Working Paper Series

Congressional Budget Office

Washington, DC

Output, Investment, and Growth

in a World of Putty-Clay

Mark Lasky

Congressional Budget Office

E-mail: mark.lasky@cbo.gov

May 2007

Working Paper 2007-07

Working papers in this series are preliminary and are circulated to stimulate

discussion and critical comment. They are not subject to CBO’s formal review

and editing processes. The analysis and conclusions expressed in them are those

of the authors and should not be interpreted as those of the Congressional Budget

Office. References in publications should be cleared with the authors. Papers in

this series can be obtained at www.cbo.gov/publications.
Abstract

This paper presents a putty-clay model of capital that proceeds from profit maximization to an analytic solution for aggregate investment that fits the data well. The

key innovation is a production function in which a geometric average of the capitallabor ratio embedded in each unit of capital replaces the aggregate capital-labor ratio

in the standard Cobb-Douglas production function. The accelerator in this model

depends on the growth rate of output in excess of labor productivity at full employment, rather than on the growth rate of output alone as in previous work. Growth of

labor hours at full employment is an important driver of investment. Expectations

of faster productivity growth boost current investment. The effect of productivity

growth on the real rate of return in steady state is smaller than in the traditional

neoclassical model. Cash flow and restrictions implied by the neoclassical model are

not significant when added to the estimating equations. The putty-clay assumption

allows growth of the capital stock to be split into its capital-widening and capitaldeepening components. The model explains the sharp rise in the gap between capital

income and business fixed investment in the early 2000s.
Economists have long had difficulty constructing a model of business investment that

both fits the data well and is grounded in the theory of the firm. In the traditional

neoclassical model first developed by Jorgenson (1963), the demand for capital is

derived from the profit-maximizing decisions of firms, and so is consistent with theory.

However, that model’s assumption that capital is putty-putty, i.e., that firms are free

to vary the number of workers required to operate any machine after it is put in

place, implies that the short-run response of investment to a change in the ratio

of output price to the cost of capital should be the same as the short-run response

to the same percentage change in output. That implication is not borne out in

empirical work (see Chirinko’s (1993) survey). In addition, the dynamics needed

to translate demand for capital into investment do not follow explicitly from firm’s

profit-maximizing decisions.

The q theory of investment, proposed by Keynes (1936) and developed by Tobin

(1969), in which investment depends positively on the ratio of the market value

of a firm to the replacement cost of its capital stock, does a better job of fitting

the data but requires the ad hoc addition of variables like cash flow and implies

unrealistically large costs to firms of adjusting the amount of capital they use. Cooper

and Haltiwanger (2006) find adjustment costs of a more realistic magnitude using

plant-level data from manufacturing, but the properties of their data set, especially

the degree of serial correlation, are quite different from those of aggregate investment.

Past work using the putty-clay approach to capital, originally developed by Johansen

(1959), Solow (1962), and Phelps (1963), has not solved that problem. Under the

assumption of putty-clay capital, firms can choose among a wide variety of possible

ratios of capital to labor when purchasing capital, but cannot alter that ratio once

capital has been delivered. Thus, each unit of capital, or “machine,” is used by a fixed

amount of labor throughout its service life. The theoretical literature has generally

assumed that new machines function independently of old machines. In that case,

the productivity of new machines cannot be inferred from aggregate productivity–

the productivity of existing capital–making empirical work difficult. The existing

emprical work (e.g., Bischoff (1971)), while providing a good fit for the data, has

required assumptions that strain the link with theory.1

This paper presents a putty-clay model of investment that is derived from the profitmaximizing decisions of firms and that provides a good fit for the aggregate data. The

key innovation in the model is an aggregate production function in which a geometric

average of the capital-labor ratio embedded in each existing machine replaces the

aggregate capital-labor ratio in the standard Cobb-Douglas production function. The

productivity of new machines is then a function of aggregate productivity, allowing

an analytic solution for investment as a function of aggregate variables. Investment

is the number of new machines multiplied by the capital-labor ratio embodied in

those machines. With the addition of a variable capturing trends in the number of

1For example, Bischoff assumes that “entrepreneurs choose their factor proportions by a simple

suboptimal rule of thumb” (p. 70).

1
machines per worker, estimated errors are small. Even without that variable, the

model captures turning points well. The results are also consistent with previous

empirical work finding that the short-run response of investment to variations in

output is much larger than the short-run response to variations in the cost of capital.

The model also has several unique implications. Because investment depends on

the number of new machines, which in turn is tied to labor hours, the accelerator–

the response of investment to changes in demand–uses the growth rate of output

less the growth rate of labor productivity at full employment. That differs from

previous work, in which the accelerator depends on the growth rate of output alone.

Higher productivity does increase the capital-labor ratio embodied in new machines,

but does not lead to an accelerator effect. Because higher productivity leaves the

number of new machines unchanged, a productivity shock cannot generate changes

in employment through its effect on investment. In addition, the impact of a change

in trend growth of productivity on the real rate of return in steady state is smaller

than in the neoclassical model.

At the same time, expectations of faster productivity growth in the future directly

raise investment, in contrast to the neoclassical model where expectations are irrelevant. Greater productivity in the future raises the marginal effect on revenue of

buying a machine of higher quality.

Businesses use labor hours at full employment (the portion of labor hours independent

of the business cycle) to help forecast the need for machines in the future, so growth of

labor hours at full employment is a key driver of investment. That helps mitigate one

of the problems of the neoclassical model: the dependence of investment on output,

which in turn depends on the capital stock.

Putty-clay capital provides a way of distinguishing capital widening–an increase in

capital that accommodates additional workers–from capital deepening–an increase

in capital that boosts the output of existing workers. While both increase the economy’s capacity to produce output, only the latter raises productivity. The model

also allows for gradual changes in capital’s coefficient in production through a variable measuring trends in machines per worker. Unusually strong investment in the

late 1970s and early 1980s and an increase in capital’s share of income at about the

same time appear to reflect an increase in that variable. Finally, because the model

has implications for both capital’s share of income and business fixed investment, it

explains most past variations of capital income less investment as a share of total

output.

Section 1 presents the putty-clay production function and derives investment starting

from the profit maximization decision of the firm. Section 2 examines the theoretical

properties of the model, comparing the short-run behavior of investment and the

rate of return in steady state to their properties in the standard neoclassical model.

Section 3 describes the data used in this paper. Section 4 presents the results from

estimating the model and tests its robustness against restrictions implied by the

2
neoclassical model and a simple version of the q theory with cash flow. Section 5

calculates estimates of capital per labor hour and the implied capital stock. Section

6 shows the implications of the model for capital income minus investment.

1 The Model

I assume that capital is produced and used in the form of “machines,” be they drill

presses, aircraft, computers, office buildings, oil wells, or software packages.2 Each

machine is associated with a fixed amount of labor when it is in use. To take an

obvious example, the number of workers it takes to drive a truck does not change

over its service life. A firm is free to choose the amount of capital in a new machine

(its “quality”) at the time of investment, but once the machine is put in place, that

amount is fixed. Ex ante, capital is putty; ex post, it is clay.

That assumption matters because the amounts of capital embodied in each existing

machine are combined into an aggregate index of the quality of existing capital using

a geometric average. The aggregate index of machine quality replaces the ratio of

capital to labor in the standard Cobb-Douglas production function. Firms maximize

profits, choosing the quality of new machines so that the present discounted value of

the after-tax revenues from an extra dollar of quality equals the after-tax cost of that

quality. In addition, I assume quadratic costs of utilization, so firms try to align the

total number of machines with labor hours. Real investment is the number of new

machines times the quality of new machines. Inventories, another form of capital

necessary to calculate the total stock of capital, are discussed in Appendix A.

1.1 The Production Function and Capital per Labor Hour

At time t, output is determined by the production function

Yt = At Lt



 ̄

kt

α . (1)

Y is total output, A is total factor productivity (TFP), L is labor hours,  ̄k is an

index of capital per labor hour, and α is the coefficient of that index in production.

Labor is homogeneous over time, so TFP includes both average labor quality and the

technology of production. The only difference between that production function and

the neoclassical production function is the treatment of capital per labor hour.

The index of capital per labor hour depends on the quality of each existing machine

(k) and on the equilibrium number of machines per labor hour (x).3 The quality of

2Although labor hours are the measure of labor used in this paper, one can think of a machine as

the amount of a specific type of capital used by one worker. In the case of an office building, an office

would be a structures “machine.” One section of a retail establishment would also be a machine. 3The actual number of machines per labor hour may differ from x because of time to build. It

takes longer to adjust the capital stock than it does to adjust the amount of labor.

3
each existing machine is fixed over its service life, so the index of capital per labor hour

changes only as new machines are added to the stock, existing machines are scrapped,

or the number of machines per labor hour changes. The first two factors encompass

most of what economists normally think of as capital deepening. Examples include

increased real investment in computers as the price of new computers declines, higher

investment in response to tax incentives or a decline in the cost of funds, and the

replacement of depreciating capital with higher-quality new capital due to a lower

ratio of the cost of capital to the cost of labor than when the depreciating capital

was installed. (That latter factor may explain at least part of the increase in labor

productivity observed by Foster, Haltiwanger, and Krizan (2002) when a new retail

establishment replaces an old one.)

Examples of a change in the number of machines per labor hour are a shift in demand

toward sectors with a higher ratio of capital to labor, such as mining, and the shifting

of tasks from people to computers as businesses adopted the personal computer. In

practice, changes in x account for a small fraction of changes in  ̄k; for example,  ̄k

nearly quadrupled between 1955 and 2005, while x rose just 4 percent. The variable

x is added to the model to control for changes in the importance of mining and for

trends in capital per labor hour that cannot be explained by the determinants of k.

If we allowed for unlimited variation of x, the equation for investment would become

an identity.

To formalize an expression for  ̄k, I assume there are M kinds of capital, denoted by

the suffix m. The quality of machines of type m installed at time t, i.e., real dollars of

investment per new machine, is kmt. Let Rmt−i,i be the number of type-m machines

installed at time t − i and remaining in service i periods later. Then the index of

capital per labor hour for machines of type m at time t is

log 

 ̄

kmt



= [∞

i=0

Rmt−i,i log (kmt−i)

NmKt

,

where NmK

t = S∞

i=0 Rmt−i,i, the total number of machines of type m in service at

time t. Thus,  ̄km is a geometric average of the quality of each type-m machine still

in service. (In this paper, N denotes the number of new machines, while NK denotes

the number of existing machines.) Each machine is a one-horse shay, remaining fully

operational until it is removed from the capital stock in its entirety.

To prevent the growth of the index of capital per labor hour from depending on the

base year chosen for the price indexes, we must express it using first differences (∆):

∆ log 

 ̄

kt



= xt−1

#SM

m=1 NmK

t−1∆ log 

 ̄

kmt



NK

t−1

+ ∆ log (xt)

$

, (2)

where NK

t−1 = SM

m=1 NmK

t−1. Logarithmic growth of capital per labor hour depends

on a weighted average of logarithmic growth of capital per labor hour for each type

of machine plus logarithmic growth of x. That sum is premultiplied by xt−1 so that a

4
given percentage change in machine quality has a larger impact on output the more

machines there are per labor hour. (For example, we want a 1.04 percent increase in

all the  ̄km when x = 1 to have the same impact on output as a 1 percent increase

in all the  ̄km when x = 1.04.) In addition, the parallel treatment of ∆ log 

 ̄

km

and ∆ log (x) in equation 2 implies that a permanent 10 percent increase in real

investment would have the same long-run effect on capital per labor hour whether it

were used to increase x or the quality of new machines.

The levels of quality of different types of machines are scaled so that machines of different types have the same importance in production. Thus, there are far fewer light

truck “machines” in a light truck than there are aircraft “machines” in a commercial

aircraft.

To solve the firm’s profit maximization problem, we need to express log 

 ̄

kt



in levels.

Equation 2 becomes

log 

 ̄

kt



= [

t

i=1

log (Xt−i) + xt

#SM

m=1 NmK

t log 

 ̄

kmt



NK

t

+ log (xt)

$

, (3)

where Xt is defined with

log (Xt) = −∆xt

#SM

m=1 NmK

t log 

 ̄

kmt



NK

t

+ log (xt)

$

−xt−1

[

M

m=1

∆NmK

t

NK

t

log 

 ̄

kmt



.

I assume that, when choosing the quality of new machines (kmt), firms expect

log (Xt+i)=0 for all positive i.

To grasp the intuition behind  ̄k, consider the special case in which Xt = xt = 1.

Index the existing machines from 1 to NK

t . Then we can rewrite equation 3 as

 ̄

kt = k

1/NK

t

1 k

1/NK

t

2 ...k1/NK

t

NK

t

. (4)

Because each ki is measured using real dollars of capital per labor hour,  ̄

k is a geometric average of the capital-labor ratio embodied in each existing machine.

