Working Paper Series

Congressional Budget Office

Washington, DC

Putty-Clay Capital and an

Index of Capital per Hour

Mark Lasky

Congressional Budget Office

Washington, DC

E-mail: mark.lasky@cbo.gov

June 2005

2005-08

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

discussion and critical comment. These papers 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/tech.cfm.
Abstract

This paper presents an index of capital per hour constructed using the assumption

of putty-clay capital. That index is based on an aggregate putty-clay production

function that both preserves the desirable empirical properties of putty-clay models

and yields a closed-form solution for aggregate investment as a function of aggregate

variables. The key to the production function is an index of capital per hour that

is a logarithmic average of the ratios of capital to labor embodied in each unit of

capital. That index separates out the capital-deepening portion of investment (more

capital per worker) from the capital-broadening portion (more capital to employ more

workers). The capital-deepening portion is identified using the cost of capital. The

putty-clay index of capital per worker shows stronger growth during 2001-2004 than a

putty-putty index, making the unusually strong growth of labor productivity during

that period somewhat less of a puzzle.

Unless otherwise noted, the figures and tables in this paper were created by the

author.
1 Introduction

Productivity has grown rapidly in the United States in recent years. From the first

quarter of 2001 to the fourth quarter of 2004, output per hour in the nonfarm business

sector grew at an annual rate of 4.1 percent, roughly double its 2.0 percent average

rate of growth during the preceding 40 years. Although productivity often grows at

above-average rates early in business-cycle expansions, its robust increase in recent

years has been unusual (Congressional Budget Office, 2004, pp. 26-27). Researchers

have suggested various explanations, including a delayed effect as firms learned how

to use new technologies first purchased in the late 1990s (Basu, Fernald, and Shapiro,

2001).

Such rapid productivity growth is even more unusual because it has coincided with

unusually weak investment in new plant and equipment. Nonresidential fixed investment averaged 10.4 percent of gross domestic product (GDP) from the second

quarter of 2001 to the fourth quarter of 2004, below the 11.1 percent of GDP it

averaged during the preceding 40 years despite an upward trend over time.

The impact of investment on productivity depends on whether the investment goes

toward pure replacement demand, i.e., replacing depreciating capital with identical

units; capital deepening, i.e., increasing the amount of capital used by an individual

worker; or toward capital broadening, i.e., increasing the stock of capital to accommodate additional workers. Although both capital deepening and capital broadening

augment the economy’s capacity to produce output, capital deepening raises productivity while capital broadening does not.

The putty-putty model of capital traditionally used in growth accounting (e.g., Solow,

1956) offers little guidance on how to split investment into its capital-deepening and

capital-broadening components. Under the putty-putty assumption, also called ex

post variable proportions, businesses can vary the ratio of capital to labor both before

and after capital is put in place. Implicitly, all capital is always fully utilized. If

one uses actual labor hours as the labor input to production, as does the Bureau of

Labor Statistics in its measure of multifactor productivity (U.S. Department of Labor,

1997), then one may misinterpret a cyclical rise in labor hours that reduces the ratio

of capital to labor hours as a reduction in capital deepening.1 On the other hand, if

one corrects for this problem by using cyclically adjusted labor hours as the input to

production, as the Congressional Budget Office (CBO) does in its projections (2001),

one may misinterpret a cyclical rise in investment that raises capital per cyclically

adjusted labor hour as an increase in capital deepening.

1Firms can adjust labor hours more rapidly than they can adjust the capital stock, so an increase in demand initially raises the ratio of labor hours to capital. In the putty-putty model,

this reduces capital per labor hour, implying less capital deepening. The economic consulting firm

Macroeconomic Advisers (2001) offsets the implied reduction in productivity by enhancing the normal procyclicality of total factor productivity, which is equivalent to abandoning the putty-putty

assumption.

1
The putty-clay model of capital, originally developed by Johansen (1959), Solow

(1962), and Phelps (1963), allows us to separately identify capital broadening and

capital deepening in a way that is not distorted by the business cycle. Under the

putty-clay assumption, also called ex ante fixed proportions, a business can choose

among a wide variety of possible ratios of capital to labor before capital is ordered

(putty) but cannot alter that ratio once capital is put in place (clay). The dependence

of the capital-labor ratio on the cost of capital allows us to use the cost of capital

to identify capital deepening, leaving capital broadening as a residual. As an added

virtue, empirical estimates of the impact of the cost of capital on investment, such as

those surveyed by Chirinko (1993), are more consistent with putty-clay capital than

with putty-putty capital.

Past research has generally assumed that the production of individual units of capital, or "machines," is independent. That assumption has limited the usefulness of

putty-clay models. Instead, this paper relies on an aggregate putty-clay production

function. The key to that production function is an aggregate index of capital per

worker equal to the logarithmic average of the ratios of capital to labor across all

machines. Holding the ratio of capital to labor embodied in each machine constant

over the machine’s service life guarantees that investment retains the useful empirical

properties of other putty-clay models. At the same time, the use of an aggregate

index of capital per worker makes total factor productivity (TFP) of new capital

equal to that for all capital, eliminating the problem in many putty-clay models that

the TFP of new capital is independent of that of old capital and thus unobservable

from aggregate data.

The production function has other important implications for investment not examined empirically in this paper. The accelerator depends on growth in output minus

growth in productivity, rather than on growth in output alone. Replacement demand

depends on changes in the cost of capital since the depreciating capital was installed.

Investment depends on the expected future growth rate of productivity.

This paper focuses on the construction of an index of capital per labor hour. That

index allows us to distinguish capital deepening from capital broadening in a way not

distorted by the business cycle and thus to obtain a better measure of the impact of

investment on labor productivity. I find that capital deepening has remained strong

since the recession in 2001 despite the sharp slowdown in investment. That helps

make the rapid growth of productivity during 2001-2004 somewhat less of a puzzle.

Section 2 presents a basic version of the putty-clay model to illustrate its basic workings and its implications for investment. Section 3 makes several modifications to the

basic model to make it more realistic, allowing for varying utilization, multiple types

of capital, changes in the capital intensity of the production function, time to build,

taxes, a distribution of service lives, and a different treatment for inventories than

for fixed capital. Section 4 discusses two other potential modifications not included

in the model: permanent changes in machines used per labor hour and endogenous

service lives of capital. Section 5 discusses the data used to construct the index of

2
capital per hour. Section 6 discusses the resulting index and its implications for

productivity. Section 7 presents conclusions.

2 The Basic Model

This section presents a basic version of the putty-clay model to illustrate how it

works. That version uses several simplifying assumptions that are relaxed later in

the paper. There are no taxes. The rate of return is expected to remain at its

current level. There is a single type of capital. There is no time to build capital,

and there are no costs of adjustment, so the number of machines always equals the

number of employed workers.

The model employs several assumptions that are not unique to putty-clay models. I

assume that production can be represented by a single production function producing

one intermediate good. However, in order for the price of capital goods to differ from

the average price of all output produced, I assume that the single intermediate good

can then be turned into a range of final goods and services whose average price is

equal to the price of the intermediate good. The amount of each of those goods and

services that can be produced from the intermediate good is inversely proportional

to its price. I assume that all workers are equally productive.

Firms are competitive and thus price-takers. All firms are identical, so that the profitmaximization decision can be analyzed using the economywide production function.

The production function exhibits constant returns to scale.

Several additional assumptions are specific to the putty-clay model. I assume that

the capital stock is composed of individual units I will call "machines." Ultimately,

the labor input into production is proportional to aggregate hours worked, so the

index described in this paper is referred to as an index of capital per hour. However,

for expositional purposes, it is easier to describe the labor input as being composed

of individual workers. Hence, the actual labor input to production is full-timeequivalent workers, or hours worked per year divided by 2000. In the basic model,

each such worker uses one machine. As discussed in the next section, short-run

changes in hours have the same effect on production whether they come from changes

in employment or changes in the workweek.

Firms choose the quality of new machines on the basis of prices, output per worker,

and expectations at the time they are purchased. Although the productivity of a

machine can change over time with overall technology and with the quality of other

machines being used, the quality of each machine and its associated labor input

remain unchanged during its service life.

