TAX AND ECONOMIC DEPRECIATION OF IIA(7HINERY AND EQUIPMENT 

A Theoretical and Empirical Appraisal 

Phase XI Report 

Economic Depreciation of the U.S. Capital Stock: 

A First Stee 

Submitted to 

The U.S. Treasury Department 

OFFICE OF TAX ANALYSIS 

Washington, D.C. 

< -

July 26, 1979 

Submitted by 

FRANK C. WYKOFF 

Eldon Smith Associate Professor 

of Economics, Pomona College and 

Claremont Graduate School 

(714) 621-8000x2970 

CWES R. HULTEN 

Economist, Public Finance Group 

The Urban Institute 

(202) 223-1950 XS19 

*Note: 	 This is a preliminary report and may not be quoted 

or referenced without consent of the authors and 

the Office of Tax Analysis. 

c 
- Acknowledgments <I 

d number of individuals provided a great deal of help, vork and 

encouragement to us as we undertook this research. We certainly could 

not have done it alone. While all errors and opinions stated here are 

our own, and are not necessarily held by anyone else or any organization, 

ve wish to thank many people by name. 

A number of private economists and Treasury economists have advised 

and helped us throughout. Special thanks to Larry Dildine. Barvey Galper. 

Hike Kaufman, Seymour Yiekawsky and Gary Robbins from Treasury and to 

Robert Coen, Martin Feldstein, Dale Jorgenson and Paul Taubman. Invaluable 

research assistance was provided by Kazim Darbandi, Janet Furman. Marc 

Hellmen, Waseem Kahn, David Kennard, Tom O'Toole and Chang Min Shin. Data 

and interpretation of data vas provided by Lloyd Somers at the Shephard 

Caterpillar Dealership. GSA assisted in obtaining auction data. Excellent 

computer prograrmning advice was given by Donald McIntyre. 

We a150 profited from comments by referees at ehe Western Economic8 

Association Heeting. Summer, 1978 and at the Eastern Economics Association 

Heetings, Spring, 1979. 
ECOKOMIC DEPRECIATION OF THE U.S. CAPITAL STOCK: A FIRST STEP

I. INTRODUCTION AND SUNWIRY: CWNTRACT ROLE AND OBJECTIVES 

A. The Role of this Report in OTA's Research Plan 

This paper consists of the second phase of a three-phase research contract 

let by the Office of Tax Analysis (OTA). OTA, within the office or. the btcretary of the Treasury, let this contract for two central purposr,. ~LZ:~ to 

provide a publicly defensible set of initial estimates of the actual rates Iof economic depreciation of the major assets which comprise the U.S. capital 

stock. As vas made clear in the contract statement, and which will be reemphasized below, many of the estimates which will be provided idehis report 

are based-more on judgment than on analysis. However, two major analytic 

contributions, which will provide the foundations for future analysis, are 

contained in this report: first, we provide a detailed methodology for 

estimating economic depreciation from 'data on used asset prices. Second,

., 

we implement this methodology for a number of specific assets which represent 

a rather large proportion of the total stock, so that a definite starting point 

is provided for subsequent measurement efforts. 

This contract also makes progress tward another major advancement for the 

Office of Tax Analysis. For many years, OTA has analyzed tax laws and proposals on a nearly case by case basis. While utilizing a data-tax-model, 

OTA has not had available to it a single major analytic model from which it 

Could draw definitive quantitative conclusions. However, under the leadership 
OTA. this situation began to change in 1976 and 1977. It became evident to 

these analysts that policy recommendations from the government should be 

based upon a coherent analytic model of the tax system so that the proposals 

presented and evaluations undertaken wer a wide range of topics and over 

a long period of time would be internally consistent with one another. The 

conviction that such a coherent and consistent framework could be built was 

tbely indeed for a number of major research breakthroughs provided just 

the model needed to meet this OTA objective. Two major strains of research 

were brought together in the late 1970's to provide OTA with the analytic 

and quantitative material necessary to develop this model. The first 

strain was launched by Arnold C. Harberger in (1962). Harberger developed 

a model in which one could determine the incidence of a corporate income 

tax imposed on one industry in an economy containing two industry groups. 

Herbert Scarf in (1969) then developed a converging computer algorithm for 

quantitatively measuring the set of general equilibrium prices for an 

economy with, at least conceptually, any number of industries based upon 

their supply and demand schedules. Applying Harberger's tax incidence 

4

--- > -_- ----,=is to Scarf's computer algorithm one could obtain quantitative measures 

of the impact by industry of a change in the tax code. While such analysis 

is still to some extent in its infancy, several of Scarf's students, especially John Shoven and John Whalley, have actually developed a general 

equilibrium model with the Scarf computer algorithm for a large set of 

industries and for a variety of types of taxes. A number of scholars have 

since been working on this type af computer algorithm in order to evaluate 

taxes. However, 8 major difficulty with these models from a practical point 

of view has been the poor data base available far the analysis. It is 

2 
ta72.: s*:cci,5 y>cnLi?.im T.: i:iij r,::! <,::A+;:i?cgn:l :~t~xt~ . ,. 0.; ;;?~;s,xc~ j.:?, z~1cxr.x 

years has been addressed. 

This second body of research began vlth the famous studies in the 

early 1960's by Dale Jorgenson. Jorgenson was one of the first econonists 

to fully appreciate the ability of economics to integrate its conceptual 

ideas with the powerful data base provided by the U.S. government. 

Jorgenson, with a number of collaborators, provided empirical estimates of 

U.S. investment demand for a number of industries. Jorgenson showed that 

one could provide reliable estimates of investment requirements using 

actual data on U.S. capital goods prices and quantities when emplophg a 

neoclassical capital demand framework. Central to Jorgenson's approach to 

investment was the notion that neoclassical economics provideb'the analytic f., 

basis for investment demand. Three major components to Joqgepon's 

investment model are essential: (I) a flexible accelerator represents the 

demand for investment, (2) an aggregate production function represents the 

underlying demand for capital and (3) a user-cost-of-,capital measure 

represents the price of capital goods. Throughout the 1960's and 1970's 

T-_.__- --33 and his collaborators consistently improved their measures of 

.I 

rho c*'?ntities and prices of capital. goods. 

Nevertheless, the central problem in Jorgenson's work, from a 

.S2zzre!nent point of view, continued to be the difficulty of measuring the 

:isintity of capital in place. Jorgenson was one the first economists to 

appreciate the importance of maintainbg an internally consistent depredation 

model. In an important SU"~TJT work, "The Economic Theory of Replacement 

and Depreciation," in (,1973)Jorgenson applied this concept of internal 

consistency to show that it is necessary in a coherent model to utilize 

a method of depreciation whi,ch is consistent with the method one uses for 

the replacement of capital. 

3 
 
?ttco~,~~,.z%t.1~ th.e velue of providl.n.q 2, data base which is as coherent and consistent as the theoretical model itself, OTA turned to Jorgenson to provide 

for the quantitative basis with sufficient quality to provide theadvanced conceptual frarrework and computer algorithm being used by Shoven. 

Employing the concepts developed by Hall. in (1968) and by Jorgenson in 

his earlier investment studies, Wykoff in (1970) developed ,rb usercost based stutly of capital depreciation. 'k'ykoff employed the theory of 

depreciation and replacement to actual empirical estimates of the depreciatio 

of automobiles in the United States. Later, under the auspi,ces of OTA, 

= 

Charles Hulten and Frank k.yk,off in (1975) and in (1977) extsnded the nethodology developed by Jorgenson, Hall and Wykoff and applied this new methodol.ogy 

to the study of economic depreciation of commercial and industrial, structures 

It became evi,dent to OTA at this time that Hulten and Wykoff could provide 

/ 

estimates of economic depreciation which could be used. In turn, by Jorgenson tc 

to develop measures of capital and'tnvestment flows by industry. This would 

provide the kind of measurement base needed by ShoGen so that OTA could Fmp1.e 

ment his model. 

Thus, in 1975 and 1976 OTA began to develop a model which brought to­

/

gether these two major branches of research. The ultimate objective wiU be 

a computer al.gorithm, based upon actual estimates of the 1J.S. capital stock, 

for evaluating various types of business taxes and for estimting the impacta 

by industry of various proposals to change the tax laws. With this capabil.it 

OTA will have an intarnally consistent conceptual model with the highest quali,ty data base available. Furthemore, OTA will be able to continually Upgrade both the data base and the conceptual franework as new breakthroughs 

are made in the economics profession. In other words, wi,th the culmination 

of this major research effort on t>e part oft OTA, it wi.11 have developed both 

4 
 
a model for analyzing all major tax questions and a foundation for building 

d .yes ...*.baairc~: (.ci,sbf:i.p*i. i3f.o 2tie jcLefjJEs>j,: Lut:.;:,:* 

We turn now to a discussion of the specific contributions of this report 

to the requirements of OTA. It will be recalled that this project addresses 

several specific tax issues in i.ts own right which are qui,te important in 

light of some of the major controversies concerning today's tax code. 

The accurate definition and measurement of the tax base 1.6 an important consideration in the administration of any tax. DistnWinn- fn ch* base 

of a tax can lead to violations of the standard canons of equi,ty and efficiency, 

and to popular dissatisfaction with the tax. Unfortunat-':, ---* ---- %..---

present some difficulty in this direction, but few present more problems than 

the taxati.on of income from capital. 

The di,fficulty in defining the base of the tax on capital income lies 

primarily io the distinction between accrual and realization. Many components 

of capital income-capital gains, depreci,ation, inventory revaluation-accrue 

during a tax period but are not realized in any market transaction. Consequently, no di.rect test of the size on these accruals is available, and indirect methods are required. In this study we focus on one particularly 

trn11hl-=ome component of capital income-economic depreciation. 

/ 

Economic depreciation 1.5 the amount of money whkh must be replaced 

in order to keep the ori,ginal capital investment intact. It arises from the 

fact that some forms of capital--notably plant and equipment-are used up or 

bEioze obsolete in the course of generathg income. The Federal Income,Tax 

Code has. since its inception in 1913, recognized the principle of all,owing 

a deduction for depreciation of capi.tal assets. Major difficulties have, 

hatever, arisen in the attempt to imp1,ement this principle. Many approaches 

have been tried and rejected, and the recent collapse of the Asset Depreciation 

Range vintage reporting system signals yet another period of controversy over 

5 
 
&pX(.cia ;ioa p ":; ta'luras ~ '!:j8; i; Cc'I>:: yo (,<&r:;y ~.~J;..:: ?,yti:, c:~-,'bcJ: ,~,'" t';! :::::;e:"iy;y;.. ~ 

of whether the Treasury and Congress should continue in their attempt to base 

depreciation allowances on actual taxpayer experience, somehow measured, or 

whether the Treasury should recognize the near impossibility of measuring 

this component of economic income and provide more or less arbitrary, but 

administratively feasible, guidelines for depreciation allowances. 

The revaluation of assets for depreciation purposes is another controversial area of tax reform. See Aaron (1976). The tax code currently allows 

depreciation deductions to be based on the original cost of an asset. The 

inflation of recent years has. however, caused the prices of new and used 

capital assets to increase. Rising asset prices lead to rising replacement 

costs which should be taken into account when defining taxable income. 

B. Primary Objectives of this Report 

Recognizing the above policy p;oblems and planning its new analytic 

tax model, OTA decided to determine the feasibility of developing empirical 

depreciation estimates for a variety of asset classes with special emphasis 

on producer durable equipment. The Contract Work Statement clearlx states 

vu= L fiht objectives of this study: 

Employing the multiple asset model of economic depreciation and 

the econometric models of estimation outlined (in the Work 

Statement) above, average relative productive efficiencies and 

average economic depreciation rates for the various classes 

of as'sets will be estimated within several broad asset categories: (A) Machine Tools, (B) Vehicles, (C) Heavy Duty 

Construction Equipment, and (D)' possibly additional asset 

classes specified in Tables 2 and 3 (of the Work Statement). 

The first purpose of this report. then, is to measure the actual depreciation and revaluation of some, but by no means all, types of plant and 

+ipQent. Our approach is based on the analysis of the market prices 

of used capital goods. The observed market prices of used (of. "Vintage") 

6 
capital should decline in valire as it ages precisely because the capital asset 

is used up in production or because it becomes irosolete, by woasuring and 

correctly interpreting the vintage price effects, insight can be obtained 

about the reasonableness of depreciation policy. The use of vintage prices 

as a means of assessing depreciation policy is hardly new. but thls approach 

has only slowly been gaining widespread acceptance among economists because 

of the long held view that used asset markets do not exist for most assets, 

and that the markets that do exist are too thin to provide meaningful data. 

(A discussion of exlsting studies appears in the Phase I report.) There 

has been, furthermore, skepticism about whether assets which do appear in 

used good markets are representative of thcse which never enter the market 

place. 

In Phase I of this contract we confronted these arguments and reached 

the following conclusion: The market data for used capital are considerably 

richer than the conventional wisdom suggests. Used buildings, autos, trucks, 

machine tools, office equipment, electrical equipment, and construction equipment are all transacted in reasonably active resale markets. Nhile this list 

hardly encompasses all fixed capital assets, it does account for a surprisingly large fraction of total fixed investment. Equipment catcsories for 

which we have found vintage price data account for 55% of 1977 investment 

expenditures in producer'durable equipment, and structure categories for which 

data exists account for 42% of 1977 investment expenditures for nonresidential 

Structures. 

Sicond1yD we argued that while some vintage prices be biased <omward, the direction of the bias favors the taxpayer at the expense of the 

Treasury. This is not hecessarily inappropriate, since recent tax 

Practice generally requires that the Treasury not disturb depreciation 

Chimt without good reason, and any bias in favor of the taxpayer provides 

a margin of error for the Treasury. 
a meaningful source of information, we now, in this report consider the econometric problem of obtaining estimates of the depreciation process and of converting these estimates into estimates of the relative productive efficiencies 

of specific assets. In the conceptual sections of this report we discuss 

difficulties associated with inflation, asset retirement, obsolescence and the 

endogeneity of depreciation. Several explicit econometric models are outlined 

and discussed in some detail. A new econometric model is also developed in 

this conceptual section. These models are than applied to thirty specific 

asset groups. These thirty types of assets represent seven classes of producer durable equipment, two classes of private nonresidential structures and 

one class of consumer durable assets. These ten asset categories contain 

nearly 50% of the entire stock of fixed capitil in the United States. The 

econometric addendum of this Phase IT Report contains in extensive detail 

the analysis of these thirty specific assets organized by the relevant asset 

classes (needed by Jorgenson and Shoven), This econometric addendum, consisting of some 1200 pages, thus represents an attempt to provide a 

defensible set of estimates of the depreciation process for the entire 

, stock of l7.S. capital assets, which embodies information obtained from 

the market for these assets. 

One major result of policy significance that follows from this 

analysis is that the pattern of economic depreciation of machinery and . 

equipment appears to be accelerated relative to the straight line pattern. 

This result suggests that accelerated forms of depreciation such as those 

now allowed in the U.S. Ta Code-dedining balance and sum of years digitsare warrented. We also found in an earlier study undertaken for OTA, TOS74-27, that accelerated forms of depreciation tare warrented for structures 

88 Well. However, it appears that the available tax deductions permitted 

0th private nonresidential structures and producer durable equipment 

5 
T'Le Lecoa< cenp:a2 p':F:>Se of tti.Jo Tfr]OZt also clearly ,>?.~,J:fij:.:.iite~ 

in the Work Statement of the contract. 

For asset classes in which data is insufficient for full 

econometric estimation, other methods, with supporting justification, will be employed for making the required estimates. ... In cases where data is insufficient, the best 

professional judgment will be used for making the required 

~ estimates of depreciation. 

While we feel that the data we have is reasonably useful for 6 producer 

durable equipment classes (hereafter referred to as PDE)and 2 private nonresidenti.al structure classes (PNS) and 2 consumer durables classes (CD) , 

we have only partial information on 2 PDE classes, 2 PNS classes and 2 CD 

classes. Furthermore, we have no actual data on the remaining asset categories-namely 14 PDE classes and 9 PNS classes and 5 CD classes. In term 

of the vol.ume of capital represented we have reasonably good estimates representing approximately 47% of the U.S. capital stock and oily partial iniv 

formation for the remaining 53% of the stock. Consequently, the second purpose of this report will be to convert the detailed estimates we have for 

specifi.c assets into depreciation estimates and productive efficiency estimates for the 22 PDE classes and the 10 PNS classes. As indicated in the 

Work Statement and again in the Phase I Report of this contract, tte depreC-LAVLI L~L~S and efficiency estimates for the asset classes for which bie did 

not have detailed data are based upon judgment. Perhaps the next step i.n 

conrinuing study of the depreciation problem should be to try to provide 

both a methodology and some actual estimates of depreciation for those classes 

not covered in detail by this study. 

In addition to the two objectives outlined above for this report, 

namely the detailed study of specific assets and the extension to estimates 

for the major PDE and PNS classes, verbal requests on the part of Treasury 

fficials indicated a desire to also obtain esti.mates for consumer durable 

ssets- "hi,s problem is somewhat more difficult than the earlier two problems 

orrs.!r-,zes ti:?: &f~:~Q~.s.y>,,-Jf:::~j::~L>, :ju:r ..:;*l.::., <<sGC?L; ;.,ru..l:ic!.::2:::: ;.is we><7­

as the depreciation rate estimates which provide the basis for the final set 

of depreciation:and efficiency measures suggested to the Treasury for its 

overall study. The third and final section of this report. entitled “Judgmental. Estimates of Depreciation and Eff iciencies for U.S. Depreciable Capital 

Stocks,“ contains a discussion of the decision roles and problems encountered 

in converting the specific asset by asset depreciation estimates into judgments of depreciation for large asset classes. 

C. A Brief Overview of the Phase I Report: Assessment of the Quality 

and Availability of Data on Vintage Prices of Machinery and Equipment 

Phase I was a report of a major data search undertaken for this contract. 

