ROBERT E. KOHN Southern Illinois University at Edwardsville

The Rate of Interest in a Stationary Economy*

A stationary economy is represented by a neoclassical production possibility frontier in which the quantity of labor is fixed and the quantity of capital is endogenous. Because the production functions of firms are assumed to have special homothe- ticity, the frontier is a straight line, whose distance from the origin increases with the marginal propensity to save. In this model of a stationary economy, which in- corporates a special assumption on the marginal propensity to consume out of wealth, the rate of interest depends only on the marginal propensity to save out of income and the life-span of capital equipment.

1. Introduction One of the basic models of microeconomic theory is the con-

cave production possibility frontier showing alternative combina- tions of two goods that can be produced with fixed total quantities of labor and non-depreciating capital. The assumption that the quantity of capital is constant, even though the gross value of out- put may vary from one combination of goods to the next, is incon- sistent with Keynesian theory that saving, which is the source of new capital, varies with income. In this paper, a production pos- sibility frontier is derived that is consistent with the Keynesian sav- ings function. The quantity of capital, which is one of the outputs of the economy, remains constant in a long run equilibrium in which the proportion of net income that consumers wish to save each pe- riod yields a sum equal to the depreciation of the capital stock in that period. As such the frontier, which is linear, characterizes a stationary economy.

In this particular stationary economy, consumers may shift from one combination of goods to another without affecting the total quantity of capital. The wage rate, relative prices, and income re- main constant. Although a linear production possibility frontier is not realistic, the model can be interpreted in the context of a quasi-

*I am grateful to an anonymous referee and to Harold Bamett, Don Elliott, and Jerry Hollenhorst for their helpful comments on an earlier version of this pa- per.

Journal of Macroeconomics, Summer 1986, Vol. 8, No. 3, pp. 373-380 373 Copyright 0 1986 by Wayne State University Press.

Robert E. Kohn

convex frontier that is piece-wise linear over relevant and reason- ably wide ranges. It is based upon production functions that exhibit increasing and then decreasing returns to scale. The linearity re- sults from the assumption that the production functions of firms in the two industries have a special homotheticity.

For simplicity, a special assumption is made on the marginal propensity to consume out of wealth (capital). An interesting and counter-intuitive result of the model is that the rate of interest in this stationary state depends only on the marginal propensity to save out of income and the life span of capital equipment. Other than the assumption of homotheticity, it is independent of the particular parameters of the production functions. The rich results of the model reflect its double foundation in microeconomics and macroeconom- ics.

2. The Technology of Production

In its more general interpretation, the neoclassical production possibility frontier represents a sequence of competitive market equilibrium allocations of inputs. It assumes that each industry con- sists of many firms, free to enter and exit and using a common production function that exhibits increasing and then decreasing re- trims to scale. A simple form of homothetic production function taken from Henderson and Quandt (1958, p. 46) is used here.

Consider an economy in which there are two competitive in- dustries, X and Y. Each firm in industry X employs k, units of cap- ital, 4, units of labor and produces x units of output according to the homothetic production function,

x = at?: k: - tTi+’ k:+I . (1)

An initial range of increasing returns to scale implies that u > l/2. The production function for firms in industry Y is

y = be; k; - 4;+’ kg”,

in which u > I/2. There is long run competitive equilibrium when the number of firms in each industry is such that for each firm, the sum of marginal products times the quantity of the correspond- ing input “exhausts” that firm’s total output. In that case, the in- dustry production functions [see Kohn (1984)] are

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The Rate of Interest in a Stationary Economy

X = AL;i2 cl2 (3)

where

A = .2(a/(2u + 1))“+“2(2u - l)u-l” (5)

B = 2(b/(2u + 1))“+“‘(20 - l)o-1’2 (6)

Note that lower case letters pertain to firms and upper case letters to industries.

In this model the total quantity of labor, L,,, is given,

L, + Ly = Lo . (7)

However, the total quantity of capital, by Hicksian aggregation,

is endogenous to the model. Following the Austrian approach, a unit of capital is defined as a bundle of c units of good X. This may be thought of as an output such as bricks, which can be costlessly assembled into one kind of kiln for manufacturing more bricks or into another kind of kiln for baking bread. We shall assume that capital, whether put together for industry X or for industry Y has a productive life of n-periods. It is a crucial assumption of this model that both industries use the same basic kind of capital equip- ment.

