by the Bureau of Economic Analysis (2023)[5], except that the BEA data allocatesgovernment investment into the general government sector whereas the APDB data includes
government investment in the APDB investment aggregate. The constant dollar APDB investment, depreciation and reproducible capital stock data are perfectly consistent with thegeometric model of depreciation.
The APDB has annual current and constant dollar estimates for 10 reproducible capital stocks
plus inventory change plus 6 types of land. These 17 constant dollar capital stocks are beginning of the year estimates but the corresponding prices are midyear prices. We convert thesemidyear prices into beginning of the year prices by taking the arithmetic average of the current
midyear price and the previous year midyear price. The APDB also has detailed information
on hourly wage rates and annual hours worked for many types of labour classified by age,sex, education and type of worker (employee or self-employed). We will not make use of thedetailed labour information: we simply used the resulting APDB aggregate quality adjusted
price and quantity of labour for year t, PLt and QtL .
The price indexes for the output aggregates for year t are defined as PCt (private consumpt(exports of goods andtion)*29 , PGt (government consumption), PIt (gross investment), PX
t30
services) and PM (imports of goods and services)* . The corresponding quantity or volume
indexes are defined as QtC , QtG , QtI , QtX and QtM . The 11 components of the APDB investment series are as follows: (1) IT hardware; (2) Communications equipment; (3) Transport
equipment; (4) Other machinery and equipment; (5) Dwelling structures; (6) Non-residentialbuildings; (7) Other structures; (8) Research and development; (9) Computer software; (10)Other intangible assets and (17) Net increase in inventory stocks.*31 Denote the year t pricetindex for investment good n by PIn
and the corresponding quantity or volume index by QtIn
for n = 1, . . . , 17 and t = 1970, . . . , 2022. The APDB price series for reproducible investments
are listed in Table A1 and the corresponding quantity series are listed in Table A2. The unitsof measurement are in trillions of 1970 constant dollars.

*28 The methodology for constructing the labour, capital and productivity accounts for the US and Asian

countries is explained in Asian Productivity Organization (2022; 165-188)[1].
*29 The price of consumption was adjusted downward by removing indirect taxes on outputs in order toobtain an approximation to the producer price of consumption.
*30 The price of imports was adjusted upwards by adding tariffs to the border price of imports.
*31 We cumulated the estimates for real inventory change to form beginning of the year inventory stocks.

We set the price of inventory change equal to the end of year price of inventory stock. These conventions
allowed us to apply user cost theory to the stock of inventories. In Tables A1 and A2, we labelled thetyear t price and quantity of inventory change as PI17
and QtI17 . Assets 11-16 are land stocks. We did
not include investments in land stocks in this study since these investments are not recognized in thecurrent international System of National Accounts.

22

Table A1: Price Indexes for US Reproducible Investments
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

PI1
1.00000
0.67587
0.559980.500390.432160.399150.33373
0.28188
0.19059
0.15771
0.12164
0.10657
0.098080.08172
0.06584
0.05582
0.048020.04034
0.037220.03474
0.03136
0.028210.023990.020270.017830.01477
0.01115
0.00859
0.006380.00495
0.00433
0.00357
0.003100.00276
0.00256
0.002260.00198
0.001770.001610.001490.001440.001390.001380.001370.001380.001380.001360.001360.001370.001340.001310.001330.00139PI2
1.00000
1.01677
1.06170
1.08037
1.18518
1.30770
1.37345
1.32475
1.37678
1.40083
1.495991.61514
1.73120
1.78463
1.834071.87439
1.88606
1.87379
1.85554
1.85899
1.857171.86775
1.84335
1.80267
1.76728
1.69672
1.64909
1.63449
1.53893
1.43565
1.382481.30871
1.23452
1.05489
0.95174
0.904440.845120.789270.700900.639410.57676
0.549530.510860.47782
0.42995
0.38385
0.342070.320790.302030.28488
0.27304
0.26205
0.25988

PI3
1.00000
1.02154
1.04238
1.05074
1.156171.30601
1.41440
1.51750
1.63484
1.76883
1.94683
2.135992.30486
2.38076
2.44665
2.509102.60075
2.60770
2.631542.725112.79755
2.903132.980523.03054
3.126413.19077
3.211313.22029
3.208783.23021
3.26157
3.28972
3.28924
3.30241
3.339903.41053
3.44258
3.45684
3.48075
3.55534
3.587933.68260
3.77482
3.79586
3.84272
3.90679
3.93791
3.953314.03456
4.087074.133644.275524.68266

PI4
PI5
PI6
PI7
1.00000 1.00000 1.00000 1.00000
1.02539 1.05098 1.09624 1.07348
1.04236 1.11280 1.17834 1.13004
1.07326 1.21141 1.26395 1.22238
1.19841 1.34337 1.40873 1.44461
1.42353 1.48421 1.56112 1.56420
1.53608 1.56882 1.62011 1.61517
1.63810 1.71554 1.73241 1.66631
1.76659 1.92480 1.90627 1.77852
1.91389 2.13733 2.11003 1.94016
2.15343 2.37790 2.35708 2.01758
2.35925 2.56150 2.55678 2.28504
2.53575 2.68465 2.73185 2.48623
2.63763 2.74432 2.83119 2.34445
2.66571 2.81691 2.92301 2.26529
2.69479 2.87259 3.02464 2.37306
2.81758 3.03598 3.14442 2.65302
2.87636 3.17485 3.24433 2.70729
2.96440 3.29146 3.37893 2.71097
3.05756 3.41345 3.50560 2.84825
3.17531 3.48722 3.62340 2.88063
3.30326 3.50771 3.72449 2.95890
3.37807 3.58406 3.75027 3.04836
3.40098 3.72808 3.85506 2.98750
3.43791 3.87175 3.97008 3.05306
3.47123 4.01020 4.07819 3.35178
3.48756 4.07233 4.12580 3.57472
3.50559 4.21007 4.26485 3.47910
3.51224 4.33512 4.43467 3.71418
3.52466 4.50442 4.62237 3.89519
3.53518 4.71390 4.81433 3.96958
3.57471 4.94473 4.98888 4.12756
3.61413 5.06443 5.12985 4.53281
3.65456 5.31084 5.23995 4.69601
3.70865 5.68881 5.51466 5.02488
3.81705 6.09701 5.98045 5.74836
3.88153 6.44831 6.44107 6.60118
3.97596 6.53946 6.86055 7.19371
4.09216 6.44872 7.10587 7.64303
4.14941 6.23955 7.28187 7.56937
4.14727 6.19757 7.12688 7.67974
4.23631 6.22240 7.28125 8.01027
4.32437 6.28879 7.45452 8.45721
4.32164 6.59165 7.56251 8.51778
4.34573 6.99714 7.79153 8.94102
4.36940 7.21251 7.95558 9.12524
4.36464 7.47320 8.09039 9.10904
4.37696 7.81323 8.27618 9.39472
4.43593 8.27303 8.62899 9.26953
4.49939 8.51296 9.12736 9.48236
4.53565 8.79649 9.46904 9.60616
4.68426 9.69699 9.86040 9.99053
5.07353 11.07624 11.85630 10.78711

23

PI8
PI9
PI10
PI17
1.00000 1.00000 1.00000 1.00000
1.04382 1.05603 1.08889 0.93258
1.07645 1.22239 1.12294 1.00000
1.12088 1.35754 1.17141 1.15217
1.22354 1.43624 1.22613 1.35922
1.32321 1.70886 1.34098 1.40000
1.38775 1.83141 1.42098 1.40164
1.46302 2.15563 1.48286 1.45752
1.54458 2.14661 1.59709 1.59586
1.64948 2.67716 1.66546 1.81351
1.80049 2.94414 1.74004 1.91254
1.94623 3.96771 1.84254 2.15531
2.08271 4.74826 1.93262 2.13523
2.17215 6.01327 2.01593 2.12408
2.24344 6.32033 2.09859 2.35766
2.29201 6.74004 2.19965 2.41811
2.33250 6.58133 2.30470 2.20688
2.37182 6.43226 2.38810 2.30719
2.44625 7.04110 2.48258 2.38453
2.47653 7.94874 2.62560 2.55637
2.50335 8.00673 2.73804 2.47809
2.54501 8.16032 2.90359 2.51844
2.58129 8.27947 2.94880 2.57584
2.59893 8.98822 2.97831 2.63679
2.65155 9.29016 3.05778 2.59610
2.75317 9.50958 3.13220 2.73869
2.75073 9.15994 3.20100 2.78156
2.81676 9.46103 3.25780 2.58470
2.82870 9.64210 3.24257 2.50680
2.87925 10.06312 3.35064 2.50096
2.97622 10.62807 3.45741 2.55675
2.99321 10.55350 3.51224 2.57458
3.03320 9.92951 3.52655 2.55607
3.10133 9.44297 3.57264 2.20047
3.16943 9.32880 3.59042 2.41006
3.23837 9.39630 3.60486 2.80335
3.28987 9.42468 3.63731 2.46026
3.36838 9.41765 3.67169 2.77872
3.45528 9.60404 3.66600 2.80756
3.46005 9.41520 3.67803 2.75002
3.51914 9.14362 3.61282 3.07708
3.59899 9.28274 3.59310 3.23853
3.67948 9.37802 3.62280 3.19538
3.65614 9.50706 3.61019 3.16564
3.70782 9.52453 3.64910 3.09468
3.76212 9.47558 3.72009 3.25672
3.69651 9.50454 3.79522 3.63563
3.76160 9.50764 3.84454 3.10563
3.86586 9.37476 3.90106 3.22573
3.93387 9.29080 3.94401 3.14483
4.10597 9.14762 4.00237 3.90540
4.21436 8.95539 4.09541 2.90687
4.37195 8.86819 4.35736 3.81354

Table A2: Quantity Indexes for US Reproducible Investments
Year
QI1
QI2
QI3
1970
0.00325 0.00782 0.01841
1971
0.00474 0.00742 0.01948
0.00707 0.00703 0.02278
1972
0.00782 0.00838 0.02724
1973
1974
0.01028 0.00865 0.02483
1975
0.01043 0.00823 0.02143
0.01533 0.00897 0.02400
1976
0.02321 0.01187 0.02889
1977
1978
0.04530 0.01386 0.03155
0.07349 0.01647 0.03308
1979
0.11827 0.01862 0.02696
1980
1981
0.18528 0.01945 0.02600
0.22937 0.01987 0.02298
1982
0.35506 0.02007 0.02590
1983
1984
0.57983 0.02240 0.02972
0.74162 0.02425 0.03025
1985
0.86994 0.02593 0.02809
1986
1987
1.11170 0.02620 0.02748
1988
1.24924 0.02881 0.03008
1.52323 0.02880 0.02677
1989
1990
1.52828 0.03022 0.02694
1.66775 0.02920 0.02785
1991
1992
2.24713 0.03098 0.02920
2.79367 0.03221 0.03391
1993
3.40045 0.03755 0.03928
1994
5.10148 0.04306 0.04068
1995
7.35439 0.04881 0.04245
1996
1997 10.63263 0.05518 0.04534
1998 15.36354 0.06365 0.04970
1999 21.86683 0.07750 0.05874
2000 26.41311 0.10028 0.05526
2001 27.25659 0.09471 0.04939
2002 29.05439 0.07867 0.04576
2003 32.66864 0.09070 0.04237
2004 36.95599 0.10334 0.05071
2005 41.69228 0.11025 0.05692
2006 52.24049 0.13074 0.06135
2007 60.17714 0.15550 0.05961
2008 66.22915 0.16809 0.04700
2009 70.77260 0.16532 0.02259
2010 79.00717 0.20634 0.04266
2011 76.41101 0.21959 0.05443
2012 82.44852 0.23668 0.06272
2013 80.66736 0.27031 0.06844
2014 79.38585 0.31051 0.07519
2015 79.18422 0.36946 0.08348
2016 79.01030 0.42905 0.07814
2017 84.12615 0.49073 0.07777
2018 95.24785 0.50362 0.08402
2019 97.26011 0.51003 0.08119
2020 108.15529 0.51269 0.05890
2021 119.20516 0.56846 0.05590
2022 124.82850 0.64307 0.05377

QI4
0.051390.04784
0.05272
0.062100.06587
0.059230.060540.06585
0.07325
0.07801
0.074120.07538
0.06863
0.06697
0.07632
0.079630.080890.08083
0.08267
0.090440.08806
0.08103
0.081130.08625
0.090640.096390.100020.103400.110350.11179
0.11930
0.116420.11571
0.121260.126130.13738
0.14598
0.150710.151460.12930
0.134510.14572
0.155190.15905
0.16693
0.16287
0.160960.168010.173500.179290.169840.178810.18309

QI5
0.04397
0.05604
0.06532
0.06472
0.05167
0.04509
0.05532
0.06733
0.071880.069190.054640.050390.04161
0.059190.06777
0.069850.077930.07897
0.07806
0.075290.06900
0.06338
0.07175
0.077140.083860.08084
0.08755
0.089440.09686
0.102810.10344
0.104480.111070.12109
0.132770.141350.13063
0.10606
0.080400.063080.06094
0.06075
0.06852
0.077090.080100.08866
0.09502
0.09886
0.09827
0.09750
0.10451
0.11672
0.10620QI6
0.033480.032900.03349
0.03594
0.03501
0.03051
0.02946
0.02886
0.03325
0.03838
0.03882
0.04106
0.04189
0.03895
0.045170.05127
0.048890.048430.04905
0.05084
0.052220.04556
0.04275
0.04180
0.043820.04900
0.05400
0.058110.061540.06204
0.06452
0.06351
0.05551
0.05384
0.05430
0.05333
0.05616
0.061450.06362
0.05258
0.04178
0.039900.041160.041440.044490.05095
0.05562
0.05575
0.05669
0.05734
0.05509
0.051540.04859

24

QI7
0.03378
0.03303
0.03306
0.03379
0.03358
0.03508
0.036410.03835
0.04267
0.044810.05055
0.052430.048020.043070.049080.049490.040740.040780.040810.03906
0.042810.041880.03959
0.042960.041690.040420.04104
0.04291
0.042810.04292
0.04804
0.05130
0.04676
0.04733
0.04606
0.046420.047990.05067
0.05306
0.04903
0.04872
0.049810.05230
0.05133
0.055140.050120.044220.045890.05029
0.05250
0.046630.04468
0.04404

QI8
0.028440.028070.029180.030220.029800.029540.031200.03266
0.03483
0.037070.03905
0.04126
0.04306
0.045650.049960.05479
0.057100.05950
0.061410.06403
0.066910.06933
0.069990.06932
0.06937
0.07160
0.076390.080430.085090.09048
0.09657
0.09934
0.09904
0.101180.103110.10826
0.114400.12005
0.123020.121290.12206
0.12569
0.126180.13260
0.13725
0.14269
0.153480.158640.167280.179420.18495
0.200010.21028QI9
0.003130.00306
0.00305
0.00326
0.003780.00388
0.003960.00355
0.004140.004250.004690.004220.004130.003830.004480.004880.005440.00623
0.00654
0.00702
0.00774
0.008170.00849
0.008480.00853
0.008860.010200.012140.01379
0.016150.017450.01775
0.018180.019430.02097
0.022310.02359
0.02563
0.02715
0.02800
0.02882
0.03111
0.03346
0.035070.037290.03926
0.043240.04765
0.05349
0.057120.06204
0.069990.07921QI100.004130.004080.004390.004200.00426
0.004120.005240.00585
0.00575
0.00659
0.005970.007220.00727
0.00767
0.00855
0.00851
0.009400.009240.00933
0.009760.010100.010210.010700.011280.011880.01286
0.01379
0.014200.014960.015310.016220.01572
0.016510.01775
0.01801
0.01952
0.01973
0.01932
0.018200.017530.019280.01930
0.019540.01959
0.02025
0.02101
0.02187
0.02257
0.02333
0.023620.02265
0.02226
0.02420QI170.00200
0.008900.009100.013800.01030-0.00450
0.012200.01530
0.016170.00993-0.00329
0.01383-0.00698-0.00273
0.02774
0.009020.00299
0.01175
0.007760.01084
0.00585-0.00016
0.00633
0.007890.02458
0.011390.011070.027430.025410.024310.02132-0.01488
0.00782
0.006410.02660
0.02051
0.02805
0.01224-0.01040-0.05484
0.017520.01430
0.02228
0.03333
0.027400.043020.010750.010530.017480.02296-0.00963
0.004020.04125

The year t overall investment price PIt is set equal to the chained Törnqvist price index*32 of the
11 reproducible investment components listed in the above tables and the year t companion
constant dollar aggregate investment is denoted by QtI . The APDB aggregate price andquantity component series of GDP (at producer prices) are listed below in Table A3 alongwith the APDB quality adjusted price and quantity of labour. The units are in trillions of1970 constant dollars.

*32 For definitions of the Törnqvist price and quantity indexes and their connection to the economic approach

to index number theory, see Diewert (1976)[13] and Diewert and Morrison (1986)[22].

25Table A3: Prices and Quantities for Labour and the Output Components of US GDP
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

t
PC
1.00001.04061.07951.13591.25281.36411.4404
1.53791.6482
1.79711.98552.14912.28692.38852.47522.56552.62982.71532.8191
2.94363.0703
3.16403.2502
3.33633.39343.46983.54643.60633.63693.69603.79353.8835
3.92724.00934.10304.22134.33394.45394.59134.59614.6711
4.78384.86844.92474.99015.00105.05805.14645.24385.32665.40915.61805.9758t
PG
1.00001.0831
1.16461.24561.3523
1.47691.5655
1.66881.78191.93002.13182.34462.50592.61182.74302.84772.90973.0042
3.1111
3.2424
3.39503.53633.66553.75363.85643.96254.05254.14094.22164.3802
4.58904.74134.88505.09615.32275.59295.83896.09556.36946.31756.51866.7262
6.83697.01357.1605
7.1527
7.16287.33657.61947.72797.91778.35758.9418PIt
1.00001.04281.08781.15231.28561.43761.51461.60681.7323
1.88522.05652.24812.40142.44952.47822.52442.61662.65902.72712.81612.86642.92252.95362.98333.03193.09233.07913.08003.07883.10423.16033.19083.21643.2406
3.3382
3.49553.6116
3.69723.74373.71863.68213.74363.81183.83673.90693.9502
3.96764.02244.09844.16484.23984.4042
4.7843t
PX
1.00001.03681.08141.22601.5102
1.6643
1.7187
1.78891.89772.12652.34192.51562.52822.53882.56252.48612.44602.50012.62862.67222.68742.71732.70402.70842.73992.80132.76392.71432.65012.63042.68082.65972.63672.67832.77602.87192.96313.07483.21483.0252
3.15153.35003.37563.38073.3824
3.2192
3.1544
3.23673.34443.32803.2436
3.61843.9724t
PM1.00001.06591.13071.32201.87752.07142.09752.28062.45072.8592
3.54263.74293.62273.4842
3.45983.34583.34683.54853.71513.7963
3.89883.86273.86653.83693.8682
3.95723.8802
3.73993.53503.5476
3.69903.60503.55903.66593.8371
4.0612
4.21604.36404.79214.26814.51284.85674.87174.81114.77564.39644.24684.33884.47154.43334.33734.65774.9893t
PL1.00001.07311.14461.21701.32291.44781.5522
1.66981.80581.96082.16002.36302.54952.67762.79662.93253.07603.19183.3641
3.46473.6381
3.82734.0073
4.09284.15554.20574.34244.48214.66564.83555.1102
5.33755.41675.61995.8675
6.07136.28246.51016.67066.7267
6.83586.9345
7.05397.12457.2941
7.47337.55127.75577.99428.20658.69349.13609.420226

QtC
0.60350.62650.66490.69780.69190.70760.74710.77860.81270.83200.82940.84100.85340.90180.94930.99841.03971.07491.11991.1526
1.17611.1784
1.22191.26461.31351.3523
1.3992
1.45201.52901.6113
1.69251.7343
1.7784
1.83481.90391.97132.02852.07822.08142.05432.09352.12912.1582
2.19592.25802.33422.39152.45462.52192.57242.50762.71862.7874

QtG
0.19280.19300.1936
0.19200.19610.20090.20090.20450.20810.20970.21280.21620.22130.22800.23140.24250.2536
0.2586
0.2637
0.27180.27930.28420.28600.28590.28730.28810.28940.29470.30030.30850.31330.3244
0.33690.34270.34790.35080.35500.36070.36950.38520.38520.37340.36790.36080.35780.36410.37040.37020.37530.39010.40140.40280.3993QtI
0.22980.24490.26550.2886
0.27280.2376
0.27250.30480.33710.35000.32390.34630.30730.33020.40890.41580.4156
0.43130.43830.45100.44790.42370.4432
0.46880.51150.5256
0.56900.62410.67580.7266
0.76810.73550.73810.76880.82900.87200.90040.88310.83000.69180.7632
0.79310.85070.8967
0.94200.99180.99001.03151.08711.12071.07671.1451
1.1775QtX
0.05970.06080.06560.07770.08390.08330.08700.0891
0.09850.10830.11990.12130.11210.10910.11800.12200.13120.14560.16910.18870.20540.21890.23410.24180.26320.29020.31390.35140.35960.37750.40890.38610.37850.38650.42380.45320.49620.53970.57090.52320.58930.63160.65700.67680.70320.70530.70870.73790.75890.76280.66290.70480.7540QtM
0.05830.06150.06830.07140.06980.06210.07420.08240.08950.09100.08500.08720.08600.09700.12050.12830.13940.14770.15350.16030.16600.16580.17750.19280.21580.23310.25340.28760.32120.35850.40510.39510.40960.43060.47790.50890.5416
0.55550.54370.47450.53580.5616
0.57530.58240.61250.64440.65370.68450.71220.72060.65600.75090.8154QtL
0.65840.6552
0.67450.70410.70650.68920.71030.73410.76690.79090.78840.79540.77930.79050.83290.85390.86600.89400.9192
0.94980.96200.94630.9572
0.97751.0092
1.04301.06291.09751.13311.1632
1.1896
1.17681.1762
1.17451.1902
1.2093
1.23671.25481.2503
1.1932
1.19861.22501.25411.27911.3072
1.33791.36041.3842
1.41191.4353
1.37151.4144
1.4682

We turn now to the APDB data on US capital stocks. The beginning of the year t capitalstocks for the above 10 reproducible investment assets are also available in the APDB. Denotet
the asset n and year t stock price index by PKn
and the corresponding quantity index byt
QKn for n = 1, . . . , 10 and t = 1970, 1971, . . . , 2022. The APDB also has current and constantdollar series for 6 types of land used by the US production sector: (11) Agricultural Land;(12) Forest Land; (13) Industrial Land; (14) Commercial Land; (15) Residential Land and (16)Other Use Land.*33 Denote the beginning of the year t price index for these land assets byt
PKn
and the corresponding quantity index by QtKn for n = 11, . . . , 16 and t = 1970, . . . , 2022.
We also constructed a beginning of the year stock of inventories by cumulating the constantdollar inventory investments, QtI17 . Denote the beginning of the year stock of inventories fortyear t by QtK17 and the corresponding beginning of the year price by PK17
. Thus Asset 17 is
Inventory Stocks. The beginning of the year prices and quantities of these 17 capital stocks
are listed in Tables A4 and A5 along with the price and quantity of a Törnqvist capital stockt
aggregate, PK
and QtK .