In this special case, NK = L, so equations 4 and 1 can be combined to express

production as

Yt = At Lt kα/Lt

1 kα/Lt2 ...kα/Lt

Lt . (5)

Now rewrite the standard Cobb-Douglas production function as

Yt = At Lt

Kt

Lt

α

, (6)

5
where K is the aggregate putty-putty stock of capital. When x = 1, the puttyclay production function is just a Cobb-Douglas production function in which the

aggregate capital-labor ratio has been replaced by a geometric average of the capitallabor ratio of each machine.

Indeed, one can consider the Cobb-Douglas production function as a special case of

the putty-clay production function in which firms are free to vary the quality of each

machine throughout its service life. Given aggregate capital Kt, the firm maximizes

output by distributing that capital equally across all machines, i.e., k1 = k2 = ... =

kLt = Kt/Lt. In that case, equation 5 collapses to equation 6.

In the putty-clay model, machines are complements: an increase in k1 raises the

marginal product of an extra dollar of k2. For example, a software upgrade is generally

more valuable the more powerful is the personal computer using it. All else equal,

the greater the quality of existing machines, the greater will be the quality of new

machines. This is not true in the neoclassical model. The greater is K, the lower is

the marginal product of an additional dollar of K.

Returning to the putty-clay production function in equation 1, output varies proportionately with labor hours in the short run. Unanticipated short-run variations of

labor hours lead to more or less intensive use of the capital stock. Output per labor

hour Y /L can even be procyclical if TFP depends on aggregate demand. A rising cost

of utilization, discussed below, prevents firms from permanently setting NK below

xL.

The assumption that output varies proportionately with labor hours in the short run

implicitly assumes that utilization rates for each machine are equal. That assumption departs from much of the putty-clay literature. Solow (1962), Phelps (1963),

Solow et al. (1966), Calvo (1976), and Gilchrist and Williams (1998) assume that

all machines above a market-clearing quality are in use while all others are idle.

In that case, a short-run rise in output and labor hours increases the utilization of

less-efficient capital, reducing average labor productivity. While that is certainly a

plausible assumption, it is unhelpful in empirical work, given that economists since

at least Fabricant (1942) have found short-run increasing returns to labor. The assumption that a machine’s utilization does not depend on its quality relative to other

machines also greatly simplifies the firm’s investment decision.

The treatment of TFP in this paper also differs from that in much of the putty-clay

literature. Johansen (1959), Phelps (1963), Solow et al. (1966), Calvo (1976), and

Gilchrist and Williams (1998) all assume that the output of a given unit of capital is

constant over its service life, i.e., that all technical progress is embodied in capital.4

To determine the contribution of a machine to output, one must know not only its

4While Ando et al. (1974) do not assume that the output of a given unit of capital is constant over

its service life, their assumption that the required labor input per unit of existing capital rises at a

constant rate m per year leaves the productivity of existing capital independent of current technical

progress.

6
quality but also its vintage. That assumption prevents easy aggregation of capital. In

this paper, once one knows the quality of a machine, its vintage provides no additional

information about its contribution to output.

Note that the production function exhibits constant returns to scale when the utilization rate is held constant. If the number of hours doubles, then an unchanged

 ̄

k implies a doubling of the number of machines and thus a doubling of the total

amount of capital. From equation 1, output would also double.

1.2 The Firm and Profit Maximization

1.2.1 The Profit Function

I assume an arbitrary number of monopolistically competitive firms. Demand for the

output of the jth firm (Y jt) is given by

Y jt = fjtYt

pjt

pt

η

,

where fj is the jth firm’s share of the market when all firms charge the same price

(

Sf j = 1), p is the aggregate price of output, pj is the price charged by the jth

firm, and η is the price elasticity of demand for each firm’s output. For simplicity, I

assume that the f j are constant over time. The implications of that assumption are

discussed below.

Businesses seek to maximize the expected present discounted value (PDV) of current

and future profits, PDV (π). For simplicity, I assume no taxation and only one type

of capital. The effect of the corporate income tax on investment, which is identical

to its effect in the traditional neoclassical model, and the effect of property taxes on

investment are developed in Appendix B and included in the empirical section below.

The impact of multiple types of capital on investment is explored later in this section.

The PDV of current and future profits of the jth firm at time t is given by

PDV (πj)t =

]∞

i=0

pjt+iY jt+ie−ridi −

]∞

i=0

w ̄t+iLjt+ie−ridi −[∞

i=0

qt+ikjt+iNjt+ie−ri

−

]∞

i=0

1

2

ξt+iw ̄t+iLjt+i

#

1 − NjK

t+i

xt+iLjt+i

ψ

$2

e−ridi, (7)

where r is the nominal cost of corporate funds (or nominal rate of return) expected at

time t, w ̄ is straight-time compensation per hour, regarded by each firm as exogenous,

q is the price index for capital goods, Nt is the number of machines installed at time

7
Table 1: Sample Values of the Cost of High Utilization

α ψ x ξ

0.25 0.6 1.05 1.68

0.24 0.6 1.05 1.58

0.25 0.5 1.05 1.73

0.25 0.6 1.16 1.97

t, ξt+i and ψ govern the cost of high utilization rates (ψ < 1), and the suffix j denotes

a variable for the jth firm.5 I assume that

ξt+i =

1 − ψ

α

ψ

xt

− 1

2



1 − ψ2

−1

. (8)

That assumption is motivated by convenience rather than by any deep economic

rationale: it generates NK = xL in equilibrium. Table 1 shows values of ξ for

possible combinations of α, ψ, and x.

6

For simplicity, I assume that A,  ̄kt−1, and the distribution of existing vintages are

identical for each firm, so that each firm faces the same production function. Only

the number of machines (NK) and market share (f) differ among firms.

The first two terms on the right-hand side of equation 7 are familiar, except that

wL ̄ only includes straight-time labor compensation. The third term is the PDV of

investment: q is the price index of new capital goods, while kjNj is real investment.

Because N is a discrete number of machines purchased, rather than a rate of investment, I assume that investment occurs at the beginning of each period, meaning the

PDV is calculated with a summation rather than an integral.

The final term in equation 7 is the cost of high utilization rates (xL/NK is the

utilization rate, indexed to 1), needed because the production function contains no

penalty to a firm for buying few machines and then operating them at a high rate. One

obvious component of that cost is higher hourly compensation for the overtime and

extra shifts needed to obtain extra production from a limited amount of capital. (The

marginal value of an extra hour of leisure rises as weekly hours worked increase, and so

overall hourly compensation–including any deferred portion–should increase with

hours per week.) In addition, part of the cost of higher utilization must fall on capital,

since the share of labor compensation in total income is not procyclical, as it would

have to be if the entire cost of increased utilization accrued to labor. One example

would be extra wear and tear on capital that results in additional maintenance in

later periods or earlier retirement. The cost of high utilization could also proxy

5Throughout the paper, rates of growth and rates of return in the equations correspond to the

unit in which time is measured. Thus, if time is measured in quarters, rates of growth and rates of

return are quarterly rates. However, in any discussion of data, I cite annual rates. 6The values for α assume that α (1 + 1/η) is 0.26 and η is −26 (for α = 0.25) or −13 (for α = 0.24).

The values for x are the miminum and maximum between 1960 and 2006.

8
for an absolute limit on the amount of production possible from a given capital

stock coupled with a permanent reduction in market share fj when a firm cannot

meet the demands of its customers, so that future demand Y j depends inversely on

current xL/NK. Whatever the costs of high utilization rates, they cannot include a

contemporaneous reduction in the efficiency of production, since that would violate

the production function. Utilization costs are expressed as a function of labor hours

rather than output so that the choice of kj, which affects output but not labor hours,

does not depend on expected future rates of utilization.

1.2.2 First-Order Conditions

Each firm maximizes profits over three variables: the quality of new machines (kj),

the number of new machines purchased (Nj), and labor hours (Lj). Given aggregate

output (or demand) Y and aggregate price p, the firm’s choice of labor hours determines its output (Y j) and the price charged to customers to sell that output (pj). To

simplify the mathematics of the firm’s investment decision, I assume identical service

life S for all machines, so that firms expect machines installed at time t to depreciate

at time t + S.

7

After a few substitutions and cancellations, the three first-order conditions for maximization of profits become

α



1 + 1

η

 ] S

i=0

pjt+i

Y jt+i

NjK

t+i

1

kjt

xt+ie−ridi = qt, (9)

] S

i=0

ξt+i

w ̄t+i

xt+i

#

1 − NjK

t+i

xt+iLjt+i

ψ

$

ψe−ridi = qtkjt, (10)

and



1 + 1

η



pjt

Y jt

Ljt

=  ̄wt

%

1 + 1

2

ξt

#

1 −



NjK

t

xtLjt

ψ

2

$& . (11)

Equation 9 states that profits are maximized when the PDV of the additional revenues

produced by an extra dollar of machine quality equals the cost of an extra dollar

of machine quality. According to equation 10, the reduction in the costs of high

utilization from adding another machine equals the cost of that machine.8 Equation

7 In the empirical section, I calculate depreciation following the treatment used by the U.S. Department of Labor (1983). Retirements of type-m capital installed at time t follow a truncated

normal distribution centered around the end of the expected service life t + Smt. 8The first-order conditions for Njt do not include the effect of N jt on kj ̄ t+i. If the quality of new

machines grows over time, then an extra machine purchased today will raise the average productivity

of the capital stock in the near term, when the quality of that machine is greater than the average

quality of existing machines, but reduce it when the machine nears retirement and is less productive

than most existing capital. The PDV of the net impact is small, roughly 1 percent of the cost of

capital for equipment and 2 percent for structures, and complicates the algebra considerably, so I

ignore it.

9
11 states that the marginal revenues from an additional hour of labor equal the costs

of that hour–straight-time pay plus the costs caused by a higher utilization rate.9

1.3 Investment: The Quantity and Quality of New Machines

1.3.1 The Number of New Machines with No Time to Build

To solve this system of equations, we must make some further assumptions. I assume

that firms expect NjK/(xL) to remain constant from time t to time t + S, so that

we can replace NjK

t+i/(xt+iLt+i) with NjK

t /(xtLt). Since NjK

t+i is fixed from t + i to

t+i+1 but Lj can vary during that interval, the assumption that NjK/(xL) remains

constant cannot hold in continuous time. Consequently, Lt+i denotes average labor

hours from t+i to t+i+ 1 and Yt+i denotes average total output from t+i to t+i+1

in the remainder of the paper. With time to build, discussed below, the firm makes

its investment decisions for time t long enough before time t that it can assume that

NjK/(xL) will be at its equilibrium level at time t when making those decisions, and

thus can remain constant from time t to time t + S. In addition, I assume that firms

expect x to remain constant from time t to time t + S. Equation 8 then implies we

can replace ξt+i with ξt.

To solve for NjK

t , we first use equation 11 to substitute for w ̄t+i and equation 8

to substitute for ξ in equation 10. After setting the left-hand side of the resulting

equation equal to the left-hand side of equation 9 (multiplied through by kjt) and

considerable algebra, we find that NjK

t = xtLjt. Summing over all firms,

NK

t = xtLt. (12)

At the time a firm plans its capital stock for time t, it desires that the number of

machines equal machines per labor hour times expected labor hours. To achieve the

desired stock, the number of new machines (N) installed by all firms at time t must

equal the desired change in the total number of machines plus the number of machines

depreciating at time t (dep):

Nt = NK

t − NKt−1 + dept.

What happens if desired N is negative? In practice, depreciation is large enough

that this does not occur at the aggregate level. However, if we allow for variations in

market share (the f j), individual firms may face a situation in which they desire to

reduce Nj but cannot. Caballero (1999) shows that irreversibility of investment leads

firms to invest less than they otherwise would. Adding irreversibility of the number

9 In practice, firms can vary the intensity of effort over the business cycle (see Appendix C). High

intensity of effort must have some cost, or firms would keep intensity high all the time. In that case,

labor productivity at full employment should replace actual labor productivity (Y jt/Ljt) in equation

11.

10
of machines to the putty-clay model would raise NK/(xL) above 1 during periods

when irreversibility is binding, thereby reducing it below 1 when irreversibility is not

binding. While that may have an important adverse effect on the equilibrium level of

investment, I do not consider irreversibility of the number of machines in this paper.

Rather, the putty-clay model assumes the irreversibility of machine quality.

