New evidence on the retirement and depreciation of machine tools.

Oliner, Stephen D.

I. INTRODUCTION

In the late 1970s and early 1980s, Charles Hulten and Frank Wykoff [1979; 1981a; 1981b] undertook a major effort to estimate economic depreciation for the U.S. capital stock. Their pioneering work, which analyzed prices for many used assets, became the primary reference for estimates of economic depreciation. However, since then research on depreciation has been limited, despite the need to examine the robustness of Hulten and Wykoff's results and to cover assets omitted from their studies. In addition, little work has been done to estimate retirement distributions, which also play an important role in the construction of capital stocks.

This paper provides new evidence on the depreciation and retirement patterns of machine tools, a grouping that includes many types of equipment that cut or form metal. The data were collected from a survey of dealers belonging to the Machinery Dealers National Association. These dealers provided information on the prices of used machine tools from their own sales records. They also estimated certain facets of the retirement distribution for machine tools, including the average service life, the direction of change in the average life since the mid-1970s, and the ages by which 25 percent and 75 percent of an investment cohort are retired.

Machine tools can be divided into two broad groups. The first consists of the conventional machines that require manual control from an operator. Although the design and safety of these conventional machines has improved over the years, the pace of technical progress has been slow. The second group consists of numerically controlled machines, which are more highly automated than their conventional counterparts.(1) American manufacturers began to install numerically controlled machines in large numbers during the 1970s, and these machines have continued to replace conventional if machines on the factory floor. However, at the time of my survey, conventional machines still represented the bulk of machine tools operating in the United States, and the estimates of depreciation and retirement in this paper refer strictly to conventional machines.

The paper is organized as follows. Section II traces the adoption of numerically controlled machines by U.S. industry and then models how this phenomenon would have affected the depreciation and retirement patterns of conventional machine tools. Section III briefly describes the data set. Section IV analyzes the information on retirement patterns provided by the dealers, while section V presents the estimates of economic depreciation. Section VI summarizes my findings and suggests avenues for further research.

II. ADOPTION OF NUMERICALLY CONTROLLED MACHINES: EVIDENCE AND IMPLICATIONS

At roughly five-year intervals since the 1920s, American Machinist magazine has conducted an in-depth inventory of the stock of machine tools used by American manufacturers. For a detailed breakdown of machine types, the inventory provides information on the number of machines in use and the age distribution of this stock. Table I uses the American Machinist surveys to characterize the adoption of numerically controlled machines in the United States since the mid-1970s. As shown on the first line, the share of numerically controlled machines in the total machine-tool stock rose from 1.8 percent in 1976-78 to 9.2 percent in 1989. This five-fold increase in the share of numerically controlled machines in the existing stock implies a considerably higher share of these machines in new investment. Indeed, as shown in the right-hand columns of the table, numerically controlled machines represented 17.1 percent and 25.2 percent of all machine tools acquired during 1979-83 and 1985-89, respectively.2 Thus, numerically controlled machines have become a significant part of total U.S. investment in machine tools.

Table I also shows that the adoption of numerically controlled machines has varied widely across machine categories. Numerical control has made the greatest inroads for boring machines, turning machines (lathes), and punches, while the penetration has been much lower for cutoff and sawing machines, grinders, and a range of metalforming machines. This variation likely has a simple economic explanation. Automation is most profitable for complex machining activities, because it reduces the number of skilled workers needed to control the operation, increases accuracy, cuts down on the movement of workpieces between various single-function machines, and eliminates the need for manual tool changes on each machine. As a rule, the machine types with the highest shares of numerically controlled machines in Table I perform relatively complex operations. Consider, for example, a job to punch out several pieces from a metal sheet. If done with a conventional punching machine, the operator might have to reposition the sheet and change the punching tool for each hole. Instead, a numerically controlled turret punch could be programmed to make the set of punches, selecting the tools automatically from an overhead storage carousel. In contrast, automation might not pay off for easier tasks, such as sawing a piece of metal, polishing a pre-cut object with a grinder, or making simple bends with many types of metalforming machines.

TABLE I
Importance of Numerically Controlled Machine Tools in
the United States(a)
 (Percent of all U.S. machine tools, based on number of units)
 Share of Existing Stock Share of Investment(b)
Type of Machine 1976-78 1983 1989 1979-83 1985-89
Total 1.8 4.7 9.2 17.1 25.2
 Metalcutting 2.3 5.4 10.5 18.9 27.1
 Boring 5.9 11.1 22.2 30.7 52.0
 Drilling 1.0 2.8 3.6 5.6 8.6
 Milling 2.1 6.8 11.3 20.7 26.9
 Cutoff and
 sawing < .1 NA NA NA NA
 Grinding <.1 .6 2.9 2.1 8.4
 Turning 2.5 9.1 18.3 37.1 52.0
 All other 7.1 NA NA NA NA
 Metalforming .3 1.9 3.9 7.4 14.6
 Punching 8.3 16.4 NA 41.8 NA
 All other .1 .8 NA 3.1 NA

(a) Author's calculations from data presented in the following articles in
American Machinist magazine: "12th American Machinist Inventory of
Metalworking Equipment 1976-78," December 1978, p 134; "13th American
Machinist Inventory of Metalworking Equipment 1983," November 1983, p. 114;
"14th American Machinist Inventory of Metalworking Equipment," Special Report
No. 808, November 1989. (b) For 1979-83, the investment share was computed as
the ratio of numerically controlled machines zero to four years old in the
1983 inventory to all machines zero to four years old in that inventorY For
1985-89, the same calculation was done using the 1989 inventory.

How has the increasing use of numerically controlled machines affected the service lives and depreciation rates of conventional machines? This question can be addressed within a simple model for the price of a durable good. Let R(A,nc) denote the income generated by a conventional machine tool of age A, conditional on the extent of adoption of numerically controlled machines (denoted by nc). I impose two assumptions on R(A,nc). The first is that the income generated by the machine declines with age ([differential]R/[differential]A <0), reflecting deterioration in the machine's condition. The second assumption is that the diffusion of numerically controlled tools depresses the income stream from conventional machines ([differential]R/[differential]nc<0). This reduction in income occurs because the use of numerically controlled machines reduces marginal production costs and thus the price of goods produced by conventional machines; in addition, the installation of numerically controlled machines could cause conventional machines to be idled except during periods of peak demand.

