Total Factor Productivity Computed and Evaluated Using Multi-Step Perturbation*

Baoline Chen 

Bureau of Economic Analysis 

1441 L Street, NW 

Washington, DC 20230 

e-mail: baoline.chen@bea.gov 

and 

Peter A. Zadrozny 

Bureau of Labor Statistics 

2 Massachusetts Ave., NE 

Washington, DC 20212 

e-mail: zadrozny_p@bls.gov 

July 5, 2005 

Additional key words: Taylor-series approximation, model selection, numerical 

solution 

 

JEL codes: C32, C43, C53, C63 

******************************************************************************* 

* Do not duplicate, circulate, or quote without permission * 

******************************************************************************* 

* The paper's analysis and conclusions represent the authors' views and do not 

necessarily represent any positions of the Bureau of Economic Analysis or the 

Bureau of Labor Statistics. 
1

ABSTRACT 

For period t, let qt = f(vt) + τt, where qt denotes measured output 

quantity, f(⋅) denotes a production function, vt = (v1t, ..., vnt)

T denotes a 

vector of n input quantities, τt denotes total factor productivity (TFP), and 

all variables are in natural-log form. Then, f(vt) = , for 0 < αit < 1 

and = 1, is a Cobb-Douglas (CD) 1st-order log-form appr oximation of a 

production function. If f(⋅) is approximated as a CD production function, the 

share parameters, αit, are set to successive two-period-averaged observed inputcost shares, and the observed input quantities are considered optimal or inputcost minimizing, then, qt - is the log of Solow-residual TFP (STFP). 

STFP could be subject to positive or negative input-substitution bias for two 

reasons. First, the CD production function restricts all input substitutions to 

one. Second, observed inputs generally differ from optimal inputs, which respond 

to input-price- substitution effects. In this paper, we test the possible inputprice-substitution bias of STFP in capital, labor, energy, materials, and 

services (KLEMS) inputs data for U.S. manufacturing from 1949 to 2001. (1) Based 

on maximum likelihood estimation, we determine a best 4th-order approximation of 

a CES-class production function. The CES class includes not only the standard 

constant elasticity of input substitution production functions but also includes 

so called tiered CES production functions, in which prespecified groups of 

inputs can have their own input-substitution elasticities and input-cost shares 

are parameterized (i) tightly as constants, (ii) moderately as smooth functions, 

or (iii) loosely as successive two-period averages. (2) Based on the best or 

optimal estimated production function, we compute the implied optimal TFP (OTFP) 

as qt - f( ), where f( ) denotes the best estimated production function 

evaluated at the computed optimal inputs, . (3) For the data in percentagegrowth (%∆) form, we obtain two main conclusions: (i) relative to the average 

values of %∆STFP and %∆OTFP of about 1%, |%∆STFP - %∆OTFP| exceeds about 100% 

about 30% of the time in the first half of the sample period, so that %∆STFP is 

frequently significantly biased relative to %∆OTFP; (ii) in the second half of 

the sample period, %∆STFP - %∆OTFP is mostly close to its average of about .1%, 

so that %∆STFP is not significantly biased relative to %∆OTFP. 

it

n

∑ 1i itv =

α

∑ =

α

n

1i it

it

n

∑ 1i itv =

α

vt

ˆ vtˆ

vt

ˆ
2

1. Introduction. 

 The paper is specifically motivated as discussed in the preceeding 

abstract, but is also more generally motivated by the desire to accurately 

compute price indexes based on explicit forms of the functions being maximized. 

There are two main, mathematically identical, but economically different 

applications: computing price indexes of production inputs based on maximizing 

output of a production function for given input costs, as here, and computing 

price indexes of consumer goods based on maximizing utility of consumed goods 

for given expenditures, as in Zadrozny and Chen (2005). Here, we consider 

standard constant elasticity of input substitution (CES) production functions, 

with one input-substitution elasticity for all inputs, and more general tiered 

CES (TCES) production functions, with a different input-substitution elasticity 

for each prespecified group of inputs. 

We are also interested in using even more general production functions, 

which we call generalized CES (GCES) production functions (see equation (6.1) 

below), in which each input can have its own price elasticity parameter, but, 

for brevity, limit the present applications to CES and TCES production 

functions. CES and TCES production functions have analytical solutions of their 

optimization problems. GCES production functions generally do not have 

analytical solutions except in special homothetic cases, such as the CES and 

TCES cases. Generally, optimization problems based on GCES production functions 

can be solved only numerically. In Zadrozny and Chen (2005), we describe in more 

detail than here the multi-step perturbation (MSP) method as a quick and 

accurate method for numerically solving the corresponding utility maximization 

problem. The MSP method, as the name implies, is a multi-step extension of the 

single-step perturbation method (Chen and Zadrozny, 2003). 

Here, we could have used analytical CES and TCES solutions, but, for two 

reasons, use numerical solutions produced by the MSP method. First, we use the 

MSP method in order to test its accuracy in solving the static optimization 

problems. In all cases, we obtained nearly double-precision or about 14-decimaldigit accuracy when we checked the numerical MSP solutions against the 

analytical solutions, which encourages us to work in the future only with 

numerical solutions of GCES production functions. Second, we are interested in 

studying TFP bias by generalizing the Cobb-Douglas (CD) production function by 

adding nonlinear log-form Taylor-series terms up to a specified order. However, 

to do this tractably we must restrict the number of estimated parameters and we 
3

do this by parameterizing the higher-order Taylor terms in terms of these CESclass production functions. 

 We proceed here entirely in log form for four reasons: (i) TFP and related 

price and quantity indexes are usually considered in log form; (ii) log-form 

variables are unit free, scaled equivalently, and, hence, lie mostly within or 

close to a unit sphere, which promotes numerical accuracy; (iii) log-form 

derivatives of the CES-class production functions are easier to derive, program, 

and compute with; and, (iv) comparisons with benchmark Solow residuals are 

easier in log form. 

As noted, q denotes the log of the quantity of observed goods and 

services, f(⋅) denotes the log of output produced by the production function, 

and, = q - q denotes the log of the level of technology or TFP of f(⋅), 

where f( ) more specifically denotes the log of optimal output produced by 

optimal log-form inputs, . Henceforth, 

t

ˆτ ˆ

vˆ

vˆ t

ˆτ , with the hat, denotes optimal TFP 

(OTFP) based on the best production-function model, where "best" is explained in 

section 3, and τt, without the hat, denotes Solow-residual TFP (STFP) based on 

the not necessarily best model and observed but not necessarily optimal inputs. 

To distinguish between q and f(⋅), we, respectively, refer to them as "goods and 

services" and "output." Let p = (p1, ..., pn)

T denote an n×1 vector of logs of 

observed or computed input prices (superscript T denotes vector or matrix 

transposition) and let v = (v1, ..., vn)

T denote an n×1 vector of logs of 

observed or computed input quantities. The context of whether input quantities 

or prices are observed or optimal-computed will be spelled out in each case. 

Whether prices are in nominal or real (deflated) units makes no difference, so 

long as real prices in a period are obtained by deflating each nominal price by 

the same value. 