1.3.2 The Quality of New Machines

To derive the quality of new machines (k), first observe from equation 10 that if

NjK

t+i/(xt+iLt+i)=1, then current and future kj is the same for all firms. The

assumption that  ̄kjt−1 and At are equal for all firms, coupled with equation 1, implies

that Y j/Lj, and thus pj, are the same for all firms now and in the future.10 Using

that equality across firms and setting xt+i/NjK

t+i equal to 1/Ljt+i, we can rewrite

equation 9 as

α



1 + 1

η

 ] S

i=0

pt+i

Yt+i

Lt+i

e−ridi = qtkt. (13)

Let y represent labor productivity (Y /L) and y ̇ represent its expected trend rate

of growth.11 I assume that time to build is long enough that firms assume that

any procyclical component of Y /L present when they pick kt will be gone by time

t, so that Y /L can be expected to grow at its trend rate thereafter. To simplify

the algebra, I assume that firms solve their maximization problem using the average

expected growth rate of output prices (p ̇) and nominal rate of return expected to

prevail over a machine’s service life.

Inserting those expectations into equation 13, solving the integral, and rearranging

terms yields

kt = α



1 +

1

η

 pt yt

vt

, (14)

where

vt = qt

tr − tp ̇ − y ̇

1 − e− (tr− tp ̇−y ̇)S (15)

and tr and tp ̇ are, respectively, the nominal cost of funds and rate of growth of output

prices expected at time t. Equation 14 states that the quality of new machines is

proportional to output per hour, y, and to the ratio of the price of output (p) to the

cost of capital (v). Workers are given more capital to work with the more productive

they are and the cheaper capital is. Productivity in turn depends both on TFP

and on the quality of existing machines ( ̄k), since machines are complementary. The

1 0To be exact, that conclusion also requires that NjK grow at the same rate across firms. 1 1 In this paper, y ̇ equals its sample average in all periods, so there is no need for a time subscript.

11
expression for the cost of capital v is similar to that found by Ando et al. (1974),

except that it substitutes r − p ̇ − y ̇ for r − p ̇ and excludes taxes.12

The cost of capital in the standard putty-putty model in the absence of taxes, vP P ,

is

vPP

t = qt



tr − tq ̇ +

1

S



,

where tq ̇ is the expected growth rate of q at time t. In both models, a higher price

of capital goods and a higher nominal rate of return raise the cost of capital, while

a longer service life (or lower rate of depreciation) reduces the cost of capital. The

sizes of the effects of higher rates of return and a longer service life on the cost of

capital differ between the models due to different functional forms. There are two

other important differences.

First, the real rate of return is the nominal rate minus the rate of overall price inflation

in the putty-clay model, but the nominal rate minus the rate of growth of capital

prices in the putty-putty model. In the putty-putty model, the marginal product

of existing computers falls as they are puttied together with new computers, raising

the cost of capital required to equate marginal revenue with marginal cost. In the

putty-clay model, however, a fall in the price of computers drives down the price of

existing computers but does not make them less productive. Instead, the marginal

revenue of an additional dollar of machine quality grows with output prices. That

rate of growth is subtracted from the nominal cost of funds.

Second, in the putty-clay model, existing machines become more productive as labor productivity rises. All else equal, the more rapidly labor productivity will rise

in the future, the more valuable is each dollar of quality added to a new machine

today. Hence, the expected rate of growth of labor productivity is subtracted from

the expected real rate of return. In the putty-putty model, the increased productivity of existing machines disappears as they are puttied together with new capital.

Consequently, productivity growth has no effect on the cost of capital in that model.

When there is more than one type of capital, the cost of type-m capital (vm) depends

on the price (qm) and service life (Sm) of such capital. In addition, if r−p ̇ is expected

to change over time, tr − tp ̇ may differ at time t for types of capital with different

expected service life. In the absence of time to build, real investment in type-m

capital at time t is kmtNmt.

1 2Equation 15 is also similar to the formula for determining the payment on a home mortgage

with a fixed rate. In that case, the term of the mortgage replaces S and the mortgage rate replaces

r − p ̇ − y ̇.

12
1.3.3 The Number and Quality of New Machines with Time to Build

The change in the number machines in a period depends on information from previous periods because of various factors collectively called “time to build,” following

Kydland and Prescott (1982). It takes time for businesses to gather information and

to translate that information into an investment plan. The existence of unfilled orders

for capital goods gives evidence of a further time lag between the time those plans

are made and the arrival of new capital. For structures, considerable time can elapse

between groundbreaking and final completion of a project. Investment will also lag

behind changes in the desired stock if capital is “lumpy,” i.e., if businesses do not

adjust their capital stock until some threshold level of adjustment is required.13

Time to build for the quality of new machines need not be the same length as time

to build for the number of new machines. Lumpy investment, adjustment costs, and

the information-gathering interval between changes in demand and the development

of a plan to meet them all suggest that the time to build for N exceeds the time to

build for k.

Time to build is usually modeled by assuming costs of adjustment, either of the

capital stock or of investment. However, adding a cost of adjusting the number of

new or existing machines greatly complicates the model. Consider the addition of

a cost term containing (Njt+i − Njt+i−1)

2 to equation 7. Equation 10 would then

include Njt − Njt−1 and Njt − Njt+1, making the number of new machines depend

on past and expected rates of investment. That in turn would cause the utilization

rate, xL/Nk, to deviate from 1. From equation 9, we can see that kj varies positively

with expected utilization. Because of the cost of adjusting the rate of purchase of

new machines, firms would satisfy part of the need for greater capacity by purchasing

machines of greater quality. To calculate the quality of new machines, we would

not only need to know labor productivity and the cost of capital, but also past and

expected N. Instead of adjustment costs, I assume a fixed schedule of planning and

deliveries.

Start by assuming that firms choose NK

t based on information available at time t−T.

Throughout this section, I assume that firms know xt with certainty at every stage

of time to build. Then, from equation 12, NK

t depends on firms’ expectations of Lt

at time t − T. Let Et−T (Jt) denote time t − T expectations of variable J at time t.

Output equals labor hours times labor productivity, so

Et−T (Lt) = Et−T (Yt)

Et−T (yt) (16)

for each firm and for the economy as a whole.

Let Y ̄ and L ̄ denote output and labor hours at full employment, i.e., output and labor

1 3For evidence that capital is lumpy, see Doms and Dunne (1998) and Cooper, Haltiwanger, and

Power (1999).

13
hours consistent with an unemployment rate stable at the full employment level.14

Assume that firms expect the difference between actual output and output at full

employment in the future to be a linear function of today’s difference between actual

output and output at full employment:

Et−T (Yt) − Et−T



Y ̄t



= θ



Yt−T − Y ̄t−T



. (17)

I assume firms expect the gap between output and output at full employment to

narrow over the time-to-build horizon, so that θ < 1.

The expected growth rate of output at full employment equals the expected growth

rate of hours at full employment (n ̇) plus y ̇, so

Et−T



Y ̄t



= Y ̄t−T e(n ̇ +y ̇)T . (18)

Inserting this into equation 17 and rearranging, we have

Et−T (Yt) = 

e(n ̇ +y ̇)T − θ



Y ̄t−T + θYt−T . (19)

Output expected at time t is a linear combination of actual output and output at full

employment at time t − T.

Assume that firms expect the procyclical component of labor productivity to disappear by the end of the time-to-build horizon and labor productivity at full employment to grow at rate y ̇, so Et−T (yt)= ̄yt−T eyT ̇ . Substituting this expression and

equation 19 into equation 16, substituting L ̄t−T for Y ̄t−T /y ̄t−T , and rearranging, we

have

Et−T (Lt) = 

enT ̇ − θ

eyT ̇



L ̄t−T +

θ

eyT ̇

Yt−T

y ̄t−T

. (20)

Expected future labor hours, and thus the desired number of machines, are a linear

combination of labor hours at full employment and the ratio of actual output to labor

productivity at full employment. Actual output has a smaller weight the greater the

expected narrowing of the gap between actual output and output at full employment,

i.e., the lower the θ.

Assume that at time t − T, each firm places an order for 1

T Ojt−T new machines to

be delivered in each period from t − T + 1 to t, a total order of Ojt−T . The standard

deviation of ∆NK is smaller than the standard deviations of ∆L ̄ and ∆Y j/y ̄, implying that firms’ response to movements in L ̄ and Y j/y ̄ is spread out over time. For

structures, which take several quarters to complete, the filling of an order over several

quarters is a natural assumption. Assume that firms place orders at time t − T that

1 4Because labor hours are slow to adjust to output, they may not be at their full employment level

if the unemployment rate is currently at its full employment rate but was higher or lower than that

rate in previous quarters.

14
optimize expected NjK

t , i.e., Et−T



NjK

t



= xt ∗ Et−T (Ljt).

15 The desired number

of machines is a linear combination of the Lj  ̄ and Y j/y ̄, which can be summed over

firms, so we can analyze aggregate orders Ot−T directly.

At time t − T − 1, the number of machines expected at time t − 1, Et−T −1



NK

t−1



,

equals orders placed at or before time t−T −1 but not yet delivered plus the expected

volume of orders placed after t − T − 1 but delivered before or at time t − 1. As an

approximation, I assume that the expected volume of orders placed after t − T − 1

but delivered before or at time t − 1 is the same at time t as at time t − 1.

16 Then

Et−T



NK

t



= Et−T −1



NK

t−1



+ Ot−T − Et−T −1 (Ot−T ) + 1

T

T

[−1

i=0

Et−T (Ot−T +i).

Rearranging terms, we have

Ot−T = Et−T



NK

t



− Et−T −1



NK

t−1



+

T − 1

T

Et−T −1 (Ot−T )

− 1

T

T

[−1

i=1

Et−T (Ot−T +i). (21)

To solve that expression, we must know how firms form expectations of future orders.

I will assume that they expect the same level of orders that they would in steady

state, i.e., if L ̄ and Y j/y ̄ grew at rate n ̇ . If O grows at rate n ̇ , then

T − 1

T Et−T −1 (Ot−T ) − 1T

T

[−1

i=1

Et−T (Ot−T+i) = T − 1

T Ot−T

%

1 −

T

[−1

i=1

(1 + n ̇)

i

&

≈ Ot−T

k

1 − enT /  ̇ 2

l

. (22)

Substituting from equation 22 into equation 21 and rearranging, we have

Ot−T = e−nT /  ̇ 2 

Et−T



NK

t



− Et−T −1



NK

t−1

 . (23)

The volume of orders made at time t − T is slightly smaller than the change in the

desired number of machines at time t because the expected volume of orders to be

made after t − T but delivered before or at t is somewhat greater than the volume of

orders made after t − T − 1 but delivered before or at t − 1.

1 5That prevents a positive shock between periods t−T and t from producing a permanent oscillation

in orders. For example, a positive shock in period t−1 would lead firms to raise orders in that period

to meet the new higher target for NK

t . However, that would leave deliveries too high at time t + 1,

leading to a reduction in orders made in period t. That zigzag pattern of orders would continue ad

infinitum.

1 6The only reason that is an approximation is the assumption that firms always expect labor hours

at full employment to grow at rate n ̇ . Consequently, if the level of L ̄t−T is 1 percent greater than

expected at time t − T − 1, the growth of L ̄ in subsequent periods will rise by 1 percent.

15
To obtain an expression for total investment, we must add in orders for replacement

machines. Replacement demand is smooth enough that we can assume that deliveries

of replacement machines equal the number of machines depreciating in every period.

Once replacement demand is factored in, Ot−T can be negative. The number of new

machines is then replacement demand plus a moving average of past orders:

Nt = 1

T

[

T

i=1

Ot−i + dept.

Using equations 12, 20, and 23 to replace Ot−i, we have

Nt = 1

T

[

T

i=1

enT /  ̇ 2 − θ

e(y ̇+n/ ̇ 2)T



∆ xtL ̄t−i



+

θ

e(y ̇+n/ ̇ 2)T ∆



xt

Yt−i

y ̄t−i

 + dept

The number of new machines is a function of lagged growth of labor hours at full

employment and lagged growth of output in excess of labor productivity plus replacement demand.

With more than one type of capital, the number of new machines of type m must

also take account of changes in the desired mix of capital and differences in time to

build between different types of capital. Ideally, separate orders Omt−Tm would be

placed for each type of capital at time t − Tm, with Tm being the time to build

type-m capital. However, if the desired ratio of type-m machines to all machines

(NmK/NK) is changing, the portion of orders Omt−Tm delivered before t −T would

cause the actual ratio to differ from the desired ratio in periods between t − T and t.

That makes it difficult to observe the desired ratio. Instead, I assume that various

aggregate orders Ot−Tm are placed, with the share of the order going to different types

of machines able to vary with desired NmK/NK. The number of new machines of

type m is then given by

Nmt =

NmK

t

NK

t

∗[

i



βmi∆ 

xtL ̄t−i



+ γmi∆



xt

Yt−i

y ̄t−i

 (24)

+ ∆

NmK

t

NK

t

∗

xt−1L ̄t−1 + depmt + mt,

where NmK

t /NKt

∗ is desired NmK/NK at time t,

S

i (βmi + γmi) > 1, and mt

is an error term.