The outcome of the Phase I Report was a body of data. to be studied here, on #? 

specific assets. To assist in our summary of Phase I, Table I lists the 

major asset classes for which estimates are required in the Jorgenson-Shoven 

analysis. From Table I depreciable assets are seen to fall into three broad 

categories: (A) Producer Durable Equipment (PDE) I (B) Private Nonresidential 

Structures (PNS), and (C) Consumer Durables (CD). PDE contains twenty-two d 

CLdbbIZb, YNS and CD have 10 and 7 respectively. The search undertaken in the 

Phase I Report consisted of studying three types of sources: (1) existing 

Library sources or bibliography in economics, business and engineering, (2) 

Cm”mrcia1 and industrial sources, or published price series used in various 

industries and (3) government agency sources (especially the General Services 

Administration and the Treasury Department itself). On the basis of i3IJ.S data 

search, the 22 PDE classes, 10 PNS classes and 7 CD classes were partitioned 

into three types of asset classes based upon the availability of data for 

research. These asset categories are referred to as Type A, Type B, and 

11 
 
Table 1 

MOR ASSET CLASSES 

(A) Producer Durable Equipment 

1. Furniture and fixtures 
 

2. Fabricated metal products 
 

3. Engines and turbines 
 

4. Tractors 
 

5. Agricultural machinery (except tractors) 
 

6. Construction machinery (except tractors) 
 

7. Mining and oilfield machinery 
 

0. Metalworking machinery 
 

9. Special industry machinery (not elsewhere classified) 
 

10. General industrial equipment 
 

11. Office, computing and accounting machinery 
 

12. Service industry machinery 
 

13. Electrical transmission, distribution and industrial apparatus 
 

14. Communications equipment 
 

15. Electrical equipment (not elsewhere classified) 
 

1-16. Trucks, buses and truck trailers 
 

17. Autos 
 

18. Aircraft 
 

19. Ships and boats 
 

20. Railroad equipment 
 

21. Instruments 
 

22. Other 
 

(B) Private Nonresidential Structures 

1. Industrial 
 

2. Commercial 
 

3. Religious 
 

4. Educational 
 

5. Hospital and institutionai 
 

6. Other1 
 

7. Public utilities 
 

0. Farm 
 

9. Mining exploration, shafts and wells 
 

10. Bther2 
 

(C) Consumer Durables 

1. Motor vehicles and parts 
 

2. Furniture 
 

3. Kitchen and household appliances 
 

4. Radio and television receivers, recorders. musical instruments 
 

5. Wheel goods, durable toys, sports equipment 
 

6. Residential structures 
 

7. Other 
 

(D) Inventories 

1. Farm 
 

2. Non-farm 
 

(E) Land 

'Consists of buildings used primarily for social and recreational activities 

fmi buildings not elsewhere classified. 

*Consists of streets, dams and reservoirs, sewer and water facilities 

12 
 
Type C asset classes. Type A asset classes are those for which we have 

extensive data and with which we app1.y our methodology to provide what we 

consider to be reasonably reliable estimates of econ0mi.c depreciation for 

those classes. Type B asset categories are those for which we have found 

some existing studies or for which we have some data but which we do not 

consi.der to be either sufficiently reliable nor suffici,ently extensive to 

warrent defensible estimates based sole1.y on the data. Type C asset cztslgories are those for which we have no data whatever. Tablc Z.zmca$;.s the 

partitioning of asset classes from Tab1,e 1 hto the three types of asset 

groups- Within the Type A categories asset classes fa1,linco Lnree SCP 

groupi.ngs. The first subgrouping consists of PDE classes 4, 5. R, '?, 16 

and 17. These asset classes' estimates are based upon the analysis reported 

in this Phase II Report, Secti.on 2. The consumer durable class 1, Autos, is 

also studied in detail in this report. The two PNS class estimates are 

based upon the extensive study undertaken in Contract TOS 74-27. The remaining 

asset-category labeled as a Type A asset is the consumer durable class of 

residential structures. We believe the two studies undertaken of depreciation of residential structures by Weston and Leigh are reliable enough to 

include this as Type A assets. / 

While of thirty-nine possible asset classes listed in Table 1, we are 

only placing ten into the Type A category. the proportion of U.S. depreciable 

capital which falls into the Type A category is quite large. Based on total 

U.S. private purchases of new assets I.n 1976, the six classes of PDE for which 

we shall provide reasonably good estimates on some assets contained nearly 

502 of the total producer durable equipment sales. Type A assets cover 42% 

of the total PNS purchases, and 66% of total 1976 CD purchases. 

A caveat was mentioned in the Phase I Report which bears repeathg here. 

Some of the PDE, PNS and CD asset categories are quite broadly defined and our 

13 
??blf< 2 

TYPES OF ASSET CLASSES BY AVAILABILITY 

OF DATA AND RELIABILITY OF ESTIMATES 

Type A 
 me B Type C 
 

.PDE 4 
 PDE 11 PDE 1-3 
 

6 
 19 5 
 

8 
 7 
 PNS 5 
 

10 
 9 
 6 
 

16 
 12-15 
 

17 
 CD 2 18 
 

4 20-22 
 

PNS 1, 
 

2 
 PNS 	 3 
 

4 
 

CD	 1 
 7-10 
 

6 
 

CD 	 3 
 

5 
 

7 
 

14 
 
-- 

data appli.es eo only a portion of the assets in these broad categories. It 

is unrealistic to think that our estimates repr.k%:-.t compre.. ai;;t!.,.ic coverage 

of the mi1l.ions of types of specific machinery employed in the U.S. and this 

is true even for the asset categories which are listed under Type A. 

We turn now to illustrate the data co'ntained i,n Phase I which forms 

the basis for the analysis in this report. We illustrate the detai.1 of the 

data provided in the Phase I Report by using as an example one subclass of 

assets-the D-7 Tractor. 

Our tractor data consists of the prices of used tractors reported i.n 

Blue Book of Current Narket Prices of Used Heavy Construction Equipment, 

Forke Brothers Blue Book Co.. Lincoln. Nebraska, 1968-1977. These prices 

reflect actual transacti.on prices of individual units sold on open aucti,ons 

in the U.S. In some instances, prices may reflect units which are not 

actually sold to a new owner but are paid back at a pre-arr&ged price to 

the original owner. These "paybacks" reflect the in-use value to the 

existing owner. Host prices, according to Forke Brothers and industry 

sources, do reflect actual sales. Units sold at auction are thought by industry sources to be representative of tractors in place. Tractors are usuaL*y oought and sold at auctions often by dealers who acquire tractors, new 4 

-__._Cmspecific projects. When projects are completed, dealers sell 

off their capital to other users in order to liquidate until they arrange a 

.. .--9-" ..--r-wd-.Lt.Used tractors are also sold by various agricultural companies 

much as used automobiles are sold by households. However, perhaps unlike 

automobile buyers, used tractor purchasers appear to be rather sophisticated 

dealers with some knowledge about machinery. Consequently atic tion prices 

do not appear to suffer from the type of lemon bias suggested by Ackerlof 

in (1970). 

15 
 
-- -- 

-- 

-- 

Tractors come in many shapes and sizes and may be used for a variety 

of purposes from farming to road construction to dam building." Tractors 

ere often sold with ancillary equipment including winches, rippers, cnble 

control units, canopies and the like. Furthermore, within a general sizeclass of tractor, say 0-7, indicating a large, heavy (25,000 to 35,000 Ibs.) 

tractor usually, though not exclusively, used on agricultural jobs, each unit 

often has unique characteristics. Major distinctions are indicated by engine 

letter types, but other distinctions are also indicated by different engine 

serial numbers. We standardized tractor prices by pricing ancillary equipment and by determining the relative prices of variouz engine types. Asset 

prices were modified so that each price represented the price of a D-7 tractor with a straight dozer and a ROPS canopy (after 1971), having standard 

equipment only. Thus ancillary equipment prices were deducted from sale 

prices. Table 3 illustrates the ratios used to standardize prices. 

Table 3 

RATIO OF STANDARDIZED TRACTOR TO 

TRACTORS WITH ANCILLARY EQUIPMENT* 

Ancillary Equipment 

(1) Type of Bulldozer 

Bare 

Straight

U 
 Ansle 

caihe 

Ripper

Winch 

0-7 Ripper 

Kelly Ripper 

#29 C.U.C. 

ROPS canopy 

Type of Engine 
 

E F G 
 

.' 

-

1.133 

1.000 1.000 1.000 

.986 .986 .988 

.997 1.002 1.017

1.017 I - .926 -
 -876 .895 

.902 .899 .912

' .965 - - 980 - - 1.148 _" 

_. 

4he prices of ancillary equipment were found in various issues of Green Guide, vel. I: The Handbook of New and Used Construction Equipment Val%, Equipment 

Guide Book Co. I Mountain View, CA. ~ and in Sale Kit 11, Caterpillar Tractor COWI 

Peoria, Illinois. 

16 
 
The actual prices used in the analysis are summarized in Table 4. 

The sample contains 5132 observations, and covers years from 1963 to 1977, 

and ages one to thirty-five. Figure 1portrays the average age-price pattern 

for the sample as a whole. Each observation is deflated by a price index of 

a new asset. 

The retirement distribution used in this study to weight the observed 

prices is taken from the Iowa Engineering Studies undertaken i,n the 1930s 

predominently by Robley Winfrey as reported in Marsten, et..:. !I?:?) F.%se

retirement distributi.ons report the percent survi.ving of an original cohort 

of assets according to a given probability distribution :I-: :!L :--:----

class Iiife. The Winfrey L5 distribution was chosen for this studv. After 

conferring with industry sources, we selected 25 years as the average retirement age for tractirs-few tractors are retired before 20 years, about lo%, IC 

then by 25 years only 47'6 of the original cohort remain. 

The Phase X Report contains data analysis of a total of 26 apecifi.c 

assets. Table 5 contains a list of these specific assets organized by asset 

cl.ass. _:1 

D. Summary of Major Results from Phase I1 

4 

'7) "-c"lts for Specific Assets: 

Part E1 of this report includes a description of four different methods 

i?r ecr!rrating economic depreciation from vintage asset prices. In this 

-___-.. LeyVlL;hese four different methodologies were applied to estimating depreciation for 26 different specific types o� assets. In addition, this same 

methodology was applied earlier for the Treasury to a study of a dozen different types of commercial and industrial structures. All combined, then, 

we have studied the economic depreciation process of over 30 different assets 

ranging from machine tools, trucks and construction equipment to commercial 

17 
Table 4 . -. .. ... -..1 ,. .. . .. 

s23.1.~~ 6374.67 I 7221.00 i 9797. 

81 

i 

3 1 51 - 1081.?1 I PRP.81 I 1503.57 I 4E64. 

8726. 

4893. 

6731 * 

6357. 

! 

3411. 

¶PO8 . PJIP. 

56Rj 

3535. 

4762 

Xl?. 
 

3791 

184'" 

lsno~ 

12315, 

18 
 

7 
.2OI318.00 

46103.90 

20972.70 

G 

10512.30 

15742.50 

13127.40 

-1.d 

I 
 

I $ 
 I . 
 I 
 

t 
 I B1 

I 
 

t 
 

I 
 

I 
 I 
 

I 
 

t ' 

I 
 

I 
 

I 
 

I 
 

I 
 1 

I 

I 
 8 

t 
 X 
 d 

t 

I 
 

I 
 

I 
 

7097.20 	 t 
 I 
 

I 
 

I 

I 
 

52n2.10 k 
 I 
 

I 
 

I 
 

t 
	 

B 

U 
Qhlc 5 

ASSETS STUDIED IN DETAIL BY ASSET CLASS 

Producer Durable Equipment 

4: 	 Tractors: D-4 Tractor 

D-6 Tractor 

D-7 Tractor 

D-8 Tractor 

D-9 Tractor 

6: 	 Construction Machinery (except tractors): 

Air Compressor 

Motor Grader 

Rubber Tired Loader ._ 

8, 10: Metalworking Machinery and General Industrial Equipment:

MPG g-Milling, drilling and boring machines, small 

MPG 12-Drilling machines and boring machines, large 

MPG 19-All other tools 

11:	 Office, Computing and Accounting Machinery: 

Remington Typewriters (electric) (GSA) 

16: 	 Trucks, Buses and Truck Trailers: 

GMC Pickup Truck (half-ton) 

Ford Pickup Truck (three-quarter ton) 

Tandem Truck Tractor (6-wheeled rig) 

Tandem Dump Truck (ten ton) 

U: Autos: GSA Chevrolet 

GSA Ford 

GSA Plymouth 

Consumer Durables 

1: 	 Motor Vehicles and Parts: 

Buick 

Cadillac (DeVil.le) 

Chevrolet (Nova) 

Chevrolet (Stationwagen 

GMC Pickup Truck 

Ford Pickup Truck 

Plymouth 

Volkswagen 

, 

Standard) 

20 
 
and industrial buildings and to consumer automobiles. Central to our study 

was a test to determiiie whether geometric or straightline depreciation is an 

appropriate form. Our statistical finding was to reject both geometric and 

straight line depreciation process. However, in general, the analysis of 

depreciation and of the productive efficiency sequences indicates an accelerated pattern relative to straight line. In other words, the age-price 

patterns tend to be distinctly convex. While this convex pattern could 

possibly be the result of biases in vintage asset prices, as discussed in 

the addendum to the Phase I report, convexity appears for asset classes 

which are not subject to severe secondary market problems. Consequently 

it is unreasonable to ascribe the convex pattern to biases in the data. We 

conclude that depreciation appears to be very generally one of convexity. 

The four basic methodologies employed in our study were: (1) the polynomial regression, (2) the Box-Cox power transformation, (3)-the Box-Cox 

model on retired prices, and (4) the Box-Cox model with a truncated distribution. Each of these methods is discussed either in the appendix to the 

Phase I Report or in Part 11 of this report. The effect of retiring assets 

and then estimating the depreciation process seems to be to lower the depreciation rate for the early years but to significantly raise the average rate 4 

02 ewIiuuuL depreciation over asset life. In other words, the average annual 

rate of economic depreciation when one accounts for the retiremeni: of assets, 

8s well as for in-place loss in value, is greater when onefailsto account for rethment. Depreciation rates typically increased from say around 9% to around 

11Xe or from 14% to 18%. In some cases the percent increases from retirement 

were dramatic. For example. In the structure classes. the average depreciation 

rates were increased by retirement from about 11/22 to about 2 112%. Usually. 

however, allowance for retirement has only modest effects because retirement 

takes place late in an assets8 life when little is left in the productive 

22 
',+rj 

process anyway. In-place loss in value plays a greater role in the depreciation process than retps-me'it 3,tsaj.f. X+3 .'_ ...,, i. ~ ,.,.._.. q Lk..,?L?.!, -.jr G1i.L;T.i ic; '?!>?Xt

aut by the empirical evidence, seems intuitively obvious, it has not been 

generally accepted by the economics profession. Many economists persist 

in arguing that assets tend to be one-hoss shay in nature and that the retirement process is the major force for depreciation. This study suggests 

that this conventional wisdom is not correct. 

In addition to retiring assets according to a predetermined retirement 

distribution, we also tried a method developed for dealing with censoredsample problems. The procedure is to treat retirement as a stochastic process. OSed assets are randomly dropped from the sample populati.on of the 

original cohort only if their prices fall below some minimum level. The 

result is a truncated distribution. 'While we have no strict test 	 procedure

I c 

for choosing between depreciation estimates with retired data and those 

derived from the truncated distribution, we can compare the two approaches. 

Truncating the distribution at some low price increases the average depreciation rate for older assets but hns.only negligible effect on newer assets. .'1 

The reason for this result is obyious, the distribution of new asset prices 

rarely dips as low as the truncated level. The effect of retiring assets 4 

is more substantial and in some cases even reverses the pattern of depreciation 

from accelerated to decelerated. 

The choice of retired or truncated depreciation must depend upon one's 

assessment of the theoretical plausibi,lity of the two stories. The reti,rement distribution approach has the advantage that it can be, as shown by 

Hulten and Wyitoff in (1976) fully integrated into the Hotelling-Hal.l.-Jor-, 

genson model of replacement and depreciation. Furthernore, the retirement 

di.stributioa is an extension of the perfect foresight assmption utilized by 

these authors. The principal advantage of the truncation approach is that 

ia 
 
+,. 

-- 

the retirement process is seen to be a_stochastic process which depends on 

&L k3 ~d.~p~~~~g -us,l+;e 9;: -iir st ~~j~~~ c,: ::;?tira.;;;p!: 2.3 :I :;c; L..,.:;;,L>v ., I 

on its age. Furthermore, it is not necessary to assume, under the truncation 

approach, that owners of assets have some prior knowledge of when their particular asset may be retired. Nor is it necessary to assume that re&ired 

assets and unretired assets all have the same characteristics while in place, 

an assumption which is probably unrealistic but which is necessafy under the 

reti,rement distri,bution approach. For purposes of this Ph=se TT Qn:--*; ve 

choose to use the predetermined retirement distribution approach. The truncation approach is rel.atively new and has only been appl?-’ - ---:’,-rC 

the assets studied. We do consider this approach to be promisina and shall 

pursue it in the future. 

Before summarizing the depreciati.on results derived from the regression 

equations, it is useful to assess the regressions in terms of goodness-of-fit. 

Whi1.e statistical detail will be presented subsequently, we present at this 

point in Figure 2 a comparison of the actual prices, when retired, to the 

fitted prices from the Box-.Cox method on retired prices. Figure 2 illustrates 

the actual price-age pattern of the D-7 Tractor, using retired prices, from 

ages 0 to 35 compared to the price-age pattern predicted by the Box-Cox pro4 
 

cedure. In both cases, prices are normalized by setting the price of a oneyear-old asset to one. The actual prices presented here are the average prices 

from the entire body of data which are deflated accordhg to the average year 

for each age group obtained from the data. The Box-Cox fitted values are not 

deflated but are predicted at the same average year for whi.ch the actual 

prices have been deflated. (1.n other words the Box-Cox procedure automatically deflates as well as depreciates assets.) It i,sevident from visual 

inspection of Figure 2 that the Box-,Cox procedure tracks the actual pri.ces 

extremely well. 

23 
 
Figure 2 

Y 
Selected rates of depreciation for a sample of assets are shown in Tables 

6A, B and C. Table 6A contains depreciation rar.h'? ::UP prodL::t!r ihiab1.e equipment; Table 6B contains rates for private non-residential structures, and the 

rates pertaining to consumer automobi.les and aome figures for residential 

structures appear in Table 6C. The general thrust of these results are not 

implausible. Structures depreciate far slower than other assets with commercial. and industrial structures depreciating at around 2-3% per year compared 

to residential structures which depreciate at around 1-1. 112%per pear. 

Consumer autos depreciate at around 20-254: per year whereas the producers' 

autos depreciate at a more rapid rate of around 30%. Tractcrc 3-d +r**&­

appear to be depreciating at around a 10-15% rate on average and office equipment and mctal working machinery appear to have depreciation rates in the 

same range. 

As noted above. retirement tends to increase the average annual rate 

of depreciation for all but very young assets. The depreciation processes . 

are quite accelerated for structures, perhaps even more so than geometric 

whereas for automobiles and some types of producer durable equipment the depreciation rates appear to,be flat or slightly decelerated vis-a-vis the geoU=LLI- &ate. Because there is some ambi.guity as to the degree of accelera­

-i_-i. inpredation, we employed the Box-Cox method to calculate declines 

in efficiency. These efficiency functions are not illustrated until later In 

:eyrtD however generally the efficiency functions are more accelerated 

&LL Jiuse produced by a semi-log (geometric) price-age pattern. In other words, 

Our procedures indicate that a flexible functional form produces a depreciation process which is more accelerated than that produced by a direct geometric pattern. 

It Is helpful at this this point to draw a brief comparison between 

our estimates for the various asset classes and existing studies. Our new 

25 
-- -- 

-- -- 

Tqbla 6A 
 

SELECI'ED RATES OF DEPRECIATION FOR PRODUCER
 

DURABLE EQUIPMENT BY AGE (UNCONSTRAINEO BOX-COX MODEL)
 

- TRACTOR (E-7) METALWORKING MACHINERY (WG 12) 

Age Unretired Retired Truncated Unretired Retired 

1 15.3% 8.8% 12.2% 13.1% 19.8Z ' 

5 11.0 9.5 9.9 7.0 11.1 

10 9.9 10.7 9.8 5.6 9.3 

.I5 9.6 12.2 10.5 5.2 9.1 

20 9.7 14.2 11.8 5.1 9.7 

25 10.0 17.1 13.9 5.2 11.0 

CONSTRUCTION MACHINERY (LOADER) OFFICE EQUIPMENT (TYPEWRITERS) 

Age Unretired Retired Truncated IJnretired* Retired 

1 13.0% 

5 11.5 

10 10.8 

15 10.4 

20 10. z 

25 l0.b 

4.8% 14.3% - - 8.3 10.1 3.8% 20.4% 

11.3 10.8 5.8 25.6 

13.9 14.8 7.8 27.4 

16.7 30.6 10.2 27.4 

19.7 _. 13.0 26.6 

"Sample did not contain observations on new assets. 