We assume f(⋅) is analytical, hence, for a sufficiently large k is 

arbitrarily well approximated by a kth-order Taylor series. Let e(x) = (exp(x1), 

..., exp(xn))T for any n×1 vector x = (x1, ..., xn)

T. We write the input-cost 

line as e(p)Te( ) = e(p)Te(v), where p and v are given, so that e(p)Te(v) 

denotes observed expenditures on inputs and optimal is computed. We consider 

the following output maximization problem: for given f(⋅), p, and v, maximize 

f( ) with respect to v , subject to e(p)Te( v ) = e(p)Te(v). Because is absent 

from the statement of the problem, it plays no role in its solution. Like Solow, 

we compute residually: first , then 

vˆ

vˆ

vˆ ˆ ˆ t

ˆτ

t

ˆτ vˆ t

ˆτ . The difference with Solow is that 

vˆ is computed as optimal and is not equated with observed v. 
4

We consider only interior solutions which satisfy the usual 1st- and 2ndorder conditions (2.1), (2.2), and (2.5). As functional forms, we consider 

standard CES production functions (Arrow et al., 1961) and more general TCES 

production functions, which are multi-level generalizations of standard singlelevel one-input-elasticity CES functions (Sato, 1967; Burnside et al., 1995), 

that allow different input groups to have different substitution elasticities. 

For each production function, we solve for optimal inputs using the MSP method. 

In the CES and TCES cases (such that the CD case is a subcase of the CES case), 

we use analytical solutions to check the MSP method's accuracy, and, given the 

present successful application of the MSP method with CES and TCES production 

functions, in the future, we shall consider more general GCES production 

functions which do not have analytical solutions. 

By a model we mean (i) a multiple-times-differentiable production 

function, f(⋅), (ii) a parameterization of f(⋅) over a data sample, and (iii) 

values of constant structural parameters which determine f(⋅) in the sample. We 

now consider three parameterizations in more detail: (a) unrestricted timevarying reduced-form parameters set every period to different values of the 

structural parameters; (b) time-varying reduced-form parameters restricted by a 

smooth function of constant structural parameters; and, (c) constant reducedform parameters equal to constant structural parameters. 

For example, f(vt) = denotes a period-t log-form CD production 

function for mean-adjusted data, whose reduced-form parameters, αit, depend on 

constant structural parameters in the vector θ. In the typical case (a) of a 

data-producing agency, reduced-form parameters are unrestricted, are set yearto-year to relative input costs, and are statistically unreliable (have infinite 

estimated standard errors), because the number of estimated structural 

parameters, dim(θ), equals the number of observations, nT: αit = θit, for i = 1, 

..., n and t = 1, ..., T, so that dim(θ) = nT. In the typical academic case (c) 

of an econometric analysis, the reduced-form parameters are constant over a 

sample in terms of structural parameters and are statistically reliable (but, 

perhaps, are not the best estimates because the reduced-form parameters are 

constant over the sample) because there are fewer estimated structural 

parameters than observations: αit = θi, so that dim(θ) = n < nT. In the 

application in section 3, we also consider the in-between case (b), in which nT 

reduced-form parameters vary smoothly according to an integrated moving-average 

(IMA) process (Gardner, 1985), such that dim(θ) < nT. 

it

n

∑ 1i itv =

α
5

What difference does the extra generality of going beyond the CD 

production function make? Normally, empirical validity is measured by residual 

size. In this case, we have output residuals, q - q , and input residuals, v - 

. However, because OTPF and output residuals are identical, judging TFP's 

empirical validity using sizes of output residuals makes no sense. For example, 

statistically ideal zero output residuals imply zero log-TFP. Thus, instead, we 

propose judging TFP's empirical validity using an information criterion (IC) 

based on input residuals. The many IC that have been proposed differ in their 

propensities for choosing models with particular numbers of parameters. For 

example, Akaike's IC (1973) often picks less parsimonious models (i.e., with 

more parameters), while Schwarz's IC (1978) often picks more parsimonious 

models. 

ˆ

vˆ

As usual, for a given data sample, we consider a model's parameter 

estimates and derived quantities like TFP as statistically reliable when the 

parameter estimates and derived quantities have finite standard errors. This 

occurs if and only if the degrees of freedom of the parameter estimates are 

positive. Among the models being considered, the one which minimizes a chosen IC 

is considered the best or empirically-most-valid model. An IC test based on input 

residuals for choosing the best model for computing TFP has several advantages. 

First, the test's justification does not depend on the method for estimating 

parameters. Second, the test can compare nonnested models. Third, the test does 

not require data on produced goods and services, q, although, of course, these 

data are ultimately needed to compute TFP. 

By setting input-cost-share parameters year-to-year to observed input cost 

shares, a Solow-residual analysis treats observed inputs as optimal, so that 

input residuals are exactly zero, degrees of freedom of estimated parameters are 

exhausted, and, strictly, the estimated parameters and implied TFP have no 

statistical reliability. By contrast, by testing with an IC based on input 

residuals, we can select the empirically-most-valid model among CES, TCES, and 

possibly other models, compute the best model's implied TFP, and compare it with 

benchmark Solow residuals. Along the way, we can check the MSP method's accuracy 

by comparing analytical and MSP-based numerical solutions in the CES and TCES 

cases which have analytical solutions. We illustrate these ideas using annual 

data on capital, labor, energy, materials, and services (KLEMS) inputs in 

manufacturing industries, from 1949 to 2001, released to the public by the Bureau 

of Labor Statistics. Thus, we provide a method for computing the empiricallymost-valid TFP, potentially more valid than STFP. In other words, we check the 
6

robustness of STFP to deviations of the production function from the CD 

approximation which underlies STFP. 

 The remainder of the paper is organized as follows. Section 2 discusses 

using the MSP method to compute optimal inputs and residual inputs. Section 3 

discusses the econometric design for choosing the best model. Section 4 compares 

the present direct-production-function approach to an indirect-cost-function 

approach, usually based on the translog cost function, and in doing so defines 

what we mean by input-substitution bias. Section 5 applies the MSP method and 

the econometric design to the KLEMS data: (1) it applies the MSP method to the 

KLEMS data to compute input residuals for the CES and TCES models being 

considered; (2) it selects the best model which minimizes the information 

criteria of Akaike (1973), Schwarz (1978), and Hurvich and Tsai (1989); and, (3) 

it computes OTFP based on the best model, computes the benchmark STFP, and 

compares the two TFP computations. Section 6 sums up the paper. 

2. Using the MSP Method to Compute Optimal Inputs. 

 We now describe the MSP method for computing a 1st-order approximation (k 

= 1) of the optimal input function for an output maximizing problem based on any 

twice differentiable production function and any number of steps (h ≥ 1). The 

extension of this MSP computation to a 4th-order approximation and any number of 

steps is discussed in detail in Zadrozny and Chen (2005). Although, we discuss 

only the 1st-order approximation here in order to minimize the technical 

details, in the application in section 5 we use the 4th-order approximation. The 

MSP method is a variant of the 1st-order method for computing a cost-of-living 

index described by Vartia (1983). 

We exploit the property for simplifying computations that maximizing a 

function in a constraint set results in a solution equivalent to the one 

obtained by maximizing a monotonic transformation of the function in the same 

constraint set. In original units of measurement denoted by upper-case letters, 

the output maximization problem is: for given F(⋅), P, and V, maximize F( ) 

with respect to , subject to PT = PTV, where F(⋅), P, V, and denote 

antilogs of f(⋅) and the elements of p, v, and . Although the original-unit 

and log-unit formulations of the problem lead to slightly different 1st-order 

conditions, they have equivalent solutions, namely, Vˆ = exp( vˆ ). As noted 

before, proceeding in log form has several advantages. 

Vˆ

Vˆ Vˆ Vˆ

vˆ
7

We want to compute optimal and residual inputs, for each period, in a 

sample of input prices and quantities, for CES and TCES production functions. 

Let {pt,vt denote a given sample of observed input prices and quantities. 

Then, for given f(⋅), pt, and vt in period t, the vector of optimal inputs, , 

which solves the output maximization problem, implies the vector of input 

residuals, vt - . 