In equilibrium, the price of a conventional machine tool at age A will be

(1) [Mathematical Expression Omitted],

where r is the discount rate, A* is the service life of the machine, and S is the machine's scrap value. Equation (1) says that the price of the machine equals the rental income it generates while in use plus its scrap value, with both components discounted to the current date. To determine A*, I need the following scrapping condition, which states that the machine is removed from service when its value in use equals its scrap value:

(2) [Mathematical Expression Omitted].

Equations (1) and (2), along with the two assumptions on R(A,nc), allow us to characterize how the adoption of numerically controlled machines affects the service life (A*) and depreciation rate ([delta]) of conventional machines. The results, which are proven in the appendix, can be summarized as follows:

PROPOSITION. Given equations (1) and (2), and assuming that [differential]R/[differential]A < 0 and [differential]R/[differential]nc < 0, the increased adoption of numerically controlled machines reduces the service life of conventional machines but has an ambiguous effect on their rate of depreciation. That is, dA*/d(nc) < 0, but d[delta]/d(nc) cannot be signed.

The intuition behind these results can be gleaned from Figure 1, which plots the price of a conventional machine tool ([P.sup.K]) against its age (A). [Mathematical Expression Omitted] is the price schedule in effect before the adoption of numerically controlled machines, while [Mathematical Expression Omitted] shows the schedule after the adoption of these machines. [Mathematical Expression Omitted] lies below [Mathematical Expression Omitted] because the use of numerically controlled machines depresses the income stream generated by the conventional machine. Now, the age at which each [P.sup.K] schedule drops to scrap value (the dashed line) represents the service life of the conventional machine. Thus, the price schedules [Mathematical Expression Omitted] and [Mathematical Expression Omitted] yield service lives of [Mathematical Expression Omitted] and [Mathematical Expression Omitted], respectively. As can be seen, [Mathematical Expression Omitted] > [Mathematical Expression Omitted], indicating that the adoption of numerically controlled machines reduces the service life of the conventional machine. This occurs simply because the market price of this machine falls to its scrap value at an earlier age. The downward shift in the [P.sup.K] schedule, and thus the decline in A*, will be relatively large when automation yields major efficiencies. Although I have no direct measure of the gains from automation, the shares in Table I reveal the machine types for which numerically controlled machines have made the greatest inroads. One would expect the decline in A* to be greatest for conventional machines in categories with high shares of numerically controlled machines, a prediction that is tested in section IV.

Figure 1 also demonstrates why the adoption of numerically controlled machines has an ambiguous effect on the depreciation rate of conventional machines. Let [delta]0 and [delta]1 denote the depreciation rates associated with the [Mathematical Expression Omitted] and [Mathematical Expression Omitted] schedules--that is, the rates at which these schedules decline with age. To compare these depreciation rates one needs to know the exact position of [Mathematical Expression Omitted] relative to [Mathematical Expression Omitted], not merely that [Mathematical Expression Omitted] lies below [Mathematical Expression Omitted]. Such a precise characterization of [P.sup.K] cannot be obtained under my weak assumptions for R(A,nc), the function that describes the income generated by conventional machines. I purposely avoided placing much structure on R(A,nc) because its actual form is unknown and likely varies widely across individual types of machine tools. Without specifying R(A,nc) in greater detail, the model yields no clear implication for the depreciation rate of conventional machines.(3)

To summarize, numerically controlled machines have become an increasingly important part of total U.S. investment in machine tools. My model analyzes the impact of this increased use of numerically controlled technology on the large stock of conventional machine tools. The model predicts a reduction in the service lives of these conventional machines, but it yields no clearcut prediction concerning the rate of depreciation. Further, the reduction in service lives should be most pronounced for the conventional machines whose functions are best suited to automation-that is, the machine types for which numerically controlled tools already represent a sizable share of the installed units.

III. THE DATA FROM MACHINERY DEALERS

The data set includes transaction prices for thirty-two models of metalcutting and metalforming machine tools drawn from the sales records of dealers belonging to the Machinery Dealers National Association. The set of models was developed with the assistance of Association officials and is intended to represent a broad crosssection of the conventional machine tools used in the United States after World War II. Table II provides a full listing of the machine models for which prices were obtained.

[TABULAR DATA II OMITTED]

The survey respondents also provided information on service lives for twenty-two models. Specifically, the survey asked dealers to estimate a model's average service life, as well as the ages by which 25 percent and 75 percent of a cohort of the model would be retired. In addition, the dealers were asked to indicate whether the model's average service life had become shorter, become longer, or remained unchanged since the mid-1970s.

The survey form was mailed to each member of the Association in late 1986, with a response deadline of 15 January 1987. The response rate to this initial mailing was relatively low, and a follow-up survey was sent out in the summer of 1988. All told, sixty-six dealers replied to the initial and follow-up surveys, providing prices for 573 separate transactions. A total of forty-nine dealers supplied information on service lives.

Some of the reports on machinery sales were incomplete or contained apparent inconsistencies. A given sale was included in the data set only if

(a) the dealer provided information on the machine's sale price, year of sale, year of manufacture, model number, and manufacturer, and

(b) the information on the machine's model number and year of manufacture was consistent with its serial number, provided the latter was known.

In all, 440 sales met both conditions and were used to estimate depreciation rates in this paper. The sales in my sample took place mainly in the early and mid-1980s, although the full range of dates spans the period 1968 to 1988.

With respect to the survey responses on retirement patterns, about 15 percent of the responses turned out to be logically inconsistent, in that the reported average service life was too low to have been the mean of any retirement distribution based on the reported 25th and 75th percentile ages. To see how this determination was made, assume that retirements occur as follows, with F(x) denoting the age by which the proportion x of the initial cohort has been retired, and [epsilon] > 0:

As [epsilon] approaches zero, retirements become more concentrated at the earliest ages consistent with the assumed 25th and 75th percentiles of the distribution, establishing the lower bound on the mean service life ([mu]). In the limit,

[Mathematical Expression Omitted]

+ (0.50 -[epsilon]) [F(0.25) + [epsilon]]

+ [epsilon]F(0.75) + 0.25[F(0.75) + [epsilon]]}

= 0.50F(0.25) + 0.25F(0.75) = [mu]*.

The variable [mu]* is the lowest average service life consistent with the reported values for F(0.25) and F(0.75). If a dealer reported the average service life of a model to be less than [mu]*, that set of values for [mu], F(0.25), and F(0.75) was classified as inconsistent and was omitted when estimating the parameters of the retirement distribution in the next section.

IV. DISTRIBUTION OF RETIREMENTS

This section analyzes the estimates of economic lives and retirement patterns reported by the Association's dealers. From this information I construct a rough estimate of the retirement distribution for conventional machine tools and then assess the average service life assumed by the Bureau of Economic Analysis for the broader category of total metalworking machinery.

