The classic dismissal Modigliani and Miller (1958) of the traditional view that there exists an optimal capital structure for a profit-maximiz- ing firm opened the floodgate of formal analytical as well as empirical studies on corporate capital structure for over this past three decades. Immediately, after the debut of the Modigliani-Miller propositions, how- ever, Durand (1959), and a few years later, Brewer and Michaelson (1965) questioned the realism and adequacy of the postulates behind the Modigliani-Miller analysis. But, soon after the appearence of those criticisms, a host of works (e.g., Hirshleifer (1966), Stiglitz (1969), Fama and Miller (1972), Baron (1974), Hamada (1969), Rubinstein (1973) surfaced to corroborate the celebrated Modigliani-Miller (1959), propo- sitions under more general and less restrictive conditions. In thier origi- nal work, Modigliani and Miller (1959), however, pointed out that if the firm's cost of borrowing fell short of investors' cost of borrowing, then value of the firm would increase with an increase in its debt. Baumol and Malkiel (1967) also contended that a firm was not leverage-indif- ferent if investors incurred transaction costs in arbitrage activities. A few years later, Stiglitz (1972) brought out that if debt capital was traded in markets in which equity investors held superior prospect on the firm's performance to that of the lenders to the firm, then a sufficiently high level of debt could induce a drop in the value of the firm. Soon also Rubinstein (1973) demonstrated that if security markets were partially segmented and if debt was traded in a separate market where traders were more risk averse than investors in the firm's equity capital, then value of the firm and its debt level were inversely related. Having in- troduced corporate income taxes, agency conflicts, bankruptcy costs, and information asymmetry, however, several authors such as Baxter (1967), Jensen and Meckling (1976), Kim (1978), Turnbull (1979, 1981), Lee and Barker (1977), Kraus and Litzenberger (1973), Scott (1969), Robichek and Myers (1965), and Bierman and Thomas (1972) have shown that there exists an optimum capital structure for a firm. The simple intuitive reason is that existence of taxes provide tax shelter and thus reduce cost of capital, but financial distress leading to bankruptcy and reorganization, information asymmetry, agency problems and the like impose additional costs, and these divergent pulls create a unique combination of debt and equity for which cost of capital becomes least and value of the firm most. More recently, Stulz (1988), Hirshleifer and Thakor (1989), Harris and Raviv (1988), Narayanan (1987), Myers and Majluf (1984), Heinkel and Zechner (1998), among many others, have extended the MM world to the forefront of research on capital structure and defined the extent of the validity of leverage irrelevance.

Merton Miller (1988), Franco Modigliani (1988), and Stiglitz (1988) have reexamined this issue in recent years obviously for its tremen- dous impact on the entire arena of corporate finance. Milton Harris and Arthur Raviy (1991), in the golden anniversary review article have brought the theory of capital structure in its relation to agency costs and conflicts, asymmetric information, output/input market interactions, corporate control considerations, and so on. Bhattacharya (1988) has explored the legacy of Miller and Modigliani, and Modigliani (1984) amplifies his thoughts on NMM -- past, present, and future. Against the backdrop of this outstanding stature of the Modigliani-Miller theory, we find that almost every aspect of this theoreical environment has been fully examined with possibly the exception of the study of this structure in a general equilibrium and in a dynamic framework, sus- tained by growth and capital accumulation. Most of the works cited thus far are couched in static analytical frameworks. Using a multiperiod model, however, Scott (1976) introduced the dynamic analysis on this issue, Later, Brennan and Schwartz (1978) used the dynamic stochastic optimization technique, adopted by Merton (1971), to analyze corpo- rate income taxes, valuation and optimal capital structure. In a recent work, Ghosh (1992) has generalized the Sau study (1969), conducted in the classical calculus of variations, and intertemporalized the Lintner works (1976) by optimum control theory a la Pontryagin (1962). In this work, we plan to extend the Ghosh model in general equilibrium wherein there are two firms producing net earnings, as Sau (1969) shows in case of one firm, by efficient utilization of debt and equity capital under constant returns to scale, and demonstrates that in com- petitive capital market optimal capital structure does in deed exist for each firm, and for the economy. Since the result is strikingly different from the Modigliani and Miller (MM) theory, an effort is made to com- pare this analysis with the classic paradigm of Modigliani and Miller. We also study in this paper the effects of resource allocations on the firms comprising the economy within the building blocks of compara- tive statics, and bring out some of the existing thoughts in relation to leverage. In section II, we sketch the theoretical model of the economy, and then attempt to deduce the results out of the strait-jacket of such a structure. In section III, we make an effort to compare our finding with the main result of Modigliani and Miller, to the exentitis feasible to do so, and then we make some observations for possible further entensions of the model under a vaiety of scenarios.