T } = 1t

vt

ˆ

vt

ˆ

Figure 1 illustrates MSP computation of in terms of two inputs, in the 

movement from points A to B. Points A and B denote start and end points of an 

MSP computation. Straight lines AA and BB, through A and B, denote start and end 

input-cost lines. Curved lines fA and fB, tangent at A to AA and tangent at B to 

BB, denote start and end isoquants. Observed input prices and quantities are p = 

(p1,p2)

T and v = (v1,v2)T. Observed v is at A and BB denotes the "observed" cost 

line defined by observed p and v: e(p)Te( ) = e(p)Te(v). The objective is to 

compute , the optimal combination of inputs on BB. The implied negative input 

residual, - v, is depicted by the vector difference B - A. 

vt

ˆ

vˆ

vˆ

vˆ

The MSP method starts at A but generally works correctly only if the 

starting point is optimal. Generally, A is not optimal on the observed cost line 

BB, because isoquant fA, which passes through A, is not tangent to BB at A. 

However, A is optimal on AA, because AA is constructed to be tangent to fA at A. 

Accordingly, AA is defined by e( )

Te( ) = e( )Te(v), where satisfies the 

1st-order conditions (2.1) and (2.2), for given f(⋅) and v. Thus, and AA are 

"optimal" at A. The MSP method computes the change in optimal inputs as they 

move from A to B in response to the counterclockwise rotation of the input-cost 

line at the initial point A, as the price vector flattens from in AA to p in 

BB. 

ˆp vˆ ˆp ˆp

ˆp

ˆp

As before, for given assumed f(⋅) and given observed p and v, the 

objective is to compute optimal . For these given quantities, the log-form 

output-maximization problem is: maximize f( ) with respect to , subject to 

e(p)Te( ) = e(p)Te(v). The Lagrangian function of the problem is l = f( ) + 

(e(p)T(e(v) - e(p)Te( v )), where denotes the Lagrange multiplier. We obtain 

the 1st-order conditions of the maximization problem by differentiating l with 

respect to and , set the results to zero, and write them as 

vˆ

vˆ vˆ

vˆ vˆ

λ

ˆ ˆ λ

ˆ

vˆ λ

ˆ

(2.1) ∇f( vˆ ) = λ

ˆe(p+ vˆ )

T, 
8

(2.2) e(p)Te( ) = e(p)T vˆ e(v), 

where ∇f( ) = [∂f( )/∂v1, ..., ∂f( vˆ )/∂vn] denotes the 1×n gradient row vector 

of first-partial derivatives of f( ). For given f(⋅), p and v, equations (2.1) 

and (2.2) can be solved for unique values of and , at least locally and 

numerically if 2nd-order conditions (2.5) hold. 

vˆ vˆ

vˆ

vˆ λ

ˆ

 As discussed before, we start the MSP method at observed inputs and need 

to treat them as optimal. Because observed inputs, v, are generally not optimal 

at observed prices, p, we first need to compute the "optimal" price vector, , 

at which v is optimal. We do this by considering the 1st-order conditions (2.1) 

and (2.2) as ∇f(v) = e( +v)T and e( ˆp )

Te(v) = e(p)Te(v), for given assumed f(⋅) 

and given observed p and v, and solving for and . Let E(x) = diag(e(x)) 

denote the n×n diagonal matrix with n×1 vector e(x) on the principal diagonal; 

because all observed inputs are positive and finite, E(v) has finite and nonzero 

diagonal elements and, hence, is nonsingular; E(v)-1e(v) = u, where u = (1, ..., 

1)T denotes the n×1 unit vector of ones; and, e( )

Te(v) = e(p)Te(v) when 

computing , because the computed input-cost line defined by and the 

observed input-cost line defined by p both pass through the observed inputs, v. 

The solution values of and are 

ˆp

λ

ˆ ˆp

λ

ˆ ˆp

ˆp

ˆp ˆp

λ

ˆ ˆp

(2.3) = ∇f(v)u/e(p)T λ e(v), ˆ

 e( ) = E(v)-1∇f(v)T ˆp / λ . 

ˆ

At this point, having computed according to equations (2.3), we now 

consider as observed and given, and relabel it as p. Thus, we now consider as 

given the same f(⋅) and v as before and the computed relabelled as p. For 

these given quantities, we now differentiate 1st-order conditions (2.1) and 

(2.2) with respect to , , and p and write the result as 

ˆp

ˆp

ˆp

vˆ λ

ˆ

(2.4) F(x)d = G(x)dp, yˆ
9

or = dp, 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

− +

∇ λ− + − +

1x1

T

2

p(e ˆ)v 0

p(E ˆ)v p(e ˆ)v

ˆ (f ˆ)v

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

λ

ˆd

dvˆ

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+ − +

λ +

T T

p(e ˆ)v p(e )v

p(E ˆ)v

ˆ

where ∇

2f(⋅) denotes the n×n Hessian matrix of 2nd-partial derivatives of f(⋅), 

F(x) is an (n+1)×(n+1) matrix function, G(x) is an (n+1)×n matrix function, x = 

( , pT)

T contains all 2n+1 variables, = ( , )T contains the n+1 

"endogenous" variables to be determined, and p contains the n given or 

"exogenous" prices. Although the hats emphasize computed values, for simplicity, 

we omit them from x because, unlike in v or y, we do not need to distinguish 

between hatted and unhatted x. 

T

yˆ yˆ

T

vˆ λ

ˆ

 The elements of x are all known, because they are either observed or 

previously computed. For given x, equation (2.4) implies the unique value dyˆ = 

H(x)dp, where H(x) = F(x)-1G(x), if and only if |F(x)| ≠ 0, which is implied by 

the 2nd-order conditions of the problem, where |⋅| denotes the determinant of a 

square matrix. For i = 2, ..., n, let Fi(x) = denote the 

(i+1)×(i+1) submatrix of F(x), where Ai(x) denotes the upper-and-left-most i×i 

submatrix of - E(p+ ) and bi(x) denotes the first i elements of 

-e(p+ ). Then, the 2nd-order conditions of the output-maximization problem are 

⎥

⎦

⎤

⎢

⎣

⎡

×11

T

i i

)x(b 0

)x(A )x(b

(f ˆ)v

2 ∇ λˆ vˆ

vˆ

(2.5) (-1)i+1|Fi(x)| > 0, 

for i = 2, ..., n (Mann, 1943; Samuelson, 1947). Thus, when x maximizes output 

and satisfies 2nd-order condition (2.5), equation (2.4) has the unique solution 

(2.6) = H(x)dp, dyˆ

where H(x) = F(x)-1G(x) is an (n+1)×n matrix function of x. Although equation 

(2.6) derives from the true y process, we write its left side as to 

emphasize that the true dy is approximated using this equation. 

dyˆ

We now consider an interaction between continuous and discrete time. Let 

[1,T+1) = U denote a continuous-time interval divided into T unitlength periods indexed by their beginning moments, t = 1, ..., T, where [t,t+1) 

= {s|t ≤ s < t+1}. Definitions of variables hold both in continuous time within 

T

1t

t,t[ )1 =+
10

a period, denoted by argument s, and in discrete time t at starting moments of 

the periods, denoted by subscript t. Thus, discrete time periods are indexed by 

their starting moments. Above, we denoted observed and given values without hats 

and computed values with hats. We now also denote true values without hats and 

continue to denote computed values with hats. For example, y(s) denotes true y 

in continuous time and denotes computed y in continuous time. Because 

computed values are meant to be optimal but are actually approximations of 

optimal values, strictly, a hat implies a value is "computed, optimal, and 

approximate," although for simplicity, we refer to hatted values only as 

computed. 

