# Internal funds and the investment function.

I. 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~.)

III. 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 firm's possession of some degree of market power on the demand side; thus, we will assume a downward sloping demand curve, p = c - dQ, with c and d positive constants. Making both substitutions, the net revenue term becomes a quadratic function of K (|Alpha~, |Beta~ |is greater than~ 0): pQ - wL = |Alpha~K - |Beta~|K.sup.2~. (6)

Finally, for the model examined in this section we make the simplest assumption for the interest rate function consistent with the requirement that it be upward sloping: r(D, K) = |Rho~ + |Psi~D. Since bankruptcy risk is more properly a function of the debt/assets or debt/equity ratio, rather than the level of debt alone, this assumption is relaxed in the next section. Making the indicated substitutions, equation (5) becomes:

|Mathematical Expression Omitted~,

subject to DIV(t) |is greater than or equal to~ 0, D(t) |is greater than or equal to~ 0, where |Mathematical Expression Omitted~.

The objective becomes to maximize the integral by choosing optimal paths of capital and debt. Following the results in control theory incorporating constraints on state variables in, for example, Arrow and Kurz |2~ or Kamian and Schwartz |18~, one determines the optimal solution by forming a Lagrangean expression (L) made up of the normal (current value) Hamiltonian (H) plus multiplier expressions for the inequality constraints.(5) Thus,

L = H + ||Mu~.sub.1~DIV + ||Mu~.sub.2~D, (8)

where:

|Mathematical Expression Omitted~,

||Mu~.sub.1~(t), ||Mu~.sub.2~(t) are the multiplier functions for the inequality constraints; ||Lambda~.sub.1~(t), ||Lambda~.sub.2~(t) are the costate variables; and

||Mu~.sub.1~(t)DIV(t) = 0, ||Mu~.sub.2~(t)D(t) = 0, ||Mu~.sub.1~(t), ||Mu~.sub.2~(t) |is greater than or equal to~ 0.

Substituting for H and DIV:

|Mathematical Expression Omitted~.

An optimal path for the firm's capital stock and debt must satisfy the following necessary conditions:

|Delta~L/|Delta~I = 0 = -q(1 + ||Mu~.sub.1~) + ||Lambda~.sub.1~ (10)

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~,

with, as noted above, ||Mu~.sub.1~(t), ||Mu~.sub.2~(t) |is greater than or equal to~ 0, ||Mu~.sub.1~DIV = 0 = ||Mu~.sub.2~D.

Irrespective of whether the constraint on dividends is binding, the equations have important implications for the relationship between the optimum capital stock and debt. Equations (10) and (11) imply that ||Lambda~.sub.2~ = -||Lambda~.sub.1~/q. Although not important, assume for simplicity that q is not a function of time; then |Mathematical Expression Omitted~. In this case, the left hand side of (12) equals -q times the left hand side of (12) equals -q times the left hand side of (13). From this we have:

(1 + ||Mu~.sub.2~)(|Alpha~ - 2|Beta~K) - ||Lambda~.sub.1~|Delta~ = q(1 + ||Mu~.sub.1~)(|Rho~ + 2|Psi~D) - q||Mu~.sub.2~. (14)

It can be shown, further, that ||Mu~.sub.2~ is always zero--i.e., even in the unconstrained problem it turns out that debt cannot violate the nonnegativity constraint.(6) Finally, since it is also true that (1 + ||Mu~.sub.1~) = ||Lambda~.sub.1~/q, equation (14) simplifies to:

|Alpha~ - 2|Beta~K = q(|Rho~ + |Delta~ + 2|Psi~D). (15)

Solving for the capital stock, equation (15) becomes:

K(t) = ||Alpha~- q(|Rho~ + |Delta~ + 2|Psi~D(t))~/2|Beta~. (16)

Except for the term containing D(t), this equation is a familiar variant of the neoclassical equation for the firm's optimal capital stock; q(p + |Delta~) is the cost of capital for this latter model. It is also clear from (16) that it must be the case that |Alpha~ - q(|Rho~ + |Delta~) |is greater than~ 0, or K(t) can never be positive. From equation (16) we can also see a familiar implication of the Jorgensonian theory of investment where no dynamic adjustment costs are present: depending on the level of debt, equation (16) may imply a jump in the stock of capital. Instead of the usual differential equation in K(t) encountered in control theory, equation (16) is a degenerate equation linking (only) optimal levels of K(t) and D(t).

The system at any point in time is completed by the inequality constraints DIV |is greater than or equal to~ 0 and D |is greater than or equal to~ 0, and the equation for dividends (the sources and uses of funds identity). Except when a jump occurs, which is discussed below, the dividend expression is:

|Mathematical Expression Omitted~.

Of the four possible combinations of dividends and debt greater or equal to zero, two can be ruled out as either impossible or irrelevant. The two cases D |is greater than~ 0, DIV |is greater than~ 0 and D = 0, DIV = 0 are impossible, as shown formally in Stevens |25, Appendix, section I~. Thus, if D |is greater than~ 0, the only possible case is DIV = 0: since the interest rate on debt is always higher than the firm's discount rate, it would be suboptimal for the firm to pay dividends with any debt outstanding. In this case, unless a jump occurs, the system becomes equation (16) and equation (17) holding with an equality; these are sufficient to determine the optimal path of K(t) and D(t). The other possible case is D = 0 and DIV |is greater than~ 0. It turns out that this is the long-run stationary equilibrium |25, Appendix, section IV~. Using equation (16) and setting debt equal to zero, one derives the long-run capital stock, K* = ||Alpha~ - q(|Rho~ + |Delta~)~/2|Beta~. Given that |Mathematical Expression Omitted~ in this state, equation (17) is sufficient to determine equilibrium dividends.

Investment and Debt: the Initial Jump

To examine the dynamic process of investment, debt expansion and reduction, and dividend payments, let us consider the effects of an exogenous shock to the system. Suppose that initially the firm is in a state of long-run equilibrium, with a stationary capital stock and debt equal to zero, as discussed above. Consider the effects of an upward shift in the firm's demand curve--causing a discrete upward shift in the parameter |Alpha~ to |Alpha~* in equation (15) or (16).

Since it can be proved that equation (16) holds at every point in time, whether there is a jump or not, the first point to notice is that either D or K or both will have to undergo a discrete jump in order that the equation be satisfied.(7) Further, at the instant of a jump, the sources and uses of funds equation (17) becomes particularly simple. Since the time period covered by the jump is infinitely short, it is intuitively clear that the net impact of flows of profits, depreciation, or interest payments is zero. (For a rigorous proof, see Appendix, section III of Stevens |25~.) Incorporating the assumption of no new equity financing and the finding that dividends are zero during the jump, all the terms in equation (17) drop out except those in |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~--i.e., in the absence of other inflows and outflows during the jump, the sources and uses of funds identity reduces to the statement that the value of the change in capital must equal the value of new debt floated. Replacing these derivatives by the discrete changes that occur during the jump, equation (17) reduces to: D+(0) - D-(0) - q|K+(0) - K-(0)~ = 0, (18)

where the "+" and the "-" superscripts refer to the levels of the indicated variable just after and before the jump. Using (18) to substitute for the variable D+(0), one can then use equation (16) to determine the optimal jump for the capital stock, K+(0). One can show further that, because of the concavity of the relevant functions, there can be a jump only at time zero.(8) The optimal jump and subsequent developments are depicted in Figure 1. In the figure, the capital stock, K(t), is measured along the horizontal axis. The level of debt and various functions of the capital stock are measured along the vertical axis; for simplicity, the price of a unit of capital stock, q, will be set at 1 at every point in time.