Machine quality must also be chosen at the time an order is made. The quality

of machines delivered at time t will roughly equal the average quality ordered from

periods t − T to t − 1. I assume that firms expect the nominal rate of return and tax

law will be the same at time t as at time t−T. In that case, the quality of new orders

will grow at rate tp ̇ + y ̇ − tq ̇. To attain the desired average quality of new machines

delivered at time t, firms order quality

α



1 +

1

η

 pt−T y ̄t−T

vt−T

e(tp ̇+y ̇− tq ̇)T /2

16
at time t − T. Consequently, the quality of delivered machines of type m is given by

kmt = α



1 +

1

η

[

i

Gmi

pt−i y ̄t−i

vmt−i

e(tp ̇+y ̇− tq ̇)i/2, (25)

where Gmi is the fraction of machines delivered at time t that are ordered at time

t − i.

17 Total real investment in type-m capital at time t is Nmtkmt.

The maximum lag length for the number of machines exceeds the maximum lag

length for the quality of machines. Gathering information on the state of demand,

necessary for determining the number of new machines but not their quality, precedes

the placing of an order for machinery or the groundbreaking for a structure. Perhaps

more important, capital is lumpy, so new orders depend partly on changes in labor

hours and output occurring in earlier periods.

2 Properties of the Model

Many of the properties of the putty-clay model can most easily be seen in comparison

to the properties of the more familiar neoclassical model. This section discusses the

behavior of investment to illustrate the short-run properties of the model and the

behavior of the rate of return in steady state to illustrate the long-run properties of

the model. I ignore taxes, since the corporate income tax has the same effect on the

cost of capital in the putty-clay and putty-putty models. In both models, a corporate

income tax raises the cost of capital unless there is expensing, while an investment

tax credit reduces the cost of capital.18 Property taxes have different effects in the

two models, but the difference is not large.

2.1 Properties of Investment

In the absence of taxes and with η = −∞ (perfect competition), the desired stock of

putty-putty capital given a Cobb-Douglas production function in Jorgenson’s (1963)

neoclassical model is K∗

t = αptYt/vP Pt , where

vPP

t = q (tr − tq ̇ + δ)

and δ is the rate of geometric depreciation of the capital stock corresponding to 1/S.

(For simplicity, I assume a single type of capital in both models, so the m suffixes

1 7Since structures are used with land, a structure “machine” is composed of both structure and

land. I assume that the cost of the land portion of such a “machine” is 32 percent as large as the

cost of the structure portion, so kmt from equation 25 is multiplied by 1/1.32 for structures and

0.32/1.32 for land. (See Appendix C for further details.) In practice, that assumption affects only x

and the estimated growth rate of overall k ̄. 1 8That result assumes a given cost of corporate funds. If the cost of corporate funds is endogenous,

then a corporate income tax can affect the cost of capital even if new investment can be expensed.

17
are eliminated.) Net investment is assumed to be a weighted average of current and

past changes in desired capital (∆K∗

t ). Gross investment IP P is then given by

IP P

t = α

[

i

χi∆

pt−i Yt−i

vpp

t−i



+ δKt−1. (26)

In the absence of taxes, with η = −∞, x = 1, and one type of capital, gross investment

in the putty-clay model is

IP C

t = α

+[

i



βi∆L ̄t−i + γi∆

Yt−i

y ̄t−i

 + dept

,[

j

Gj

pt−j y ̄t−j

vP C

t−j

, (27)

where19

vP C

t = q

tr − tp ̇ − y ̇

1 − e− (tr− tp ̇−y ̇)S .

Several important differences between the putty-clay and putty-putty models can be

seen by comparing the two investment equations. Perhaps the most characteristic

difference between the models is that investment depends on the level of the real cost

of capital in the putty-clay model but on growth of the real cost of capital in the

putty-putty model. In a putty-clay world, the quality of existing machines is fixed

once they are in place, so firms cannot go back and remold existing machines when

interest rates, stock prices, or tax rates change. In a putty-putty world, however,

they can.

In the putty-clay model developed in this paper, the accelerator term depends on both

the growth of labor hours at full employment and the growth of output in excess

of labor productivity at full employment.20 That is, firms invest in both capital

and labor when they cannot meet the growth of demand solely by increasing the

productivity of their existing capital and labor. In the putty-putty model, however,

the accelerator term depends on the growth of real output, regardless of the rate of

growth of labor productivity. In that model, if real output is growing at a 3 percent

rate, firms feel the same need to increase physical capacity whether productivity is

also growing at a 3 percent rate or not growing at all. Another way of putting this

is that the impact of faster growth of output on investment in the putty-clay model

depends on whether that extra growth comes from faster growth of hours or from

faster growth of productivity, while in the putty-putty model the source of that extra

growth is irrelevant. Previous empirical work based on putty-clay capital has defined

the accelerator using the growth of output, the same as in the putty-putty model.

However, it would be incorrect to say that investment is independent of productivity

in the putty-clay model. The presence of y ̄ in the second summation in equation 27

1 9For simplicity, I leave the exp [(p ̇t + y ̇ − q ̇t)i/2] term out of equation 27. That means that Gj is

actually the product of the Gj from equation 25 and exp [(p ̇t + y ̇ − q ̇t)i/2]. 2 0The growth of output in excess of labor productivity at full employment is similar to the growth

of labor hours, which is the growth of output in excess of actual labor productivity.

18
means that investment is proportional to the level of productivity as long as higher

productivity is reflected in higher output. If demand keeps pace with technology,

investment depends on the level of productivity in the putty-clay model and on the

growth of productivity in the putty-putty model. If demand growth fails to keep pace

with productivity, improved technology can actually slow putty-clay investment in

the short run, a result found empirically by Basu, Fernald, and Kimball (2004).

The presence of the growth of labor hours at full employment in the putty-clay

investment equation reflects the assumption that firms expect actual labor hours to

gradually converge to labor hours at full employment. Hence, changes in labor hours

at full employment provide information about future employment and output, and

thus influence investment.

The models’ treatment of replacement demand differs in two ways. First, replacement

demand in the putty-putty model is a constant fraction of the existing stock while

in the putty-clay model it depends on the mix of vintages in the existing stock.

Geometric depreciation of the capital stock, as in the putty-putty model, is impossible

in the putty-clay model if productivity rises over time. Eventually, the productivity

of new machines becomes so much larger than that of existing machines that firms

willingly discard existing machines.

Second, replacement demand in the putty-putty model is tied to the original value

of the capital being replaced. If depreciating capital cost $1 billion in 2000 chained

dollars when it was purchased, then replacement demand is $1 billion in 2000 chained

dollars. In the putty-clay model, however, replacement demand is tied to the current

quality of new machines, not to the quality of depreciating machines. If productivity

has grown or the real cost of capital has fallen since the depreciating machines were

purchased, replacement demand will exceed the original value of the depreciating

machines.

The cost of capital also differs between the models. Geometric decay in the puttyputty model allows the rate of depreciation to be added linearly to the rate of return.

Fixed service lives in the putty-clay model lead to a more complex functional form

for the cost of capital. They also lead to a lower cost of capital for the same service

life, all else equal. Although the expected service lives are the same, the present

discounted value of the marginal output from an extra dollar of investment (in the

putty-clay case, an extra dollar of machine quality) is greater in the case of fixed

service lives, meaning a lower cost of capital is needed for the firm to break even.

In the putty-clay cost of capital, the expected rate of growth of nominal labor productivity (p ̇ + y ̇) is subtracted from the nominal corporate cost of funds (r). The

marginal revenue of an extra dollar of machine quality rises over time with the price

of output and with labor productivity. Expectations of future growth have an important effect on investment. In a putty-putty world, however, the future value of

an extra dollar of investment today depends on the future price of new investment.

Thus, the marginal revenue of an extra dollar of investment grows at the rate of the

19
price index for investment.

Changes in the desired mix of capital can have important effects on investment by

type of machine in the putty-clay model. An increase in the desired share of typem machines in overall capital produces a large transitory increase in investment in

type-m capital (from a larger ∆

 NmK

t

NK

t



in equation 24) and a smaller permanent

increase (from a larger NmK

t

NK

t

and depmt in equation 24).

2.2 The Rate of Return in Steady State

To examine the properties of the putty-clay and putty-putty models along the steadystate growth path, assume exogenous saving rate s, p ̇ = q ̇, no taxes, η = −∞, and

x = 1. In steady state, Y/L and  ̄k grow at the same rate y ̇ in both models. In the

standard neoclassical model of Solow (1956), the real rate of return in steady state is

rrP P = α

s (y ̇ + δ + n ̇) − δ.

The rate of return in steady state for the model of this paper differs from that result

for two reasons: the putty-clay nature of capital and the use of a fixed service life

instead of geometric depreciation. To illuminate the source of differences with the

standard model, I solve the putty-clay model under the assumptions of both fixed

service life and constant failure rate.21

2.2.1 Putty-Clay: Constant Failure Rate

With a constant failure rate, the fraction of machines purchased at t and still in use

at time t + i is e−δi

. Firms choose kt such that

kt = α y ̄t

(rr + δ − y ̇)

. (28)

Saving equals investment, so sYt = Ntkt or

kt = s Yt

Nt

. (29)

With x = 1, labor hour equals the number of machines in operation, so

Lt =

] ∞

i=0

Nt−i e−δidi = Nt

δ + n ̇

. (30)

2 1As noted earlier, geometric depreciation is unrealistic in a putty-clay model with technological

progress, whether the depreciation is from reduced efficiency of machines or a constant failure rate.

Geometric depreciation is assumed only for illustrative purposes.

20
Combining equations 28 and 29, substituting for Nt using equation 30, replacing

Yt/Lt with y ̄t, and rearranging, we have

rrCF = α

s (δ + n ̇) − δ + y. ̇ (31)

In the absence of technological progress, the real rate of return is identical to that

in the putty-putty model. Firms have no desire to change the quality of existing

machines, so putty-putty and putty-clay are the same in steady state. However, if

there is technological progress and α/s > 1, a higher rate of return is needed to

equate investment with saving for putty-putty capital than for putty-clay capital.

As y ̇ increases for a given r, the additional investment to increase the quality of old

machines in the putty-putty framework exceeds the additional investment to build

higher-quality new machines to take advantage of expected rising productivity in the

putty-clay framework.

2.2.2 Putty-Clay: Fixed Service Life

With no taxes, fixed service life, and the price of output equal to the price of new

capital, equations 14 and 15 can be combined as

kt = α y ̄t

1 − e−(rr−y ̇)S

rr − y ̇ .

In steady state with x = 1,

Lt =

] S

i=0

Nt−i di =

] S

i=0

Nt e−ni ̇ di = Nt

1 − e−nS ̇

n ̇ . (32)

Following the same steps as in the case of a constant failure rate, we obtain the

expression

rrP C − y ̇

1 − e−(rrP C −y ̇)S = α

s

n ̇

1 − e−nS ̇ . (33)

2.2.3 A Numerical Example

Since an analytic solution for rrP C is impossible, this section uses a numerical example

to examine steady-state properties of the model. I look at five changes that each

produce a 1.5 percentage point increase in rrPP : an increase in y ̇, an increase in n ̇ ,

a reduction in S (equivalent to an increase in δ), a reduction in s, and an increase in

α.

In the base case, y ̇ = 0, S = 20, n ̇ = 0.01, s = 0.2, and α = 0.3. With no technological

progress, rrPP is identical to rrCF (see Table 2). A higher rate of return is needed

21
Table 2: The Rate of Return in Steady State

Rate of Return (rr, in percent)

Parameters Putty-Clay

Constant Fixed

y ̇ n ̇ S (1/δ) s α Putty-Putty Failure Rate Service Life

0 0.01 20 0.2 0.3 4.00 4.00 5.55

0.01 0.01 20 0.2 0.3 5.50 5.00 6.55

0 0.02 20 0.2 0.3 5.50 5.50 6.73

0 0.01 12.5 0.2 0.3 5.50 5.50 8.17

0 0.01 20 0.1714 0.3 5.50 5.50 7.50

0 0.01 20 0.2 0.35 5.50 5.50 7.50

in the putty-clay model with fixed service life to equate investment with saving. As

long as rr is positive, the present discounted value of the additional revenues from an

extra dollar of machine quality today is greater for fixed service life than for constant

failure rate, i.e.,

] S

i=0

e−rr∗idi > ] ∞

i=0

e−(rr+1/S)idi for rr > 0.