Unretired Retired 

., 

26 
 
--- 

"CL.:e 33 
 

SELECTED RATES OF DEPRECIATION MR PRIVATI? 
 

NONFSSXDENTIAL STRUCTURES (BOX-COX MODEL)^ 
 

Retail 

Box-Cox 

lb 3.54 

5 2.77 

10 2:47 

15 2.32 

20 2.22 

30 2.10 

40 2.03 

50 1.99 

60 1.96 

70 1.94 

Office Ware- Factory Retail Office Ware- Factory 

house house 

(Transformed) Box-Cox (Untransformed) 

4.32 5.57 3.02 5.39 5.72 6.81 3.00 

2.85 3.68 2.99 2.41 2.66 3.23 2.02 

2.64 3.05 3.01 1.63 1.84 2.26 1.68 

2.43 2.74 3.04 1.29 1.48 1.83 1.50 

2.30 2.55 3.07 1.09 1.27 1.57 1.39 r< 

- 2.15 2.32 3.15 0.86 1.02 1.27 1.25 

2.08 2.19 3.24 0.73 0.88 1.10 1.17 

2.04 2.11 3.34 0.64 0.79 0.98 1.11 

2.02 2.05 3.45., 0.57 0.72 0.90 1.06 

2.02 2.01 3.57 0.53 0.66 0.83 1.03 

cl apercentage decline 

7.. 
 Yactes the age of a new asset. 

27 
 
Table 6C 
 

SELECTED RATES OF DEPRECIATION FOR CObSUMER DWLSS 
 

BUICK 

Automobiles 

CHEVROLET (NOVA) - Age Unretired Retired Truncated Unretired Retired Truncated 

0 16.5% 15.8% 15.1% 19.0% 18.9% 15.5% 
 

1.	 17.0 18.1 14.3 15.5 17.3 15.1 

2 17.8 19.6 15.0 15.1. 17.4 15.2 

3 18.6 21.2 16.3 15.2 18.0 15.5 

5 20.7 24.7 20.4 16.0 20.0 16.8 

7 23.5 29.2 28.4 17.6 22.9 28.8 

PLYWUTH CHEVROLET STATION WAWN 

Age Unretired Retired Truncated Unretired Retired Truncated 

0 23.2% 23.2% 23.4% 19.0% 18.4% 19.0% 

1 20.9 20.9 19.4 18.3 19.6 18.3 

2 21.1 21.1 19.0 18.6 20.7 18.6 

3 21.8 21.8 19.2 19.1 21.8 19.1 

5 24.2 24.2 20.8 20.5 24.5 20.5 

7 28.0 28.0 23.4 22.4 27.8 22.4 

Residential Structuresa 
 Rafael Weston Wilhelmina Leigh (1950-1970)
 Owner-Occupied Tenant-Occ'd. Unadjusted Starts Adjusted Starts 
 

Average 1.6% 1.5% 1.06% .954; 
 

r 

'-- Leigh, Wilhelmina A. "Economic Depreciation for the Residential 

Housing Stock of the U.S.. 1950-1970," Harvard University,

Dept. of City and Regional Planning, March, 1979. 

28 
 
~ 

studies of airtomobiles and tractors follow very closely the results of existing studies by WykoSL’, I-, Ackem:~ aici Griilciies, X U ~ ~iie ~ :.ii:por ~ ~ ~~ 2~ 2

difference should be stressed: the retirement process has a substantial effect 

on the depreciati.on pattern for older assets. The retirement pattern used 
 

for automobiles was based upon actual registration fi,gures from R.L. Polk 
 

& Co. Constructing an actual retirement pattern from these registration 

figures, and retiring the vintage prices accordingly. produced rloFroriation 
 

patterns which were larger on average than the depreciativn i-&:i: --:---

in the literature. 
 

Table 6B illustrates the depreciation rates produL=c ;> =_; 

Treasury study of comercial and industrial structures. A”!-. *x.- ---**- ~ture 

depreciation patterns do not appear surprising and they are consistent across 

r e 

a wide variety of methodologies and a wide variety of assets. These results 

di.ffer sharp1.y. however, from earlier and rather weak ecorometri,c analysis 

by Taubman and Rasche in (1969). We have explained elsewhere why we 

disagree with the Taubman and Rasche results. Specifically, they dealt 

with only five data points and employed a methodology which imposed more 

stringent assumptions about the workings of the economy than were imposed 

here. Furthermore, their econometric testing procedure was of more... luted flexibility than ours. 

Fially, we also undertook analysis on machine tool data made 

available by the Office of Industrial Economics through the work of 

Professor Carl Beidleman. Our results are consistent dth Beidleman’s 

except for the fact that once one allows for retirement and once one 

uses a flexible estimation procedure such as the Box-Cox, the depreciation 

Patterns on average are somewhat more rapid. 

29 
(2) Summary Results of Average Depreciation Rates for All Asset Classes 

While the results presented for specific assets were based upon careful procedures and formal statistical hypothesis tests, the method of deriving 

average rates and representative efficiencies for broad classes of assets such 

as those listed in the BFA Statistical Tables involved considerable judgsent 

and ad hoc method. We shall briefly sumarfze these ad hoc methods and judgmental procedures here and then present the estimates by asset class. 

Our analysis begins with the Type A classes, from Table 2 above: PDE 

classes 4. 6, 8, 10, 16 and 17; PNS classes 1and 2, and CD classes 1and 6. All 

classes, except CD6,were studied in detail either in Report TOS-74-27 or 

here. The average rates presented here were obtained by calculating the best 

geometric approximation to the predicted Box-Cox prices on retired data. 

These best geometric approximations (hereafter BGA rates) are the minimum 

variance averages to those presented in Table 6. These average rates are 

in Table 7. These rates are averages over the BGA rates of the specific 

assets in each class. The specific assets listed by class appeared in Table 

5. The average BGA rates are our best judgments as to the aver5ge rates for 

= these classes. As mentioned above, these estimates are based on considerable 

econometric research and they apply to depreciation of assets which comprise 
 

approximately 55% of the total. stock of producer durable equipment. 42% of 

the total. stock of private nonresidential structures and 66% of the total 
 

Stock of consaer durable goods. 
 

The remainder of the U.S. capital stock falls into either Type B or 

TYPe C asset categories. For assets in these latter categories. our esrimtes 

of the average rates are based more on judgment than on analysis. Severthe30 
 
. +, TYi)&l! ; 

BGA DEPRECIATION RATES TYPE A ASSET CLASSES 

Asset Class, 

Producer Durable Equipment 

4 Tractors 

6 Construction Machinery 

8 Metalworking Machinery 

10 General industrial equipment 

16 Trucks, bases and truck trailers 

17 Automobiles 

Private Nonresidential Structures 

1 Industrial ,. 

2 Commercial 

Consumer Durables 

1 Hotor vehicles and parts 

6 Residential structures 

,BGARate 

16.3% 

17.2 

12.3 

12. 

25.4 

33.3 

3.6 

2.5 

27.3 

1.3 

31 
 
9 55, 3% ?;:??.:!.E‘?% th,,? i::E c~c-J@.llti~>l*.:!?t:rP7im?.Ff .of those ::~n?.i~.?.!’2;,??‘iCl::: 

should be modified. This conclusion follows from the relatioriship between 

our estimates of the Type A asaets and the conventional treatment of the 

Type A asseta and from existing atudies of economic depreciation of other 

assets. In the case of Class B assets. we analyzed each asset case by case 

and brought into our judgment (1) ancillaq studies undertaken by others, 

(2) the treatment of depreciation by BEA, Dale Jorgenson, BLS and Jack 

Faucett Associates, as well as (3) some judgmental analysis on our part. 

For the Type C assets in which we had no specifi.c data available, we drew .- . ., -

our inferences from similar assets within the Type A categories and from adjustments implied by our analysis to the conventional wisdom. These procedures 

and judgments are described in detail later in this report. The average BGA 

and judgmental depreciation rates for the Type B and Type C a&et classes 

are presented in Table 8. 

In order to appreciate the implications of our study for depreciation 

esti&tion, we present Tables 9A and B in which our depreciation estimates 

for each asset class are compared to >our alternative treatments of depreciation. The firat three alternatives are based upon asset lives used by a 

number of research institutions including Jack Faucett Associates.; the Bureau 

of Labor Statistics, and Dale Jorgenson, Inc. The rates presented in Tables 

9A and B are calculated by applying a double declining balance scheme in the 

first rowI a 1.5 declining balance scheme in the second row. and in the third 

row a straight declining balance scheme, each applied to the lives given to 

Us by Professor Jorgenson. Thus the first yardstick for comparison are rates 

based on Bulletin F lives. The fourth columns of Tables 9A and B represent 

rates implicit in the published figures on investment flows and capital stocks 

Of the Bureau of Economic Analysis (BEA). We calculated these BEA implicit 

average rates, and wewill discuss our procedures later. The final Column 

of Tables 9A and B contains our estimates. 

32 
r

2s 3 z ? 
 

BGA AND JUDWNTAL DEPRECIATION RATES BY ASSET CLASS 
 

TYPE B AND C ASSET CLASSES
 

Asset Class Rate Asset Class Rate 

Producer Durable Equipment Private Nonresidential Structures 

1 11.00% 3 1.882 

2 9.17 4 1.88 

3 7.86 5 1..88 

5 9.71 6 2.90 

7 16.50 7 3.16 

9 10.31 8 2.37 

11 27.29 9 5.63 

12 16.50 10 2.-90 

13 11.79 

14 11.79 Consumer Durables 

15 11.79 2 10.00 

18 18.33 3 15.oa 

19 7.50 4 15.00 d 

20 6.60 5 15.00 

21 14.73 7 15.00 

22 14.73 

33 
 
1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

-- 

Class 

PDE 

L'b 

-1 

1[I 

15 

15 

I? 

18 

19 

20 

21 

22 

Table. !.*A 

DEPRECIATION RATES BY ASSET CLASS 

COMPARISON OF RESULTS 

Jorgenson Lives Implicit BEA 

DDB 1.5 DB DB @CAI 

.1333 * 1000 .OM7 .lo92 

-1111 .OB33 ,0556 .0803 

.0952 .0714 .0476 .Of346 

2500 - 1875 .1250 .2564 

.1176 .0882 .OS88 .1516 

* 2222 -1667 .1111 .3388 

e 2000 .1500 .IO00 .2118 

.I250 -0938 .0625 .13CO 

.I250 * 0938 -0625 e 1424 

,1429 .lo71 .0714 .1676 

2500 -1875 .izsa .0330 

.zoo0 .1500 e 1000 .1311 

-1429 .lo71 a 0714 .1565 

.1429 -1071 .0714 .I565 

1429 .lo71 .0714 -1565 

.2941 2206 .1471 .1298 

.2941 2206 .I471 .4057 

D 2222 .1667 .1111 .2276 

.0909 .0682 .0455 a 1078 

.0800 .0600 .0400 .1362 

.LE18 e I364 .0909 .1282 

* 1818 .1364 a 0909 .1740 

Hulten-Wykoff 

(BGA) -_"-

.I100 

.0917 

.0786 

. .1633 

.0971 

.I722 

-1650 

.1225 

.LO31 

.1225 

.2729 

-1650 

.1179 , 

.1179 

.1179 

.2537 

.3333 

.1833 

.0750 

.0660 

-1473 

.1473 

34 
 
-- 

1 

2 

3 

4 

5 

6 

7 

1 

2 

3 

4 

5 

6 

7 

COMPARISON OF RESULTS I Continued 

Jorgenson Lives 

Class DDB 1.5. DB 

PNS .0741 

-0556 

.0417 
 

.0417 
 

.0417 
 

.0645 
 

.0741 
 .0667 
 

8 .OX6 

9 .1250 

10 .0645 

Goldsmith 

DDB 

CD 	 .1333 

-1333 

-1667 

.2000 

.zoo0 

-0556 

.0417 
 

-031.3 
 

0313 
 

* 0313 

.0484 

-0556 

.0500 
 

.0395 
 

-0938 
 

-0484 
 

DB 

.0667 
 

.0667 
 

.0833 
 

.loo0 

e 1000 

Implicit BEA Bulten-Wykoff

DB (BGA) (BGA) 

-0370 .OB35 -0361 

.0278 .0409 .0247 

-0208 .0430 .0188 

.0208 .0430 .0x88 

.0208 .0430 .0233 

.0323 0640 .0454 

.0370 

.0333 -0567Gr: 

.0263 .0237 

.06 25 .0563 

.0323 .0590 .0290 

.lo16 t%* .0316 

Flow of Funds Hulten-Wykoff 

DDB DB -IBGA) 

.2500 .1250 .2725 

* 2000 1000 .1000 

.2500 e 1250 .1500 

.2500 .2500 .15008 

.2500 .2500 .1500 

.0128 

.1500 

(.0110) 
 

35 
 
With a few exceptions, BW rates are more rapid compared to 

the corresponding rates of our analysis. Also, double declining 

balance, which has been so popular in econometric research, is too rapid. 

Thus, even though the depreciation patterns which we observed for all of 

our specific assets are accelerated. the rates are considerably less than 

the double-declining balance scheme popularly used in the tax code and in economic research. Unfortunately, the recent adjustments made by BEA actually 

tend to operate in the wrong direction. Specifically, BEA capital stock 

figures imply depreciation rates which are even more rapid than Jorgenson's 

double declining balance rates. The analysis of this report and our earlier research Implies that economic depreciation is not as rapid as double 

declining balance of the Jorgenson lives. At the same time, the depreciation 

process for producer durable equipment is more rapid than 1.5 declining balance. 

We settled as our best estimate on a 1.65 declining balance scheme applied 

to the Bulletin F lives for the asset classes for which we had no independent 

information. In general, then, our analysis suggests that the appropriate 

average depreciation rate would be obtained by calculating a 1.65 declining 

balance method on the lives provided by Jorgenson. 

In the case of structures our results are sometihat diffeGent. Again, 

the double declining balance method is too rapid and again the BEA estimates 

are even larger than the double declining balance method. Our study of industrial. and commercial structures indicates that depreciation should be quite 

a bit slower than double declining balance. On average, our estimates for 

Private nonresidential structures imply, if one were to use the Jorgenson 

lives, a declining balance scheme calculated at .9 rather than 2 times the 

declining balance rate. For the additional structure classes then, we imPosed a depreciation method which vas .9 declining balance on the Jorgenson 

asseir lives. 

36 
 
Om treatment of consumer durables is based on the Flw of Funds Account 

of the Federal Reserve and by early capital. stock studies of Raymond Goldsmith. We reproduce these as the conventional wisdom in Table 9B. [These 

sources were discussed in the Phase 1 Report.) In the case of automobiles 

our estbates are more rapid than either the Flow of F~zk?cz:.z.tc. calculated at a double declining balance method, or the Goldsmith rates. 

. ., ., . Since our estimates are based on far more actual infor .-. -1.... . . ".. c.ri 

that fhey should be employed. The central cause of these TP'=+<,,-'~ rtpid 

depreciation rates appears to be that the retirement process is combined 

with vintage price data. 

.<: 

In the case of private residential structures, we based our best 

professional judgment rate of 1.3% upon an average of four rates, two obtained by a study of Rafael Weston as part of his PM. thesis at Harvard 

under Dale Jorgenson and two provided by Professor Wilhemina Lei.gh from 

_!, 

her PM. thesis studied under Professor Charles R. Hulten at Johns Hopkins 

~,,~..~~.si.ty. These estimates are reasonably close to the only a>ternative

- . -.- . . available, the earlier study by Goldsmith. 

Later in this report we will discuss in more detail the methods 

--+: tc derive the estimates of depreci.ation which appear in Table 9. Furthermore, we shall present the relative effici.ency sequences for assets 

in addi,tion to the BGA estimates of the average rates. 

This concludes our sunnary discussion of the results of thi.s Phase 

11 study. In the next section, Part I1 of this report, we present in detail 

our theoretical, econometri.c, and empirical analysi,s of the specific zsscts. 

37 
In the fol1,wing section, Pert IXX2 we discuss the development of the ac?un?. 

depreciation and efficiency function estiibates bas& ~ponthe detail.ed analysis described in Part 11 of this report. 

3 

38 
 
11. 	 THEORETICAL AND ECONOMETRIC ANALYSIS OF THE 

KNTAGE PRICES OF INDIVIDUAL ASSETS 

In the pages which follow we present an overview of t!x gcr.cr5l 

theoretical framework employed in the analysis of economic drplrcidciuu and 

asset efficiency. In the addendum to Phase I we presented the general 

theoretical model employed here and discussed extensively tire CLUILUU~CLL-L 

problems involved. Here we will briefly sketch those theore+’--’ ?-a ?cowmetric issues. Several new issues are brought to the fore in this report. 

First, we discuss at some length the role of capital taxes in the deriva­ r 

tion of productive efficiencies from vintage prices. This analysis includes 

a discussion of the incidence of taxes implicit in the analysis of Harberger 

(.l962), Jorgenson (1967), Stiglitz (1972), and Feldstein and Rothchild (1974). 

The objective of this discussion is to place in perspective our treatment of 

tax incidence in the construction of economtc depreciation and relative effidnnr‘nc from vintage asset prices. We show that our treatment of taxes is 4 
 

consistent rJith that employed by Jorgenson and that our procedure for estimating depreciation from vintage prices rests on the basic notion of duality 

commonly employed in microeconomic theory. 

Second, a number of very thorny and subtle econometric problems will 

be discussed. We comment briefly on the choice of flexible functional forms 

which were discussed in Phase I. We then introduce a new method for dealing 

with asset retirements. Because vintage prices represent only assets which 

have survived to a particular age, we employed an asset retirement pattern 

suggested by early studies of Robley Winfrey. However here we apply a new 

39 
 
?:rt'-P,d 4,z.r;c:.:: nyi ?.pi;% S,,Ar: U!. l:>sf:,I::: t :: (?:Y.::K?C:Y:; :To ne;. 6 2,); 'iI. 0C~:;Si.?i: i/ c;:.ci 

that retirement strikes those assets which are least va1.uabl.e (lowest in 

price) at any particular age. This "truncati,on" approach is discussed in 

detail in this section. We shall also comment on several other problems 

such as the method used to deflate vintage pri,ces in calculating depreciati.on. The final conceptual contribution in this section is to illustrate 

the calculation of depreciation and asset inflation from the econometrically 

estimated vintage asset prices. 

The final section of Part I1 consists of an example of the actual empirical analysis of individual. assets. The full details comprise the appendix to this report. The appendix itself consists of over 1200 pages of 

econometric analysis of individual assets, organized by asset class. Those 

who wish to examhe the econometric analysis in detail may do so with the 

use of this appendix. We turn now to discussion of the theoretical basjls (-

for estimating economic depreciation from vintage prices and for utilizing 

these estimates for purposes of tax analysis. 

