The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Vintages

  • Dale W. Jorgenson
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1402

Abstract

Vintages are durable goods acquired at different points of time. The acquisition prices for capital goods of each vintage at each point of time together with investments of all vintages at each point of time constitute the basic data on quantities and prices. These data can be employed in generating the complete vintage accounting system.

Keywords

Capital measurement Capital services Depreciation Durable goods model of production Investment Jorgenson, D. W. Price–quantity duality Production functions Rate of return Vintage accounting Vintages 

JEL Classifications

E21 

Investment represents the acquisition of capital goods at a given point of time. The quantity of investment is measured in the same way as the durable goods themselves. For example, investment in equipment is the number of machines of a given specification and investment in structures is the number of buildings of a particular description. The price of acquisition of a durable good is the unit cost of acquiring a piece of equipment or a structure.

By contrast with investment, capital services are measured in terms of the use of a durable good for a stipulated period of time. For example, a building can be leased for a period of years, an automobile can be rented for a number of days or weeks, and computer time can be purchased in seconds or minutes. The prices of the services of a durable good is the unit cost of using the good for a specified period.

Aggregation over Vintages

We can refer to durable goods acquired at different points of time as different vintages of capital. The flow of capital services is a quantity index of capital inputs from durable goods of different vintages. Under perfect substitutability among the services of durable goods of different vintages, the flow of capital services is a weighted sum of past investments. The weights correspond to the relative efficiencies of the different vintages of capital.

The durable goods model of production is characterized by price-quantity duality. The rental price of capital input is a price index corresponding to the quantity index given by the flow of capital services. The rental prices for all vintages of capital are proportional to the price index for capital input. The constants of proportionality are given by the relative efficiencies of the different vintages of capital.

We develop notation appropriate for the intertemporal theory of production by attaching time subscripts to the variables that occur in the theory. We can denote the quantity of output at time t by yt and the quantities of J inputs at time t by xjt (j = 1, 2, …, J). Similarly, we can denote the price of output at time t by qt and the prices of the J inputs at time t by pjt (j = 1, 2, …, J).

In order to characterize capital as a factor of production, we require the following additional notation:
  • At – quantity of capital goods acquired at time t.

  • Kt,τ – quantity of capital services from capital goods of age τ at time t.

  • pA,t – price of acquisition of new capital goods at time t.

  • pKt,τ – rental price of capital services from capital goods of age τ at time t.

To present the durable goods model of production we first assume that the production function, say F, is homothetically separable in the services of different vintages of capital:
$$ {y}_t=F\left[G\left({K}_{t,0},{K}_{t,1}\dots {K}_{t,\tau}\dots \right),{x}_{2t}\dots {x}_{Jt}\right]. $$
(1)
Where Kt is the flow of capital services, we can represent this quantity index of capital input as follows:
$$ {K}_t=G, $$
where the function G is homogeneous of degree one in the services from capital goods of different ages.
If we assume that the quantity index of capital input Kt is characterized by perfect substitutability among the services of different vintages of capital, we can write this index as the sum of these services:
$$ {K}_t=\sum \limits_{\tau =0}^{\infty }{K}_{t,\tau }. $$
Under the additional assumption that the services provided by a durable good are proportional to initial investment in this good, we can express the quantity index of capital input in the form:
$$ {K}_t=\sum \limits_{\tau =0}^{\infty }{d}_{\tau }{A}_{t-\tau }. $$
(2)
The flow of capital services is a weighted sum of past investments with weights given by the relative efficiencies {dτ} of capital goods at different ages.
Under constant returns to scale we can express the price of output as a function, say Q, of the prices of all inputs. The price function Q is homothetically separable in the rental prices of different vintages of capital:
$$ {q}_{t\kern0.5em =\kern0.5em }Q\left[P\left({p}_{K,t,0},{p}_{K,t,1}\dots {p}_{K,t,\tau}\dots \right),{p}_{2t}\dots {p}_{Jt}\right].\kern0.5em $$
(3)
Where pK,t is a price index of capital services, we can represent this index as follows:
$$ {p}_{K,t}=P, $$
where the function P is homogeneous of degree one in the rental prices of capital goods of different ages.
Under perfect substitutability among the services of different vintages of capital, we can write the price index of capital input P as the price of the services of a new capital good:
$$ {p}_{K,t}={p}_{K,t,0}. $$
Under the additional assumption that the services provided by a durable good are proportional to the initial investment, we can express the rental prices of capital goods of different ages in the form:
$$ {p}_{K,t,\tau }={d}_{\tau }{p}_{K,t},\kern0.72em \left(\tau =0,1,\dots \right). $$
(4)
The rental prices are proportional to the rental price of capital input with constants of proportionality given by the relative efficiencies {} of capital goods of different ages.