ˆ )s(y

For each period t, an MSP computation proceeds as follows. We think of 

starting computations at the start of a period, at the moment s = t, and ending 

them at the end of the period, at the moment s = t+1. We think of the observed 

input quantities, vt, as occuring at the start of the period and think of the 

observed input prices, pt, as occuring at the end of the period. For given 

assumed f(⋅) and given observed pt and vt, we first compute the "optimal" 

starting price vector, , which makes vt optimal. We assume the price vector 

moves continuously from its "optimal" starting value, , to its observed 

ending value, pt. Then, given (t1) = vt, we compute the remaining optimal input 

quantities at h equidistant points along the optimal input path in response to 

the price movements. We compute 

pt

ˆ

pt

ˆ

vˆ

ti

∆yˆ ≡ (ti+1) - ≡ , for i = 1, ..., 

h, and pick ≡ (th) - vt as the top n×1 subvector of 

yˆ tiyˆ ∫

+

=

t 1i

ts i

dˆ )s(y

vt ∆ˆ vˆ yt ∆ˆ = , so 

that = vt + . Figure 1 depicts the computations as the movement from 

points A to B along the curved line with arrowheads. 

∑ =

∆

h

1i ti

yˆ

vt

ˆ vt ∆ˆ

The implicit function theorem (Apostol, 1974, p. 374), upon which the MSP 

method is based, implies that if the production function is twice differentiable 

and satisfies the 2nd-order conditions, so that its optima are interior points, 

then, the exact solution path is differentiable, and, in each computational 

subperiod s ∈ [ti,ti+1), for i = 1, ..., h, has the 1st-order polynomial Taylorseries approximation of the true y(s), 

(2.7) = + (s-t ˆ )s(y i), ti

yˆ ti∇yˆ
11

where is an (n+1)×1 coefficient to be computed in terms of observed input 

prices and quantities. We state an analogous polynomial price process in 

equation (2.10). 

ti

∇yˆ

The approximation = of ∆yt ≡ y(t+1) - yt ≡ has the 

theoretical approximation error ε = | ∆yt - 

yt ∆ˆ ∑ =∆

h

1i ti

yˆ ∫

+

=

1t

ts

)s(dy

yt ∆ˆ |. We partition each period 

[t,t+1) into h subperiods of length h-1, as [t,t+1) = [ti,ti+h-1), where 

[ti,ti+h-1) = [t+(i-1)h-1,t+ih-1), for i = 1, ..., h. For each subperiod i in 

period t, we compute the coefficient of the approximate process (2.7), 

h U =1i

ti

∇yˆ , 

and, then, compute the subperiod increments, ti

∆yˆ , as 

(2.8) = h

-1. ti

∆yˆ ti∇yˆ

The theoretical approximation error of a kth-order approximate solution is 

on the order of h-k. The approximation error can be controlled by setting k and 

h, although, in the discussion in this section, we consider only h > 1 and k = 

1. See Zadrozny and Chen (2005, table 1) for values of h and k which predict 

achieving particular orders of magnitude of accuracy. 

 We now describe the MSP method for k = 1. For i = 1, ..., h and s ∈

[ti,ti+h), we differentiate the approximate y process (2.7) with respect to s 

and obtain 

(2.9) (s) = . dyˆ ti

∇yˆ

We compute the coefficient, , so that it is equal to the first differential 

of the true y process (2.6). 

ti

∇yˆ

Analogous to approximate y process (2.7), we assume prices follow a 1storder polynomial process, for s ∈ [t,t+1) and t = 1, ..., T, 

(2.10) p(s) = + ˆpt ∇pt(s-t), 

with n×1 coefficient ∇pt. Although the price coefficient remains at its initial 

value, indexed at t1 = t, throughout computations in period t, the y 

coefficient, , is indexed by ti and updated at each iteration i within 

period t. From price process (2.10), we require only that it passes through the 

ti

∇yˆ
12

computed start-of-period prices p(t) = , given by equations (2.3), and the 

observed end-of-period prices p(t+1) = pt, because in discrete time firms care 

only about starting and ending prices and do not care about within-period 

prices. 

pt

ˆ

In the following, we distinguish between differentiating with respect to s 

and differencing with respect to t. For s ∈ [t,t+1), differentiating price 

process (2.10) with respect to s, we obtain 

(2.11) dp(s) = ∇pt. 

Then, differencing price process (2.10), we obtain the price coefficient, ∇pt, 

in terms of differenced prices, ∆pt, as 

(2.12) ∇pt = ∆pt. 

where the differenced prices are given in terms of the computed and observed 

prices, and p ˆpt t, as 

(2.13) ∆pt = pt - . pt

ˆ

As required, the price coefficient set by equations (2.12) and (2.13) implies 

that the price process (2.10) passes through the computed and observed prices. 

 We now describe computing { using the MSP method. We sequence the 

computations in an outer loop, for periods t = 1, ..., T, and an inner loop, for 

subperiods i = 1, ..., h. For each period t, we des cribe the inner-loop 

computations in four steps. Within the four steps, we take as given an assumed 

f(⋅) and observed pt and vt. 

vt

ˆ

T

} = 1t

Step 1: Initialize x, Prices, and Their Differentials. 

 For given f(⋅), pt, and vt and for i = 1, hence, for s = t1 = t, we first 

compute and according to equations (2.3). We set xt = ( , )

T = ( , 

, )

T. Following equation (2.11), we set the price differential as dp(t) = 

∇pt. Following equations (2.12) and (2.13), we compute the price coefficient, 

∇pt, in terms of the computed and observed prices, and pt. 

t

ˆλ pt

ˆ

T

yt

ˆ

T

pt

ˆ

T

vt

t

ˆλ

T

pt

ˆ

pt

ˆ
13

Step 2: Compute 1st-Order y Coefficient. 

For s = t1 = t, equations (2.6), (2.9), and (2.11) imply that

(2.14) H(xt) = F(xt)

-1G(xt), 

 = H(x ∇yˆt t)∇pt. 

For k = 1 and i = 1, equation (2.8) implies that 

(2.15) = h

-1 ∆yˆt ∇yˆt , 

so that = + t2

yˆ yt

ˆ yt ∆ˆ . 

Step 3: Update Prices, x, and y. 

For i = 2 and, hence, for s = t2 = t+h-1, equations (2.10) and (2.11) imply 

that we update prices and their differentials as 

(2.16) p(t2) = + ∇pth

-1 ˆpt , 

 dp(t2) = ∇pt, 

such that the price coefficient, ∇pt, remains at its initial t1 = t computed 

value. We set = ( , )

T. We repeat step 2 and, thereby, update the y 

coefficient to . Following equation (2.8), we compute 

t2

x

T

t2

yˆ

T

t2

p

t2

∇yˆ t2

∆yˆ = h

-1 and 

= + . 

t2

∇yˆ t3

yˆ

t2

yˆ t2∆yˆ

Step 4: Repeat Steps 2 and 3. 

For i = 3 and, hence, for s = t3 = t+2h-1, we update prices and their 

differentials as 

(2.17) p(t3) = + 2∇pth

-1 ˆpt , 
14

 dp(t3) = ∇pt. 

We set = ( , )

T. We repeat step 2 and update the y coefficient to . 

We compute ∆ = h

-1 and update y as = + . We repeat these 

steps for i = 4, ..., h and, hence, for s = t4 = t+3h-1, ..., th = t+1-h-1. 