A. Taxes and the Relative Efficiency Function 

d 

Tr. this section, we develop the model of capital prices with special 

emphasis on taxes and relative efficiencies. Beginning with the capital 

pLLeLiLcorY of Hotelling (1925), Hall (1968) and Jorgenson (1973), we assume 

tnat in a world with no taxes, an optimizing capital user, operating with 

perfect certainty in an efficient and competitive capital market, will equate 

the purchase price (or acquisition cost) of a capital asset to the present 

discounted value of the future flow of user-costs (or service prices) on 

the asset: 

40 
 
whefe q(s,t) and c(6.t) denote the respective purchase price and user cost 

of an asset age-s in year-t, and where r is the coristant discount rate and 

L is the asset's life. 

Now assuming that the capital user is taxed on his income from production. eq. lmust be extended to allow for the tax structure. Following 

Ball and Jorgenson in (1967) and in Fromm (1971), Feldstein and Rothschild 

(1974) and our own analysis in Hulten and Wykoff (l?77), we shall assume 

that the tax falls on the capital user and that the rate of return is exogenously determined. 

Since these assumptions are by no means widely accepted, and since 

a numher of different models have been constructed to suggest that the incidence o!c' a tax on the use of a particular type of capital may be shifted 

elsewhere, we shall discuss this assumption. Harberger in (1962) and Fromm 

in (1971) argued that a tax on one type of capital will lower its return 

relative to other rates, thus inducing resource re-allocation away from the 

taxed capital. This resource flow lowers returns on untaxed capital and 4 
 

raises returns on taxed capital, which diffuses the tax burden. Ballentine 

and Eris (1975) provide empirical support for the position that all capital 

bears the full burden of the tax. 

Along a completely different line of analysis, Stiglitz in (1973) 

contends that capital users evade the tax burden altogether by resorting to 

debt finance. Debt finance, which generally enjoys tax deductible interest 

payments, avoids the tax on capital. This theoretical argument is by no 

means resolved however. King in (1974) shows, under assumptions slightly 

different from those of Stiglitz, that the cost of capital will change 

41 
 
under some financing methods (use of retained earnings or new stock issues) 

but not other8 (Jebt finance). Both Stiglitz and King andlyze incidence in 

a certain, partial equilibrium analysis setting. King points out, in (1973), 

that: "To discuss the incidence of the tax requires, in general, a fully­ - fledged general equilibrium model,. ..'I1 Furthemore, since. as King also 

notes, the "raison d'etre of incorporation and the advantages of limited liability, however, are to be seen only in a world of uncertaintypVb2it would seem 

premature to accept an incidence conclusion based on debt finance in a certain 

world. The assumption that the capital user pays the tax would rule out his 

ability to shift the tax either by resource reallocation or by debt finance. 

,- The tax structure may be represented by the following Spb015: 

u: marginal tax rate 

Ti: tax life for depreciation purposes under rule i 

r/­

i 1,2.3 where i: Bulletin F lives 

2: Revenue Procedure 62.21 lives 

3: a.lives 

D

13(s): tax depreciation deductiog at age-s on an asset valued at $1.00 

when new, given tax life Ti and depreciation method j, 

j 8 1,2,3 where 1: straight line . 2: double declining balance 

3: sum of years digits 

Zij: present value of tax depreciation deductions on a new $1.00 asset 

given life Ti and method 1. -

1. Ring (1974) p. 277. 

2. King (1975) p. 279. 

42 
 
k: investment tax credit 

a: proportion or' k deducted in calcui,aa;;$ depre:,.::1:..Liit basis. 

For a capital user subject to tax rate u. given deductions initially 

valued at 2

13 and tax credit k, eq. 1 

L 

2. qat) = (1-L z c(s,tis) . [l-k-(l*k) 	 uzil] s=O s+l 

(l+r) 

The many changes which the tax code has undergone since 1954,4 are summarized, 

for machinery and equipment, in Table 10. (See Hulten and Wvkoff (1977) for 

corresponding rules on structures.) 

Table 10 

TAX PAWETER VALUES 1952-1979 

- - U i i k a -

Source: 

Pre-1954 

1954-61 

1962-63 

1964 

1965-70 

1971 

1972-77 

1978 

w79
 

Sc. 1250, U.S. 

.52 1 1 0 -

-52 1 3 0 -

.52 2 3 .07 1 

.50 2 3 .07 0 

.48 2 3 .07 -.a 

.48 3 3 0 0 

.48 3 3 -10 0 

.46 3 3 .10 0 

.4s 3 3 .10 0 

Tax Code. 

3. See Hall. and Jorgenson (1968) 

4. See Prentice-Hall (1972) 

. ,, 43 
Setti.nr: a4 for convenience, eq. 2 may be written BS: 

3.	 q(0,t) - Jl-u)c(O,tl+ 

(l+r) 

I. 

.= 

In eq. 3 the price of new capital is decomposed into two parts. The first 

two terms on the right hand side of eq. 3 comprise the user-cost and tax 

liability on a new asset, and the second two terms, therefore, consist of 

the period-t present value of a one year old asset in the next period. 

Thus, we have: 

4. 	 q(O,t) (l-u)c(O,t) +T(O)q(O,t) + q(l,t+l) 

(l+r) 

where ~(0)- (l+r)k+uDij(0). Solving 4 for the user-cost of capital yields: 

=% 

Eq. 5 depicts the u5er cost of new capital which is seen to depend upon 

the asset acquisition price when new q(0,t) the rate of return, r, the 

price after the first period, q(l,t+l) and the parameters u. k. D 

i j(01, 

of the tax structure. The user-cost can be estimated from data on q(6.t) 

and r and details of tax law: up k and D

ij (6). A similar expression to e%, 

5 for c(s,t) may be derived as well. 

The optimizer equates the marginal rate of substitution between 

44 

I 
or, using 5 and the corresponding expression for c(s,t>: 

where T(s) - uDij (s) e 

Jorgenson in (1973) calls $(s.t) the asset relative efficiency 

function, because it represents the in-use productivity of an age-s asset 

relative to that of a new asset in period-t. 3orgenson'a econometric 

analysis contains the assumption that $(s,t) is stationary and geometric. 

i.e. I 

-6s'" 

8. $(s,t) = +(SI = e s = 0,1,2,..., m 

where 6is the constant (geometric) rate of 106s in productive efficiency. 

In (1974) Feldstein and Rothschild argue that relative asset efficiency is neither geometric nor stationary. They present two analytic 

cases in which an optimizing durable goods producer is seen to alier his 

asset technology on the basis of changes in tax rules or in rates of return. 

In one case, producers alter the lives of one-hoss-shay assets and in another they alter the in-use productivity of perpetuities. In each case, 

they optimize the present value of the future stream of after-tax usercosts. If Feldstein and Rothschild are correct, then the capital-user pays 

the after-tax user-cost, (l-u)c(O.t) C T(0)q(fltt), not c(0,t). Eq. 7, 

employed by Hall and Jorgenson, aasumes that the capital user pays c(0,t). 

The questions of whether the relative efficiency function, $(s,t) is 

45 
 
srationaq a,nd Seometrir., rr?.cht.n the context: of .Sorr,enson's analytic frc-mcwork, must be addressed empirically wi.th the use o'f eq. 7; not by an alternative theoretical model. 

Some have argued that the Jorgenson framework is hopelessly rigid 

in its assumptions. Robert Hall in (1977) argues that of all the controversi.al assumptions in Jorgenson's theory of optimal capital. accumulation, 

however, only the one that @(s,t)is geometric is essential. Hall's 

argument is as follows: a rational, well-informed decision to acqui.re a 

long-lifed asset requires that the producer act as if he knows the asset's 

relative productivity in the future as well as in the current period. But 

the productivity of the asset in the future, when say 3 years ol.d, relative 

to that of a new asset 3 years hence will depend, in general, on the quantities of capital acquired in the intervening years as wel.1 as on the current l&el of acquisitions. Consequently, current and (all) future investment decisions to be made rationally must be made simultareously. To 

Hall the problem so framed "appears hopelessly complex." 

-6s 

The assumption that he perceives @(s,t)to equal e allows Jorgenson's 

I . -r---&tzr* Rall argues, to ignore future investment levels in scheduling 

/ 

.- ..l ? hvestment as long as a positive level of gross investment is planned 

in each period. Thus, i.n Jorgensbn's world, the choice of schedul.ing an 

i:--r:t=xt either now or in the future depends only on th8 relation between 

c'nr marginal value product of new capital and its user-cost. In other words, 

any errors made in term of flows beyond the current period can be corrected 

i.n the future provided that future marginal rates of substi.tution are be1i.eved to be b.om ex ante. 

Hall argues that future components of asset price can be ignored 

if +(s,t) is geometric in the future. In fact, +(apt) need only be 

46 

i 
-

StdCiOMry, and any stationary schedule will do. Of course, e 

-68 Is 

very simple aaaiytically and greatly facilitates aggregation. It is not 

however essential.. %e Jorgenson investment story does depend though on 

Q(s,t) being perceived to be stationary. 

Hall also argues that models not ushg ex ante stationarity of $(s,t) 

are intractable. Wi.lliam Schwann (1977), following Treadway (1969) and 

Brechling (l975), developed a model in which relative asset efficiencies u 

are endogenously determined. Schwom's producer simultaneously sets main- , 

tenance requirements, utilization rates and the I.wel of net new investment. 

Schworm's producer must be able to forecast prices far into the future and 

make5 all present and future investment decisions si.multaneously. Although 

possible, it seems somewhat implausible that producers take into account 

long forecasts of all future investment decisions in determining current 

-

acquisitions. Thus we agree with Hall's concl.usi.on that. "as a practical 

,matter. then, a model that assumes a simple predetermined relation between 

the future marginal values of different vintages seems a good guide �or 

investment." We conclude this seition by noti.ng that the central assumption 

of Jorgenson's model, that $(s,t) = $(s) for all t, requires examination 

within the context of Jorgenson's conceptual frmework. The dwality relation 

between the physical loss in productivity $(s,t) and relative asset usercosts c(s,t)/c(O,t)* expressed in eq. 7 allows the study of @(s,t) from vintage price data, however these assumptions are not needed in the calculation 

of economic depreciation. 

B. Econometrics of Estimating Vintage Acouisition Prices 

From a conplete sartPle of asset vintage prices on a homogeneous class I 

of assets, one can arrange a rectangular array by age and date of the prices. 

47 
 
From a rate of retern nnd ?.5n ?it:: ii::!:”:s:~:tre az i..,:.?, .. ’\.;: ,:JO icim ,zirqr

s x t of user costs c(s,t) and estimate relative efficiencies directly from 

the array. See for example Wykoff in (1970). Often, however, we do not 

have complete price arrays nor perfectly homogeneous assets. Producer durable equipment is qui,te specialized and various pri,ces represent slightly 

different types of equipment. We must, therefore, rely on statistical lnference to estimate the average pattern of vintage asset prices. That i,s, 

we can fit a regression pl,ane to prices to obtain a set of fitted prices 

by age and date. These fitted prices may be combined with after-tax rates 

of return to produce relative asset efficiencies. 

To avoid imposing a priori a specific form on the price-age pattern 

of assets. we employ flexible functional. forms in our regression anal.ysis. 

Two forms are used: The Box-Cox power transformation and the po1,ynomial 

regression. 

The Box-Cox power transformation, an intrinsically nonlinear procedure discussed at some length by Zarembka in (1974) and in Treasury Contract TOS-74-27, permits joint estimation of (a) parameters which determine 

a specific functional form within the Box-Cox class and (b) parameters which 

determine the slope and intercept. Since certain restrictions on .the unKnown form parameters produce one-hoss-shay, linear and geometric forms. 

one may employ classical hypothesis testing procedures to evaluate the suit&­

bility of these patterns. Estimation of the Box-Cox parameter is undertaken 

using a non-linear maximum likelihood procedure. Asymptotic likelihood 

ratio tests at 95% levels of significance are used on joint restrictions 

and asymptotic normal tests are used on individual parameter restrictions. 

The polynomial regressions are intrinsically linear and include 

one-hoss-shay, linear and accelerated patterns as special cases. Linear 

48 
variables in a straightforward way (i.e. by using multiple regression). 

The degree of polynomial, and hence the functional form, is determined by 

starting with fourth degree polynomials and deleting successive powers by 

age and year until the residual regression variance is minimized. This 

procedure produces the appropriate specification within the polynomial 

class. Attempts to compare the best polynomial form with semi-log forms. 

e 

implying geometric decay, here undertaken as well. 

(1) A Truncated Distribution to Allow for Scrappage 

In section A of Part I1 above, the model 05 capital 7rir-C a-4 q'ser 

costs refers to the prices of individual assets. For several reasons, our 

interest is in groups, or cohorts, of assetsnot in individual units. First. 

investment decisions ordinarily involve acquisitions of cohorts of assets and 

firms must consider the productive prospects of the average asset in the cohort not merely of one unit. Even if assets are homogeneous in terms of 

their built-in productivity, different units last for different lengths of 

time. Second, on a more pragmatic level, as we dell with vintage price 

A.*= 7-11 have only prices of those vintage assets not yet retired. The 4 

*wrap ased asset price of the original cohort should reflect the prices 

of retired units as well as of the survivors. Thus to reflect the average 

p-rformance of the original cohort. used prices must be modified to 

allow for retirements. 

In (1977) we studied the retirement problem as an extension of the 

perfect certainty assumption of the Hotelling-~11-Jorgenson model outlined 

above. Each asset in the cohort is assumed to be identical while in place, 

but each has a different, yet certain, retirement date. Thus all assets of 

49 
.. a ~,?v+:,ii ?,!:'kc;:": .:,gil,J. >:$yF:"bee r;&);v;: i::J.;tivf c:..tricia.n,:y.g~i;s.~e~~e p'I..bpi)

while in place, so that assets retired early are not assumed to deteriorate more rapidly than long-lifed assets. Purthermorer'since all retirements 

are antici,pated with perfect foresight, unintentional casualty losses and 

errors are ruled out. Under these assumptions, calculations of the average 

cohort used pri,ce merely corresponds to premultiplying observed vintage 

prices by their probability of having survived to that age. Given prices, 

adjusted for the retirement process. one can proceed to estimate the priceage performance of a cohort of assets using classicaletesting procedures. 

We implemented this method in TOS-74-27 using a retirement distribution 

from Marston et.al. (1952) on structures. The same approach is employed 

in thi.s study of PDE and CD assets. 

Here we introduce a new analysis of the retirement problem from a 

quite different perspective. Isstead of viewing retirements of indivi.dua1 

units as known with certainty, we assume the retirement process to be 

stochastic. This approach is suggested by the work on censored samples 

of Amemiya (1973) and Tobin (1958) and by the work on truncated samples 

of Berndt, Ball, Hall and Hausman (1974) and Hausman and Wise (1977).

.' 

We thi& of asset vintage prices as behaving as follows. (For 

convenience of exposition we think in terms of a specific asset: tractors.) 

Tractor prices fall. with age because of wear and tear and obsolescence. 

At each age, however, the prices of individual tractor units will vary 

about the average price of the cohort due to differences in intensity of 

use, the variety of tasks performed, differences in polides with respect 

to maintenance and repair and so forth. A typical scatter diagram for 

a given class of tractors might look something like Figure 3 following. 

50 
-- 

F?"Sure 3 

Hypothetical Scatter of Tractor 

Prices by Age (in one year) 

price-age curve 

.. 

.. si 

In this simple example suppose the solid line depicts the trLs i,;:;,,,, 

on average, of tractor prices with age. The specification which is assumed 

to generate the scatter in Figure 3 is: 

.. 

where A and a are paskiveunknown Constants * qi is the pri.ce of tractor i, SI 

the age of tractor i and ui are independently distributed random variables 

assumed to be normal with mean zero and constant variance a'. (We assume 

this simple form for ease of exposition. Later, we will. introduce flexible 

4

functional forms.) We would like to estimate -u from the scatter. -a Is 

the true unknown parameter representing the percentage rate of price decline 

per year. 

An actual scatter of points like those in Figure 3 is generated from 

a sample of vintage asset prices. In the case at hand, tractors, the prices 

are taken from public auctions of used equipment. Not all tractors are 

Included in the sample-only used tractors available for resale enter the 

data base. We may ask how this sampling procedure might fail to represent 

all assets of a given vintage? One possible problem is that assets up for 

51 
. 25 
?ig!!,rp b<$ 

Hypothetical Scatter of Tractor 

Prices by Age (in one year) 

Effects on Data of Scrappage 

rr 

The magnitude of the bias in -a of -0 can be shown to depend upon 

the scrappage level price qC the true slope a, the variance of ui, 0’. and 

the age distribution of the sample Si. See Hausman and Wise (1977) for 

details. The solution to the problem of dealing with a truncated sample 

is to view the stochastic terms u as having a truncated normal distribution. 

i 

Before solving the problem we should point out that our specification of the 

4 

true line is more complex than the simple semi-log form used for illustrative 

purposes above. Therefore we turn now to the flexlble Box--Cox power transformation form. 

Following our work in (l977), we assume that the form of the true 

price-age curve falls within the class of Box-Cox power functions:* 

10. q*% 5 BS*% + Ui 

where q*i = qil- e 1 and s*% siz­ e 1

- ~ 

e 01 2 

“we assume here for simpLicity that’B11 variables are variations from the mean. 

53 
T2il,?rii .?

f. 'I>$ )'ii QT!? the prl'ce. ra.d -ye ,Llf :;x2ct.:>:,: :~.~o;:cc"ioe~yGJLA wi,ar* 

6, 61 and 6, are unknown parameters, 0 may be thought of as the "slope" parameter and (el,e2) as the form parameters. If (fl,,6z) - (1.1) then the priceage curve Is lincar; and if (e,,62)J(O.O), then the price-age curve becomes 

log-linear. Finally, (e .e )+(0,1) implies the semi-log form used in the 

1 2 

above example. 

We are now prepared to deal with the exclusion of non-sunrivors in 

the sample. We assume 8

i to be non-stochastic and ui. to be a truncated 

normal distri.bution. The normal distributi.on of the ui has a mean of zero 

and a canstant vari.ance 0'. Thus the distribution of qfi is normal mean' 

BS*~ variance 8and is truncated at price q 

*. 

or, visually: 

Figure 5 

Truncated Distribution of q*i 

q i* 'bN@6*jdS2) 

Figure 5 implies that at any given age sit tractor prices will be observed 

to be distdbuted normally about a value BS"~ but that prices below some 

critical value Qc wfll not be observed because these tractors 65111 have been 

6 

scrapped. The truncation value in Figure 5 is q*c = (q, '"1)/6 Recall 

a 

54 
For estimation purposes we must study the distribution of the u Since i' * 

u - q 

* 

i-~from eq. 10. ui. is truncated at uci - q 

* ci-Bs * f, therefore, 

e1 02 

* From ui=q i-k*i, we may calculate the change i,n ui from a ~qven2 oh-nge 

in qi as: 

The likel.ihood function for the sample of observati.ons ql, q2...# q, is: 

n 

or following Zarembka (1974): 

Our next problem is to determine the frequency function of the truncated 

normal. ui. Let F(u i

) be defined as the cumulative distribution function for 

each value of u

i" That is, F(ui) is the area to the left of ui remaining 

in ehe truncated distribution: 

. 55 

i 
$4 

I, a: .~ <ep i:U Zuci J for ui"uci --h-i 16. F(ui) 

for uiIuci 

or 

17. F(ui) - The conditional density function f(ui) can be calculated from F because 

f(ui) F*(ui): 

18. �f(U,) = g(ui) /rg (ui)dui for ui>uci 

Uci a 

0 othendse 

Where g(ui) is a normal density function wf.th mean 0 and variance u*: 

* where exp. is the exponential, i'-ui q*i-6s i9 and where P"g(ui)dui is a 

Uci 

standardized unit normal distribution function G[u c i/O] and uci is given in 

eq. 12 above. 