Because the machines are integral units, investment is assumed to take place in

discrete periods. However, to make the math easier, production is assumed to be

3
continuous.

2.1 The Production Function

At time t, the production function is

Yt = AtLt

 ̄

kα

t , (1)

where Y is total output, A is total factor productivity, L is the number of workers,

and  ̄k is the index of capital per hour. The workweek is constant over time, so

each machine is used a fixed number of hours. Technical progress is Hicks-neutral,

affecting the productivity of labor and capital equally. The production function

in equation 1 is no different from the standard putty-putty production function.2

The key difference between putty-clay and putty-putty lies in the specification of the

capital input.

Let the workers at time t and the machines they use be indexed 1 to Lt. I assume

that the aggregate index of capital per worker is a logarithmic average of the quality,

or size, of each machine ki used by worker i at time t. Thus,

log 

 ̄

kt



= [

Lt

i=1

log (ki)

Lt

, (2)

or

 ̄

kt = k1/Lt

1 k1/Lt2 ... k1/Lt

Lt .

Using this latter expression, we can rewrite the production function as

Yt = AtLtkα/Lt

1 kα/Lt2 ... kα/Lt

Lt .

From that equation, one can see that the putty-clay production function is actually

a huge Cobb-Douglas production function with each machine as a separate input to

production.

Since the determinants of the quality of each machine of a single vintage are the same,

we will instead differentiate machines by the time at which they were installed. In

that case, the index of capital per worker can be expressed as

log 

 ̄

kt



=

S

[−1

i=0

Nt−i log (kt−i)

SS−1

i=0 Nt−i

, (3)

where S is the service life of capital and Nt−i is the number of machines put in service

at time t−i. With full employment, the total number of machines in service at time

t

SS−1

i=0 Nt−i equals the number of workers Lt.

2 If we define Kt as k ̄tLt, then AtLtk ̄α

t = AtL1−α

t Kα

t .

4
Note that the production function exhibits constant returns to scale. If the number of

workers doubled, then a doubling of the number of each type of machine would double

the amount of capital being used, but  ̄k would remain constant. From equation 1,

it is obvious that output would also double.

2.2 Quality of New Machines

To determine the quality of machines of different vintages, we must examine a business’s investment decision. The present discounted value of the portion of a firm’s

profits that depends on the quality of new machines over their service life, multiplied

by the number of firms, is

πnt =

] S

i=0

pt+i Yt+i e−(tr+ tp ̇)i di − Nt qt kt, (4)

where p is the price of output, tr is the expected real rate of return over the service

life of capital put in place at time t, tp ̇ is the expected rate of increase of output

prices over the service life of capital put in place at time t, and q is the price index

of capital, or price per unit of quality. The price of a new machine is thus the price

index of capital times the quality of new capital qtkt. Each firm chooses the quality

of new machines kt to maximize profits.

Maximizing profits with respect to quality of new machine kt yields first-order conditions:

0 = α

] S

i=0

pt+i

Yt+i

 ̄

kt+i

Nt

SS−1

j=0 Nt+i−j

 ̄

kt+i

kt

e−(r+ tp ̇)i di − Nt qt.

3 (5)

Let y denote output per worker, and let ty ̇ denote the expected rate of growth of

output per worker over the service life of capital put in place at time t. Then,

because the number of machines in service equals the number of workers, yt+i =

Yt+i/

SS−1

j=0 Nt+i−j . After solving out the integral and rearranging terms, equation

5 can be rewritten as

kt = α pt yt

vt

, (6)

where

vt = qt

r − ty ̇

1 − e−(r− ty ̇)S . (7)

The quality of new machines is proportional to output per worker y, to capital’s

coefficient in the production function α, and to the ratio of the price of output to the

3The quality of new machines purchased today influences the quality of new machines purchased

tomorrow and so has a second-order effect on future output. However, by the envelope theorem,

that effect does not affect profits and so can be ignored.

5
cost of capital v. Quality is negatively related to the real rate of return and positively

related to the expected rate of growth of output per worker and to the service life.

A higher nominal rate of return increases the discounting of future revenues from a

machine, while higher rates of growth of output prices and output per worker raise

those revenues. The more rapid the expected rate of growth of output per worker,

the more a marginal unit of quality contributes to future output, and thus the greater

is the quality of new machines. The expression for the cost of capital in equation 7

is similar to that in Ando, Modigliani, Rasche, and Turnovsky (1974), except for the

inclusion of expected growth in productivity.

2.3 Investment

Real investment I equals the quality of new machines times the number of new

machines:

It = Nt kt. (8)

In this basic model, the number of new machines equals growth in the number of

workers plus the number of machines discarded, or

Nt = ∆Lt + Nt−S. (9)

We can also invert equation 8 to solve for the number of depreciating machines Nt−S

in terms of investment and machine quality in period t − S

Nt−S = It−S

kt−S

= vt−S It−S

α pt−S yt−S

(10)

Substituting equations 9 and 10 into equation 8, real investment is

It =



∆Lt +

It−S

kt−S



kt. (11)

Using the definition of y, we can also express real investment as

It =



∆

Yt

yt



+

It−S

kt−S



kt. (12)

Substituting for k from equation 6, real investment can also be expressed as

It = ∆

Yt

yt

 α pt yt

vt

+ It−S

pt yt /vt

pt−S yt−S /vt−S

,

where vt is defined by equation 7.

This basic model illustrates several properties of investment in a putty-clay world:

6
• Investment depends positively on growth of the number of workers employed

(from equation 11), or equivalently on growth of output in excess of productivity

(from equation 12).

• Nonetheless, investment depends positively on the level of productivity, which

is a determinant of the size of new machines.

• Replacement demand is not simply equal to the original value of the machines

currently depreciating (It−S), but also takes account of growth in the desired

size of new machines since the machines currently depreciating were put in place

(kt/kt−S).

• If investment has varied in the past, the replacement component of current

investment will exhibit replacement cycles, peaking S periods after past peaks

of investment.

• As in the standard investment model, investment is inversely related to the real

rate of return, and real investment is inversely proportional to the ratio of the

price of new capital to the price of output.

• Investment depends positively on the expected rate of future productivity growth.

3 Enhancements to the Basic Model

Estimating an index of capital per worker for the U.S. economy requires several

modifications of the basic model just presented. The number of hours a machine is

used per year may vary in the short run, and the number of machines utilized may

differ from the total number of machines. There are many different kinds of capital,

and their relative importance in production varies over time. The capital intensity

of production can vary over time with the importance of capital-intensive sectors.

Because of adjustment costs and the time it takes to plan and build new capacity,

the actual capital stock rarely equals the desired capital stock. Retirements of a

given vintage of capital do not all occur simultaneously at a fixed amount of time

after installation. The cost of capital must take account of taxes. Finally, inventories

must be handled differently than other capital.

3.1 Utilization of Machines

In the real world, the number of machines utilized is generally smaller than the total

number of machines. To address that issue, I normalize the quality of machines so

that, on average, the number of machines equals the number of full-time-equivalent

workers. Alternatively, I could normalize the quality of machines so that, on average,

the number of utilized machines equals the number of full-time-equivalent workers.

7
Neither desired investment nor the index of capital per hour would be affected by

that change.

Given that firms rarely use all their capital, which machines are idle? I assume that

the distribution of machines used is identical to the distribution of all machines, and

thus that capital per worker for utilized machines is identical to capital per worker for

all machines. That assumption differs from the more common assumption in puttyclay models, e.g., Solow (1962) and Gilchrist and Williams (1998), that firms use

only the most productive capital available. Although that assumption is perfectly

reasonable, it implies that, all else equal, capital per worker should be countercyclical.

In fact, however, labor productivity is procyclical. Incorporating the assumption

that firms only use their most productive capital just makes it harder to explain the

procyclicality of TFP.

The amount of output produced by a given stock of capital may vary in the short run

for reasons other than changes in TFP. Because capital adjusts more slowly to demand shocks than does employment, demand shocks will cause procyclical variations

in the ratio of workers to machines and thus in output per machine. For a given

number of workers, an increase in the average workweek will also increase output per

machine. Firms may also raise the output of a given stock of capital by increasing

the number of shifts.