Given the quantity of capital input Kt, representing the flow of capital services, and the price of capital inputs pK,t, representing the rental price, capital input plays the same role in production as any other input. We next derive the prices and quantities of capital inputs from the prices and quantities for acquisition of durable goods pA,t and At.

Vintage Accounting

We begin our description of the measurement of capital input with the quantities estimated by the perpetual inventory method. Taking the first difference of the expression for capital stock in terms of past investments (2), we obtain:
$$ {K}_t-{K}_{t-1}={A}_t+\sum \limits_{\tau =1}^{\infty}\left({d}_{\tau }-{d}_{\tau -1}\right){A}_{t-\tau },={A}_t-{R}_t, $$
where Rt is the level of replacement requirements in period t. The change in capital stock from period to period is equal to the acquisition of investment goods less replacement requirements.
We turn next to a description of the price data required for the measurement of the price of capital input. There is a one-to-one correspondence between the vintage quantities that appear in the perpetual inventory method and the prices that appear in our vintage price accounts. To bring out this correspondence we use a system of present or discounted prices. Taking the present as time zero, the discounted price of a commodity, say qt, multiplied by a discount factor:
$$ {q}_t=\prod \limits_{s=1}^t\frac{1}{1+{r}_s}{p}_t. $$
The notational convenience of present or discounted prices results from dispensing with explicit discount factors in expressing prices for different time periods.
In the correspondence between the perpetual inventory method and its dual or price counterpart the price of acquisition of a capital good is analogous to capital stock. The price of acquisition, say qA,t is the sum of future rental prices of capital services, say qK,t, weighted by the relative efficiencies of capital goods in all future periods:
$$ {q}_{A,t}=\sum \limits_{\tau =0}^{\infty }{d}_{\tau }{q}_{K,t+\tau +1} $$
(5)
This expression may be compared with the corresponding expression (2) giving capital stock as a weighted sum of past investments.
Taking the first difference of the expression for the acquisition price of capital goods in terms of future rentals (5), we obtain:
$$ {a}_{A,t}-{q}_{A,t-1}=-{q}_{K,t}-\sum \limits_{\tau =1}^{\infty}\left({d}_{\tau }-{d}_{\tau -1}\right){q}_{K,t+\tau }=-{q}_{K,t}+{q}_{D,t}, $$
where qD,t is depreciation on a capital good in period t. The period-to-period change in the price of acquisition of a capital good is equal to depreciation less the rental price of capital. Postponing the purchase of a capital good makes it necessary to forgo one period’s rental and makes it possible to avoid one period’s depreciation. In the correspondence between the perpetual inventory method and its price counterpart, investment corresponds to the rental price of capital and replacement corresponds to depreciation.
We can rewrite the expression for the first difference of the acquisition price of capital goods in terms of undiscounted prices and the period-to-period discount rate:
$$ {p}_{K,t}={p}_{A,t-1}{r}_t+{p}_{D,t}\hbox{--} \left({p}_{A,t}-{p}_{A,t-1}\right),\kern0.5em $$
(6)
where pA,t is the undiscounted price of acquisition of capital goods, pK,t the price of capital services, pD,t depreciation, and rt the rate of return, all in period t. The price of capital services pK,t is the sum of return per unit of capital pA, t − 1rt, depreciation pD,t, and the negative of revaluation, pA, tpA, t − 1. To apply this formula we require a series of undiscounted acquisition prices for capital goods pA,t, rates of return rt, depreciation on new capital goods, pD,t, and revaluation of existing capital goods pA, tpA, t − 1.