Eq. 15, the likelihood function, may be written as: 

The correspondhg log-likelihood function becomes: 

- n lnG[uci/ul 

i=1 

56 
Haxk,m likelihood estimators of 6, e.,. and 8may he chr-.si::ied .;:a> ,?WY~~­

mization of eq. 21 using procedures set out in Berndt,lhll. Hall and HausPian 

(1974). Similar log Xkelihood functions can be constructed for different 

specificati,ons Of the true price-age curve than the Box-Cox forms in eq. 10. 

Before presentiag the data and empiri.ca1 results, we shall now bri,efly summarize our treatment of three econometric problems: (1) the choice of functi.ona1 forms, (2) the treatment of asset reti,rements, and (3) the treatment 

of capital gains: 

(2) Econometric Problems Summarized: 

The first problem in estimating the price-age pattern U; CS+_L& is 

to specify a model flexible enough to determine the patterns from the evidence rather than to use a predetermined functional form which restricts 

the shape of the price-age curve a priori. In (19771, we applied two flexible functional forms, the polynomial regression and the Box-Cox power transformation, as well as two more conventional functional forms, linear and 

semi-log. to vintage prices of structures. The latter two specifications 

each represent a commonly assumed price-age pattern, straight-line price-age 

.-**-ye ="xinst a higher order polynou&al alternative using a straig5t forward f-test. 

The semi-log form can also be compared to the polynomial using a 

test suggested by Theil (1971). However, the results of the Theil test are 

ustiallv ambiguous. As noted above, the Box-Cox power transformation includes both the sed-log and linear forms as special cases and again one 

can test these restrictions using classical hypothesis testing procedures. 

Table 11 depicts the four specifications to be studied. In (1977) 

we show that the Box-Cox and polynomial forms themselves are members of a 

57 
XQrn ;L,enP.ml C!.PFE Qf8OX-cOX fXrtt.t$ OVifl?. t3Z!Xl~ TcX?Wu%'., ilS ;? yX>.i2riC.al. 

matter, the general form i,s too complex for conventional non-linear esti.­

mation procedures, so that we are unable to discriminate statistically 

between the Box-Cox and polynomial fonns. 

Table 11 

Specifications for Empirical Work 

Lm: 

= ac8si+Yti+ui i=lr2r...rn .. . qi 

SF;MT-LOG: 

lnqi = M.BSi+Yti+Ui i=lp2,...,n 

POLYNOMIAL: 

i=1,*2*...,n 

Box-cox: 

qF: auction price of asset i (either adjusted or non-adjusted for retirements 

Si: age of asset i at auction 

ti: year of auction on asset i 

ui: random disturbance term assumed to be N(o,U21) 

a,BDy,BJpyJD8j:unknown parameters 

A second problem in estimating price-age curves of used assets is to ._ 

58 
r2113i).fur asset retirements. As noted above, one approach is to treat retirements as non-stochastic. Observed used-asset prices are pre-multiplied 

by a survivor probability based on a retirement distribution for the particular class of assets. The result is essentially a new set of vintage priceseach price in the old set pre-multiplied by the probability of having surviwed to that age. This approach as shown in (1977) implies that scrapped 

assets were worth the same 86 survivors when scrapped. This implication is 

not terribly plausible for some assets such as tractors. Those scrapped are 

likely to be worth considerably less than survivors. At one extreme scrapped 

assets areworth zero. If scrapped assets are worth nil, then, as shown in 

(l971), estimation can proceed on observed prices without explicit regard 

for scrappage, i.e., the appropriate procedure is to use unretired data. 

Here we report on estimation results using unadjusted prlces, retirement 

adjusted prices, and the truncated distribution approach outlined here. 

A third problem warranting comment Involves the treatment of Inflation. The equations in Table 11 imply that asset prices depend upon ages 

and upon date. The inclusion of date reflects the fact that, given age, 

asset prices vary from year to year as a result of inflation and possible 

4 

intertemporal shifts in supply or demand. However, the forms in Table 11 

may not be an entirely satisfactory way of treating these capital gains and 

losses. See, for example. Taubman's remarks on Hulten and Wykoff in (1977). 

Consequently, we select a price-deflator relevant to producers duxable equipment to deflate asset prices spriori in order to capture the general infla-' 

tion aspect of the capital gain phenomenon and present results for deflated 

and undeflated data. 

(3) Calculation of Depreciation and Revaluation from Estimated Vintage Prices: 

The year-to=-year changes f.n the present value of an asset can be de59 
~:03~~3~1i~d htu 2x1 r:f,xt due 00 increasing age and an effect due to the paasing of time. Formally, 

Equation 22 indicates that the difference between the precent value of an 

EI year old asset in year t and its value in the following year, when it is 

an &-I. year old asset, can be thought of as (a) the difference between the 

present value q(6,t) and the value of s+l year old asset c in the same year,

and (b) the difference between the price of an 64-1 year old asset In year 

t and Ftl respectively. The effect (a) is economic depreciation, the decUne in asset price due to age. The effect (b) is revaluation, the capi­ + 

tal gain or loss due to other inflationan factors influencing the trend 

of asset prices. 

Figure 6 

La. I a fi .-

a a 3 6 5 

60 
Figure 6 portrays the effects cleezrrtherl hy fir:* X, ':Kxc. e:.:za;.; vii:l.ie :;E 

viztai;a asset:: ii.1 yasr t is descri,bed by the curve EAB, while the average 

value of assets in year t+lis FDC. The curve FDC is drawn above FAB to 

indicate that asset hflation has occurred between t and t+l, and is skewed 

to indicate that age-.price relations hi,^ may not be uniformly affected by 

the general revaluation of assets. The value of a three year old asset in 

year t is denoted by the point A; in year t+l,this same asset is lncated 

at point C. Equation 22 indicates that the change in asset prke is composed of economic depreciation, the movement along the curve EAB from A to B, 

and asset revaluation, the shift in the curve from B to C. AS arawn, kigure 

6 shows that revaluation ourweighed depreciation, and that Lite yrit:r vi the 

asset actually increased despite the downward effect due to depreciati.on. 

What factors determine the shape of the curves EAB and FDC? This 

can be determined by substituting the present value eq. 2 into the price 

< -

decomposition eq. 22, Changes i.n asset vahes can then be seen to depend 

on changes in the expected quasi-rent, the expected-li.fe, the expected 

rate of discount, and the expected tax treatment of the asset. The implicit 

rent will, in general, decline with age because of deterioration in the 

Z-... ..I "output decay" and "input decay", to use the terminology 05 Felds+n-tn --A Rothschild (1974). Output decay results when the machine generates 

less output due to deteriorated condition or more "down time," and input 

30ray rpfers to the need to use more hputs of labor, materials, and main-

*-....--- L-Iy..c%to maiatain the same flow of output. Deterioration and impending 

retirement will, in general, cause the curves FXB and m)C to slope downward from left to right. Obsolescence, less generous tax benefits, and 

increased uncertainty about future income and costs can also contribute 

to the downward sloping pattern. The year-.to-year shift in age-price 

63 
,;:;,(l"t*p; w,aulrs frca laflation, changes In expectations, changes in the 

discount rate and tax treatment., and changes in optimal utilization and 

economic life. 

Whatever the factors influencing the age-price patterns of assets. 

the important point is that these patterns can be observed for certain 

types oE assets. Curves like FAB and FDC can in principle be constructed 

for each year and used to evaluate the actual experience of that class 

of assets. Average rates of depreciation can then be calculated which 

can be compared with the corresponding tax treatment, and issues like the 

reasonab1,eness of accelerated depreciation can be evaluated. The observed 

age-,price patterns can also be used as a framework for judging the reasonableness of various methods of revaluing assets for depreciation purposes. 

And, since the age-price patterns are based on direct observations of vintage asset prices, this approach does not depend on assumptions about how 

the vi.ntage prices are actually determined (for example, whether the present value formul.ation of eq. 1 is in fact the way used prices are actually 

formed) 

C, Illustrative Results for D-7 Tractor 

We have applied the methodology outlined in the preceding sections 

fo a =de variety of assets. In addition to several classes of commercial 

and industrial structures studied in an earlier report, we present here 

new analysis applied to twenty-six assets which fall in the classes of 

producer durable equipment and consumer durables. The full statistical 

and econometric detail appears in the appendix where the evidence is listed 

by asset categories beginning with Producer Durable Equipment class 4-

Tractors and ending with Consumer Durable class 6d-Residential Structures. 

62 
The bulk of the bDidySi,S applies to the Type A asset classes (referred to 

in the introduction to this report). In this section we provide a comparatively detailed descripti,on of the analysis of one subclass of assets, the 

0.7 Tractor. 

The nature of the D-7 Tractor data was outlined in the introductory 

section earlier. This section contains only the econometric re%lts fcr 

estimathn of depreciation and calculations of effi.ciency s~quar~ceskr the 

D7 Tractor. 

Table 12 gi.ves parameter estimates for the case in wi.:, As =mulik 

prices are weighted by the probability of survival, i.e., ret!-?' prtccs. 

The first two l,ines show the resu1.ts of the linear and sd-log regression, 

while the next three lines give the maximum likelihood estimates for the 

Box-Cox model, with the most general form of the model, appeari.ng on line 5. .I, 

The polynomial results are shown on the bottom line. The standard errors, 

given in parentheses, indicate that all estimates are si,gnificant at conventional levels. The estimates of a, B and y also have the expected sign: 

negative for the depreciatl-w paxameter, 8. 

Table 13 provides test statistics for determining the most I.ikely 

A n .-

functional, form. X - -2ILog (IO) - Log (Q)] is approximately chi-square for 

large N when the null hypothesis is true, the results of Table 13 indicate 

that all constraints failed to be accepted at the 95% level of significance. 

This implies that neither the linear nor geometric forms are likely to have 

generated the observed sample. The case against these two depreciation 

pattens is strengthened by the fact that the asymptotic normal. test of the 

Unconstrained esthte of 4 cannot accept the hypothesis that 61 I or that 

63 -+ 0. This implies that the dependent variable is neither linear nor 

1,ogarithmic. . 

63 
--- 

Linear 

Semi-Log 

BOX-Cox 

2 Constraints 

1 Constraint 

0 Constraint 

Table 12 

TRACTOR �

(Model D-7) 

Undeflate and Retired Pr :e Data 

(nm582) 

8 e 8 CI 6 Y 1 2 3 Log L 

1 1 1 21907.6 -1543.4 2305.3 -5865.4 

(685.6) (39.4) (92.6)

-. -

0 1 1 10.203 -.141 .140 -5592.3 

(.040) (-002) (.005) 

.455 .455 .455 247.83 -32.406 23.219 -5567.6 

(.004) C.004) (.004) (8.917) (1.1h2) (1.161) 

.212 1.077 1.077 35.102 -.?I31 .952 -5486.0 

(.004) (.044) C.044) (1.094) (.098) C.085) 

.2?2 .998 1.296 41.278 -1.147 .809 -5483.2 

(.004) c.046) (.131) (1.397) (.148) C.179) 

PolynomLal Equation 

a 8 B 8 Y 

1 2 3 y, y2 3 d 

45594.3 -4737.0 186.17 -2.818 -5097.3 1475.2 -86.95 

64 
-- 

Table 13 

TRACTOR (D-7) 

Table : Box-Cox Hypothesis Tests 

(Likelihood Ratio Tests) 

h(w) -2IL*(w) - L*(r2))’ 

Constraints Log Likelihoods 

&n L* (w) L*(Q) x n x2(n) 

I e =e -5486.0 -5483.2 5.6 1 3.84 

2 3 

11 6 -5567.6 -5483.2 168.8 2 5.99 

1 2 s 

III e + o e =e 1 -5592.3 -5483.2 2m.2 3 7.81 

I 2 3 CI 

LV e =e=e=1 -5E65.4 -5483.2 764.4 3 7.81 

1 2 3 

’The terms are defined in the Table for D-4 Tractors. 

65 
What ahape: ?,h.sn!; ?.o I!\F?! ~ ~ i ~ . ~ ~?,t~x-Cox i ~ , ~ ii:gcj~~ases ~ ~ , ~ ~ Icoply? : ~ i ~ This ~ ~ ~

can Le determic'ed by investigating the fi,rst and second order partial derivatives of the non-stochastic part of the Box-Cox funct�onal fonn (eqs. 1 and 

2): The partial derivatives are, respectively: 

and 

= 

A uniformly negative first order partial derivative and a uniformly positive 

second order partial derivative indicates that Box-Cox age-price pattern is 

strictly convex. Thus, while the age-price pattern is apparently not geometric, it does have tho same general shape. This result is eminently reasonable in view of the actual agc?Lprice pattern shown in Figures l and 2 

of the introductlon. 

Table 14 compares the parameters of the unconstrained Box-Cox model 

estimated under different assumptions about the retirement process. The 

estimates vary across retirement assumptions. It is plausible for �the unre8 

tired case to differ from the other cases, but the divergence between the 

truncated and survival-weighting approaches is relatively modest. 

Finally, the test of the polynomial regression against the semi-log 

regression pzoved ambiguous. Since, however, the unconstrained Box-Cox 

likelihood was always larger than either the geometric or the polynomial, 

the Box-Cox model appears preferable for this class of assets. 

We have applied the methods described in the preceding sections to 

a vide variety of assets: commercial and industrial buildings, automobiles, 

66 
Table 14 

TRACTOR 

(Model D-7) 

Comparison of Transformed Prices to Truncation Form 

Unconstrained Box-Cox (Undeflated Prices) 

Form Parameters Slope Parameters 

Form E E e a B Y Log L 

1 2 3 

Unretired .217 .640 1.266 38.714 -2.082 .758 -5519.2 

(.004) C.046) (.128) (1.31) (.254) C.163) 

Retired .232 .998 1.296 41.278 -1.147 .gag -5483.2 

(.0043 c.046) (.131) (1.397) (.148) (. 179) 

Truncated* .344 .713 1.242 96.404 -6.213 2.799 -5473.4 

(.004) (.048) (.124) (3.994) (.812) (.579) 

“9,. P $2047 (Taken from &.n {qi: i - 5821) 

, 

67 
trucks machine tools ~ cnnstruction. ~?.cw~,.;,iv,,?.nt . .. wd ?.?y c-~ir:.::.?!'c.E. %!A:::r,?,o~s,: 

findings are roughly consistent with the results reported for D-7 tractors. 

Geometric bd atraight-line are almost uniformly rejected, and the estimated 

age-price patterns are almost always accelerated relative to straight-line. 

The final step in studying depreciation and productive efficiency of an individual asset is to construct the relative efficiency function from the estimated vintage asset pri,ces. It wi1.l be recalled that productive effici,encies of physical assets are derived from vintage prices 

by employing the concepts of the user cost of capital and of duality. Duali,ty 

establishes the linkage between the marginal rates of substitution of two 

pieces of capital and the ratio of their respective user costs. We constructed 

the user cost formulas utl.li,zing the theoretical model. outlined earlier in 

Secti,on A of Part I1 of this report. We normalized the efficiency functions 

on the user cost of a new asset. In ordef to compare the efficiency sequences .,-

produced by the Box-Cox prices, we plotted the efficiency function derived 

from the Box-Cox approach and in the same graph an efficiency function derived 

from a Semi-log equation. These graphs appear in the appendi,x. Figure 7 

produces this figure for the D-7 tractor utilizing retired data. 

The actual efficiency sequences calculated for every year io.which 

L.,....L;ts fzr each asset we studied appear in the sppendiz as well. Table 

15 contains the annual Box-Cox efficiency sequences based on retired data 

iur me G-7 Tractor from 1968 through 1977. Because it may be necessary 

eventually to construct capital stock estimates outside the sample period, 

we have produced the average Box-Cox efficiency sequency over the years. 

Table 15 also contains the average Box-Cox efficiencies ages new to 32. 

By comparison, the efficiency functions one would obtain by assuming a 

constant geometric rate derived from the semi-log equations (see Table 12 

68 
- Figure 7 
U 

0 

Table 15 i 

BOX-COX EFFICIENCIES 

.Asset: Tractor D-7 Years: 1968-1977 (1975-1977)

Retired Prices 

Age 1975 

0 1 

1 0,92322 

2 0 ,85199 

3 0.70510 

4 a, 72249 

5 0,66368 

6 0.60856 

7 0,55696 

0 0.E" 

9. 0, Y6367 

10 0 ,$2168 

11 0,3026 

12 0,34620 

13 0,31259 

14 0 * 2814 

15 0,25258 

16 '0,22602 

17 0.20150 

10 0.17915 

19 0,15063 

20 0,1399 

21 0.12205 

22 0.10730 

23 0,093399 

24 o 

,000~03 

25 0.06'?501 

26 o. 059lrn3 

27 0,050424 

20 0,042481 

29 0 I 035492 

30 0 I 029302 

31 0.024075 

32 0.019503 

1976 1977 Box-Cox Sem

1 1 

0,92573 0,0

0.95676 0 I 7

0 I 79193 0.6

%,13(195 0.5

0.67361 0.4

0 ,61972 0 I 3

0 I 56914 0 I 3

a I 52172 0.2

0 47'731 0,2

D * 43578 0 2

0.3Y701 0.1

0 I 360% 0.1

0,3272 0,1

0 I 29594 0,1

0 I 26694 0.0

0.2401 

0,2153 

n, 0

0.0

0,192'45 0.0

0 I 17144 0,0

0,15217 0.0

0,13451t 0,03

0.11846 0,0

0 10303 

n, o~o"J.~ 

0, 0

0,0

0 0706Ql+ 0,0

0, 067832 o, 0

o nsma 0.0

0,04Y57 0, 0

0, 041928 0,00

0.035181 0 * 0

0.029259 0.00

0.024096 0,0
U 

ITable 15 

BOX-COX EFFICIENCIES 

Asset: Tra :tar E-7 (Can't.) Years: '1968-1974 

Retired Prices 

1968 1969 1970 

1 L 

0 .91.08 0.91106 

n ,82079 0,5282 

0 - 75266 0,75322 

n 65116 0,60267 

0.61639 0 I 61726 

n .55565 0 55669 

0 I 4995 0,50069 

0 44768 0, 44903 

0 I 39996 o .ttni47 

0,35612 n ,35777 

0 I 31594 0.31?72 

0 :27921 0.28111 

0.24573 0 I 24774 

0,21531 n ,2im 

0.18775 0 I 18991 

n. 16288 0,16508 

0.14051 0 14274 

0 I 120b8 0,1227': 

0 I 10262 U, in485 

0,086782 0,080979 

0.07280& D I 0749W 

n .ooom2 0,062616 

0 % 049856 0 I051842 

0.04U611 U 0424.93 

0 I 032677 0. n3w+i 

n. 025~3 0 # 027564 

o,n20:?53 u, n::i71+-7 

0 IO15532 0,016879 

0 I 01166 0 * 012855 

0 I 0085354 n I 0095763 

0.0060624 0.0069495 

n. 00411199 n ,now~71 

1971 1972 1973 

1 1 

0.91834 

0,04278 

0,77221 

0,70627 

n ,6v.~72 

0 < 58732 

0 I 53386 

O,lrBL&16 

0.43801 

0.39525 

0 3557 

0.31919 

0,28555 

0.25463 

0 ,22629 

n ,20036 

n, 17671 

0.1552 

0.1357 

0 111807 

o,inxi 

0,087975 

0 I 075264 

0.063961 

n I 0539~

0, 045i63 

n, o:wt6? 