*33 This is mainly government owned land used for various purposes.

27

Table A4: Beginning of the Year Prices for US Capital Stocks
t
t
Year
PK
PK1
1970 1.00000 1.00000
1971 1.03507 0.83828
1972 1.10345 0.61853
1973 1.18431 0.53068
1974 1.30856 0.46660
1975 1.48624 0.41584
1976 1.63924 0.36663
1977 1.75868 0.30796
1978 1.92623 0.23635
1979 2.15475 0.17425
1980 2.38301 0.13976
1981 2.57194 0.11417
1982 2.74364 0.10239
1983 2.83876 0.08995
1984 2.85056 0.07382
1985 2.89506 0.06086
1986 3.01874 0.05195
1987 3.18727 0.04421
1988 3.36871 0.03880
1989 3.53491 0.03600
1990 3.65974 0.03307
1991 3.68149 0.02980
1992 3.65890 0.02611
1993 3.63326 0.02214
1994 3.64411 0.01906
1995 3.72662 0.01631
1996 3.83155 0.01297
1997 3.92026 0.00987
1998 4.10025 0.00749
1999 4.35702 0.00567
2000 4.59168 0.00464
2001 4.85769 0.00395
2002 5.10694 0.00334
2003 5.37056 0.00293
2004 5.72530 0.00266
2005 6.33815 0.00241
2006 7.00678 0.00212
2007 7.31432 0.00188
2008 7.32293 0.00169
2009 6.97863 0.00155
2010 6.45953 0.00147
2011 6.25588 0.00142
2012 6.40211 0.00139
2013 6.68223 0.00138
2014 7.06991 0.00138
2015 7.43901 0.00138
2016 7.72553 0.00137
2017 8.04857 0.00136
2018 8.33643 0.00137
2019 8.59752 0.00136
2020 8.94065 0.00133
2021 9.62338 0.00132
2022 10.55863 0.00136
2023
0.00139t
PK21.00000
1.00838
1.039211.071021.13276
1.246421.34056
1.349091.35076
1.38880
1.44840
1.55554
1.67315
1.757901.80933
1.854211.880211.87991
1.86465
1.85725
1.85806
1.862441.85553
1.82299
1.78496
1.73198
1.67289
1.64177
1.58669
1.48728
1.40905
1.34558
1.27160
1.14469
1.003310.92808
0.87477
0.817180.745070.670140.608070.563140.530190.49433
0.45388
0.406900.362950.33143
0.311410.293450.278960.267540.260960.25987t
PK31.00000
1.01077
1.03196
1.04656
1.10345
1.231081.36020
1.46594
1.576161.70182
1.85782
2.041402.220422.34280
2.41369
2.477862.554912.604212.61960
2.678312.761322.85033
2.941813.00552
3.07846
3.15858
3.20103
3.21579
3.21452
3.219483.245883.275633.28947
3.295813.321133.37520
3.426543.44969
3.46877
3.518023.571623.63525
3.72870
3.78532
3.81927
3.87474
3.92233
3.94559
3.99392
4.06080
4.11034
4.20456
4.479074.68263

t
t
t
t
t
PK4
PK5
PK6
PK7
PK8
1.00000 1.00000 1.00000
1.00000 1.00000
1.01270 1.02549 1.04812 1.03674 1.02191
1.03388 1.08189 1.13729 1.10176 1.06013
1.05781 1.16210 1.22115 1.17621 1.09867
1.13583 1.27739 1.33634 1.33350 1.17222
1.31097 1.41379 1.48493 1.50441 1.27338
1.47981 1.52651 1.59062 1.58969 1.35549
1.58709 1.64218 1.67626 1.64074 1.42539
1.70235 1.82017 1.81935 1.72242 1.50381
1.84024 2.03106 2.00815 1.85934 1.59704
2.03366 2.25761 2.23356 1.97887 1.72499
2.25634 2.46970 2.45694 2.15131 1.87336
2.44750 2.62307 2.64432 2.38563 2.01448
2.58669 2.71448 2.78152 2.41534 2.12744
2.65167 2.78061 2.87710 2.30488 2.20780
2.68025 2.84475 2.97382 2.31918 2.26773
2.75619 2.95429 3.08453 2.51304 2.31226
2.84697 3.10542 3.19438 2.68016 2.35217
2.92038 3.23315 3.31164 2.70913 2.40905
3.01098 3.35245 3.44227 2.77961 2.46140
3.11644 3.45033 3.56450 2.86444 2.48995
3.23929 3.49746 3.67395 2.91977 2.52419
3.34067 3.54588 3.73739 3.00363 2.56316
3.38953 3.65607 3.80267 3.01793 2.59012
3.41945 3.79991 3.91257 3.02028 2.62525
3.45457 3.94097 4.02414 3.20242 2.70237
3.47940 4.04126 4.10200 3.46325 2.75196
3.49658 4.14120 4.19533 3.52691 2.78376
3.50892 4.27259 4.34977 3.59664 2.82274
3.51845 4.41977 4.52853 3.80469 2.85399
3.52992 4.60916 4.71836 3.93239 2.92775
3.55495 4.82931 4.90161 4.04858 2.98473
3.59442 5.00458 5.05937 4.33019 3.01322
3.63435 5.18763 5.18491 4.61441 3.06728
3.68161 5.49982 5.37731 4.86045 3.13539
3.76285 5.89291 5.74757 5.38663 3.20391
3.84929 6.27266 6.21077 6.17477 3.26413
3.92875 6.49388 6.65082 6.89744 3.32914
4.03406 6.49409 6.98322 7.41837 3.41184
4.12079 6.34413 7.19388 7.60620 3.45768
4.14834 6.21856 7.20438 7.62456 3.48961
4.19179 6.20998 7.20407 7.84501 3.55908
4.28034 6.25559 7.36789 8.23375 3.63925
4.32301 6.44022 7.50852 8.48750 3.66782
4.33369 6.79439 7.67703 8.72941 3.68199
4.35757 7.10482 7.87356 9.03314 3.73498
4.36703 7.34285 8.02299 9.11715 3.72933
4.37081 7.64321 8.18329 9.25189 3.72907
4.40645 8.04312 8.45260 9.33213 3.81374
4.46766 8.39299 8.87819 9.37595 3.89988
4.51752 8.65472 9.29821 9.54427 4.01993
4.60996 9.24674 9.66473 9.79835 4.16018
4.87890 10.38661 10.85837 10.38883 4.29317
5.07353 11.07624 11.85632 10.78712 4.37197

28

t
t
t
t
t
t
t
Year
PK9
PK10
PK11
PK12
PK13
PK14
PK15
1970 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
1971 1.02806 1.04442 1.04569 1.03841 1.02560 1.02560 1.10341
1972 1.13928 1.10588 1.16544 1.11461 1.09190 1.09190 1.25098
1973 1.29004 1.14714 1.40807 1.19651 1.16534 1.16534 1.35133
1974 1.39695 1.19872 1.68306 1.30862 1.24831 1.24831 1.55640
1975 1.57257 1.28351 1.91557 1.49106 1.40287 1.40287 1.96862
1976 1.77016 1.38096 2.25512 1.68524 1.56443 1.56443 2.32907
1977 1.99358 1.45187 2.61475 1.78678 1.66694 1.66694 2.63734
1978 2.15117 1.53994 3.02686 1.88809 1.79607 1.79607 3.15812
1979 2.41192 1.63125 3.59217 1.97162 1.98392 1.98392 3.90813
1980 2.81071 1.70271 4.10491 2.07364 2.20040 2.20039 4.43091
1981 3.45597 1.79125 4.30774 2.24286 2.40080 2.40080 4.52062
1982 4.35807 1.88755 4.17254 2.36509 2.68945 2.68945 4.36832
1983 5.38089 1.97424 4.05451 2.40745 2.90741 2.90741 4.33089
1984 6.16690 2.05721 3.76294 2.39445 2.84557 2.84557 4.66055
1985 6.53028 2.14907 3.21130 2.38952 2.77969 2.77969 5.20344
1986 6.66081 2.25213 2.81405 2.39676 2.75306 2.75306 6.07430
1987 6.50696 2.34635 2.73662 2.41339 2.87630 2.80761 7.24808
1988 6.73684 2.43529 2.86115 2.50361 3.17995 2.96241 8.40756
1989 7.49507 2.55403 2.97620 2.67445 3.43780 3.05146 9.36191
1990 7.97792 2.68176 3.09466 2.87956 3.65152 3.10456 9.82555
1991 8.08371 2.82076 3.17470 3.03667 3.82909 3.03975 9.53577
1992 8.22006 2.92613 3.25202 3.15246 3.98418 2.75380 9.13590
1993 8.63403 2.96349 3.43081 3.36844 4.13513 2.35852 8.87841
1994 9.13938 3.01798 3.62183 3.60974 4.30336 2.09527 8.74019
1995 9.40005 3.09491 3.78845 3.78642 4.51136 2.08314 8.63247
1996 9.33494 3.16654 3.96694 3.93643 4.78022 2.14650 8.58164
1997 9.31067 3.22934 4.16517 4.07082 5.12684 2.24448 8.79453
1998 9.55177 3.25011 4.37084 4.23043 5.56081 2.72568 9.38040
1999 9.85281 3.29653 4.60353 4.36673 6.08130 3.26726 10.42696
2000 10.34583 3.40395 4.93527 4.46973 6.67444 3.44520 11.90367
2001 10.59102 3.48475 5.28086 4.56644 7.31174 3.71645 13.51222
2002 10.24173 3.51932 5.45543 4.57454 7.95061 3.76246 15.36732
2003 9.68646 3.54952 5.51233 4.56731 8.53766 3.75776 17.67116
2004 9.38608 3.58145 6.03167 4.63239 9.01516 3.96208 20.50270
2005 9.36275 3.59756 7.18595 4.75009 9.33032 4.45552 24.56166
2006 9.41070 3.62101 8.36558 4.93447 9.44589 5.33774 27.71376
2007 9.42137 3.65442 8.87795 5.13866 9.34992 6.14649 27.34450
2008 9.51105 3.66876 8.94943 5.33659 9.06169 6.88237 23.64786
2009 9.50982 3.67194 8.98778 5.50437 8.63181 6.52851 19.00617
2010 9.27960 3.64534 9.37018 5.52552 8.13549 4.53200 16.25553
2011 9.21338 3.60287 10.27922 5.48306 7.65966 3.61624 14.27961
2012 9.33059 3.60787 11.69926 5.48505 7.28683 4.19682 13.09658
2013 9.44274 3.61641 13.18125 5.54368 7.07956 4.48792 14.74880
2014 9.51600 3.62956 14.11635 5.66080 7.06933 5.05679 17.04571
2015 9.50026 3.68451 14.45474 5.79841 7.25248 5.87177 18.56820
2016 9.49027 3.75757 14.55009 5.90648 7.59409 6.47944 20.08394
2017 9.50629 3.81980 14.91295 5.96136 8.03723 6.97179 21.78121
2018 9.44140 3.87272 15.23799 6.00715 8.51739 7.47011 23.45716
2019 9.33298 3.92245 15.42207 6.05945 8.96974 7.68706 24.79480
2020 9.21941 3.97310 15.80484 6.09480 9.34980 8.36316 26.44699
2021 9.05170 4.04880 16.95692 6.11076 9.60509 9.71709 29.35484
2022 8.91198 4.22629 18.65843 6.18476 9.75741 11.21533 33.78415
2023 8.86838 4.35726 19.49921 6.25259 9.82352 12.07903 36.36039

29

t
PK16
1.00000
1.08980
1.25088
1.440031.709142.095592.49828
2.879933.43998
4.23965
4.902815.15616
5.06171
5.041415.23408
5.51337
6.149147.18919
8.34143
9.37472
9.969649.85503
9.64679
9.57953
9.591399.640149.74616
10.10489
10.89924
12.22069
14.06072
16.23182
18.71792
21.66917
25.46636
31.0113035.68362
36.07793
32.09902
26.59893
23.43098
21.48031
20.72792
23.44144
26.77158
28.97868
31.18148
33.72247
36.22275
38.26933
40.67494
44.99057
51.78277
55.73730t
PK171.00000
1.002430.929431.00270
1.15279
1.36087
1.39290
1.40565
1.46189
1.59806
1.81456
1.92802
2.159542.14079
2.11065
2.36233
2.424172.201892.31500
2.386292.559622.49128
2.46724
2.584432.643102.599992.742232.78756
2.590722.51246
2.50650
2.56327
2.57780
2.55926
2.20298
2.413212.809542.46424
2.77903
2.81705
2.75460
3.08338
3.24133
3.20238
3.17072
3.10056
3.263433.64123
3.10892
3.23321
3.149173.908652.90857
3.82082

Table A5: Beginning of the Year US Capital Stock Quantities
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

QtK
3.19747
3.268613.356653.45911
3.57489
3.65371
3.702623.78480
3.88794
4.01444
4.14166
4.21806
4.309644.350624.42874
4.57180
4.70776
4.82847
4.93892
5.05997
5.164385.251635.303715.369295.445995.55905
5.66495
5.804445.97484
6.15790
6.342316.552476.686216.81927
6.94936
7.10395
7.257617.424917.529897.586827.55260
7.59068
7.627197.69440
7.77324
7.87649
7.994708.08877
8.19136
8.310648.42130
8.457318.59638

QtK1
0.01349
0.01229
0.012710.01498
0.017240.02092
0.023650.029760.040780.067390.110160.17852
0.28388
0.39579
0.58179
0.90467
1.27020
1.63784
2.10333
2.54907
3.095023.48668
3.88137
4.647675.65727
6.882169.2000312.75841
18.06154
25.83353
36.85800
48.5210257.45963
65.3011073.8997883.54085
94.16128110.96673
129.58735
147.76124
164.57125
183.40342
194.40635
207.23516
214.96042
219.34074
222.23249
224.06239
229.26620
242.85332
254.21506
271.41805
292.82907

QtK2
0.020220.022100.023170.023620.025160.02655
0.027230.028400.03182
0.03615
0.04169
0.04775
0.05306
0.057430.06088
0.065530.07066
0.076020.08030
0.085820.089900.09428
0.09668
0.09991
0.10350
0.11082
0.121250.134200.149540.16852
0.19498
0.23495
0.260160.265110.279340.300850.322360.35770
0.40606
0.45317
0.486620.54767
0.605100.663120.73695
0.827940.948141.09075
1.250131.38366
1.49027
1.57238
1.68237

QtK3
0.07947
0.07847
0.078640.08168
0.088070.090900.09009
0.09176
0.097420.104210.11084
0.11058
0.10954
0.106030.10587
0.109170.112200.11267
0.11249
0.11468
0.11334
0.112540.11271
0.11387
0.119070.12783
0.13601
0.144030.15285
0.163620.18008
0.189900.192290.190940.18683
0.19085
0.199050.210140.217240.211350.185210.182610.19100
0.204850.22096
0.23958
0.26147
0.27375
0.28246
0.295360.30299
0.28898
0.27522QtK40.232450.237850.23906
0.244390.25743
0.27156
0.27715
0.28298
0.292590.30725
0.323520.33338
0.34269
0.34416
0.34378
0.35205
0.361810.371100.37867
0.38658
0.399840.40888
0.409850.410110.41535
0.42338
0.435540.44902
0.46313
0.481030.49706
0.51724
0.531100.54187
0.55566
0.570850.592150.61969
0.64662
0.66883
0.667470.67091
0.68371
0.702590.72230
0.74575
0.76124
0.77201
0.78574
0.80353
0.82364
0.831230.8452330

QtK50.55234
0.56894
0.59680
0.632310.66572
0.68438
0.69563
0.716610.74855
0.78349
0.81373
0.828020.837470.837630.855160.88057
0.906640.93982
0.97238
1.002451.027421.04569
1.05728
1.07502
1.09824
1.12626
1.15072
1.18118
1.21189
1.248451.28913
1.32886
1.36740
1.41070
1.46157
1.52006
1.58040
1.63288
1.658171.65498
1.636101.61555
1.59527
1.58317
1.58182
1.58371
1.59388
1.609451.62488
1.642101.65813
1.67974
1.71163

QtK6
0.32395
0.33003
0.335020.34004
0.34724
0.35287
0.35357
0.35318
0.352230.35575
0.36399
0.37201
0.38165
0.391280.39706
0.40866
0.425290.438290.449730.46082
0.472390.48474
0.489310.489930.49000
0.49173
0.498890.51065
0.52549
0.54247
0.55851
0.57587
0.590620.596170.59945
0.602290.602420.607700.617800.62863
0.62806
0.61655
0.60393
0.593410.58466
0.57976
0.58167
0.58792
0.59260
0.59897
0.60560
0.609200.60864

QtK7
0.50843
0.528850.54801
0.566440.585390.60358
0.622850.64301
0.66458
0.68992
0.716500.74820
0.781070.808610.830290.85757
0.884390.902070.919310.936150.95002
0.96804
0.98451
0.99673
1.01292
1.02669
1.03962
1.05329
1.06836
1.08289
1.09714
1.11651
1.13839
1.15524
1.17198
1.18576
1.19660
1.213291.232191.25182
1.26801
1.283111.29843
1.31535
1.332141.352431.36701
1.37492
1.38155
1.395101.41086
1.41980
1.42583

QtK80.07483
0.08635
0.09535
0.10365
0.111310.117130.121620.126750.13223
0.138630.14584
0.153480.16167
0.169930.178960.190180.20364
0.21663
0.229330.24134
0.253440.265850.278090.28861
0.29652
0.302970.310210.320420.33234
0.346210.36233
0.380900.39844
0.412390.425620.438080.452840.470360.489650.50797
0.52124
0.53269
0.54526
0.555880.57029
0.586170.60396
0.628130.65237
0.67984
0.713080.74500
0.78449

Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

QtK9
0.00657
0.00701
0.00726
0.007410.00768
0.00830
0.008800.009200.009130.00957
0.009960.01060
0.010620.010560.010280.010630.011200.01205
0.013280.014360.015480.01683
0.018100.019220.019960.02050
0.021130.02267
0.02533
0.02849
0.03257
0.036390.039200.041450.04399
0.04698
0.05011
0.05327
0.057090.06092
0.064200.067080.07092
0.075450.079840.084630.08948
0.09606
0.104140.11443
0.124370.135130.14898

QtK10
0.01482
0.014390.01404
0.01404
0.01389
0.01382
0.013650.01450
0.015640.016390.01766
0.01806
0.019420.02046
0.02157
0.023140.02425
0.02583
0.02684
0.02766
0.02863
0.029630.03046
0.03149
0.03274
0.034170.03606
0.038250.040200.042280.044100.046210.04733
0.048820.05099
0.05279
0.055410.05751
0.05868
0.05857
0.05791
0.05894
0.059710.060480.06108
0.062100.06350
0.06526
0.067150.069190.070930.071360.07133

QtK11
0.169020.166180.16369
0.161860.160970.160540.160710.160930.160610.160130.15898
0.157540.156190.155180.154480.15392
0.153360.152650.15192
0.151080.150290.14975
0.14965
0.149680.150080.15059
0.15100
0.151110.151220.150890.150240.149390.148510.14795
0.147460.14699
0.14649
0.14595
0.145260.14492
0.144800.14478
0.14475
0.14458
0.14434
0.14400
0.14324
0.142480.142310.14201
0.14201
0.14201
0.14201

QtK12
0.12774
0.128020.128310.12858
0.128870.129150.129150.129150.128960.129400.129260.129120.12898
0.12798
0.12784
0.127700.12756
0.127420.12853
0.128480.128420.12836
0.12835
0.128700.12906
0.129420.129780.130140.130170.130200.130240.130270.130480.130560.130640.13072
0.130800.13088
0.131420.131960.132510.13305
0.13349
0.13361
0.13373
0.133850.133970.133970.13384
0.13384
0.13384
0.13384
0.13384

QtK13
0.227810.22700
0.225930.224160.222500.220420.218640.21732
0.215860.213610.21092
0.20767
0.204220.200540.19792
0.196190.19473
0.19287
0.190810.18779
0.182560.175620.167390.16026
0.154430.151210.150410.15174
0.15258
0.152760.15232
0.15135
0.14987
0.148500.147460.146450.14487
0.143500.14080
0.13906
0.13768
0.137170.137390.139390.140620.141120.141210.14153
0.14158
0.14071
0.141400.142100.1419731

QtK14
0.538390.54546
0.55832
0.574970.59037
0.592790.60084
0.61725
0.631180.655240.67062
0.67361
0.683270.685040.705250.729550.75098
0.773780.781210.80666
0.829440.84234
0.84467
0.86473
0.87799
0.896150.909540.93449
0.96423
0.994161.01901
1.04771
1.05961
1.07610
1.09532
1.11509
1.13964
1.15652
1.165221.16973
1.15305
1.16927
1.17963
1.195211.20240
1.222091.245691.264441.28670
1.30863
1.328171.29850
1.34401

QtK15
0.227920.23201
0.23762
0.24226
0.24698
0.25066
0.25575
0.259450.264910.26865
0.278530.284160.28857
0.290800.295850.301290.30724
0.31134
0.31726
0.32338
0.326080.33009
0.33526
0.338700.34193
0.34868
0.35182
0.357170.36557
0.37357
0.380090.39476
0.40224
0.41253
0.41904
0.42752
0.435130.44466
0.445630.445410.44504
0.45101
0.453080.455800.458170.46235
0.466130.46738
0.47155
0.474580.474190.47656
0.48259

QtK16
0.03999
0.04158
0.04287
0.043780.044040.04405
0.04378
0.04352
0.04367
0.043610.04415
0.04503
0.045880.04708
0.047480.04776
0.04803
0.048450.04808
0.048490.04898
0.049270.049210.048890.04833
0.04757
0.04699
0.04651
0.046140.04607
0.04627
0.046410.046640.04663
0.046700.04669
0.04673
0.04674
0.046850.04638
0.04615
0.045640.04533
0.04527
0.045280.045290.04559
0.04605
0.046100.046200.046210.046140.04593

QtK17
0.040020.042020.050900.05998
0.073760.08404
0.079540.09172
0.10699
0.123130.13304
0.12976
0.14356
0.13659
0.13387
0.161560.170560.173540.18527
0.193020.20383
0.20967
0.20951
0.21583
0.22371
0.24824
0.259610.27066
0.29803
0.323390.34766
0.36894
0.35409
0.36189
0.368290.39484
0.415310.44330
0.45552
0.445140.390410.40789
0.422160.444400.47767
0.50501
0.54795
0.55868
0.569190.586640.60955
0.59994
0.60395

Year t nominal Gross Domestic Product (GDP), VYt , is defined as follows:tt
VYt ≡ PCt QtC + PGt QtG + PIt QtI + PX
QtX − PMQtM ;t = 1970, . . . , 2022.