Average Service Lives

Table III summarizes the information on average service lives from the survey. The first column presents the sample mean of the average service lives reported for all machine models. The second and third columns show the same statistic when the sample is split into two groups on the basis of numerically controlled machine shares. All "High-NC" models belong to a machine category in Table I for which the numerically controlled share of the installed stock in 1976-78 was more than 1 percent; all other models were placed in the "Low-NC" group. The 1976-78 inventory was used for this partition because it has more extensive information on shares of numerically controlled machines than the later inventories.

[TABULAR DATA III OMITTED]

For the sample as a whole, the mean value of the average service lives reported by dealers was 30.6 years; using only the ninety-seven responses with full and consistent information on the mean service life, F(0.25), and F(0.75), the estimate of [mu] rises slightly to 31.6 years. The estimate of the average service life for models with a high share of numerically controlled machines is only a bit shorter than that for models with a low share. The lack of a sharper contrast may appear at odds with the predicted effect of numerically controlled technology on service lives. However, differences among the models in the sample--concerning, say, the intensity of use--make it difficult to compare these lives across the two groups. Changes in service lives over time for each group (discussed below) are more relevant for assessing the model predictions derived in section II.

The American Machinist inventories provide some support for my survey-based estimate of the average service life of conventional machine tools. In the 1983 inventory, the breakdown of non-numerically controlled metalcutting and metalforming machines by age group was as follows: 11.4 percent of the machines were zero to four years old, 19.3 percent were five to nine years old, 34.8 percent were ten to nineteen years old, and 34.6 percent were twenty years or older. Assuming that the machines in each age group had average ages, respectively, of 2 years, 7 years, 14.5 years, and 25 years, the weighted average age for the total inventory of conventional machines would be 15.3 years. To derive an average service life consistent with this estimate, assume that each cohort entering the inventory had the same number of units and that all machines were retired at a single age. Under these assumptions, the service life common to all cohorts would be twice the average age, or 30.6 years, a figure very close to those shown in Table III. Applying this technique to the 1989 American Machinist inventory yields a service life of 28.8 years-again consistent with the results in Table III.(4)

Two factors can explain the long service lives for conventional machines indicated by the American Machinist inventories and by the Association's dealers. First, the rate of technical advance in conventional machines since World War II has been relatively slow. As a result, obsolescence has not provided a strong reason to retire older machines, at least not until the increased adoption of numerically controlled machines in recent years. Second, conventional machines typically have had low utilization rates, resulting in a slow pace of deterioration. Cook [1975, 28] estimates that "the average machine tool in a conventional shop is cutting metal only between 3 and 10 percent of the time." Such low rates reflect the considerable time spent moving the metal workpiece from one machine to another, positioning the metal in each machine, and mounting the cutting tool for the job. A machine running only 3 percent of the time would accumulate less than one year of active service over a thirty-year period.

Changes in Average Service Lives

The survey also asked dealers whether they perceived any change in average service lives between the mid-1970s and the date of the survey response. The first row of Table IV summarizes all responses to this question. The responses were split almost evenly between those indicating no change in average lives over the period and those reporting a reduction. Only 5.3 percent of the respondents said that average service lives had lengthened.

TABLE IV
Changes in Average Service Lives since the Mid-1970s
as Reported by Survey Respondents

 Percent Reporting
 Shorter No Longer Number of
 Lives Change Lives Responses
All models 45.7 49.0 5.3 151
High-NC Models(a) 55.4 41.9 2.7 74
Low-NC Models(a) 36.4 55.8 7.8 77

(a) High-NC models are those that belong to a machine type in Table I with a
numerically controlled share in 1976-78 of more than 1 percent. All other models
in the sample are classified as low-NC models.

To test whether the asymmetry in the response distribution is statistically significant, let [n.sub.s], [n.sub.L], and [n.sub.U] denote, respectively, the number of responses in the sample indicating that average service lives for conventional machines had become shorter, longer, or remained unchanged since the mid-1970s. The total number of responses, n, equals [n.sub.s] + [n.sub.L] + [n.sub.U]. In addition, let [p.sub.j] = [n.sub.j]/n (j = S,L,U) denote the proportion of sample responses in each of the three categories; [P.sub.j] represents the corresponding proportion in the population of dealers. The null hypothesis is that the share of the dealer population perceiving a rise in average service lives equals that perceiving a decline [[P.sub.s] = [P.sub.L] = 0.5(1 - [P.sub.U])] Because [P.sub.U] is unknown, I replace [P.sub.U] with its sample value [P.sub.U] = 0.49, which implies the following null:

[H.sub.0]:[P.sub.S] = [P.sub.L] = 0.255 and [P.sub.U] = 0.49.

If all observations in the sample are independent, the number of respondents in each category ([n.sub.S], [n.sub.L], [n.sub.U]) has a multinomial distribution. Using this fact, Mood, Graybill, and Boes [1974, Theorem 9, 446] develop a statistic (Q) for testing hypotheses about the underlying population probabilities. Specialized to my three-category application,

(3) [Mathematical Expression Omitted]

The statistic Q has a small value if the sample probability for each group ([P.sub.j]) is close to that expected under the null hypothesis ([P.sub.j]). As n approaches infinity, Q has a limiting [chi square] distribution, with one degree of freedom in this case. Substituting the values for n, [P.sub.S], [P.sub.L], and [P.sub.U] from Table IV into equation (3), along with the values for [P.sub.S], [P.sub.L], and [P.sub.U] under [H.sub.0], Q equals 48.33, compared with the critical value of 3.84 at the 5 percent level of significance. Thus, [H.sub.0] can be rejected with a very high degree of confidence, indicating that the population of machinery dealers has perceived a decline in average service lives for conventional machine tools since the mid-1970s.

If the diffusion of numerically controlled tools were responsible for this reduction in average service lives, one would expect the greatest asymmetry in the distribution for machine types with a high share of numerically controlled machines. The second and third rows of Table IV present the evidence on this issue. For the machine models in the high-share group, 55.4 percent of the responses indicated a reduction in average service lives, while only 2.7 percent indicated a lengthening of lives. For the models in the low-share group, the asymmetry is less pronounced. This contrast is consistent with the view that the shorter average lives stem, at least in part, from increased use of numerically controlled technology.

