ROBERT E. KOHN Southern ZUinois University at EdwardsMe

Edwardsville, Zllinois

A Capital Budgeting Mode/ of the Supply and Demand for Loanable Funds*

In the general case, machines have a finite life span and are funded, explicitly or

implicitly, by a series of equal annual payments for interest and repayment of prin- cipal. The demand curve for loanable funds by neoclassical firms and the corre- sponding supply curve are derived in this paper. It is advantageous to graph both curves against the annual repayment per dollar of loanable funds rather than against

the rate of interest alone. Although supply and demand curves are often related to the rate of interest, the area defined by points on such curves has meaning only in special cases.

1. Introduction This paper examines the demand curve of the neoclassical firm

for loanable funds to purchase machinery when the loan is repaid by a stream of equal annual payments for interest and amortization over the finite life of the machines. A compatible supply curve of loanable funds in which households receive a series of equal annual repayments is generated from an indifference map between con- sumption in the present and equal quantities of consumption in each of the future years in which the loan is repaid.

Although supply and demand curves for loanable funds are commonly characterized as functions of the rate of interest alone, the rectangular area defined by the origin and the intersection of the supply and demand curves does not have the familiar “total payments” interpretation. The model developed here may be con- trasted, at one extreme, with the one period models of supply and demand for loanable funds in Hirshleifer (1988, 426-33) and Mans-

*I am grateful for the referee’s many helpful suggestions for improving this pa- per. The term “capital budgeting” refers to the schedule of cash flows over the life of an investment. There are three often confused meanings of the word “capital” in economic writings (see Hirshleifer 1970, 153), all of which are implied in the present paper. To avoid confusion, the word “capital” is used only to call “capital budgeting” and the *‘capital-recovery factor” by their familiar names.

Journal of Macroeconomics, Summer 1990, Vol. 12, No. 3, pp. 427-436 427 Copyright Q 1996 by Louisiana State University Press 0164-0704/90/$1.50

Robert E. Kohn

field (1988, 528-38), and at the other extreme with Friedman’s (1976, chap. 17) inter-temporal model based on permanent, nondepreciat- ing buildings. In both of these extreme cases, the rate of interest on the vertical axis times the corresponding quantity of loanable funds on the horizontal axis does equal total interest per year. In the case of depreciating machines, however, it is more meaningful to plot the supply and demand curves for loanable funds against the annual repayment per dollar rather than against the interest rate. The annual repayment per dollar times the dollar amount of loanable funds measures the annual machinery cost to firms, as well as the annual repayment of principal plus interest to households.

2. The Competitive Firm’s Demand Curve for Loanable Funds The demand curve of the firm for loanable funds to finance

the purchase of machinery is derived from a special marginal rev- enue product curve for machinery. Fama and Miller (1972, 117) and Hirshleifer (1988, 329-31) explain this special curve as follows. Consider a perfectly competitive firm employing labor, L, and ma- chines, K, to make a good, X, according to a production function, X(L, K), that exhibits increasing and then decreasing returns to scale. The price of the good, p, and the wage rate, w, are taken as given. The special marginal revenue product curve for machines is a long- run planning curve in which both inputs are variable. In the in- teresting case in which the cross marginal product, XLK, is nonzero, there is, for each quantity of K, an optimally adjusted quantity of labor, L*, whose conventional marginal revenue product, pX,, equals the given wage rate. Hirshleifer (1988, 330-31) demonstrates that this special marginal revenue product of machinery curve slopes downward to the right and is flatter than the ordinary marginal rev- enue product curve in which the quantity of labor is held constant. This special curve, labeled pXg , is characterized in Figure 1, where the horizontal axis measures the quantity of machines, K, and the vertical axis, labeled $/K/ n, measures dollars of marginal revenue product of machines in each of their n years of useful life.