(A1)

The year t price index for GDP is defined as a Törnqvist price index of the 5 component priceand quantity series for consumption, government output, investment, exports and (minus)
imports, which we denote by PYt . The companion quantity index for GDP is denoted by QtY .
VYt , PYt and QtY are listed in Table A6 below.
Now that we have estimates for the US capital stock, we can calculate real and nominal capitalt, for each year t. The real capital output
output ratios for the US economy, QtKY and VKYtttratio, QKY ≡ QK /QY , is an indicator of partial efficiency while the nominal capital outputtratio, VKY≡ VKt /VYt , is a partial indicator of the international competitiveness of the US
economy: it indicates how many dollars are required to purchase a sufficient amount of capitaltin order to produce one dollar of output in year t (on average). The lower VKY
is, the easiertt
it is to start a business in the US in year t. VKY and QKY are listed in Table A6 below.
In order to indicate the importance of land as a component of the total capital stock, wetconstruct a year t Törnqvist capital stock aggregate price PKR
that excludes the 6 land
stocks. Denote the companion year t reproducible capital stock quantity by QtKR and definett
the corresponding year t value by VKR≡ PKR
QtKR . Define the year t reproducible real andtttt
nominal capital output ratios as QKRY ≡ QKR /QtY and VKRY
≡ VKR/VYt . These variables
are also listed in Table A6 below.

32

Table A6: US GDP Aggregates and Nominal and Real Capital Output Ratios
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

t
VKY
3.11203.03753.02173.00103.16433.3652
3.37733.32823.31103.41943.59403.53183.6772
3.53173.2512
3.17003.21973.28693.29653.28783.28843.26243.0921
2.95332.83412.81872.79292.75572.80722.89012.9467
3.11363.23833.3142
3.37933.5831
3.8226
3.89163.87043.78033.3556
3.15443.11423.16003.23953.32303.40543.44183.43273.44713.65293.57983.66183.2903QtKY
3.11203.07332.99582.92363.04233.12193.00492.93072.85232.85412.95012.9276
3.05872.96212.81022.79112.79202.76622.71642.68812.69062.73992.67802.63752.57172.55762.50812.45932.42602.38922.36762.42552.43512.41622.37122.34482.32982.33832.36742.44982.37462.35162.31132.28202.2495
2.21682.21302.18572.15032.12912.20922.10032.09982.5802

t
VKRY
QtKRY
1.8167 1.8167
1.7804 1.8132
1.7645 1.7854
1.7532 1.7630
1.8696 1.8609
1.9853 1.9344
1.9489 1.8685
1.9020 1.8354
1.8795 1.8065
1.9216 1.8312
2.0241 1.9150
2.0340 1.9195
2.1731 2.0260
2.0994 1.9737
1.9417 1.8769
1.9236 1.8819
1.9731 1.8979
1.9931 1.8931
1.9599 1.8755
1.9309 1.8649
1.9368 1.8797
1.9578 1.9271
1.9095 1.8919
1.8747 1.8690
1.8394 1.8324
1.8556 1.8315
1.8589 1.8071
1.8298 1.7811
1.8194 1.7666
1.8238 1.7517
1.8310 1.7513
1.8936 1.8040
1.9447 1.8223
1.9546 1.8138
1.9401 1.7891
1.9947 1.7798
2.0900 1.7790
2.1529 1.7977
2.2322 1.8363
2.3120 1.9100
2.2034 1.8523
2.1634 1.8306
2.1415 1.7995
2.1286 1.7780
2.1260 1.7578
2.1346 1.7361
2.1481 1.7380
2.1408 1.7217
2.0995 1.6966
2.1031 1.6863
2.2017 1.7587
2.1163 1.6840
2.1038 1.6812
1.9913 1.8223

t
PYt
PK
QtY
1.0000 1.0000 1.0275
1.0473 1.0351 1.0635
1.0940 1.1035 1.1205
1.1538 1.1843 1.1832
1.2581 1.3086 1.1751
1.3788 1.4862 1.1704
1.4585 1.6392 1.2322
1.5487 1.7587 1.2914
1.6593 1.9262 1.3631
1.7985 2.1548 1.4066
1.9561 2.3830 1.4039
2.1319 2.5719 1.4408
2.2822 2.7436 1.4090
2.3809 2.8388 1.4688
2.4640 2.8506 1.5759
2.5491 2.8951 1.6380
2.6177 3.0187 1.6862
2.6824 3.1873 1.7455
2.7759 3.3687 1.8182
2.8900 3.5349 1.8824
2.9944 3.6597 1.9194
3.0919 3.6815 1.9167
3.1689 3.6589 1.9805
3.2448 3.6333 2.0358
3.3066 3.6441 2.1177
3.3815 3.7266 2.1735
3.4409 3.8316 2.2587
3.4986 3.9203 2.3602
3.5435 4.1002 2.4628
3.6019 4.3570 2.5774
3.6894 4.5917 2.6788
3.7842 4.8577 2.7015
3.8403 5.1069 2.7458
3.9155 5.3706 2.8223
4.0174 5.7253 2.9307
4.1478 6.3382 3.0296
4.2705 7.0068 3.1151
4.3949 7.3143 3.1753
4.4793 7.3229 3.1806
4.5224 6.9786 3.0969
4.5712 6.4595 3.1806
4.6638 6.2559 3.2279
4.7516 6.4021 3.3000
4.8257 6.6822 3.3717
4.9093 7.0699 3.4555
4.9627 7.4390 3.5531
5.0205 7.7255 3.6126
5.1113 8.0486 3.7007
5.2221 8.3364 3.8094
5.3104 8.5975 3.9033
5.4071 8.9406 3.8119
5.6460 9.6234 4.0268
6.0547 10.5586 4.0939
3.3758 4.6218 2.3854

33

QtK
3.1975
3.26863.35673.4591
3.57493.65373.7026
3.78483.88794.01444.1417
4.21814.30964.35064.42874.57184.70784.82854.93895.06005.16445.25165.30375.36935.44605.55905.66495.80445.97486.15796.34236.55256.68626.8193
6.94947.10397.25767.42497.52997.58687.55267.59077.6272
7.69447.77327.8765
7.99478.08888.19148.31068.42138.45738.59645.8789VYt
1.0275
1.11381.22581.36511.4784
1.61371.79712.00002.26192.52972.7462
3.07173.21553.49703.88304.17534.41394.6821
5.04715.44025.74755.92626.27596.60567.00247.34977.77188.25738.72709.28359.883010.2228
10.5445
11.0505
11.7738
12.5662
13.303113.9551
14.2470
14.0055
14.5389
15.054215.6800
16.270916.964217.6328
18.1370
18.9154
19.8929
20.7281
20.611622.7353
24.78719.3025t
VK
3.19753.38333.70394.09674.67805.43036.06956.65637.48918.65019.869610.8486
11.8241
12.350312.6244
13.2356
14.211515.3896
16.6378
17.8865
18.900319.3339
19.4057
19.5080
19.8458
20.7165
21.7055
22.7549
24.4983
26.8301
29.1218
31.829934.1461
36.6233
39.7872
45.0259
50.852554.3082
55.140952.945648.786347.4864
48.830151.4157
54.9561
58.5933
61.763365.1030
68.2866
71.450975.2918
81.3879
90.7660
31.238t
VKR
1.8666
1.98302.16292.39322.76403.20363.50243.80404.25124.86125.55866.24796.98777.34157.53968.03198.70909.33229.891810.5042
11.1315
11.6025
11.9839
12.3833
12.8801
13.6384
14.4473
15.109315.8779
16.931618.0958
19.3582
20.5064
21.5995
22.842925.065427.803630.044431.802232.3805
32.0357
32.5689
33.5791
34.6347
36.0651
37.6390
38.960740.4937
41.7653
43.592945.381648.114652.1475
19.121

The real capital output ratio, QtKY , fell from 3.12 in 1975 to 2.10 in 2022. This is probably areflection of the growth in services relative to goods: services typically require less capital toproduce a dollar of output. However, the nominal capital output ratio grew from 3.11 in 1970
to 3.95 in 1980 and then fell to 2.75 in 1997. It rose to 3.89 in 2007 (due to rising land prices)
when the great recession hit the US economy. The capital output ratio fell to 3.11 in 2012
and then rose as a new property price bubble occurred, to finish at 3.66. However, overall, theUS economy has not varied too much from the average nominal capital output ratio of 3.29 sothat the economy has remained competitive over the years. The average share of reproducible
capital in the total capital stock was 61.2% so the average share of land was 38.8%. Thus land
is a huge missing input in most macroeconomic production models of the US economy.
We turn our attention to the construction of user costs for the US economy. User costs requiredepreciation rates for assets. Assets 11-17 are non-depreciable assets so we set the period t
depreciation rate for these assets, δnt , equal to 0 for n = 11, . . . , 17 and t = 1970, . . . , 2022.
Since the APDB uses the geometric model for depreciation for reproducible assets 1-10, the
APDB depreciation rates satisfy the following equations:
t
t
t
Qt+1
Kn = (1 − δn )QKn + QIn ;n = 1, . . . , 10; t = 1970, . . . , 2022.

(A2)

We can solve equations (A1) for the depreciation rates δnt for the years 1970-2022 for assets
1-10. The resulting nonzero depreciation rates are listed below in Table A7.

34

Table A7: Geometric Depreciation Rates for Reproducible US Capital Stocks
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

δ1t
0.32975
0.351480.377370.37153
0.38259
0.368080.389810.409440.458370.45576
0.45305
0.44766
0.41377
0.427150.44165
0.41571
0.395450.39455
0.38201
0.383390.367240.365120.38152
0.38386
0.38456
0.40447
0.41261
0.417720.42032
0.419700.400190.37753
0.36918
0.36860
0.36962
0.37194
0.37632
0.374500.37083
0.365200.36565
0.356640.358110.35198
0.348930.34783
0.34730
0.35223
0.35618
0.353700.357780.360310.360310.38427δ2t
0.293760.287450.283890.28976
0.288480.284170.286500.297390.29955
0.302340.301120.29613
0.292180.28933
0.291390.29183
0.291110.288390.290100.288080.28733
0.284240.287110.286420.292090.29446
0.29574
0.296900.29872
0.30294
0.309280.295820.28337
0.28847
0.292930.29497
0.29594
0.299540.29794
0.29100
0.29857
0.296090.29525
0.296290.297870.30106
0.302110.30378
0.29604
0.29156
0.28892
0.29158
0.29158
0.29356

δ3t
0.244270.246050.251020.25535
0.24979
0.244630.247880.253200.254160.253810.24557
0.244530.241830.245830.249620.249400.246120.245430.24798
0.245100.24475
0.24598
0.248790.252100.256310.254240.253130.253520.254700.258400.252360.247450.244960.243420.249890.255280.25253
0.249870.243450.230590.244380.252140.255820.25546
0.256030.257050.251910.252290.25179
0.249060.240640.241030.241030.24909δ4t
0.197860.19606
0.198230.200740.20099
0.19751
0.19744
0.198730.200270.20095
0.198620.198190.19596
0.19569
0.19793
0.198480.19789
0.197390.197420.19965
0.19761
0.195810.19733
0.19750
0.19891
0.19896
0.19869
0.19884
0.19960
0.19908
0.19941
0.198290.19758
0.19832
0.19966
0.203340.20002
0.19974
0.19989
0.195360.19637
0.198130.19937
0.19833
0.19864
0.19764
0.197290.19984
0.19817
0.198110.19699
0.19828
0.19828
0.19840

δ5t
0.04956
0.049520.049970.04949
0.04958
0.04946
0.049360.049390.049360.04972
0.04959
0.049430.049510.04974
0.049540.04972
0.049360.04938
0.049360.050200.04938
0.049540.051080.050150.050850.05006
0.049620.049710.049760.049760.049420.049620.04956
0.049770.05082
0.053290.049450.049460.050410.049520.049800.050160.050540.049540.049440.04956
0.04984
0.051840.049880.049610.05000
0.05050
0.050500.0498835

δ6t
0.084590.08456
0.08500
0.084530.084610.084500.084400.084420.084390.08475
0.084620.084460.084540.084770.08457
0.08475
0.084390.084410.084390.085230.084410.08457
0.086110.085180.085880.085090.084650.08474
0.08479
0.08479
0.084450.08466
0.084590.084810.08586
0.088320.084480.08450
0.085440.08455
0.08483
0.085190.08557
0.08458
0.084470.08460
0.08487
0.08687
0.084910.084640.085030.085530.085530.08491

δ7t
0.02627
0.026240.02668
0.026210.02630
0.02618
0.02608
0.026100.02607
0.026430.026310.02615
0.026220.026450.02625
0.026430.02608
0.02609
0.02607
0.02691
0.02609
0.02625
0.027800.02687
0.02757
0.02678
0.02633
0.026430.02647
0.026480.026130.02634
0.02628
0.02649
0.027540.03000
0.026160.02618
0.027130.026240.02652
0.026880.02725
0.02626
0.026160.02628
0.02656
0.02855
0.02660
0.02633
0.02672
0.027220.027220.02660δ8t
0.22606
0.22084
0.219030.217650.215390.21392
0.21433
0.21443
0.21498
0.215360.215400.215500.21526
0.21548
0.216480.21733
0.21659
0.21605
0.215400.215160.21504
0.21473
0.21387
0.21278
0.21218
0.212410.21335
0.213800.214280.21479
0.215280.21473
0.21357
0.213270.212970.21344
0.213960.21420
0.21383
0.212640.212200.212370.21194
0.212620.21282
0.21308
0.214100.21395
0.21432
0.215030.21460
0.21546
0.21546
0.21479

δ9t
0.40849
0.402020.39933
0.40258
0.411140.407160.404210.39372
0.40475
0.403290.40774
0.395640.394190.389870.40196
0.405800.410220.415370.411280.41067
0.412520.410070.40743
0.402850.400570.40135
0.40962
0.41837
0.41986
0.42356
0.418400.41047
0.40653
0.40735
0.40867
0.40838
0.40767
0.409400.40849
0.405840.40408
0.406540.40785
0.406700.40708
0.406540.40974
0.411850.41475
0.41238
0.41232
0.41547
0.415470.40769t
δ100.307820.308440.312420.31057
0.31158
0.31043
0.32201
0.32460
0.319810.324480.315800.324140.320700.32077
0.32368
0.319800.322480.318450.317120.317780.317810.31667
0.31760
0.31853
0.319150.32098
0.321800.320270.320410.31905
0.319810.316090.31725
0.319240.317860.320080.318220.315510.31204
0.310560.315120.314360.31433
0.313880.314920.315850.31665
0.316860.31706
0.31626
0.313260.312280.312280.31707

We now explain the user cost of capital methodology developed by Jorgenson and his coworkt
ers.*34 Suppose the beginning of the year t price of a new unit of capital stock n is PKn
andta producer faces an annual cost of capital at the beginning of year t of r . Thus the cost to
the producer is the total of the purchase of one unit of capital stock n at the beginning oftyear t plus the interest cost of financing the purchase, which is equal to (1 + rt )PKn
. Supposetfurther that asset n in year t has a geometric depreciation rate equal to δn . Various levels ofgovernment may charge an annual tax on the use of each unit of asset n at the tax rate τnt att
the beginning of year t so that the total tax charged on asset n during year t is τnt PKn
QtKn .
We think of firms buying their capital stocks at the beginning of year t, using the servicesof these stocks during period t and then selling the used assets at the end of the period at
prevailing prices at the end of the period. The year t user cost Unt of asset n is the resulting
per unit net cost.*35 Using the geometric model of depreciation, we can use the price of at
new unit of capital stock n at the beginning of year t, the PKn, listed in Table A4 as theappropriate price of the year t beginning of the year capital stock quantity QtKn for asset n.
tDefine the year t ex post inflation rate for asset n, itn , as follows where PKn
is the beginningof year t price for a unit of asset n:t+1t(1 + itn ) ≡ PKn;/PKn

n = 1, . . . , 17; t = 1970, . . . , 2022.

(A3)

Thus the year t user cost for asset n, Unt , times the initial year t capital stock, QtKn , is equal
to the following expression:
Unt QtKn ≡ Beginning of period cost − end of period benefit
n = 1, . . . , 17; t = 1970, . . . , 2022
t+1tt
(1 − δnt )QtKn
QtKn } − PKn= {(1 + rt )PKn
QtKn + τnt PKntt= [1 + rt + τnt ]PKn
QtKn − PKn(1 + itn )(1 − δnt )QtKn

= [r

t

t
+ τnt − itn + δnt (1 + itn )]PKn
QtKn .

using (A3)(A4)

tThus the year t Jorgensonian user cost for asset n is Unt = [rt + τnt − itn + δnt (1 + itn )]PKn
.*36
t
t
t
t
Note that the end of period benefit part of the user cost formula, PKn (1 + in )(1 − δn )QKn ,decreases as the depreciation rate δnt increases and increases as the asset inflation rate itn
increases. For land assets, where the depreciation rate is zero, the annual asset inflation rate
could be so great that the overall user cost becomes negative instead of being positive. In thiscase, the user cost becomes a user benefit; i.e., instead of the purchase of the asset, using it
and then selling it at the end of the period being a cost, it becomes an addition to revenues.
In order to calculate user costs, we require information on the cost of capita rt , asset tax
rates τnt , asset inflation rates itn , depreciation rates δnt and beginning of the year asset pricest
PKn. Alternative methods for estimating the cost of capital will be discussed below, asset
tax rates will be listed in Table A8 below, ex post asset inflation rates as well as asset prices
can be calculated using the information in Table A4 on beginning of the year asset prices anddepreciation rates are listed in Table A7 above.

*34 See Jorgenson (1963)[29],Jorgenson and Griliches (1967)[31], (1972)[32], Christensen and Jorgenson(1969)[7] and Jorgenson (1989)[30].
*35 This is the method used by Diewert (1980, 472–473)[14] (2014)[16] to derive a user cost formula.
*36 This user cost formula was also derived by Christensen and Jorgenson (1969)[7] using a different methodof derivation. Jorgenson was a firm advocate of using ex post asset inflation rates in a user cost formula
so we term the user costs defined by (A4) as Jorgensonian user costs.

36

Table A8: Asset Tax Rates (Percentages)
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

τ1t
0.653
0.529
0.579
0.580
0.521
0.502
0.478
0.469
0.359
0.348
0.317
0.354
0.378
0.373
0.386
0.403
0.404
0.391
0.409
0.420
0.427
0.465
0.474
0.478
0.504
0.478
0.460
0.461
0.421
0.415
0.423
0.390
0.405
0.397
0.386
0.348
0.338
0.331
0.316
0.330
0.356
0.335
0.323
0.325
0.327
0.322
0.324
0.314
0.306
0.307
0.323
0.302
0.293
0.401

τ2t
0.653
0.661
0.653
0.620
0.588
0.548
0.538
0.503
0.454
0.388
0.376
0.393
0.408
0.417
0.439
0.444
0.438
0.427
0.424
0.435
0.450
0.492
0.513
0.515
0.533
0.517
0.527
0.528
0.480
0.458
0.445
0.419
0.423
0.388
0.381
0.362
0.350
0.338
0.312
0.327
0.344
0.332
0.313
0.315
0.309
0.304
0.307
0.304
0.296
0.301
0.319
0.294
0.286
0.426

τ3t
0.493
0.499
0.487
0.465
0.444
0.418
0.412
0.400
0.348
0.301
0.287
0.299
0.309
0.315
0.331
0.336
0.335
0.324
0.323
0.333
0.344
0.377
0.394
0.396
0.412
0.402
0.405
0.400
0.372
0.359
0.344
0.327
0.329
0.318
0.305
0.283
0.274
0.271
0.253
0.267
0.281
0.260
0.249
0.252
0.256
0.254
0.257
0.245
0.242
0.243
0.250
0.232
0.231
0.331

τ4t
0.762
0.774
0.752
0.727
0.692
0.662
0.636
0.616
0.539
0.467
0.449
0.462
0.477
0.489
0.508
0.515
0.521
0.505
0.504
0.515
0.535
0.584
0.609
0.610
0.632
0.619
0.626
0.620
0.577
0.555
0.530
0.506
0.511
0.494
0.472
0.439
0.426
0.411
0.384
0.382
0.391
0.373
0.356
0.354
0.354
0.350
0.338
0.328
0.320
0.326
0.341
0.319
0.312
0.501

τ5t
2.105
2.165
2.118
2.065
1.906
1.768
1.739
1.724
1.519
1.304
1.235
1.267
1.301
1.339
1.414
1.430
1.446
1.413
1.398
1.426
1.466
1.586
1.681
1.714
1.768
1.731
1.738
1.738
1.617
1.560
1.496
1.423
1.422
1.389
1.339
1.238
1.200
1.250
1.367
1.560
1.674
1.782
1.788
1.698
1.656
1.591
1.583
1.571
1.532
1.529
1.485
1.406
1.301
1.565

τ6t
2.151
2.257
2.180
2.095
1.951
1.808
1.760
1.742
1.537
1.330
1.264
1.299
1.341
1.377
1.448
1.472
1.466
1.434
1.431
1.457
1.506
1.638
1.705
1.740
1.799
1.762
1.772
1.775
1.660
1.597
1.525
1.445
1.455
1.401
1.356
1.272
1.236
1.223
1.192
1.342
1.512
1.488
1.485
1.489
1.449
1.431
1.384
1.346
1.327
1.369
1.304
1.231
1.167
1.532

τ9t
0.247
0.255
0.260
0.245
0.219
0.215
0.206
0.210
0.168
0.162
0.144
0.165
0.163
0.174
0.168
0.172
0.163
0.161
0.169
0.175
0.171
0.188
0.197
0.206
0.207
0.202
0.199
0.204
0.189
0.184
0.177
0.163
0.160
0.155
0.151
0.141
0.137
0.139
0.124
0.135
0.144
0.133
0.131
0.137
0.142
0.141
0.143
0.131
0.129
0.128
0.134
0.112
0.114
0.169