In each of those cases, output varies proportionately with hours worked in the short

run. So, I assume that the labor input into production Lt is the number of full-timeequivalent workers and that variations in the workweek and the number of shifts

have the same impact on production as variations in the number of workers. I also

assume that firms expect all variations in the workweek and in the number of shifts

to be temporary. The next section explores the potential impact of relaxing that

assumption below.

3.2 More Than One Type of Capital

In the real world, there are many broad types of tangible capital–equipment and

software, nonresidential structures, residential structures, land, and inventories–

as well as intangible capital. Within most of those broader categories, in turn,

there are many different kinds of capital. This paper focuses on the nonresidential

tangible capital stock used by the nonfarm business and nonprofit institution sectors.

Thus, the paper analyzes four broad categories of capital: equipment and software,

nonresidential structures, (nonresidential) land, and inventories, indexed below as

J. Equipment and software and nonresidential structures are further broken into

numerous subcategories, indexed below as Mj for broader category j.

4

4The Bureau of Economic Analysis also publishes data on several different kinds of inventory.

This paper looks at the overall stock of real inventories and does not examine the effect of changes

in the composition of those inventories on capital per hour.

8
For simplicity of exposition, I assume that each worker uses one "machine" of each

of the broad categories: one piece of equipment or software, one structure, one plot

of land, and one unit of inventory. In reality, at a point in time, several workers

operate one airplane, while a worker who uses software also uses the computer running

that software. However, the model would behave identically if each worker used some

fraction of several different machines, as long as those fractions stayed constant during

the service life of that machine.

For machines of type m with service life Sm, the logarithm of capital per worker is

log 

 ̄

kmt



=

Sm

[−1

i=0

Nmt−i log (kmt−i)

SSm−1

i=0 Nmt−i

, (13)

where Nmt−i is the number of machines of type m put in place in period t and kmt−i

is the quality of those machines. We can then express the overall index of capital

per worker as

log 

 ̄

kt



= [

j∈J





αjt

αt

[

m∈Mj

S

Nkmt

m∈Mj Nkmt

log 

 ̄

kmt





, (14)

where Nkmt is the total number of machines of type m at time t,

SSm−1

i=0 Nmt−i.

Equation 14 only holds at a point in time. If the relative importance of different

types of capital changes, i.e., if the αj or the Nkmt/

S

m∈Mj Nkmt vary over time,

equation 14 cannot be used to calculate a time series for  ̄k. The reason is obvious if

we consider shifts in the relative importance of two types of machine, one for which

 ̄

km grows more rapidly than for the other because of different rates of growth in the

prices of the two types of machine. Depending on the year used to index the prices

used to calculate  ̄km, a shift in relative importances in the two types of machines

could add or subtract from the growth rate of the overall index of capital per worker.

The relative usage of different types of machines changes over time because firms have

the choice of many different production technologies given the same stock of capital.

At the margin, firms switch from one technology to another when the output from the

two technologies, given the existing stock of capital, is the same. Thus, in calculating

capital per worker, I assume that changes in the distribution of machines and in the

importance of capital in production (changes in αt) have no effect on output at the

time they occur.

To keep changes in the mix of machines and the importance of capital in production

from affecting output, we must add an extra "transition" term to equations 1 and

14. Equation 1 becomes

log (Yt) = log (At) + log (Lt) + αt log 

 ̄

kt



+ tranYt. (15)

If the change in output is

∆ log (Yt) = ∆ log (At) + ∆ log (Lt) + αt−1∆ log 

 ̄

kt



,

9
then

∆tranYt = −∆αt log 

 ̄

kt



.

Likewise, equation 14 becomes

log 

 ̄

kt



= [

j∈J





αjt

αt

[

m∈Mj

S

Nkmt

m∈Mj Nkmt

log 

 ̄

kmt





 + tranKt. (16)

If the change in the index of capital is

∆ log 

 ̄

kt



= [

j∈J





αjt−1

αt−1

[

m∈Mj

S

Nkmt−1

m∈Mj Nkmt−1

∆ log 

 ̄

kmt





,

then

∆tranKt = −

[

j∈J



∆

αjt

αt

 [

m∈Mj

S

Nkmt

m∈Mj Nkmt

log 

 ̄

kmt







−

[

j∈J





αjt−1

αt−1

[

m∈Mj

∆

#

S

Nkmt

m∈Mj Nkmt

$

log 

 ̄

kmt





.

The change in the overall logarithmic index of capital per worker is a weighted average

of the change in the logarithmic indexes of capital per worker for each type of capital.

3.3 Changes in the Mix of Goods Produced

The importance of capital in production can change either because of general changes

in the importance of capital in production, as discussed above, or because the mix

of output shifts toward goods and services whose production is more or less capitalintensive than average.5 An increase in the importance of capital in production

produces an increase in the desired quality of new machines, while a shift in the mix

of output changes the number of machines used per worker. The latter option is

particularly useful for modeling the mining sector, which uses much more capital per

worker than other sectors. A desire to raise mining output, for example when oil

prices are high, results in an increase in the number of mining "machines" rather

than a generalized increase in investment in all capital.

5Strictly speaking, shifts in the mix of output are impossible in the one-good model presented in

this paper. However, the actual modification introduced is not inconsistent with the model, and it

helps capture a genuine feature of the economy.

10
Let xj denote the number of excess machines per worker for broad category of capital

j due to output of the capital-intensive sector. For example, if mining uses 100

percent more nonresidential structures per dollar of output than other sectors and

mining accounts for 5 percent of output, then xj for nonresidential structures is

1.00*0.05, or 0.05. Equation 16 becomes

log 

 ̄

kt



= [

j∈J

%

αjt

αt

#

log (1 + xjt) +

S

m∈Mj Nkmt log 

 ̄

kmt



S

m∈Mjx Nkmt

$& + tranKt, (17)

where Mjx excludes mining. The (1 + xjt) term causes capital per worker  ̄

kt to

increase as xjt rises, i.e., as the number of machines per worker rises. The exclusion

of mining machines from the denominator of the second term on the right-hand side

magnifies the impact of an increase in the individual  ̄kmt on overall  ̄

kt because an

increase in the number of machines of type j per worker raises proportionately the

impact of an increase in the quality of all machines of type j. The change in the

index of capital is

∆ log k ̄t



= [

j∈J

%

αjt−1

αt−1

#

∆ log (1 + xjt) +

S

m∈Mj Nkmt−1∆ log 

 ̄

kmt



S

m∈Mjx Nkmt−1

$& , (18)

and the equation for tranKt changes correspondingly.

3.4 Time to Build

The basic model assumes that investment depends on changes in the desired number

of machines and their cost at the time that capital is put in place. In reality,

important lags exist between the time at which investment is determined and the

time it takes place, collectively called "time to build," following Kydland and Prescott

(1982). It takes time for businesses to determine the need for more capital and to

plan for how to meet that need. The existence of unfilled orders for capital goods

shows a gap between the time that such plans are made and the time equipment is

delivered. For structures, the time between ground-breaking and final completion

is considerable. Investment will also lag behind the changes in demand driving it

if capital is “lumpy,” i.e., if businesses do not adjust their capital stock until some

threshold level of adjustment is required.6 Lags also may be added to the investment

process if integrating several new factories into production during a short period of

time is more costly than integrating those same factories into production over a longer

period.

As a first step, assume that the interval between the beginning of the planning stage

and the final delivery of capital–the time to build–is T periods and that the economy

6For evidence that capital is lumpy, see Cooper, Haltiwanger, and Power (1999) and Doms and

Dunne (1998).

11
is always at full employment. Then the right-hand-side variables in equation 6 for

the quality of capital are expectations at time t − T.

The number of new machines is also determined by expectations at time t − T. The

total number of machines at time t, Nkt will equal the time t − T expectation of the

number of workers at time t

Nkt = t−TLt.

Let η represent expected growth in workers per period. Then

t−TLt = ηTLt−T .

Substituting the equation for t−TLt into the equation for Nkt and taking first differences, we have

∆Nkt = ηT ∆Lt−T .

