A note on physical depreciation and the capital accumulation process

Journal

of Development

Economics

34 (1991) 385-395.

A note on physical capital accumulation

North-Holland

depreciation process

and the

Craig S. Hakkio” Federal Reserve Bank of’ Kansas City. Kansas City. MO 64798-0001.

USA

Bruce C. Petersen Federal Reseme Bank qf’Chicago. Chicago. IL 60690, USA Washington Univrrsny, St. Louis, MO 63130, LISA Received

May 1987. final version

received

November

1988

Abstraci; This paper examines the effect of different depreciation patterns on an economy’s capital accumulation process, We model the transition path and the new steady-state growth rate following a change in either the economy’s saving rate or its capital-output ratio. We show that the growth path of the capital stock can be very jagged, with pronounced declines in the growth rates of the capital stock. While a new steady state is eventually reached, an economy may go through periods of very uneven growth. Our simulation results have interesting implications for both developing and developed economies that have experienced a change in their saving rate.

1. Introduction In models of economic growth, physical depreciation of the capital stock is either ignored or assumed to occur exponentially. The assumption of exponential depreciation greatly simplifies models of economic growth since depreciation is then independent of the age profile of the capital stock. Unfortunately, from the standpoint of analytical tractability, the available evidence indicates that either linear or one-hoss-shay depreciation is a better approximation of reality. This paper examines the effect of different depreciation patterns on an economy’s capital accumulation process. We consider the transition path to a new steady state following a change in either the economy’s savings rate or its capital-output ratio. We show that the growth path of the capital stock can be very jagged, with pronounced declines in the growth rates of the *The authors wish to thank Robert Coen, R. Glenn Hubbard, and Mike Maresse for helpful comments, and Steven Goldman for particularly insightful comments which greatly improved the revised version of the paper. The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Banks of Kansas City and Chicago, or the Federal Reserve System. 0304_3878/9O~SO3SO


Science Publishers

B.V. (North-Holland)

386

C.S. Hakkio

and B.C. Petersen.

Depreciation

and capital

accumulation

capital stock. While a new steady state is eventually reached, an economy may go through periods of very uneven growth. Our simulation results, while only suggestive, have interesting implications for both developing economies and developed economies that have experienced a change in their saving rate. Previous studies by Domar and Harrod have also stressed the importance of the pattern of depreciation on economic growth. Domar (1953, p. 25) however, makes a quite restrictive assumption concerning the growth rate of gross investment which greatly simplifies the problem and permits analytical solutions. This assumption also makes his model inappropriate for examining the growth process of developing economies. Harrod (1970, p. 29) recognizes the added complexity of modeling the growth path of replacement investment for a developing economy, but he does not pursue the problem.

2. The specification

of depreciation

The evidence on the pattern of physical depreciation is surprisingly sparse. Feldstein and Rothschild (1974) state that, to their knowledge, a monograph by Winfrey (1935) is the most comprehensive study of the distribution of service lives of capital in the United States. This study has been widely applied,’ but apparently never superseded or updated. Winfrey calculated the mortality curves for 176 different capital assets. An examination of the frequency curves reported in his study indicate that only one out of 176 resembles an exponential distribution2 The other 175 have mortality rates which first increase and then eventually decrease. Assets in Winfrey’s study with short average lives tend to have sharply peaked mortality curves; they resemble the one-hoss-shay pattern. Assets with very long lives tend to have quite flat mortality curves; they resemble the constant, linear pattern. Together, the linear and the one-hoss-shay patterns can be thought of as providing upper and lower bounds on the mortality curves reported by Winfrey. We are aware of only one other study, Coen (1975), that reports measurements of physical depreciation for a large number of asset categories3 Coen estimates the depreciation patterns for equipment and for structures for all 20 two-digit Census manufacturing industries. He found that the depreciation pattern for equipment in 10 of the 20 industries, and for structures in 15 of the 20 industries was best approximated by either linear or one-hoss-shay depreciation. ‘The Offtce of Business Economics still uses a modilication of Winfrey’s S-3 curve (a Pearson type II curve with parameters due to Winfrey) to calculate the dtstribution of lives of all capital goods in the Ollice of Business Economics’ Capital Stock project. *The mortality curve in question (number 105) is for telephone switchboards. jHulten and Wykoff (1981) is an additional, comprehensive study of economic depreciation (the decline in asset prices). as opposed to physical depreciation.