Consider an economy with two types of available capital—equity (E) and debt (D)--sustaining two firms, with different degrees of leverage, each producing net income under conditions of perfect competition. Assume that each net earnings production function exhibits constant returns to scale. More specifically, the functional relationships are de- fined as follows:

(1) X, =FE,D),

(2) X, = FE, D,)

where X,, and E, denote, respectively, net earnings and the (utilized) equity capital of the firm i (i= 1, 2), all measured in the same physical units'; D, represents debt capital of firm i, and it is expressed in nu- merical terms (underlying a value, of course, in the units in which X, and E,are denominated). Obviously, then:

(IN) x, / X,/D = (D/D).£(E,/D,). (2N) x, /X,/D = (D/D).£(E/D)), where f,(E,/D,)/F(CE/D, 1), £(0) = 0, (4) =4, (N(E/Di) > 0, fO(E/ D) <0 for 0 < E/D, <4. All these mean that both types of capital - equity and debt - are essential to each firm. This postulate is made to structure the building block of our analysis consistent with realistic situ- ations of firms using both capitals and thus avoiding the extremum of fully levered or fully unlevered conditions, and yet having different de- grees of leverage. Because of cost and risk minimization and portfolio diversification, firms often do in deed use less than perfect measures of leverage (that is, 0 < E/D, <4). Here we are making that point only. x, the earnings per unit of debt, is the function of equity debt ratio, and all the restrictions on this function simply signify that xirises with every rise in E/D,, but at decreasing rates.

Now, let the allocations of component capitals be as follows:

(3) E,+E,=E, and (4) D,+D,=D.

The relations (3) and (4) state that sum total of each capital in both firms equals that capital asset for the entire economy. The expressions (3) and (4) can easily be rewritten:

ai (3N) Di) (£1) 4 (22) (f2j;= and DD DD D (4N) Di, D2} D D

By algebratically solving the simultaneous equations (3N) and (4N) one gets the following:

Corporate Ceverage And Growth: A General Equifiénum Analysis

££ (30) Di D Di ang Ei _ Es Di D: E, E (40) D:D D D EE D D:

EE (10) — »P_D: Ei Xi ‘Er E }. fit ) D; D3; (20) E: po 5 Ei DD, Si D D D:

One can see now that x, is dependent upon not only its equity debtratio, but also on its relative leverage, Assume that firm 1 is rela- tively more levered (that is, (E,/D,) - (E,/D,) < 0). Soon after complet- ing the analytical structure, we will examine the implications of this postulated difference in leverage for these firms.

So, for now, let us build the remaining structure of the model. In the competitive conditions, as we all know, returns to each capital must be equal across firms, which then means:

(5) k, = PAN(E,/D,) = {N(B/D,)

(6) k, = P.{£(B/D,) - (E/D,).f,N(B/D,} = = {£(E/D,) - (EY D,).£,N(E,/D,)} where k,, and k, are the returns to equity and debt, respectively and P is the price of the first firm's product in terms of the second firm's product. From (5) and (6) then:

E (=) (7) jae a D Ei ke fey Di

, and

Fit) (3) p= 22. Ki) Since

Uk! ke) — LOLMO. 9 de /D) [fOr one can express, in view of the monotonic relation as above, that equity debt ratio is a direct function of component costs of capital. That is,

(0) E/D,=hikp/kz), hN(e)>0 for i= 1, 2.

Next, consider the dynamics of the economy. First of all, let us envisage the equity accumulation as follows:

(0) ay ab, dt where a is the rate of equity dilution (ifa > 0; ifa <0, itis a case ofnew equity issuance). Here we are assuming that out of the total carnings of the economy, x (where x = x, + Px,), [3/ Px, /x is the proportion of income reinvested into the economy, In other words, we assume that the second firm's income is fully absorbed for current consumption of the investors in the economy, and they retain the first firm's earnings for current investment (that is, equity build up) in the economy. We make this simplifying assumption without any loss of generality in true sense for simple income distribution within the economy amongst in- vestors”. Next, assume that debt grows exogenously at the rate b, and so:

a) Den di

Therefore differentiation of E/D with respect to time (t), and then the substitution of equations (10) and (11) into it results in the following expression:

ye D

E (12) = x(a Bie.