37

t
τ10
0.247
0.259
0.246
0.238
0.218
0.207
0.205
0.198
0.175
0.149
0.141
0.148
0.153
0.159
0.167
0.170
0.169
0.165
0.164
0.169
0.174
0.191
0.197
0.198
0.207
0.202
0.205
0.203
0.187
0.183
0.175
0.165
0.165
0.160
0.152
0.141
0.138
0.140
0.123
0.136
0.145
0.132
0.131
0.136
0.142
0.142
0.144
0.132
0.131
0.130
0.136
0.114
0.118
0.168

t
τ11
1.085
1.136
1.129
1.143
0.993
0.925
0.955
0.898
0.805
0.692
0.636
0.628
0.636
0.684
0.662
0.672
0.688
0.730
0.722
0.736
0.762
0.820
0.873
0.896
0.912
0.899
0.908
0.905
0.839
0.814
0.783
0.738
0.727
0.704
0.721
0.675
0.638
0.637
0.664
0.752
0.818
0.788
0.734
0.662
0.645
0.628
0.650
0.661
0.650
0.677
0.658
0.617
0.585
0.773

t
τ13
1.034
1.064
1.050
1.001
0.926
0.889
0.862
0.833
0.738
0.641
0.605
0.622
0.671
0.654
0.667
0.699
0.681
0.717
0.704
0.712
0.731
0.793
0.832
0.841
0.871
0.858
0.874
0.872
0.817
0.787
0.752
0.713
0.717
0.688
0.650
0.595
0.574
0.593
0.618
0.723
0.835
0.842
0.832
0.808
0.790
0.765
0.763
0.752
0.734
0.743
0.702
0.662
0.595
0.764

t
τ14
1.299
1.336
1.318
1.257
1.163
1.116
1.082
1.046
0.927
0.805
0.760
0.781
0.843
0.822
0.837
0.878
0.856
0.881
0.865
0.872
0.902
0.947
0.953
0.950
1.038
1.076
1.070
1.096
1.126
0.987
0.910
0.913
0.822
0.880
0.802
0.818
0.774
0.773
0.781
0.820
0.838
1.222
1.123
1.055
1.047
0.984
0.947
0.924
0.893
0.884
0.883
0.805
0.721
0.953

t
τ15
1.875
2.058
1.899
1.837
1.768
1.700
1.569
1.587
1.404
1.228
1.063
1.090
1.090
1.213
1.298
1.343
1.364
1.339
1.299
1.305
1.295
1.361
1.468
1.467
1.553
1.489
1.553
1.542
1.491
1.439
1.401
1.303
1.344
1.288
1.247
1.169
1.068
1.084
1.137
1.358
1.612
1.610
1.768
1.767
1.601
1.536
1.505
1.490
1.426
1.433
1.391
1.295
1.196
1.434

t
τ16
0.278
0.302
0.287
0.283
0.260
0.247
0.238
0.234
0.209
0.181
0.161
0.163
0.163
0.179
0.187
0.194
0.199
0.198
0.193
0.195
0.193
0.204
0.220
0.220
0.231
0.224
0.231
0.230
0.222
0.214
0.208
0.195
0.199
0.192
0.186
0.175
0.160
0.163
0.141
0.171
0.153
0.158
0.166
0.184
0.173
0.190
0.148
0.116
0.116
0.124
0.139
0.122
0.102
0.193

t
τ17
0.248
0.268
0.260
0.255
0.225
0.190
0.213
0.208
0.180
0.137
0.124
0.131
0.130
0.153
0.169
0.148
0.161
0.158
0.152
0.150
0.161
0.172
0.193
0.190
0.204
0.179
0.181
0.197
0.181
0.171
0.158
0.145
0.159
0.160
0.157
0.132
0.141
0.130
0.114
0.127
0.147
0.124
0.125
0.131
0.135
0.125
0.108
0.118
0.117
0.120
0.096
0.113
0.102
0.160

In order to save space, the tax rates listed in Table A8 are in percentage terms; thus theactual 1970 tax rate for Asset 1 is 0.00653 instead of 0.653. The tax rates for assets 7 (Other
Structures), 8 (Research and Development) and 12 (Forest Land) were zero for all years.
The economy wide ex post rate of return in year t, rJt , can be calculated using Jorgensonian
user costs: simply set the value of output in year t, VYt , less the value of labour input, VLt ,equal to the sum of the year t user cost values defined by (A4):
VYt = VLt +

∑17

t
t
t
t
t
t
t
n=1 [rJ + τn − in + δn (1 + in )]PKn QKn ;t = 1970, . . . , 2022.

(A5)

The rJt are listed in Table A9 below. Once the ex post rates of return are available, Jorgensonian user cost can be calculated. However, the resulting user costs for land assets were
extremely volatile and all land assets had negative Jorgensonian user costs.*37 But when weaggregated the user costs over all 17 assets using a Törnqvist quantity index to do the aggregation, we found that the resulting capital services aggregate was fairly smooth and well
behaved. The aggregate user cost of capital was always positive. Thus the problem of negative
user costs seems to arise in a visible manner only if we want to estimate production functions
that distinguish different types of capital where land is included as an asset.
We would like user costs to approximate rental prices for assets. But rental prices are always
positive so Jorgensonian user costs are not good approximations to rental prices when land
assets are in scope. Owners of assets who rent them to producers and determine the rental
prices at the beginning of the year are not able to accurately forecast the end of year prices fortheir assets. Thus they form expectations about year end asset prices when determining rental
prices. This suggests that we should use expected inflation rates in the user cost formulae inplace of actual ex post prices. The problem is that there are many methods for forming
expectations. Our approach to this problem is to simply smooth the actual ex post asset
inflation rates by a more or less arbitrary method and use these smoothed rates in place of
the actual rates. We use a centered 7 year moving average of actual asset inflation rates as an
approximation to expected asset inflation rates. Denote this smoothed inflation rate for asset∗
n in year t as itn .*38 These smoothed inflation rates are listed in Table A9 below. Now replace∗
the itn in equations (A5) by their smoothed counterparts itn , replace rJt by rt and solve theresulting equation for rt . The rt are also listed in Table A9.

*37 Here is a list of the land and inventory assets (11 to 17) and the number of negative user costs: 11: 10;

12: 4; 13: 0; 14: 8; 15: 16; 21; 17: 9. Thus there was a total of 68 negative user costs. The user costs
for assets 1-10 were all positive.
∗∗*38 For 1970, set i1970 = (1/2)(i1970 + i1971 ), for 1971, use the 3 year centered moving average i1971 =nnnn) and for year 3, use a 5 year centered average. Similar adjustments to the 7+ i1972+ i1971(1/3)(i1970nnnyear centered moving average rule were made for the final 3 years in our sample.

38

Table A9: Alternative Rates of Return and Smoothed Asset Inflation Rates
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

∗

∗

∗

∗

∗

∗

∗

∗

t
rt
rJ
it1
it2
it3
it4
it5
it6
it7
it8
0.1002 0.0854 -0.2119 0.0195 0.0159 0.0168 0.0402 0.0666 0.0497 0.0297
0.1138 0.1216 -0.1886 0.0232 0.0153 0.0189 0.0515 0.0690 0.0557 0.0319
0.1396 0.1307 -0.1591 0.0455 0.0432 0.0569 0.0721 0.0825 0.0858 0.0498
0.1436 0.1633 -0.1534 0.0442 0.0569 0.0694 0.0737 0.0768 0.0740 0.0521
0.1450 0.1857 -0.1635 0.0432 0.0661 0.0780 0.0856 0.0821 0.0758 0.0569
0.1504 0.1527 -0.1636 0.0428 0.0745 0.0866 0.0943 0.0848 0.0782 0.0604
0.1569 0.1258 -0.1716 0.0446 0.0856 0.0983 0.0996 0.0903 0.0778 0.0666
0.1559 0.1497 -0.1805 0.0469 0.0920 0.1034 0.0989 0.0911 0.0711 0.0694
0.1488 0.1748 -0.1797 0.0434 0.0880 0.0935 0.0925 0.0861 0.0683 0.0678
0.1366 0.1589 -0.1801 0.0398 0.0808 0.0832 0.0861 0.0833 0.0620 0.0666
0.1198 0.1249 -0.1829 0.0431 0.0741 0.0764 0.0787 0.0806 0.0509 0.0646
0.1104 0.1154 -0.1748 0.0465 0.0671 0.0676 0.0665 0.0732 0.0447 0.0606
0.0924 0.0778 -0.1581 0.0445 0.0602 0.0601 0.0555 0.0637 0.0453 0.0546
0.0915 0.0548 -0.1512 0.0383 0.0498 0.0498 0.0469 0.0527 0.0456 0.0456
0.0980 0.0754 -0.1425 0.0266 0.0366 0.0378 0.0393 0.0437 0.0347 0.0368
0.0983 0.1042 -0.1380 0.0152 0.0272 0.0301 0.0357 0.0384 0.0228 0.0291
0.0971 0.1144 -0.1323 0.0080 0.0238 0.0270 0.0349 0.0361 0.0254 0.0228
0.0949 0.1122 -0.1208 0.0042 0.0241 0.0290 0.0334 0.0355 0.0347 0.0193
0.0914 0.1038 -0.1134 0.0001 0.0249 0.0320 0.0321 0.0332 0.0379 0.0177
0.0855 0.0923 -0.1142 -0.0044 0.0235 0.0300 0.0310 0.0304 0.0267 0.0164
0.0751 0.0620 -0.1128 -0.0073 0.0242 0.0266 0.0293 0.0294 0.0173 0.0158
0.0680 0.0484 -0.1159 -0.0104 0.0271 0.0244 0.0287 0.0283 0.0243 0.0166
0.0695 0.0513 -0.1349 -0.0147 0.0258 0.0210 0.0271 0.0254 0.0323 0.0161
0.0758 0.0675 -0.1574 -0.0175 0.0220 0.0167 0.0265 0.0236 0.0305 0.0161
0.0880 0.0925 -0.1778 -0.0226 0.0174 0.0115 0.0290 0.0244 0.0306 0.0161
0.0986 0.1003 -0.1948 -0.0310 0.0130 0.0074 0.0320 0.0278 0.0348 0.0155
0.1086 0.0982 -0.1989 -0.0360 0.0111 0.0058 0.0337 0.0313 0.0389 0.0177
0.1167 0.1218 -0.2003 -0.0395 0.0089 0.0056 0.0349 0.0328 0.0430 0.0185
0.1164 0.1335 -0.2020 -0.0431 0.0058 0.0057 0.0348 0.0333 0.0443 0.0157
0.1180 0.1219 -0.1900 -0.0525 0.0042 0.0062 0.0363 0.0341 0.0420 0.0156
0.1188 0.1210 -0.1689 -0.0674 0.0046 0.0074 0.0414 0.0361 0.0470 0.0172
0.1213 0.1095 -0.1480 -0.0734 0.0070 0.0100 0.0471 0.0407 0.0597 0.0183
0.1277 0.1098 -0.1307 -0.0726 0.0090 0.0129 0.0514 0.0463 0.0723 0.0194
0.1288 0.1251 -0.1211 -0.0745 0.0087 0.0154 0.0503 0.0505 0.0842 0.0185
0.1219 0.1656 -0.1138 -0.0807 0.0082 0.0182 0.0435 0.0521 0.0908 0.0193
0.1042 0.1615 -0.1036 -0.0872 0.0097 0.0197 0.0350 0.0518 0.0845 0.0199
0.0827 0.0979 -0.0941 -0.0862 0.0116 0.0191 0.0269 0.0484 0.0755 0.0186
0.0643 0.0529 -0.0857 -0.0791 0.0130 0.0187 0.0181 0.0431 0.0720 0.0183
0.0532 0.0036 -0.0751 -0.0767 0.0144 0.0186 0.0090 0.0365 0.0636 0.0184
0.0515 -0.0205 -0.0588 -0.0782 0.0143 0.0167 0.0040 0.0277 0.0471 0.0168
0.0674 0.0337 -0.0426 -0.0805 0.0147 0.0141 0.0068 0.0208 0.0344 0.0145
0.0791 0.0926 -0.0284 -0.0827 0.0159 0.0111 0.0133 0.0174 0.0286 0.0130
0.0901 0.1147 -0.0171 -0.0837 0.0157 0.0083 0.0214 0.0158 0.0263 0.0109
0.1049 0.1301 -0.0101 -0.0829 0.0143 0.0075 0.0301 0.0184 0.0281 0.0096
0.1124 0.1219 -0.0050 -0.0810 0.0135 0.0072 0.0378 0.0231 0.0252 0.0100
0.1105 0.1052 -0.0031 -0.0808 0.0123 0.0061 0.0429 0.0270 0.0188 0.0100
0.1079 0.1064 -0.0051 -0.0782 0.0118 0.0063 0.0432 0.0311 0.0170 0.0132
0.1081 0.0979 -0.0057 -0.0724 0.0138 0.0089 0.0451 0.0335 0.0167 0.0177
0.1155 0.0950 -0.0018 -0.0611 0.0211 0.0164 0.0562 0.0475 0.0203 0.0202
0.1179 0.1027 0.0022 -0.0463 0.0258 0.0218 0.0609 0.0579 0.0245 0.0230
0.1147 0.1294 0.0035 -0.0353 0.0325 0.0287 0.0666 0.0705 0.0296 0.0277
0.1396 0.1549 0.0157 -0.0232 0.0446 0.0396 0.0860 0.0849 0.0417 0.0284
0.1438 0.1257 0.0259 -0.0144 0.0554 0.0491 0.0948 0.1077 0.0493 0.0252
0.1074 0.1068 -0.1139 -0.0237 0.0298 0.0313 0.0470 0.0487 0.0487 0.0281

39

Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

∗

∗

it9
it10
0.0681 0.0516
0.0895 0.0469
0.0954 0.0513
0.1041 0.0548
0.1114 0.0571
0.1133 0.0572
0.1180 0.0581
0.1390 0.0591
0.1583 0.0567
0.1739 0.0524
0.1767 0.0511
0.1738 0.0488
0.1594 0.0472
0.1325 0.0469
0.1047 0.0449
0.0835 0.0442
0.0592 0.0447
0.0402 0.0461
0.0342 0.0451
0.0385 0.0401
0.0502 0.0367
0.0492 0.0349
0.0321 0.0313
0.0226 0.0270
0.0244 0.0205
0.0265 0.0172
0.0264 0.0200
0.0214 0.0208
0.0127 0.0186
0.0059 0.0165
0.0018 0.0149-0.0022 0.0147-0.0060 0.0135-0.0130 0.0102-0.0150 0.0074-0.0103 0.0061-0.0060 0.0038-0.0026 0.0009-0.0004 0.0004
0.0006 -0.0002
0.0015 -0.0010-0.0001 0.0006-0.0002 0.0034
0.0035 0.0068
0.0035 0.0104
0.0001 0.0120-0.0034 0.0135-0.0071 0.0157-0.0091 0.0198-0.0096 0.0214-0.0124 0.0239-0.0128 0.0313-0.0102 0.0374
0.0441 0.0286∗

it11
0.08010.12280.14040.14840.16440.17470.16530.14450.12030.09090.05780.0144-0.0300-0.0543-0.0548-0.0446-0.0349-0.0209
0.00350.02900.04090.04100.04200.04340.04680.05090.05330.05540.05350.04830.05460.07490.09080.08920.08040.07630.08080.08120.0736
0.06830.06970.07190.07230.06980.05900.04100.02640.02670.0376
0.0431
0.05100.07280.07270.0599∗it12
0.05590.06180.08370.08700.08960.08540.08230.0806
0.06850.05240.04300.03460.02880.02240.01600.01800.02640.03500.04070.05010.05930.06100.05690.05080.04860.04780.04130.03420.02750.02160.01870.01680.01770.02020.02270.02690.02770.02450.02100.01690.01400.01200.01020.01100.01320.01430.01370.01100.00930.00820.00800.00860.01150.0353∗it13
0.0451
0.05250.07050.07620.08360.08930.0953
0.0981
0.0976
0.09270.08030.06590.04960.04040.04250.03690.03420.0442
0.05330.06010.05950.05130.04830.04970.05490.06250.07100.07880.08440.08640.08410.07700.06540.05000.03190.0127-0.0061-0.0224-0.0344-0.0402-0.0390-0.0309-0.0174-0.00080.01610.03060.04070.04480.04340.03760.02910.01660.01130.044440∗

it140.0451
0.05250.07050.07620.08360.08930.0953
0.0981
0.0976
0.09270.08030.06590.0496
0.03690.03170.01890.00980.0098-0.0003-0.0194-0.0382-0.0469-0.0469-0.0428-0.0092
0.03260.06090.08810.09070.08620.08740.07460.07450.08830.09420.0851
0.04160.00490.0101-0.0083-0.0118-0.0059
0.01620.07070.10980.09100.0937
0.0987
0.0977
0.09390.10200.13100.1156
0.0531∗

it15
0.11860.10580.14680.14990.16340.17820.18580.16700.1244
0.09700.08900.07740.06740.07590.09580.11690.12520.11010.08750.05950.02970.0051-0.0120-0.0155-0.00180.0201
0.04440.06590.08730.10960.12890.14770.15010.12790.08930.0417-0.0004-0.0407-0.0808-0.0811-0.0570-0.0249
0.01480.04760.07590.09590.08740.08090.08970.08890.09220.11240.11360.0750∗it16
0.11880.12960.16040.16380.17880.19090.19160.17230.1374
0.1094
0.09300.07280.05610.05790.0734
0.09370.1034
0.09630.0856
0.06820.04420.02200.00590.00210.01490.03530.05780.07970.10080.12180.14160.16140.16560.14570.10790.06150.0220-0.0149-0.0510-0.0539-0.0351-0.00760.0277
0.05640.07890.09190.08220.07710.08690.08690.09060.1111
0.11370.0827∗

it17-0.0352
0.00280.0677
0.05300.05840.08210.09020.07780.0691
0.0645
0.06120.07250.06300.03050.02890.01620.02780.02600.00760.01060.02700.01730.02070.01270.00640.0035-0.0037-0.0037-0.0005-0.0094-0.0316-0.0079
0.01990.00270.01770.01880.0167
0.05360.04730.02210.03830.01690.02240.04210.00420.00260.0006
0.03640.00310.04040.0626
0.09960.02890.0283

The mean ex post rate of return on US assets over the 53 year sample period is 10.68%. The
mean rate of return on assets when we used smoothed asset inflation rates in the user cost
formulae is 10.74%. Both average rates of return are quite high. Note that the ex post rate of
return in 2009 was −2.05%. The smoothed rates of return rt are probably much closer to arealistic cost of capital for US firms. Note that the average smoothed asset inflation rates for
assets 11 (Agricultural Land), 14 (Commercial Land) and 15 (Residential Land) are 5.99%,
5.31% and 7.50% respectively. These rates of return on holding land are substantial but well
below the overall average rate of return on US assets.*39∗t
The smoothed year t user cost for asset n is defined as Unt ≡ [rt + τnt − itn + δnt (1 + itn )]PKn
and is listed in Table A10.

*39 However, our accounting framework does not include monetary holdings and natural resource stocks asassets and this omission means that our ex post rates of return on assets are too high.

41

Table A10: User Costs Using Smoothed Asset Inflation Rates
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

U1t
0.578510.49704
0.384630.32763
0.295720.260690.24059
0.208380.167350.12092
0.094480.075140.06171
0.054780.04599
0.036440.02995
0.02504
0.021240.01957
0.017130.015240.014080.012430.011190.010180.00833
0.006470.004930.003700.002900.002350.00195
0.001690.001510.001310.001100.000930.000800.000710.000680.000650.000640.000640.000640.000640.000630.000640.000650.000640.000630.000650.00067
0.06931U2t
0.38672
0.394640.41305
0.437200.462900.510280.55899
0.573850.57067
0.57643
0.571410.58765
0.597620.629020.67838
0.711590.727360.723020.719020.70765
0.691450.67906
0.69063
0.69235
0.716400.727590.72771
0.733200.714270.687310.67515
0.636480.594280.542770.477220.430900.38738
0.345320.304090.268890.25895
0.24594
0.237240.228700.213400.19170
0.16975
0.154210.142480.13065
0.12050
0.12056
0.11701
0.48285

U3t
0.337390.357150.37479
0.378070.385800.422170.468640.504930.537200.566930.580260.627210.64766
0.70973
0.78079
0.819130.839540.847490.848390.84676
0.842110.847410.89079
0.94776
1.03269
1.09647
1.144431.18186
1.19104
1.21350
1.20479
1.20144
1.214421.215411.22425
1.19869
1.12858
1.05934
1.00015
0.963101.084021.17038
1.25560
1.333111.37828
1.39874
1.38663
1.39066
1.41360
1.421091.369421.468101.54572
0.96780

U4t
0.29218
0.30626
0.30988
0.313290.33009
0.373710.417080.441160.476160.50750
0.53209
0.58452
0.599210.652130.717780.744470.767610.78024
0.783120.801660.80003
0.80997
0.855430.901630.97094
1.02870
1.07484
1.10928
1.11324
1.11767
1.121171.12559
1.15024
1.16185
1.14734
1.11472
1.04597
0.994420.976430.97766
1.063271.14059
1.22566
1.30003
1.33823
1.33658
1.32543
1.32904
1.338221.348021.31929
1.42627
1.491960.91151

U5t
0.132540.139480.15392
0.16700
0.169030.18089
0.19689
0.211130.228290.238880.241310.269910.26801
0.29900
0.345650.365140.37728
0.393490.40161
0.40395
0.38386
0.37100
0.39587
0.43111
0.489890.53433
0.58046
0.62373
0.638110.65794
0.66304
0.678230.713930.75052
0.79646
0.806030.74379
0.70768
0.70630
0.71605
0.79273
0.83506
0.864420.919700.968030.96053
0.973421.015641.02425
1.04851
1.00694
1.13291
1.21777
0.5723042

U6t
0.145310.165410.19443
0.218380.232500.260440.280390.292280.30877
0.318120.32003
0.346170.34923
0.394540.45191
0.48349
0.50304
0.51471
0.52869
0.54208
0.526120.52562
0.558450.596220.663120.707540.74787
0.79367
0.81503
0.849630.875100.89800
0.933110.940710.93404
0.90853
0.839900.808110.81809
0.892811.06831
1.17671
1.29759
1.40788
1.45993
1.45397
1.429491.45496
1.43878
1.44872
1.378811.54439
1.547130.76582