C.S. Hakkio

and B.C. Petersen, Depreciation

and capital accumulation

387

While the evidence on depreciation patterns for individual assets is quite limited, it does not appear that exponential depreciation is the most common pattern. The assumption, then, that the aggregate capital stock of an economy depreciates exponentially, has little support. We now turn to the mathematical expressions for exponential, linear, and one-hoss-shay depreciation that will be employed in this paper. In general, the amount of depreciation in any period t, D,, can be written as

where T is the life of the investment (T may be infinite, as in the case of exponential depreciation), I,_j is the amount of gross investment made in period t-j, and yI_ j is the fraction of l,_j which depreciates in period t. If depreciation is exponential, then the pattern of y’s can be written as ytmj=y( 1 -y)j- ‘. Under this specification eq. (1) reduces to

(2)

and the capital

K,=

c

stock at time t can be expressed

(1 ->)‘I,_,

mj.

as

(3)

j=O

If depreciation is linear instead of exponential, Under this specification, eq. (1) reduces to

D,=(1/T)[1,_,+1,_2+...I,_T],

and the capital

K,=

stock can be expressed

T;i

i i=l

i

1

l,_i.

the y’s are all equal to l/T.

(4)

as

(3

388

C.S. Hakkio

and B.C. Petersen,

Depreciation

and capital

If depreciation is one-hoss-shay, then depreciation amount of gross investment t- T periods ago:

and the capital K,=

i

accumulation

in period

t equals

the

stock equals I,mi.

i=l

It is apparent from eqs. (2) and (3) that under exponential depreciation, the rate of depreciation is independent of the pattern of previous investment: D,/K, is always equal to 7. This is not true for linear and one-boss-shay depreciation. For linear depreciation, using eqs. (4) and (5) DJK, can be written as

If the economy has been at a zero-growth steady state for the last T years, it is easy to show that D,/K,=2/T. However, if the growth rate of capital is positive, then D,/K, must be less than 2/T. This is apparent since depreciation is a weighted sum of investments where the weights equal (l/T), while the current capital stock is a weighted sum of past investments where the weights are inversely related to the age of past investment. This means that the greater the growth rate of capital, the lower the rate of depreciation. For one-hoss-shay depreciation, the current capital stock, eq. (7) is an unweighted sum of investment made in the previous T years. Depreciation in period t, however, is equal to investment made T years prior to r. Hence, if the capital stock is not growing, D,/K,= l/T. (This is one-half the depreciation rate that arises for the case of linear depreciation under a zerogrowth steady state). For one-hoss-shay depreciation, as is true for linear depreciation, the greater the growth rate of capital, the lower is the depreciation rate, since depreciation is identical to investment made T + 1 years prior and the capital stock is the unweighted sum of all investment made in the past T years.

3. Depreciation

in the Harrod-Domar

model

To examine the effects of linear and one-hoss-shay depreciation on the capital accumulation process and the transition path between steady states, we choose to work with a variant of the seminal Harrod-Domar model of

C.S. Hakkio

and B.C. Petersen,

Depreciution

and capital

accumulation

389

economic growth. While the Harrod-Domar model abstracts from many important considerations in the study of economic growth,4 it is well-suited to study the effect of depreciation on the capital accumulation process. As Domar notes (1953, p. 14), such a model is ‘unexcelled for use in a first examination of a new relationship in the theory of growth’. We note that the phenomena that we observe should arise under quite general conditions. The distinctive assumptions of the Harrod-Domar model are: (1) a constant portion (s) of income (Y) is devoted to savings, and (2) the capitalassumpoutput ratio, u, is exogenously given.5 Using the Harrod-Domar tions, and denoting capital depreciation by D,, the capital accumulation process can be written as6 K

t+1-K, K,

sY,-D,

=-y=2;-K,’

