Internal funds and the investment function.

• Author: Stevens, Guy V.G.

1. Introduction

   For more than two decades, researchers have discovered repeatedly a statistically significant relationship between a firm's fixed investment expenditures and its cash flow or retained earnings. The cumulative evidence makes it hard to reject an impact of the firm's internal financial situation on its capital spending.(1)

   To many who have followed the theory of investment over the years, this development seems like an echo from a bygone era--the revival of a theory once thought defunct. In the 1960s and early 1970s, work developing stock-adjustment models and the neoclassical model of investment seemed to dominate, if not disprove, theories containing liquidity and financial variables |7; 16; 4~. Thus, for example, in a notable article, Jorgenson and Siebert |17~ tested alternative theories on the same body of (microeconomic) data and found liquidity or finance-based theories dominated by the neoclassical theory of investment.

   The paradox of this revival can be partially explained by noting that financial variables have returned to the explanation of investment in an eclectic form, often embedded in models which also contain the determinants associated with neoclassical and stock-adjustment models. Thus, in a key article, Coen |5, 164~, relying on concepts introduced by Greenberg |11~ and Hochman |14~, proposed and tested an investment function where the speed of adjustment to the neoclassically-determined optimal capital stock was a function of internal funds:

   |I.sub.t~ = {|b.sub.0~ + |b.sub.1~(|F.sub.t - 1~ - |Delta~|K.sub.t - 1~)/(|K*.sub.t~ - |K.sub.t - 1~)}||K*.sub.t~ - |K.sub.t - 1~ + |Delta~|K.sub.t - 1~, (1)

   where I is the level of capital expenditures; K and K*, the actual and the desired level of the capital stock; F, the firm's cash flow; and |Delta~, the rate of depreciation.

   In such models the capital stock adjusts dynamically to K*, the optimum capital stock defined by the neoclassical theory; what is added is a variable speed of adjustment dependent on some measure of the firm's level of internal funds. The empirical work noted above has demonstrated that these financial effects are a statistically significant additional influence on fixed investment, both in the United States and Western Europe.

   The central concerns of this paper are the theoretical justification of these eclectic theories of investment and the dynamic implications of the investment functions that result. Plausible heuristic stories have been provided to support the inclusion of cash flow variables in the investment function, but so far no such investment functions have been rigorously derived from an underlying theory of the firm. The results presented below indicate that, under certain conditions, investment functions similar, but not identical to Coen's equation (1) can be justified theoretically. Moreover, the investment functions that result exhibit unusual dynamic properties; in particular, the speed of adjustment increases monotonically until the long-run equilibrium is attained. II. Issues in Linking Investment to Finance

   The idea that has been used most frequently in heuristic justifications for including financial variables in the speed of adjustment is the notion that the rate of interest on outside borrowing will be an increasing function of the level of borrowing or the ratio of debt to assets or equity. Coen |5, 150~ and Nickell |20~ saw this as one of the most promising explanations of the empirical regularity. Such an upward sloping supply function for debt can, in turn, be based on lenders' perceptions of a positive relationship between the debt/assets ratio and the risk and costs of bankruptcy |26; 15; 20; 27, 260-73~. Although an upward sloping supply curve for debt is firmly accepted,(2) little has been done to derive its dynamic implications for the firm's investment function. This, along with the associated question of the conditions under which Coen-type investment functions are justified, is, of course, the goal of this paper. To pursue this end, we will embed alternative versions of an upward sloping supply schedule for debt into the well-known neoclassical model of the firm.