A statistical test for differences in the distribution of responses across high- and low-share groups can be constructed using a variant of the Q statistic from equation (3). As above, let j index the category into which a response falls (j = S,L,II). In addition, let i index the group (high or low share) of the model for which the response was provided (i=HIGH, LOW) denotes the proportion of (sample) responses for models in group i falling into response category j; [p.sub.i,j] represents the corresponding probabilities in the population of dealers. The hypothesis to be tested is

[H.sub.1]: ([p.sub.i,s], [p.sub.i,L], [p.sub.i,U]) = ([p.sub.s], [p.sub.L], [p.sub.U]) = (0.457, 0.053, 0.490)

for each group. As before, ([p.sub.s], [p.sub.L], [p.sub.U]) denotes the vector of probabilities estimated from the full sample.

Mood, Graybill, and Boes [1974, equation (30), 449] develop a statistic ([Q.sup.*]) for testing the equality of independent multinomial distributions, which can be used to test [H.sub.1]. Under the null hypothesis of equality across the high- and low-share groups,

(4) [Mathematical Expression Omitted]

The statistic [Q.sup.*] has a limiting [chi square] distribution with degrees of freedom equal to the number of response categories minus one (d.f. = 2 in this case). Intuitively, [Q.sup.*] has a small value--leading one not to reject [H.sub.1]--if the proportion of group i's responses in category j ([p.sub.i,j]) is similar to this proportion for the full sample ([p.sub.ij]). Using equation (4), [Q.sup.*] equals 6.34, slightly above the 5 percent critical value of 5.99. Accordingly, with some confidence, the null hypothesis can be rejected in favor of the view that the decline in service lives has been concentrated in the high-share group of machines. This pattern lines up with the implications of the model in section II.(5)

Full Retirement Distribution

Thus far, the discussion has focused on the mean of the survey-based retirement distribution. To provide information about the spread of the distribution around the mean retirement age, I now consider the survey estimates of the 25th and 75th percentile ages. Obviously, with estimates of only [mu], F(0.25), and F(0.75), the retirement distribution cannot be described in much detail. The objective is merely to see whether the distribution looks symmetric and, if so, to find the best fitting normal approximation.

The estimates presented here are based on the ninety-seven survey responses for retirement patterns that were logically consistent according to the test described above. For this set of responses, the estimate of [mu] the average service life, was 31.6 years (see Table III), while the sample averages for the 25th and 75th percentile ages of the retirement distribution were 25.3 years and 38.0 years, respectively. Because the estimate of [mu] is virtually centered between F(0.25) and F(0.75), the three observed points are consistent with a symmetric distribution.

To find a normal approximation to these points, a value for the standard deviation ([sigma]) is needed to go with the estimate of [mu]. As a first step to derive 6, I adjusted down both the 25th and 75th percentile ages by 0.05 years to center 1 exactly between them and denoted the adjusted 75th percentile age by [F.sup.*](0.75). If [F.sup.*](0.75) is assumed to come from a normal distribution with a mean of 31.6 years and unknown 6, the 75th percentile age from the corresponding standard normal distribution is

(5) z(0.75) = [[F.sup.*](0.75) - 31.6] /[sigma] = 6.35/[sigma].

Tables for the standard normal distribution indicate that z(0.75) = 0.675, which yields [sigma] = 9.4 years from equation (5). Thus, the normal approximation to the retirement distribution is N([mu],[sigma].sup.2])=N(31.6, 88.4).

Based on this normal approximation, the first column of Table V shows the proportion of a given investment cohort that remains in service at selected ages. For comparison, column 2 presents the analogous series implied by a Winfrey S-3 distribution with a thirty-two-year average service life. The Winfrey S-3 is a symmetric retirement distribution used by the Bureau of Economic Analysis to construct capital stocks for virtually all types of business equipment. The table uses the S-3 distribution with a thirty-two-year average life in order to approximate the estimated mean service life of my distribution.(6) Comparing columns 1 and 2 shows that the pattern of retirements in the S-3 distribution is quite close to that in my normal approximation. For example, twenty-five years after installation, the normal approximation indicates that 75.8 percent of the cohort is still in service, compared with 81.3 percent for the S-3 distribution. At other ages, the proportion of the cohort still in service is even closer across the two distributions. Accordingly, the retirement distribution estimated in this paper can be viewed roughly as a Winfrey S-3 with a thirty-two-year average service life.

[TABULAR DATA V OMITTED]

Columns 3 and 4 of the table present the retirement hazard rates implied by my survey-based retirement distribution and the Winfrey S-3 distribution. The hazard rate is the probability of retirement over a given period, conditional on having survived to the beginning of the period. For example, the hazard rate shown in the table for age fifteen years is the probability of retirement between ages fifteen and twenty, conditional on survival to age fifteen; it is calculated simply as the difference between the survival probabilities at ages fifteen and twenty, divided by the survival probability at age fifteen. As shown in the table, both the survey-based distribution and the Winfrey S-3 imply hazard rates that increase with age. That is, a forty-year-old machine tool is much more likely to be retired in the subsequent five years than is a ten-year-old machine, a reasonable assumption for assets like machine tools that gradually deteriorate with continued use. The rising hazard rate implied by the survey-based retirement distribution influences my estimates of depreciation, as shown in section V.(7)

BEA's Average Service Life for Metalworking Machinery

When computing capital stocks for metalworking machinery, the Bureau of Economic Analysis assumes that the average service life for this asset depends on the industry in which it is used. For example, as shown in U.S. Department of Commerce [1993, table B, M-17], metalworking equipment used by producers of primary metals is assumed to have a twenty-seven-year average life, while such equipment used by producers of motor vehicles is assumed to have only a fourteen-year average life. To derive the average service life for the aggregate of all metalworking equipment, I averaged the industry-specific average lives, with each industry's weight equal to its share of total constant-dollar investment in metalworking machinery. Investment weights are more appropriate than capital-stock weights for this average because I am seeking the average service life that the Bureau of Economic Analysis implicitly applies to an economywide investment cohort. Because the industry-level investment shares vary somewhat over time, so does the calculated average service life. This value of hovers between 18 1/2 and 19 1/4 years until 1980 and then edges down to about 17 3/4 years by 1990.

The Bureau of Economic Analysis's assumed value of the average service life thus appears to be much shorter than the roughly thirty-one-year estimate from my sample. This inference, however, is not valid because of differences in asset coverage. In particular, my sample covers only metalcutting and metalforming machine tools, while the Bureau of Economic Analysis's definition of metalworking machinery also includes rolling mill machinery, other types of metalworking machinery, special dies and tools, machine tool accessories, and power-driven hand tools. Table VI provides a breakdown of total constant-dollar business investment in metalworking machinery for selected years, based on unpublished Bureau of Economic Analysis data. As indicated on line 3, outlays for metalcutting and metalforming machine tools represented from 28.6 percent to 47.4 percent of total outlays for metalworking machinery during the years shown, leaving sizable investment shares for the other components in the table.