s D,

If there is no depreciation, then it is evident that the capital stock and output will grow in any period r+ 1 at a rate equal to s/u. We consider the transition path of the capital stock under various depreciation patterns following a change in either s or c. The former could occur for any number of reasons. In a command economy, the government might decide to levy higher taxes and increase the economy’s rate of savings. On the other hand, a decrease in c could occur if innovation made new capital more productive than existing capital. Since there is no quantitative distinction between looking at an increase in s versus a decrease in u, in the remainder of the paper we focus on the transition path following an increase in the savings rate. Consider the case of exponential depreciation. Substituting eq. (2) into eq. (9) the capital accumulation equation becomes %ee Hahn and Matthews (1964) for a summary of the Harrod-Domar model. Many other models of economic growth could be used. Alternative specifications might include: (I) different theories of savings behavior, (2) a variable capital-output ratio. or (3) optimizing behavior by consumers and firms. While these models may add more ‘realism’ to the results. the basic issues concerning depreciation would be unchanged. sit is an empirical fact that the capital-output ratio has been approximately constant in a number of countries for which there is evidence. For a review of the evidence. see Solow (1970, pp. 2-8). 6We assume that capital is homogeneous. Within the confines of the Harrod-Domar model. very little changes if this assumption is relaxed. Suppose instead that there are N kinds of capital, each having a different capitalloutput ratio, r,, and thus the Harrod-Domar t’ is a weighted average of these ratios. In addition, suppose a constant fraction .s. of income is saved and devoted to capital of each type n. In this case, Y= K , ‘lr, +

‘. + K,,Jr,

+

‘. + K,&vv

and

K,,

, +, -

K,,

, = s,I: -D,,.

The results which follow are not affected by this type of heterogeneity of capital, Moving away from the Harrod-Domar model would complicate the treatment of heterogeneity of capital, but as we argue elsewhere. it would not change the basic insights of the paper.

390

C.S. Hakkio

and B.C. Petersen, Depreciation

and capital accumulation

(K , +1- WK, = (s/u)-Y. That is, the growth rate of the capital stock and of output is s/u--y. If s> ~7, growth in t+ 1 will be positive. Suppose s increases to s’. Then the growth rate of capital will immediately jump to its new level: (s’/D)-7. The important point here is that under exponential depreciation, because the depreciation rate is independent of the economy’s investment history, there is no transition path from one steady state to the next. Next, consider the capital accumulation process under linear and one-hossshay depreciation. Contrary to the case of exponential depreciation, knowledge of s, r, and y, is not sufficient to determine the growth rate of the capital stock at t+ 1; it is necessary to know the economy’s investment history for the previous T years. In other words, a difference equation of order T is required to describe the capital accumulation process. Because of the mathematical complexity, only a brief verbal discussion of the process is given below; the transition path of capital is simulated in the next section. Consider first the case of one-hoss-shay depreciation because it generates a highly unusual transition path from one steady-state to the next. Suppose an economy is initially at some steady-state rate of growth. For ease of discussion, we choose the initial condition to be zero steady-state growth. The depreciation rate consistent with this assumption is D/K= l/T; the implied savings rate is s = c/T. Now suppose s increases to s’ in period t + 1. This will cause the growth rate of capital to jump to s’/c- l/T in this period. However, unlike the case of exponential depreciation, the new steady state is not reached in period r+ 1. This follows because depreciation will remain unchanged for the next T years. Thus, the rate of depreciation will monotonically decline, causing the rate of growth of the capital stock to monotonically increase, until period t+T. A sharp discontinuity in the transition path of depreciation rates occurs in period t + T + 1. In this year, the depreciation rate will sharply increase because depreciation now equals investment in year t + I which equals in the prior year. After t+ T+ 1, the (.s’/t’)K, as opposed to (s/u)K,_, depreciation rate will continue to fall again. However, because of the dynamics of the capital accumulation process, the next discontinuity occurs prior to t + 2T + 1. This process repeats itself, although it dampens out as the new steady state is approached, with a depreciation rate lower than l/T The transition path of the capital stock must exhibit the same discontinuities that show up in the transition path of D/K. This growth rate of the capital stock will accelerate from period t + I to t+ T, sharply fall in f + T + 1, accelerate from t + T + 2 to some time prior to t + 2T, and then fall again. This process will be very apparent in the simulation results in the next section.