The firm is assumed to be initially at a point of long-run equilibrium, point |E.sub.0~ on the K axis: a point with no debt and an equilibrium capital stock of K-(0). As an equilibrium point, it must satisfy the optimality condition (16)--plotted in the figure as the straight line MRMC(|Alpha~) (which intercepts the D axis at ||Alpha~ - (|Rho~ + |Delta~)~/2|Psi~, with a negative slope of -|Beta~/|Psi~). (Recall that q is set to 1.)

Consider the effects of an increase from |Alpha~ to |Alpha~* of the intercept of the marginal revenue product of capital--a shift of the MRMC curve parallel to itself to the northeast. The new (short-term) equilibrium is determined by the intersection of the MRMC(|Alpha~*) curve and the sources and uses of funds identity (18). Since equation (18) shows that changes in debt and capital during the jump must be proportional to each other, and given the assumption that q = 1, equation (18) is drawn as a 45 degree line through the original equilibrium point |E.sub.0~. The optimal jump is determined by the intersection of the 45 degree line and the MRMC(|Alpha~*) curve, at the new (short-run) equilibrium |E.sub.1~ (K+(0), D+(0)); this implies an optimal jump of K+(0) - K-(0) for capital and D+(0) for debt.

Determining the Optimal Paths of Capital and Debt after the Jump

Unlike the original Jorgenson model, the determination of the optimal jump is not the end of the story, but only its beginning. The key to the movement to the long-run equilibrium is that the firm's financial resources change over time endogenously as the firm operates and generates profits. These profits are linked to further capital stock growth, changes in debt, and dividend payouts through the more general flow of funds constraint (17).

In Stevens |24~ considerable space is devoted to solving the system defined by equation (16) and the differential equation (17), and describing the dynamic paths implied for K(t) and D(t) after the jump. However, the important points about the path can be demonstrated without the full derivation. First, the equilibrium condition (16) still holds at every point along the dynamic path and implies a negative linear relationship between capital and debt; in (K, D) space in Figure 1, this adjustment path falls along the MRMC (|Alpha~*) curve between |E.sub.1~ and the new long-run equilibrium, E*, where K* = ||Alpha~* - q(|Rho~ + |Delta~)~/2|Beta~ and D* = 0.

Taking the derivative of both sides of (16), we get:

|Mathematical Expression Omitted~.

Using this expression to substitute for |Mathematical Expression Omitted~ in (17), and noting that DIV(t) equals zero as long as D(t) is greater than zero, equation (17) becomes:

|Mathematical Expression Omitted~.

From the optimality conditions, the right hand side of (20)--the firm's profits after interest and depreciation deductions--must be positive on the segment between |E.sub.1~ and E*; hence |Mathematical Expression Omitted~ must be positive along the optimal path and, from (19), |Mathematical Expression Omitted~ must be negative. Thus, over time, starting from point |E.sub.1~, the capital stock, K(t), will increase and debt will decrease along the optimal path--the segment |E.sub.1~E*. At E* the long-run equilibrium will be reached as debt falls to zero.

The Investment Function and Cash Flow

A key aim of this paper is to examine the firm's investment function in the light of the empirical results of Coen |5~, Artus et al. |3~, and others, which successfully linked the speed of adjustment to cash flow. Using equation (16), it is possible to relate the firm's investment both to cash flow and to the "gap" between actual capital and the long-run equilibrium level, K* (defined above):

K* - K(t) = q|Psi~D(t)/|Beta~. (21)

Differentiating (21) leads to the same linear relationship between |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ as derived above for equation (19). Combining equation (19) with equation (21) leads to: |Mathematical Expression Omitted~.

Equation (22) shows that the investment function can be written as a flexible accelerator with a variable speed of adjustment--the latter equal to the absolute value of the percentage rate of change of debt.

One can go further, however, and, using equation (20), relate the above speed of adjustment to the firm's cash flow.(9) Note that the right hand side of (20) equals net revenues (|Alpha~K - |Beta~|K.sup.2~) minus depreciation charges (q|Delta~K), minus interest costs on the debt, (|Rho~ + |Delta~D)D. The right hand side, therefore, is a measure of net profits; denote it by |Pi~(t). Further, since |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ are linearly related as shown in (19), the left hand side of (20) can be expressed as a linear function of |Mathematical Expression Omitted~. Thus, equation (20), a version of the sources and uses of funds constraint, reduces to:

|Mathematical Expression Omitted~.

Using (23) to substitute for |Mathematical Expression Omitted~ in the investment function (22), we derive a final investment function that features a variable speed of adjustment that is a function of the firm's net profits:(10) |Mathematical Expression Omitted~.

We have in equation (24) an investment function that is remarkably similar to the suggestive, but ad hoc, formulation of equation (1) used successfully in the empirical work noted above in section I. For the relevant range, optimal investment can be described by a flexible accelerator with a variable speed of adjustment. Moreover, the speed of adjustment is a function of the firm's level of profits, proportional to the ratio of net profits to the level of outstanding debt.

IV. A More Realistic Model: Introducing Corporate Taxes and a Borrowing Rate a Function of the Debt/Assets Ratio

In this section we will see that the general lessons of the previous section carry over to more realistic models. One of the most appealing modifications of the preceding model is to let the firm's borrowing rate be a function of the debt/equity or the debt/assets ratio, rather than the level of debt alone. We have alluded to the risk of bankruptcy as the preferred theoretical reason supporting the upward sloping supply curve of debt. With this justification, the probability of the firm falling into the bankruptcy state, where it cannot cover its interest costs, can be shown to be a function of the debt/assets ratio and not of the absolute level of debt |20; 22; 26~.

Despite this more realistic supply schedule for debt, without more, the optimal long-run level of debt would still be zero. An obvious modification that assures a positive long-run debt/assets ratio is the introduction of a corporate income tax along with the usual deductibility of interest.(11) These two new assumptions result in predictable changes in the problem of section III, equations (5) through (16). The borrowing rate, r(t), will now be a function of the debt/assets ratio, |Phi~:

r(t) = |Rho~ + |Psi~|Phi~ = |Rho~ + |Psi~|D(t)/qK(t)~. (25)

Corporate taxes reduce potential dividends payable to stockholders. We shall assume the simplest possible case: a constant tax rate, |Tau~, assessed on total revenue minus interest and depreciation charges:

Tax = |Tau~||Alpha~K - |Beta~|K.sup.2~ - (|Rho~+ |Psi~|Phi~)D - |Delta~qK~. (26) Incorporating these modifications into equation (7), the expression for the value of the firm that is to be maximized, leads to the following modified Lagrangean expression:(12)

|Mathematical Expression Omitted~

subject to: K = I - |Delta~K, ||Mu~.sub.1~(t)DIV(t) = 0, and ||Mu~.sub.1~(t) |is greater than or equal to~ 0. The new necessary conditions for an optimum are: |Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~,

and with,

|Mathematical Expression Omitted~.