TABLE VI
Composition of Investment in Metalworking Machinery(a)
(Percent, based on data in 1987 dollars)
 1977 1980 1985 1990

1. Total 100.0 100.0 100.0 100.0
2. Heavy machinery 55.9 61.7 48.9 51.3
3. Metalcutting and metalforming 42.6 47.4 34.9 28.6
 machine tools
4. Other metalworking machinery 13.3 14.3 14.0 22.8
5. Power-driven hand tools 15.6 14.6 14.7 12.2
6. Special dies and tools, and 28.6 23.7 36.4 36.5
machine tool accessories
(a) Author's calculations from unpublished Bureau of Economic Analysis
tables. Components may not sum to totals due to rounding.

To assess the Bureau's average service life for total metalworking machinery, the average service life must be estimated for each of these components. Accordingly, I applied the estimate of [mu] from my sample of machine tools, 31.6 years, to all heavy metalworking machinery (line 2 in the table). For special dies and tools, I calculated the average service life from estimates of the "mean retention period" from studies conducted by the Treasury Department's Office of Industrial Economics in the 1970s, the results of which are summarized in Brazell, Dworin, and Walsh [1989, appendix table 1]. Five such studies estimated mean retention periods for special tools, with the average of these estimates at 5.8 years. I used this average as a proxy for the average service life of special dies and tools. These tools likely wear out quickly because they actually do the work of cutting or forming the workpiece. For power-driven hand tools and machine-tool accessories, I assumed the same 5.8-year average lifetime, as no separate estimates were available.

The weight attached to the 31.6-year average lifetime was 0.545, the average investment share for heavy machinery over the years shown in Table VI. With this weight, the average service life for total metalworking machinery is 19.9 years (0.545 [multiplied by] 31.6 + 0.455 [multiplied by] 5.8), which nearly matches the eighteen- to nineteen-year average life implicit in the Bureau of Economic Analysis's capital stocks. That is, my sample of conventional machine tools produces a value of [mu] quite consistent with the Bureau's assumed average service life for total metalworking machinery.

This consistency, however, should not be viewed as an endorsement of the Bureau's average service life for this equipment. The 31.6-year average lifetime that I assumed for heavy machinery took no account of numerically controlled machines. Because these machines incorporate computer technology, their service lives would be expected to be shorter than those for conventional machines. Thus, after accounting for numerically controlled machines, the actual value of [mu] for total metalworking machinery likely would be below the Bureau's assumed value. With numerically controlled machines estimated to represent one-quarter of all machine tool units purchased in the U.S. during 1985-89 (see Table b, the bias in the Bureau's assumption could be substantial. At this point, the following conclusion seems fair: the Bureau's assumed average service life for metalworking machinery appears reasonable for an economy without numerically controlled machines, but probably is too long for the U.S. economy as it now stands.

V. ESTIMATES OF ECONOMIC DEPRECIATION

Methodology

This paper estimates depreciation by following the usual methodology in the literature, which "lets the data speak" with a minimum of a priori restrictions. The general equation to be estimated is

(6) p = P(A,t,z)

which relates the observed sale price of a used asset to its age (A), the date of sale (t), and a vector of the asset's characteristics (z). In their work, Hulten and Wykoff used the Box-Cox power transformation as a flexible representation of P(A,t,z). The Box-Cox function nests several popular hypotheses for the form of the depreciation function, including the straight-line and geometric forms. However, the Box-Cox function is highly nonlinear in its parameters, which complicates the estimation of equation (6). I adopted a simpler approach, regressing the natural log of P on third-order polynomials in age and time, an interaction of age and time, and a vector of characteristics (z):

(7) [Mathematical Expression Omitted]

Equation (7) allows for a wide range of depreciation patterns at a given time. It also permits the entire depreciation function to shift over time (for y [not equal to] 0). The hypothesis of geometric depreciation at a time-invariant rate is nested as a special case ([[beta].sub.2] = [[beta].sub.3] = [gamma] = 0)

Given estimates of the parameters of equation (7), the depreciation function is computed as the percent change in price due to a unit increase in age, holding fixed time and characteristics [[differential](lnP)/[differential]A]. By fixing time, the estimated depreciation rate abstracts from differences in price levels over time that are unrelated to aging--that is, from general inflation. By controlling for the characteristics z, the estimated rate abstracts from price differences due solely to the heterogeneity of assets in the sample.

Two sets of variables appear in my specification of z. The first set indicates whether the machine was sold with tooling--the attachments to the machine, such as drill bits, that actually cut or form the workpiece. The inclusion of tooling should raise the machine's sale price. Because the information regarding tooling was missing for some observations, I included one dummy variable that equaled unity when tooling was included and zero otherwise and a second dummy that equaled unity when the information was missing and zero otherwise. The other variables in z consisted of thirty-one model-specific dummies to capture the sharp differences in average price levels across the models in the sample.(8)

The final issue in specifying equation (7) concerns sample selection. The sample necessarily omits the machines from a given cohort that already have been retired from service. To measure the full effect of aging on a cohort of machine tools, one must account--as in Hulten and Wykoff [1981a; 1981b]--for the implicit zero prices for machines no longer in service. This can be done by forming the adjusted price P' as a weighted average of the observed prices and the unobserved zero prices, with the weights reflecting the probability of remaining in service at a given age. That is,

(8) P'(A) = P(A)[1-F(A)] [multiplied by] 0 F(A) = P(A)S(A),

where F(A) is the probability that a machine will have been retired by age A and 1 - F(A) [equivalent] S(A) is the probability of survival to age A. Equation (8) implies that

(9) -[differential][lnP'(A)]/[differential]A = - {[differential][ln P(A)] /[differential]A + S[ln S(A)] /[differential]A}.

In addition, [differential]-[ln S(A)]/[differential]A [equivalent] H(A), the hazard rate for retirement at age A. Accordingly,

(10) [delta](A) = [delta](A) + H(A),

where [delta]' and [delta] denote the depreciation rates based on ln P' and ln P, respectively.