An increase in y ̇ from zero to 0.01 raises r by only 1.0 percentage point in the puttyclay models, no matter what type of depreciation is used. With α, s, n ̇ , and S

unchanged, rr − y ̇ is unchanged, so any increase in y ̇ produces a comparable increase

in rr. Firms cannot change the quality of existing machines, so a larger expected

increase in y ̄ increases only the quality of new machines. With putty-putty capital,

however, firms want to increase the quality of every existing machine as y ̄ rises. A

larger increase in rr is needed to crowd out the extra investment in that model.

The smaller positive impact of productivity growth on the real rate of return in the

putty-clay model helps to explain why low real interest rates coincided with rapid

productivity growth in the early 2000s. Kroszner (2006) sketches the standard rationale for effects of productivity growth on the real interest rate in a putty-putty

framework and then notes that interest rates have been very low in the early 2000s

given rapid productivity growth. That result is less surprising in a putty-clay framework, especially if above-average growth is not expected to persist over the service

life of new capital.

In every other case, the results for the putty-putty model are identical to those for

the putty-clay model with constant failure rate. In those cases, it is the assumption

of fixed service life that causes differences between the putty-putty and putty-clay

models. An increase in n ̇ raises rr by 1.18 percentage points with fixed service lives,

less than with constant failure rate. With constant failure rate, the ratio of replacement machines to the total is always 1/S. However, with fixed service lives, the ratio

of replacement machines–machines put in place S years ago–to the total number of

22
machines declines as n ̇ increases.22 Because replacement demand is relatively smaller,

so is rr.

Note that the effect of higher output growth in the putty-clay model depends on

whether it comes from more rapid growth in labor hours or in productivity. In the

putty-putty model, the source of faster growth is irrelevant.

A reduction in S and equivalent rise in δ raise rr significantly more with fixed service

life than with constant failure rate. In either model, a more rapid rate of depreciation

raises replacement demand, and thus raises rr. However, that initial increase in rr

increases the disparity in the PDV of additional revenues from an extra dollar of

machine quality between fixed service life and constant failure rate, requiring a larger

increase in rr for fixed service life.

A reduction in s and an increase in α have about the same effect on rr in each model.

While the absolute rise in rr is somewhat larger in the putty-clay model than in the

putty-putty model, the percentage increase is slightly smaller.

3 Data

This section provides an overview of the data used in estimating the equations for investment and a more detailed discussion of the estimated time series for x. Appendix

C discusses the other data in more depth.

I estimate equations for five categories of private nonresidential fixed investment:

computers and peripheral equipment; communication equipment; software; other producers’ durable equipment excluding agricultural equipment and mining and oilfield

machinery; and nonresidential structures excluding farm and mining exploration,

shafts, and wells. The model’s assumption that the difference between actual output

and output at full employment is correlated from one year to the next does not hold

for farming, so investment in agricultural machinery and farm structures is not included. Excluding mining-specific capital (mining and oilfield machinery and mining

exploration, shafts, and wells), the ratio of capital to output in mining is about the

same as in the other industries using nonresidential fixed capital. Consequently, investment in mining-specific capital is not included in the investment equations and

is instead treated as a change in x. Such investment fits the model poorly anyway,

since the desired output of this industry is determined by the relative price of mining

output rather than by aggregate demand.

The sectoral coverage of output and labor hours is defined to be consistent with

investment. Labor hours at full employment (L ̄) and output per hour at full employment (y ̄) are not directly observable, and are estimated with Kalman filters, as

2 2Mathematically, equation 32 implies that the ratio of replacement machines to total machines,

Nt/Lt − n, equals n

1−e−nS − n with fixed service lives. That expression is decreasing in n.

23
discussed in Appendix C.

The real rate of return is a weighted average of the cost of debt and the cost of

equity for nonfinancial corporate business. The cost of equity is calculated using

the dividend-discount model. That method is less than ideal but necessary, given

that other measures rely on estimates of profits and thus of depreciation that are

inconsistent with the putty-clay model. The expected growth rate of y ̄ (y ̇) at every

time t is set equal to 1.91 percent, the coefficient of time in a regression of ln (y) on

time during 1949-2005. Hence, this paper does not estimate the impact of variations

in y ̇ on investment.

The pattern of retirements of capital follows the treatment used by the Bureau of

Labor Statistics (BLS). In U.S. Department of Labor (1983), BLS assumes that

retirements of a given vintage of fixed capital follow a truncated normal distribution

centered on the expected service life and ranging from 0.02 to 1.98 times that service

life. The standard deviation of the density function is 0.49 times the service life.

In contrast to BLS, however, retirements are measured in numbers of machines, not

constant dollars.

I assume that the observed values for xo and for each ratio NmK/NoK, where NoK

denotes all machines except mining-specific capital and inventories, equal a desired

value, used by firms to determine the number of new machines in equation 24, plus

an error. Observed xo equals actual NoK divided by a rough estimate of businesses’

forecast of labor hours when orders for those machines were placed.23 I assume

α



1 + 1

η



equals 0.26. (A larger value for α would result in a proportionately smaller

estimate of xo.) The desired values of NmK/NoK are estimated by applying a

Hodrick-Prescott filter to the raw data for 1954 to the end of the sample. To isolate

long-run trends, the filter for xo uses more smoothing than is standard.24

Raw and filtered data for xo from 1929 through the third quarter of 2006 are shown

in Figure 1. The most striking feature of the data is the large drop in xo during

the Great Depression and World War II and the rebound of xo back near its 1929

level in the postwar period. An obvious explanation is that businesses extended the

service lives of capital beyond their normal length from the early 1930s through the

early 1950s. During the Great Depression, firms may have used hoarded labor to

perform extra maintenance on existing plant and equipment. During World War II,

shortages likely prevented businesses from purchasing as much new capital as they

would have liked, forcing them to further extend the service lives of existing capital.

With shortages at an end after World War II, businesses aggressively replaced the

obsolete capital they had been unable to replace earlier. Because of the unusual

2 3That estimate is mov13 

L ̄∗ 0.6+0.4 ∗ mov13 

L∗/L ̄∗, where mov13(y) denotes a 13-quarter

moving average of y and L∗ and L ̄∗ are two-quarter moving averages of L and L ̄, respectively. The

coefficient of 0.4 on mov13 L∗/L ̄∗

is broadly consistent with estimates of the γmi. 2 4For xo, λ equals 6400, twice the normal amount of smoothing. Changing the amount of smoothing

for the Nkm/Nko affects the individual investment equations, but has little impact on the overall

fit.

24
character of business investment during the early postwar period, I begin the sample

period for the estimation of investment in 1957.

One other important feature of xo is a rise from about 0.95 in the late 1970s to

more than 1.0 by the late 1980s. The adoption of the personal computer, which

occurred during that period, can explain much of the increase, but the amount spent

on computers is too small to explain the entire change in xo.

One interesting possibility is that xo depends on the expected volatility of demand.

The derivation of investment earlier in this paper assumes that, under uncertainty, the

PDV of profits calculated using expected values of the variables equals expected PDV

of profits. That assumption is reasonable for revenues, labor costs, and investment

costs. However, the costs of utilization are highly nonlinear. An increase in expected

volatility raises the expected cost of utilization, because the increase in the cost of

high utilization is greater than the reduction in the cost of low utilization. That will

cause firms to increase both the ratio of machines to labor hours (NK/L), which the

model interprets as an increase in x, and the ratio of p to w ̄.

Figure 2 shows total x excluding mining and computers. That measure of machines

per hour rose steadily from the late 1960s until the late 1980s, perhaps reflecting

an increase in expected volatility of output, and has fallen steadily since the early

1990s as expected volatility has declined. Under that interpretation, greater expected

volatility caused firms to target a higher average level of excess capacity in order to

avoid the costs of periods of high utilization. The cost of maintaining that excess

capacity also led firms to raise the ratio of p to w ̄, putting upward pressure on

inflation.

4 Estimation

The model fits the data well, and the estimated parameters generally conform to the

theory. However, the handling of machines per labor hour has important effects on

the fit of the model. We can reject the implications of the neoclassical model for

the role of the cost of capital and the form of the accelerator. A basic q model of

investment has little effect on the results.

4.1 Results of Estimating the Investment Equations

The model to be estimated is the system composed of equations 24 to 25 with xo

substituted for x and NoK for NK. The goal is to determine the βmi, γmi, and Gmi

that produce the best fit. Estimating those parameters jointly in a single equation

for each type of capital proves troublesome. The sum of the Gmi should be 1 for each

type of capital, but is significantly less than 1 for each type of capital when freely

estimated. Hence, the Gmi must be treated separately from the βmi and γmi.

25
The estimation procedure for each type of capital is to first choose a set of Gmi,

then to solve equation 25 for kmt, then to divide real investment by kmt to get Nmt,

and finally to estimate equation 24 to obtain the βmi and γmi. The objective is to

minimize the error in equation 24. I constrain the Gmi to sum to 1 for each type of

capital. To impose some structure on the choice of the Gmi, I use moving averages.

The fact that contemporaneous xot is multiplied by various lags of the L ̄t−i and

Yt−i/y ̄t−i rules out polynomial distributed lags for the βmi and γmi. Instead, I use

moving averages. No constraint is imposed on the sums of the βmi and γmi.

Equation 24 can be estimated using ordinary least squares (OLS) if the dependent

variable, Nmt−∆

 NmK

t

NoK

t



L ̄t−1 xot−1 −depmt, is stationary. While one might expect

this expression to drift upward over time, it has in fact been negative in recent years,

so I test for a unit root using an augmented Dickey-Fuller test assuming a constant

and no drift. Using one-sided p values, the probability of a unit root is 0.5 percent for

software, 1.6 percent for computers and peripherals, 3.6 percent for communications

equipment, 16.0 percent for other equipment, and 38.7 percent for structures. Table

3 shows the results using OLS. Corrections for autocorrelation are discussed below.

The top panel of Table 3 shows the Gmi used in equation 25 to calculate the quality

of new machines (km). The remainder of the table shows the results from estimating

equation 24 for the number of new machines (Nm) using least squares. Consistent

with the theory, those equations are estimated without a constant.25 Because of

their volatility, the variables L ̄t and Yt/y ̄t are replaced by their two-quarter moving

averages.

The results generally confirm two implications of the theory. First, the lag length for

the number of new machines exceeds the lag length for the quality of new machines.

Second, in general, the longer the time to build, the greater the weight on labor

hours at full employment (S

i βmi) and the less the weight on actual labor hours

(

S

i

S

γmi). Thus, hours at full employment receive more than 70 percent of estimated

i βmi + γmi for investment in structures, but have no role in the equation for

computers. Software is an anomaly, with a long time to build but with a zero weight

on labor hours at full employment. That equation has a low R2, however, warranting

some scepticism.

Based on the discussion above, we expect S

i βmi + γmi to be slightly greater than

1. The estimated coefficients for communications equipment, other equipment, and

structures are consistent with that expectation. However, the sum of coefficients

appears too large for computers and software.

Taken together, the equations fit the data well (see Figure 3).26 The model generally

2 5Estimates of standard errors in the table are calculated assuming that xot and the Gmi are

given. Because they are instead estimated, the standard errors in the table may be biased. 2 6 In Figure 3, the fitted value of nominal investment for each type of capital equals the fitted

value for Nm times the actual value for km times the actual price index (qm). Those fitted values

for nominal investment are then summed. For the categories of investment created by aggregating

26
Table 3: Least Squares Results for Investment Equations

Telecom. Other

Computers Software Equipment Equipment Structures

km (Eq. 25)

Lags (i) 0 0 to 3 0 to 3 0 to 8 1 to 4

Sum of Gmi 1 1 1 1 0.7

Lags (i) 5 to 8

Sum of Gmi 0.3

Nm (Eq. 24)

Lags (i) 1 to 5 0 to 9 0 to 7

Sum of βmi 0.276 0.611 0.426

(t-statistic) (5.3) (25.6) (19.5)

Lags (i) 6 to 10 8 to 14

Sum of βmi 0.285 0.342

S

(t-statistic) (5.5) (17.2)

i βmi 0 0 0.561 0.611 0.767

Lags (i) 0 to 1 2 to 6 2 to 7 0 to 6 0 to 1

Sum of γmi 0.159 0.388 0.278 0.260 0.015

(t-statistic) (2.6) (7.3) (12.6) (20.5) (3.1)

Lags (i) 2 to 5 7 to 13 7 to 14 7 to 14 2 to 8

Sum of γmi 0.734 0.398 0.199 0.180 0.148

(t-statistic) (9.6) (6.6) (7.8) (12.6) (18.1)

Lags (i) 6 to 9 14 to 18 9 to 16

Sum of γmi 0.280 0.353 0.112

S

(t-statistic) (4.4) (7.2) (12.6)

i

S

γmi 1.173 1.139 0.477 0.440 0.276

i βmi + γmi 1.173 1.139 1.038 1.051 1.043

R2 0.646 0.612 0.814 0.911 0.959

D.W. 0.65 0.18 0.42 0.41 0.29

Sample 1965:q1- 1965:q1- 1957:q1- 1957:q1- 1957:q1-

2006:q3 2006:q3 2006:q3 2006:q3 2006:q3

27
captures the timing and magnitude of changes in investment. One notable exception

is a growing overestimate after 2004. In the model, the shortfall of investment after

2004 is the counterpart to the decline in raw xo during that period shown in Figure

1.