As was the case for the preceding model, the costate variables can be eliminated from equations (28) to (31) and, irrespective of whether the constraint on dividends is binding, one can solve for the capital stock in terms of the determinants of the firm's marginal revenue product (|Alpha~,|Beta~) and its marginal cost of capital |q(|Rho~ + |Delta~ + 2|Psi~|Phi~ - |Psi~||Phi~.sup.2~)~:

K(t) = ||Alpha~- q(|Rho~ + |Delta~ + 2|Psi~|Phi~ - |Psi~||Phi~.sup.2~)~/2|Beta~. (33)

The solution of this system is qualitatively quite similar to that of the preceding section. Equation (33) shows that shifts in |Alpha~ or any other parameter of the model will lead to jumps in K(t) and D(t)--and the ratio of the two, |Phi~. Finally, one can demonstrate that if |Phi~ is greater than its long-run equilibrium (|Phi~*), DIV falls to zero; moreover, when the debt/assets ratio is worked down to |Phi~*, both debt and capital attain their long-run equilibrium values and dividends becomes positive |25, Appendix, section II~. Similar to the approach taken in the earlier model, one can determine the long run equilibrium values K*, |Phi~*, and D* by examining the solution to the system when ||Mu~.sub.1~ becomes zero (dividends become non-negative). Noting that ||Lambda~.sub.1~ and ||Lambda~.sub.2~ are in this case equal to q and -1, respectively, from equations (30) and (31) we derive the following:

|Phi~* = D*/qK* = |Tau~|Rho~/(1 - |Tau~)2|Psi~, (34)

K* = {|Alpha~ - q||Rho~/(1 - |Tau~) + |Delta~ - |Psi~||Phi~*.sup.2~~}/2|Beta~. (35)

We will assume that the parameters |Rho~, |Tau~, and |Psi~ take on values such that |Phi~* is less than 1.(13)

Since we are primarily interested in the form of the associated investment function, we can be brief about the dynamic paths of the endogenous variables. Assuming once again an upward shift of |Alpha~ to |Alpha~* for the system initially in long-run equilibrium, again there must be a jump in K and D to satisfy equation (33); and again the modified sources and uses of funds equation (18) will apply during the jump, providing a second equation to determine both K and D. Since D and K increase in proportion, the debt/assets ratio increases above the long-run equilibrium level |Phi~* and dividends fall to zero. From this point, after the initial jump, the dynamics of the system are governed by equation (33) and the modified dividend equation (32). Since both equations are nonlinear because of the debt/assets ratio, it is impossible in this case to derive an explicit solution for the paths of debt and capital. However, as shown for the previous model, capital increases monotonically from its initial value to the long-run equilibrium K*; moreover, the debt/assets ratio, |Phi~, decreases monotonically to |Phi~*.

It should be noted that this model, like the previous one, leads to ever-increasing rates of investment, |Mathematical Expression Omitted~, along the optimal path. The monotonic fall of the debt/assets ratio, combined with the increasing generation of after-tax profits, leads to continuously increasing funds for investment.

A New Relationship between Investment and Financial Variables

Let us once again return to the ultimate question: How do the complications introduced in this particular model affect the validity of the variable speed-of-adjustment investment function developed by Coen and others? As might be suspected, the nonlinearities introduced by making the debt/assets ratio the key variable in the interest rate function severely complicate the form of the investment function. Nevertheless, we still have the elements that link the variables |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and after-tax profits, |Pi~: the dividend equation (32) and the fact that optimal dividends are zero during the period when net investment is positive. It should be noted again that equation (32) holds only after the initial jump. In deriving the investment function, I shall take the same approach as in the preceding section: first deducing the level of capital in terms of its deviations from the long-run equilibrium, K*, and then differentiating this expression with respect to time. Equation (33) relates the capital stock at any given time to a single endogenous variable: in this case the debt/assets ratio, |Phi~. Subtracting the expression for K(t) in (33) from K*, defined in equation (35), gives:

2|Beta~(K* - K)/q = 2|Psi~|Phi~ - |Tau~|Rho~/(1 - |Tau~) - |Psi~(||Phi~.sup.2~ - ||Phi~*.sup.2~). (36)

From the expression (34) for the optimal debt/assets ratio, |Phi~*, it can be seen that the second term of the right hand side of (36) equals -2|Psi~|Phi~*, so K* - K may be expressed compactly solely as a function of the difference between the actual and the long-run equilibrium debt/assets ratio:

K* - K = q|Psi~|2(|Phi~ - |Phi~*) - (||Phi~.sup.2~ - ||Phi~*.sup.2~)~/2|Beta~. (37)

Differentiating this nonlinear function with respect to time leads to one form of the investment function:

|Mathematical Expression Omitted~.

This is obviously more complicated than the simple relation, |Mathematical Expression Omitted~, derived for the previous model. However, after the initial jump, the dividend equation still leads to |Mathematical Expression Omitted~, where |Pi~ is now defined as after-tax profits. Substituting |Mathematical Expression Omitted~, equation (38) becomes:

|Mathematical Expression Omitted~.

Dividing (39) through by the expression for K* - K, we arrive at a significantly more complicated form of the variable speed-of-adjustment investment function, but one which retains the key property that the speed of adjustment depends on the firm's profits:

|Mathematical Expression Omitted~.

V. Conclusions and Caveats

This paper demonstrates that it is possible to provide theoretical support for the eclectic investment functions tested successfully by Coen |5~, Artus et al. |3~, and others. Models of the firm featuring intertemporal profit maximization subject to increasing costs of external debt lead to investment functions that, in important ways, are close to those estimated empirically. In particular, the theoretically-implied investment functions are a combination of the concepts of a flexible accelerator and a variable speed of adjustment, with the latter dependent on the supply of internal funds.

The dynamic adjustment mechanism implied by investment functions of this class is unusual with respect to both its cause and pattern. Unlike the typical dynamic-cost explanation of investment over time, costs in the above models are static, independent of the rate of change of any variable. However, borrowing costs are increasing functions of variables--debt or the debt/assets ratio--that change dynamically because of the firm's operations. Thus, these static costs also change over time as a result of the firm's actions and, as the costs change, investment is generated.

Not only is the causal mechanism generating investment different, but so is the dynamic pattern of the investment. Rather than having the rate of investment decline monotonically as the firm's capital stock approaches the equilibrium level, investment in this model increases monotonically as an increasing supply of internal funds is divided optimally between debt reduction and investment. Although the models presented in this paper serve to provide theoretical underpinnings for this important class of empirical investment function, one must note the caveats. We have pointed out that the investment function does not hold for points where capital and debt jump. Moreover, the underlying structure of the theoretical model suggests extensions to more general models and investment functions. The class of investment functions examined in this paper depends on a model with a single dynamic element determined by the cost of outside debt and the supply of internal funds. Models with additional dynamic factors should imply different investment functions. An illustration is provided by a small modification of the first model studied in section III: letting the demand-curve parameter |Alpha~ be a continuous function of time, instead of a constant. It is easily shown that the same key equations (16) and (17) hold to determine the optimal paths of capital and debt. However, from (16) the capital stock now is a function of two dynamic variables, |Alpha~ and D; and investment, |Mathematical Expression Omitted~, equals |Mathematical Expression Omitted~, instead of the previous |Mathematical Expression Omitted~. Both terms now enter the variable speed of adjustment. Despite the presence of this second term, the key relationship between |Mathematical Expression Omitted~ and profits (|Pi~) is still operational, so the speed of adjustment will continue to be a function of the firm's cash flow. Now, however, cash flow will be only part of the adjustment story.

1. See Coen |5~, Gardner and Sheldon |10~, Eisner |8~, Artus et al. |3~, Fazzari et al. |9~ and, for earlier work, Greenberg |11~ and Hochman |14~.

2. However, the existence of significant costs associated directly or indirectly with the state of bankruptcy is still a matter of debate. (See Haugen and Senbet |12~.)