C.S. Hakkio

and B.C. Petersen,

Depreciation

and capifal

accumulation

391

The case of linear depreciation and the transition path from one steady state to the next can be described more briefly. Suppose the economy is initially at a zero-growth steady state (s= 2vJT and DJK =2/T) and suppose s increases to s’ in period t + 1. As a result, the growth rate of capital rises to [(d/v)-(l/T)]. This is not the new steady state because once again there exists a downward adjustment path in the rate of depreciation. This is obvious from a comparison of the relative weights on new versus old investment for D and K in eq. (8). An increase in the saving rate must cause K to initially rise faster than D, forcing down the depreciation rate during the transition period. In turn, this will drive up the growth rate of the capital stock until the new steady state is reached.

4. Simulating the transition path and the new steady state In this section we further study the behavior of the capital accumulation process following a change in the savings rate. The transition path is complicated because eq. (8) is a Tth order difference equation. For large values of T it is difficult to obtain analytical solutions. Therefore, we choose to analyze the capital accumulation process under linear and one-hoss-shay depreciation by simulating the equations. It is assumed that the capitaloutput ratio, v, equals 3.’ We note that we could just as easily look at changes in v and hold s fixed. It is also assumed that the economy begins in a zero-growth steady state. For linear depreciation, this means s= 2vJT; for one-hoss-shay depreciation, this means s=v/T. In order to begin with the same saving rate in both economies, we assume that the life of capital, T, is 40 years for linear depreciation and 20 years for one-hoss-shay depreciations. That is, s= 15 percent in the initial zero-growth steady state. Fig. 1 shows the transition path following a 10 percent increase in the saving rate, from 15 percent to 16.5 percent. The growth rate of capital is plotted on the vertical axis. As the figure shows, under one-hoss-shay depreciation the growth rate of capital follows a jagged path which dampens out over time. In the first year following the 10 percent increase in the saving rate, the growth rate of capital equals [(Y/v) -( l/T)] =0.5 percent. As previously discussed, this is not the new steady-state growth rate since l/T is not the steady-state depreciation rate when g>O. The depreciation rate must fall for the next T years, reaching its minimum value (4.3 percent) in the 20th year, causing the growth rate of capital to reach its peak (1.2 percent). Given the dynamics of one-hoss-shay depreciation, in year T+ 1 (year 21) there must be a sharp increase in the depreciation rate, forcing down the growth rate of 7This has been approximately the capital-output ratio in the twentieth century. See, for example, Samuelson (1980, p. 690).

United

States

during

the

0.2

/--

0.10

f.i ’

__L____-

-__..IL.I_-~.’ 0 Fig. I. Transition

25

50 Time

path for linear and one-hoss-shay

Note: 7‘=40 for linear depreciation and savings rate rises IO percent. from s=O.l5

T=20 for one-hoss-shay to s=O.165.

75

-_I-_

i

100

depreciation. depreciation.

At time 0. the

capital (in our example, to 0.82 percent). This rachetting effect repeats itself, but dampens out over time; the growth rate of capital eventually converges to 0.93 percent with a corresponding depreciation rate of 4.6 percent. It is important to note that an increase in the saving rate of greater than 10 percent would result in a sharper increase in the depreciation rate in year TS 1 and a lower steady-state rate of depreciation. The transition path under linear depreciation is similar, but the ratchetting effect is not as sharp. The growth rate jumps to 0.5 percent in the first year, rises slowly to 0.76 percent in year 34 (less than T), falls to 0.74 percent in year 41, and then rises and finally converges to 0.85 percent. Unlike onehoss-shay depreciation, the fall in the growth rate occurs before year T because depreciation kicks in immediately (and accumulates) under linear depreciation. Fig. 2 plots the steady state growth rate of capital as a function of the saving rate. The initial condition is zero growth; s=2c/T and T =40 under linear depreciation and s = q/T and T = 20 under one-hoss-shay depreciation. Steady state growth rates were calculated for various values of s. Fig. 2 shows that the growth rate of capital under one-hoss-shay

Percent 5

r----

o1

--.

--.--

.._

-_1__1__d~_.._1~.__L

0.15

0.156

0.162

__I-_

0.166

0.174

0.16

0.166

‘_

0.192

_.,~.___L

0.196

_.__‘_._I_

0.204

0.21

__L

0.216

0.222

Saving Rate Fig. 2. Steady state growth rate as a function of the saving rate. Note: T=40 for linear depreciation and 7=X growth rates are plotted for each value of s.

for one-boss-shay

depreciation.