When the equations for other equipment and structures are freely estimated in first

differences, the estimated S

i βmi + γmi are an unrealistically low 0.48 and 0.59,

respectively. Hence, I constrain those sums to equal the sums of coefficients obtained

using OLS. The resulting estimates of γmi and βmi are each within 0.04 of the OLS

estimates for other equipment and within 0.02 of the OLS estimates for structures.

When the equations are estimated with an AR1 correction, the estimated βmi and

γmi are not significantly different than the OLS estimates for computers and communications equipment. However, estimated S

i βmi+γmi is less than 0.78 for the other

three categories of investment. When I constrain S

i βmi + γmi for those categories

to be the same as for OLS, most individual coefficients are not significantly different

from the OLS estimates.

4.2 Effect of Holding Machines per Labor Hour Constant

Estimating the model while holding xo constant at its sample average worsens the

model’s performance. Estimates of S

i βmi+γmi are well above 1 for each category of

investment, ranging from 1.114 for communications equipment to 1.259 for computers.

In addition, the model does not fit the data as well (see Figure 4). Fitted values are

well below actual investment during periods when xo is estimated to have risen–the

late 1950s and 1980-1986–and are even further above actual investment during the

period after 2002, when xo is estimated to have been falling. Even so, the model still

picks up turning points fairly well.

4.3 Tests of Key Implications of the Neoclassical Model

The most characteristic difference between the putty-clay model and the standard

neoclassical model is the treatment of ∆ (p/vm), the change in the reciprocal of

the real cost of capital. That term has no effect on investment in the putty-clay

model, but a large impact on investment in the neoclassical model. We can test that

implication of the neoclassical model by adding

NmK

t

NoK

t

[

i

φi L ̄t xot

∆ (pt−i/vmt−i)

pt−i/vmt−i

to equation 24. That is the additional term one gets after dividing a variant of equation 26 containing NmK/NoK and xo by equation 25. According to the neoclassical

finer detail–other equipment and structures–I use an aggregate km. That variable is created by

inserting an aggregate vm into equation 25 and then multiplying the resulting km∗ by the coefficient

obtained from a regression of Im/Nm on km∗.

28
model, S

i φi should be roughly 1. According to the putty-clay model, it should be

zero.

The results are generally inconsistent with the neoclassical model. For computers and

software, S

i φi is negative no matter what specification is used. For communications

equipment and other equipment, S

i φi is small and statistically insignificant. For

structures, S

i φi is statistically significant, but quite small (0.019 for an eight-quarter

moving average). That structures fall so far from the neoclassical model is perhaps

surprising. Although the labor requirement of many types of equipment, for example

motor vehicles, is obviously fixed during their service life, the labor requirement of

a building can be changed through remodeling. However, the very small estimate

of S

i φi indicates that such remodelings in response to changes in the real cost of

capital play little role in explaining investment in structures.

The neoclassical model also implies that the accelerator term should depend on

changes in output alone rather than on changes in the ratio of output to labor productivity. We can test that implication by adding

NmK

t

NoK

t

[

i

+

βm

i

∆ L ̄t−i y ̄t−i xot



y ̄t−i

+ γm

i

∆ [Yt−i xot]

y ̄t−i

,

to equation 24. If the neoclassical hypothesis about the form of the accelerator is

correct, the quantity S

i βm

i +γmi should roughly equal 1 and S

i βmi +γmi should

equal zero. Instead, the estimated S

i βm

i+γmi are negative for each type of capital.

Consequently, we can reject the neoclassical hypothesis that the accelerator depends

on changes in output and not on changes in the ratio of output to labor productivity.

4.4 Tests of a Basic q Model of Investment with Cash Flow

Fazzari, Hubbard, and Petersen (1988) found that cash flow is an important determinant of investment, especially for firms that face financing constraints.27 They

proposed adding cash flow to a model of investment based on Tobin’s q:

It/Kt = μ +[

i

gi Qt−i +[

i

hi (CFt−i/Kt−i), (34)

where I is nonresidential fixed investment, K is the capital stock, Q is Tobin’s q, and

CF is cash flow gross of dividends. K is measured as tangible assets at current cost.

Q is defined as (V + B − MI) /K, where V is the market value of equity, B is the

market value of debt, and MI is the market value of inventories.

I estimated equation 34 using aggregate variables, with I defined to exclude capital

specific to mining and farming.28 Consistent with q theory, S

i gi is positive. In

2 7Although Kaplan and Zingales (1997) argue that cash flow is not a good measure of financing

constraints, they still find it highly significant in determining investment. 2 8 I estimate equation 34 using a lagged eight-quarter third-degree polynomial distributed lag for

both Tobin’s q and cash flow.

29
addition, S

i hi equals 0.69, meaning that each dollar of additional cash flow increases

investment by 69 cents. One might expect some of the effects of cash flow to spill over

to investment in mining capital and inventories, but those variables have an incorrect

positive sign when added to the right-hand side of equation 34.

To account for the putty-clay model, I added Ipc

t /Kt to equation 34, where Ipct is the

fitted value of nominal investment from the putty-clay model. (To be consistent, I,

K, and CF are all nominal.) After that variable is added, S

i gi becomes negative so

Q is removed from the regressions. With or without Q, cash flow fades in importance

once fitted values for investment from the putty-clay model are added. In every

specification, the estimated sum of coefficients on cash flow, S

i hi, is less than 0.025.

F-tests indicate that cash flow is significant at the 95 percent confidence level, but

that effect is perverse: rising cash flow significantly reduces the level of investment.

Thus, this simple version of the cash flow model adds little explanatory power to

the putty-clay model of investment, and Q has the wrong sign. If a firm does not

need more capacity, it does not build more capacity, no matter what cash flow or the

market value of existing capacity is.

There are a few caveats to those results. First, recent papers examining the q model

with cash flow have used firm-level data, not the aggregate data used in this section.

Second, the fitted values for investment in the putty-clay model take x as exogenous.

To the extent that cash flow may help to explain x, or to the extent that I have

not incorporated a variable like x in the cash flow model, cash flow may still have

a role in explaining investment. Finally, although I do not develop it, the model

in this paper implies that changes in market share (f j) have important effects on

which firms invest. (For an individual firm, a 1 percent increase in market share has

the same effect as a 1 percent rise in aggregate demand.) Cash flow may thus be

significant in investment equations using firm-level data as a proxy for changes in a

firm’s market share.

5 Estimates of Capital per Labor Hour and the Capital

Stock

The ratio of capital to labor plays an important role in determining labor productivity

in Solow’s (1957) well-known framework for growth accounting. Traditional measures

of the ratio of capital to labor hours based on homogeneous (putty-putty) capital

cannot distinguish between capital deepening–an increase in the amount of capital

used by a given worker that adds to labor productivity–and capital widening–

an increase in the capital stock to accommodate an increased number of workers.

In contrast, the index of capital per hour in the putty-clay model excludes capital

widening, and thus can be used to isolate the true impact of investment on labor

productivity.

30
The failure of traditional measures of the ratio of capital to labor to exclude changes

unrelated to capital deepening leads to counterintuitive movements in that ratio.

Consider the response to a positive shock to demand. In the very short run, labor

hours rise, albeit less than proportionately to output because of procyclical productivity, while capital is nearly unaffected. Thus, the shock raises labor productivity

(output per hour) but reduces standard measures of the ratio of capital to labor.

After a year or two, labor hours have adjusted to the higher level of output for the

most part, eliminating the procyclical gain in productivity. The capital stock is also

larger than it would be without the shock, but it has not yet risen in proportion

to the increase in output or hours, so the capital-output ratio remains lower than

what it would have been in the absence of the shock to demand. Thus, a positive

shock to demand reduces the growth rate of standard measures of the ratio of capital

to labor. At the same time, dividing the capital stock by hours at full employment

overestimates capital deepening because the shock boosts investment but not hours

at full employment.

Those patterns are visible in the data on labor productivity and capital per labor hour

published by the BLS for the nonfarm business sector (see Figure 5).29 For 1949-2005,

there is actually a slight negative correlation between the growth of labor productivity

and the growth of what theory says should be one of its primary determinants, capital

per labor hour. While growth of labor productivity slows during recessions, capital

per labor hour grows at its most rapid pace during and immediately after recessions.

The putty-clay index of capital per labor hour instead isolates capital deepening. The

primary component of that index,  ̄k, depends only on the growth of the quality of

machines. The other component, x, is constructed to focus on trends in the number

of machines per labor hour at full employment. Consequently, growth in the index

of capital per labor hour does not show any obvious cyclical movements (see Figure

6).30 The strongest growth of capital per labor hour occurs during the late 1990s in

response to rapid growth of TFP and a low cost of corporate funds due to the boom

in stock prices.

The putty-clay model can be used to construct a capital stock consistent with labor

hours at full employment. To convert the putty-clay index of capital per labor hour

into a capital stock, I multiply by a two-year moving average of labor hours at full

employment. For the most part, that measure of the capital eliminates the slowdown

in growth after every recession in the BLS measure of the capital stock (see Figure 7).

The more rapid growth of the putty-clay capital stock than of the putty-putty capital

2 9At present, BLS has published data for the nonfarm business sector on a NAICS basis only for

1987-2004. To this series, I splice data for the business sector in 2005 and data for the nonfarm

business sector on an SIC basis for 1948-1986. BLS estimates include the output, capital, and

labor hours of nonfarm tenant housing, while the putty-clay estimates do not. The impact of those

differences in coverage on growth rates is small.

3 0As mentioned above, the figure excludes data from before 1957 because of the uncertainty of

how to measure x during a period when businesses were still rebuilding their capital stocks after

constraints on investment during World War II.

31
stock during 2002-2003 makes it somewhat less difficult to explain the unusually

strong growth of labor productivity during those years.

6 Capital Income and Investment

Weak to moderate business investment in the presence of strong corporate profits

since 2003 has puzzled many analysts. According to the q model of investment

with cash flow, unusually high levels of profits and cash flow should translate into

unusually strong business investment, limiting the rise in business saving. Privatesector economists and other researchers (e.g., Cardarelli and Ueda (2006), Hatzius et

al. (2006), and Loeys et al. (2005)) have suggested that business investment instead

remains moderate relative to GDP due to excessive investment in the late 1990s.

However, Desai and Goolsbee (2004) argue that capital overhang was not a factor in

the collapse of investment in the early 2000s.

An alternative view, based on the model in this paper, is that investment is driven

primarily by the need to expand the stock of productive resources, not by the profitability of resources already in use. In that view, any factor that reduces the need

to increase the number of machines, such as a slowdown in the growth of the labor

force at full employment, will reduce investment relative to capital income.

6.1 Capital Income

Variations in x can affect capital’s share of income. Modifying equation 14 for type-m

capital and multiplying through by NmK

t vmt yields

NmK

t vmt kmt = α



1 + 1

η



pt yt NmK

t

= α



1 +

1

η



pt

Yt

Lt

NmK

t .

In the putty-putty model, the left-hand side of the equation exactly equals the income earned by type-m capital. In the putty-clay model, that result is only approximate, but it is empirically useful. Summing over all types of capital and replacing

S

m NmK

t /Lt with xt, we have

income of tangible capital ≈ α



1 +

1

η



xt ptYt.

If total income is ptYt and income from intangible capital is −1

η ptYt, then

labor income ≈



1 +

1

η



[1 − αxt] pt Yt.

32
Dividing through this equation by pt Yt produces

wt

ptyt

≈ 1 +

1

η − α



1 +

1

η



xt, (35)

where w is (labor income) /L, or compensation per hour. The left-hand side of the

equation is labor’s share of income.

Because equation 35 is only approximate and because we do not know 1/η, the

equation must be estimated. If firms set prices using y ̄ rather than y, the left-hand

side of equation 35 should be w/ (py ̄). (The ratio y/y ̄ is then added as an independent

variable.) If real compensation per hour is slow to react to changes in productivity,

wt will also depend on current and lagged yt/yt−1. In addition, because a large share

of mining income goes to capital, high energy prices reduce real compensation per

hour. I find that mining investment as a share of nominal output is a better proxy for

the impact of energy prices on w than are energy prices, having a strongly significant

negative coefficient in the estimated equation.

The fitted value for capital’s share of income equals 1 minus the fitted value of

equation 35.31 The equation captures most of the important movements in capital’s

share of income but underpredicts in the early 1970s, possibly due to price controls,

and overpredicts in the mid 1990s and the early 2000s (see Figure 8).