0.03078 

0,025011 

0 020076 

n ,o 15893 

0,01.?3O3 
for the semi-log coefficients) also appear in Table IS. Thin cnmp1-m~;~;YT 

Giocwsion GP the econometric analysis of the E-7 Tractor. 

A similar set of procedures was undertaken for each of the 26 assets 

studied in detail. Al.1 of these resu1,ts appear in the append1,x to thi.s report. 

Each asset group is organized by its asset classifi,catfon. Preceding each 

asset class’s actual. econometric results i,s a brief outline of the information 

available on that particular asset. This should make it more convenient 

to study the actual ana1ysi.s of the assets. We turn now to the problem of 

deriving from these individual asset studies estimates of depreciation for 

the entire 1J.S. capital stock. c 

72 
- - - - - 

111. DEPRECIATION FATES AND EFFICIENCY FUNCIXONS TOR 

TEE U.S. CflITAL STOCK: METHODOLOGY LYD RESULTS 

The objective of this section is to produce our best ycfo-ej-ril 

estimates of economic depreciation and relative productive efficiencies 

for the U.S. capital stock organized into 22 PDE, 10 PXZ .. ~ 

.--- -. ~ -

The flnal estimtes are built upon three types of informat+nn' '1) cur 

detailed econometric investigation of the 30 specific capital assets. The 

econometric results appear in two sources: (a) The results for PDE classes 

and for the consuner automobile appear in the appendix. (b) The evidence 

? 

for private non-residential structures appears in the Phase IT Report of 

Contract TOS-74-29. (2) The existing literature on economic depreciation: 

the bibliography search, in the Phase I report. produced over 180 studies 

with direct bearing on the theory, measurement and policy issues involving 

economic depreciation. Several dozen of these studies actually produced

c 

aepreciarion estimates of specific types of capital equipment. (3) the 

conventional treatment of depreciation in existing literature: the point 

of departure for measuring economic depreciation for producer durable eqrrlpmerit and private non-residential structures must begin with the seminal work 

of Dale Jorgenson and his followers as well as with the recent capital stock 

studies of the Bureau of Economic Analysis (BEA). For consumer durables two 

Sources have been located which suggest the convenrjonal treatment for these 

%Sets: (a) the well-known work of Raymond Goldsmith and (b) the Flew nf Funds 

73 
Accounts of $he U.S. Federal Reserve System. 

The y :.::':Y:.CGI confronted in thi,s section is to convert these three 

sources of information into specific depreciation and efficiency function 

estimates for each of &he 39 classes of assets which comprise the U.S. 

capital. stock. This conversion is accomplished in three stages as now reported, In section A below we convert our estimates, the literature search 

and the conventional wisdom into more useable forms. Thi.s primarily consists of simplifying the econometric results obtained for the indivi.dua1 

assets as discussed above. The second step, reported in section B. is to produce average depreciation and efficiency function estimates for the asset 

clasself in which we have considerable data, Type A asset classes. The third 

and final step involves inferring our best estimates of depreciation and productive efficiency for the Class B and C assets for which we have only scat- r c: 

tired or no data. The final outcome wilX cpnsist of two tables containing 

our best professional judgment (BPJ) average rates of depreciation for each 

of the 39 asset classes a1,ong with two sets of efficiency sequence estimates 

for each asset class. One set of efficiency sequences is derived directly !, 

from the Box-Cox power transformation and the other from the best geometric 

approxunation (BGA) to that functional form. / 

A. Conversion of Micro Estimates into Useable Form for Macro Approximations: 

The first specific problem to be solved in the construction of depreciation estimates by asset class is to convert the detailed econometric analysis 

of individual assets into forms which can be easily averaged for purposes of 

constructing macro estimates. The first step in accomplishing this conversion 

is to consider the following question: If the depreciation pattern produced 

by a Box-Cox equation were to be approximated by one smooth, constant geometric 

74 
pattern, what: would be the Reometri,c rate and. hmt7 c?me .?n.ild he tl:e fir. cf 

the geometric pattern wh5.ch approximates the Box-Cox to the latter form? 

To answer this question, we employ the following equation: 

The left hand side of the above equation consists of the logs of the predicted asset prices, by age and date, from the Box-Cox power transformation. 

Based on our hypothesis tests of various functional forms we selected the 

unconstrained Box-Cox power transformation form. It is the maximum lilcelihood set of parameter estimates from among the four Box-Cox power transformations tried. While the resultant predicted prices are our best guesses 

as to the prices of assets by age and date, the functional form is extremely 

complex and impossible to easily aggregate. To resolve this problem, we 

utilize the predicted Box-Cox prices in the equation above to estimate the 

approximate average rate of decline. The coefficient 8 in the equation above 

will represent the average rate of price decline with age according to the 

Box-Cox power transformation. We label this coefficient the best geometric 

approximation to the Box-Cox depreciation process. sese rates are hereafter

referred to as BGA rates.) , 

The BGA depreciation and revaluation rates from the unconstrained BoxCox form appear in Table 16. Briefly, these depreciation and revaluation 

rates are derived by estimating the above geometric equation using as a 

dependent variable the predicted prices from the unconstrained Box-Cox power 

transformation estimated on retired asset prices. As an indication of the 

closeness of the BGA rates to the Box-Cox predicted prices we include in 

Table 16 the coefficient of determination or R 2 for the above equation. 

For each asset class, the depreciation rate estimates fall wi.thin 

75 
-- 

-- 

-- 

-- 

-- 

-- 

Table 16 

BGA Depreciation and Revaluation Rates 

and R2-Values for Specific Assets 

(Unconstrained Box-Cox on Retired Prices) 

Asset - 6 Y R2 Asset 6 '

Tractor 

D-4 

D-6 

D-7 

D-8 

11-9 

Compressor 

Grader 

Loader 

WG 9 

WG 12 

WG 19 

GMC Truck 

Ford Truck 

Truck 

Tractor 

Dump Truck 

~ Y R2 

PDE 4 (-16.33%) PDE 11 (-27.37%) 

-12.04 17.92 .954 Typewriter -24.58 I .923 

-18.05 25.23 .911 PDE 17 (-33.33%) 

-16.22 19.52 .957 Chevrolet -39.76 .978 

-17.32 20.76 .966 Plymouth -31.03 .945 

-18.05 18.37 .978 Pickup -29.19 .948 

PDE 6 (-17.22%) PNS 1 (-3.61%) 

-16.76 2.55 .833 Factory - 3.61 3.08 .997 

-19.69 15.68 '' ,962 PNS 2 (-2.47%) 

-15.22 14.67 .966 Office . -2.47 3.84 .989 

PDE 8, 10 (-12.25g) Retail -2.20 4.17 .993 

-14.08 1 .991 Warehouse -2.73 2.99 .995 

-10.51 .977 CD 1 (-27.25%) . -12.02 .958 Buick -27.54 4.06 .970 

PDE 16 (-25.37%) Cadillac -29.54 2.86 -975 

-22.25 5.12 .952 Chevrolet -27.75 2.40 .962 

-23.70 5.06 .967 Plymouth -30.50 3.00 .981 

-31.34 -7.61 .779 Wagon -29.48 0.00 -994 

-24.18 12.03 -922 

'Noycoefficient appears for data based on one or two years only. 

76 
a comparatively narrow ~.ZII%~~For kx?.%pZ.e7 >;,:irk .~;wj.iicl: L+L:,::~Y::; w 5;. :imates for the five types of tractors is frOm-12% to -18%. The range for 

consumes automobiles is from -27% to -30%. The range for PDE class 16 is 

from 22% to 31%. The range for private non-residential structures commercial is from 2.2% to 2,7%. h we shall see later these ranges are quite 

nasrow in comparison to the total range of depreciation values reported in 

the conventional treatment of assets by othex analysts. The R2 values indicate that the EiGA appxoach is very close to the underlying, unconstrained 

Box-Cox estimation procedure. Of the 29 assets studied and reported in Table 

16, 26 have R2 values above .9. (We only report here the BGA rates for assets 

which are used in the subsequent analysis. In particular we do not report 

estimates of the remaining structure classes which do not fit into the PNS 

categories, nor evidence for a number of MPG classes which were studied in 

some detail, nor evidence for a few asset classes for which the Box-Cox power 

transformation failed to converge. ) 

We conclude from Table 16 that the best geometric approximation to the 

Box-Cox power transformation produces a set of estimates of depreciation 

and revaluation which are quite close to the true Box-Cox rates. Furthermore, 

within each asset class, the range of estimates is very narrow, soshat 

-.-.Io- .-~esby class should be reasonably reliable. Referring again to 

Table 16, the number in parentheses next to the title of each asset class 

L ,L, -“*rage BGA rate which will be employed to represent the average depreciation rate for that class. For example, for PDE Class 4 the average 

is 16.33%. For PDE Class 16 the average rate 5.5 25.37% and for the PNS 

Class 2 the average rate is 2.47%. These average BGA rates for each class 

will be used in subsequent analysis. 

77 
The li.terature search reported in the Phase T Rcpsrr: has senrcd. 2, 

number of purposes i.n this study. In addition to acquainting us with the state 

of the art in depreciation research, several. studies which reported actual 

depreciation estircates gave us some notion of the reasonab1,enes.s of our results, In particul.ar, the studies of automobiles by Wykoff ~ Ramm and Ackerman confirm the reasonableness of our automobile depreciation estimates. 

Our esti,ma.tes are somewhat higher than these earlier studies, because we 

have introduced the retirement of automobiles,as well as the loss of inplace value. Additional confirmations were derived from the earlier study 

of tractors by Griliches and from the study of trucks by Robert Hall. These 

studies confirm the rank ordering of depreciation rates across these various 

asset classes as well as the general magnitudes. The only major study which 

appears to be quite far out of 1i.ne with-our estimates is the work of Taubman 

and Rasche on office buildings. As we noted earlier, we have reason to 

disagree with the Taubman and Rasche results, Fhdlr, we note that 

Robert Coen reports similas patterns of depreciation but disagrees as to 

the appropriate rate. 

In addition to providing an independent basis for judgment of the 

quality of our estimates, the literature search produced several studies 

which report rates of depreciation for assets not covered in our own 

econometric analysis. Two of these studies are by Rafael Weston (1972) 

and Wilhemina Leigh (1977) are of a very important consumer asset 

class--6, residential structures. In addition, the literature search 

produced two other studies which we will employ in the development of 

our best guesses as to depreciation rates by asset class. Table 17 contains 

the depreciation rate values we have derived from these other studies in 

addition to the rates we have derived from Weston and Leigh. The two re70 
aaS,nl.nz ?,ti\?.:lea ccwzz~:j.cl.: cf J. ctii:!y 6:: ski;:;^ ;,:r :$I,;u Sci~Le.: ax1 02 

furniture and radios by Garcia dos Santos. We feel that these latter studies 

are somewhat less reliable than the others primarily because they do not 

deal with data from the U.S. capital stock. 

As we noted above, depreciation of the U.S. capital stock must begin 

with the work undertaken by Dale W. Jorgenson, BEA, BLS, Raymond 

Goldsmith, and the Federal Reserve System. Just as our econometric 

analysis had to be modified to make it conformable to the construction 

of depreciaiton estimates by asset class, some modificiation is necessary 

for the evidence provided by these conventional sources. We begin first 

,-

with the asset Liferimes provided to us by Dale W. Jorgenson. Tables 

18A and 18B contain the llfetimes provided to us by Professor Jorgenson 

rC 

for 22 PDE classes and for 19 non-residential structure classes. It 

is our understanding that these service lives are esentially those use 

in the BEA capital stock study. However, there are some modest exceptions 

based on wotk undertaken in recent months at BLS and by Jack Faucett 

Associates. We used these lives to construct depreciation estimates Cor 

each of the PDE and PNS classes required in this study by calculating 

the double declining balance, 1.5 declining balance and straight &kclining 

balance methods from them. These declining balance rates appeared earlier 

in this report in Tables 9A and 9B. It should be recalled that the 

methodology employed by Jorgenson and others in constructing capital stock 

estimates has involved utilizing the double declining balance method 

applied to asset lives. Thus the double declining balance column 

of Tables 9A and 9B employed with the Jorgenson lives is the point of 

departure for our estimation procedure. 

79 
m5 'Li; ?.7 

Selected Estimates by Asset Class from Literature Search 

- Assets Studied 

PDE 19 Steel Hull 

Wood Hull 

(Japanese) 

cD2 Furniture 

ea 4 Radio 

(50-500 eon) 

(50-500 ton) 

Evidence Average Authors 

-13.4 to -14.3 -13.8 Lee 

- 3.8 Garcia dos 

Santos 

- 7.1 Garcia dos 

Santos 

-1.6 - 1.55 Weston 

-1.5 

-1.06 -1.01 Leigh 

-95 

06 Owner Occupied 

Tenant Occupied 

Census (Unadjusted) 

Census (Adjusted) 

80 
7,151c :e. 

Producers Durable Equipment 

(see NZPA Table 5.6) 

lifetime 

1. Furniture and fixtures 15 

2. Fabricated metal products 18 

3. Engines and turbines 

4. Tractors 

5, Agricultural machinery, except tractors 

6. Construction machinery, except tractors 

7. Mining and oilfield machinery 

8. Metalvorkini: machinery 

25 

8 

17 

9 

LO 

16 

16 

14 

8 

10 

14 

14 

14 

6.8 

6.8 

9 

22 

1 

25 

11. 

15 

, 

9, 

10. 

11. 

12. 

13 

14. 

15 

16. 

17. 

18* 

19. 

20. 

21* 

22 (I 

Special industry machinery, N.E.C. 

General industrial, incl.uding materials 

handling, equi.pment 

Gffice, computing, and accounting machinery 

Service industry machinery 

Electrical transmlssion, distribution and 

industrial apparatus 

Communication equipment 

Electrical equipment, N.E.C. 

Trucks, buses and truck trailers 

Autos 

Mrcraf t 

Ships and boats 

Railroad equipment 

Instruments 

Other equipment 

81 
Non-residential Structures 

(see Interindustry Transactions in New Structures and Equipment,

1963 and 1967, Volume I,p. v, vi) 

1. Industrial buildings 

2. Office buildings 

3. Warehouses 

4. Garages and service stations 

5. Stores and restaurants 

6. Religious buildings 

7. Education buildings 

8. Eospital buildings 

9. Other nonfarm buildings 

10. Telephone and telegraph facilities 

11. Railroads 

12, Electric utility facilities 

13, Gas utility facilities 

14. Petroleum pipelines 

15. Farm residential buildings 

16. Farm service facilities 

17. Oil and gas wells 

18. Oil and gas exploration 

19. Other nonbuilding facilities 

lifetime 

27 

36 

36 

36 

36 

48 

48 

48 

31 

27 

51 

30 

30 

c 

20 

.. 50 

38 

16 

16 

31 

82 
4>.~"a7.t:erantj.lre rioiirrci of ba.o.c~~nr'costi.ma.tes for depreciation 

rates for the U.S. capi,tal stock comes from capitat stock studies 

undertaken by BEA. BEA has produced capital stock studles for some 

time, and we have used the estimates reported in the April 1970 Issue 

of the Survey of Current Business. The BEA procedures for estimating 

depreciation were described in the October 1966 issue of the Survey of 

Current Business and involve utilizing both straight line and double 

declining balance deprecfation methods combined with the Winfrey-3 

retirement distribution. However, BEA emphasizes the estimates based 

on their straight line variant. By employing a perpetual inventory 

equation and utilizing the gross investment data from the national 

accounts and the capital stock data produced by the BEA study, we were 

able to derive average annual rates of depreciation implicit in the 

BEX studies. The formula employed is as follows: 

As with the rates from Jorgenson's lives, the implicit BEA rates are 

in Tables 9A and 9B. With the exception of the office equipment'class, 

the depreciation rates produced from the BEA capital stock studies are 

not completely unreasonable. However, as we shall see later, these 

iates are in general higher than those of our study. 

For purposes of comparison, we turned to two studies 

83 
57kich .;-?l?y ?e.?re.c?-SC 5.x. TI~Z E . .,.',,.,.v7v ,,,.!<. - (1 3 7, :,u:!;. 

of housing by Raymond Goldsmith based on data produced in a 1937 Financial 

Survey of Urban Housing. The second source for consrrmer.durabl,e benchmark 

estimates are service lives assumed by the Federal Reserve System in the Flar 

of Funds Accounts which is discussed in Phase I. The Federal Reserve also 

reports service 1,ives derived from the Goldsmith studies in the 1962 -NBER

Reports (also discussed in Phase I). Employing the same method for constructing rates from Jorgenson's servtce lives, we constructed doub1.e declining 

balance and decl.ining balance depreci,ation rates from the Goldsmith and Flow c 

of Funds service lives. These depreciation rates also appear in Table 9B. /r 

B. Derivation of Depreciation Rates and Efficiency Functions by 

r 8 Asset Class for the U.S. Capital Stock 

As noted in the introduction to this Phase II Report, the 22 PDE, 

10 PNS and 7 CD asset classes are partitioned into three types. The type A 

asset classes are those for which we,have done extensive research in this 

project and in previous Treasury work or for which we have reliable estimates 

from other studies. The type B asset classes are those for which we have r 

partial evidence and type C asset cLasses are those for which we have no 

micro studies. The asset classes are listed by type earlier in the report 

in Table 2. For the type A asset classes, 10 altogether, we utilize the average BGA depreciation rates reported in Table 16. These rates are based upon 

the extensive analysis undertaken in this study and in the study presented 

to the Treasury under Contract TOS-74-27. The consumer durable class 6--

residential structures rate employed is an average of the rates estimated 

by Weston and Leigh. These rates are listed in the last colwm of Tables 

9A and B opposite the appropriate asset class number. We consider our 

e4 
estimates in Tables 9A and B to be more reliabLe than any esti.mates roade to 

date on these asset classes. Consequently, it is instructive eo I .:erve tt 

these estimates, carefu1l.y constructed by historical standards, compare 

to the conventional treatment of assets by Jorgenson and BEA. These comparisons will serve as a basis for inferences about the appropriate depreciation patterns we should use for both type B and type C classes. 

Inspection of Tables 9A and B indicates that of the 8 assit classes 

for which we have reliable estiffiates, 7 of our estimates produce rates siEaller 

than the BEA rates and 7 produce rates smaller than the Jorgenson rates. 