Finally, we compute and pick 

t3

x

T

t3

yˆ

T

t3

p t3

∇yˆ

t3

yˆ t3∇yˆ t4

yˆ t3

yˆ t3∆yˆ

th

∆yˆ th

∆vˆ as the top n-dimensional subvector of 

the computed . th

∆yˆ

3. Econometric Design. 

 We now discuss the econometric design of the empirical application. As 

noted before, a model is a production function, a parameterization of the 

production function over a sample, and particular numerical values of the 

constant structural parameters in the vector θ. The structural parameters could 

determine time-varying processes of parameters more directly in the production 

function. For example, in the IMA models in section 5, production-function share 

parameters follow integrated moving averages (IMA) defined by elements of θ. The 

ultimate goal is maximum likelihood estimation of several models and choosing as 

the best one the model which minimizes one or more information criteria (IC). 

However, because we do not yet have all the necessary computer programs 

completed, for now we apply a coarse version of maximum likelihood estimation. 

That is, for each considered class of models (CES and TCES), we pick a best 

model from a set of models defined over a relatively small and discrete grid of 

numerical parameter values. When the additional computer programming is done, we 

shall be able to implement the maximum likelihood estimation more fully over 

continuous intervals of the parameters. 

For a particular model, the log-likelihood function and an IC are computed 

as follows. Suppose we have a sample of observations on input prices and 

quantities, in log form {pt,vt , for periods t = 1, ..., T. Period-t log-form 

input residuals are observed input quantities minus computed optimal input 

quantities, ξt = vt - . Suppose the residuals are distributed normally, 

identically, independently, with zero means, and covariance matrix Σξ or ξt ~ 

NIID(0, Σξ). Let L(θ) denote -(2/T) × log-likelihood function, except for terms 

independent of parameters. Then, L(θ) = ln| |, where ln|⋅| denotes the natural 

T

} = 1t

vt

ˆ

Σξ

ˆ
15

logarithm of a determinant and = (1/T) , where the residuals, ξt, are 

evaluated at a particular values of θ. An IC = L(θ) + P(dim(θ)), where P(dim(θ)) 

denotes a penalty term which depends on the number of estimated parameters, 

dim(θ). Each structural parameter is estimated, so that dim(θ) = number of 

structural parameters. In section 5, we consider Akaike's (1973) information 

criterion (AIC), Hurvich and Tsai's (1989) bias-corrected AIC (BCAIC), and 

Schwarz's (1978) Bayesian information criterion (BIC). For example, for AIC, 

P(dim(θ)) = (2/T)dim(θ). In each model class, we choose as the maximum 

likelihood estimate the model which minimizes L(θ) in the class over the 

parameter grid. The theory behind ICs says that they can choose a best model 

among nested or nonnested models. As the best overall model, we choose the one 

which minimizes one or more ICs. 

Σξ

ˆ ∑ =

ξξ

T

1t

T

tt

4. Comparing with Translog Cost Function and Defining Substitution Bias. 

 We now discuss the advantages of the present direct production-function 

approach compared with an indirect cost-function approach, in particular, 

compared with the translog cost-function approach, the most commonly used 

indirect cost function. In doing so, we define "substitution bias." Notation is 

as before; in particular, lower-case letters denote logarithms. 

The direct output-maximization problem is: for a given production function 

and observed input prices and quantities, f(⋅), p, and v, maximize output, 

f( ), with respect to input quantities, vˆ , subject to the cost line, e(p)Te( vˆ ) 

= e(p)Te(v). Under 1st- and 2nd-order conditions, the problem has a unique 

solution for given p and v, = g(p,v). The application in section 5 is based 

on a purely numerical 4th-order approximation of g(p,⋅), for varying p and 

constant v, 

vˆ

vˆ

(4.1) = g(p, vˆ ⋅) ≅ v + ∇g(p - p′) + (1/2)[(p - p′)

T ⊗ In]∇2g(p - ) p′

 + (1/6)[(Π2⊗(p - p′)

T ⊗ In]∇3g(p - p′) 

 + (1/24)[(Π3⊗(p - p′)

T ⊗ In]∇4g(p - p′), 

where denotes the computed initial input-price vector in a period, as in 

section 2, ∇g, ..., ∇

4g denote matrices of 1st- to 4th-order partial 

p′
16

derivatives of g with respect to p evaluated at p′ and v, Πk⊗(p- )

T denotes k-1 

successive Kronecker products of (pp′

p′)

T, ⊗ denotes a single Kronecker product, 

In denotes the n×n identity matrix, and n = dim(p). Chen and Zadrozn y (2003, 

appendix A) discuss this notation in more detail. 

The corresponding indirect cost-minimization problem is: for given f(⋅), 

p, and = q-τ, minimize input costs, c = ln[e(p)Te( )], with respect to input 

quantities, , subject to the production function, f( ) = . Under 1st- and 

2nd-order conditions, the problem has a unique solution, = h(p, ), so that 

the minimized indirect cost function is a function of p and , (p,

q′ vˆ

vˆ vˆ q′

vˆ q′

q′ cˆ q′) = 

ln[e(p)Te(h(p, ))]. The indirect-cost-function approach was introduced to 

circumvent the inability to solve analytically (i.e., explicitly and in closed 

form) direct problems based on more general production functions than the CES 

production function, which was considered empirically too limited (see also 

Berndt, 1991, ch. 9, pp. 449-506). 

q′

The most frequently used indirect cost function is the translog cost 

function, a 2nd-order Taylor-series approximation of c (p, ) (Christensen et 

al., 1971, 1973). Like equation (4.1), the translog approximation of (p,

ˆ q′

cˆ q′) 

only in terms of prices is 

(4.2) (p, cˆ ⋅) ≅ ( , cˆ p′ q′) + ∇cˆ (p - p′) + (1/2)(p - p′)

T ∇2 cˆ (p - ), p′

where ∇cˆ and ∇

2 cˆ are 1×n and n×n matrices of 1st- and 2nd-partial derivatives 

of (p, cˆ q′) with respect to p evaluated at p′ and v. The envelope theorem (also 

called Shepard's lemma and Roy's theorem) implies that the 1st-partial 

derivatives of cˆ (p,⋅) with respect to p are equal to the optimal input function. 

Thus, differentiating equation (4.2) with respect to p and using the symmetry of 

∇

2 cˆ implies that 

(4.3) = h(p,⋅) ≅ ∇cˆ

T + ∇2 vˆ cˆ (p - p′), 

a 1st-order approximation of the optimal input function, so that equation (4.3) 

corresponds to the first two terms on the right side of equation (4.1). Thus, 

the 4th-order approximate optimal input function used here is more general, at 

least in certain dimensions, than the 1st-order function (4.3). 
17

 We now define substitution bias. Taylor-series theory says that errors in 

approximations (4.1) and (4.3) of the optimal input function are, respectively, 

on the order of ||p- ||5 p′ and ||p- p′||2, where ||⋅|| denotes a vector norm (Golub 

and Van Loan, 1996, pp. 52-54). Because optimal inputs are known to be 

homogeneous of degree zero in p and c, the approximation errors should be a 

concern only when individual prices change in different proportions. In any 

period, the difference between equations (4.1) and (4.3) is 

(4.4) δ = (1/6)[(Π2⊗(p- )

T⊗In]∇3 p′ g(p- p′) + (1/24)[(Π3⊗(p- )

T⊗In]∇4 p′ g(p- p′). 

The quantity δ assumes positive and negative values over a sample of periods. We 

say significant input-substitution bias exists in a sample if δ has a 

significant nonzero mean, a significant variance, or both, where "significance" 

could be interpreted according to the subject matter of the application. 