Equation (10) states that the depreciation rate computed from adjusted prices equals that computed from observed prices plus the retirement hazard rate. Recall from Table V that the hazard rate from my survey-based retirement distribution increased monotonically with age. As shown in equation (10), such a hazard rate will cause the cohort depreciation rate ([delta]') to rise with age even if the depreciation rate computed from observed prices ([delta]) is constant. In fact, I find exactly this pattern for [delta]' and [delta] when estimating equation (7).

Results

The top part of Table VII presents OLS regression estimates of equation (7) using first the observed prices and then the adjusted prices as the dependent variable. For brevity, the table reports the coefficients only on the age-related terms in equation (7). Columns 1 and 3 report the results when equation (7) is estimated without restrictions, while columns 2 and 4 report the results when [A.sup.2], [A.sup.3], and A [multiplied by] t are omitted from the regression, which forces depreciation to be geometric. For all the columns, the coefficients on [A.sup.2], [A.sup.3], and A [multiplied by] t have been multiplied by [10.sup.2], [10.sup.4], and [10.sup.2], respectively, to remove leading zeroes. The bottom part of the table presents the annual rates of depreciation implied by these coefficients. That is, the depreciation rates shown in columns 1 and 3 equal

(11) -([[beta].sub.1] + 2[[beta].sub.2]A + 3[[beta].sub.3][A.sup.2] + [gamma]t),

while those in columns 2 and 4 are simply -[[beta].sub.1]; the signs on the coefficients are reversed to present the depreciation rates as positive numbers. The depreciation rates calculated from expression (11) vary over time, and the rates shown in columns 1 and 3 are computed at the mean pricing date in the sample, 1982.

Table VII indicates that the observed dealer prices appear consistent with a constant annual depreciation rate of about 3.5 percent. This is the depreciation rate estimated in column 2, where the rate is restricted to be constant. The results in column 1 provide little evidence against this restriction: none of the higher-order age terms has a significant coefficient at the 5 percent level, and a formal F test of the null hypothesis that [[beta].sub.2] = [[beta].sub.3] = [gamma] = 0 has a p-value of 0.11. Even taking the estimates of [[beta].sub.2], [[beta].sub.3], and [gamma] at face value, the implied age-varying depreciation rates differ only slightly from 3.5 percent, as shown at the bottom of column 1. Thus, before adjusting for retirements, the dealer prices imply a slow pace of depreciation over the life of a conventional machine tool.

[TABULAR DATA VII OMITTED]

Using the adjusted prices substantially alters the results, with the changes due entirely to the hazard rate for retirement now embedded in the price variable. First, column 3 shows that [[beta].sub.3] is negative and significant at the 1 percent level, which provides strong evidence that the rate of depreciation becomes more rapid with age. The estimated annual depreciation rate for a forty-year-old machine (shown at the bottom of column 3) is 18.1 percent, more than six times the 2.9 percent rate for a ten-year-old machine. I should stress, however, that the depreciation rate reaches double-digits only after the investment cohort has lost about three-quarters of its initial value. As a result, the cohort depreciation function largely reflects the slower depreciation rates that prevail early on, with the faster rates mainly affecting the shape of the right-hand tail. Column 4 shows that the best geometric approximation to the age-varying depreciation rate is 9.5 percent, well above the 3.5 percent rate estimated from observed prices; the difference is the average annual hazard rate.

Comparison to previous estimates

The rates of depreciation for conventional machine tools shown in Table VII are broadly consistent with the estimates from Hulten and Wykoff [1981a] and Hulten, Robertson, and Wykoff [1989]. The latter study estimated Box-Cox price functions for four types of machine tools--milling machines, turret lathes, grinders, and presses. Using Hulten, Robertson, and Wykoff's parameter estimates, I calculated the implied depreciation rate for each type of machine at its mean age and price.(9) These estimates do not embed an adjustment for retirement, and thus are comparable to my estimate of a 3.5 percent annual rate of depreciation. For milling machines and turret lathes, the depreciation rates from Hulten, Robertson, and Wykoff were 4.97 percent and 3.27 percent, respectively, while the depreciation rate for presses was essentially zero.(10) These estimates confirm the slow depreciation rate I found before adjusting for retirements.

Hulten and Wykoff [1981a] estimated depreciation for conventional machine tools after adjusting the prices for retirements. When restricting their Box-Cox function to take the geometric form, they estimated a constant annual depreciation rate of 12.25 percent. Although this estimate is somewhat above my geometric rate of 9.5 percent, the two estimates are not different in any meaningful sense, owing to uncertainty about the proper retirement distribution with which to adjust the observed prices. In fact, if my observed prices are adjusted by the Winfrey S-3 distribution with a thirty-two-year mean lifetime (rather than by the survey-based distribution), the estimated geometric rate of depreciation rises to 10.6 percent; using the Winfrey S-3 with a thirty-year mean lifetime boosts the estimated depreciation rate even further to 13.6 percent, a bit above Hulten and Wykoff's estimate.

The only other study of depreciation for conventional machine tools appears to be Beidleman [1976]. Technically, Beidleman estimated depreciation not for machine tools per se, but for the capital equipment used by manufacturers of these machines. In fact, this industry makes heavy use of machine tools in production, and most of the nineteen asset groups included in Beidleman's study consist of machine tools. Averaging across these groups, he estimated a geometric depreciation rate of 7.48 percent per year without making an adjustment for retirements. This rate is well above the comparable depreciation rates estimated in this paper and by Hulten, Robertson, and Wykoff. Because Beidleman's sample and estimation procedure differ substantially from those in this paper, it is difficult to pinpoint why the results differ.

VI. CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH

This paper has estimated the retirement pattern and the rate of economic depreciation for a broad set of conventional machine tools, using data collected from a survey of machinery dealers. The average service life of these machines was estimated to have been a bit more than thirty years at the time of the survey (the late 1980s). This estimate, which I corroborated with separate data on stocks of conventional machine tools by age group, indicates that these assets have had longer service lives than most other types of business equipment. The long service lives likely reflect the slow pace of technical advance in conventional machine tools, as well as the relatively low utilization rates for these machines on the factory floor.

At the same time, the survey responses provided strong evidence that the average service life of these conventional machine tools has become shorter since the mid1970s. Although the dealers did not explain why this had occurred, economic theory points to the increased use of numerically controlled technology as a leading candidate. In a simple model of capital goods prices, I showed that substitution toward numerically controlled machines would reduce the service lives of conventional machines by causing the income stream that they generate to shift down. The pattern of survey responses also linked the diffusion of numerically controlled technology to the shift toward earlier retirement. Specifically, the proportion of dealers reporting a reduction in average service lives was relatively high for the conventional machines that faced the greatest competition from numerically controlled technology.