Corresponding to any given set of prices, p and w, are two values of K that are of interest. There must be in general a mini- mum of K machines in place for the marginal revenue product of labor to equal W, and hence the pX2 curve begins at K. However, the competitive firm would never operate in the region of increas- ing returns to scale. The minimum quantity of machines that it would use is k, together with the corresponding quantity of labor, at which

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I I I

I I I

\

I I I I I I I I I I I I I I

\ Px;

Figure 1.

the sum of marginal physical products times respective input quan- tities exhausts the total product.’ The pX$ curve to the right of k

‘Sher and Pinola (1981, 543) illustrate the derivation of the conventional mar- ginal revenue product curve for input K using a Cobb-Douglas production function. A more appropriate production function for a competitive firm, because it exhibits increasing and then decreasing returns to scale, is the following function from Hen- derson and Quandt (1958, 46):

x = 15L2K” - L3K3.

Assuming that p = $1.00 and w = $36.00, it can be shown that

pX, = 30L’K - 3L3KZ,

L* = 5[1 + (1 - 0.48/K)“*]/K,

and therefore that

pXX = 180[1 + (1 - 0.48/K)“‘]/K”.

It follows from the equation for L* that K = 0.48 and from the condition for locally constant returns to scale (see Kohn 1988, 134) that k = 4/3. Observe that in this case the conventional marginal revenue product, pX,, eventually becomes negative as K increases, whereas pXg does not.

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may be interpreted as the firm’s demand curve for machines.’ The intersection of the vertical line at k and the special marginal rev- enue product curve, which determines the upper-left boundary of this demand curve, is the long-run analogue of the intersection of the average revenue product curve and the ordinary marginal rev- enue product curve in the short run.

Assuming that the capital budgeting approach correctly de- scribes the way in which firms finance their purchase of machinery having an initial cost of c dollars per machine and a life span of n years with no subsequent scrap value, the annual repayment per machine is c{i/[l - (1 + i)-“I}, where i is the rate of interest (Thuesen and Fabrycky 1984, 42, 43). The cofactor of c, which is the familiar equal-payment-series-capital-recovery factor represents the combined interest and amortization cost per dollar of machin- ery. It is assumed that there is perfect certainty regarding the pro- spective cash flow and that revenues are received and wages, in- terest, and amortization of principal paid at the end of each year.

When the firm is ready to invest, at time zero, its market demand curve for loanable funds is characterized by the demand curve, pX$, in Figure 2. Here the annual return to the marginal dollar invested in machinery is a function of the total dollar value of loanable funds, F. This curve is derived from the pX$ curve as follows. First, the variable K is replaced by F/c, and second, the entire function is divided by c. This changes the horizontal axis from machines, K, to loanable funds, F, and the vertical axis may now be used to measure the annual repayment per dollar of loan- able funds, which is their unit cost to the firm.3 The lowest possible

“If, in the context of the preceding numerical example, the annual repayment per machine is $182.25, the firm would purchase 4/3 machines, employ 6.75 units of labor, and its total revenue of $486 would equal its total cost. As the annual repayment per machine declines, the firm would substitute machines for labor and increase output. These are the “substitution effect” and the “scale effect” that Hir- shleifer (1988, 325) identifies. Hirshleifer (1988, 329) also observes the case in which the lowering of an input price causes increased employment of both inputs; this result can be numerically demonstrated with a production function such as

x = (15.L - L2 + 15K - KZ + LK)” ,

where L, K s 15. This is an increasing then decreasing returns-to-scale version of a production function in Allen (1947, 288-9).

‘The demand tknction for loanable funds based on the technology and costs in footnote 1 simplifies to

pX$ = 18Oc[l + (1 - 0.48c/F)““]/F*.

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A Capital Budgeting Model

$/F/n

P$

G

icl ----- - l-(l+io)-n I If5 I

l/n I I I

0 ck F,, F

Figure 2.

annual repayment per dollar of loanable funds, given n, is the limit of i/[l - (1 + i)-“1 as i approaches zero, which is l/n. This makes good sense, for if the interest rate is zero, the equal annual repay- ment per dollar of loanable funds in each of the n years is simply $1/n.