U7t
0.078030.08901
0.091250.115070.13000
0.151070.17055
0.185040.18653
0.19092
0.190990.20023
0.177810.17782
0.208540.237650.247260.23375
0.218050.240260.24160
0.205930.197930.220080.259050.29317
0.33633
0.357110.35887
0.394100.389930.36265
0.36186
0.33813
0.29695
0.28159
0.218630.14038
0.13706
0.242880.46052
0.61330
0.755310.88083
0.995011.07039
1.07546
1.11397
1.14147
1.128481.07523
1.237071.278230.41470U8t
0.30329
0.31661
0.339020.352130.370170.40350
0.432280.450230.46701
0.478710.49071
0.521480.53347
0.577070.63069
0.664020.684020.695830.70561
0.708390.691440.68079
0.69382
0.714580.754590.807470.847800.879400.89874
0.914850.93879
0.96024
0.98240
1.00449
1.002200.967740.92068
0.879190.861780.86767
0.93577
1.00100
1.06793
1.13689
1.16845
1.17932
1.16203
1.14899
1.19735
1.22772
1.23634
1.38457
1.457520.80560

U9t
0.470850.47791
0.55168
0.62755
0.688310.77458
0.87249
0.931960.99168
1.05592
1.19257
1.39159
1.70687
2.16496
2.70736
2.978963.15736
3.178023.26192
3.561533.668423.64493
3.77968
4.03373
4.35018
4.569524.71076
4.884405.070485.32077
5.56525
5.66338
5.524165.28280
5.07699
4.86965
4.66140
4.49004
4.405424.359224.380234.48753
4.659544.82416
4.93666
4.925174.94515
4.994625.06859
5.013414.93869
5.10271
5.047123.58490

Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

t
U10
0.37474
0.40988
0.46366
0.480440.502810.54355
0.60982
0.64256
0.66493
0.696880.68455
0.722020.722150.754290.808490.83758
0.880420.90003
0.92376
0.96443
0.991121.02306
1.07589
1.11989
1.19276
1.26867
1.32646
1.37193
1.38488
1.40989
1.46448
1.495131.53919
1.57132
1.56227
1.51665
1.44738
1.39074
1.343421.33501
1.402111.42090
1.45553
1.50256
1.53018
1.54587
1.56595
1.587091.627781.65040
1.64051
1.747211.82368
1.14940t
U110.030930.00250
0.012310.00945-0.01584-0.02876
0.00259
0.053370.110700.189140.280270.44079
0.53705
0.61900
0.600030.480440.39069
0.33700
0.27201
0.189890.129310.11175
0.11779
0.14171
0.182150.214710.25530
0.292900.31153
0.358430.355500.284100.241170.25700
0.29378
0.249510.06917-0.09423-0.12346-0.08288
0.05546
0.155070.294400.550240.84420
1.09565
1.27985
1.31156
1.28555
1.25690
1.110481.23861
1.43530
0.37721t
U120.044290.054070.06237
0.06780
0.07253
0.09689
0.12572
0.13458
0.15165
0.16604
0.159100.16998
0.15049
0.16657
0.196230.19180
0.169360.144710.12679
0.094700.045410.02134
0.039700.084120.14206
0.19249
0.264900.33574
0.37644
0.421230.447440.47756
0.503210.495800.45958
0.36736
0.271710.204140.172100.19047
0.294840.368090.438310.52063
0.56161
0.55762
0.55669
0.57858
0.638160.664640.650150.80095
0.81787
0.29258t
U130.065400.073810.08695
0.090290.08825
0.098220.109940.110240.105210.09984
0.100120.12174
0.133030.16760
0.17683
0.189960.19175
0.16651
0.143390.11173
0.08356
0.094170.11765
0.142410.179800.201670.221740.23874
0.223620.24006
0.28230
0.37662
0.552700.73169
0.869400.90977
0.893180.86599
0.849740.85446
0.933380.907190.843640.805420.73655
0.63493
0.56806
0.56859
0.67656
0.786150.86630
1.24490
1.350500.4204243t
U140.06805
0.07660
0.08988
0.09327
0.091210.10141
0.113390.113800.10860
0.10309
0.10353
0.12556
0.13765
0.18284
0.21233
0.245130.26388
0.26367
0.29713
0.34675
0.37972
0.378070.34660
0.302110.22536
0.15989
0.125440.08875
0.10091
0.136420.139670.207770.231230.185120.141420.12179
0.261070.412160.350620.44425
0.397130.35183
0.357240.20051
0.065800.172140.153410.12963
0.199530.251860.180070.16182
0.396850.20555t
U150.00036
0.03158
0.01480
0.01635-0.00103-0.02119-0.03075
0.01252
0.121400.202870.18356
0.19856
0.156860.120350.07051-0.02694-0.08790-0.01281
0.141960.36515
0.573230.72969
0.878220.94060
0.92000
0.80574
0.68444
0.581810.413080.23833
0.04678-0.17990-0.13719
0.24271
0.92288
1.82375
2.60047
3.16591
3.437812.77929
2.28360
1.715461.21747
1.105850.89409
0.556470.714180.917070.94071
1.07373
0.96465
1.18079
1.424480.71533t
U16-0.01584-0.01391-0.02233-0.02498-0.05324-0.07959-0.08069-0.04043
0.04643
0.123360.139210.202280.191990.17859
0.13837
0.03568-0.02660
0.004540.063810.18050
0.32732
0.473220.63455
0.72697
0.722390.63182
0.51800
0.39708
0.19429-0.01962-0.29025-0.61908-0.67272-0.32488
0.40318
1.378202.22434
2.91600
3.39132
2.849162.43864
1.897641.32720
1.18008
0.943290.59346
0.847221.08404
1.079271.23153
1.03926
1.33675
1.612120.63055t
U170.13786
0.11397
0.06925
0.093400.10243
0.09552
0.09589
0.112770.119070.117420.10850
0.07560
0.066360.133940.149360.197460.171850.15530
0.197500.182240.127300.13061
0.125070.167930.220920.251970.31287
0.340900.307700.324370.38105
0.33497
0.28203
0.326900.23297
0.209290.18952
0.029420.019540.086420.08426
0.19586
0.22343
0.20515
0.34728
0.33855
0.35375
0.265090.353210.254450.167210.16072
0.337140.19214

Moving to the use of smoothed asset inflation rates in the user cost formula did not eliminate
the earlier negative user costs but it did reduce the number: there are 5, 8 and 14 negative user
costs for assets 11 (Agricultural Land), 15 (Residential Land and 16 (Other Land) respectively.
t
We use the data for the smoothed user costs, U1t , . . . , U17, along with the data on the beginttning of the year asset stocks, QK1 , . . . , QK17 , to form 4 capital services aggregates. The firstsubaggregate is a Machinery and Equipment aggregate consisting of assets 1-4, the second is
a Structures subaggregate consisting of assets 5-7, the third is an Other subaggregate which
consists of assets 8-10 and 17 (Research and Development, Computer Software, Other Intangibles and Inventory Stocks) and the fourth is a Land subaggregate consisting of assets 11-16.
Denote the prices and quantities for these aggregate capital services by PUt M , PUt S , PUt O and
PUt L and QtU M , QtU S , QtU O and QtU L . These aggregate user costs are listed in Table A11 (withprices normalized to equal 1 in 1970).*40 The four subaggregates were then aggregated into
an overall capital services aggregate using Törnqvist direct quantity aggregation. The year t
value, price and quantity of the capital services aggregate are denoted by VUt , PUt and QtU .

*40 These capital services input subaggregates were formed using Törnqvist direct aggregation of input

quantities; i.e., bilateral chained quantity indexes were formed first (quantities were always positive) andthen the corresponding aggregate price indexes were calculated by deflating total subaggregate value by
the quantity index.

44

Table A11: Prices and Quantities of Four Capital Services Subaggregates
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

t
PU
1.00000
1.08313
1.16234
1.25503
1.28506
1.409031.56385
1.697641.85720
1.985442.02606
2.24890
2.252212.491232.74284
2.83386
2.851642.87525
2.97713
3.169073.21591
3.217713.35499
3.516483.71368
3.81287
3.94083
4.015443.963704.01973
3.97127
3.91420
4.01220
4.17592
4.38270
4.640074.770414.82905
4.81727
4.80782
5.107515.23770
5.40376
5.579445.693675.73352
5.77818
5.89187
6.07416
6.16574
5.84925
6.50474
7.11141

t
t
t
t
PU
PU
PU
PU
QtU
M
S
O
L
1.00000 1.00000 1.00000 1.00000 0.36904
1.03597 1.09951 1.01687 1.16595 0.37920
1.04721 1.21456 1.04646 1.30444 0.39040
1.05477 1.38372 1.12108 1.35412 0.40492
1.09630 1.46556 1.18792 1.19002 0.42314
1.21704 1.62842 1.27334 1.23802 0.43705
1.34589 1.78646 1.36808 1.48236 0.44421
1.41718 1.90740 1.44961 1.82235 0.45601
1.49972 2.00490 1.51045 2.40931 0.47220
1.56750 2.07522 1.54956 2.92053 0.49305
1.60895 2.08786 1.57443 3.05638 0.51493
1.72733 2.27049 1.61651 3.81838 0.53012
1.75311 2.19204 1.66973 3.97121 0.54553
1.88233 2.38235 1.95414 4.56370 0.55411
2.03177 2.75739 2.18078 4.65848 0.56647
2.07426 2.98456 2.37799 4.28176 0.58979
2.09597 3.09574 2.40715 3.95130 0.61371
2.08769 3.12696 2.41093 4.06416 0.63603
2.06138 3.12571 2.52260 4.88096 0.65663
2.06988 3.23373 2.55036 6.12367 0.67831
2.03294 3.13708 2.43889 7.17369 0.69893
2.02263 2.96704 2.42620 7.85437 0.71619
2.09770 3.08730 2.47619 8.33654 0.72735
2.16494 3.35742 2.63595 8.34745 0.74080
2.28640 3.82189 2.86885 7.66913 0.75636
2.37694 4.17902 3.08175 6.60094 0.77719
2.41335 4.56684 3.30574 5.86780 0.80098
2.41650 4.87639 3.45623 5.16699 0.83132
2.35375 4.97204 3.46774 4.54040 0.86799
2.29037 5.22207 3.57517 4.25861 0.91021
2.22925 5.27010 3.77848 3.43361 0.95787
2.16638 5.27796 3.75926 3.14927 1.00696
2.13402 5.47148 3.69325 3.85525 1.04022
2.09130 5.54995 3.77135 5.53649 1.06567
2.02325 5.58836 3.57261 8.76703 1.09307
1.93015 5.53727 3.41956 13.08549 1.12583
1.78397 4.98457 3.24401 18.40643 1.15993
1.65948 4.54980 2.85301 22.87290 1.19821
1.57651 4.54928 2.77749 23.51953 1.22614
1.52587 5.02662 2.88408 21.39923 1.24361
1.62739 6.28166 3.02047 18.75001 1.24242
1.70513 7.10030 3.31386 15.45404 1.25238
1.79647 7.85808 3.51452 13.10178 1.26462
1.87690 8.65426 3.65634 11.00654 1.28291
1.90983 9.30205 3.94162 8.61347 1.30478
1.89269 9.51683 3.94753 8.39752 1.33152
1.85610 9.54956 3.94952 9.20232 1.36111
1.84398 9.88348 3.81218 10.05129 1.38832
1.84720 9.98701 4.04441 11.26479 1.41677
1.83940 10.05374 3.94408 12.80517 1.45150
1.78350 9.60846 3.81624 11.13304 1.48542
1.90310 10.89250 4.10471 12.98331 1.50862
1.97545 11.37361 4.43145 17.82376 1.54067

45

QtU M
0.110360.11168
0.11269
0.116310.123770.13053
0.132800.13666
0.144450.15566
0.168240.17698
0.18568
0.18992
0.195630.20784
0.220120.230410.23984
0.250070.26095
0.268810.272880.279280.290280.30559
0.326800.35146
0.37957
0.413700.45432
0.495760.520530.53372
0.54923
0.572200.60135
0.642110.68267
0.710450.70563
0.722890.74933
0.78392
0.820420.86167
0.901480.93152
0.96266
1.00020
1.03225
1.038481.05234

QtU S
0.15996
0.164630.170440.177080.18379
0.188450.19162
0.195820.201310.208200.215380.221130.226560.230280.23506
0.242220.25024
0.25784
0.265140.27203
0.278110.283810.287200.29038
0.294450.299280.30452
0.31133
0.31879
0.32728
0.336100.345450.35443
0.362200.370770.380390.38982
0.39935
0.40566
0.407890.40624
0.403010.40001
0.39866
0.39887
0.40053
0.40349
0.40691
0.40992
0.414160.41855
0.42233
0.42630

QtU O
0.036860.04069
0.04430
0.04772
0.051370.05437
0.055700.058850.062200.06605
0.069850.07277
0.07661
0.079140.08168
0.08798
0.09360
0.099220.10565
0.11133
0.11741
0.123670.129360.134770.139100.144140.14872
0.15545
0.16567
0.17700
0.190150.203380.211390.21960
0.228220.23846
0.24951
0.26138
0.27406
0.28536
0.292360.301120.31128
0.32218
0.33485
0.34833
0.36399
0.381180.400390.423880.449160.47315
0.50300

QtU L
0.06187
0.06229
0.063170.064240.065280.06530
0.06572
0.06676
0.06756
0.06866
0.06967
0.06987
0.070150.07004
0.07083
0.071960.073080.074370.07482
0.07649
0.077840.078770.079270.080290.08095
0.082110.08274
0.083880.08535
0.08673
0.087770.088780.088910.08951
0.09034
0.09153
0.09280
0.09435
0.09455
0.09447
0.09399
0.09496
0.095420.09617
0.09659
0.09724
0.09788
0.09823
0.09891
0.09950
0.09987
0.09970
0.10098

The price of aggregate capital services PUt increased 7.11 fold over the sample period. The
price of Machinery and Equipment Services PUt M increased 1.98 fold, the price of Structures
Services PUt S increased 11.37 fold, the price of Other Capital Services PUt O increased 4.43 fold
and the price of Land Services PUt L increased 17.82 fold.
Denote the values of the 4 subaggregates in year t by VUt M , VUt S , VUt O and VUt L . Denote thecorresponding year t value shares of aggregate capital services by stU M , stU S , stU O and stU L .
These variables are listed in Table A12 below.

46

Table A12: Values and Shares of Four Capital Services Subaggregates
Year
VUt
VUt M
1970
0.36904 0.11036
1971
0.41072 0.11570
0.45378 0.11801
1972
0.50819 0.12268
1973
1974
0.54376 0.13569
1975
0.61582 0.15886
0.69468 0.17873
1976
0.77414 0.19367
1977
1978
0.87697 0.21664
0.97893 0.24400
1979
1.04329 0.27068
1980
1981
1.19219 0.30570
1.22864 0.32551
1982
1.38042 0.35750
1983
1984
1.55373 0.39748
1.67139 0.43111
1985
1.75009 0.46136
1986
1987
1.82875 0.48102
1988
1.95486 0.49440
2.14961 0.51762
1989
1990
2.24770 0.53050
2.30449 0.54370
1991
1992
2.44024 0.57241
2.60502 0.60462
1993
2.80889 0.66371
1994
2.96331 0.72637
1995
3.15652 0.78869
1996
1997
3.33812 0.84929
3.44043 0.89342
1998
3.65878 0.94752
1999
2000
3.80395 1.01280
3.94143 1.07400
2001
4.17358 1.11083
2002
4.45017 1.11618
2003
4.79062 1.11124
2004
2005
5.22393 1.10442
5.53336 1.07279
2006
5.78623 1.06557
2007
5.90667 1.07623
2008
2009
5.97906 1.08406
6.34566 1.14833
2010
2011
6.55958 1.23262
6.83371 1.34615
2012
7.15793 1.47134
2013
2014
7.42901 1.56687
2015
7.63428 1.63088
7.86474 1.67325
2016
8.17979 1.71770
2017
8.60569 1.77823
2018
2019
8.94954 1.83976
8.68861 1.84103
2020
2021
9.81320 1.97633
2022 10.95630 2.07883
Mean 3.87830 0.84691

VUt S
0.159960.181020.20700
0.245030.26936
0.30688
0.34232
0.37351
0.40360
0.43205
0.44969
0.50208
0.496620.54862
0.648170.72292
0.77468
0.806260.82874
0.87967
0.872440.84209
0.88667
0.97492
1.12535
1.25071
1.39071
1.518171.58501
1.70908
1.77130
1.82330
1.939282.01020
2.07202
2.10629
1.94307
1.81697
1.845482.05033
2.55189
2.86152
3.14335
3.45008
3.710313.81180
3.853174.021644.09391
4.16382
4.02165
4.60023
4.84857
1.70270

VUt O
0.03686
0.04138
0.04636
0.05349
0.06103
0.06923
0.076200.08530
0.09395
0.10235
0.10998
0.117630.12792
0.15466
0.178130.209230.225310.239220.26650
0.28394
0.286340.30004
0.32032
0.35525
0.39904
0.444200.491620.53726
0.57449
0.63282
0.71849
0.764540.78071
0.828210.81535
0.815440.809420.74573
0.761210.82299
0.883070.99788
1.09399
1.17801
1.31983
1.37506
1.43757
1.453141.61932
1.67183
1.71411
1.94215
2.22903
0.63580

VUt L
0.06187
0.072620.082400.08698
0.077680.08084
0.097420.12166
0.162780.20053
0.21294
0.26678
0.27858
0.31964
0.32995
0.308130.288740.302250.365210.46838
0.558420.61866
0.660840.670230.62079
0.542030.48551
0.433390.38750
0.369360.30135
0.27959
0.34276
0.49559
0.79201
1.19777
1.708072.157962.22375
2.02168
1.76238
1.46756
1.250221.05851
0.83200
0.816540.90075
0.98731
1.11423
1.27413
1.111811.29450
1.79988
0.69288

47

stU M
0.29903
0.281710.260070.241410.24955
0.257970.257280.250180.247030.249250.259450.256420.26494
0.25898
0.255820.25794
0.26362
0.263030.252910.24080
0.236020.235930.234570.232100.236290.245120.249860.254420.25968
0.25897
0.26625
0.27249
0.266160.250820.231960.211420.19388
0.184160.182210.181310.180960.18791
0.19699
0.205550.21091
0.213630.21275
0.209990.206630.20557
0.211890.201390.189740.23556

stU S
0.433440.44073
0.45618
0.482170.495360.49833
0.49278
0.482480.460230.44135
0.43103
0.421140.404200.397430.417170.43252
0.442650.44088
0.42394
0.409220.388150.365410.36335
0.374250.400640.42206
0.44058
0.454800.460700.467120.46565
0.46260
0.46466
0.45171
0.43252
0.403200.351160.314020.31244
0.34292
0.402150.436240.45998
0.48199
0.499440.49930
0.489930.49166
0.47572
0.46526
0.46287
0.46878
0.442540.43617

stU O
0.09988
0.10075
0.102160.105260.112230.112420.109700.110190.107130.10455
0.105410.09867
0.104120.11204
0.11465
0.12518
0.12874
0.130810.13633
0.13209
0.127390.130200.131270.13637
0.14206
0.149900.155750.16095
0.16698
0.172960.18888
0.19398
0.18706
0.186110.170200.156100.14628
0.12888
0.12887
0.13765
0.139160.152130.16009
0.16457
0.17766
0.180120.18279
0.177650.188170.186810.19728
0.19791
0.203450.14449

stU L
0.167640.176820.181590.171160.14286
0.131280.140240.157150.185620.204850.204100.223770.226740.23156
0.21236
0.18436
0.16499
0.165280.186820.217890.248440.26846
0.270810.257280.22101
0.18291
0.153810.12983
0.112630.10095
0.079220.07094
0.082130.11136
0.16533
0.229290.30869
0.37295
0.37648
0.338130.27773
0.22373
0.18295
0.147880.11199
0.106960.114530.120700.129480.14237
0.127960.13191
0.164280.18378

There is a considerable amount of variation in the shares of the four capital services subaggregates in total capital services. The average share of Structures was 43.6%, of Machinery andEquipment was 23.6%, of Land was 18.4% and of Other Capital (mostly R&D and Inventory
services) was 18.4%. All of the subaggregate user costs were positive and three of them werefairly smooth but the share of Land services was quite volatile.*41
We turn now to the construction of alternative indexes of output, input and Total Factor
Productivity (TFP) for the US economy.
In Table A6, the GDP output index QtY and the companion output price index PYt werelisted. The GDP price index PYt is a chained Törnqvist price index that uses the pricesttPCt , PGt , PIt , PX
, PM
and the quantity indexes QtC , QtG , QtI , QtX , −QtM as inputs into the priceindex PYt . The quantity index QtY was defined residually by deflating the year t value of GDP,VYt by the year t GDP price index; i.e., QtY ≡ VYt /PYt . We are now in a position to define acompanion index of the year t quantity of inputs QtZ as a chained Törnqvist quantity index
that uses the prices of labour and the four types of capital services PLt , PUt M , PUt S , PUt O , PUt L
and the quantity indexes QtL , QtU M , QtU S , QtU O , QtU L as inputs into the quantity index QtZ and
the companion input price index PZt is defined residually by deflating the year t value of GDP,VZt = VYt by the year t GDP input quantity index; i.e., PZt ≡ VZt /QtY . The year t level of Total
Factor Productivity is defined as the year t output quantity QtY divided by the year t input
quantity QtZ ; i.e.,
TFPt ≡ QtY /QtZ ;t = 1970, . . . , 2022.
(A6)
TFP growth is defined as TFPGt ≡ TFPt / TFPt−1 for t = 1971, . . . , 2022.*42 The series
QtY , QtZ , TFPt and TFPGt are listed in Table A13.
Recall that Jorgensonian balancing rates of return rJt were listed in Table A9 above. These
rates of return can be used with actual ex post asset inflation rates to construct Jorgensonian
user costs. We used these user costs in place of our smoothed user costs to construct Jorgensonian estimates of year t Total Factor Productivity, TFPtJ and year t estimates of TFP
growth, TFPGtJ . These alternative GDP based estimates of TFP are also listed in Table A13.
In the main text, we did not use GDP as our output concept. Instead, we used Gross Output(GO) as our output concept. Gross Output simply regards imports as an input instead of asta negative output. Thus the value of Gross Output in year t, VGY, is defined as follows:tt
VGY≡ PCt QtC + PGt QtG + PIt QtI + PX
QtX ;t = 1970, . . . , 2022.