6.2 Capital Income Minus Investment

Subtracting the model’s prediction for investment from its prediction for capital income (the nonlabor income of private nonfarm business plus nonprofits less tenant

housing) yields the model’s prediction for capital income minus investment (see Figure 9).32 I focus on capital income less investment rather than on retained earnings

less net investment, a more common measure of the financial surplus of business,

because the putty-clay model implies nothing about profits’ share of total capital

income or about the share of profits paid out as dividends.

The model explains high levels of capital income less investment in the 1960s, low

levels in the late 1970s and early 1980s, and the sharp rise in the level during the early

2000s. Most of that rise during the early 2000s stemmed from a sharp slowdown in

the growth of labor supply, measured as labor hours at full employment (L ̄). Figure

10 shows the impact on capital income less investment of letting L ̄ grow at its average

3 1Labor income is measured as compensation of employees less that of general government, government enterprises, farming, and housing, plus 65 percent of nonfarm proprietors’ income. That

sum is multiplied by the ratio of GDP to gross domestic income in order to make income consistent

with output used to estimate y ̄. The independent variables are a constant, yt/y ̄t, nominal mining

investment divided by pty ̄tLt, and a 12-quarter fourth degree PDL of yt/yt−1. 3 2Values for mining investment are actuals rather than fitted values from a model. However, since

mining investment is also the proxy for the effect of high energy prices on capital income, the net

effect of mining investment on capital income less investment is not large.

33
growth rate over 1954-2006.33 Rapid growth of L ̄ added to investment in the late

1970s and early 1980s, reducing the financial surplus of business, while slow growth

of L ̄ raised that surplus in the early 2000s. In fact, differing rates of growth of labor

hours at full employment between 1978-1982 and 2003-2004 explain almost the entire

5.5 percentage point difference in capital income less investment as a share of output

between those two periods.

Why has the uninvested portion of capital income remained so high since 2004 when

the model indicates it should have declined? Part of the explanation lies in a widening

gap between actual investment and the level of investment predicted by the model.

More important, however, has been a sharp rise in capital’s share of income. The

putty-clay model can explain why that increase in income has not directly aided

investment, but cannot explain the rise in capital income itself.

While it is difficult to explain the abrupt rise in capital’s share of income between

2003 and 2006, the general rise in capital’s share of income since the early 1990s

may be due to an increasing importance of intangible capital, such as patents and

firm-specific know-how. The model would capture this as a reduction in the absolute

value of η. Intangible capital may have grown in importance with the increased usage

of information technology. For example, in its fiscal 2006, Microsoft Corporation accounted for more than 0.6 percent of capital income (pretax profits plus depreciation)

but less than 0.1 percent of services from tangible capital.34 Other explanations for

a rise in capital’s share of income are also possible.

3 3The interval for calculating the growth rate begins in 1954 because lagged growth of L ̄ affects

capital income less investment. I also change the growth rate of Y /y ̄ by the same percentage amount

as the growth rate of L ̄. 3 4Microsoft’s depreciation of tangible capital was 0.08 percent of total depreciation of nonfarm

nonresidential capital.

34
Appendix A: Inventories

Although inventories have a very short service life and their cost is closely tied to the

cost of funds, the real stock of inventories has a much larger short-term response to

output than to the cost of funds, just like fixed capital. Inventories seem to have a

putty-clay quality that lasts much longer than the service life of an inventory.

Even so, inventories are modelled somewhat differently than fixed capital. In contrast

to a “machine” of fixed capital, the quality of which is fixed over its service life, an

inventory “machine” varies in quality with labor productivity at full employment (y ̄)

during its service life. Conceptually, a warehouse and the shelves in the warehouse

are of a fixed size during their service life, but the value of each item on those shelves

rises with labor productivity. Thus, an inventory “machine,” or unit of inventory

technology, consists of a certain amount of inventories held per real dollar of output

per labor hour, denoted k@y, rather than an amount of inventories per labor hour.

The amount of inventories corresponding to that “machine” is k@y · y ̄.

If the cost of holding inventories is the after-tax real cost of funds times the value of

inventories, then the PDV of the portion of current and future profits that depends

on the type of inventory technology adopted at time t (k@yt) is

PDV (π)t =

] Sm

i=0

(1 − ut+i) pt+i Yt+ie−ridi

−

] Sm

i=0

(1 − ut+i) Nmt qmt+i k@yt y ̄t+i rt+i e−ridi,

where Sm is the expected service life of new inventory technology, Nmt is the number

of new units of inventory technology adopted at time t, and qmt+i is the price index

for inventories at time t + i. We can solve for k@yt by: maximizing PDV (π)t over

k@yt; making the same assumptions about u, Y , and r as for fixed capital; assuming

that the expected rate of growth of qm is the same as that for p; substituting a

constant r for rt+i; and solving the integrals. After rearranging terms,

k@yt = αpt

r qmt

.

Inventory technology is proportional to the ratio of the price of output to the price

of inventories and inversely proportional to the real cost of funds expected over the

life of the technology.

35
Appendix B: The Cost of Capital with Taxes

With taxes, the present discounted value of future after-tax profits for the jth firm is

PDV (πj)t =

]∞

i=0

(1 − ut+i) pjt+iY jt+ie−ridi −

]∞

i=0

(1 − ut+i)  ̄wt+iLjt+ie−ridi

−[∞

i=0

qt+ikjt+iNjt+i



1 − Ct+i − Zu

t+i



e−ri

−[∞

i=0

(1 − ut+i) propjt+ie−ridi

−

]∞

i=0

1

2 (1 − ut+i) ξt+iw ̄t+iLjt+i

#

1 − NjK

t+i

xt+iLjt+i

ψ

$2

e−ridi, (36)

where u is the tax rate on corporate income, C is the investment tax credit for new

capital, and prop is the PDV of property taxes to be levied on new structures.35

The price of output pj excludes all taxes on production and imports except property

taxes, since those other indirect taxes are not part of the after-tax revenue of the

firm and do not affect the relative costs of capital and labor. Zu

t , the PDV of the tax

value of depreciation allowances per dollar of capital installed at time t, is given by

Zu

t ≡

]

St

i=0

ut+i Dt,i (1 − Bt Ct) e−ridi,

where Dt,i is the share of depreciation allowances taken i years after time of purchase

t, and Bt is the share of the investment tax credit that is deducted from the allowable

base for depreciation. (The service lives of some types of capital have changed over

time, so the service life of capital installed at time t is denoted St.)

The PDV of property taxes levied on new structures installed at time t, propjt, is

given by

propjt =

]

St

i=0

uxt+iNjt qt kjt



1 − i

St



e(p ̇+y ̇) i

e−ridi,

where ux is the property tax rate. The expression assumes that the property tax

base of a structure declines linearly over its service life subject to a rising trend as

nominal output per labor hour rises. Property taxes are deductible from corporate

income taxes, and so are multiplied by 1 − ut+i in the equation for after-tax profits.

3 5The equations in this appendix assume a single type of capital. If there are many types of capital,

the formula for the cost of capital remains the same except that the variables C, Zu, D, and S are

specific to the type of capital.

36
The treatment of property taxes in this paper differs from that in the literature.

Empirical studies of investment–e.g., Hall and Jorgenson (1967), Bischoff (1971),

and Tevlin and Whelan (2003)–generally ignore taxes on production and imports

altogether. On the other hand, in U.S. Department of Labor (1983), the Bureau of

Labor Statistics includes all indirect taxes in constructing the cost of capital.

For structures, the first-order conditions for kj and Nj become

α



1 + 1

η

 ] St

i=0

(1 − ut+i) pjt+i

Y jt+i

NjK

t+i

1

kjt

xt+ie−ridi (37)

= qt (1 − Ct − Zu

t )

+

] St

i=0

(1 − ut+i) uxt+iqt



1 − i

St



e(p ̇+y ̇) i

e−ridi

and

] St

i=0

(1 − ut+i) ξt+i

w ̄t+i

xt+i

#

1 − NjK

t+i

xt+iLjt+i

ψ

$

ψe−ridi (38)

= qtkjt (1 − Ct − Zu

t )

+

] St

i=0

(1 − ut+i) uxt+iqtkjt



1 − i

St



e(p ̇+y ̇) i

e−ridi.

Combining equations 37 and 38 gives the same result as combining equations 9 and

10, because the added tax terms cancel out, so it is still true that NK

t = xtLt.

To obtain the cost of capital, we solve equation 37 for kjt (equal to kt) using the

same assumptions as above. (The equation for machine quality is unchanged.) The

cost of capital is

vt = qt

tr − tp ̇ − y ̇

1 − e−(tr− tp ̇−y ̇)St

1 − Ct − utZt

1 − ut

for equipment and software (on which property taxes are not assessed), where

Zt ≡

]

St

i=0

Dt,i (1 − Bt Ct) e− tri di,

and for structures is

vt = qt

tr − tp ̇ − y ̇

1 − e−(tr− tp ̇−y ̇)St

1 − Ct − utZt

1 − ut

+qt uxt

 1

1 − e−(tr− tp ̇−y ̇)St − 1

(tr − tp ̇ − y ̇) St



.

For land, the present discounted value of profits includes the discounted value of land

when it is sold. (If the firm reuses the land, that notional sales value cancels out the

37
future purchase price of land.) Assuming that the sales price of land is expected to

grow at the rate tp ̇ + y ̇, equation 36 has an added term:

+[∞

i=0

(1 − ut+i) qt+ikjt+iNjt+ie−(r−p ̇−y ̇)St e−ri.

In addition, the base for property taxes does not decline during the service life of

land, so

propjt =

]

St

i=0

uxt+iNjt qt kjt e(p ̇+y ̇) ie−ridi.

Solving the first-order conditions for maximization of profits with respect to kjt, we

find

vt = qt

tr − tp ̇ − y ̇

1 − e−(tr− tp ̇−y ̇)St

1 − Ct − utZt

1 − ut

+ qt uxt

−qt e−(tr− tp ̇−y ̇)St tr − tp ̇ − y ̇

1 − e−(tr− tp ̇−y ̇)St

.

The final right-hand-side term comes from the present discounted value of the sales

price of the land.

Appendix C: Data

Price indexes and real and nominal quantities for 49 detailed categories of private

nonresidential fixed investment are taken from the National Income and Product

Accounts of the Bureau of Economic Analysis (BEA) for 1929 on, and from BEA’s

fixed asset tables for earlier years. (The latter data are necessary for calculating the

number of machines depreciating during the sample period.) For years before 1901,

necessary for calculating retirements of certain long-lived assets, I use 1901 stocks

and assume a steady-state growth path before 1901 using growth rates from Gordon

(1999). Price indexes and real investment for the two aggregate investment categories

are obtained using the formula for chain weighting.

Nominal output is gross value added for the nonfarm business and nonprofit institution sectors less gross value added of nonfarm tenant housing, output of government

enterprises, and nonproperty taxes on production and imports, all from BEA. Nonfarm tenant housing and government enterprises are excluded because their investment is not part of nonresidential fixed investment. Taxes on production and imports

other than property taxes are not part of the after-tax revenue of the firm and do

not affect the relative costs of capital and labor. Property taxes are excluded from

the equation used to chain real output, meaning firms treat higher property taxes as

a reduction in the net price they receive for output.

38
Table 4: Kalman Filter Estimate of Labor Hours at Full Employment

Parameter Coefficient z-Statistic

c1 -31.0 -19.5

c2 -11.6 -7.0

c3 11.6 7.0

c4 0.0086 2.8

c5 0.48 2.4

The sample is quarterly, from 1947:q2 to 2006:q1.

The sectoral coverage of labor hours is the same as that for output. Labor hours for

nonfarm business and nonprofit institutions come from the Bureau of Labor Statistics.

Hours for government enterprises are from BEA. I use compensation and proprietors’

income in housing sector output to calculate labor hours for nonfarm tenant housing.

Labor hours at full employment, an unobservable variable, are estimated using a

Kalman filter combining two equations. First, the difference between the unemployment rate (ru) and the unemployment rate at full employment (ru ̄ ) depends

negatively on the current and lagged ratios of actual labor hours to L ̄ (increases in

labor demand) and positively on growth in L ̄ (increases in labor supply):

rut − ru ̄ t = c1 ln Lt

L ̄t



+ c2 ln Lt−1

L ̄t−1



+ c3 ln  L ̄t

L ̄t−1



, (39)

where c1<0, c2<0, and c3>0. Second, labor hours at full employment follow

ln L ̄t



= SVt + c4 t + c5 ln (NPt), (40)

where the state variable SV is a random walk, t is a time trend rising by 0.25 per

quarter, and NP is the population aged 16 and older. Data for ru ̄ are from the

Congressional Budget Office (CBO). Coefficients obtained in estimating the Kalman

filter defined by equations 39 and 40 are shown in Table 4.