3. One can defend the assumption of all-equity financing by noting that with a perfect market for riskless debt (at a constant rate of interest), the firm's real decisions for investment and production are independent of whether financing is from equity or riskless debt.

4. The explicit incorporation of uncertainty and costs of bankruptcy into the firm's objective function would severely complicate the analysis, particularly in view of the sources and uses of funds constraint introduced into the model. Even if tractable, the analysis of such a model would probably not change the conclusion of this paper that financial variables can be shown to account for some of the variability in the speed of adjustment under some, but not all circumstances.

For discussions of the complications with intertemporal maximization under uncertainty, see Abel |1~, Nickell |20~, and Stevens |23~.

5. Later in this section and in more detail in the Appendix to Stevens |25~, we analyze the complications introduced when there can be jumps in the state variables, K and D. It is shown that equations (10) through (16), below, hold whether a jump occurs or not.

6. Probably the easiest way to show that ||Mu~.sub.2~ always equals zero is to prove that ||Mu~.sub.2~ = 0 when D = 0; thus the constraint is inactive at the boundary. In Stevens |25 Appendix, section I.B~, it is shown that D = 0 implies that DIV |is greater than~ 0; thus, when D = 0, ||Mu~.sub.1~ = 0; moreover, using the arguments developed in the text, equation (13) becomes: |Rho~ = |Rho~ + 2|Psi~D - ||Mu~.sub.2~. Since D = 0 by assumption, ||Mu~.sub.2~ = 0. That there is no tendency for debt to become negative can be seen intuitively as follows: negative levels of debt, like negative borrowing, is equivalent to lending; but because of the linear interest rate function, the rate of return on such lending is less than |Rho~. Hence, since paying dividends has the implicit return |Rho~, it will never be optimal to carry on such lending.

7. It is proved in the Appendix (section III) of Stevens |25~, following work by Vind |28~ and Arrow and Kurz |2~, that equation (16) continues to hold in the presence of a jump.

8. This is proved by Arrow and Kurz |2, Proposition 12, 57~.

9. It should perhaps be noted again that equation (20) and the implied relationship between cash flow and |Mathematical Expression Omitted~ does not hold during the jump (rather, equation (18) does).

10. Note that, from equation (20), the measure of profits, |Pi~(t), is defined as net of depreciation and interest expenses. Depreciation is subtracted because equations (22) or (24) are functions for net investment, not gross. For an equation for gross investment, one just adds |Delta~K(t) to equation (24). 11. The following treatment of taxation is meant only to be a suggestive way to get a positive long-run debt/assets ratio; a realistic approach would require, among a number of things, the integration of corporate taxation with the personal taxation faced by stockholders.

12. Since it was shown that ||Mu~.sub.2~(t) always equals zero (footnote 6), the term ||Mu~.sub.2~D that appeared in equation (9) is dropped.

13. Realistic values for these parameters in the U.S. context seem to lead to values for |Phi~* well under 1; for example, assume the interest rate (|Rho~) for a firm with no debt is .10; then, if the firm's borrowing rate were to double to .20 as the debt/assets ratio moved from 0 to 1, the implied value for |Psi~ would be .10. If one assumes, finally, that the tax rate, |Tau~, is .25, the implied value for |Phi~* is 1/6 = 17 percent.

It can be shown that if the firm starts from a long-run equilibrium at |Phi~* |is less than~ 1, the debt/assets ratio, |Phi~, will be |is less than~ 1 throughout the optimal path. (See Stevens |25~, Appendix, section II.B.)

References

1. Abel, Andrew B., "A Stochastic Model of Investment, Marginal q and the Market Value of the Firm." International Economic Review, June 1985, 305-22.

2. Arrow, Kenneth J. and Mordecai Kurz. Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins, 1970.

3. Artus, Patrick, Pierre-Alain Muet, Peter Palinkas, and Peter Pauly, "Economic Policy and Private Investment Since the Oil Crisis." European Economic Review, May 1981, 7-51.

4. Bischoff, Charles W. "The Effects of Alternative Lag Distributions," in Tax Incentives and Capital Spending, edited by Gary Fromm. Washington: Brookings, 1971.

5. Coen, Robert M., "The Effect of Cash Flow on the Speed of Adjustment." in Tax Incentives and Capital Spending, edited by Gary Fromm. Washington: Brookings, 1971.

6. Duesenberry, James S. Business Cycles and Economic Growth. New York: McGraw-Hill, 1958.

7. Eisner, Robert, "A Distributed Lag Investment Function." Econometrica, January 1960, 1-29.

8. -----. Factors in Business Investment. Cambridge: Ballinger, 1978.

9. Fazzari, Steven M., R. Glenn Hubbard, and Bruce C. Petersen, "Financing Constraints and Corporate Investment." Brookings Papers on Economic Activity, 1:1988, 141-95.

10. Gardner, Roy and Russell Sheldon, "Financial Conditions and the Time Path of Equipment Expenditures." Review of Economics and Statistics, May 1975, 164-70. 11. Greenberg, Edward, "A Stock Adjustment Investment Model." Econometrica, July 1964, 339-57.

12. Haugen, Robert A. and Lemma W. Senbet, "The Irrelevance of Bankruptcy Costs to the Theory of Optimal Capital Structure." Journal of Finance, May 1978, 383-94.

13. Hirschleifer, Jack, "On the Theory of Optimal Investment Decision." Journal of Political Economy, August 1958, 329-52.

14. Hochman, Harold M., "Some Aggregative Implications of Depreciation Acceleration." Yale Economic Essays, Spring 1966, 217-74.

15. Jensen, Michael C., and William H. Meckling, "Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure." Journal of Financial Economics, October 1976, 305-60.

16. Jorgenson, Dale W., "Capital Theory and Investment Behavior." American Economic Review, May 1963, 247-59.

17. ----- and Calvin D. Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior." American Economic Review, September 1968, 681-712.

18. Kamian, Morton I. and Nancy L. Schwartz, Dynamic Optimization. Amsterdam: North-Holland, 1981.

19. Myers, Stewart C. and Nicholas S. Majluf, "Corporate Financing and Investment Decisions When Firms Have Information that Investors Do Not Have." Journal of Financial Economics, June 1984, 187-221.

20. Nickell, Stephen J. The Investment Decisions of Firms. Cambridge: Cambridge University Press, 1978.

21. -----, "Comments." European Economic Review, May 1981, 57-59.

22. Scott, James H., "Bankruptcy, Secured Debt, and Optimal Capital Structure." The Journal of Finance, March 1977, 1-19.

23. Stevens, Guy V. G., "On the Impact of Uncertainty on the Value and Investment of the Neoclassical Firm." American Economic Review, June 1974, 319-36.

24. -----. "Internal Funds and the Investment Function: Exploring the Theoretical Justification of Some Empirical Results." Special Studies Paper #199, Federal Reserve Board, June 1986.

25. -----. "Internal Funds and the Investment Function." International Finance Discussion Paper #450, Federal Reserve Board, August 1993.

26. Stiglitz, Joseph E., "Some Aspects of the Pure Theory of Corporate Finance: Bankruptcies and Take-Overs." Bell Journal of Economics and Management Sciences, Autumn 1972, 458-82.

27. Van Horne, James C. Financial Management and Policy, 6th ed. Englewood Cliffs: Prentice-Hall, 1983.

28. Vind, Karl, "Control Systems with Jumps in the State Variables." Econometrica, April 1967, 273-77.

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~.)