(A7)t
The GO price index for year t, PGY, is a chained Törnqvist price index that uses thetttt
output prices PC , PG , PI , PX and the corresponding quantity indexes QtC , QtG , QtI , QtX astinputs into the price index PGY. The corresponding year t GO quantity index QtGY istdefined residually by deflating the year t value of Gross Output, VGY
by the year t GOttt
price index; i.e., QGY ≡ VGY /PGY . The companion index of the year t quantity of inputs QtGZ is a chained Törnqvist quantity index that uses the prices of imports, labourt
and the four types of capital services PM, PLt , PUt M , PUt S , PUt O , PUt L and the quantity indexes
QtM , QtL , QtU M , QtU S , QtU O , QtU L as inputs into the quantity index QtGZ and the companionttt
input price index PGZ
is defined residually by deflating the year t value of GO, VGZ
= VGY
t
t
t
by the year t GO input quantity index; i.e., PGZ ≡ VGZ /QGZ . The year t level of GO Total*41 When land prices increase, land inflation rates increase even more on a proportional basis, so the user

cost of land tends to decrease when there is a property bubble. When the bubble ends, the user cost ofland tends to increase.
*42 This precise methodology for measuring TFP is due to Diewert and Morrison (1986)[22] and Kohli(1990)[39] but it is a variation on the methodology pioneered by Jorgenson and Griliches (1967)[31] andChristensen and Jorgenson (1969)[7].

48Factor Productivity is defined as the year t output quantity QtGY divided by the year t input
quantity QtGZ ; i.e.,
TFPtG ≡ QtGY /QtGZ ;t = 1970, . . . , 2022.
(A8)
Gross Output TFP growth is defined as TFPGtG ≡ TFPtG / TFPt−1
for t = 1971, . . . , 2022.
G
The series QtGY , QtGZ , TFPtG and TFPGtG are listed in Table A13.

49

Table A13: Alternative GDP and Gross Output Measures of Input, Output and Productivity
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

QtY
1.0275
1.06351.1205
1.1832
1.1751
1.17041.2322
1.29141.36311.40661.40391.44081.40901.46881.57591.63801.6862
1.74551.8182
1.8824
1.9194
1.91671.98052.03582.11772.17352.25872.36022.46282.57742.67882.70152.74582.82232.93073.0296
3.1151
3.17533.1806
3.09693.18063.22793.30003.37173.45553.5531
3.61263.70073.80943.90333.81194.02684.09392.3854QtGY
1.08581.12501.18881.25461.24451.22871.30631.37551.45551.49991.4876
1.52661.4942
1.56831.70581.7771
1.83871.90771.98672.05862.10242.09942.17682.25002.35832.43392.54212.68142.82002.9737
3.12343.1365
3.19573.29393.4501
3.58113.70043.77543.7685
3.61373.75903.83303.91993.9996
4.11464.24474.31444.43284.56954.67424.52414.8282
4.95322.7426QtZ
1.02751.03441.06491.10901.12961.12591.15411.18971.23861.2835
1.30251.32431.3223
1.34201.3971
1.4412
1.4767
1.52671.57231.62441.65651.6556
1.6773
1.71101.75871.8134
1.85641.92091.99212.06232.13272.15982.18662.20592.24642.29532.35462.40692.42492.37422.37932.41712.45982.50292.55252.60912.65902.70852.76322.81842.77312.84132.92841.9055QtGZ
1.08581.09591.13311.1803
1.1987
1.1842
1.2283
1.27391.3312
1.37711.38641.41081.4072
1.4414
1.5254
1.5783
1.62611.68531.7372
1.7967
1.83501.83391.86791.9176
1.98852.06072.12382.22082.32292.42612.53712.55712.59642.63272.71112.78602.87312.9383
2.94742.83962.89172.95123.00643.05683.1315
3.2141
3.27363.34693.42373.48833.39843.53143.66592.2279TFPt
1.00001.0281
1.0522
1.06691.0402
1.0395
1.06771.08551.10051.09591.07791.08791.0655
1.09451.12801.1366
1.14191.1433
1.1564
1.15881.15871.1577
1.18081.18981.2041
1.1986
1.21671.2287
1.2363
1.24981.25611.25081.25571.27941.30461.31991.32301.3193
1.31171.3044
1.33681.3354
1.3416
1.34711.35381.36181.35861.3664
1.37861.38491.3746
1.4172
1.39801.216450

TFPtJ
1.00001.02791.0514
1.0662
1.04031.0395
1.06751.08481.09891.09291.07441.0842
1.06141.0895
1.12211.13091.13541.13581.14801.15021.1504
1.1495
1.17251.18101.1952
1.18951.2071
1.21841.2246
1.23671.2422
1.23661.24151.26511.28971.30431.3065
1.30201.29581.28981.32221.32081.32691.3316
1.3371
1.34501.3424
1.34981.36171.36801.35801.39951.38001.2065

TFPtG TFPGt TFPGtJ TFPGtG
1.0000 1.0000 1.0000
1.00001.0266 1.0281 1.0279
1.02661.0492 1.0234 1.0228
1.02201.0629 1.0140 1.0141
1.01311.0382 0.9750 0.9757
0.97671.0376 0.9993 0.9992
0.99941.0635 1.0271 1.0270
1.02501.0798 1.0167 1.0161
1.0153
1.0934 1.0138 1.0131
1.01261.0892 0.9958 0.9945
0.9962
1.0730 0.9836 0.9830
0.98511.0820 1.0094 1.0092
1.0084
1.0618 0.9794 0.9790
0.98131.0881 1.0272 1.0264
1.02471.1182 1.0306 1.0299
1.02771.1259 1.0076 1.0078
1.00691.1307 1.0047 1.0040
1.0043
1.1320 1.0012 1.0004
1.0011
1.1436 1.0115 1.0107
1.01031.1458 1.0021 1.0020
1.00191.1457 0.9999 1.0001
0.99991.1448 0.9991 0.9993
0.9992
1.1653 1.0199 1.0200
1.01791.1733 1.0076 1.0073
1.00691.1860 1.0120 1.0120
1.01081.1811 0.9954 0.9953
0.99591.1969 1.0151 1.0148
1.01341.2074 1.0099 1.0093
1.0087
1.2140 1.0062 1.0051
1.00551.2257 1.0109 1.0099
1.00961.2311 1.0050 1.0044
1.00441.2266 0.9958 0.9955
0.99631.2308 1.0039 1.0040
1.00351.2512 1.0189 1.0190
1.01651.2726 1.0197 1.0194
1.01711.2854 1.0117 1.0113
1.01011.2880 1.0023 1.0017
1.00201.2849 0.9972 0.9966
0.9976
1.2786 0.9942 0.9952
0.9951
1.2726 0.9945 0.9953
0.9953
1.2999 1.0248 1.0252
1.0214
1.2988 0.9990 0.9989
0.99911.3039 1.0046 1.0046
1.00391.3085 1.0041 1.0035
1.00351.3139 1.0049 1.0042
1.00421.3206 1.0059 1.0059
1.00511.3180 0.9977 0.9981
0.99801.3245 1.0057 1.0055
1.00491.3347 1.0090 1.0088
1.00771.3400 1.0046 1.0046
1.00401.3313 0.9926 0.9926
0.99351.3672 1.0310 1.0305
1.02701.3512 0.9864 0.9861
0.9883
1.1946 1.0065 1.0063
1.0059

The average TFP growth rate using GDP as the output concept was 0.65% per year usingsmoothed user costs and 0.63% per year using Jorgensonian user costs. Smoothing the user
costs did not greatly affect our measures of GDP Total Factor Productivity Growth. Using
Gross Output as our output concept, the mean TFP growth rate fell to 0.59% per year.
This drop was expected because regarding imports as an input rather than a negative output
increases the overall input volume measure by about 10% so we could expect a 10% drop in
measured TFP growth using Gross Output in place as GDP as the output measure.
We conclude this data construction Appendix by listing the year by year main diagonal elasticities that correspond to the average elasticities that were listed in Tables 2 and 1 in the maintext. These elasticities were generated by differentiating our estimated Joint Cost Function
over the years 1970-2022.

51

Table A14: Main Diagonal Elasticities of Inverse Output Supply 1970-2022
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

Ep11
0.15360.15130.14660.14280.14540.13450.13320.13570.13290.12950.12370.12380.11410.1126
0.11790.11370.10910.10480.0976
0.09350.08880.08270.07950.07740.07570.0716
0.06950.06740.06630.06490.06180.0596
0.05870.05750.0561
0.05430.05150.04750.04400.04090.03920.03640.03510.03340.03150.03040.02860.02710.02580.02500.02460.02230.0204Ep22
0.17950.17150.16210.15330.16730.18810.17670.16590.15520.15230.16360.15470.16300.15690.14050.14100.14460.14210.13940.13650.13650.13860.13660.13230.12750.12640.12000.11380.10870.10410.10040.10260.10290.1001
0.0961
0.09250.08960.08900.0901
0.09460.08970.08540.08190.07850.07510.07350.07280.07070.06900.06950.07370.0701
0.0675Ep33
0.10010.10390.10980.1176
0.11360.10640.11350.11750.11980.1196
0.11460.11500.10810.11110.12030.12100.12000.11990.12070.12070.11850.11460.11680.11970.12390.12550.12970.13400.13640.13850.13980.13590.13400.13440.13630.13730.13720.13470.12960.12150.12520.12510.12670.12760.12770.12790.12550.12460.12320.12150.11700.11570.1149Ep44
0.01990.01920.01910.02060.02040.01850.01860.01840.01930.01960.01990.02000.01830.01780.01940.01950.02000.02080.02240.02360.02440.02540.02620.02630.02750.02850.03040.03350.03460.03540.03730.03530.03380.03390.03540.03580.03750.03780.03640.03350.03490.03510.03620.03680.03790.03920.03890.03920.04000.03990.03850.03750.0383

The above inverse elasticities are all small.*43*43 If there were sectoral production functions for C, G and I (the no joint production hypothesis), then allof the elasticities listed in Table A14 would be equal to zero. It appears that production is “almost”
Nonjoint.

52

Table A15: Main Diagonal Elasticities of Input Demand 1970-2022
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

Ex11
-0.5103-0.5010-0.4858-0.5021-0.6814-0.7562-0.6791-0.6695-0.6306-0.6639-0.7697-0.7262-0.7238-0.6539-0.5651-0.5287-0.5140-0.5197-0.5238-0.5201-0.5290-0.5225-0.5109-0.4815-0.4517-0.4375-0.3998-0.3588-0.3321-0.3209-0.3182-0.3133-0.3074-0.3024-0.2939-0.2915-0.2850-0.2872-0.3118-0.2955-0.2955-0.3101-0.3045-0.2974-0.2911-0.2694-0.2633-0.2638-0.2654-0.2616-0.2667-0.2551-0.2435

Ex22-0.2418-0.2464-0.2501-0.2555-0.2676-0.2737-0.2795-0.2848-0.2850-0.2869-0.2915-0.2908-0.2883-0.2874-0.2920-0.2942-0.2997-0.2987-0.2937-0.2901-0.2869-0.2833-0.2864-0.2889-0.2950-0.3015-0.3037-0.3067-0.3050-0.3022-0.2982-0.2917-0.2914-0.2884-0.2888-0.2878-0.2822-0.2798-0.2783-0.2702-0.2819-0.2876-0.2931-0.2986-0.2987-0.2962-0.2941-0.2958-0.2932-0.2927-0.2772-0.2916-0.3061

Ex33-0.6548-0.6595-0.6325-0.5879-0.6274-0.7422-0.7488-0.7356-0.7171-0.7049-0.7380-0.7714-0.8464-0.9063-0.9123-0.9525-0.9951-1.0084-0.9642-0.9848-0.9783-0.9935-1.0264-1.0786-1.1649-1.2608-1.2401-1.1910-1.1179-1.0472-0.9467-0.9349-0.9637-0.9187-0.8488-0.7819-0.6898-0.6089-0.5638-0.5666-0.5928-0.6063-0.6287-0.6548-0.6418-0.6228-0.6128-0.5960-0.5750-0.5703-0.5382-0.5523-0.558853

Ex44-0.0396-0.0419-0.0450-0.0487-0.0486-0.0492-0.0509-0.0520-0.0520-0.0515-0.0501-0.0510-0.0489-0.0504-0.0558-0.0594-0.0620-0.0634-0.0628-0.0634-0.0621-0.0595-0.0612-0.0657-0.0728-0.0786-0.0853-0.0921-0.0986-0.1058-0.1103-0.1130-0.1170-0.1173-0.1163-0.1129-0.1014-0.0922-0.0915-0.1016-0.1227-0.1379-0.1524-0.1677-0.1814-0.1898-0.1930-0.1996-0.2012-0.2032-0.2036-0.2159-0.2126

Ex55-0.4416-0.4438-0.4705-0.5352-0.4758-0.4095-0.4265-0.4255-0.4269-0.3993-0.3463-0.3278-0.3067-0.3475-0.3979-0.4208-0.4083-0.3843-0.3870-0.3734-0.3353-0.3162-0.3094-0.3190-0.3401-0.3575-0.3835-0.4102-0.4073-0.4107-0.4259-0.3950-0.3690-0.3622-0.3329-0.3032-0.2802-0.2351-0.2155-0.2133-0.2202-0.2365-0.2467-0.2525-0.2686-0.2651-0.2585-0.2401-0.2452-0.2284-0.2061-0.2057-0.2121Ex66-0.0117-0.0129-0.0137-0.0134-0.0114-0.0110-0.0124-0.0143-0.0178-0.0206-0.0210-0.0244-0.0248-0.0263-0.0253-0.0225-0.0208-0.0216-0.0253-0.0308-0.0362-0.0382-0.0398-0.0391-0.0351-0.0301-0.0265-0.0233-0.0212-0.0202-0.0168-0.0158-0.0193-0.0274-0.0426-0.0619-0.0865-0.1068-0.1107-0.1019-0.0870-0.0720-0.0607-0.0508-0.0399-0.0393-0.0433-0.0471-0.0522-0.0590-0.0538-0.0583-0.0750

The trends in the above own elasticities of input demand generated by our estimated joint
cost function are quite interesting. The magnitudes of the Import, Machinery and Equipment
and Other Capital Services (inputs 1, 3 and 5) own elasticities of demand trended downward
over the sample period, indicating that the magnitude of substitution possibilities for these
inputs declined over time.*44 On the other hand, the magnitudes of the Structures and Land
Services (inputs 4 and 6) own elasticities of demand trended upward over the sample period
but the size of these elasticities was small over the entire sample period. During land pricebubbles, the own elasticity of demand for land tended to increase in magnitude. Finally, theown elasticity of demand for labour remained relatively constant over the entire sample period.

*44 Perhaps this phenomenon is due to the growth of multinationals and the resulting internal international

supply chains. Another possible explanation is the growth of services relative to goods production inmost countries: services inputs may be less substitutable than goods inputs.

54Appendix B: Proof of the Flexibility of the Normalized Quadratic
Joint Cost Function
A flexible functional form for a joint cost function C(y, w) has enough free parameters so thatit can provide a second order Taylor series approximation to an arbitrary twice continuously
differentiable joint cost function C ∗ (y, w) at an arbitrary point (y ∗ , w∗ ). We assume that bothC(y, w) and C ∗ (y, w) are linearly homogeneous in the components of y holding w constant*45
and are linearly homogeneous in the components of w holding y constant.*46 It turn out thatthe linear homogeneity assumptions and the assumption of twice continuous differentiability
imply that C(y, w) will be a flexible functional form if it has enough parameters to satisfy
the following conditions:*47∇2yw C(y ∗ , w∗ ) = ∇2yw C ∗ (y ∗ , w∗ );M N restrictions;(B1)∇2yy C(y ∗ , w∗ ) = ∇2yy C ∗ (y ∗ , w∗ );∇2ww C(y ∗ , w∗ ) = ∇2ww C ∗ (y ∗ , w∗ );M (M − 1)/2 independent restrictions;(B2)

N (N − 1)/2 independent restrictions.

(B3)

There are M 2 restrictions in the matrix equation (B2) but Young’s Theorem in calculus implies
that the upper triangle of matrix elements in the matrix of second order partial derivatives ofC(y ∗ , w∗ ) is equal to the lower triangle; i.e., [∇2yy C(y ∗ , w∗ )]T = [∇2yy C(y ∗ , w∗ )] and similarly,[∇2yy C ∗ (y ∗ , w∗ )]T = [∇2yy C ∗ (y ∗ , w∗ )]. Thus there are only M (M + 1)/2 independent restrictions on the second order partial derivatives of C(y ∗ , w∗ ) in the matrix equation (B2). But
due to the linear homogeneity of C(y, w) in the components of y, Euler’s Theorem on homogeneous functions implies the following M restrictions on the second order partial derivativesof C(y ∗ , w∗ ) and C ∗ (y ∗ , w∗ ):∇2yy C(y ∗ , w∗ )y ∗ = 0M ; ∇2yy C ∗ (y ∗ , w∗ )y ∗ = 0M .

(B4)Since the M by M matrices ∇2yy C(y ∗ , w∗ ) and ∇2yy C ∗ (y ∗ , w∗ ) are symmetric, equality of theupper diagonal elements in equations (B2) plus the 2M equations in (B4) will imply equalityof all M 2 elements in the matrix equation ∇2yy C(y ∗ , w∗ ) = ∇2yy C ∗ (y ∗ , w∗ ).
Similarly, due to the linear homogeneity of C(y, w) in the components of w, Euler’s Theorem
on homogeneous functions implies the following N restrictions on the second order partial*45 This assumption implies that the N partial derivative functions, ∂C(y ∗ , w ∗ )/∂wn for n = 1, . . . , N , arealso linearly homogeneous in the components of y. Thus by Euler’s Theorem on homogeneous functions,
∑M
∗ 2
∗
∗
∂C(y ∗ , w∗ )/∂wn =m=1 ym ∂ C(y , w )/∂wn ∂ym for n = 1, . . . , N . These equations imply that∇w C(y ∗ , w∗ )T = y ∗T ∇2yw C(y ∗ , w∗ ). Similarly, ∇w C ∗ (y ∗ , w∗ )T = y ∗T ∇2yw C ∗ (y ∗ , w∗ ). Thus if∇2yw C(y ∗ , w∗ ) = ∇2yw C ∗ (y ∗ , w∗ ), then ∇w C(y ∗ , w∗ ) = ∇w C ∗ (y ∗ , w∗ ).
*46 This assumption implies that the M partial derivative functions, ∂C(y ∗ , w ∗ )/∂ym for m = 1, . . . , M ,are also linearly homogeneous in the components of w. Thus by Euler’s Theorem on homogeneous
∑
∗ 2
∗
∗
functions, ∂C(y ∗ , w∗ )/∂ym = N
n=1 wn ∂ C(y , w )/∂ym ∂wn for n = 1, . . . , N . These equations imply
that ∇y C(y ∗ , w∗ )T = w∗T ∇2wy C(y ∗ , w∗ ). Similarly, ∇y C ∗ (y ∗ , w∗ )T = w∗T ∇2wy C ∗ (y ∗ , w∗ ). Thusif ∇2yw C(y ∗ , w∗ ) = ∇2yw C ∗ (y ∗ , w∗ ), then ∇2wy C(y ∗ , w∗ ) = ∇2wy C ∗ (y ∗ , w∗ ) and ∇y C(y ∗ , w∗ ) =∇y C ∗ (y ∗ , w∗ ). This last equality implies that C(y ∗ , w∗ ) = C ∗ (y ∗ , w∗ ) since Euler’s Theorem implies
that C(y ∗ , w∗ ) = y ∗T ∇y C(y ∗ , w∗ ) and C ∗ (y ∗ , w∗ ) = y ∗T ∇y C ∗ (y ∗ , w∗ ).
*47 Note that the total number of free parameters that C(y, w) must have in order to be a flexible functional
form for a joint cost function is M N + (1/2)M (M − 1) + (1/2)N (N − 1) = (M + N )(M + N − 1)/2.
This is the same number of free parameters that is required in order for f (y2 , . . . , yM , x1 , . . . , xN ) to bea flexible functional form for a constant returns to scale production function.

55

derivatives of C(y ∗ , w∗ ) and C ∗ (y ∗ , w∗ ):∇2ww C(y ∗ , w∗ )w∗ = 0N ; ∇2ww C ∗ (y ∗ , w∗ )w∗ = 0N .

(B5)Since the N by N matrices ∇2ww C(y ∗ , w∗ ) and ∇2ww C ∗ (y ∗ , w∗ ) are symmetric, equality of theupper diagonal elements in equations (B3) plus the N equations in (B5) will imply equalityof all N 2 elements in the matrix equation ∇2ww C(y ∗ , w∗ ) = ∇2ww C ∗ (y ∗ , w∗ ).
∗Let y ∗ ≡ [y1∗ , . . . , yM
] ≫ 0M be a positive reference output vector and let w∗ ≡∗] ≫ 0N be a positive vector of reference input prices. Let α ≡ [α1 , . . . , αN ] ≫ 0N
[w1∗ , . . . , wN
and β ≡ [β1 , . . . , βM ] ≫ 0M be positive vector of predetermined constants and that satisfy
the following linear restrictions:
αT w∗ = 1; β T y ∗ = 1.

(B6)

The basic Normalized Quadratic Joint Cost Function, C(y, w) is defined as follows:*48
C(y, w) ≡ (1/2)(wT Aw)(αT w)−1 (β T y) + (1/2)(y T By)(αT w)(β T y)−1 + wT Dy.

(B7)

The M by N matrix D is unrestricted. Assume that the matrix A has the following properties:
A is a negative semidefinite N by N matrix;(B8)

A is symmetric so that A = AT ;(B9)

Aw = 0N .

(B10)

∗

Assume that the matrix B has the following properties:
B is a positive semidefinite M by M matrix;
T

(B11)

B is symmetric so that B = B ;(B12)

By = 0M .

(B13)

∗

Now compute the first and second order partial derivatives of C(y, w) defined by (B7) andevaluate them at the point (y ∗ , w∗ ). Using the restrictions (B6), (B10) and (B13), we obtainthe following first and second order partial derivatives:∇y C(y ∗ , w∗ ) = DT w∗ ;

(B14)

∗

∇w C(y , w ) = Dy ;(B15)

∇2yy C(y ∗ , w∗ ) = B;
∇2ww C(y ∗ , w∗ ) = A;
∇2yw C(y ∗ , w∗ ) = DT .

(B16)

∗

∗

(B17)
(B18)

To prove the flexibility of the Normalized Quadratic Joint Cost Function defined by (B7) withthe restrictions (B6)-(B13), we need to find matrices A, B and D that lead to the satisfactionof equations (B1)-(B3). Using equations (B16)-(B18), this is very simple: define A, B and D
as follows:A ≡ ∇2ww C ∗ (y ∗ , w∗ );(B19)

B ≡ ∇2yy C ∗ (y ∗ , w∗ );(B20)

D ≡ [∇2yw C ∗ (y ∗ , w∗ )]T .

(B21)*48 This functional form is analogous to the Normalized Quadratic Value Added Function Π(p, x) that was

defined in Diewert and Fox (2021)[20].