The number of new machines at time t, Nt, then equals the desired change in machines

plus retirements at time t (which I assume are known at time t − T), or

Nt = ηT ∆Lt−T + Nt−S. (19)

Now assume that the economy is not always at full employment. In addition, assume

that the expected gap between actual employment and full employment L ̄ is a linear

function of the current gap between actual employment and full employment, i.e.,

t−TLt − t−TL ̄t = θ ηT 

Lt−T − L ̄t−T



,

where θ < 1. Also, let

t−TL ̄t = ηTL ̄t−T .

Then equation 19 becomes

Nt = ηT 

θ∆Lt−T + (1 − θ)∆L ̄t−T



+ Nt−S.

Finally, assume that the number of machines put in place at time t is determined

by expectations formed during periods t − T + 1 through t. Then, as a rough

approximation, we can say that

Nt =  ̄η



θ

Lt − Lt−T

T + (1 − θ)

L ̄t − L ̄t−T

T



+ Nt−S, (20)

where η ̄ is close to an average of the ηt+i

.

7

7 In equation 20, N is expressed at the same frequency as T. So if investment is measured at an

annual rate but the data are quarterly, the denominators of the fractions become T /4.

12
3.5 Retirements of Capital

The basic model assumes that all capital of a given vintage is retired at the same

time, a fixed period after it is purchased. In the real world, service lives vary by

type of capital and are spread over a long period of time.

I follow the treatment used by the Bureau of Labor Statistics (BLS). 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 (U.S. Department of Labor, 1983). BLS chose the value 0.98 because

Hulten and Wykoff (1981a and 1981b) found that some assets are considerably older

than the service lives estimated by the Bureau of Economic Analysis (BEA), and

also to take account of assets that are accidentally destroyed when new. In BLS’s

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

3.6 Taxes and the Cost of Capital

Corporate taxation affects the costs of all categories of fixed capital. In addition,

state and local property taxes are levied on land and structures.8 Each broad type

of capital has distinct features that require separate treatment.

3.6.1 Equipment and Software

Firms choose the quality of new equipment machines of type m, km, that maximizes

expected after-tax profits. The present discounted value of the portion of after-tax

profits that depends on the quality of new machines of type m purchased at time t is

πmt =

Sm

] t

i=0

(1 − ut+i) pt+i Yt+i e−(trm+ tpm ̇ )i di−Nmt qmt kmt (1 − Cmt − Zmu

t ),

(21)

where Smt is the average service life of capital of type m put in place at time t, u

is the combined federal and state and local tax rate on corporate income, trm is the

expected real rate of return over the service life of capital of type m installed at time

t, tpm ̇ is the expected rate of output price inflation over the service life of capital

of type m installed at time t, and Cm is the investment tax credit for new type-m

capital. The present discounted value of depreciation allowances per dollar of capital

8 Indirect taxes levied on output fall on both capital and labor, and thus I do not consider them

as part of the cost of capital.

13
of type m installed at time t, Zmu

t , is

Zmu

t ≡

Sm

] t

i=0

ut+i Dmt,i (1 − Bmt Cmt) e−(trm+ tpm ̇ )i di, (22)

where Dmt,i is the share of depreciation allowances taken i years after time of purchase t, and Bm is the share of the investment tax credit that is deducted from the

allowable base for depreciation.

To simplify the first-order conditions for maximization of 21, I assume that the expected rate of corporate taxation equals the current rate, so that ut replaces ut+i.

Rearranging the first-order conditions as in the basic model, we find that quality of

new machines of type m is

kmt = αjt pt y ̄t

vmt

, (23)

where

vmt ≡ qmt

trm − tym ̇

1 − e−(trm−tym ̇ )Smt

1 − Cmt − ut Zmt

(1 − ut) , (24)

Zmt ≡

Sm

] t

i=0

Dmt,i (1 − Bmt Cmt) e−(trm+ tpm ̇ )i di,

y ̄t is cyclically adjusted output per worker, αjt is the coefficient of equipment and

software in the production function, and tym ̇ is the expected rate of growth of output

per worker y during the service life of type-m capital installed at time t.

9 I use

cyclically adjusted output per worker to remove variations in output per worker that

firms know are temporary and thus ignore in making their investment decisions.

Adding time to build turns the right-hand side of equation 23 into a moving average.

For equipment, equation 24 must be modified to account for scrappage value, which

BEA counts as a subtraction from investment in equipment. Assuming that the

present discounted scrappage value as a fraction of the purchase price is Sc, equation

24 becomes

vmt ≡ qmt

trm − tym ̇

1 − e−(trm−tym ̇ )Smt

1 − Cmt − ut Zmt

(1 − ut) − Sc

. (25)

I use equation 24 for software, which has no scrappage value.

9 I assume firms expect the production function coefficient αjt to remain constant in the future.

If one relaxed that assumption, the −tym ̇ in equation 24 would become −tym ̇ −t α ̇ j.

14
3.6.2 Land

The fact that land does not depreciate but the structures on it do means that land

per worker is not permanently set at the rate at which the land is first put into service

but can change whenever an old structure depreciates and a new structure replaces

it. When land is first put into service, the expected profit function for that land

during the service life of the structure with which it is put into service is

πlt =

] Sl

i=0

(1 − ut+i) pt+i Yt+i e−(trl+ tpl ̇ )i di − Nlt qlt klt (1 − Clt − Zlu

t )

−

] Sl

i=0

uxt+i (1 − ut+i) Nlt qlt klt eΦl∗i e−(trl+ tpl ̇ )i di, (26)

where the suffix l denotes land, ux is the property tax rate (multiplied by 1 − ut+i

because property taxes are deductible against corporate income taxes), and Φl is the

rate of price appreciation of land over its service life.

Equation 26 must be modified to take account of land’s infinite service life. I assume

that land is depreciated for tax purposes only when it is first put into service. After

its first use, firms expect the marginal contribution of land to nominal revenues to

rise by the expected increase in the price of land, as in a putty-putty model, rather

than by the expected increase in nominal output per worker. Thus, the variable tyl ̇

is a function of expected growth in productivity during the service life of the first

structure and expected growth in the real price of land thereafter. To simplify the

expression for the cost of land, I assume that tΦl equals tyl ̇ +t pl ̇ .

Solving the first-order conditions, we find that the cost of capital for land is

vlt ≡ qlt (trl − tyl ̇ +  ̄uxt ∗ (1 − ut)) 1 − Clt − ut Zlt

(1 − ut) , (27)

where ux ̄ t is the expected property tax rate. Without the depreciation term, the cost

of land would be the real rate of return less real growth of the land’s productivity

plus property taxes.10

3.6.3 Structures

In calculating the expected value of property taxes, I assume that firms expect the

assessed value of a structure to rise with nominal output per worker but fall proportionately with the number of years remaining in its service life. Thus, in equation

26, the term eΦm∗i is replaced by (1 − i/Smt) e(ty ̇+tp ̇)i

. In addition, to account

for brokers’ commissions, I assume that purchasers of new structures pay brokers a

1 0The cost of capital for land is only used to calculate the importance of land in production αj. I

calculate the index of nonresidential land per worker using data from BLS.

15
commission at rate b, which is the same for all structures. The cost of capital for

structures is then

vmt = (1 + b) qmt

trm − tym ̇

1 − e−(trm−tym ̇ )Smt

1 − Cmt − ut Zmt

(1 − ut) (28)

+ (1 + b) qmt

 1

1 − e−(trm−tym ̇ )Smt − 1

Sm (trm − tym ̇ )



ux ̄ t.

3.7 Inventories

Because inventories are held only for a short period of time, it is tempting to assume

that firms can adjust the "quality" of inventories, i.e., the desired level of inventories

held per unit of output, very quickly. However, a look at the inventory-sales ratio

suggests that, once one removes the effects of the business cycle on sales, the ratio

responds very slowly, if at all, to changes in interest rates, which are the cost of

holding inventories. Instead, the desired level of inventories per unit of output is a

function of existing technology and capital and does not respond quickly to changes

in the rate of return.