III. 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 firm's possession of some degree of market power on the demand side; thus, we will assume a downward sloping demand curve, p = c - dQ, with c and d positive constants. Making both substitutions, the net revenue term becomes a quadratic function of K (|Alpha~, |Beta~ |is greater than~ 0): pQ - wL = |Alpha~K - |Beta~|K.sup.2~. (6)

Finally, for the model examined in this section we make the simplest assumption for the interest rate function consistent with the requirement that it be upward sloping: r(D, K) = |Rho~ + |Psi~D. Since bankruptcy risk is more properly a function of the debt/assets or debt/equity ratio, rather than the level of debt alone, this assumption is relaxed in the next section. Making the indicated substitutions, equation (5) becomes:

|Mathematical Expression Omitted~,

subject to DIV(t) |is greater than or equal to~ 0, D(t) |is greater than or equal to~ 0, where |Mathematical Expression Omitted~.

The objective becomes to maximize the integral by choosing optimal paths of capital and debt. Following the results in control theory incorporating constraints on state variables in, for example, Arrow and Kurz |2~ or Kamian and Schwartz |18~, one determines the optimal solution by forming a Lagrangean expression (L) made up of the normal (current value) Hamiltonian (H) plus multiplier expressions for the inequality constraints.(5) Thus,

L = H + ||Mu~.sub.1~DIV + ||Mu~.sub.2~D, (8)

where:

|Mathematical Expression Omitted~,

||Mu~.sub.1~(t), ||Mu~.sub.2~(t) are the multiplier functions for the inequality constraints; ||Lambda~.sub.1~(t), ||Lambda~.sub.2~(t) are the costate variables; and

||Mu~.sub.1~(t)DIV(t) = 0, ||Mu~.sub.2~(t)D(t) = 0, ||Mu~.sub.1~(t), ||Mu~.sub.2~(t) |is greater than or equal to~ 0.

Substituting for H and DIV:

|Mathematical Expression Omitted~.

An optimal path for the firm's capital stock and debt must satisfy the following necessary conditions:

|Delta~L/|Delta~I = 0 = -q(1 + ||Mu~.sub.1~) + ||Lambda~.sub.1~ (10)

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~,

with, as noted above, ||Mu~.sub.1~(t), ||Mu~.sub.2~(t) |is greater than or equal to~ 0, ||Mu~.sub.1~DIV = 0 = ||Mu~.sub.2~D.

Irrespective of whether the constraint on dividends is binding, the equations have important implications for the relationship between the optimum capital stock and debt. Equations (10) and (11) imply that ||Lambda~.sub.2~ = -||Lambda~.sub.1~/q. Although not important, assume for simplicity that q is not a function of time; then |Mathematical Expression Omitted~. In this case, the left hand side of (12) equals -q times the left hand side of (12) equals -q times the left hand side of (13). From this we have:

(1 + ||Mu~.sub.2~)(|Alpha~ - 2|Beta~K) - ||Lambda~.sub.1~|Delta~ = q(1 + ||Mu~.sub.1~)(|Rho~ + 2|Psi~D) - q||Mu~.sub.2~. (14)

It can be shown, further, that ||Mu~.sub.2~ is always zero--i.e., even in the unconstrained problem it turns out that debt cannot violate the nonnegativity constraint.(6) Finally, since it is also true that (1 + ||Mu~.sub.1~) = ||Lambda~.sub.1~/q, equation (14) simplifies to:

|Alpha~ - 2|Beta~K = q(|Rho~ + |Delta~ + 2|Psi~D). (15)

Solving for the capital stock, equation (15) becomes:

K(t) = ||Alpha~- q(|Rho~ + |Delta~ + 2|Psi~D(t))~/2|Beta~. (16)

Except for the term containing D(t), this equation is a familiar variant of the neoclassical equation for the firm's optimal capital stock; q(p + |Delta~) is the cost of capital for this latter model. It is also clear from (16) that it must be the case that |Alpha~ - q(|Rho~ + |Delta~) |is greater than~ 0, or K(t) can never be positive. From equation (16) we can also see a familiar implication of the Jorgensonian theory of investment where no dynamic adjustment costs are present: depending on the level of debt, equation (16) may imply a jump in the stock of capital. Instead of the usual differential equation in K(t) encountered in control theory, equation (16) is a degenerate equation linking (only) optimal levels of K(t) and D(t).

The system at any point in time is completed by the inequality constraints DIV |is greater than or equal to~ 0 and D |is greater than or equal to~ 0, and the equation for dividends (the sources and uses of funds identity). Except when a jump occurs, which is discussed below, the dividend expression is:

|Mathematical Expression Omitted~.

Of the four possible combinations of dividends and debt greater or equal to zero, two can be ruled out as either impossible or irrelevant. The two cases D |is greater than~ 0, DIV |is greater than~ 0 and D = 0, DIV = 0 are impossible, as shown formally in Stevens |25, Appendix, section I~. Thus, if D |is greater than~ 0, the only possible case is DIV = 0: since the interest rate on debt is always higher than the firm's discount rate, it would be suboptimal for the firm to pay dividends with any debt outstanding. In this case, unless a jump occurs, the system becomes equation (16) and equation (17) holding with an equality; these are sufficient to determine the optimal path of K(t) and D(t). The other possible case is D = 0 and DIV |is greater than~ 0. It turns out that this is the long-run stationary equilibrium |25, Appendix, section IV~. Using equation (16) and setting debt equal to zero, one derives the long-run capital stock, K* = ||Alpha~ - q(|Rho~ + |Delta~)~/2|Beta~. Given that |Mathematical Expression Omitted~ in this state, equation (17) is sufficient to determine equilibrium dividends.

Investment and Debt: the Initial Jump

To examine the dynamic process of investment, debt expansion and reduction, and dividend payments, let us consider the effects of an exogenous shock to the system. Suppose that initially the firm is in a state of long-run equilibrium, with a stationary capital stock and debt equal to zero, as discussed above. Consider the effects of an upward shift in the firm's demand curve--causing a discrete upward shift in the parameter |Alpha~ to |Alpha~* in equation (15) or (16).

Since it can be proved that equation (16) holds at every point in time, whether there is a jump or not, the first point to notice is that either D or K or both will have to undergo a discrete jump in order that the equation be satisfied.(7) Further, at the instant of a jump, the sources and uses of funds equation (17) becomes particularly simple. Since the time period covered by the jump is infinitely short, it is intuitively clear that the net impact of flows of profits, depreciation, or interest payments is zero. (For a rigorous proof, see Appendix, section III of Stevens |25~.) Incorporating the assumption of no new equity financing and the finding that dividends are zero during the jump, all the terms in equation (17) drop out except those in |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~--i.e., in the absence of other inflows and outflows during the jump, the sources and uses of funds identity reduces to the statement that the value of the change in capital must equal the value of new debt floated. Replacing these derivatives by the discrete changes that occur during the jump, equation (17) reduces to: D+(0) - D-(0) - q|K+(0) - K-(0)~ = 0, (18)

where the "+" and the "-" superscripts refer to the levels of the indicated variable just after and before the jump. Using (18) to substitute for the variable D+(0), one can then use equation (16) to determine the optimal jump for the capital stock, K+(0). One can show further that, because of the concavity of the relevant functions, there can be a jump only at time zero.(8) The optimal jump and subsequent developments are depicted in Figure 1. In the figure, the capital stock, K(t), is measured along the horizontal axis. The level of debt and various functions of the capital stock are measured along the vertical axis; for simplicity, the price of a unit of capital stock, q, will be set at 1 at every point in time.