56Under our regularity conditions on the production possibilities set S, it can be shown that∇2ww C ∗ (y ∗ , w∗ ) is a symmetric negative semidefinite matrix which satisfies ∇2ww C ∗ (y ∗ , w∗ )w∗= 0N and hence, the matrix A will satisfy the restrictions (B8)-(B10). It also can beshown*49 that ∇2yy C ∗ (y ∗ , w∗ ) is a symmetric positive semidefinite matrix which satisfies∇2yy C ∗ (y ∗ , w∗ )y ∗ = 0M and hence, B will satisfy the restrictions (B11)-(B13). This establishes the flexibility of the basic Normalized Quadratic Joint Cost function.

*49 See Diewert (2022)[17].

57Appendix C: Nonparametric Estimates for Technical Progress
In this section, we will calculate nonparametric estimates for gross output Total Factor Productivity Growth and compare these estimates with our Appendix A index number estimates
TFP growth and our main text estimates of technical progress. The analysis in this section isbased on Diewert and Fox (2018)[19].*50 Their nonparametric methodology has the advantage
of giving a decomposition of TFP growth into technical progress and inefficiency components.
There are two key concepts that their analysis is based on:• An approximation to the aggregate production possibilities set for an economy can beformed by using linear multiples of past output and input vectors and• The Cost Constrained Gross Output Function can be used to form measures of efficiency,
output price change, input price change, input quantity change and technology change.
Denote the year t observed output and input quantity vectors by y t and xt and the corresponding price vectors by pt and wt . Following Diewert and Fox (2018)[19], we assume thatthe year t national technology set can be approximated by assuming it consists of past observed output and input vectors, (y s , xs ), and linear multiples of these vectors for past years
and the current year t. Let S t denote the resulting year t production possibilities set. Thus
S 1 ≡ {(y, x) : y = λy 1 , x = λx1 ; λ ≥ 0}, S 2 ≡ {(y, x) : y = λ1 y 1 , x = λ1 x1 ; λ1 ≥ 0, y =λ2 y 2 , x = λ2 x2 ; λ2 ≥ 0}, . . ., S t ≡ {(y, x) : y = λs y s , x = λs xs ; λs ≥ 0, s = 1, 2, . . . , t}.
These definitions for the S t mean that we are assuming that S t is a constant return to scaletechnology set for each period.
The year t Cost Constrained Gross Output Function for the US economy, Gt (p, w, x), is
defined as follows for the positive price vectors p and w, positive input vector x and using
the period t production possibilities set S t :*51
Gt (p, w, x) ≡ maxy,z {p · y : (y, z) ∈ S t ; w · z ≤ w · x};

t = 1970, . . . , 2022.

= maxs {p · y s w · x/w · xs : s = 1, 2, . . . , t};= w · x maxs {p · y s /w · xs : s = 1, 2, . . . , t}.

(C1)

Given the period t technology set S t , output prices p, input prices w and the constraint thatprimary input costs should not exceed cost w · x, we assume that producers choose the output
vector y and input vector z to maximize national gross output, p · y, subject to total input
cost w · z to be equal to or less than the given input cost w · x.
Due to our assumptions on the year t national production possibilities set S t , the year t cost
constrained value added function Rt (p, w, x) can be calculated for any hypothetical p, w andx by solving the very simple maximization problem, maxs {p · y s /w · xs : s = 1, 2, . . . , t}.,which involves taking the maximum of t numbers.
The Cost Constrained Gross Output Function defined by equation (C1) can be used to decompose gross output growth from year t − 1 to year t, pt · y t /pt−1 · y t−1 , into various explanatory
growth factors. The explanatory factors are as follows:• efficiency changes,• changes in real output prices,• changes in primary inputs,*50 The only difference is that we use the Cost Constrained Gross Output Function in place of the Cost

Constrained Value Added Function.
*51 This function is a variant of Diewert’s (1983; 1086)[15] balance of trade restricted value added function.

58

• changes in real input prices, and• technical progress.
We now define the above explanatory factors using the observed data and the function
Gt (p, w, x) defined by equation (C1). Following the example of Balk (1998; 143)[2], we define
the gross output efficiency of the production unit for year t, et , as follows:et ≡ pt · y t /Gt (pt , wt , xt ) ≤ 1;

t = 1970, . . . , 2022.

(C2)

where the inequality in equation (C2) follows using definition (C1).*52 Thus if et = 1, thenproduction is allocatively efficient in year t and if et < 1, then production for the sector during
year t is allocatively inefficient. Note that the above definition of value added efficiency is agross output counterpart to Farrell’s (1957; 255)[26] cost based measure of overall efficiency.
Define an index of the change in gross output efficiency εt for the production sector over theyears t − 1 and t as follows:εt ≡ et /et−1 = [pt · y t /Gt (pt , wt , xt )]/[pt−1 · y t−1 /Gt−1 (pt−1 , wt−1 , xt−1 )];t = 1971, . . . , 2022.

(C3)

Thus if εt > 1, then gross output efficiency has improved going from year t − 1 to t whereas
it has fallen if εt < 1.
We turn our attention to defining nonparametric measures of output price change going fromyear t − 1 to t. Following the example of Konüs (1939)[46] in his analysis of the true costof living index, it is natural to single out two special cases of a family of real output pricetindexes: one choice is αL
where we use the year t − 1 technology and set the reference input
prices and quantities equal to the year t − 1 input prices wt−1 and primary input quantitiest
xt−1 (which gives rise to a Laspeyres type output price index ) and another choice is αP
where
we use the year t technology and set the reference input prices and quantities equal to theyear t input prices and quantities wt and xt (which gives rise to a Paasche type real output
price index ). We then define an overall measure of real output price change αt by taking thegeometric mean of these two indexes. These indexes are defined as follows:tαL≡ Gt−1 (pt , wt−1 , xt−1 )/Gt−1 (pt−1 , wt−1 , xt−1 );tαP≡ Gt (pt , wt , xt )/Gt (pt−1 , wt , xt );t t 1/2αt ≡ [αLαP ] ;

t = 1971, . . . , 2022;

t = 1971, . . . , 2022;

t = 1971, . . . , 2022.

(C4)
(C5)
(C6)

Two natural measures of input quantity change are the Laspeyres and Paasche input quantity
indexes. Denote the year t indexes as βLt and βPt . Again, it is natural to take the geometricaverage of these two indexes which gives rise to the Fisher ideal input quantity index, β t .
These indexes are defined as follows:
βLt ≡ wt−1 · xt /wt−1 · xt−1 ;βPt ≡ wt · xt /wt · xt−1
β t ≡ [βLt βPt ]1/2 ;t = 1971, . . . , 2022;

t = 1971, . . . , 2022;

t = 1971, . . . , 2022.

(C7)
(C8)
(C9)

We now consider indexes which measure the effects on cost constrained gross output ofa change in real input prices going from year t − 1 to t. Thus we consider measures of*52 Use the fact that (y s , xs ) is a feasible solution for the year t maximization problem defined by (C1).

59the change in cost constrained gross output of the form Gs (p, wt , x)/Gs (p, wt−1 , x). SinceGs (p, w, x) is homogeneous of degree 0 in the components of w, it can be seen that wecannot interpret Gs (p, wt , x)/Gs (p, wt−1 , x) as an input price index. If there is only one
primary input, Gs (p, wt , x)/Gs (p, wt−1 , x) is equal to Gs (p, 1, x)/Gs (p, 1, x) = 1 and thismeasure of input price change will be independent of changes in the price of the single input. In the case where the number of primary inputs is greater than 1, it is best to interpret
Gs (p, wt , x)/Gs (p, wt−1 , x) as measuring the effects on cost constrained gross output of achange in the relative proportions of primary inputs used in production or in the mix of inputs used in production that is induced by a change in relative input prices when there is morethan one primary input. We consider two special cases of this family of input mix indexes,
Case 1 and Case 2. The first case index, γ1t , will use the year t Cost Constrained Gross Output
function and the year t − 1 reference vectors pt−1 and xt−1 while the second case index, γ2t ,will use the year t−1 Cost Constrained Gross Output function and the year t reference vectors
pt and xt . We take the geometric mean of these two indexes γ t to provide a measure of theoverall effects of a change in input prices.*53
γ1t ≡ Gt (pt−1 , wt , xt )/Gt (pt−1 , wt−1 , xt );γ2t ≡ Gt−1 (pt , wt , xt−1 )/Gt−1 (pt , wt−1 , xt−1 );γt≡ [γ1t γ2t ]1/2 ;t = 1971, . . . , 2022;
t = 1971, . . . , 2022;

t = 1971, . . . , 2022.

(C10)
(C11)
(C12)

Finally, we use the Cost Constrained Gross Output function in order to define measures oftechnical progress going from year t−1 to t. These measures hold p, w and x constant and onlychange the technology from the year t − 1 technology to the year t technology.*54 Thus, thesemeasures are of the form Gt (p, w, x)/Gt−1 (p, w, x). If there is positive technical progress
going from year t − 1 to t, then the production possibilities set S t will be larger than the
period t − 1 set, S t−1 , and thus Gt (p, w, x) will be equal to or greater than Gt−1 (p, w, x)
and our measures of technical progress will be equal to or greater than 1. Our measures oftechnical progress cannot fall below 1.
We consider two measures of technical progress, a Laspeyres measure τLt and a Paasche measureτPt . However, the Laspeyres case τLt will use the year t input vector xt as the reference input
vector and the year t − 1 reference output price and input price vectors pt−1 and wt−1 while
the Paasche case τPt will use the year t − 1 input vector xt−1 as the reference input and theyear t reference output and input price vectors pt and wt .*55 As usual, we take our overall
year t measure of technical change τ t to be the geometric mean of the Laspeyres and Paasche
measures of technical change.
τLt ≡ Gt (pt−1 , wt−1 , xt )/Gt−1 (pt−1 , wt−1 , xt );τPt ≡ Gt (pt , wt , xt−1 )/Gt−1 (pt , wt , xt−1 );τt≡ [τLt τPt ]1/2 ;t = 1971, . . . , 2022;
t = 1971, . . . , 2022;

t = 1971, . . . , 2022.

(C13)
(C14)
(C15)

*53 These choices of the reference vectors will make our decomposition of gross output growth an exact one.

Usually, these input mix growth factors are close to one for all periods.
*54 These measures of technical progress measures were defined by Diewert and Morrison (1986; 662)[22]

using the country’s GDP function. A special case of the family was defined earlier by Diewert (1983;
1063)[15]. Balk (1998; 99)[2] also used this definition and Balk (1998; 58)[2], following the example ofSalter (1960)[52], used the joint cost function to define a similar family of technical progress indexes.
*55 In our case where the reference technology is subject to constant returns to scale, τ t turns out to beLindependent of xt and τPt turns out to be independent of xt−1 . These “mixed” indexes of technicalprogress turn out to be true Laspeyres and Paasche type indexes.

60Using the above definitions, it can be shown that the following exact decomposition of year
t nominal gross output growth (relative to the 1970 level of gross output) into explanatory
growth factors holds:
GOtG ≡ pt · y t /pt−1 · y t−1 = εt αt β t γ t τ t ;t = 1971, . . . , 2022.

(C16)

Table C1 lists GOtG and the growth components on the right hand side of equation (C16).

61

Table C1: A Nonparametric Decomposition of Gross Output TFP Growth
Year
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
Mean

GOtG
1.0862
1.10481.12011.10281.08251.12091.1203
1.1341
1.12441.0922
1.11511.03801.0872
1.12131.07091.05991.06681.07901.07681.0572
1.02691.0602
1.0551
1.06701.0555
1.0584
1.06601.05681.0703
1.0783
1.02341.03051.0522
1.07751.0754
1.0652
1.05091.02890.9512
1.05781.04861.03951.03191.04281.02901.02191.04651.05451.03660.98051.11831.10001.0657εt1.00001.00001.00000.97750.99881.02411.00011.00000.99650.98661.00850.98151.02371.00391.00001.00001.00001.00001.00001.00000.99951.00051.00001.00000.99581.00421.00001.00001.00001.00000.99681.00321.00001.00001.00001.00000.99800.9961
0.99361.01250.99921.00081.00001.00001.00000.99811.00201.00001.00000.99431.00570.98840.9998αt1.04831.0455
1.06141.1115
1.09711.0543
1.06391.0717
1.09111.10131.08681.06131.03541.03071.02791.02441.0282
1.03611.0392
1.0352
1.0283
1.02251.02081.01791.02271.01351.01061.00481.01501.02661.01891.01131.02091.02871.03601.03081.03001.0304
0.99241.01611.02831.01641.01141.01370.99751.00531.01861.02291.01341.01281.0482
1.0722
1.0348βt
γt1.0093 1.0000
1.0340 1.0000
1.0417 1.0000
1.0156 0.9995
0.9879 1.0001
1.0373 1.0006
1.0371 1.0000
1.0450 1.0000
1.0345 0.9998
1.0067 0.9986
1.0176 0.9997
0.9974 0.9991
1.0243 1.0014
1.0583 1.0009
1.0347 1.0000
1.0303 1.0000
1.0364 1.0000
1.0308 1.0000
1.0343 1.0000
1.0214 0.9999
0.9994 0.9997
1.0186 1.0000
1.0266 1.0000
1.0370 1.0000
1.0363 1.0001
1.0306 0.9999
1.0457 1.0000
1.0460 1.0000
1.0445 1.0000
1.0457 1.0000
1.0079 0.9997
1.0154 0.9999
1.0140 1.0000
1.0298 1.0000
1.0276 1.0000
1.0313 1.0000
1.0227 0.9997
1.0031 0.9994
0.9635 1.0013
1.0184 1.0006
1.0206 1.0000
1.0187 0.9999
1.0167 1.0000
1.0244 1.0000
1.0264 1.0000
1.0185 1.0000
1.0224 0.9999
1.0230 1.0000
1.0189 1.0000
0.9742 0.9994
1.0391 1.0003
1.0381 0.9999
1.0238 0.99998

62

τt
NTFPtG
1.0266
1.02661.02201.02201.01311.01311.00000.97701.00000.99891.00011.02491.01531.01531.01261.01261.00000.99621.00000.98521.00001.00821.00000.98071.00001.02511.02301.02801.00691.00691.00431.00431.00111.00111.01031.01031.00191.00191.00000.99991.00000.9992
1.01751.01801.00691.00691.01081.01081.00000.99591.0092
1.01331.00871.00871.00551.00551.00961.00961.00441.00441.00000.99651.00041.00361.01651.01651.01711.01711.01011.01011.00201.00201.00000.99761.00000.99551.00000.99491.00901.02221.00000.9992
1.0032
1.00391.00351.00351.0042
1.0042
1.00511.00511.00000.99801.00311.00491.0077
1.00771.00401.00401.00000.99371.0206
1.02671.00000.9883
1.00611.0059TFPtG
1.02661.02201.01310.97670.99941.02501.01531.0126
0.99620.98511.00840.98131.02471.02771.00691.00431.00111.01031.00190.99990.9992
1.01791.00691.01080.99591.01341.00871.00551.00961.0044
0.99631.00351.01651.01711.01011.00200.9976
0.99510.9953
1.02140.99911.00391.00351.0042
1.0051
0.99801.00491.00771.00400.99351.02700.98831.0059κt-0.0097-0.0098-0.0097-0.0094-0.0079-0.0080-0.0082
0.00340.0032-0.0007-0.0009-0.0009-0.0117-0.0118-0.0118-0.01200.0006-0.0034-0.0027-0.0027-0.0024-0.0123-0.0075-0.0074-0.0072-0.0072-0.0080-0.0081-0.0080-0.0076-0.0062-0.0064-0.0065-0.0076-0.0074-0.0037-0.0026-0.0027-0.0026-0.0039-0.0037-0.0036-0.0036-0.0034-0.0034-0.0035-0.0037-0.0036-0.0037-0.0030-0.0119-0.0117-0.0057et
1.00001.00001.00000.97750.97630.99991.00001.00000.99640.98300.9914
0.9731
0.99611.00001.00001.00001.00001.00001.00001.00000.99951.00001.00001.00000.99581.00001.00001.00001.00001.00000.99681.00001.00001.00001.00001.00000.99800.99400.98771.00000.99921.00001.00001.00001.00000.99801.00001.00001.00000.99431.00000.9884
0.9970On average, US nominal gross output grew 6.57% per year. The explanatory factors for thisaverage growth rate are as follows: efficiency growth subtracted 0.02% per year, output priceinflation contributed 3.48% per year, input quantity growth contributed 2.38% per year, theinput mix effect on average was negligible, and technical progress contributed 0.61% per year.
The level of efficiency et was negative for 17 years: 1974-1976 (first oil shock recession), 19791983 (second oil shock recession), 1991, 1995, 2007-2009 (financial crisis), 2011, 2016, 2020
(covid shock) and 2022. For these years, production was in the interior of our estimatedproduction possibilities set.
A new nonparametric measure of TFP growth for the economy going from year t − 1 to t canbe defined (following Jorgenson and Griliches (1967)[31]) as an index of output growth divided
by an index of input growth. An appropriate index of output growth is the gross output ratio
pt · y t /pt−1 · y t−1 divided by the gross output price index αt . An appropriate index of input
growth is β t . Thus define the nonparametric year t Gross Output TFP growth rate, NTFPtG ,for the US economy as follows:
NTFPtG ≡ {[pt · y t /pt−1 · y t−1 ]/αt }/β t = εt γ t τ t ;t = 1971, . . . , 2022.

(C17)

where the last equality in equation (C17) follows from equation (C16). The nonparametricyear t TFP growth, NTFPtG , is equal to the product of period t efficiency change εt , the period
t input mix index γ t (which typically will be small in magnitude) and the period t measureof technical progress τ t . The nonparametric measure of TFP growth, NTFPtG , and our old
index number measure of TFP growth, TFPtG from Table A13 in Appendix A, are listed
above in Table C1. Note that the index number and nonparametric estimates of TFP growth
are almost identical. Note also that on average TFP growth (measured by both methods) isequal to 0.59% per year over the years 1971-2022. The average rate of technical progress isslightly higher at 0.61% per year. On average, inefficiency dragged down technical progress by
0.02% per year. Preventing recessions is important. Using the Diewert and Fox nonparametricmethodology, technical progress can never be negative.
Table C1 also lists the Joint Cost Function based estimates of cost saving technical charge foreach year t, κt . Taking the negative of these measures and adding one should lead to measuresthat are approximately equal to our measures of TFP growth. From viewing Table C1, it canbe seen that this approximation is not very close. As was mentioned in the main text, theeconometric estimates of technical progress are not able to capture year to year shocks to
TFP.*56 However, as was noted in the main text, the long term trends in cost saving technicalchange are fairly close to the long term trends in TFP growth.
We follow the example of Kohli (1990)[39] in order to obtain a levels decomposition for theobserved level of nominal gross output in year t, pt · y t , relative to its observed value in year
1 of our sample, p1970 · y 1970 . Define the cumulated explanatory variables as follows:
E 1970 ≡ 1; A1970 ≡ 1; B 1970 ≡ 1; C 1970 ≡ 1; T 1970 ≡ 1.

(C18)

For t = 1971, . . . , 2022, define the above variables recursively as follows:E t ≡ εt E t−1 ; At ≡ αt At−1 ; B t ≡ β t B t−1 ; C t ≡ γ t C t−1 ;T t ≡ τ t T t−1 ;t = 1971, . . . , 2022.

(C19)*56 There is another problem with our joint cost function methodology: it assumes optimizing behavior on

the part of producers, which is not consistent with observed outputs and inputs being in the interior of the
production possibilities set. The Diewert and Fox nonparametric methodology allows for nonoptimizing
behavior on the part of producers but it has the disadvantage that the year t reference technologythat their methodology utilizes may not be a very close approximation to the actual year t production
possibilities set.

63

Using the above definitions, it can be seen that we have the following levels decomposition forthe level of year t observed gross output relative to its level in 1970:
GOt ≡ pt · y t /p1970 · y 1970 = At B t C t E t T t ;t = 1970, . . . , 2022.

(C20)

The components of the levels decomposition of gross output given by (C20) are listed in Table
C2.

64

Table C2: The Levels Decomposition of Gross Output for the US 1970-2022
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022

GOt
1.00001.08621.20001.3442
1.48241.60471.79862.01502.28522.56962.80663.1296
3.24863.53193.96044.24104.49504.79515.17385.57095.88976.04786.4121
6.7652
7.2182
7.61868.06348.59559.08339.721510.4823
10.7270
11.0542
11.6316
12.5328
13.4772
14.3553
15.0854
15.521314.7643
15.6176
16.377017.0230
17.5663
18.3180
18.849419.2614
20.1568
21.2546
22.0333
21.6040
24.1606
26.5764Et
1.00001.00001.00001.00000.97750.97630.99991.00001.00000.99640.98300.9914
0.9731
0.99611.00001.00001.00001.00001.00001.00001.00000.99951.00001.00001.00000.99581.00001.00001.00001.00001.00000.99681.00001.00001.00001.00001.00000.99800.99400.98771.00000.99921.00001.00001.00001.00000.99801.00001.00001.00000.99431.00000.9884At
1.00001.0483
1.09601.16331.29291.4184
1.4954
1.59101.70521.86052.04882.22682.36322.44692.52202.59242.65562.73042.82892.93973.04313.1292
3.19973.26613.32473.40013.44593.48243.49913.55153.64593.71473.7567
3.8351
3.94524.0872
4.21324.33944.47114.43724.50894.63644.71264.7661
4.83124.81904.84454.93445.04745.11515.18065.43055.8224Bt
1.00001.0093
1.04361.08711.10401.0906
1.1313
1.17331.22611.2684
1.27691.2994
1.29601.32751.40491.45371.49771.55221.60001.65481.69011.68901.7204
1.7661
1.8314
1.89801.95612.0455
2.13952.23462.33682.35522.39142.42482.49702.56602.64622.70632.71472.61542.66342.71822.76912.81542.88422.96043.01513.0826
3.15343.21293.1301
3.25263.376465

Ct
1.00001.00001.00001.00000.99950.99951.00021.00011.00010.99990.99850.99820.99730.99870.99970.99970.99970.99970.99970.99970.99960.99930.99930.99930.99930.99930.99920.99920.99920.99920.99920.99890.99890.99890.99890.99890.99890.99850.99800.99920.99980.99980.99970.99970.99970.99970.99960.99950.99950.99950.99890.99930.9992Tt
NTFPt
1.0000 1.0000
1.0266 1.0266
1.0492 1.0492
1.0629 1.0629
1.0629 1.0385
1.0629 1.0373
1.0631 1.0631
1.0793 1.0794
1.0929 1.0930
1.0929 1.0889
1.0929 1.0728
1.0929 1.0816
1.0929 1.0607
1.0929 1.0873
1.1181 1.1177
1.1258 1.1254
1.1305 1.1302
1.1318 1.1314
1.1435 1.1431
1.1456 1.1452
1.1456 1.1451
1.1456 1.1443
1.1657 1.1648
1.1737 1.1728
1.1863 1.1855
1.1863 1.1806
1.1972 1.1963
1.2077 1.2067
1.2143 1.2133
1.2260 1.2250
1.2313 1.2304
1.2313 1.2261
1.2318 1.2304
1.2522 1.2508
1.2737 1.2722
1.2865 1.2850
1.2891 1.2876
1.2891 1.2846
1.2891 1.2788
1.2891 1.2722
1.3007 1.3005
1.3007 1.2995
1.3049 1.3045
1.3095 1.3091
1.3150 1.3146
1.3217 1.3213
1.3217 1.3186
1.3258 1.3251
1.3360 1.3354
1.3413 1.3407
1.3413 1.3323
1.3689 1.3679
1.3689 1.3519

Define the period t level of Nonparametric Gross Output TFP relative to 1970, NTFPt , asfollows:
NTFP1970 ≡ 1; NTFPt ≡ (NTFPtG )(NTFPt−1 );t = 1971, . . . , 2022

(C21)

where NTFPtG is defined by equation (C17). Using definitions (C18)–(C21), it can be seenthat we have the following levels decomposition for Nonparametric TFP using the cumulated
explanatory factors defined by definitions (C18) and (C19):
NTFPt ≡ GOt /[At B t ] = C t E t T t ;t = 1970, . . . , 2022.