Assume that each year, a fraction 1/Sj firms set the inventory to output ratio  ̃ky

that they will use for Sj years. Then those firms choose  ̃ky to maximize the profit

function

πjt =

] Sj

i=0

(1 − ut+i) pt+i Yt+i e−(trj+ tpj ̇ )i di

−

] Sj

i=0

(1 − ut+i) qjt+i  ̃

kyt

Yt+i

Sj rt+i e−(trj+ tpj ̇ )i di.

where the suffix j denotes inventories.11 I assume that the value of inventories rises

with the price of output during the time they are held, implying that the pre-tax cost

of inventories is the real interest rate times the value of inventories held. Similarly,

I assume that firms expect the price of inventories qj to rise at the same rate as the

price of output p over the period for which the inventory technology is being chosen.

Rearranging the first-order conditions, we find that

 ̃

kyt = αjt pt

trj qjt

. (29)

The desired ratio of inventories to output is proportional to the importance of inventories in production and to the ratio of the price of output to the price of inventories,

1 1As an approximation, I assume that firms expense inventories. Also, I use 1/Sj as an approximation for the share of workers using the inventory technology chosen at time t. Assuming the labor

force grows over time, the actual share would be somewhat larger at first and somewhat smaller just

before the inventory technology is replaced.

16
and inversely proportional to the expected real rate of return. The index of inventories per worker  ̄

kj is found by multiplying an Sj-year moving average of the desired

ratio of inventories to output  ̃kyt by output per worker yt.

3.8 Summary

Equation 18 is used to calculate the index of capital per worker. For equipment,

software, and structures, indexes of type-m capital per worker  ̄km are calculated

using equation 13, modified so that retirements of machines installed at time t are

determined by a truncated normal distribution centered on t + S. The indexes of

machine quality km used to construct those  ̄km are calculated using

kmt = 1

Tm

Tm

[−1

i=0

αjt−i pt−i y ̄t−i

vmt−i

.

12

The number of new machines Nm is found by modifying equation 8 for multiple types

of capital and inverting it:

Nmt = Imt/kmt,

where Im is real gross investment in type-m capital. The variable xj equals the

number of mining-specific machines divided by the total number of nonmining-specific

machines for equipment and software and for structures.13

The cost of capital vmt is defined using equation 25 for equipment, equation 24 for

software, and equation 28 for structures. I use BLS data to calculate land per

worker, as discussed below. Inventories per worker equal cyclically adjusted output

per worker y ̄t times a 15-year moving average of the desired ratio of inventories to

output ky ̃ t defined using equation 29.

4 Other Potential Modifications

Two other potential modifications to the basic model were not incorporated in the index of capital per hour: permanent changes in machines used per full-time-equivalent

worker and endogenous service lives of capital. This section discusses the theoretical

impacts of those modifications.

1 2 I assume that Tm is 4 for equipment and software and 5 for structures. 1 3Using investment data for 1997 from Meade, Rzeznik, and Robinson-Smith (2003), the mining

industries use about the same amount of nonmining-specific capital per dollar of output as other

industries. Thus, mining-specific investment, defined as mining and oilfield machinery and mining

exploration, shafts, and wells, captures the extra capital due to the greater capital intensity of mining.

17
4.1 Machines Used per Full-Time-Equivalent Worker

In the short run, a 1 percent rise in the average workweek, a 1 percent rise in the

number of shifts worked per machine, and a 1 percent rise in the number of workers

each have the same impact on output. However, once a firm expects those changes to

be permanent, they have different effects on investment. Changes in the workweek

and the number of shifts per machine affect the desired quality of new machines but

not the desired number, whereas changes in the number of workers affect the desired

number of new machines but not their quality.

For this section, we return to the assumptions of the basic model except for the

assumption that the number of machines equals the number of full-time-equivalent

workers. Let Wt be the number of workers, ht be hours per worker per year divided

by 2000, and γt be the number of shifts, or the number of workers per machine. Then

Lt = htWt and the number of machines in use is Wt/γt.

The labor input to production is the number of machines in use, or Wt/γt, times

annual hours of usage per machine (divided by 2000), or htγt. But that is just Lt,

the number of full-time-equivalent workers. So the production function is identical

to that in equation 1. The index of capital per hour is identical to that in equation

3. The only difference is that now the total number of machines need not equal the

number of full-time-equivalent workers.

As in the basic model, the firm desires to maximize the present discounted value of

profits as given by equation 4. The first-order conditions are identical to those in

the basic model, except that the desired number of machines is now Lt/ (htγt) rather

than Lt. Solving for the optimal quality of new machines, we have

kt = α pt yt ht γt

vt

,

where

vt = qt

r − ty ̇ − th ̇ − tγ ̇

1 − e−(r− ty ̇− th ̇ − tγ ̇)S ,

and th ̇ and tγ ̇ are the expected exponential rates of growth of the workweek and of

the number of shifts. The optimal quality of new machines is equal to that in the

basic model multiplied by the intensity of usage of machines as given by htγt.

The equation for real investment becomes

It = Nt kt = [∆ (Lt/ (htγt)) + Nt−S] kt.

Note that if current and expected h and γ equal 1, then investment collapses to that

in the basic model.

To illustrate the impact of those changes to the basic model, consider the impact of

a permanent reduction in the workweek ht that leaves total hours worked Lt and the

18
number of shifts γt unchanged and thus raises employment Wt. (Those assumptions

were presumably the intent of France’s introduction of a 35-hour workweek in 1998.)

The optimal quality of new machines kt falls proportionately to the change in ht. The

shorter the period of time a machine will be used, the less the incremental revenue

from using machines of higher quality. At the same time, the desired increase in the

number of machines ∆ (Lt/ (htγt)) is sharply higher in period t. The net result is

higher investment in period t but lower investment in subsequent periods. (In the

real world, time to build would spread the initial positive impact on investment over

a few years.)

In a steady state, however, the reduction in h reduces output and investment. To

simplify the steady-state analysis, assume that relative prices, TFP, the real rate

of return, labor hours, the workweek, and the number of shifts are all expected to

remain constant. Then kt =  ̄

kt. The number of new machines each period equals

the number of replacement machines, or Lt/ (htγt) /S. Real investment is

It = Lt

(htγt) S kt = α pt Ytvt S .

If output Y was unaffected by the reduction in hours, real investment would also

be unaffected. The increased number of new machines would exactly offset the

reduction in the quality of new machines. But the reduction in the workweek would

reduce  ̄k, which would in turn reduce output and thus investment. A given dollar of

investment is less effective in producing output as it is spread over more workers.

The results would be the same if we considered a reduction in the number of shifts γ

instead of a reduction in the workweek. The number of machines would rise, giving

a short-run boost to investment, but their average quality would decline more than

proportionately in the new steady state.

I do not incorporate hours per week or the number of machines per worker in the

model because the two variables are probably inversely correlated, and I have no data

on the latter. An inverse correlation would occur if the downward trend in weekly

hours is related to the increase in employment at retail establishments, including

restaurants. Such establishments frequently use part-time workers and employ multiple shifts. Taken alone, the downward trend in weekly hours implies that the rate

of growth in the number of machines is more rapid than estimated and the rate of

growth in capital per hour is slower than estimated.

4.2 Endogenous Service Lives

This paper assumes that service lives are exogenous. However, service lives could

be endogenous in two ways. First, the service life of a machine may depend on the

number of hours it has been used. A period of increased utilization would increase

wear and tear on capital and thus force firms to accelerate replacements. A period

of lower utilization would allow firms to stretch out the replacement schedule.

19
The assumption that service lives depend on utilization would not have large impacts on the index of capital per hour. Because firms expect that utilization after

installation will equal its historical average, the quality of new machines would be

unchanged by assuming that service lives depend on future utilization. However, the

pattern of retirements and thus the average age of the existing stock would be affected

somewhat. More important would be the implications for business-cycle theory. If

service lives depend on wear and tear, then the marginal cost of production is higher

than if service lives are exogenous.

Second, the service life could be determined by endogenous replacement, as suggested

by Feldstein and Rothschild (1974). If the optimal quality of new capital rises enough

relative to that of existing capital, the differential between the productivity of new

and old machines can become so great that it becomes economical for a firm to discard

existing machines and replace them with new ones. In economic terms, the rental

value of the old machines (which is different from the cost of capital in the putty-clay

model) becomes negative. A firm would actually have to be paid to continue to use

old machines rather than replace them. That is probably the situation for computers

and software.