The firm is assumed to be initially at a point of long-run equilibrium, point |E.sub.0~ on the K axis: a point with no debt and an equilibrium capital stock of K-(0). As an equilibrium point, it must satisfy the optimality condition (16)--plotted in the figure as the straight line MRMC(|Alpha~) (which intercepts the D axis at ||Alpha~ - (|Rho~ + |Delta~)~/2|Psi~, with a negative slope of -|Beta~/|Psi~). (Recall that q is set to 1.)

Consider the effects of an increase from |Alpha~ to |Alpha~* of the intercept of the marginal revenue product of capital--a shift of the MRMC curve parallel to itself to the northeast. The new (short-term) equilibrium is determined by the intersection of the MRMC(|Alpha~*) curve and the sources and uses of funds identity (18). Since equation (18) shows that changes in debt and capital during the jump must be proportional to each other, and given the assumption that q = 1, equation (18) is drawn as a 45 degree line through the original equilibrium point |E.sub.0~. The optimal jump is determined by the intersection of the 45 degree line and the MRMC(|Alpha~*) curve, at the new (short-run) equilibrium |E.sub.1~ (K+(0), D+(0)); this implies an optimal jump of K+(0) - K-(0) for capital and D+(0) for debt.

Determining the Optimal Paths of Capital and Debt after the Jump

Unlike the original Jorgenson model, the determination of the optimal jump is not the end of the story, but only its beginning. The key to the movement to the long-run equilibrium is that the firm's financial resources change over time endogenously as the firm operates and generates profits. These profits are linked to further capital stock growth, changes in debt, and dividend payouts through the more general flow of funds constraint (17).

In Stevens |24~ considerable space is devoted to solving the system defined by equation (16) and the differential equation (17), and describing the dynamic paths implied for K(t) and D(t) after the jump. However, the important points about the path can be demonstrated without the full derivation. First, the equilibrium condition (16) still holds at every point along the dynamic path and implies a negative linear relationship between capital and debt; in (K, D) space in Figure 1, this adjustment path falls along the MRMC (|Alpha~*) curve between |E.sub.1~ and the new long-run equilibrium, E*, where K* = ||Alpha~* - q(|Rho~ + |Delta~)~/2|Beta~ and D* = 0.

Taking the derivative of both sides of (16), we get:

|Mathematical Expression Omitted~.

Using this expression to substitute for |Mathematical Expression Omitted~ in (17), and noting that DIV(t) equals zero as long as D(t) is greater than zero, equation (17) becomes:

|Mathematical Expression Omitted~.

From the optimality conditions, the right hand side of (20)--the firm's profits after interest and depreciation deductions--must be positive on the segment between |E.sub.1~ and E*; hence |Mathematical Expression Omitted~ must be positive along the optimal path and, from (19), |Mathematical Expression Omitted~ must be negative. Thus, over time, starting from point |E.sub.1~, the capital stock, K(t), will increase and debt will decrease along the optimal path--the segment |E.sub.1~E*. At E* the long-run equilibrium will be reached as debt falls to zero.

The Investment Function and Cash Flow

A key aim of this paper is to examine the firm's investment function in the light of the empirical results of Coen |5~, Artus et al. |3~, and others, which successfully linked the speed of adjustment to cash flow. Using equation (16), it is possible to relate the firm's investment both to cash flow and to the "gap" between actual capital and the long-run equilibrium level, K* (defined above):

K* - K(t) = q|Psi~D(t)/|Beta~. (21)

Differentiating (21) leads to the same linear relationship between |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ as derived above for equation (19). Combining equation (19) with equation (21) leads to: |Mathematical Expression Omitted~.

Equation (22) shows that the investment function can be written as a flexible accelerator with a variable speed of adjustment--the latter equal to the absolute value of the percentage rate of change of debt.

One can go further, however, and, using equation (20), relate the above speed of adjustment to the firm's cash flow.(9) Note that the right hand side of (20) equals net revenues (|Alpha~K - |Beta~|K.sup.2~) minus depreciation charges (q|Delta~K), minus interest costs on the debt, (|Rho~ + |Delta~D)D. The right hand side, therefore, is a measure of net profits; denote it by |Pi~(t). Further, since |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ are linearly related as shown in (19), the left hand side of (20) can be expressed as a linear function of |Mathematical Expression Omitted~. Thus, equation (20), a version of the sources and uses of funds constraint, reduces to:

|Mathematical Expression Omitted~.

Using (23) to substitute for |Mathematical Expression Omitted~ in the investment function (22), we derive a final investment function that features a variable speed of adjustment that is a function of the firm's net profits:(10) |Mathematical Expression Omitted~.

We have in equation (24) an investment function that is remarkably similar to the suggestive, but ad hoc, formulation of equation (1) used successfully in the empirical work noted above in section I. For the relevant range, optimal investment can be described by a flexible accelerator with a variable speed of adjustment. Moreover, the speed of adjustment is a function of the firm's level of profits, proportional to the ratio of net profits to the level of outstanding debt.

IV. A More Realistic Model: Introducing Corporate Taxes and a Borrowing Rate a Function of the Debt/Assets Ratio

In this section we will see that the general lessons of the previous section carry over to more realistic models. One of the most appealing modifications of the preceding model is to let the firm's borrowing rate be a function of the debt/equity or the debt/assets ratio, rather than the level of debt alone. We have alluded to the risk of bankruptcy as the preferred theoretical reason supporting the upward sloping supply curve of debt. With this justification, the probability of the firm falling into the bankruptcy state, where it cannot cover its interest costs, can be shown to be a function of the debt/assets ratio and not of the absolute level of debt |20; 22; 26~.

Despite this more realistic supply schedule for debt, without more, the optimal long-run level of debt would still be zero. An obvious modification that assures a positive long-run debt/assets ratio is the introduction of a corporate income tax along with the usual deductibility of interest.(11) These two new assumptions result in predictable changes in the problem of section III, equations (5) through (16). The borrowing rate, r(t), will now be a function of the debt/assets ratio, |Phi~:

r(t) = |Rho~ + |Psi~|Phi~ = |Rho~ + |Psi~|D(t)/qK(t)~. (25)

Corporate taxes reduce potential dividends payable to stockholders. We shall assume the simplest possible case: a constant tax rate, |Tau~, assessed on total revenue minus interest and depreciation charges:

Tax = |Tau~||Alpha~K - |Beta~|K.sup.2~ - (|Rho~+ |Psi~|Phi~)D - |Delta~qK~. (26) Incorporating these modifications into equation (7), the expression for the value of the firm that is to be maximized, leads to the following modified Lagrangean expression:(12)

|Mathematical Expression Omitted~

subject to: K = I - |Delta~K, ||Mu~.sub.1~(t)DIV(t) = 0, and ||Mu~.sub.1~(t) |is greater than or equal to~ 0. The new necessary conditions for an optimum are: |Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~,

and with,

|Mathematical Expression Omitted~.