It follows from Klein (1946, p. 95) that the marginal rate of technical substitution of the firm is identical to that of the industry; therefore, when all firms are using least-cost combinations of in- puts,

It can be tentatively noted that the production possibility frontier for this economy is

X/A + Y/B = (ZCL,,)“2 (10)

It remains to render K endogenous.

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Robert E. Kohn

3. Capital and Saving The number of units, c, of good X that comprise a unit of

capital, times the price of good X, which is p,, is the price of cap- ital, [cp,]. Assuming that (l/n)th of the capital stock wears out each period, the value of capital that must, in a steady state equilibrium be replaced each period, is [cp,]K/n.

The Keynesian consumption function in this economy is as- sumed to be

c = (1 - s)Z + f-[cp,]K ) (11)

where C is consumption per period, Z is net income per period, and [cp,]K is the total value of capital or wealth. The marginal pro- pensity to save out of income is s and the marginal propensity to consume out of wealth is r. For simplicity, it is assumed here that if net income were zero, households would consume that portion of capital that would depreciate in the current period, in which case the value of r is (l/n). Given the Keynesian identity,

c+s=z, 059

where S is savings per period, it follows that

s = sz - [cp,]K/n . (13)

Alternatively, S is the net, after depreciation, addition to the capital stock.

In a steady state competitive market equilibrium, the price of good X is equal to the wage rate, w, divided by the marginal prod- uct of labor in industry X. Substituting the marginal product, XL, which can be derived from Equation (3), into the expression for the purchase price of capital yields

[cP,I = c[w/&l = cWWW3~~'")l . (14)

The price per unit of capital per period of time is [cp,]T, where T is the equal-payments capital recovery factor for the case in which all payments and receipts occur at the end of each period [see Thuesen and Fabrycky (1984), pp. 42-431,

T = i/(1 - (1 + i)-“) , 05)

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The Rate of Znterest in a Stationary Economy

and i is the interest rate per period. The marginal condition for cost minimization by firms can be expressed in terms of aggregates:

It follows that gross national income, which is wLa + [cp,]TK, sim- plifies to 2wLa.

The stationary state, in which the portion of net income not consumed in the period equals the depreciation of capital, is mod- elled by setting S in Equation (13) equal to zero. Substituting gross national income minus depreciation for Z in that equation yields

@wL, - [wWn) = [w,lWn .

Substituting 2wZ$‘“/(AK1’“) for px and simplifying yields

(17)

K = (Ans/(c(l + a)))“& . (18)

The production possibility frontier, derived from (10) and (18), is

X/A + Y/B = (Ans/(c(l f s)))& , (19)

and is a continuous, straight line if the number of firms is divisible as well as variable. The frontier of stationary state, consumable quantities, X, and Y,, which is derived by substituting Y, for Y and (X, + cK/n) for X in equation (19) and using (18), is

X,/A + Y,/B = (Ans/(c(l + s)‘))L, . w-v

Because the production functions of the firms yield Cobb-Douglas industry production functions with identical factor shares, the locus of tangencies between industry isoquants lie along the diagonal of an Edgeworth box. Accordingly the prices of inputs and outputs are constant and independent of the combinations of outputs produced. Using Equation (18), the prices of the goods can be expressed as

p, = 2~41 + s)/(A’ns) (21)

p, = 2~41 + s)/(ABns) (22)

Given the market prices of the two goods, it may be noted that Equation (19) is equivalent to

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Robert E. Kahn

p,x + p,Y = Z(1 + 8)) (24

where, Z(l + S) is gross national product in the stationary state, and that Equation (20) is equivalent to

PJ, + PJC = z 3 (24)

where Z is net income in the stationary state.

4. Results of the Model From equations (16), (la), and (21) comes

T = (1 + s)/(2ns) (25)

The steady state equilibrium rate of interest, i*, for any parameter, s, is obtained by solving equations (15) and (25) simultaneously,’

i*/(l - (1 + i*)-” = (1 + s)/(2ns) (26)

It follows that in the stationary state defined by this model, the equilibrium rate of interest is based only on the marginal propen- sity to save and the life-span of capital equipment. The larger the