One solution, not implemented in this paper, is to use a mix of those assumptions.

One might assume endogenous replacement for computers and software, service lives

that vary with usage for other equipment, and exogenous service lives for structures.

5 Data

This section describes the data used to construct an index of capital per worker.

Land and inventories are handled somewhat differently than equipment, software,

and nonresidential structures.

5.1 Fixed Investment

Constructing an index of capital per worker requires quarterly data for investment

and the price index of investment for each type of fixed capital shown in the appendix.

BEA publishes nominal investment and price indexes back to 1959 for most of the

required series. Real investment is obtained by dividing nominal investment by the

price index. BEA classifies investment in structures differently in the years after

1997 than in the years before. I use the pre-1997 classifications and splice them with

the post-1997 classifications using the 1997 data, which are available for both systems

of classification. For a few series that are not available separately all the way back

to 1959, such as the two categories of trucks, I apportion nominal investment in the

broader category (trucks, buses, and truck trailers for the two categories of trucks)

using the shares in the last year for which the components are available separately. In

20
those cases, I also assume the price indexes of the components move proportionately.

From 1947 to 1958, annual data are available for most of the detailed types of capital,

but quarterly data are only available for the broader categories listed in the appendix. Within each year, the price index and real investment of each subcategory is

assumed to move in the same way as the price index and real investment for the

broader category. Detailed series that are only available summed with other series

are disaggregated as described in the previous paragraph.

Annual data are available from BEA for 1929 to 1946. To create investment data

for the period before 1929, which are not currently available from BEA, I determined

the steady-state growth path of investment required to reproduce BEA estimates of

the stock of each type of capital in 1929, assuming depreciation rates as in Bureau

of Economic Analysis (2003) and annual real growth of 3.11 percent of output in the

nonfarm business sector from Gordon (1999).14

5.2 Output

Output and the labor input are chosen to match the capital input and thus cover

nonfarm business plus nonprofit institutions less value added for nonfarm tenant

housing. Data for real and nominal output (used to construct the price index pt) are

from BEA. To calculate real value added in nonfarm tenant housing prior to 1990,

I use the price index for consumption of nonfarm tenant housing.

5.3 Workers

BLS provides data on labor hours for the nonfarm business and nonprofit institution

sectors back to 1947. I extend labor hours back to 1929 using data on full-timeequivalent employees from BEA, and before 1929 using growth rates from Gordon

(1999). Labor hours for rental housing (a small number) are subtracted out using

data on compensation and proprietors’ income. The number of workers is the number

of full-time-equivalent workers, found by dividing labor hours by 2000.

As indicated by equation 20, investment is determined by a combination of growth

in workers and growth in the number of workers at full employment. The latter

variable is not directly observable, so I estimate it using a Kalman filter exploiting

the relationship between the unemployment rate and the demand for labor. Part

of the drop in unemployment resulting from increased demand for labor occurs with

a lag, because businesses initially obtain part of the desired increase in hours by

extending the workweek, hiring more workers later.

1 4For the steady-state growth rate of investment to equal the steady-state growth rate of output

requires that the cost of capital grow at the same rate as the price of output.

21
Table 1: Kalman Filter Estimate of Labor Hours at Full Employment

Parameter Estimated Coefficient z-Statistic

c1 0.0052 1.5

c2 0.77 3.5

c3 -30.2 -19.2

c4 -11.1 -6.2

c5 0.370 -4.9

The sample is quarterly, from 1946:q2 to 2004:q3.

I assume that labor hours at full employment are determined by

ln L ̄t



= SVt + c1t + c2 ln (NPt), (30)

where SV is the state variable, defined to be a random walk, t is a time trend equal

to 1 in the first quarter of 1901 and rising by 0.25 in each subsequent quarter, and

NP is the population aged 16 and older. The Kalman filter is estimated by assuming

that the difference between the unemployment rate ru and the unemployment rate at

full employment ru ̄ is determined by current and lagged ratios of actual labor hours

L to labor hours at full employment L ̄

rut − ru ̄ t = c3 ln Lt

L ̄t



+ c4 ln Lt−1

L ̄t−1



+ c5 (SVt − SVt−1), (31)

where c3>0, c4>0, and c5>0. The final term captures the temporary rise in unemployment stemming from a jump in labor force participation. Data for ru ̄ is from the

Congressional Budget Office (CBO). Coefficients obtained in estimating the Kalman

filter defined by equations 30 and 31 are shown in Table 1.

5.4 Output per Worker

In making their investment decisions, firms ignore variations in output per worker

that they know to be temporary. I assume that TFP At has a procyclical component

U, intensity of effort, that depends on current and lagged values of the ratio of output

to output at full employment:

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

, (32)

where Y ̄ is output at full employment, b1>0, b2<0, and b3<0. The assumption that

demand affects productivity has a long history in the literature (e.g., Kuh, 1965).

Initially, businesses meet a positive shock to demand by boosting both labor hours

and worker effort. Over time, a greater share of the increase in output is met through

additional labor hours. I assume that effort reverts back to pre-shock levels within

a year.

22
Table 2: Kalman Filter Estimate of Productivity at Full Employment

Parameter Estimated Coefficient z-Statistic

b1 0.432 25.0

b2 -0.226 -14.9

b4 0.0123 8.4

The sample is quarterly, from 1947:q1 to 2004:q3.

Intensity of effort enables the ratio of output to output at full employment to deviate

from the ratio of labor hours to labor hours at full employment:

Yt

Y ̄t

= Ut

Lt

L ̄t

.

Substituting for Ut from equation 32, replacing Y ̄t−i with L ̄t−iy ̄t−i, and rearranging,

we find

ln Lt

L ̄t



= (1 − b1) ln  Yt

L ̄ty ̄t



− b2 ln  Yt−1

L ̄t−1y ̄t−1



(33)

−b3 ln  Yt−2

L ̄t−2y ̄t−2



− (1 − b1 − b2 − b3) ln  Yt−3

L ̄t−3y ̄t−3



.

Using values of L ̄t obtained in the previous section, I estimated the unobservable y ̄t−i

employing a Kalman filter, where

ln ( ̄yt) = svt + b4t.

The state variable svt is assumed to follow a random walk. I assume that the

coefficient on the final term on the right-hand side of equation 32 is half as large as

the second-to-last term, i.e., (1 − b1 − b2 − b3) = b3/2. Estimated coefficients are

shown in Table 2.

5.5 Cost of Capital

The cost of capital vm has several determinants: the price index of new investment

(discussed above), the rate of return, expected growth of productivity, the investment

tax credit, the corporate tax rate, the treatment of depreciation allowances, the

property tax rate, the cost of brokers’ commissions, and the value of scrappage.

5.5.1 Rate of Return

Conceptually, the nominal rate of return is the rate of payments by business to capital

(including retained earnings), net of corporate and property taxes, plus the rate of

23
revaluation of assets due to inflation. The real rate of return is the nominal cost

of funds 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 real rate or return is just net

payments to capital.15

For corporations, the real rate of return is a weighted average of the after-tax yield on

debt and the after-tax yield on equity. I assume that the after-tax rate of return for

all legal forms of organization is the same as the after-tax rate of return for nonfarm

nonfinancial corporations.

During the period for which it is available (1970 to 2002), Moody’s average corporate

bond yield roughly equals the average of the yields on Aaa-rated corporate debt and

Baa-rated corporate debt, also from Moody’s. So, the after-tax yield on debt is

calculated as an average of the yields on Aaa-rated and Baa-rated corporate debt

times 1 − ut. The overall corporate tax rate ut is

ut = uft + uslt ∗ (1 − uft),

where 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 CBO and from Gravelle (1994). The state and local tax rate equals

state and local corporate tax collections divided by profits before tax for corporate

business.

The after-tax yield on equity is calculated using the dividend-discount model. According to that model, the after-tax yield on equity 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 the second quarter of

2004, 0.507.

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 each quarter. For years prior to 1952, when

flow-of-funds data are unavailable, the market value of equity is calculated using the

index for the S&P 500, while the market value of debt is calculated as a function

of nominal GDP and the ratio of business interest payments to the corporate bond

1 5The 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 meets that condition.