As was the case for the preceding model, the costate variables can be eliminated from equations (28) to (31) and, irrespective of whether the constraint on dividends is binding, one can solve for the capital stock in terms of the determinants of the firm's marginal revenue product (|Alpha~,|Beta~) and its marginal cost of capital |q(|Rho~ + |Delta~ + 2|Psi~|Phi~ - |Psi~||Phi~.sup.2~)~:

K(t) = ||Alpha~- q(|Rho~ + |Delta~ + 2|Psi~|Phi~ - |Psi~||Phi~.sup.2~)~/2|Beta~. (33)

The solution of this system is qualitatively quite similar to that of the preceding section. Equation (33) shows that shifts in |Alpha~ or any other parameter of the model will lead to jumps in K(t) and D(t)--and the ratio of the two, |Phi~. Finally, one can demonstrate that if |Phi~ is greater than its long-run equilibrium (|Phi~*), DIV falls to zero; moreover, when the debt/assets ratio is worked down to |Phi~*, both debt and capital attain their long-run equilibrium values and dividends becomes positive |25, Appendix, section II~. Similar to the approach taken in the earlier model, one can determine the long run equilibrium values K*, |Phi~*, and D* by examining the solution to the system when ||Mu~.sub.1~ becomes zero (dividends become non-negative). Noting that ||Lambda~.sub.1~ and ||Lambda~.sub.2~ are in this case equal to q and -1, respectively, from equations (30) and (31) we derive the following:

|Phi~* = D*/qK* = |Tau~|Rho~/(1 - |Tau~)2|Psi~, (34)

K* = {|Alpha~ - q||Rho~/(1 - |Tau~) + |Delta~ - |Psi~||Phi~*.sup.2~~}/2|Beta~. (35)

We will assume that the parameters |Rho~, |Tau~, and |Psi~ take on values such that |Phi~* is less than 1.(13)

Since we are primarily interested in the form of the associated investment function, we can be brief about the dynamic paths of the endogenous variables. Assuming once again an upward shift of |Alpha~ to |Alpha~* for the system initially in long-run equilibrium, again there must be a jump in K and D to satisfy equation (33); and again the modified sources and uses of funds equation (18) will apply during the jump, providing a second equation to determine both K and D. Since D and K increase in proportion, the debt/assets ratio increases above the long-run equilibrium level |Phi~* and dividends fall to zero. From this point, after the initial jump, the dynamics of the system are governed by equation (33) and the modified dividend equation (32). Since both equations are nonlinear because of the debt/assets ratio, it is impossible in this case to derive an explicit solution for the paths of debt and capital. However, as shown for the previous model, capital increases monotonically from its initial value to the long-run equilibrium K*; moreover, the debt/assets ratio, |Phi~, decreases monotonically to |Phi~*.

It should be noted that this model, like the previous one, leads to ever-increasing rates of investment, |Mathematical Expression Omitted~, along the optimal path. The monotonic fall of the debt/assets ratio, combined with the increasing generation of after-tax profits, leads to continuously increasing funds for investment.

A New Relationship between Investment and Financial Variables

Let us once again return to the ultimate question: How do the complications introduced in this particular model affect the validity of the variable speed-of-adjustment investment function developed by Coen and others? As might be suspected, the nonlinearities introduced by making the debt/assets ratio the key variable in the interest rate function severely complicate the form of the investment function. Nevertheless, we still have the elements that link the variables |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and after-tax profits, |Pi~: the dividend equation (32) and the fact that optimal dividends are zero during the period when net investment is positive. It should be noted again that equation (32) holds only after the initial jump. In deriving the investment function, I shall take the same approach as in the preceding section: first deducing the level of capital in terms of its deviations from the long-run equilibrium, K*, and then differentiating this expression with respect to time. Equation (33) relates the capital stock at any given time to a single endogenous variable: in this case the debt/assets ratio, |Phi~. Subtracting the expression for K(t) in (33) from K*, defined in equation (35), gives:

2|Beta~(K* - K)/q = 2|Psi~|Phi~ - |Tau~|Rho~/(1 - |Tau~) - |Psi~(||Phi~.sup.2~ - ||Phi~*.sup.2~). (36)

From the expression (34) for the optimal debt/assets ratio, |Phi~*, it can be seen that the second term of the right hand side of (36) equals -2|Psi~|Phi~*, so K* - K may be expressed compactly solely as a function of the difference between the actual and the long-run equilibrium debt/assets ratio:

K* - K = q|Psi~|2(|Phi~ - |Phi~*) - (||Phi~.sup.2~ - ||Phi~*.sup.2~)~/2|Beta~. (37)

Differentiating this nonlinear function with respect to time leads to one form of the investment function:

|Mathematical Expression Omitted~.

This is obviously more complicated than the simple relation, |Mathematical Expression Omitted~, derived for the previous model. However, after the initial jump, the dividend equation still leads to |Mathematical Expression Omitted~, where |Pi~ is now defined as after-tax profits. Substituting |Mathematical Expression Omitted~, equation (38) becomes:

|Mathematical Expression Omitted~.

Dividing (39) through by the expression for K* - K, we arrive at a significantly more complicated form of the variable speed-of-adjustment investment function, but one which retains the key property that the speed of adjustment depends on the firm's profits:

|Mathematical Expression Omitted~.

V. Conclusions and Caveats

This paper demonstrates that it is possible to provide theoretical support for the eclectic investment functions tested successfully by Coen |5~, Artus et al. |3~, and others. Models of the firm featuring intertemporal profit maximization subject to increasing costs of external debt lead to investment functions that, in important ways, are close to those estimated empirically. In particular, the theoretically-implied investment functions are a combination of the concepts of a flexible accelerator and a variable speed of adjustment, with the latter dependent on the supply of internal funds.

The dynamic adjustment mechanism implied by investment functions of this class is unusual with respect to both its cause and pattern. Unlike the typical dynamic-cost explanation of investment over time, costs in the above models are static, independent of the rate of change of any variable. However, borrowing costs are increasing functions of variables--debt or the debt/assets ratio--that change dynamically because of the firm's operations. Thus, these static costs also change over time as a result of the firm's actions and, as the costs change, investment is generated.

Not only is the causal mechanism generating investment different, but so is the dynamic pattern of the investment. Rather than having the rate of investment decline monotonically as the firm's capital stock approaches the equilibrium level, investment in this model increases monotonically as an increasing supply of internal funds is divided optimally between debt reduction and investment. Although the models presented in this paper serve to provide theoretical underpinnings for this important class of empirical investment function, one must note the caveats. We have pointed out that the investment function does not hold for points where capital and debt jump. Moreover, the underlying structure of the theoretical model suggests extensions to more general models and investment functions. The class of investment functions examined in this paper depends on a model with a single dynamic element determined by the cost of outside debt and the supply of internal funds. Models with additional dynamic factors should imply different investment functions. An illustration is provided by a small modification of the first model studied in section III: letting the demand-curve parameter |Alpha~ be a continuous function of time, instead of a constant. It is easily shown that the same key equations (16) and (17) hold to determine the optimal paths of capital and debt. However, from (16) the capital stock now is a function of two dynamic variables, |Alpha~ and D; and investment, |Mathematical Expression Omitted~, equals |Mathematical Expression Omitted~, instead of the previous |Mathematical Expression Omitted~. Both terms now enter the variable speed of adjustment. Despite the presence of this second term, the key relationship between |Mathematical Expression Omitted~ and profits (|Pi~) is still operational, so the speed of adjustment will continue to be a function of the firm's cash flow. Now, however, cash flow will be only part of the adjustment story.

1. See Coen |5~, Gardner and Sheldon |10~, Eisner |8~, Artus et al. |3~, Fazzari et al. |9~ and, for earlier work, Greenberg |11~ and Hochman |14~.

2. However, the existence of significant costs associated directly or indirectly with the state of bankruptcy is still a matter of debate. (See Haugen and Senbet |12~.)

3. One can defend the assumption of all-equity financing by noting that with a perfect market for riskless debt (at a constant rate of interest), the firm's real decisions for investment and production are independent of whether financing is from equity or riskless debt.