The BEA tnick rates are small and the Jorgenson auto rate is a bit mall, 

but these can both be attributed probably to our allowance for retirements. 

Consequently our estimation process implies that both the BEA and the Jorgenson depreciation method, double declining balance applied to the lives reported in Tables 18, are too large. 

Taking the 6 PDE classes first, we are interested in drawing a general 

inference about the Jorgenson methodology of deriving a depreciation rate from 

asset service lives. If we can establish a general pattem of the relationship between our depreciation estimates and those derived from the Jorgenson 

service lives, then we can apply the same procedure to deriving hew depreciation estimates from the service lives for the remaining 2sset classes. 

In other words, we hope to infer a method of deriving a rate frcm the Jorgznson lives by comparing our methodology to the rates produced by Jorgenson. 

Ke proceed by using the follotdng equation 

8 =X/L 

A 

This fomula is used for deriving a depreciation rate 6 from a senice life 

If the double declining balance lethod is used then the X is 

replaced'by 2. If EL 1.5 declining balance schene is used then 
J! br%%orEs 1,,:5. "1, SO!.?*, for the l:¶lr.7.E.Z> "Tt et:. r-iT,hnc h?X3. :r:.::.a::. >: '?!!x! 

equation by employing our depreciation rates for each of the six classes 

for which we had evidence and by using Jorgenson's service lives. The result i.s the appropriate number to which one will then apply a declining 

0al.ance method with Jorgenson's service lives and derive a new depreciation 

rate for other asset classes. 

For Producer Durable Equi.pment, the appropri.ate decl,ini.ng balance 

pattern, based on our new depreci.ation estimates are invariably larger than 

1.5 but less than double declining balance. In the six cases studied, with 

the exception of the auto class, the declining balance rates ranged from 

1.3 to 1.9. The auto rate we bel.ieve can be explained by the fact that our 

auto estimate is based on automobile prices derived from GSA data. These e 

automobile prices reflect the depreciation of autos used by industry rather 

than by households. Jorgenson's rates are probably based on depreciation 

studies of consumer autos. (This conclusion is supported by the fact that 

our consumer auto rates are somewhat lover than Jorgenson's doub1,e declining 

rates and therefore consi,stent with the trend observed in these other PDE 

-:----;.; The average declining balance method implied by our estimates, 

6 

r?-.-:. .': 1.61-dedining balance. In other words, if one had no infomation 

other than the service lives on each asset class, then the appropriate de­

;--c'.+T?n method to use for each of the additi.onal classek would be to 

app;j a 1.65 declining balance scheme to the Jorgenson service lives. 

For Private Non-residential, Structures, as r;e have di.sccssed at some 

length in (1977) and in the addendum to the Phase I Repott, the implicit 

declining balance method applied to Jorgenson's lives for structures sould 

be l.ess than B straight declining balance rate. In particular, for the fom 

largest structures studied, factories, offices, retai.1 stores, and warehouses, 
the implicit declining balance method ranter? from ,.79to -9%. %:+ ::-.%r”-ge 

rate was .91. Again, if one had no other information for the other structure 

classes than the Jorganson service lives, one would impose a .9 declining 

balance scheme to the lives in order to derive the appropriate deprecistion 

rate. 

Because automobiles and residential structures are really unique 

consumer durable goods, it is unwarrented to derive inferences for the other 

consumer durable classes such as furniture, radios, toys and the like. We 

shall have to use other methods in drawlng estimates for these assets. 

We are now prepared to produce our judgmental estimates for type B 

asset classes, These classes are: PDE 11--Office Computing and Accounting 

Nachinery, PDE 19-Ships and Boats, PNS 5-Hospital and Institutional Buildings, PNS 6--Other Buildings (mainly social and recreational), CD 2-

Furniture and CD &-Radio, TV9 Recorder and Musical Instruments. We shall 

discuss each of these asset classes case by case. We employ the following 

pieces-of information in deriving these judgmental estimates: (a) the evldence from type A assets, (b) the conventional treatment of these assets 

by Jorgenson. BEA, Goldsmith or the Federal Reserve, and (c) any general 

information we feel should have bearing on these particular asset classes. 

The evidence we have for PDE 11, Office Computing and Accounting 

Hachinery, consists of our econometric estimates of the depreciation rate 

implicit in vintage price data on Royal Typewriters made available by GSA. 

The typewriter depreciation rate is 27.29%. Jorgenson’s double declining 

balance scheme implies a rate of 25%. The BEA capital figures evidently 

c0ntai.n a typographical error because their rate suggests 3%. The 1.65 

declining balance scheme applied to the Jorgenson service lives, suggested 

by the type A asset information, suggests a depreciation rate of 20.6%. 

87 
Thus the ru~eof values which ve might cdnsider runs from 20.6% to 27.3%. 

Even though the typewriter obviously does not repreoent the largest category 

of office computing and accounting machinery, we believe that it is closer 

to the true depreciation value than any of the other possibilities. The 

reason for this belief is that the malor types of assets in this category 

are probably computers and accounting machinery. These types of assets have 

undergone substantial technological changes over the past several decades 

and there are Indications, from the computer industry especially, that such 

changes are likely to continue for some time. Since obsolescence has played 

such a major role in depreciating these types of assets. we feel that the 

typewriter estimate is not out of line and therefore we have settled on a 

rate of 27.3% for Class PDE 11. 

The evidence for Class PDE 19--5hips and Boats comes from the Lee 

study of the Japanese fishing fleet. The average rate we derived from Lee's 

study is 13.8%. The conventional estlmates for PDE 19 depreciation are 9.Z 

by Jorgenson and 10.8X by BEA. The inference from our type A assets, 1.65 

declining balance applied to the Jorgenson service lives, suggests a depreclatlon rate of 7.5%. Thus the range of values is from 7.5% to 13.8X. Be- r 

cause the Lee study deals vith Japanese vessels and because the U.S. commercial fleet tends to be quite a bit older than the Japanese commercial fleet 

wft believe that the 7,5% rate, based on the other PDE type A asset classes 

is closer to correct and we employ this rate here. 

The two PNS classes listed as Class B asset classes, PNS 5-Hospital 

and 'Itlstitutional Buildings and PNS 6-Other Buildings (Social and Recreational 

especially) are listed as type B asset categories because we did study some 

somewhat similar buildings In Contract TDS-74-27, We employ the BGA rates 

to the unconstrained BosCox estimates for these two asset classes. For 

88 
meC I:q' -.*!ldings the depreciation rate is 2.33% and for recreational buildings the BGA rate was 4.54%. These rates will be used here. 

We turn now to our judgmental estimates for Consumer Durable Classes 

2 and 4. These suggested depreciation rates are based on even sketchier 

information than the ones above. The season for this is that we have no 

solid factual evidence for U.S. consumer durable classes 2 and 4. The only 

study we have available was undertaken by Garcia dos Santos from which we 

obtained estimates of 3.8% for CD 2 and 7.1% for Ca 4. These rates compare 

respectively with Goldsmith's 13.3% and 20% rates and the Federal Reserve 

Systea's 20% and 25% rates. While the rank orderings are the same, the 

Garcia dos Santos estimates, from British data, seem quite far out of line 

when compared to the conventional wisdom. The Garcia dos S&os rates are 

quite a bit lower. Nevertheless, ouz general evidence suggests that the 

conventional treatment has been to depreciate assets too rapidly. Consequently? we have decided to lower our estimates of the depreciation of these 

two clasees of consumer durables. Vc'. employ a 10% depreciation rate for CD 

2 (furniture) and a 15% rate for CD 4 (radios, etc.). 

The final step in developing average depreciation rates for the asset 4 

classes required in this study is to provide estimates for the type C asset 

classes. Because we have no information other than the inferences we cm 

draw from the asset classes which we have studied intensely. we employ the 

etxggested declining balance scheme to the Jorgenson service lives. For Producer Durable Equipment classes we employ a 1.65 declining balance scheme to 

Jorgenson's senrice lives. For Private Non-residential Structure classes, 

We &pose a -9 declining balance scheme to the Jorgenson servfce lives. For 

the Consumer Durable classes 3, 5 and 7, we maintain the rank ordering Suggested by the Federal Reserve System but lower the average rate to be coni-
C. Best Professional Estimates of Depreciation and Relative 

Efficiencies of the U.S. Caoital Stock by Asset Class 

The estimates of depredation and relative efficiencies which follow 

involve a high degree of judgment as well as econometric analysis. There I.s 

little doubt in OUT minds thar eventually improved egtimates will be derived. 

At the same time, however. it is our conviction that thi.s study represents 

an attempt to provide a comprehensive econometric base for the derivation 

of depreciation estimates for the entire U.S. capital stock. 

The preceding section discussed the derivation of the average depreciation rates to be used for each class. Essentially. two methodologies 

were employed in deriving these average rates. One involved approximating 

average rates from the more sophisticated non-constant unconstrained Box-Cox 

power transformation applied to retired asset prices. The second involved 

using constant rates inferred from other studies. These two procedures 

suggest that we have available two possible sources for the derivation of 

efficiency sequences. Relative asset efficiencies, it will. be recalled, 

may be derived from vintage asset prices when one enploys the user-cost-ofcapital model and the principal of price-quantity duality implied by neoclassical economic theory. We have two possible sets of vintage asset 

prices, both based on estimation procedures, from vhich efficiency sequences 

may be derived. We shall make available here both sets of efficiency sequences. (Rowever, we have the specialized Sox-Cox "variable rate" cfficiency sequences only for the type A classes. It would seem to us unrea90 
Tables 19 and 20 contain our best professional judgment estimates of 

average depreciation by asset class and of the efficiency sequences correaBonding to those asset classes. Xn Table 19 we present the BGA depreciation 

rates I and the average Box-Cox efficiency sequences. These Box-Cox ef ficiency 

functions are averages over the data. The statistical appendix to this report contains the Box-Cox efficiency sequences for each year for each of 

the individual assets studied. The efficiency sequences produced in Table 

19 for the type A asset classes are those which were derived for the specific 

asset for which the BGA rate was as close to the class average. These selections are as follows: 

X C 

Asset Class 

PDE 4 

PDE 6 

PDE 8, 10 

PDE 1.6 

PDE 17 

PNS 1 

PNS 2 

CDI 

CD6 

Specific Asset 

D-7 

Loader 

MPG-19 

Ford Pickup 

c

GSA Plymouth 

Factory 

Off i.ces 

Buick 

BGA 

While the efficiency sequences presented f.n Table 19 vary across ages and 

widle they reflect the general nature of the Box-Cox patterns better than 

91 
T;?%1? 1.9 

Depreciation (BGA) and Efficiencies (Box-Cox) 

Producer Durable Equipment 

(Type A Asset Classes Only) 

Class .-

Item 4 6 8 10 16 17 

BGA Rate -.1633 -.1722 -.1225 -.I225 -.2537 -.3333 

Efficiency Sequences $(s) 

0 1 

1, 0 I 91841 

2 0 .t342?9 

3 0,77256

4 0 I 70675 ?=-. 0.64531 

6 0.58801 

7 0,53464

8 0 I40501 

9 0.43093 

10 0,39622 ii 0,35671

12 0 a 32023 

13 0 - 28662 

14 0.25573 

15 0.2274 

16 0.20148 

1( 0 I 17784 

18 0.15634 A, , 0. f36e5 

20 0,11923 

21 C. 1.0336 

22_- 0.089131 

ia 0.076418 

T'l+ 0 065113 

25 0.055100 

26 o. 046302 

27 0.0385?6 

28 0,031895 

29 0.02611 

30 0,021154 

31 0. (\16?l+5

32 0.0134O5 

33 

34 

35 

36 

?? 

1 

1.1719 

1,2303 

1.2419 

1.2239 

1 ,1856 

1.133 

1.0704 

1.0012 

0.92801/1

0,85311 

0.77813 

0.70447 

0.63322 

0.5652 

0.50102 

O.lr411,2 

1 

1.1075 

I. 1331 

1.1304 

1.111 

1 * 0805 

I. 0423 

0.998.45 

0.95059 

0 I 89996 

U ,80755 

0.79415 

0 I 74043 

0 I 68695 

0.63417 

0.58251 

0.5323 

0.38574 I 0.48381 

0.33503 1 0.43728 

0.28899 0.39291 

0.24755 i 0.35084 

0.21056 : 0.31121 

0,1778 0,2741 

O.Ib904 0.23956 

0 12398 0,20763 

0..10234 0.17832 

0.083785 0.15161 

0.068023 0.12746 

0.054745 0.10583 

0 I OM36W 

92 

1 1 1 

1 I 1075 1.3361 1.3959 

1 I 1331 1.4265 1.5456 

1 I 1304 1.3978 2.5554 

1.111 1.295 1.4694 

1 I 0805 1.1476 1..321 

1 I 0473 0.97792 1.1377 

0 I 99845 0.8029 0.94178 

0.95059 0.63517 0.75046 

0 ,R9996 0.48346 0.57571 

O18b75!5 0.35304 0.42481 

0.79415 0.24626 0.30096 

0,74043 0.16308 0.20415 

0 I 68695 0.10165 0.1321 

0.63417 

0. SEI251 

0.5313 

0,48381 d

0 * 43718 e 

0.39791 

0.35084 

0.31171 

0.2741 

0.23956 

0,10763 

0,17832 

0 I 15161 

0.12746 

0 I 10583 
c,. I 7 iy (Cot,? .;, j &aoIc. 

pepreciation Rates (BGA) and Efficiency Sequences 

(BOX-COX) 

(Type A Asset Classes) 

- Private Non-resgential Structures 

Item 

Consumer Durables 

Industrial Automobiles Residential Structures

EGA rate -.0361. 

0 1.0 

1 .9595 

2 .9274 

3 .a979 

4 .8699 

5 e 8431 

6 8174 

7 .7926 

8 7688 

9 , .7455 

10 " 7231 

11 .7013 

12 .6803 

13 -6598 

14 .6400 

15 .6207 

16 e 6020 

17 5838 

18 .5662 

19 ,5490 

20 5324 

21 5162 

22 .5005 

23 4853 

24 e 4705 

25 -4561 

26 -4421. 

27 -4285 

28 .4153 

29 a 4024 

30 3900 

31 D 3779 

32 -3661 

33 D 3547 

34 II 3436 

35 3328 

36 e 3224 

37 3122 

38 a 3023 

39 e 2928 

40 o 2834 

Commercial 

-.0247 

Efficiency 

1.0 

.7234 

.6416 

.5896 

.5509 

.5198 

.4937 

.4711 

.4521 

.4331 

.416 7 

.40l.7 

.3878 

-3748 

.3627 

e 3513 

.3406 

,.3304 

e 3208 

.3216 

e 3028 

-2949 

2865 

2788 

2714 

a 2644 

2576 

,2510 

2441 

2386 

2327 

.2270 

2214 

2161 

2109 

.2059 

* 2010 

.1963 

.1917 

-1873 

(I1829 

a2 

-.2725 

Sequences +(s) 

1.0 

.915O 

.7853 

-6559 

.5352 

,4264 

-3308 

-2490 

1806 

.1251 

.0817 

.0494 

.O268 

.0126 

.0049 

-.0128 

_II 

9

1.0 

.9872 

.9746 

.SULl 

.9498 

-9376 

.9256 

.9138 

.9021 

8905 

.a791 

.E679 

.a568 

-8458 

.E350 

,8243 

.a137 

.e033 

.7930

- 7829 

.f729

- 7630 

e 7532 

7436 

e 7341

- 7247 

.7154 

.7062 

.6972 

.6883 

.6795 

.6708 

-6623 

.6537 

a 6453 

-6371 

.6289 

.6209 

.6129 

.6051 

.5973 
h 

.__ ...... ............. .............................. iCQc00~00c~00u0~00c0o~~0ooc0cou 

...... 

oooc203 
-- 

-. 

Table 20 (Con't.) 

Producer Durable Equipment 

- Item 7 8 9 10 11 12 

BGA Rate .1650 .1225 .lo31 .1225 .2729 1650 

Efficiency Sequences d (6) 

0 1 1 1 1 1 

1 0.035 0 ,a775 0.8769 0 ,87'75 0,835

2 0, 69723 0.77003 0.804Lt3 0.77001 0,69723 

3 0 .SB:ll8 0 .c?7558 ,72149 0 I 67568 D ,58218 

4 0. 4861,2 0,57291 ii I 64.713. 0, 5929.1. !I .'I8612 

5 0.405:-'1 0 a 52028 0,58039 0.52028 (! .405?1 

6 0 .33871.1. 0 , lt565't 0,52055 0 ,4Sh51l !) 33894 

7 0.28301 0 ,40062 0 I 4.6588 0 *400h2 0.28301 

8 0 I 23432 01351Y+ 0.41875 0,35154 0 I 23632 

9 0.19732 0 ,30840 0,37557 0,3081t8 @.1?732 

10 0 I 16475 0,27069 0.33685 0.27O6O 0.16476 

11 0.13758 0,23753 0.30212 0,23753 !!. 1.3758 

12 0 * 11488 0,20843 0 .2'7[! V'7 0.2051.t3 !J I 11488 

13 0,09592 0 I 1829 0.2430Lt 0 * 1.823 0 * 095923 

14 0 ,08009 0 I 21798 0,16049 0. OOOO96 

15 0 I 06608 ",R"8; 0.39553 0. 1Lt083 0.06688 

16 a omeit 0 I 12358 0 I 1'7535 0,1235(! 0 * 055845 

I? 0.04663 0 ,1084.4 0 I 15717 0 ,1 Ok?'tLt 0.046631 

18 0 I 03893 0 I i19515' 0. 3Lt106 0 I 09515 0 I 038936 

I? OI03251 0.093SO 0. 12551 0,08350 !i.032512 

20 (1 IO27111 0,0732; 0 I 1131t7 0,07327 0,027147 7.1 0, 06'I-27 0.10177 0,0642'7 

77) 0.05642 0,09127' 0,05642 

27 0 * 04Y50 0. @8186' 0,04950 

7Lt 0, 041141.t 0.0 731+2 0 I 04344 '-5 r:. i..l 0 ,0381 .i 0. L!c.585 0, n,7t112 

95 
-- 

Item 13 14 __ 15 16 17 18- -

BGA Rate -1179 -1179 .1179 .253? .3333 .1833 -

Efficiency Sequences d(5) 

0 1 

1 0,8821 

2 O177E1 

3 0,68436 

4 0,605'44 Lr' 0.53406 

6 0.4710Y 

7 0, %I555 

8 0,36656 

9 0,32334 

10 0,28522 

11 0 I 25159 

12 0, 22193 

1,3 0,19576 

14 @ ,172S43 

15 0,15232 

16 0,13436 

17 0.11852 

18 0.10455 

19 0,09222 

20 0,08135 

21 0,0717fi

22 0,06329 

23 O1055l!ti 

24 0.011925 

25 0,0'k344 

25 

27

2a 

29 

30 

1 1 1 1 

0,9321 0 ,8821 0 .7M3 o ,8167 

0,7781 0, i7Sl 0.556C6 0.667 

0.68636 0,60636 0 ,%lSb 0 .54.474 

0 ,605'14 

0,53405 

0,47301 

0.4.1555 

n..,36656 

0,505W 

0,53406 

0 ,147 10 5' 

0 ~ 1.k 1555 

0.35556 

0 ,3102:1 

0 , 23 1.5 1 

0 , 1727'7 

0,12894 

0 I 09622 

(1 ,t.1 '.l.4.85) 

!!,36334 

0 ,29674 

0.24235 

0 ~ 19792 

0,32334 0 32534 0,07101 0 I 16165 

0,78522 0,28522 0 * 05359 0,13202 0,25151 O12f.159 0.03977 0.10782 

0.22113 0, 1,2113 0, oms? 0.088051-1 

0 I 1%i6 0,19576 0 ,02227 0. 071914 

0,172m 0,3'72hP 0 ,0 1id? 0.058732 

0,15232 0,15232 0 I 0 124 I! 0,047767

0,13455 0 I 13436 0,039174 

e, 11852 0.11852 0 I 031994 

0,1Of.t55 0 10455 0 ,026129 

0,09222 0,09222 0.0213% 

0.08135 0 I09135 0,017428 

0 I 071.75 0.07175 

0,06329 0 * 06329 

0,05583 0 I 05583 

0, 04925 0 * 04925 

0,04344t 0.04344 

96 
Item 19 

BGA Rate .0750 

0 1 

1 0,925

2 0,85562

3 0 I 79145 

4 0 ,73209 c

J 0 I 6771,9

6 0 I 6264 

7 0.57942 

8 0 I53594 

9 0 I 49576 

10 0.45658 

11 0 ,42419

12 0 I39237 

13 0.36295 

14 0 I33573 

15 0.31055 

16 0 I 28726 

17 0 I 26571 

18 0 I 24578 

19 0822735 

20 0.2103 

21 0 19453

22 0 I 179YQ

23 0.16644 

iLt 0,15396 

25 0.14241 ",

-.- 0.13173 

27 0.12185 

28 0 I 11271 

29 0 1.0426 

20 

.Ob60 

1 

0,934 

0.87236 

0 I E1478 

0.761 

0.71078 

0 ,66387 

0.42005 

0.57913 

0 I 54091 

0.50521 

0 I 47186 

0.440-72 

0 ,&11.63 

0 381+1+6 

0.35909 

0.33539-

0.31325 

0 I 29250 

0 I 27327 

0 I 25,523 

0 I 23839 

0 * 22265 

0 I 20796 

0,19423 

0,18141 

0.16Y44 

0 * 15826 

0 * 14761 

21 

.1473 

1 

0,8527 

0.7271 

0.62 

0.52867 

0.4508 

0 * 38tt3P 

0.32777 

O12'7"t9 

0,23832 

0 I 20322 

0,17328 

0, 3.4776 

0 ,12519 

0, 107!ttt 

0 I0ClSl 

0 I (178116

0.06661 

U.O567?E! 