24
yield.

The rate of return as calculated above is probably not the correct rate to use for

long-lived capital. In particular, the rate of return on stocks is the average rate

expected to prevail over the average remaining service life of the existing stock of

tangible and intangible capital. For new capital with an expected service life beyond

that average, businesses could reasonably assume that the real cost of funds would

return to its historical average. For capital with expected service life greater than

10 years, the expected real rate of return is

trm = rt +



1 − 10

Smt

[1 + ln (Smt/10)]( ̄r − rt),

where rt is the current real rate of return and r ̄ is the historical average real rate of

return.16 Given the sharp and apparently permanent drop in real rates of return in

the early 1950s, I assume r ̄ is 7.65 percent (the 1921-1950 average) through 1950 and

then falls linearly to 5.60 percent (the 1951-2004 average) from 1955 on. For land, I

used the same real rate of return as for office buildings, including medical buildings.

For inventories, the expected real rate of return (the real cost of holding inventories)

over the 15-year lifetime of a new inventory technology is two-thirds of the current

real rate of return plus one-third of the long-term real rate of return.

The expected nominal rate of return used to discount depreciation allowances equals

the expected real rate plus the expected rate of growth of output prices. I assume

that expected inflation is a function of current and past inflation and the difference

between the unemployment rate and the full-employment unemployment rate. The

more positive the latter, the more people expect inflation to decelerate from past

rates. I first estimated a relationship with consumers’ expectations of inflation

from the University of Michigan survey of consumers as the dependent variable, and

then adjusted that relationship for the average historical difference between output

inflation and consumer price inflation:

tp ̇ = 0.48 + 0.41p ̇t + 0.21p ̇t−1 + 0.10p ̇t−2 + 0.09p ̇t−3 − 0.28(rut − ru ̄ t)

For long-lived capital, I made the same adjustment to expected inflation for reversion

to historical rates as I made to the real rate of return.

5.5.2 Tax Treatment

The methods of depreciation (straight-line, accelerated, sum of digits, and expensing), tax lifetimes, and declining-balance parameters used to calculate the timing of

depreciation allowances Dmt,i are taken from Gravelle (1994), as are the basis adjustment Bm and investment credit rates Cm. Data for the methods of depreciation

used are adjusted to account for the temporary increases in the amount of expensing

1 6The same formula is used to determine tr in equation 29, setting Sm equal to 15.

25
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

the same tax treatment for land as for commercial property.

I assume that nonprofit institutions and tax-exempt cooperatives use expensing,

which is mathematically equivalent to setting the corporate tax rate for those entities to zero. I assume that all new religious, educational, and vocational structures

are purchased by nonprofits and that 75 percent of new hospitals and special-care

structures and medical equipment and instruments are purchased by nonprofits.

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. I assume that the expected property tax

rate at time t ux ̄ t is an average of the current property tax rate and its 20-year

moving average.

5.5.3 Other

Service lives for most assets are taken from Bureau of Economic Analysis (2003). The

service life for autos is taken from U.S. Department of Labor (2001). The service life

for computers is derived by calculating the depreciation rate implied by BEA data on

computer investment and the real net stock of computers, adjusted for an assumed

declining balance parameter.

I assume that businesses expect output per worker to grow at the average over the

sample period of 2.2 percent per year. For land, I use 1.5 percent, the average

of expected productivity growth and the average real growth of the price of land.

Brokers’ commissions as a share of other investment in structures b is 0.7 percent, its

average during 1959-1997. The present discounted value of sale of scrap as a fraction

of the purchase price Sc is 0.6 percent.

5.6 Stock of Land

Annual data on the real input of land into nonfarm business production are available

from BLS. Those data include land associated with tenant-occupied housing. I assume that the nonresidential portion of total land varies with the ratio of BLS’s index

of nonresidential structures to the sum of that index and 0.25 times the index of rental

residential capital. That ratio is roughly consistent with both a regression of annual

changes of the index for land on annual changes of the indexes for nonresidential and

residential structures, and with the nominal share of the productive capital stock of

nonresidential structures in the productive stock of all nonfarm business structures.

26
The price index for land is the price deflator for land from BLS.

5.7 Importance of Capital in Production (αj)

Although one can calculate time series for the various αj series using equation 23,

those series are much too erratic to be realistic estimates of technologies that change

slowly over time. Instead, for equipment and software, I developed a smooth series

that kept the implied ratio of nonmining machines to workers close to 1.0 over most

of the 1947-2004 period (see Figure 1 on page 31).17

Extending αj for equipment and software back to 1929, the estimated ratio of machines to workers falls sharply from the middle of the Great Depression until the

end of World War II and then bounces back strongly in the late 1940s and early

1950s. That pattern suggests that businesses retired equipment more slowly than

normal from 1935 to 1945, at first because they could shift labor hoarded during the

Great Depression from production to maintenance and repair of equipment, and later

because shortages prevented firms from purchasing as much new equipment as they

would have liked during World War II. Then, during the late 1940s and early 1950s,

firms increased the number of machines back to desired levels.

It proved impossible to develop a smooth αj for structures that kept the ratio of

nonmining-structure "machines" to workers at realistic levels over the entire available

period. A lower αj before the mid-1950s would raise the ratio of machines to workers

closer to 1.0 in the 1940s and 1950s but would push the ratio in 1929 even further

above 1.0. As a result, I assumed that the sum of the αjs for equipment and software

and for structures was constant at a level that produced a ratio of structure machines

to workers of about 1.0 in recent years (see Figure 2 on page 31). (The two αj series

are discontinuous between 1929 and 1930 because the 1929 values are used for the

entire period up to 1929.)

It is unclear how to compute an αj for land, since the method by which BLS constructs

the amount of land in service is not necessarily consistent with the putty-clay model.

Given that the ratio of the nominal value of land to nominal output appears to be

reasonably stationary, I assume that αj for land is constant at 0.0287. That figure

equals the sample average of the ratio of the nominal value of nonresidential land to

output at full employment (0.349) times the sample average of the cost of capital for

land (8.23 percent).

The relevant αj for an increase in inventories per worker is the average αj prevailing

when all existing inventory technologies were put in place, i.e., a 15-year moving

average. Because the αj for inventories is not necessary to construct the stock of

1 7Because of the effect of time to build, the ratio of machines to workers is procyclical. To account

for that, Figure 1 is constructed using a weighted moving average of actual workers and workers at

full employment. The moving average is 12 quarters for equipment and software and 16 quarters for

structures. The weight on actual workers is 0.5 for equipment and software and 0.4 for structures.

27
inventories per worker, it can be calculated by inverting equation 29 and taking a

15-year moving average.

Variations in the estimated overall importance of capital in production stem both

from variations in the importance of inventories and in the variation of mining-specific

capital (see Figure 3 on page 32). The average over the 1947-2004 period is 0.253.

6 Results

The putty-clay index of capital per hour shows some important differences with measures of capital per labor hour based on the putty-putty model. The putty-clay

index eliminates movements in capital per worker stemming from cyclical variations

in employment and investment. Even after accounting for those cyclical differences,

however, the putty-clay and putty-putty measures of capital per hour show some

important differences, most notably faster growth in the putty-clay measure during

2002-2004. Those differences stem from different assessments of the portions of

investment devoted to capital deepening and capital broadening. The more rapid

growth in the putty-clay measure of capital per hour since 2001 helps explain continued rapid growth in productivity despite weak investment. Two important factors

driving recent capital deepening are improvements in computers and software and a

real rate of return that is still low relative to its historical average.

6.1 Differences Between Putty-Clay and Putty-Putty Measures of

Capital per Hour

The differences between capital per labor hour in the putty-clay and putty-putty

models can most easily be seen by comparing the growth rate of the putty-clay

index of capital per hour with the BLS measure of capital per labor hour used to

calculate multifactor productivity in the nonfarm business sector (see Figure 4 on

page 33). Because labor hours are easier to adjust than the capital stock, the puttyputty measure of capital per worker is strongly countercyclical, rising sharply as

employment is reduced in recessions and rising slowly or even falling as workers are

rehired. The putty-clay measure of capital per hour depends on the cost of capital

and so leaves out those cyclical movements. Growth in capital per hour is slightly

procyclical because recessions are often triggered at least partly by a rise in the real

rate of return.