Steady

state

depreciation is greater than the growth rate of capital under linear depreciation, for each saving rate. This is due to differences in the steady state rate of depreciation. Fig. 2 also shows that the difference in the growth rates of capital between the two depreciation patterns increases with the saving rate. The reason is that the steady-state depreciation rate declines faster under one-hoss-shay depreciation, at least over the range of saving rates we consider. For example, when s = 18 percent, the steady-state rates of depreciation for the one-hoss-shay and linear patterns are 4.2 and 4.4 percent. However, when s=20 percent, the steady-state rates of depreciation are 3.8 and 4.2 percent. The difference clearly increases with s. There is one final important point not addressed in our simulations, which should be made about the capital accumulation process under linear and one-hoss-shay depreciations. The phenomena that we have observed works in reverse following a decline in the savings rate. Suppose an economy is at a positive steady-state growth rate and consider the case of one-hoss-shay depreciation. Following a decline in the saving rate. the transition path is the same as that described above, only in reverse. The depreciation rate must initially rise, then fall sharply, before reaching a new, higher. steady-state rate

394

C.S. Flakkio

und B.C. Petersun,

Deprecirrtion

nnd capital

awrmulntion

of depreciation. This of course causes the growth rate of capital to also follow a very jagged adjustment path towards a new steady state. Such a transition path. of course, cannot arise under exponential depreciation since the entire decline in growth occurs immediately. Thus, the simulation results in fig. 1 may have important implications for industrialized economies as well as developing economies.

5. Conclusions This paper has analyzed the effect of linear and one-hoss-shay depreciation on the capital accumulation process. These depreciation patterns have at least as much empirical support as the commonly assumed exponential pattern. Much progress in modeling economic growth has been made by working with exponential depreciation and adding such complications as a neoclassical production function, a variable savings rate and technological progress. Because of the complications involved in dealing with depreciation patterns other than the exponential, we have taken the opposite tack. The phenomena that we observe, however, should arise under more general settings. This is not to say, however, that our quantitative results will not change in a different setting. We have demonstrated that under either linear or one-hoss-shay depreciation, the rate of depreciation depends on the economy’s past investment history, and that the transition path between steady states can be expressed as a difference equation of an order equal to the maximum life of capital. Under linear and one-hoss-shay depreciation, a change in the economy’s saving rate causes the rate of depreciation to follow a jagged pattern - falling then rising. As a result, the transition path for capital follows a similar jagged pattern. This pattern is particularly pronounced for unlike the case of exponential one-hoss-shay depreciation. In addition, depreciation, the steady state rate of depreciation is negatively related to the economy’s rate of growth.

References Burmeister, E.. 1980. Capital theory and dynamics (Cambridge University Press, Cambridge). Burmeister, E. and A.R. Dobell, 1970. Mathematical theories of economic growth (MacMillan. New York). Coen. R.. 1975, Investment behavior. The measurement of depreciation and tax policy. The American Economic Review 65. 59-74. Domar. E.D., 1953. Depreciation. replacement and growth, The Economic Journal 63. lL.12. Feldstein. M. and M. Rothschild, 1974, Toward an economic theory of replacement investment, Econometrica 42, 393-423. Hahn. F.H. and R.C.O. Matthews. 1964. The theory of economic growth: A survey. The Economic Journal 74. 781-902.

C.S. Hakkio

and B.C. Petersen.

Depreciation

and capital

ammularion

395

Harrod, R.F., 1970. Replacements, net investment, amortization funds, The Economic Journal 80, 24-31. Hulten. C.R. and F.C. Wykoff. 1981. The measurement of economic depreciation, in: C.R. Hulten, ed., Depreciation. inflation and the taxation of income from capital (Urban Institute Press, Washington. DC) 81-123. Rostow. W.W.. 1960, The stages of economic growth (Cambridge Unicerstty Press. Cambridge). Rostow, W.W., 1963. Economics of take-off tnto sustained growth (Macmillan. London). Samuelson. P.A., 1980, Economics, 1I th ed. (McGraw-Hill, New York), Solow. R.M.. 1970. Growth theory, an exposttton (Oxford University Press. New York). Winfrey, R.. 1935, Statistical analysis of industrial property retirements. Iowa Engineering Experiment Station, Bulletin 125 (Iowa State College, Ames, IA).