(C22)

The series NTFPt is listed in Table C2 above and the decomposition given by (C22) is plottedon Figure 2.

Figure 2 Decomposition of Nonparametric Gross Output TFP into Explanatory Factors

It can be seen that the oil shocks in the 1970s led to large amounts of inefficiency which led
to TFP declines. The financial crisis of 2007 and the effects of Covid in 2020 and 2022 also
led to inefficiency but the resulting declines in TFP were not as severe.

66

Appendix D: Converting Cost Function Elasticities into Gross Output
Function Elasticities
It is useful to be able to convert the elasticities of inverse output supply and input demand
generated by the estimation of a joint cost function into elasticities of output supply andinverse input demand elasticities. We indicate how this can be done in this Appendix.*57
Suppose we have estimated a joint cost function C(y, w) for a particular year where C is twice
continuously differentiable with respect to the components of y and w at the year t values fory and w. Equations (6) and (7) in section 2 give us an output price vector p and an input
demand vector x as functions of the derivatives of the cost function; i.e., p = ∇y C(y, w)
and x = ∇w C(y, w). These equations implicitly define y and w as functions of p and x,
say y = y ∗ (p, x) and w = w∗ (p, x) so we have the following system of 10 equations for ourparticular example:
p = ∇y C(y ∗ (p, x), w∗ (p, x));(D1)

x = ∇w C(y (p, x), w (p, x)).

(D2)

∗

∗

Denote the 4 by 4 matrix of derivatives of the output supply functions y ∗ (p, x) with respect
to the components of p as ∇p y ∗ (p, x) and the 4 by 6 matrix of derivatives of the functionsy ∗ (p, x) with respect to the components of x as ∇x y ∗ (p, x). Denote the 6 by 4 matrix ofderivatives of the inverse input demand functions w∗ (p, x) with respect to the componentsof p as ∇p w∗ (p, x) and the 6 by 6 matrix of derivatives of the inverse demand functionsw∗ (p, x) with respect to the components of x as ∇x w∗ (p, x). If the 10 by 10 matrix of second
order partial derivatives of the joint cost function C(y, w), SGO , has an inverse, then the firstorder partial derivatives of the functions y ∗ (p, x) and w∗ (p, x) can be defined in terms of
the elements of the inverse of the matrix of second order partial derivatives of the joint costfunction as follows:*58
[
] [ 2
]−1
∇yy C(y, w) ∇2yw C(y, w)∇p y ∗ (p, x) ∇x y ∗ (p, x)== [SGO ]−1 .
(D3)∇p w∗ (p, x) ∇x w∗ (p, x)∇2wy C(y, w) ∇2ww C(y, w)
It can be shown that the 4 by 4 matrix of Gross Output supply functions with re∗spect to output prices, ∇p y ∗ (p, x) = [∂ym(p, x)/∂pk ], is a positive semidefinite matrix with rank equal to or less than 3. For our 2022 data, this matrix has rank
equal to 2. It can also be shown that the 6 by 6 matrix of derivatives of input
prices with respect to input quantities, ∇x w∗ (p, x) = [∂wn∗ (p, x)/∂xj ], is a negative
semidefinite matrix with rank equal to or less than 5. For our 2022 data, this matrix has rank equal to 4. These derivatives can be converted into elasticities. Thus∗∗
define Eym pk ≡ [∂ym(p, x)/∂pk ][pk /ym ], Eym xn ≡ [∂ym(p, x)/∂xn ][xn /ym ], Ewn xj ≡∗∗[∂wn (p, x)/∂xj ][xj /wn ] and Ewn pm ≡ [∂wn (p, x)/∂pm ][pm /wn ] for n = 1, . . . , 6; j =1, . . . , 6; m = 1, . . . , 4 and k = 1, . . . , 4. These estimated elasticities for the US Gross Output
function for 2022 are listed in Tables D1-D4 below.

*57 This line of research was initiated by Kohli (1981b)[35] (1983b)[38] (1991)[40].
*58 This follows from the Inverse Function Theorem in advanced calculus; see Rudin (1953; 177)[51].

67Table D1: Gross Output Elasticities with Respect to Changes in Gross Output Prices for 2022
Eym pk
y1
y2
y3
y4

C
G
I
X

p1
p2
p3
p4
C
G
I
X
0.778 -0.416 -0.517 0.154-1.949 1.962 1.211 -1.224-1.557 0.779 1.038 -0.260
0.947 -1.602 -0.530 1.185

Note: Consumption (C), Government (G), Investment (I), Export (X).

Note that all main diagonal elasticities are positive; if an output price increases, output supply
also increases.
Table D2: Gross Output Elasticities with Respect to Changes in Input Quantities for 2022
Eym xn
y1
y2
y3
y4

C
G
I
X

x1
x2
x3
x4
x5
x6
M
L
KM&E KS
KOKL-0.885 1.722 0.621 1.509 -0.369 -1.598-0.640 1.239 -2.719 -1.716 -2.433 7.269
3.363 -3.551 0.210 -3.745 3.564 1.158
0.925 0.004 0.106 2.388 -1.011 -1.412

Note: Consumption (C), Government (G), Investment (I), Export (X),
Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

Table D3: Input Price Elasticities with Respect to Changes in Input Quantities for 2022
Ewn xjw1
M
w2
L
w3 KM&E
w4 KS
w5 KO
w6 KL

x1
x2
x3
x4
x5
x6
M
L
KM&E
KS
KO
KL
-5.1797.258 -0.973 1.831 -8.220
5.283
2.197 -4.035 0.696 -0.734
4.153 -2.278-1.8744.432 -2.309 -1.280 -5.556
6.588
1.552 -2.056 -0.564 -5.341
0.992
5.417
-15.605 26.052 -5.477 2.221 -29.136 21.945
11.884 -16.930 7.693 14.368 26.001 -43.015

Note: Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

Note that all main diagonal elasticities are negative; if economy wide amount of an input
increases, then the demand clearing price for that increased input quantity decreases. Notethat the own (inverse) elasticities of demand for Other Capital Services and Land Services(inputs 5 and 6) are very large in magnitude. This reflects the fact that the (direct) elasticitiesof demand for these inputs are very small in magnitude.

68

Table D4: Input Price Elasticities with Respect to Changes in Gross Output Prices for 2022
Ewn pmw1
M
w2
L
w3 KM&E
w4 KS
w5 KO
w6 KL

p1
p2
p3
p4
M
L
KM&E KS
-3.612 -0.558 4.555 0.615
2.128 0.327 -1.456 0.001
4.879 -4.565 0.549 0.136
5.222 -1.269 -4.302 1.348-2.862 -4.027 9.166 -1.278-14.668 14.252 3.529 -2.114

Note: Imports (M), Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

Instead of working with the Gross Output function to model aggregate production in an
economy, it is of interest to work with the GDP function. This model of production dates
back to Samuelson (1953)[54]*59 and the model assumes that the prices p ≡ [p1 , p2 , p3 , p4 ]and w1 (the price of imports) are fixed for outputs y ≡ [y1 , y2 , y3 , y4 ] and imports x1 and theeconomy’s endowment vector of primary inputs x◦ ≡ [x2 , x3 , x4 , x5 , x6 ] is also viewed as being
fixed.*60 Thus in this model of production, world prices are fixed and the country’s endowmentof labour and capital are fixed in the short run. Denote the vector of input prices that
corresponds to x◦ (the new vector of fixed factors of production) by w◦ ≡ [w2 , w3 , w4 , w5 , w6 ].
Using our new notation explained in the previous paragraph, the 10 estimating equations forour (old) joint cost function model for the year 2022 can be written as follows:p = ∇y C(y, w1 , w◦ );(D4)

x1 = ∂C(y, w1 , w )/∂w1 ;(D5)

x = ∇w◦ C(y, w1 , w )

(D6)

◦

◦

◦

where C(y, w1 , w◦ ) ≡ C 2022 (y, w1 , w◦ ) = C 2022 (y, w). We now shift imports x1 from being
an input in the Gross Output model of production to imports being a negative output whichleads to the GDP model of production where GDP(p, w1 , x◦ ) is defined as the maximum ofp · y − w1 x1 (with respect to x1 and the components of y), subject to (y, x1 , x◦ ) ∈ S 2022 .
The solution functions are denoted by y(p, w1 , x◦ ), x1 (p, w1 , x◦ ), The vector of equilibrium
primary input prices is denoted by w◦ (p, w1 , x◦ ) = ∇x◦ GDP(p, w1 , x◦ ).*61
The first order partial derivatives of the functions y(p, w1 , x◦ ) ≡ [y1 (p, w1 , x◦ ), . . . ,y4 (p, w1 , x◦ )]T and w◦ (p, w1 , x◦ ) ≡ [w2◦ (p, w1 , x◦ ), . . . , w6◦ (p, w1 , x◦ )]T can be obtained bydifferentiating both sides of the following 9 equations*62 with respect to the 9 components of*59 Kohli (1978)[33] (1982)[36] (1991)[40] (1993a)[41] also worked extensively with this model of production.
*60 In order to define the US GDP function for 2022, the p and y in definition (2) are replaced by(p2022f, . . . , p2022f, w12022 ) and (y12022 , . . . , y42022 , −x2022f), the w and x in definition (2) are replaced
1
4
1
2022f
2022f
by [x2, . . . , x6
] ≡ x◦ and [w22022 , . . . , w62022 ] ≡ w◦ and S is the year 2022 production possibilities

set for the US economy. The cost function that is used in this Appendix is the estimated US Joint Cost
Function for the US for the year 2022, C(y, w) ≡ C t (y, w) where t = 2022. The partial derivatives of C
are evaluated at (y, w) = (y 2022 , w2022 ). When we form elasticities, we use predicted values for prices
and quantities that are used to convert estimated derivatives into elasticities.
*61 For the properties of joint cost functions and the relationship between the joint cost function and the
GDP function, see the section on joint cost functions in Diewert (2022)[17].
*62 These equations follow from equations (D4) and (D6). The price of imports w ◦ is held constant and so
1equation (D5) is dropped for now.

69

p and x◦ :p = ∇y C(y(p, w1 , x◦ ), w1 , w◦ (p, w1 , x◦ ));(D7)

x = ∇w◦ C(y(p, w1 , x ), w1 , w (p, w1 , x )).

(D8)

◦

◦

◦

◦

If the 9 by 9 matrix S defined below has an inverse, the matrices of first order derivatives ofthe functions y(p, w1 , x◦ ) and w◦ (p, w1 , x◦ ) with respect to the 9 components of p and x◦
are defined as follows:*63
] [
[
]−1
∇2yy C(y, w1 , w◦ )∇2yw◦ C(y, w1 , w◦ )∇p y(p, w1 , x◦ )∇x◦ y(p, w1 , x◦ )=∇p w◦ (p, w1 , x◦ ) ∇x◦ w◦ (p, w1 , x◦ )∇2w◦ y C(y, w1 , w◦ ) ∇2w◦ w◦ C(y, w1 , w◦ )≡ [S]−1 .

(D9)In order to determine the first order partial derivatives of y(p, w1 , x◦ ) and w◦ (p, w1 , x◦ ) withrespect to w1 , differentiate both sides of (D7) and (D8) with respect to w1 and obtain thefollowing matrix equation:
]
[
]
[
2
◦
∇
C(y,
w
,
w
)
∇w1 y(p, w1 , x◦ )1
yw
1
= −[S]−1.
(D10)∇w1 w◦ (p, w1 , x◦ )∇2w◦ w1 C(y, w1 , w◦ )Finally, we need to calculate the first order partial derivatives of x1 (p, w1 , x◦ ) with respect to
w1 and the components of p and x◦ , Define x1 (p, w1 , x◦ ) as follows:
x1 (p, w1 , x◦ ) ≡ ∂C(y(p, w1 , x◦ ), w1 , w◦ (p, w1 , x◦ ))/∂w1 .

(D11)

Differentiate (D11) with respect to w1 where all cost function partial derivatives are evaluatedat (y, w1 , w◦ ):
]
[
◦
]
[
∇
y(p,
w
,
x
)
w
1
1
∂x1 (p, w1 , x◦ )/∂w1 = ∇2w1 w1 C + ∇2w1 y C ∇2w1 w◦ C
∇w1 w◦ (p, w1 , x◦ )
[
]
2
]
[
∇
C
yw1
= ∇2w1 w1 C − ∇2w1 y C ∇2w1 w◦ C S−1(D12)∇2w◦ w1 C
where the second equality follows using (D10). Now differentiate (D11) with respect to thecomponents of p and x◦ . We obtain the following formula for the partial derivatives ofx1 (p, w1 , x◦ ) where ∇Tp x1 denotes the transpose of the column vector ∇p x1 :
]
[
◦
◦
[ T
[
]
]
◦ y(p, w1 , x )
∇
y(p,
w
,
x
)
∇
p
1
x
∇p x1 ∇Tx◦ x1 = ∇2w1 y C ∇2w1 w◦ C
∇p w◦ (p, w1 , x◦ ) ∇x◦ w◦ (p, w1 , x◦ )
]
[
(D13)
= ∇2w1 y C ∇2w1 w◦ C S−1

where the second equality follows using (D9). Comparing (D13) to the partial derivatives
defined in (D10) and using the symmetry of the second order derivatives of the joint costfunction, we see that:−∇p x1 (p, w1 , x◦ ) = ∇w1 y(p, w1 , x◦ ) and − ∇x◦ x1 (p, w1 , x◦ ) = ∇w1 w◦ (p, w1 , x◦ ). (D14)*63 Apply the Inverse Function Theorem to (D7) and (D8) to get this result.

70

The above derivatives can be converted into elasticities. Equations (D9) are used to define the partial derivatives for the elasticities listed in Tables D5-D8. Define Eym pk ≡[∂ym (p, w1 , x◦ )/∂pk ][pk /ym ], These estimated output price elasticities for the US GDP function for 2022 are listed in Table D5 below.
Table D5: GDP Output Elasticities with Respect to Changes in Output Prices for 2022
Eym pk
y1
y2
y3
y4

C
G
I
X

p1
p2
p3
p4
C
G
I
X
1.396 -0.321 -1.295 0.049-1.503 2.031 0.648 -1.300-3.902 0.417 3.996 0.139
0.302 -1.702 0.283 1.295

Note: Consumption (C), Government (G), Investment (I), Export (X).

Comparing the own elasticities in Table D5 with the corresponding own elasticities in Table
D1, it can seen that the new own elasticities, Eym pm for m = 1, 2.3.4, are all bigger than thecorresponding Table D1 entries. This follows from Samuelson’s Le Chatelier Principle:*64 thequantity of imports is no longer fixed when we switch from the Gross Output Function to the
GDP Function so the economy has more flexibility to respond to an increase in the price ofan output and thus the output supply response to an increase in price will tend to be bigger.
Define the elasticity of output m with respect to an increase in the economy’s endowment of
inputs 2-6 as Eym xn ≡ [∂ym (p, w1 , x◦ )/∂xn ][xn /ym ] for m = 1, . . . , 4 and n = 2, . . . , 6. Theseelasticities are listed in Table D6.
Table D6: GDP Output Elasticities with Respect to Changes in Input Quantities for 2022
Eym xn
y1
y2
y3
y4

C
G
I
X

x2
x3
x4
x5
x6
L
KM&E KSKO
KL
0.482 0.787 1.196 1.035 -2.500
0.342 -2.599 -1.943 -1.417 6.616
1.162 -0.421 -2.556 -1.773 4.589
1.299 -0.067 2.715 -2.478 -0.469

Note: Consumption (C), Government (G), Investment (I), Export (X),
Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

Define the elasticity of input price n with respect to an increase in the endowment of input
j as Ewn xj ≡ [∂wn (p, w1 , x◦ )/∂xj ][xj /wn ] for n = 2, . . . , 6 and j = 2, . . . , 6. These (inverse)
input demand elasticities are listed in Table D7.

*64 See Samuelson (1947; 36-38)[53] (1960)[55], Diewert (1974a; 146-150)[11] and Kohli (1983b)[38].

71Table D7: GDP Inverse Input Demand Elasticities with Respect to Changes in Input Quantities
Ewn xjw2L
w3 KM&E
w4 KS
w5 KO
w6 KL

x2
x3
x4
x5
x6
L
KM&E
KS
KO
KL
-0.956 0.205 0.005 12.970 -0.037
1.806 -1.957 -0.183 -10.532
4.676
0.120 -0.855 -4.793 -34.581
7.000
0.215 -0.624 -0.140 -4.365
0.164-0.276 5.461 18.569 262.729 -30.891

Note: Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

Note that the main diagonal elements in the above 5 by 5 matrix are less negative than thecorresponding diagonal elements in the last 5 rows and columns of the matrix defined in Table
D3. The GDP model of production allows for more substitution between domestic inputs
and imports than the Gross Output model of production, which treats imports as a fixedfactor of production. Thus if the quantity of labour or a capital service input increases in the
GDP model, the corresponding price of that factor of production will decrease less than thecorresponding change in the input price for the Gross Output model.
Define the elasticity of input price n with respect to an increase in the price of output m as
Ewn pm ≡ [∂wn (p, w1 , x◦ )/∂pm ][pm /wn ] for n = 1, . . . , 6 and m = 1, . . . , 4.
Table D8: Input Price Elasticities with Respect to Changes in GDP Output Prices for 2022
Ewn pmw2L
w3 KM&E
w4 KS
w5 KO
w6 KL

p1
p2
p3
p4
C
G
I
X
0.596 0.037 0.001 0.296
6.187 -1.699 -0.100 -1.145
4.139 -0.129 -0.454 -0.653
8.023 -0.241 -0.034 -0.211-0.056 0.525 1.122 3.165

Note: Consumption (C), Government (G), Investment (I), Export (X),
Labour (L), Machinery and Equipment (KM&E ), Structures (KS ),Other Capital (mainly R&D and Inventories: KO ) and Land (KL ).

We use (D13) to calculate the derivatives of the import demand function, x1 (p, w1 , x◦ ) withrespect to the components of p and x◦ . These derivatives are then transformed into theelasticities of import demand with respect to output prices, Ex1 pm for m = 1, . . . , 4 and withrespect to input endowments, Ex1 xn for n = 2, 3, . . . , 6. These elasticities for the year 2022
are listed in Table D9.
Table D9: Elasticities of Import Demand with Respect to Changes in Output Prices
and Input Quantities for the GDP Model of Production
Ex1 p1 Ex1 p2 Ex1 p3 Ex1 p4 Ex1 x2 Ex1 x3 Ex1 x4 Ex1 x5 Ex1 x6
C
G
I
X
L
KM&E
KS
KO
KL
-0.698 -0.108 0.880 0.119 1.401 -0.188 0.354 -1.587 1.020
Note: Consumption (C), Government (G), Investment (I), Export (X), Labour (L),Machinery and Equipment (KM&E ), Structures (KS ), Other Capital (mainly R&D
and Inventories: KO ) and Land (KL ).

Using (D14), we can calculate the derivatives of ym (p, w1 , x◦ ) for m = 1, 2, 3, 4 and the72derivatives of wn (p, w1 , x◦ ) for n = 2, . . . , 6 with respect to w1 , the price of imports. These
derivatives can be turned into elasticities and they are reported in Table D10.
Table D10: Elasticities of Output Supply and Input Price with Respect to the Price of Imports
Ey1 w1 Ey2 w1 Ey3 w1 Ey4 w1 Ew2 w1 Ew3 w1 Ew4 w1 Ew5 w1 Ew6 w1
C
G
I
X
L
KM&E
KS
KO
KL
0.171 0.124 -0.649 -0.179 -0.424
0.362 -0.300
3.013 -2.295
Note: Consumption (C), Government (G), Investment (I), Export (X), Labour (L),Machinery and Equipment (KM&E ), Structures (KS ), Other Capital (mainly R&D
and Inventories: KO ) and Land (KL ).

Finally, we can use (D12) to calculate the derivative of import demand with respect to anincrease in the price of imports for 2022, ∂x1 (p, w1 , x◦ )/∂w1 . This derivative can be multiplied by w12022 and divided by the fitted value for imports in 2022 from our estimatedcost function model to construct the GDP model elasticity Ex1 w1 = −0.119. This elasticity can be compared to the own price elasticity of import demand which is generated byour estimated cost function for 2022. This cost function based import demand function isx1 (y t , wt ) = ∂C t (y t , wt )/∂w1 where t = 2022. The corresponding cost function based importdemand own price elasticity of demand turned out to equal −0.028, which is less than −0.119
in magnitude. Using the Joint Cost function model of producer behavior, outputs y are held
fixed as are domestic input prices w◦ . Using the GDP model of production, output prices
p and domestic endowments x◦ are held fixed. Thus when the price of imports increases,
producers are free to substitute other inputs for imports using the Joint Cost function model
which evidently leads to a larger reduction in import demand compared to the corresponding
response using the GDP function model where the quantities of other inputs are held fixed
in response to the increase in the price of imports (but outputs are free to vary). Thus themagnitude of producer responses to changes in prices and quantities can be very different depending on what is being held constant. Different models should be used to answer different
questions.
A final note of caution: for this conversion of elasticities method to work successfully, theunderlying joint cost function should provide a good fit to the data.

73

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