Adjusting the putty-putty measure of capital per hour for cyclical movements in labor

hours, as CBO does in preparing its estimates of potential GDP, eliminates a substantial portion of the difference between the putty-clay and putty-putty estimates of

capital per hour (see Figure 5 on page 33). However, a new difference emerges. The

putty-putty measure of capital per hour tends to grow more slowly early in economic

28
expansions and more rapidly late in expansions than the putty-clay measure. The

putty-clay model interprets the acceleration of investment during an expansion as

an increase in the number of machines necessary to accommodate unemployed labor

rejoining the workforce, i.e., as capital broadening. The putty-putty model instead

interprets that acceleration as an increase in the growth rate of capital per worker,

or capital deepening.

Beyond those cyclical differences, two other differences between putty-clay capital

per hour and CBO’s putty-putty capital per hour are worth noting. First, puttyclay capital per hour grows much more slowly during the early 1950s than puttyputty capital per hour. In the putty-clay model, the high cost of capital during the

late 1940s and early 1950s meant that much of the investment during that period

was capital broadening. Businesses were continuing to make up for the shortfall

in investment during World War II. The putty-putty model assumes any growth in

capital in excess of the growth in labor hours at full employment is capital deepening.

Second, putty-clay capital per hour has grown much more rapidly than putty-putty

capital per hour since 2001. The putty-putty measure interprets the slowdown in

investment, coming at a time when labor hours at full employment continued to grow,

as a reduction in capital deepening. However, capital deepening in the putty-clay

model is determined by the cost of capital, which has remained relatively low. Thus,

the sharp slowdown in investment implies a sharp reduction in capital broadening.

6.2 Sources of Growth in Putty-Clay Capital per Hour

The putty-clay model allows us to divide investment into pure replacement demand–

the cost of replacing depreciating machines with new machines of the same quality;

capital deepening–the extra cost of replacing depreciating machines with new machines of optimal quality; and capital broadening–the cost of increasing the number

of machines (see Figure 6 on page 33). Pure replacement demand as a percent of

nominal output at full employment rises slowly over time as a result of the shift of

the capital stock toward items with shorter service lives. Capital deepening varies

with the increase in the quality of new machines compared with those they replace.

Capital broadening is primarily a lagged function of growth in demand in excess of

growth in labor productivity. In addition, as discussed above, capital broadening

occurred during the late 1940s and early 1950s to make up for a shortfall in investment during World War II. (Alternatively, one might classify that as replacement

demand.)

One important factor in capital deepening in recent years has been the rapid increase

in the use of computers and software (see Figure 7 on page 34). The price indexes

of computers and software have generally declined relative to the price indexes for

other types of capital, so a shift of the capital stock toward computers and software

tends to boost the rate at which the quality of new machines rises. Improvements in

29
the quality of computers and software accounted for 44 percent of the total growth

in capital per worker in 2003, even though those two types of capital accounted for

just 23 percent of gross fixed investment and a smaller fraction of net investment.

Another important determinant of capital deepening, or growth in capital per worker,

is the real rate of return. The lower the real rate of return, the higher the quality

of new machines and thus the more rapid the growth in capital per hour. Figure 8

shows this inverse correlation between growth in capital per hour and the real rate

of return. Capital deepening has been strongest during periods when the real rate

of return was low–the 1960s and early 1970s, and again in the period since the mid1990s. On the other hand, high real rates of return during the late 1940s and early

1950s held growth in capital per hour to a minimum. High real rates during the late

1970s and 1980s also held down growth in capital per hour, but their adverse effect

was partially offset by strong investment in petroleum structures and accelerated

depreciation for structures.

7 Conclusions

The putty-clay model of capital offers a more useful way of thinking about the impact

of investment on productivity than the more commonly used putty-putty model. It

allows one to calculate a measure of capital per labor hour that is not distorted by

cyclical fluctuations in employment and investment. The first-order conditions for

optimal size of new machines allow one to identify capital deepening using the cost

of capital and labor productivity adjusted for intensity of usage.

Although the putty-clay model cannot explain why productivity growth accelerated

during 2001-2004, it at least removes the mystery of why the slowdown in investment

did not cause productivity growth to slacken. That slowdown was the result of the

near-disappearance of capital broadening, reflected in minimal employment growth

over the same period. Businesses were investing only to replace depreciated capital,

not to add new workers. Of the nearly 1.9 percentage-point acceleration in the

growth of output per hour in private business less housing between the first quarter

of 2001 and the third quarter of 2004 compared with the prior 10 years, I find that

0.2 percentage points was due to cyclical factors, between 0.1 percentage point and

0.2 percentage points was due to increased capital deepening, and the remaining 1.5

percentage points is unexplained.

30
1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

1.30

1.20

1.10

1.00

0.90

0.80

0.70

0.60

* Workers are an average of actual workers and workers at full employment.

Figure 1. Ratios of "Machines" to Full-Time-Equivalent Workers*

(Private nonfarm less housing)

Equipment and software Structures

1

1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.130

0.120

0.110

0.100

0.090

0.080

0.070

0.060

Figure 2. Production-Function Coefficients for Nonresidential Capital

(Private nonfarm less housing)

Equipment and software Structures

31
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.265

0.260

0.255

0.250

0.245

* Equal to the coefficient of capital in production adjusted for mining.

Figure 3. Importance of Nonresidential Capital in Production*

(Private nonfarm less housing)

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

10

8

6

4

2

0

-2

-4

Sources: Author; Bureau of Labor Statistics.

Figure 4. Growth Rates of Capital per Hour

(Percentage change from previous year)

Notes: Putty-clay capital per hour is for private nonfarm less housing.

Putty-putty capital services per hour of all persons, from BLS, is for nonfarm 

 business.

Putty-clay Putty-putty

32
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

6

5

4

3

2

1

0

-1

Sources: Author; Congressional Budget Office.

Figure 5. Growth of Capital per Hour

(Percentage change from previous year)

 

Notes: Putty-clay capital per hour is for private nonfarm less housing.

Putty-putty capital per hour, from CBO, is for private nonfarm.

Putty-putty

Putty-clay

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

18

16

14

12

10

8

6

4

Figure 6. Components of Business Fixed Investment

(Excludes farming; percentage of nominal output at full employment)

Total Excluding capital broadening

Excluding capital broadening and deepening

33
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

5

4

3

2

1

0

-1

Figure 7. The Contribution of Computers and Software to Capital Deepening

(Percentage change from previous year)

Overall index of capital per hour Excluding contribution of computers and software

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

5

4

3

2

1

0

-1

9

8

7

6

5

4

3

Figure 8. Capital Deepening and the Rate of Return

Growth in capital per hour (percentage change from year ago, left scale)

Real rate of return (percent, four-quarter moving average, right scale)

34
Appendix: Types of Capital

Nonfarm, nonresidential equipment and software comprises the following

types of capital:

Information-processing equipment and software

Computers and peripheral equipment

Software

Communication equipment

Medical equipment and instruments

Nonmedical instruments

Photocopy and related equipment

Office and accounting equipment

Industrial equipment

Fabricated metal products

Steam engines

Internal combustion engines

Metalworking machinery

Special-industry machinery, not elsewhere classified

General industrial, including materials-handling, equipment

Electrical transmission, distribution, and industrial apparatus

Transportation equipment

Light trucks (including utility vehicles)

Other trucks, buses, and trailers

Autos

Aircraft

Ships and boats

Railroad equipment

35
Other equipment

Household furniture

Other furniture

Construction tractors

Other construction machinery

Mining and oilfield machinery

Service-industry machinery

Household appliances

Miscellaneous electrical

Other

Nonfarm, nonresidential structures comprises the following types of capital:

Commercial and health care

Office, including medical buildings

Other commercial structures

Hospitals and special-care facilities

Manufacturing

Power and communication

Electric power

Other power

Communication

Mining exploration, shafts, and wells

Petroleum and natural gas

Mining

Other structures

Religious

Educational

36
Other buildings

Railroads

Other

37
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