Equation (4.4) implies input-substitution bias occurs if and only if ||p- p′|| is 

sufficiently large. Significant input-substitution bias carries over through the 

production function to TFP. In section 5, we conclude that STFP, based on a 1storder production-function approximation, is frequently significantly biased 

relative to OTFP based on a 4th-order production-function approximation. 

 The direct approach is preferred because it is easier to use for 

parsimoniously generalizing a model. In empirical work, we want a model to be 

general, so that it fits data well, but also want it to be parsimonious, so that 

the estimated parameters, the estimated model, and any quantities derived 

therefrom are statistically significant. Thus, we seek a balance between model 

generality and parsimony. The direct approach is easier to use for this purpose. 

For example, section 5 illustrates successful parsimonious generalization of the 

standard constant-parameter CES model to TCES models with time-varying 

parameters. By contrast, although cost function (4.2) is straightforwardly 

extended to the 4th order, doing so effectively is not easy. The extension adds 

many new unrestricted coefficients in the derivative matrices ∇

3 cˆ and ∇4 , even 

after imposing homogeneity restrictions, which cannot effectively all be treated 

as new parameters to be estimated; the new coefficients must somehow be 

parameterized more tightly. Moreover, curvature restrictions implied by the 2ndorder conditions must also be maintained. It is not clear how this should be 

done. Generalizing a model using the combined direct and MSP methods seems to be 

easier. 

cˆ
18

5. Application to KLEMS Data. 

We now discuss the application to annual data for U.S. manufacturing from 

1949 to 2001 from the Bureau of Labor Statistics (2002). The data are prices and 

quantities of capital (K), labor (L), energy (E), materials (M), and services 

(S) used by U.S. manufacturing firms to produce output. The raw data are indexes 

of input quantities (with 1996 values being 100), expenditures on inputs in 

billions of current dollars, and the value of output in billions of current 

dollars. Prices of inputs are computed as expenditures divided by input quantity 

indexes. As noted before, the scale of prices makes no difference, in 

particular, whether they are in current- or constant-dollar form. 

 STFP is based on a 1st-order CD approximation of any differentiable 

production function. Here, a production function parameterized in a certain way 

is a model. We consider CES and TCES models of the five KLEMS inputs, such that 

in unit-elastic cases a CES model reduces to a CD model. The parameters are 

input-cost shares, α1, ..., α5, and input substitution elasticities, σ1 in the 

CES models and σ1 and σ2, for σ1 > σ2, in the TCES models. For the 53 years, we 

consider "constant" αi's estimated as sample means, "IMA" αi's equal to oneperiod ahead forecasts of estimated IMA(1,1) models of the cost shares, and 

"T rnqvist" αi's set to .5 × period t's observed input-cost shares + .5 ×

period t-1's observed input-cost shares. We estimate the IMA parameters by 

applying maximum likelihood estimation (MLE) to the observed cost shares. In 

each case, because the cost shares must sum to one, we set the αi's of the four 

largest LMKS-cost shares and set the remaining E-cost shares residually, as one 

minus the sum of the other αi's. For both CES and TCES models, we consider σ1

and σ2 ∈ {.1, .18, .5, .67, 1, 1.5 2, 5.9, 10}. Thus, we do a coarse MLE over a 

small and discrete grid of σ1's, conditional on the estimated α's. We do not 

consider joint MLE of parameters, because this usually results in implausible 

estimates of αi's. For example, until he introduces utilization rates (an 

extension which is beyond the scope of this paper), Tatom (1980) obtains the 

estimates αL > 1 and αK < 0, which contradict diminishing and positive marginal 

productivities of labor and capital. 

o&&

We evaluate the estimated models in terms of AIC, BCAIC, and BIC. We are 

especially concerned about degrees of freedom (DF) of estimated parameters and, 

for a particular IC, consider as the best one the model which minimizes that IC 

for positive DF. We are concerned with DF because a model with zero DF implies 

that the model's estimated parameters and any quantities such as TFP derived 
19

therefrom have infinite variances and, hence, strictly have no statistical 

reliability. To varying extents, the ICs considered here account for DF in their 

penalty terms. Among the ICs in table 1, BCAIC most effectively accounts for DF, 

because it is the only IC that approaches +∞ as DF approach zero from above. 

Thus, we set BCAIC = +∞ when DF are nonpositive. An IC is parsimonious if it 

selects as the best one the model with the fewest parameters. The ICs in tables 

1 are ordered in increasing parsimony according to AIC, BCAIC, and BIC. 

5.1. Results from CES-Class Models. 

We considered nine classes of models determined by three classes of 

production functions (CES, TCES1, TCES2) and three input-cost-share 

parameterizations (constant, IMA, T rnqvist). For each model class, table 1 

reports MLEs of σi over the discrete parameter grid, DF, -(2/T)L(θ), AIC, BCAIC, 

and BIC. In the cases of σ1 = 1, CES models reduce to CD models. 

o&&

The DF in table 1 are obtained as follows. Each model has five KLEMS 

inputs. Because cost shares sum to one, there are four free cost shares in each 

of the 53 years. Each model also has one or two elasticity parameters, σ1 and 

σ2. Thus, constant-cost-share models 1, 4, and 7 have 5 and 6 estimated 

parameters, hence, have 47 and 46 DF. Each IMA process has two estimated 

parameters, a moving-average coefficient and a white-noise disturbance variance. 

Thus, IMA-cost-share models 2, 5, and 8 have 9 and 10 estimated parameters, 

hence, have 43 and 42 DF. Finally, T rnqvist-cost-share models 3, 6, and 9 have 

213 and 214 estimated parameters, hence, have zero DF. Figure 2 depicts the 

largest cost-share inputs, L, M, K, and S. That is, the smallest cost shares of 

E are not graphed. In figure 2, each panel contains time plots of constant, IMA, 

and T rnqvist cost shares for each of the LMKS inputs. Strictly each panel has 

three cases, but practically each panel has two cases, because the IMA and 

T rnqvist graphs are nearly identical. Thus, the IMA and To rnqvist models 

differ significantly only in their DF. 

o&&

o&&

o&& &&

Summarizing the results in table 1 for the CES models: in the constantcost share case, σ1 = .5 yields the best minimal IC values; IMA-cost-share model 

2 is the best CES model, because it has the lowest ICs for positive DF; and, 

although To rnqvist-cost-share model 3 (and models 6 and 9) has lower ICs than 

model 2, we consider it inferior because it has zero DF. 

&&

5.2. Results from TCES Models
20

Even if we limit the TCES model search to two-tiered models, this results 

in more models than we could evaluate in practice. Thus, we first looked at 

figures 3 and 4 to obtain guidance about which input groups to form. Figure 3 

depicts the 10 pairwise scatter plots of the KLEMS inputs in log form. In the 

figure, all pairwise plots except those involving L follow clear, noiseless, 

mostly upward, straight or curved lines. Plots involving L are quite noisy. 

Thus, figure 3 suggests that all non-L inputs move in close to fixed proportions 

and have low substitutability. That is, figure 3 suggests a two-tiered TCES 

model with an outer group of L and KEMS, with relatively high input substitution 

σ1, and an inner group of K, E, M, and S, with relatively low input substitution 

σ2. Thus, we consider the L-KEMS two-tiered CES model, denoted TCES1, written in 

original unlogged form as 

(5.1) Q = [α1L

ρ + α2(β1Kγ + β2Eγ + β3Mγ + β4Sγ)ρ/γ]

1/ρ, 

where αi, βi > 0, α1 + α2 = β1 + β2 + β3 + β4 = 1, and γ < ρ < 1; the outer group, 

L and KEMS, has σ1 = (1 - ρ)

-1 and, the inner group, K, E, M, and S, has σ2 = (1 

- γ)

-1, so that σ1 > σ2. 