Equation (12) states that the rate of growth of the equity per unit of debt in the economy equals the rate of internal reinvestment (that is, the retained earnings) minus equity debt ratio times the rate of equity loss (or gain) plus the rate of growth of debt capital in the economy. Here (a + b). E/D measures the equity per unit of debt needed to keep the equity debt ratio maintained at its existing level. Since x, =(D,/D). £,(E,/D,), and £,\(.) > 0 and f,°(.) < 0, one can visualize now that d(E/ D)/dtrises initially, reaches its maximum value, and then drops with the increasing values of E/D. Our enquiry will soon be to find that value of E/D which will make economy-wide equity debt ratio and the firm- specific capital structure optimal in this general equilibrium framework. That is, given all these structural parameters of a general equilibrium, the economy must choose the time profiles of E/D, and D/D such that the following maximand is satisfied:

ry (13) max U= Jes x, at subject to:

D di

, E =x)-(a TOA initial (E/D) = (E/D)0, - a constant, and equations (LN), (2N), (3N) and (4N).

Here U represents the utility that is derived from absorption of the current earnings, c is the rate of time preference, and T is the end of time horizon under consideration. This utility function (U) with the fol- lowing characteristics, UN > 0 and UO < 0, (that is, utility increases at diminishing rates) which is well-founded in the literature Lintner (1976, 1963), Sau (1969), Willingford (1972), Shell (1989), to mention a few defines the optimality when it is maximized subject to the constraints under consideration.

A. The Results

Having the structural framework as exhibited, it is instructive for us to examine some to the equations expressing the basic underlying rela- tionships. First, postulate that the first firm is relatively more debt fi- nanced than the second firm. That is:

Now, a simple set of partial differentiations of (1N) and (2N) with respect to E/D yields:

(14 Oa SCD) ED) (E1/ Di)-(E2/ D2)

(15) Ox? + St ( E2/ D2)

O(E/D) — (E)/ Dj)-( B2/ Di)

Because of the postulated leverage ranking of firms ((E,/D,) < (E,/ D,)), itis clear from (14) and (15) that as equity level relative to debt level rises, the relatively more levered firm (first) shrinks and less lev- ered firm (second) expands. Alternatively, if debt relative to equity rises, the relatively more levered firm (first) expands and the relatively less levered firm shrinks. Note that we observe here the change in economy-wide E/D ratio, but the constancy in the firm-level leverage is intact. Since the firm's earnings change (which thereupon signifies the change in the values of the firms) without any change in debt equity ratio of the firm, one can easily state that capital structure of a firm is independent of its value, which is the inversion of the celebrated propo- sition of Modigliani and Miller [25a, b]. Let us dwell on this a bit more with a geometric structure. The diagram (Figure 1) exhibits the debt equity ratio of first and second firm as postulated already. Here, the original equity debt endowment defined by the co-ordinates of A pro- duces the outputs of the first and the second firm represented by the isoquants Iq’ (x, = 250 units, say) and 1Q°, (= 200, say). When the endowment changes from A to B (which means the same debt level. but an increase in equity capital), the first firm produces Iq) (x, = 150) and the second firm produces IQ), (x? = 300), as in this illusirative diagram. One can easily visualize through Figure 1 that if the equity debt endowment ratio moves from A to C (signifying thereby an in- crease in debt but no change in equity), first firm's earnings will rise and the less levered firm's earnings will fall. A further inspection of the diagram brings out the fact that a movement of equity debt ratio within the cone MAN increases the earnings of both the firms, and a move- ment of equity debt ratio within the cone CAM (cone BAN) brings about an increase (a decrease) in x, and a decrease (an increase) in x,. A movement of E/D along the ray AR simply means an equiproportionate increase the output of both the first and the second firm. In the existing literature, in the partial equilibrium framework, there is a long-held impression that growth implies a fater growth of the relatively levered firm and slower growth of the less levered firm. Here, then we are at odds with that claim. In fact, we find that growth may trigger all possible scenarios: growth of both firms, growth of one firm and the shrinkage of the other firm, depending upon the instrument of growth and leverage ranking of firm. We thus obtain the results obtained by Ghosh (1993, June) rehabilitated. One can goa step a step further at this point. Since we note that an increase in debt capital in the economy increases the growth of the (more) levered firm and in- duces a decay of the (less) levered firm, one can conclude that a leveraged buyout is an economic booster for a levered firm acquiring a levered firm, but the result may be weakening for a relatively less lev- ered firm. All these results, of course, presuppose the constancy of product prices, which, in view of (7) and (8), is implied by constant E,/ D, and k,/k,.