My results permit an assessment of the average service life assumed by the Bureau of Economic Analysis to construct capital stocks for metalworking machinery, the category that includes machine tools. Averaging across all industries that use metalworking machinery, the Bureau's assumed average service life for this asset has hovered around eighteen to nineteen years since 1960. Although this range is far below the roughly thirty-year average life I estimated for conventional machine tools, metalworking machinery includes many other assets with short service lives. After adjusting for this difference in coverage, the Bureau's average service life looks quite consistent with the longer average life I found for conventional machine tools. However, this assessment does not account for numerically controlled machine tools, which--because they incorporate computer technology--likely have shorter service lives than conventional machine tools. Accordingly, the Bureau's assumed average service life for metalworking machinery appears to have been appropriate before numerically controlled machines started to displace conventional machines in the 1970s, but now it is probably too long.

Consistent with the relatively long service lives for conventional machine tools, my sample of transaction prices yielded a slow rate of economic depreciation for these machines: about 3Y2 percent per year of aging, with little variation over the life of the asset. This sample necessarily omits the implicit zero prices for machines that already have been retired from service. When I adjust for this censoring problem, the resulting depreciation rate for a full cohort of conventional machines is estimated to rise sharply with age, ranging from about 3 percent annually for machines less than ten years old to nearly 20 percent annually for forty-year-old machines. This pattern reflects the rising hazard rate for retirement implied by the retirement distribution I estimate from the survey responses. However, the rapid depreciation rates are of limited practical importance; they occur after the cohort already has lost most of its initial value and thus mainly affect the right-hand tail of the depreciation function.

Looking ahead, two issues dominate the research agenda in this area. First, nothing is currently known about the depreciation and retirement patterns of numerically controlled machine tools. This is a serious problem, as numerically controlled machines have displaced conventional machines in a variety of applications. Indeed, numerically controlled machines accounted for about one-quarter of the machine tool units purchased in 198589 and likely were even more important as a share of total expenditures on machine tools. Second, although this paper presents evidence of a decline in service lives for conventional machines since the mid-1970s, it does not measure the extent of this decline. Future work should attempt to fill this gap.

APPENDIX

Proof of Proposition in Section II

MODEL.

As discussed in the text, the price of a machine in equilibrium will equal its discounted rental income plus its discounted scrap value:

(A.1) [Mathematical Expression Omitted]

where r is the discount rate, [A.sup.*] is the service life of the machine, and S is the machine's scrap value. Scrappage occurs when the value of the machine in continued use equals its scrap value:

(A.2) [Mathematical Expression Omitted]

ASSUMPTIONS.

(A.3) Rental income declines with age: [differential]R(A,nc) /[differential]A [equivalent] [R.sub.A](A,nc) < 0.

(A.4) Rental income declines with greater diffusion of numerically controlled machines: [differential]R(A,nc)/[R.sub.nc](A,nc) < 0.

PROPOSITION. Given equations (A.1) and (A.2), and assumptions (A.3) and (A.4), the increased adoption of numerically controlled machines reduces the service life of conventional machines but has an ambiguous effect on their rate of depreciation. That is, d[A.sup.*]/d(nc)<O, but d[delta]/d(nc) cannot be signed.

Proof. To begin, note that

(A.5) [Mathematical Expression Omitted]

where the inequality results because

R([A.sup.*],nc) > R(s,nc)

for all s > [A.sup.*] by virtue of (A.3).

Equation (A.5) is needed to sign d[A.sup.*]/d(nc). Now, totally differentiate (A.2) with respect to [A.sup.*] and nc:

[Mathematical Expression Omitted]

which, using (A.2), is equivalent to

[Mathematical Expression Omitted]

Rearranging (A.6), we obtain

[Mathematical Expression Omitted]

Assumption (A.4) implies that the numerator of (A.7) is negative, while (A.5) implies that the denominator is positive. Therefore, d[A.sup.*]/d(nc) < 0, as the proposition had asserted.

Next, I will show that d[delta](nc) cannot be signed. By definition, [delta]=[[differential][P.sup.K]/[differential]A]/[P.sup.K]. Therefore,

(A.8) d[delta]/d(nc) = [([P.sup.K]).sup.-2] [[P.sup.K]([[differential].sup.2][P.sup.K]/[differential]A[differential]nc) - ([differential][P.sup.K] /[differential]A)([differential][P.sup.K] /[differential]nc)].

Before computing the derivatives of [P.sup.K] that appear in (A.8), note that (A.2) can be substituted into (A.1) to yield

(A.9) [Mathematical Expression Omitted]

Using (A.9), the derivatives that appear in (A.8) are

[Mathematical Expression Omitted]

(since [R.sub.nc] < 0), and

[Mathematical Expression Omitted]

Substituting these expressions into (A.8) and rearranging terms, we obtain

(A.10) [Mathematical Expression Omitted]

The signs shown for the terms in (A.10) indicate that the sign of d[delta]/d(nc) is ambiguous. A stronger result would require more structure on R(A,nc) than simply assumptions (A.3) and (A.4).

Instantaneous Cumulative
Proportion Retired Proportion Retired At Age
0.25 - [epsilon] 0.25 - [epsilon] 0
[epsilon] 0.25 F(0.25)
0.50 - [epsilon] 0.75 - [epsilon] F(0.25) + [epsilon]
[epsilon] 0.75 F(0.75)
0.25 1.00 F(0.75) + [epsilon]