4. The explicit incorporation of uncertainty and costs of bankruptcy into the firm's objective function would severely complicate the analysis, particularly in view of the sources and uses of funds constraint introduced into the model. Even if tractable, the analysis of such a model would probably not change the conclusion of this paper that financial variables can be shown to account for some of the variability in the speed of adjustment under some, but not all circumstances.

For discussions of the complications with intertemporal maximization under uncertainty, see Abel |1~, Nickell |20~, and Stevens |23~.

5. Later in this section and in more detail in the Appendix to Stevens |25~, we analyze the complications introduced when there can be jumps in the state variables, K and D. It is shown that equations (10) through (16), below, hold whether a jump occurs or not.

6. Probably the easiest way to show that ||Mu~.sub.2~ always equals zero is to prove that ||Mu~.sub.2~ = 0 when D = 0; thus the constraint is inactive at the boundary. In Stevens |25 Appendix, section I.B~, it is shown that D = 0 implies that DIV |is greater than~ 0; thus, when D = 0, ||Mu~.sub.1~ = 0; moreover, using the arguments developed in the text, equation (13) becomes: |Rho~ = |Rho~ + 2|Psi~D - ||Mu~.sub.2~. Since D = 0 by assumption, ||Mu~.sub.2~ = 0. That there is no tendency for debt to become negative can be seen intuitively as follows: negative levels of debt, like negative borrowing, is equivalent to lending; but because of the linear interest rate function, the rate of return on such lending is less than |Rho~. Hence, since paying dividends has the implicit return |Rho~, it will never be optimal to carry on such lending.

7. It is proved in the Appendix (section III) of Stevens |25~, following work by Vind |28~ and Arrow and Kurz |2~, that equation (16) continues to hold in the presence of a jump.

8. This is proved by Arrow and Kurz |2, Proposition 12, 57~.

9. It should perhaps be noted again that equation (20) and the implied relationship between cash flow and |Mathematical Expression Omitted~ does not hold during the jump (rather, equation (18) does).

10. Note that, from equation (20), the measure of profits, |Pi~(t), is defined as net of depreciation and interest expenses. Depreciation is subtracted because equations (22) or (24) are functions for net investment, not gross. For an equation for gross investment, one just adds |Delta~K(t) to equation (24). 11. The following treatment of taxation is meant only to be a suggestive way to get a positive long-run debt/assets ratio; a realistic approach would require, among a number of things, the integration of corporate taxation with the personal taxation faced by stockholders.

12. Since it was shown that ||Mu~.sub.2~(t) always equals zero (footnote 6), the term ||Mu~.sub.2~D that appeared in equation (9) is dropped.

13. Realistic values for these parameters in the U.S. context seem to lead to values for |Phi~* well under 1; for example, assume the interest rate (|Rho~) for a firm with no debt is .10; then, if the firm's borrowing rate were to double to .20 as the debt/assets ratio moved from 0 to 1, the implied value for |Psi~ would be .10. If one assumes, finally, that the tax rate, |Tau~, is .25, the implied value for |Phi~* is 1/6 = 17 percent.

It can be shown that if the firm starts from a long-run equilibrium at |Phi~* |is less than~ 1, the debt/assets ratio, |Phi~, will be |is less than~ 1 throughout the optimal path. (See Stevens |25~, Appendix, section II.B.)

References

1. Abel, Andrew B., "A Stochastic Model of Investment, Marginal q and the Market Value of the Firm." International Economic Review, June 1985, 305-22.

2. Arrow, Kenneth J. and Mordecai Kurz. Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins, 1970.

3. Artus, Patrick, Pierre-Alain Muet, Peter Palinkas, and Peter Pauly, "Economic Policy and Private Investment Since the Oil Crisis." European Economic Review, May 1981, 7-51.

4. Bischoff, Charles W. "The Effects of Alternative Lag Distributions," in Tax Incentives and Capital Spending, edited by Gary Fromm. Washington: Brookings, 1971.

5. Coen, Robert M., "The Effect of Cash Flow on the Speed of Adjustment." in Tax Incentives and Capital Spending, edited by Gary Fromm. Washington: Brookings, 1971.

6. Duesenberry, James S. Business Cycles and Economic Growth. New York: McGraw-Hill, 1958.

7. Eisner, Robert, "A Distributed Lag Investment Function." Econometrica, January 1960, 1-29.

8. -----. Factors in Business Investment. Cambridge: Ballinger, 1978.

9. Fazzari, Steven M., R. Glenn Hubbard, and Bruce C. Petersen, "Financing Constraints and Corporate Investment." Brookings Papers on Economic Activity, 1:1988, 141-95.

10. Gardner, Roy and Russell Sheldon, "Financial Conditions and the Time Path of Equipment Expenditures." Review of Economics and Statistics, May 1975, 164-70. 11. Greenberg, Edward, "A Stock Adjustment Investment Model." Econometrica, July 1964, 339-57.

12. Haugen, Robert A. and Lemma W. Senbet, "The Irrelevance of Bankruptcy Costs to the Theory of Optimal Capital Structure." Journal of Finance, May 1978, 383-94.

13. Hirschleifer, Jack, "On the Theory of Optimal Investment Decision." Journal of Political Economy, August 1958, 329-52.

14. Hochman, Harold M., "Some Aggregative Implications of Depreciation Acceleration." Yale Economic Essays, Spring 1966, 217-74.

15. Jensen, Michael C., and William H. Meckling, "Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure." Journal of Financial Economics, October 1976, 305-60.

16. Jorgenson, Dale W., "Capital Theory and Investment Behavior." American Economic Review, May 1963, 247-59.

17. ----- and Calvin D. Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior." American Economic Review, September 1968, 681-712.

18. Kamian, Morton I. and Nancy L. Schwartz, Dynamic Optimization. Amsterdam: North-Holland, 1981.

19. Myers, Stewart C. and Nicholas S. Majluf, "Corporate Financing and Investment Decisions When Firms Have Information that Investors Do Not Have." Journal of Financial Economics, June 1984, 187-221.

20. Nickell, Stephen J. The Investment Decisions of Firms. Cambridge: Cambridge University Press, 1978.

21. -----, "Comments." European Economic Review, May 1981, 57-59.

22. Scott, James H., "Bankruptcy, Secured Debt, and Optimal Capital Structure." The Journal of Finance, March 1977, 1-19.

23. Stevens, Guy V. G., "On the Impact of Uncertainty on the Value and Investment of the Neoclassical Firm." American Economic Review, June 1974, 319-36.

24. -----. "Internal Funds and the Investment Function: Exploring the Theoretical Justification of Some Empirical Results." Special Studies Paper #199, Federal Reserve Board, June 1986.

25. -----. "Internal Funds and the Investment Function." International Finance Discussion Paper #450, Federal Reserve Board, August 1993.

26. Stiglitz, Joseph E., "Some Aspects of the Pure Theory of Corporate Finance: Bankruptcies and Take-Overs." Bell Journal of Economics and Management Sciences, Autumn 1972, 458-82.

27. Van Horne, James C. Financial Management and Policy, 6th ed. Englewood Cliffs: Prentice-Hall, 1983.

28. Vind, Karl, "Control Systems with Jumps in the State Variables." Econometrica, April 1967, 273-77.

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Author: | Stevens, Guy V.G. |
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Publication: | Southern Economic Journal |

Date: | Jan 1, 1994 |

Words: | 6821 |

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