To estimate the unobservable variable y ̄, I start by assuming that

Yt

Y ̄t

= Lt

L ̄t

Ut, (41)

where U is cyclical intensity of effort and Y ̄ ≡ y ̄L ̄. Intensity of effort depends on

current and lagged values of Y /Y ̄ :

Ut =

Yt

Y ̄t

b1 Yt−1

Y ̄t−1

b2 Yt−2

Y ̄t−2

b3 Yt−3

Y ̄t−3

1−b1−b2−b3

, (42)

where b1>0, b2<0, and b3<0. Intensity of effort reverts to unity within a year after a

shock to Y /Y ̄ absent additional shocks. Combining equations 41 and 42, substituting

y ̄L ̄ for Y ̄ , and rearranging, we can express ln Lt/L ̄t

 as a function of current and

lagged ln Yt/



L ̄t y ̄t

.

39
Table 5: Kalman Filter Estimate of Productivity at Full Employment

Parameter Estimated Coefficient z-Statistic

b1 0.448 26.2

b2 -0.230 -15.2

b3 -0.127 -7.5

b4 0.021 8.9

The sample is quarterly, from 1948:q1 to 2006:q2.

A signal equation for the Kalman filter is created by substituting for y ̄t in that

expression using

ln ( ̄yt) = svt + b4 t,

where the state variable svt follows a random walk. Estimated coefficients are shown

in Table 5.

The cost of capital is calculated as if all investment is done by nonfinancial corporate

business. The methods of depreciation (straight-line, accelerated, sum of digits, and

expensing), tax lifetimes, and declining-balance parameters used to calculate the

Dmt,i are taken from Gravelle (1994), as are the basis adjustment B and investment

credit rates Cm. The methods of depreciation used also account for the temporary

increases in the amount of expensing allowed under the Job Creation and Worker

Assistance Act of 2002 (JCWAA) and the Jobs and Growth Tax Relief Reconciliation

Act of 2003 (JGTRRA).

I assume that nonprofit institutions use expensing, which is equivalent to setting the

corporate tax rate for those entities to zero. To implement that assumption, I treat

65 percent of medical equipment and instruments, all new religious, educational, and

vocational structures, and 75 percent of new hospitals and special-care structures as

purchased by nonprofits and thus expensed. For mining, depreciation depends on the

value of depletion, which is based on the output of the mine independent of its tax

lifetime. Consequently, I substitute the service life for the tax lifetime. Also, since

the real output of a well declines over time, I assume accelerated depreciation with a

150 percent declining balance.

The numerator of the property tax rate is total state and local property taxes less

taxes on production and imports for farms and housing, from BEA. The denominator,

taken from the flow-of-funds accounts, is the value of nonresidential real estate held

by nonfarm, noncorporate business plus the nonresidential share of real estate in

nonfarm, nonfinancial corporate business.

Conceptually, the numerator of the nominal rate of return is payments by business to

owners of debt and equity (including retained earnings) after taxes plus the revaluation of assets due to inflation, while the denominator is the market value of debt and

40
equity.36 The real rate of return is the nominal rate of return less the rate of growth

of output prices. In a putty-clay world, assets are revalued according to the change

in the value of what they produce, i.e., the rate of growth of output prices. Thus, the

revaluation of assets due to inflation and the rate of growth of output prices are the

same, and the numerator of the real rate of return is payments to debt and equity

after taxes, including retained earnings.37 Consequently, the real rate of return is a

weighted average of the after-tax bond yield and the yield on equities.

The pretax bond yield is an average of the Moody’s seasoned corporate bond yields

on Aaa- and on Baa-rated corporate debt. Interest payments are deductible from

corporate taxes, so the after-tax bond yield is the pretax yield times 1 − ut, where

ut = uft + uslt ∗ (1 − uft),

uft is the federal statutory corporate income tax rate, and uslt is the effective average

state and local corporate tax rate, multiplied by (1 − uft) because such taxes are

deductible from federal corporate income taxes. Data for the federal statutory rate

come from Gravelle (1994) and CBO. The state and local tax rate equals state and

local corporate tax collections divided by corporate profits before tax for domestic

industries. The same ut is used in calculating the cost of capital vt.

The after-tax yield on equity is calculated using the dividend-discount model. That

yield equals the dividend yield for the nonfarm, nonfinancial business sector divided

by the average historical ratio of dividends to after-tax profits in that sector from

1947 through 2005, 0.514. To adjust for a surge in earnings repatriated from abroad

in 2005, which count as negative dividends, I splice nonfarm, nonfinancial dividends

with net corporate dividends.

The weights on debt and equity are constructed from data in the flow-of-funds accounts of the Federal Reserve Board. Debt is the value of credit-market instruments

of nonfarm, nonfinancial business, whereas equity is the market value of equities

of nonfarm, nonfinancial business. I use two-quarter moving averages because the

flow-of-funds data are for the end of the quarter.

I assume that businesses do not discount the entire future stream of revenues from

a long-lived asset using only the current rate of return. Instead, the expected real

rate of return for type-m capital (trrm) is a weighted average of the current rate of

return (rrt) and the historical average rate of return rr ̄ t, with the weights depending

on the service life of type-m capital. For capital with service life of five years or less,

3 6The expression “after taxes” is necessary because interest payments are deductible from the tax

base for corporate income. 3 7The equality of the real rate of return with expected payments to capital net of taxes is consistent

with the Bureau of Labor Statistics’ methodology for creating the cost of capital, as described in

Appendix C of U.S. Department of Labor (1983). Similarly, Harper, Berndt, and Wood (1989) show

that using an “internal own rate of return model” defined using expected payments to capital net of

taxes “yields the same rental prices ... as would the nominal internal rate of return model provided

average capital gains were employed.” The putty-clay model satisfies that condition.

41
trrm = rrt. For longer-lived capital,

trrm = rrt +



1 − 5

Smt



1 + ln Smt

5

 ( ̄rrt − rrt).

The weight on rrt is 0.85 when Sm = 10, 0.60 when Sm = 20, and 0.33 when

Sm = 50. Given the sharp and apparently permanent drop in rrt in the early 1950s,

I assume rr ̄ t is 7.35 percent (the average for 1921-1950) through 1950 and then falls

linearly to 5.56 percent (the average for 1951-2005) by the end of 1954, where it

remains from 1955 on.

This paper does not estimate an equation for inventory investment, but the number

of units of inventory technology and their quality are necessary to calculate the index

of capital per labor hour and the capital stock. I assume that the current inventory

technology is a geometric average of technologies chosen over the previous 15 years.38

Then  ̄km for inventories is found by multiplying that expression by y ̄, so that

log 

 ̄

kmt



= log ( ̄yt) +[

59

i=0

log (k@yt−i)

60 .

The expected real cost of funds used to determine k@yt is a weighted average of the

current and average real cost of funds, 1

3 rrt + 23 rr ̄ t.

From 1978 on, tp ̇ equals the expected inflation rate on consumer purchases one year

ahead from the University of Michigan’s surveys of consumers less the average difference over that period between that rate and the growth rate of p. That expected

inflation rate is regressed on current and lagged inflation and on ru − ru ̄ . (Rational

consumers expect inflation to accelerate when unemployment is low, giving ru−ru ̄ a

negative coefficient.) That equation is used to calculate tp ̇ before 1978. The variable

tpm ̇ equals a weighted average of tp ̇ and the historical average growth of p using the

same weights as for trrm. The relationship between tp ̇ and unemployment seems

different before 1952, so ru − ru ̄ has a coefficient of zero during that period.

Service lives for most assets are from the Bureau of Economic Analysis (2003). The

service life for autos is taken from the U.S. Department of Labor (2001). The service

life for computers is derived by adjusting the depreciation rate implied by BEA data

for an assumed declining balance parameter.

The number of new machines, necessary for calculating retirements of machines, is

estimated by using equation 25 to solve for kmt and then dividing real investment

by that kmt. The lag structure in equation 25 is the same as that ultimately chosen

for each category of capital in the investment equations.

3 8The service life for inventory technology is likely related to the service life for fixed capital. I

choose 15 years as a round number that is somewhat larger than the service life of equipment but

well below the service life of structures. Ideally, the timing of the introduction of new inventory

technology should correspond to the timing of fixed investment, rather than be a simple moving

average. However, given the uncertainty of the service life of inventory technology, further refinement

will not necessarily lead to a more accurate answer.

42
Each structure “machine” includes both a structure and the land used with it. The

putty-clay model requires knowledge of the value of land associated with each structure when it was put in place, data we do not have. I assume that λ, the ratio of

the cost of a new structure to the cost of the accompanying land (abstracting from

expectations, the ratio of vm ∗ km for structures to vl ∗ kl for land), is constant over

time.39 Nm = Nl implies that Il/kl = Im/km, and so Il = Imλvm

vl . Given historical data for Im, vm, and vl, we can calculate the implied value of nonresidential

land in use for a hypothetical λ and compare it with estimates from BLS. The data

support a value for λ of roughly 0.32.

To calculate costs of capital for the two aggregate investment categories, we have

the choice of combining the individual vm or of using aggregated rm, Sm, etc. to

solve for an aggregate v. To more easily isolate the impacts of changes in the real

rate of return and tax law on investment, I choose the latter. I use contemporaneous

weights (i.e., chain weighting) to calculate qm. For other variables, however, the

use of contemporaneous weights can introduce large quarter-to-quarter movements,

especially for the tax variables. Instead, each weight is calculated by inserting the

H-P-filtered NmK/NoK and trend values for L ̄ and Y /y ̄ into equation 24, solving to

obtain a raw weight, and dividing by the sum of the raw weights. For tax lives and

service lives, weights are exponential rather than arithmetic.40 The weights on rr ̄ t

and rrt do not vary over time, but are set to the sample average. The desired number

of inventory “machines” per labor hour, used in the calculation of the index of capital

per hour, equals the filtered values of the ratio of the number of such machines to

1

5

S4

i=0 Lt−i/4 e(n ̇ +y ̇)/2 for quarterly data, where n ̇ and y ̇ are annual rates.

3 9 I assume that the price index for land is the same as that for new structures. Given that

assumption, the cost of capital for land is the same as the cost of capital for structures, with two

exceptions. First, because land does not depreciate, firms expect the basis for property taxes to grow

at a rate of tpm ̇ + y ̇. Second, the cost of land is reduced by the PDV of the after-tax sales value of

land when the structure on it is retired. 4 0For tax lives, each weight is multiplied by one minus the share of investment expensed, since the

tax life is irrelevant when there is expensing.

43
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46
1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

1.05

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

Figure 1. Machines per Labor Hour, Excluding Mining and Inventories, 1929-2006

Raw data Filtered data

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

1.040

1.030

1.020

1.010

1.000

0.990

0.980

0.970

Figure 2. Machines per Labor Hour, Excluding Mining and Computers, 1960-2000

47
1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

13

12

11

10

9

8

7

* Potential GDP is taken from unpublished data used for Congressional Budget Office (2007).

Figure 3. Business Fixed Investment, Excluding Mining and Farming, 1957-2006

(Percent of Nominal Potential GDP*)

Actual Fitted values from model

1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

13

12

11

10

9

8

7

6

* Potential GDP is taken from unpublished data used for Congressional Budget Office (2007).

Figure 4. Business Fixed Investment, Excluding Mining and Farming, 1957-2006

(Percent of Nominal Potential GDP*)

Actual Fitted values from model

Fitted values assuming constant machines per labor hour

48
1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

10

8

6

4

2

0

-2

-4

Figure 5. Putty-Putty Labor Productivity and Capital per Hour, 1949-2006

(Percent growth, nonfarm business)

Output per labor hour (BLS) Capital per labor hour (BLS)

0

1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

10

8

6

4

2

0

-2

-4

Figure 6. Two Measures of Growth of Capital per Labor Hour, 1957-2006

(Percent change)

Putty-clay Putty-putty (BLS)

0

49
1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

7

6

5

4

3

2

1

Figure 7. Two Measures of Growth of the Capital Stock

(Percent change)

Putty-clay Putty-putty (BLS)

1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

33

32

31

30

29

28

27

26

Figure 8. Capital's Share of Income

(Percent, private nonfarm business plus nonprofits less tenant housing)

Actual Fitted Value

50
1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

20

18

16

14

12

10

8

Figure 9. Capital Income Minus Investment

(Percent of private nonfarm business output plus nonprofits less tenant housing)

Actual Fitted values

1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

20

18

16

14

12

10

8

Figure 10. Effect of Labor Supply on Capital Income Minus Investment

(Percent of private nonfarm business output plus nonprofits less tenant housing)

Actual With labor hours at full employment growing at its trend rate

51