Figures 4a-b suggest a two-tiered TCES model with L-E-KMS input groups. 

Figure 4a depicts the following broad input-price movements: all input prices

except E prices follow the same upward trend, exhibit relatively minor 

differences about the trend, and E prices are relatively constant from 1949 to 

1972 and from 1982 to 2001 and rise sharply from 1973 to 1981. Figure 4b depicts 

the following broad input-quantity movements: L is relatively constant; K, M, 

and S follow each other very closely along an upward trend; and, E rises 

significantly until 1973 and thereafter grows very slowly. In particular, figure 

4b suggests a two-tiered TCES model with an outer group of L, E, and KMS, with 

relatively high input substitution σ1, and an inner group of K, M, and S, with 

relatively low input substitution σ2. Because figure 4b shows that K, M, and S 

move in close to fixed proportions, we expect σ2 to be relatively small. The 

relative constancy of L in figure 4b could also be interpreted as indicating 

nonneutral L-saving technical change, but we limit the analysis to homothetic 

production functions, hence, limit it to the neutral technical change of the 

STFP. Thus, we consider the L-E-KMS two-tiered CES model, denoted TCES2, written 

in original unlogged form as 
21

(5.2) Q = [α1L

ρ + α2Eρ + α3(β1Kγ + β2Mγ + β3Sγ)ρ/γ]

1/ρ, 

where αi, βi > 0, α1 + α2 + α3 = β1 + β2 + β3 = 1, and γ < ρ < 1; the outer group, 

L, E, and KMS, has σ1 = (1 - ρ)

-1 and, the inner group, K, M, and S, has σ2 = (1 

- γ)

-1, so that σ1 > σ2. 

 Table 1 reports MLEs of σ1 and σ2, DF, -(2/T)L(θ), AIC, BCAIC, and BIC for 

the TCES models. Because there are outer and inner elasticities of input 

substitution in the TCES models, DF equals 46 in the constant-cost share models, 

42 in the IMA-cost share models, and remains zero in the T rnqvist-cost share 

models. In the TCES1 models, IMA-cost-share model 5, with σ1 = 1 and σ2 = .67, 

has the lowest ICs for positive DF. Similarly, in the TCES2 models, IMA-costshare model 8, with σ1 = 1 and σ2 = .67, has the lowest ICs for positive DF. 

o&&

Table 1 implies that TCES1 IMA-cost-share model 5 is the best model among 

those being considered, because it has the lowest ICs for positive DF. Thus, 

table 1 rejects a single elasticity of input substitution for all KLEMS inputs. 

Table 1 also implies that the best model 5 dominates the CD T rnqvist-costshare model underlying STFP. Model 3 in the table is also a CD To rnqvist-costshare model, but differs from the STFP model because its ICs are based on 

optimal inputs, not on observed inputs, which generally differ from optimal 

inputs. Thus, the STFP model's ICs are greater (inferior; actually infinite) 

than those of models 3 and 5 and, consequently, STFP is statistically less 

appropriate than the OTFP of models 3 or 5. Strictly, model 3 also differs from 

the STFP model because its cost shares are IMA, not To rnqvist. However, because 

IMA and To rnqvist cost shares follow each other very closely, as figure 2 

indicates, the input-cost-share differences between these models should not 

cause their ICs to differ much. 

o&&

&&

&&

&&

The MSP method was accurate for all models and sample periods. There are 

six 1st-order conditions (FOC), five marginal productivity conditions of the 

KLEMS inputs and the cost line. Ideally, each computed FOC is zero, but, in 

practice, the best we can do is to compute each FOC up to a small remainder, 

called the FOC residual. The overall accuracy of the computational method, 

whether MSP or any other method, can be measured by the largest absolute FOC 

residual. In the application, the MSP method computed absolute FOC residuals no 

larger than about 10-14. Because the data contained no more than 6 decimal

digits, computed FOC residuals no larger than about 10-14 represent very accurate 

computations. 
22

5.3. Optimal TFP Compared with Solow-Residual TFP.

We used the best TCES1 IMA-cost-share model 5 to compute year-to-year 

percentage growth in OTFP or %∆OTFP = ∆q - ∆f( vˆ ), where ∆q and ∆f( ) denote 

year-to-year percentage growth in observed output and computed optimal output 

based on model 5. Similarly, %∆STFP = ∆q – αk∆k – ... – αS∆s denotes year-toyear percentage growth in STFP, where αk, ..., αs denote To rnqvist input-cost 

shares and ∆k, ..., ∆s denote year-to-year percentage growth in observed KLEMS 

inputs. We compare OTFP and STFP as %∆OTFP and %∆STFP because percentage growth 

rates abstract from trends and better reveal differences in the two TFPs. Let rt

= (r1t, ..., r4t)

T = ( ∆p2t - ∆p1t, ..., ∆p5t - ∆p1t)T denote a 4x1 vector of 

differences in percentage growth rates of KLEMS input prices and let ||rt|| = 

vˆ

&&

∑ =

4

1i

2

rit denote the Euclidian norm of rt. When relative input prices change 

significantly, ||rt||, the 2nd- to 4th-order terms in approximate optimal input 

function (4.1) which underlies OTFP, and %∆OTPF - %∆STFP are all significantly 

nonzero. Figure 5a graphs %∆OTPF - %∆STFP and figure 5b graphs ||rt||, from 1949 

to 2001. 

Figure 5a shows a slightly negative average, a slightly upward trend, a 

relatively large variance from 1949 to the mid 1970s, a declining variance over 

the whole period, and a relatively small variance from the mid 1970s to 2001. 

Table 2 summarizes the distribution of %∆OTFP - %∆STFP in figure 5a: minimum = 

-.013, maximum = .019, average a = -.001, and stand ard deviation s = .006. A 

negative average is expected because %∆STFP is based on nonoptimal inputs. 

Confidence intervals increase in absolute value when translated to levels. For 

example, if OTPF and STFP are both normalized to one in 1949 and over the 53 

years a-s = -.007 ≤ %∆OTFP - %∆STFP ≤ a+s = .005, then, based on a normal 

distribution, with 68% probability, in 2001, |OTFP - STFP| ≤ .01 

%∆OTFP - %∆STFP has frequently been significant relative to the average 

values of %∆OTFP and %∆STFP. From 1949 to 2001, average %∆OTFP = .0112, average 

%∆STFP = .0114, and |%∆OTFP - %∆STFP| > .01 in 8 out of the first 26 years in 

the period. Thus, relative to the average values of %∆OTFP and %∆STFP, |%∆OTFP 

- %∆STFP| exceeded about 100% about 30% of the time in the first half of the 

period. According to equation (4.4), large values of |%∆OTFP - %∆STFP| are 

caused by large values of ||rt||. This appears to be the case somewhat from 1949 

to the mid 1970s, strongly in the mid 1970s, but not so much thereafter. Thus, 
23

although the average value of %∆OTFP - %∆STFP has been small from 1949 to 2001 

(about .001), in certain years in the first part of the period, |%∆OTFP - 

%∆STFP| has been large. We conclude that STFP has frequently been significantly 

biased relative to %∆OTFP, but not systematically. 

An economic argument questions whether any residual can correctly measure 

TFP. The argument is that TFP represents knowledge and technology that 

accumulate slowly over time, mostly as a result of conscious investment 

decisions. That is, TFP moves trendlike with very little noisy variation, unlike 

the residual measures whose noisy variations presumably mostly reflect shorterterm cyclical variations in observed output. This viewpoint, which suggests 

developing a structural model in which TFP accumulates as a result of endogenous 

investment decisions, has been implemented by Chen and Zadrozny (2004). 