0, 0481132 

0.041298 

0.@35214 

0 I 030027 

0,025604 

0 021833 

0.01P,if i 

22 

.1473 -

1 

0 * 8527 

0,7271 

0,62 

0 ,5284'7 

0 ~ 45G8 

0 ,381-IJS 

0.32777 

0 ,279'+? 

0.23832 

0, 20322 

0 ,17328 

0 , 1477.5 

0,125$': 

0 I 107114 

ti 0Clbl 

0, 0'iSllX 

0, Ob661 

0 * 036798 

U.048l.tL72 

0, 0I.tI29tl 

0,035214 

0,030027 

0,025b0't 

0, 0?1833 

0, 018617 , 

0.13E)nh 7 n 4Y 0 .0961+39 ti * 128P5 

.. 96a 
Table 20 (Con't.) 

Pri.vate Non-Residential Structures 

-.Item 1 2 3 4 5 

BCA Rate .0361 .0247 .0188 .0188 .0188 

Efficiency Sequences 0 (6) 

0 1 1 1 

1 

2 

0 9,537 

0 92?l 

0 C753 

0,9E;121 

0.9812 

0 96275 

3 0, RPm! 0,72772 0 I 94455 

4 n ,86323 0.90118 0. P2hfI9 

L-J 0,83207 0.88245 0 .?@5'k1.7 

6 0,80203 0, 86065 D I 89237 

7 0.77308 0 ,839tC 0 I E37559 

8 0.74517 0 ,81866 0.85913 

9 D ,71827 0 ,7984Y. 0,84298 

10 

11 

0,59231.1-

0 * 66735 

0 I 77872 

0 ,75?4.? 

0.82713 

0 I 81158 

12 0 ,61\-326 0.74073 0.77633 

13 0,62003 0,72243 0 * 78135 

14 0,59765 O170LtE?9 0,75566 

15 0 I 57608 0.68718 0 ,752xl 

16 0,555213 0.67021 0.73811 

17 0 I 53523 0 65366 0,72423 

1e 0,51591 0.63751 0,71062 

19 

20^. 

-. 

0 I 49729 

0.4793% 

0.46203 

0,6,2176 

0,60641 

0 I 591tk3 

0,59726 

0.68415 

0 67129 

22 0, 44535 0 I 57682 0 65867 

?-? 0.42927 0 # 56257 0 I 64628 

24 

25 

0.4B78 

o ,390aw 

0.511060 

0,53513 

0,63413 

0 I 62221 

26 

77 

0,30444 

0 e 37056 

0,52191 

0,5090? 

0 .h1051 

0,5C?U4 

2P 

7u 

30 

F a35717 

0,31+42': 

0.33186 

0 , 48 'I.1 !I 

0 , 1.1. i27:7 

0 , n 58777 

0,57672 

0 ,:;658R 

__ 

1 1 

0,9812 0,9812 0 96275 0,96275

0 I 94465 0,94465

0 I 9,7689 0.92689 

0.90947 U ,90947 0 * 89237 0 * 89237 

0 * 87559 0.87559 

0.85913 0.85913 

0.84298 0,84298

0 8,3713 0.82713 

0#81158 0.81198 

0.79633 0.79633 

0 78135 0.78135 

0 * 76666 0.76666 

0 * 75225 0,75225 0.73811 0.73811 0.72423 0.72423 

0 .?io62 0,71062 0,69726 0 I 69726 0.68415 0 I 68C15 0.67129 0.67129 0.65867 0.659-67

0.6Q628 0.6Y628 0.63413 0.63413 0 a 62221 0,62721 0.61051 0.61051

0.5990% 0,5990C o 59777 0.50'777

0 I 57672 0.57672 0,56580 0.56588 
Item 1 

31 0 I3178EI 

32 0 I 30831t 

33 0,2972 

34 0.28648 

35 0,27613 

36 0 ,26617 

37 0 25656 

38 0,24729 

39 1- n ,23837 

40 0.22976 

41 0 s 223,47 

42 0 ,21347 

11-3 0 I 20577 

4 4 0.19834 

Lt5 0,19118 

46 @ I1842Ei 

47 0.17762 

48 0.17123 

43 0,16503 

50 0,15907 

2 - 3 4 , 

Efficiency Sequences 6(8) 

0 ,q&[ls(; 

0.11-4915 

I:c C. 7, ct I 1111; 4 

[t ,54481 

0,55524 

0 WtR1 

0 I 4.3809 U I 534.56 0.53456 

0. 42127 0 * 5245s 0 I 52451 

0 I 41,671 

0.40641 

0, ?;I4.65 

0,513498 

0.51465 

0 I 50498 

0,37638 13 ,(1.95413 0,49548 

0.38657 0 a ttO617 0 .LID617 

0,37704 

0.36773 

n I 4 7'703 

0.46906 

0 I 47703 

0 I 46806 

0, 35865 

0,34115 

0.33272 

0 ,311.9;~ 

0,45926\ 

0 ,45063 

0.44215 

0 * 4338.14 

0 I 45926 

0.45063 

0,44715 

0,43384 

0,3245 

0 .316tl? 

0.42569 

0.41769 

0 I 42569 

0 I 41768 

0.30867 

0.30105 

0,40963 

0.40213 

0,110983 

0 I 40213 

0 I 29361 0 * 35'457 0 I 39457 

0 ,28636 0.38715 0.36715 

5 Con't. 

(I,55;f;^7'! 

0.54481 

0. s34;rj 

0,52451 

0.5146'5 

0 I 50498 

u .L(-7J'iU 

0.48617 

(1 11 1,707 

0.46806 

0 I 45926 

0.GO63 

v;4421 5 

0 I 43384 

0 I 42569 

0.42768 

0 I (1.0983 

0.40213 

0.39457 

0 a 38715 

978 
Table 20 (Con't.) 

Private Non-Residential Structures 

Item 6 

BGA Rate .0290* 

0 1 

1 0.971 

2 0,94284

3 0.9155 

4 0.88895 c

J 0.86317 

6 0.83814 

7 0.81383 

8 0.79023 

9 0 ''76731 

10 0.74506 

11 0,72346

12 0.70247 

13 0 I 6821 

14 0,66232

15 0,64311

16 0.62446 

17 0 I 60635 

18 0,58877 

19 0.5717 

20 0,55512

21 0.53902 

22 0,52339

23 0.50821 

24 0 * 49347 

L3 0.47916 

26 0.46526 57 0.45177 

28 0.43867 

29 0.42595 

30 0.4136 

- 7 8 9 10 

.0316 .0237 .0563 .0290 

Efficiency Sequences d(S) 

1 1 1 1 

0,9684 0 9763 0.9437 0.971 

0 9378 0,95316 0,89057 0.94284 

0 I 90816 0 93057 0, EN043 0.9155 

0,87?47 0.90852 0,79311 0,88895 

0 I 85168 0 I 88699 0,74846 0 I 8.6317 

0.82476 0.86596 0 .?0632 0 I 83814 

0.7987 0 * 845rt4 0,66656 0.82383 

0,77346 0,8254 0,62903 0.79023 

0.74902 0 I 80584 0,59362 0.76731 

0.72535 0.78674 0 I 5602 0.74506 

0.7033 0 I 7681 0.52866 OI7Z!346 

0, ma23 0 I 74989 0.49889 0.703+7 

0,65874 0,73212 0.47081 0 I 6821 

0 + 63792 0.71477 0 ,4443 0,66232 

0.61776 0,69783 0 ,Yi929 0 I 64311 

0 I 59824 0,68129 0 I 39568 0,62446 

0.57934 0.66514 0.3734 0.60635 

0.561 03 0.64938 0.35238 0,58877 

0 * 5433 0.63399 0,33254 0.5717 

0 I 5261.3 0 I 61896 0.31382 0,55512 

0.50951 0.6043 0.29615 0.53902 

0.49341 0,58997 0 ,279It8 0.52339 

0 * 47782 0 ,57599 0.26374 0.50821' 

0.46272 0,56234 0.24889 0 * 49347 

0, W809 0 ,54901 0.23488 0.47916 

0 e 43393 0 ,5.36 0.22166 0.46526 

0.42022 0 # 5233 0,20916 0.Y5177 

0 ,40694 0.5109 0 I 1974 0.43867 

0.39408 Or99879 0 I 10629 0 8 4.2595 

0.38163 0 48697 0.1758 0 ,4136 
Table 20 (Con't.) 

Private Non-Residential Structures 

I 

Item 6 

31 0.4016 

32 0.38996 

33 0.37865 

34 0.36'767 

35 0.357 

36 0.3466s 

37 0 I 3366 

38 0.32684 

39 0 I 31736 

40 o. 30815 

41 0.29922 

42 0 2905% 

,43 OI28212 

44 0 I27393 

45 0 %26599 

46 0 * 25828 

47 0,25079 

%�I0.24351 

49 0 I 23645 

50 0 * 22959 

7 8 

Efficiency Sequences d(s) 

9 10 Continued 

0.36957 

0 a 35789 

0 I 34658 

0 I 33563 

0.32503 

0.31473 

0,30481 

0 I 29518 

0.28585 

0.27682 

0,26807 

0 2596 

0.25139 

0.24345 

0 I 23576 

0 I 22831 

0.22109 

0.21411 

0.20734 

0,20079 

0 * 47542 

0 ~46416 

0,45316 

0 8 44242 

0.43193 

0.42169 

0,4117 

0.40194 

0,3Y242 

0.38312 

Os37&0k 

0.36517 

0,35652 

0.34807 

0.33982 

0 I 33177 

0 I3239 

0.31623 

0.30873 

0.30141 

0.1659 

0.15656 

0.14775 

0.13943 

0.13158 

0 I 12417 

0 -11718 

0.11058 

0,10&36 

0 I 0981+82 

0,092938 

0 I 087705 

0.082768 

o.078108 

0. 073'71 

0.06956 

(1.065644

0 I 061948 

0.058461 

0.055169 

0 ,4016 

0.3B996 

0,37865 

0.36767 

0.357 

0 I 34665 

0 3366 

0.32681t 

0.31736 

0 I 30815 

0,29922 

0.29054 

0,28212 

0.27393 

0,26599 

'0 * 25928 

0 I 25079 

0 I 24351 

0,23645 

0.22959 

9 7c 
Item 

BGA Rate 

0 

1 

2 

3 

11 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

Table 20 (Con't.) 

Consumer Durable8 

_I_-

1 2 3 

.2725 - .loo0 .1500 

Efficiency Sequences 0 (6) 

1 

0 0 7275 

0.32926 

0 I 35503 

0.28011 

0 I 20378 

0 I 14825 

0 .10'785 

0,075463 

0,057082 

0,041527 

0~03U211 

0.021978 

0.01!5959 

0,011632 

0 I 0084624 

1 

0.9 

0 .81 

0,729 

0.6561 

0.59OV9 

0.53144 

0,4783 

0 ,43047 

0,38742 

0.34855 

0,31351 

0,28243-

0.25419 

0.22877 

0 ,20559 

0,1853 

0,16677 

0.15009 

0,13309 

0 I 12158 

0.10942 

1 

0.55 

0 ,7225 

0.61413 

0.52201 

0 I Q(1.371 

0.37715 

0,32058 

a * 27249 

0 I 23162 

0.19687 

0 I 16734 

0 I 14224 

B I I2091 

0,10277 

0 I057334 

OIO7(1.?Sl

0 I 063113 

0,053646 

0 I 045599 

0,03876 

0,032946 

4 

.1500 II 

1 

0,85 

0.'7225 

0 I 61q3.3

0.52201 

0 I lrY371 

0,37715 

0 I 32055 

0.27249 

0.23162 

D.lY687 

0 I 16734 

0,111224 

0.12091 

0 10277 

0.087354 

0.074251 

0 I 063113 

0,053646 

0,01l3599 

0.03876 

0.032946 

0. 028004 

0.023803 

0.020233 

0.017l98 

0,0981177 0,028004 

0.088629 0 I 023803 

0,079766 0,020233 

a ,073'79 o 017198 
Table 20 (Con't.1 

Consumer Durable; 

I ten 5 

BGA Rate .I500 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

28 

25 

26 

27 

LU 

29 

30 

31 

bL 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

4s 

Efficiency 

1 

0.85 

0 * 7225 

0,61413 

0,52201 

0 ,44372 

0,37715 

0.3Zl58 

0,27249 

0.23162 

0,19687 

0, 16734 

0,14224 

0,12091 

0.10277 

0.087354 

0,074251 

0 063113 

n ,053646 

0,045599 

0.03876 

0,032946 

0 * 028004 

0.023803 

0,020233 

0.017198 

6 

LO128 

1 

0,9872 

0 I 97456 

0 I 96209 

0 I 94977 

0,93762 

0.92562 

0,91377 

0,90207 

0,89053 

0 I 87913 

0.86787 

0.85677 

0 ,8458 

0,831197 

, 0.82428 

0.81373 

o ,00332 

0 I 79304 

0,78288 

0,77286 

0 I 76297 

0,75321 

0,74356 

0.73405 

0,72465 

0 I 71538 

0,70622 

0.69718 

7. __c 

,1500 

1 

0 .85 

0 ,7225 

0,61413 

0,52201 

0 .4\r371 

0.37715 

0,32058 

0,27249 

0.23162 

0.19687 

0.16734 

0 * 1922% 

0.12O91 

0.10277 

D * 08735% 

D 8 07$251 

D ,O631i3 

0.053646 

0,045599 

0 I 03876 

0,032w6 

OIO280O4 

0.023803 ~. 

0,020233 

0,017198 

0,60829 ' 

0.67945 

0,67075 

0.66216 

0,65369 

0.64532 

0.63706 

0.62~1 

0,62086 

0.61291 

0.60506 

0.59732 

0.5896 7 

0,58212 

0 I57467 

0 I 56732 

0,56006 

98e 

G e,'t 

46 0.55289 

47 0.54581 

48 0,53882 

49 0 I 53193 

50 0,52512 
a geometric .approximation, it is 0u.r viev7 tht. thtv m.3.~be too 4??~.L'.,ccl 

for reasonable application to large asset studies. 'It appears to us that 

the BGA rates perform rather well in approximating the Box-Cox form and 

consequently little should be lost in terms of accuracy in using efficiency 

functions hplied by tl constant rate of depreciation rather than the aonconstant.Box-Cox rates. Since the constant depreciation base efficiency 

functions ehould be far easier to deal with, it is our recommendation that 

they be used in subsequent research. Table 20 contains the depreciation 

estimates and efficiency estimates based on the best geometric approximation to the unconstrained Box-Cox estimates imposed on retired prices. It 

is Table 20 which contains what we consider to be the outcome of this study 

for purpases of 3argenson's capital stock study and Shoven's study of tax 

impacts by industry. 

99 
;<&,:iL"i:hra'qCES 

(Phase I1 Report) 

(Supplemental to Phase I References*) 

Aaront Henry J. (Ed.), Inflation and The Income Tax, Tbe Brookings 

Tnstitutioa, Washington, D.C., 1976. 

Brechliug, Frank, Investment and Employment Decisions, Manchester UnYversity 

Press, Manchester, 2975. 

Colddth, Raymond W., The National Wealth of the United States in the 

Postwar Period, Princeton University Press for NBER 7; Princeton. 

1962, p. 252. 

(Ed.), Institutional Investors and Corporate Stock, 

NBER, N,Y., 1973, Appendix I, "Basic Statistiral n-*-.'' 5;- ::tien 

Stone Tice and Virginia A. mtff p. 277. 

Barberger, Arnold C., "The Incidence of the Corporation income Tax, " 

The Journal of Political Econoq, 70, June 1962, pp. 215240. 

Heal, Geoffrey, "Do Bad Products Drive out Good?", Quarterly Journal of 

Economics, AugustP 1976, pp. 499-502. 

LeLgh, Wilheljnina A. "Economic Depreciation for the Residential Housing 

Stock of &heIJ.S., 1950-1970,'' forthcoming. 

Scarf, &E., "An Example of an Algorithm for Calculating General Equilibrium 

Prices." The American Economic Review, Sept. 1969, pp. 669-677. 

Treadway, "On Rational Entrepteneurial Behavior and the Demand for Investment 

Review of Economic Studies, April, 1969. 

Wasson, Robert C. ~ John. 'C- Nusgrave and Claudia tiarkins, "Alternative Estfmates of Fixed Business Capital in the United States, 1925-1968,'' 

Survey of Current Business, Bureau of Economic Analysis, U.S. 

Commerce Dept., April, 1970, pp. 18-36. # 

*All other arti,cles referred to in this Report are Il.sted in the Phase E 

Report either in the "Bfbliography" section or in the references Co 

"Theory" section.