‘The following is a numerical example. For the case in which a = 15, u = 2, b = 8, u = 3/2, c = 324, s = 2/7, n = 4, and & = 648, the quantity of capital is 128 units and the production possibility frontier is (X/81 + Y/8 = 576). One combination of outputs on this frontier, (X = 23328, Y = 2304), would be produced by 48 firms in industry X, each using 6.75 units of labor and 4/3 units of capital to produce 486 units of output and by 72 firms in industry Y, each using 4.5 units of labor and 8/9 units of capital to produce 32 units of output. Assuming that l/n of the firms in each industry replace their totally depreciated plant each year, it would be possible to produce (X = 29160, Y = 1728) the following year as a consequence of 18 firms leaving industry Y and 12 additional firms entering in- dustry X to use the available inputs. Because the firms’ production functions are homothetic, each will use the same combination of inputs as before. The compet- itive market prices are p, = w/36, p, = 9w/32, and i = 42.676%. The gross na- tional product is 1296w, of which 288~ is spent on replacement capital. The net national income is 1008~1, which consists of 648~ of labor income and 360~ of interest income. The latter is the sum of approximately 123w, 106w, 83~1, and 48~ in interest payments on one through four-period old capital, respectively. [The for- mula for interest on the remaining balance of an equal-payments loan is given by Thuesen and Fabrycky (1984), p. 991.

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The Rate of Interest in a Stationary Economy

value of s, the greater the equilibrium quantity of capital and the lower the rate of interest.2

The simple explanation for the rate of interest in this model follows from the assumption of a stationary economy and homothet- ic production functions. Equation (16) confirms that labor income must equal capital income, so that

T = ~Ld([cp,l~) (27)

Both px and K must adjust to satisfy the value of T dictated by n and s in Equation (25). Accordingly, i* is simply the rate of dis- count that equates capital income to labor income.3 There is also a multiplier interpretation of the right-hand-side of Equation (26). When saved income, sZ, in the stationary economy increases by one dol- lar, the income of either labor or capital increases by (1 + s)/(2ns) dollars.

Finally, it should be stressed that, although the rate of inter- est has a simple aggregative interpretation, it can be shown that the same rate of discount, i*, equates the present value of the n- period stream of marginal revenue products of capital received by each individual firm to the total capital investment by that firm.4

‘If the propensity to save, s, in the numerical example in footnote 1 increases to (l/2), the equilibrium quantity of capital increases to 288 units and the interest rate declines to 18.5%. The prices of the two goods decline more than net national income, so that real income rises.

‘Keynes (1936, p. 220) surmised that the abundance of capital in a stationary economy “. . . ought to be able to bring down the marginal efficiency of capital in equilibrium approximately to zero.” It is of interest to interpret this conjecture in the context of the present model. The value of n may be estimated by dividing the total value of equipment and structures owned by U.S. nonfinancial corpora- tions in 1981, which was $3,481.9 million, by the corresponding straight-line de- preciation in that year, which was $184.5 million. [These data are taken from the Survey of Current Business (1982), pp. 33, 37. J Substituting the rounded estimate, n = 20, into equation (26) indicates that the marginal propensity to save out of income would have to exceed .6 for the stationary rate of interest to be 2%.

For n = 20, the coefficient of total wealth in Equation (13), which is l/n, is 0.05. This compares to a corresponding estimate of 0.046 by Kormendi (1983, p. 999), which provides support for the simplifying assumption that r = l/n.

‘The marginal product of capital in an individual firm producing, say, good Y, is +X-i - (u + l)fi+‘e. Replacing P, with b(2u - 1)/((2u + I#,) to satisfy the Clark-Wicksteed condition, multiplying by the equivalent of pv from Equation (22), substituting the value of k, obtained from the equality of the corresponding mar- ginal revenue product of labor and the wage rate, and finally dividing by the equiv- alent of [cp,], and replacing B with the expression in Equation (S), yields (1 + s)/ (2ns), which is T. Division of the marginal revenue product of capital by T is equiv- alent to taking the present value of n equal periodic receipts.

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Robert E. Kohn

Received: November 1984 Final version: October 1985

References Henderson, J.M. and R.E. Quandt, Microeconomic Theory, New

York, McGraw-Hill, 1958. Keynes, J.M., The General Theory of Employment, Interest and

Money, New York: Harcourt Brace, 1936. Klein, L. R., “Macroeconomics and the Theory of Rational Behav-

ior,” Econometrics (1946): 93-108. Kohn, R.E., “Industry Isoquants When the Number of Firms is a

Variable,” Revista lnternazionale Di Science Economiche e Com- merciali 31(1984): 258-63.

Kormendi, R. C., “Government Debt, Government Spending, and Private Sector Behavior,” The American Economic Review 73(1983): 994-1010.

Survey of Current Business, October 1982, 62(10). Thuesen, G.J. and W. J. Fabrycky. Engineering Economy, Engle-

wood Cliffs: Prentice Hall, 1984.

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