   Increasing Costs of Debt and the Neoclassical Model

   To better understand how to proceed, let us initially explore how finance is handled in the original version of the neoclassical model pioneered by Jorgenson |16~. Ignoring tax considerations for the moment, the (present) value, V(|t.sub.0~), at a given time, |t.sub.0~, of the neoclassical firm can be written as:

   V(|t.sub.0~) = |integral of~ |e.sup.-|Rho~t~ DIV(t)dt between limits of |infinity~ and |t.sub.0~ = |integral of~ |e.sup.-|Rho~t~|pQ(t) - wL(t) - qI(t)~dt, between limits of |infinity~ and |t.sub.0~, (2)

   where DIV(t) is the level of firm dividends at time t; Q, L, I, are output, labor input and real investment expenditures, with p, w, and q their respective prices; |Rho~ is the firm's discount rate.

   Given Jorgenson's assumption of a Cobb-Douglas production function, the maximization of the value of this firm leads to the familiar relationship between the optimal capital stock (K*) and other endogenous and exogenous variables (with, in addition to the symbols defined above, |Mathematical Expression Omitted~ and |Gamma~ equal to the output elasticity of capital):

   |Mathematical Expression Omitted~.

   So far nothing has been indicated about the financing of the optimal capital stock. The standard approach to finance in the neoclassical model is implied by the substitution of the expression (pQ - wL - qI) on the right hand side of equation (2) for dividends (DIV) on the left hand side. Since the substitution is derived from the abbreviated sources and uses of funds identity, qI = pQ - wL - DIV, and since no debt or interest variables appear in that identity, the implication is that the firm's investment is financed fully by the difference between operating revenues (pQ - wL) and dividends. When dividends are positive, this would be called financing investment out of retained earnings.(3) However, it is important to note that dividends cannot be constrained to be positive in this model. The optimal investment policy may very well imply that for some periods the value of investment, qI, will be greater than operating revenues. During such periods, since debt finance is excluded, the above identity implies that dividends will be required to be negative, i.e., the firm assesses its shareholders for new infusions of capital. When debt financing is allowed, flotations of debt (|Mathematical Expression Omitted~) and the consequent interest payments at rate r (-rD) are incorporated into the firm's objective function by adding these terms to the right hand side of equation (2). The sources and uses of funds identity becomes: |Mathematical Expression Omitted~.

   The upward sloping supply curve for debt will be represented by the interest rate, r, being an increasing function of either the level of debt, r(D), or, alternatively, the debt/assets ratio, r(D/qK).

   This paper will not attack the question of the optimal mix of external sources of finance, so other types of external finance, such as equity flotations in excess of retained earnings, will be ruled out. This will be accomplished by constraining dividends to be nonnegative and by prohibiting the issuing of new shares of stock. The exclusion of such new equity is meant to mirror the view that it is a high-cost source of finance, requiring the incurring of substantial transaction or other costs |6; 19~.)

3. A Neoclassical Model with a Linear Interest Rate Function

   In the next two sections we consider the paths of capital and debt that maximize the following generalized value function for the neoclassical firm:

   |Mathematical Expression Omitted~,

   subject to DIV(t) |is greater than or equal to~ 0, D(t) |is greater than or equal to~ 0; |Mathematical Expression Omitted~; r = r(D) or r(D/qK), |Delta~r/|Delta~D |is greater than~ 0.

   The use of equation (5) as the firm's objective function requires that the managers of the firm view the risks of bankruptcy differently from lenders. While the upward sloping supply curve for debt presupposes that lenders envisage both a risk and costs of bankruptcy, the use of equation (5) assumes that the firm either sets its subjective probability of bankruptcy at zero or does not believe that significant costs are associated with this state.(4) A model based on similar differences of opinion between lenders and firm managers is analyzed by Stiglitz |26~.

   The models that follow also make two fairly innocuous simplifications. Assuming a linear homogenous production function, one can express the optimal labor input (L) as a linear function of capital and the ratio of wage and capital costs; as a result, given fixed input prices, the net revenue term above, pQ - wL, can be written as the function paK - bK, where a and b are positive constants. Further, for a determinate equilibrium to exist, marginal revenue exclusive of investment costs must be a decreasing function of output or capital. Given the assumption of constant returns to scale on the production side, this must come by virtue of the...