If the interest rate is i,, a total of F, dollars of loanable funds would be demanded by the firm. The area of the rectangle rep- resenting the product i,/[l - (1 + &,)-“I times F, denotes the total interest and amortization cost (explicit or implicit) in each of the n years of useful machine life. Over time, however, the portion of this fixed amount allocated to interest decreases, whereas that for the amortization of principal increases. The sum of all n amortiza- tion payments equals F,.

The curve pX$ in Figure 2, between the vertical line ck and the horizontal line l/n, is the firm’s demand curve for loanable funds repayable in n equal annual installments. Horizontally adding

(In terms of the numerical example in Kohn 1986, 378, in which c = $324 and F = $432, the annual repayment per dollar of loanable funds is $0.5625.) This is

actually an inverse demand function because it yields an annual repayment per dollar for any given value of F.

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the demand curves of all firms in all industries for n-year funds (assuming away the problems of aggregation anticipated by Metzler 1950; Lemer 1953; Harris 1981; and Hirshleifer 1988, 331-33) yields the market demand curve for loanable funds repayable in equal an- nual installments over the n years.

3. The Supply of Loanable Funds by Households Leontief (1958) devised a model of intertemporal preferences

that Hirshleifer (1970, 69) calls the “discrete-perpetuity model” in which there is “a choice between current consumption . . . [and] a level annual sequence of consumptions . . . beginning . . . one year from the present and extending to infinity.” The model used here is similar, except that there are only n future years. Consider the intertemporal indifference curves U’ and U2 in Figure 3, in which the consumer has an income of I, in the present and If, which for simplicity equals I,, in each of the n future years. If the consumer does not save in the present, he or she will consume C, = I, = C, = Z, in each year, including the present year. If the interest rate is i,, the intertemporal consumption constraint is the

Figure 3.

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line AB, whose slope is i,/[l - (1 + &,)-“I; this consumer will save S, and consume Cg in the present and CT in each future year.4

As i increases, the budget constraint becomes steeper, and a supply curve of savings as a function of i/[l - (1 + i)-“1 is gen- erated. Similar supply curves of all households wishing to save in the present and be repaid in n equal annual payments of interest and principal are horizontally summed to yield a supply curve of loanable funds that is compatible with the above market demand curve.

4. Conclusions In the model presented here, competitive firms demand cur-

rent loanable funds to purchase machines for generating future rev- enues, while households supply these funds in exchange for a stream of future repayments out of those revenues. If, for simplicity, we assume that all machines in the economy have a useful life of n years and households lend funds only for that same length of time, the market demand and supply curves derived above can be su- perimposed, with loanable funds, F, on the horizontal axis and the annual repayment per dollar borrowed, i/[ 1 - (1 + i)-” 1, on the vertical axis. The intersection of the two curves at F* determines the equilibrium annual rate of repayment per dollar.

What is new in the model presented here is that machines depreciate in a finite number of years. In the models of supply and demand for loanable funds of Hirshleifer (1988, 426-33) and Mans- field (1988, 528-38), depreciation occurs in a single year (n = l), whereas in the model of Friedman (1976, chap. 17) there is no de- preciation (n = 03). None of these models readily generalizes to an economy in which there are machines having different life spans. This is a regrettable limitation of the present model because it would be more powerful if it extended to an economy in which there are supply and demand curves for loanable funds of different durations, especially one satisfying the capital budgeting assumption (Bierman and Smidt 1984, 22) of a single time value of money for all dura- tions.

From the compound-interest formula for the annual repay- ment per dollar of loanable funds, the equilibrium rate of interest, i*, corresponding to F* can be inferred. Taking this a step further,

‘Intertemporal utility fbnctions characterizing preferences over C, and C, are numerically illustrated in Kohn (1989).