6. Conclusion. 

In the paper, we used the multi-step perturbation (MSP) method to estimate 

CES and TCES production-function models of KLEMS inputs in U.S. manufacturing 

from 1949 to 2001. For each estimated model, we computed AIC, BCAIC, and BIC and 

chose as the best one TCES1 model 5 with positive degrees of freedom (DF), which 

minimized one or more of these information criteria for the sample. By 

marginally choosing model 5 as the best model, the principal of minimum IC 

slightly rejects a CES production function, with a single input-price elasticity 

of substitution for all KLEMS inputs, in favor of a TCES1 model with unitary 

outer and less than unitary inner input-price elasticities of substitution. 

Then, we computed the year-to-year percentage growth of optimal TFP (%∆OTFP) 

based on the best model and compared it with the percentage growth of standard 

Solow-residual TFP (%∆STFP). Because the model underlying the %∆STFP has zero 

DF, strictly, in contrast to %∆OTFP, %∆STFP should be considered as having no 

statistical reliability. However, to the extent that %∆STFP differs from 

statistically-reliable %∆OTFP by less than the average value of %∆STFP - %∆OTFP 

(about .001), %∆STFP can be considered statistically reliable. Nevertheless, 

relative to the average values of %∆STFP and %∆OTFP, |%∆STFP - %∆OTFP| exceeded 

about 100% about 30% of the time from 1949 to the mid 1970s. According to 

Taylor-series theory %∆OTFP - %∆STFP should be accounted for by the 2nd- to 

4th-order terms absent from the 1st-order approximate input function (4.3) which 

underlies %∆STFP. 
24

 For given estimated input-cost-share parameters, αi, we estimated 

elasticity parameters, σi, by minimizing -(2/T) × log-likelihood function over a 

coarse grid of values. In the future, we shall consider estimating the σi's more 

fully over continuous intervals, but still conditional on prior estimates of 

αi's because, unless the production function includes a measure of capacity, 

estimating the αi's and σi's jointly tends to result in implausible estimates of 

the αi's (Tatom, 1980). We shall also consider using the more general GCES 

production function, which in log form is 

(6.1) f(v) = (1/γ)⋅ln( ), ∑ =

ρ α

n

1i

ivi

ie

where each input i has its own share parameter (0 < αi < 1) and its own 

elasticity parameter (ρi < 1). When the ρi's are unequal, the GCES production 

function is globally nonhomothetic and the 1st-order conditions (2.1) and (2.2) 

generally have no analytical solution. Because the MSP method produced accurate 

solutions for the CES and TCES applications here, it should similarly produce 

accurate solutions for GCES applications. 
25

Figure 1: Illustration of Multi-Step Perturbation. 

v2

 fA fB

A 

 B 

 BB 

 AA 

 v1

Input-cost lines AA and BB are, respectively, defined by e( )

Te( ) = e( )Te(v) 

and e(p)Te( vˆ ) = e(p)Te(v), for given precomputed "optimal" and given observed 

p and v. 

ˆp vˆ ˆp

ˆp
26

Figure 2: Constant, IMA, and T o&& rnqvist LMKS Input Cost Shares, 1949-2001. 

a. Labor Cost Share

1950195419581962196619701974197819821986199019941998

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50 CCL

ICL

TCL

b. Materials Cost Share

1950195419581962196619701974197819821986199019941998

0.216

0.234

0.252

0.270

0.288

0.306

0.324

0.342

0.360 CCM

ICM

TCM

c. Capital Cost Share

1950195419581962196619701974197819821986199019941998

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

0.21

CCK

ICK

TCK

d. Services Cost Share

1950195419581962196619701974197819821986199019941998

0.050

0.075

0.100

0.125

0.150

CCS

ICS

TCS

CCL, ... CCS denote constant cost shares of labor, ..., constant cost shares of 

services; ICL, ..., ICS denote IMA cost shares of labor, ..., IMA cost shares of 

services; and, TCL, ..., TCS denote To rnqvist cost shares of labor, ..., 

T rnqvist cost shares of services. 

&&

o&&
27

Figure 3: Scatter Plots of Pairwise Log of KLEMS Input Quantities. 

a. ln(K) vs. ln(L)

-1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

-0.00

0.05

0.10

b. ln(K) vs. ln(E)

-1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

c. ln(K) vs. ln(M)

-1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

d. ln(K) vs. ln(S)

-1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

e. ln(L) vs. ln(E)

-0.27 -0.18 -0.09 -0.00 0.09

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

f. ln(L) vs. ln(M)

-0.27 -0.18 -0.09 -0.00 0.09

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

g. ln(L) vs. ln(S)

-0.27 -0.18 -0.09 -0.00 0.09

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

h. ln(E) vs. ln(M)

-1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

i. ln(E) vs. ln(S)

-2.0 -1.5 -1.0 -0.5 0.0 0.5

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

j. ln(M) vs. ln(S)

-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

The first and second variables listed at the top of each graph refer, 

respectively, to the vertical and horzontal axes. 
28

Figure 4: Log of KLEMS Input Prices and Quantities, 1949-2001. 

a. Log of KLEMS Prices

19491954195919641969197419791984198919941999

-2.4

-1.6

-0.8

-0.0

0.8

1.6

2.4

PK

PL

PE

PM

PS

b. Log of KLEMS Quantities

19491954195919641969197419791984198919941999

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

QK

QL

QE

QM

QS
29

Figure 5: %∆OTFP - %∆STFP and Norms of Differences in %∆ of Relative Input 

Prices, 1949 to 2001. 

a. % Growth OTFP - % Growth STFP

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

1945 1955 1965 1975 1985 1995 2005

b. Norms of Differences in % Growth of Relative 

Input Prices

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1945 1955 1965 1975 1985 1995 2005
30

 

 

Table 1: Summary Statistics of the Best Estimated Models. 

1 2 3 4 5 6 7 8 9 

Model αit σ1 σ2

DF 

 

 

-(2/T)L AIC BCAIC BIC 

Best CES Models 

1 Const .50 --- 47 -26.66 -26.47 -26.45 -26.29 

 

 

2 IMA 1.0 --- 43 -32.62 -32.28 -32.21 -31.94 

3 Tornq 1.0 --- 

 

 

0 -42.22 -34.18 +∞ -26.26 

Best TCES1 Models 

4 Const 

 

.50 .17 46 -27.69 -27.50 -27.48 -27.32 

5 IMA 

 

 

1.0 .67 42 -33.54 -33.21 -33.13 -32.87 

6 

 

Tornq 1.0 .67 0 -37.26 -26.98 +∞ -19.06 

Best TCES2 Models 

7 Const .50 .10 46 -24.09 -23.90 

 

 

-23.88 -23.71 

8 IMA 1.0 .67 42 -33.24 -32.90 

 

 

-32.82 -32.56 

9 Tornq 1.0 .67 0 -40.35 -32.32 +∞ -24.40 

CES, TCES1, and TCES2 production functions are, respectively, Q = (α1K

ρ + α2Lρ + 

α3E

ρ + α4Mρ + α5Sρ)

1/ρ, Q = [α1L

ρ + α2(β1Kγ + β2Eγ + β3Mγ + β4Sγ)ρ/γ]

1/ρ, and Q = [α1L

ρ + 

α2E

ρ + α3(β1Kγ + β2Mγ + β3Sγ)ρ/γ]

1/ρ. 
31

 

 

 

 

Table 2: Summary Statistics of the Distribution of %∆OTFP - %∆STFP. 

Min. -.013 

Max. .019 

Mean -.001 

Std. dev. .006 
32

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