Note now that the insertion of (9) into (8) easily results in the following expression: (16) dp (Bi/ Di)-(B:/ Dz)

Uko/ ke) — [(B:/ D2) + ko! ke! D) (kp! ke) The postulated leverage of the firms immediately suggests then that the sign of (16) is positive, and that means that if the return on debt increases (decreases) more relative to the return on equity, price of the relatively levered firm's product rises (falls). If, however, we admit of changes in the returns on component capitals, then leverage levels of the firm change as a consequence, and that in turn accentuates the directions of change of the firm's earnings.

utility level is intertemporally maximized. Here we deal with this dy- namic maximization and determine the optimum capital structure for the economy and for individual firms. The maximization problem, as already pointed out, is as follows:

P max U = fe x2 dt subject to:

d=) D _ BOX) 744) 2 = - —,0¢ </ fe <teT, a Pi (a 5 <B(<l for O<tsT initial (E/D) = (E/D)o, and E/ Dj -r > (EVD ); «

The maximization problem is then reduced to the following Hamilto- nian a /a Pontryagin er a/ [28]:

E H=e"\(1-B)xta Be, a+b)— e“{(-B)xtal P (a+b) A! } where A(t) is the Lagrangean multiplier (shadow price of equity build- up). The necessary conditions for maximality are spelled out by the following equations":

ad Do (anta

(19 EDI. (HD -(E/D)

E . e)-(a+b)—, and dt (B,/Di-(B/ Dy) OD

Equation (18) describes the intertemporal path of the shadow price of equity accumulation, equation (19) portrays the time profile of equity debt mix for the economy, and equation (20) shows the functional rela- tion between shadow price of equity accumulation and cost ratio of component capital assets in condition of dynamic optimality. Now, note that since (E,/D,) < (E,/D,), P in equation (20) is an increasing func- tion of k,,/k,, (which is clear from equation (16)) and, therefore, spe- cific value of A uniquely determines k,/k,, which then uniquely deter- mines E,/D, and E,/D, by virtue of (9). Therefore, we have already obtained the following: P and (k,/k,) are positively (functionally) re- lated (as shown in equation (16)); for every k,/k,, E,/D, > E,/D,, when, of course, both the firms operate in the economy. Figure 2 exhibits these relationships, and, as (3N) shows that E/D is the convex combi- nation of E/Di's (i= 1, 2), E/D must lie between the equity debt ratios of these two firms, which is also depicted in Figure 2. However, this diagram also shows that for a given E/D (say, (E/D)*) if P #P.,,, (and correspondingly, (k,/k, #k,/k,),,,, in Figure 2), the economy slides into one-firm economy in which D,/D = 1 and D/D = 0, and that means that firm 2 only survives. Similarly, for the same equity debt ratio, (E/D)*, if PS P,,,, (and correspondingly, k,,/k,, (k,/k,),,,,)» D,/ » = land D,/D =0, and then only firm 1 operates. For different values of E/D then one can different P,,, and P.,,, values. The locus of those P iy and P,,., Values are the P,,, and P_ curves in Figure 3. Its coor- dinate axes are E/D and A, respectively. The two-firm economy obvi- ously must lie within the interior space between P_,, and P,,,, curves, and beyond this space are the zones for one firm operations. Since in the zone northwest of P,,, (inclusive of the P,,, curve) the dynamic optimality conditions a la Pontryagin [28] are:

(18.1) “ = {(atbte) - f\N(E\/D))}.A,

(191) MED) = = 5(EyD))-a+b+6) a and in the southwest zone of P optimality is defined by:

(inclusive of P,,, curve) the dynamic min

(18.2) D = (atb+c)a - fN(E/D) t

(19.2) 4B) 1 # dt D the maximality conditions of (18) and (19), along with the ones just brought out, can be pictured, as one can see in Figure 3, by the loci of da/dt = 0 and d(E/D)/dt = 0. The autonomous differential equations (18) and (19), (18.1) and (19.1), and (18.2) and (19.2) are portrayed through the phase diagram (Figure 3):

The curves dA/dt = 0 and d(E/D)/dt = 0 intersect at point S, in the illustrative diagram (Figure 3), and therefore, the coordinates of the point S defines the optimum value of (E/D) and i. Here (E/D)* and A* are the optimal values, and that means (E/D)* is the optima] economy- wide capital structure since at this equity debt mix the utlility is maxi- mum possible. As already noted, once optimum value of ? (the shadow price of equity accumulation) is determined, optimum k,/k, is deter- mined and that specifies the optimum firm?wise equity debt ratios. Let us dwell on this point a bit further.

Differentiation of (20)-—-that is A(t) with respect to time (t) and substitu- tion of (16) into that yields:

Q1) ko ted (at bt lk? ke)

di LB DIABA DI (BA DI Ha ke) HB D) Mkn! kad)