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the demand and supply curves for loanable funds can also be graphed, and often are, as functions of the rate of interest alone. ,When the one-period supply and demand curves of Hirshleifer (1988, 433) and Mansfield (1988, 537) for loanable funds are graphed against the rate of interest, the product i*F* represents total interest paid and received in the single period. The supply and demand curves in the Friedman (1976, 309) model can only be graphed against the rate of interest, and here the product i*F* represents total interest paid each year in perpetuity. When the supply and demand curves for loanable funds in the present model, in which 1 < n < CQ, are graphed against the rate of interest by itself, the product i*F* has no economic interpretation.

Received: April 1989 Final version: October 1989

References Allen, Roy G.D. Mathematical Analysis for Economists. London:

Macmillan, 1947. Bierman, Harold, Jr., and Seymour Smidt. The Capital Budgeting

Decision. New York: MacMillan, 1984. Fama, Eugene F., and Merton H. Miller. The Theory of Finance.

Hinsdale: Dryden, 1972. Friedman, Milton. Price Theory. Chicago: Aldine, 1976. Harris, Donald J. “Profits, Productivity, and Thrift: the Neoclassical

Theory of Capital and Distribution Revisited.” Journal of Post Keynesian Economics 3 (Spring 1981): 359-82.

Henderson, James M., and Richard E. Quandt. Microeconomics Theory. New York: McGraw-Hill, 1958.

Hirshleifer, Jack. Investment, Interest and Capital. Englewood Cl&: Prentice-Hall, 1970.

-. Price Theory and Applications. 4th ed. Englewood Cliffs, Prentice-Hall, 1988.

Kohn, Robert E. “The Rate of Interest in a Stationary Economy.” Journal of Macroeconomics 8 (Summer 1986): 373-80.

-. “The HQ Production Function.” The Economic Record 64 (June 1988): 133-35.

-. “Variable Outputs Along a Production Possibility Frontier with Inputs Constant by Choice.” Rivista Znternazionale de Scienze Economiche e Commerciali 36 (September 1989): 837-46.

Leontief, Wassily. “Theoretical Note on Time-Preference, Produc-

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tivity of Capital, Stagnation and Economic Growth.” American Economic Review 68 (March 1958): 105-11.

Lerner, Abba P. “On the Marginal Product of Capital and the Mar- ginal Efficiency of Investment.” Journal of Political Economy 61 (February 1953): 1-14.

Mansfield, Edwin. Microeconomics. 6th ed. New York: Norton, 1988. Metzler, Lloyd A. “The Rate of Interest and the Marginal Product

of Capital.” Journal of Political Economy 58 (August 1950): 289- 306.

Sher, William, and Rudy Pinola. Microeconomic Theory. London: Edward Arnold, 1981.

Thuesen, G. J., and W.J. Fabrycky. Engineering Economy. Engle- wood Cliffs: Prentice-Hall, 1984.

Appendix Variables and terms and their definitions

c = initial cost per machine (a constant). C, = consumption of the household in each of the

n future years. C, = consumption of the household in the present

year. F = quantity of loanable funds in dollars. i = annual rate of interest.

i/[ 1 - (1 + i)-” ] = equal-payment-series-capital-recovery factor. Zf = income of the household in each of the n fu-

ture years. I, = income of the household in the present year. K = quantity of machines purchased by an indi-

vidual firm. k = the quantity of machines with which, given

the corresponding optimal quantity of labor, the firm experiences locally constant returns to scale.

K = the minimum quantity of machines necessary for the marginal revenue product of labor to equal the given wage.

L = quantity of labor employed by an individual firm.

n = life span of machinery in years (a constant). p = market price of good X (a constant).

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pX, = marginal revenue product of labor. pX$ = marginal revenue product of loanable funds

(dollars of revenue per year per dollar of loanable funds).

pX2 = marginal revenue product of machinery (dol- lars of revenue per machine per year) when used in combination with an optimally ad- justed quantity of labor.

X = quantity of output produced each year by an individual firm.

w = wage rate (a constant).

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