(1.) "Numerical control" is a generic term that subsumes a range of methods for automating machines. Strictly speaking, machines controlled by instructions stored in computer memory are called CNC machines (for "computer numerical control"). When a single computer controls a bank of machines, the system is termed DNC (for "direct numerical control"). In this paper, I use the numerical control designation to cover all forms of automated control. See Cook [1975] for an introduction to the methods and economics of numerical control. (2.) The American Machinist surveys do not provide data on investment. The investment shares in the table for 1979-83 equal the ratio of numerically controlled machines that were zero to four years old in the 1983 inventory to all machines that old in that inventory. The investment shares for 1985-89 were calculated by the same method using the 1989 inventory. This method assumes that retirements do not occur during the first four years of use, which enables the stock of machines age zero to four years to proxy for investment over the preceding four-year period. (3.) This ambiguous result depends critically on the exclusion of numerically controlled machines from the comparison of depreciation rates. Suppose, instead, that we compared the depreciation rate of conventional machines before the introduction of numerically controlled machines with the average depreciation rate of conventional and numerically controlled machines after the latter were introduced. Because the computer technology built into numerically controlled machines has a short economic life, these machines would be expected to depreciate more rapidly than conventional machines. Thus, the depreciation rate for a full cohort of machine tools after the introduction of numerically controlled machines probably would be faster than the depreciation rate of conventional machines alone during an earlier period. (4.) The 1970-78 inventory could not be used for this calculation because it had less detailed reformation on capital stocks by age group. (5.) The distribution theory for the Q and [Q.sup.*] statistics hinges on the assumption of independent observations. This assumption may not be valid for my sample. Many dealers provided information for two or three machine models, with each response entering the sample as a separate observation. If these multiple responses are not independent, the effective size of my sample will be smaller than its nominal size, causing the hypothesis tests to reject too often. This issue is of no practical importance for the first test reported because the computed Q statistic was so large as to reject Ho at any reasonable significance level. However, the rejection of [H.sub.1]--that the reported changes in average service lives are the same in low- and high-share groups--was less decisive, and the true marginal significance level may be greater than 0.05. Thus, the evidence of a disproportionate shift toward shorter economic lives in the high-share group, while suggestive, is not overwhelming. (6.) As discussed below, the Bureau of Economic Analysis actually assumes a considerably shorter average service life when constructing capital stocks for metalworking machinery, the category of equipment that subsumes machine tools. (7.) Rising hazard rates characterize a large class of retirement distributions. Indeed, any continuous, single-peaked retirement distribution will yield higher hazard rates for relatively old machines than for relatively young machines. To prove this result, let f(A) denote the probability of retirement at age A and let S(A) denote the probability of survival to age A; also, let A' denote the age associated with the peak of the retirement distribution. The instantaneous hazard rate at age A, H(A), is simply f(A)/S(A). Because I have assumed that f(A) rises monotonically top f(A) and then falls monotonically, there exist two ages, [A.sup.+] and [A.sup.-], such that [A.sup.+] > A' > [A.sup.-], and f([A.sup.+]) =f([A.sup.-]). Further, because the probability of survival to [A.sup.+] is less than the probability of survival to [A.sup.-], S([A.sup.+]) < S([A.sup.-]), implying that H([A.sup.+]) =f([A.sup.+]) /S([A.sup.+]) >f([A.sup.-])/S([A.sup.-]) = H([A.sup.-]). Thus, the hazard rate for each age [A.sup.-] to the left of A' maps to a higher hazard rate for some age [A.sup.+] to the right of A'. As a result, the average hazard rate for ages greater than A' must exceed the average hazard rate for ages less than A'. (8.) I use z to control for price differences that arise simply because the sample pools very diverse machines. The variables in z could have been expanded to include performance characteristics for each homogeneous class of machines in the sample. However, augmenting z in this way would have changed the interpretation of the depreciation rate estimated from equation (7). When (7) includes such performance characteristics, [differential](lnP)/[differential]A measures the age-related change in price after controlling for quality differences across models within a homogeneous class--that is, after controlling for factors that induce obsolescence. The resulting measure of depreciation would capture the effects of deterioration but not of obsolescence. By omitting such performance characteristics from z, the depreciation estimates in this paper embed the full effects of aging on price, the measure of depreciation typically estimated in the literature. However, see Oliner [1993; 1994] for a discussion of alternative measures of depreciation and their uses in constructing capital stocks. (9.) Specifically, I used the results in their Tables 5.2 and 5.3 for the post-1973 sample. (10.) For grinders, the same procedure yielded an implausible 72 percent annual rate of depreciation, which suggests that their tables contain a typographical error. This suspicion is buttressed by noting that the fitted age-price profile for grinders in their Figure 5.4 indicates a (post-1973) depreciation rate no more rapid than that for turret lathes.

REFERENCES

Beidleman, Carl R. "Economic Depreciation in a Capital Goods Industry." National Tax Journal, 29(4), 1976, 379-90.

Brazell, David W., Lowell Dworin, and Michael Walsh. "A History of Federal Tax Depreciation Policy." OTA Paper No. 64, Department of the Treasury Office of Tax Analysis, May 1989.

Cook, Nathan H. "Computer-Managed Parts Manufacture." Scientific American, February 1975, 2229.

Hulten, Charles R., James W. Robertson, and Frank C. Wykoff. "Energy, Obsolescence, and the Productivity Slowdown," in Technology and Capital Formation, edited by Dale W. Jorgenson and Ralph Landau. Cambridge, Mass.: MIT Press, 1989, 225-58.

Hulten, Charles R., and Frank C. Wykoff. "Tax and Economic Depreciation of Machinery and Equipment. " Phase II of Economic Depreciation of the U. S. Capital Stock: A First Step, submitted to U.S. Treasury Department, Office of Tax Analysis, 1979.

--. "The Measurement of Economic Depreciation," in Depreciation, Inflation, and the Taxation of Income from Capital, edited by Charles R. Hulten. Washington, D.C.: Urban Institute Press, 1981a, 81-125.

--. "The Estimation of Economic Depreciation Using Vintage Asset Prices." Journal of Econometrics, April 1981b, 367-96.

Mood, Alexander M., Franklin A. Graybill, and Duane C. Boes. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill, 1974.

Oliner, Stephen D. "Constant-Quality Price Change, Depreciation, and Retirement of Mainframe Computers," in Price Measurements and Their Uses, edited by Murray E Foss, Marilyn E. Manser and Allan H. Young. National Bureau of Economic Research, Studies in Income and Wealth, vol. 57. Chicago: University of Chicago Press, 1993, 1961.

--. "Measuring Stocks of Computer Peripheral Equipment: Theory and Application." Working Paper, Federal Reserve Board, Washington, D.C., 1994.

Department of Commerce, Bureau of Economic Analysis. Fixed Reproducible Tangible Wealth in the United States, 1925-89. Washington, D.C.: U.S. Government Printing Office, 1993.

STEPHEN D. OLINER, Board of Governors of the Federal Reserve System. This project would not have been possible without the assistance of the Machinery Dealers National Association, whose member firms provided the data used in this paper. Among the individuals with the Association, special thanks are due to Clyde Batavia, Dennis Hoff, and Darryl McEwen. I also would like to thank John Musgrave, Bill Wascher, Frank Wykoff, and an anonymous referee for helpful comments. Jeff Crawford of the Bureau of Economic Analysis kindly provided unpublished data on the composition of investment in metalworking machinery. The opinions expressed in this paper are mine alone and do not necessarily represent those of the Board of Governors or its staff.