To calculate the rate of return in each period we set the formula for the rental price pK,t times the quantity of capital Kt−1 equal to property compensation. All of the variables entering this equation – current and past acquisition prices for capital goods, depreciation, revaluation, capital stock and property compensation – except for the rate of return, are directly observable. Replacing these variables by the corresponding data we solve this equation for the rate of return. To obtain the capital service price itself we substitute the rate of return into the original formula along with the other data. This completes the calculation of the service price.

In the perpetual inventory method data on the quantity of investment goods of every vintage are used to estimate capital formation, replacement requirements and capital stock. In the price counterpart of the perpetual inventory method data on the acquisition prices of investment goods of every vintage is required. In the full price–quantity duality that characterizes the vintage accounts, capital stock corresponds to the acquisition price of durable goods and investment corresponds to the rental price of capital services.

Conclusion

The distinguishing feature of capital as a factor of production is that durable goods contribute capital services to production at different points of time. The services provided by a given durable good are proportional to the initial investment. In addition, the services provided by different durable goods at the same point of time are perfect substitutes. The weights correspond to the relative efficiencies of the different vintages of capital. The durable goods model of production was originated by Walras (1954) and is discussed in greater detail by Jorgenson (1973) and Diewert (1980).

The durable goods model is characterized by price–quantity duality. The rental price of capital input is a price index corresponding to the quantity index given by the flow of capital services. The rental prices for all vintages of capital are proportional to the price index for capital input. The constants of proportionality are given by the relative efficiencies of the different vintages of capital. The dual to the durable good model of production was introduced by Hotelling (1925) and Haavelmo (1960). The dual to this model has been further developed by Arrow (1964) and Hall (1968).

The acquisition prices for capital goods of each vintage at each point of time together with investments of all vintages at each point of time constitute the basic data on quantities and prices. These data can be employed in generating the complete vintage accounting system originated by Christensen and Jorgenson (1973) and described by Jorgenson (1980). Price and quantity data that we have described for a single durable good are required for each durable good in the system. These data are used to derive price and quantity indexes for capital input in the theory of production presented in the entry on production functions.

See Also

Bibliography

  1. Arrow, K.J. 1964. Optimal capital policy, the cost of capital, and myopic decision rules. Annals of the Institute of Statistical Mathematics 16: 16–30.CrossRefGoogle Scholar
  2. Christensen, L.R., and D.W. Jorgenson. 1973. Measuring economic performance in the private sector. In The measurement of economic and social performance, ed. M. Mess. New York: Columbia University Press for the National Bureau of Economic Research.Google Scholar
  3. Diewert, W.E. 1980. Aggregation problems in the measurement of capital. In The measurement of capital, ed. D. Usher. Chicago: University of Chicago Press.Google Scholar
  4. Haavelmo, T. 1960. A study in the theory of investment. Chicago: University of Chicago Press.Google Scholar
  5. Hall, R.E. 1968. Technical change and capital from the point of view of the dual. Review of Economic Studies 35 (1): 35–46.CrossRefGoogle Scholar
  6. Hotelling, H.S. 1925. A general mathematical theory of depreciation. Journal of the American Statistical Association 20: 340–353.CrossRefGoogle Scholar
  7. Jorgenson, D.W. 1973. The economic theory of replacement and depreciation. In Econometrics and economic theory, ed. W. Sellekaerts, 189–221. New York: Macmillan.Google Scholar
  8. Jorgenson, D.W. 1980. Accounting for capital. In Capital, efficiency, and growth, ed. G. von Furstenberg, 251–319. Cambridge: Ballinger.Google Scholar
  9. Walras, L. 1874. Elements of pure economics. Trans. W. Jaffé. Homewood: Irwin, 1954.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Dale W. Jorgenson
    • 1
  1. 1.