Leverage and asset allocation under capital market distortion.

Ghosh, Dilip K. ; Prakash, Arun J. ; Ghosh, Dipasri 等

I. INTRODUCTION

The criticism by Durand (1959) of the Modigliani-Miller (1958) results on capital structure has fostered two strands of thought in the literature. On the one hand, several pieces of research (e.g., Stiglitz (1969), Rubinstein (1973), Hamada (1969), Baron (1974), Fama and Miller (1972), and others) have proven that the Modigliani-Miller claims are valid even under more general conditions than the authors originally envisaged. However, in their original piece itself, Modigliani and Miller (1958) observe that if a firm's cost of borrowing is less than the investors' cost of borrowing, the value of the firm increases with increase in debt. Baumol and Malkiel (1967) argue that a firm is not leverage-indifferent if investors incur transaction costs in arbitrage activities. Later, Rubinstein (1973) also demonstrate that if security markets are partially segmented and if debt is traded in a separate market where traders are more risk-averse than investors in the firm's equity capital, the value of the firm and its debt level are inversely related. But a second line of research--by Kim (1978), Baxter (1967), Lee and Barker (1977), Scott (1977), Barnea, Haugen and Senbet (1981), Chen (1978), Chen and Kim (1979), and many others--has established more formally that unique level of optimum capital structure does indeed exist for a firm if market distortions caused by taxes, bankruptcy costs, agency problem, informational asymmetry, etc. are admitted of. In dynamic environment, sustained by equity accumulation and change in debt, Ghosh (1991) shows that optimum capital structure for a firm is unique even under frictionless and perfectly competitive capital market.

The existing literature has waxed eloquent on a related issue,--the issue involving various facets of allocation in both macroeconomic decisions at a firm level, and in macro-structures at economy-wide aggregate level,--sometimes in static framework, and at times in inter-temporal dynamic setting. Sharpe (1987), and Perold and Sharpe (1988, 1995) discuss strategies for asset allocation in an integrated framework. Grossman (1995), in his Presidential Address, examines dynamic asset allocation and efficiency of markets. Black and Litterman (1991) study the issue by combining investor views with market equilibrium. Scarf (1994), Hurwicz and Majumdar (1988), Craine (1988) explore allocation questions in dynamic environment through structured mathematical logic. Choi and Han (1991) attempt to ascertain wealth effect and macro uncertainty in allocation between stocks and bonds. Prakash, Dandapani and Karels (1988), and Anderson and Prakash (1990) examine allocation principle in capital budgeting context. The issue is brought out by Bolten and Besley (1991) under dynamic interaction of earnings and interest rates. Ghosh and Sherman (1993) employ a general equilibrium structure to analyze resource allocation in competitive capital market where this interaction has been played out in comparative static framework. Krasa and Yanelis (1994), Leibowitz et al (1994), Sarnat (1974), Scarf (1994), and Stapleton and Subrahmanyam (1977) analyze market imperfection in various paradigmic set-ups. Here, in this paper, an attempt is made to use a two-firm framework of general equilibrium with distortion in capital market to study how market imperfection caused by such distortion affects asset allocation, firm value, and component cost of capital under the assumption of different corporate leverage of the constituent firms.

It is shown, in Section II, that in imperfect capital market, a firm that is more levered in physical sense is not necessarily so levered in financial sense, and if that is the case, then the firm which is more debt (equity) financed may respond to a change in capital structure, cost of component capital, and price structure in the economy in perverse fashion. Corporate growth, discussed also by Gup (1980), and Roll (1973) somewhat differently, is brought out here in our analytical framework that follows, and at some point later we discuss the differences in the conclusions arrived here and earlier in the existing literature. Specifically, we show that the earnings of the more levered firm rises faster than the rise in debt/equity ratio if the firm is more levered in physical sense, and the least levered firm's rises at the slowest rate (or even drops), which confirms the long-standing results in the existing literature. However, if the firm is only more levered in physical sense, and not in financial sense, the existing results are compromised or even reversed. The rate of change in distortion is also a modifier, and therefore the existing literature needs a more critical re-examination for further corroboration or reversal of the hithertoaccepted findings.

II. THE ANALYTICAL STRUCTURE Consider an economy with two types of capital--debt capital (D) and equity capital (E) sustaining two firms, each producing a different output in condition of different capital market imperfection. Here, D represents the number of bonds (debt instruments), and E stands for the number of shares of common stock (equity instruments) outstanding in the economy. The structure of our economy is defined by the following sets of equations:

[a.sub.D1] [X.sub.1] + [a.sub.D2] [X.sub.2] = D, (1)

[a.sub.E1] [X.sub.1] + [a.sub.E2] [X.sub.2] = D, (2)

[a.sub.D1] [1.sub.D1] + [a.sub.E1] [1.sub.E1] = [P.sub.1] (3)

[a.sub.D2] [1.sub.D2] + [a.sub.E2] [1.sub.E2] = [P.sub.2] (4)

where

[r.sub.Di] = [A.sub.i][r.sub.D], (5)

and

[r.sub.E1] = [B.sub.i][r.sub.E], (6)

Here [a.sub.Di] and [a.sub.Ei] measure, respectively, units of debt and equity capital needed to produce one unit of output of firm i, [X.sub.i] represents the total net output of firm i (which can be construed as the earnings before interest in terms of product units) (1). The symbols [r.sub.Di,] and Ei r stand, respectively, for return on debt instrument (that is, interest rate), and return on equity capital for firm i, and Pi the price of the product of firm i (i = 1, 2) and rDi is the cost of debt in the absence of distortion in debt market. The [i.sub.th] firm's rate of interest ([r.sub.Di]) and rate of return on equity ([r.sub.Ei)] are assumed to be, respectively, proportional to their respective market counterparts [r.sub.D] and [r.sub.E] with constant of proportionality, Ai, and Bi. To simplify the analytical structure, we assume [B.sub.i] = 1. Note that [a.sub.Di] and [a.sub.Ei] are not assumed constant, as differentiations and later discussions will clearly spell that out, and so these should not be construed as fixed coefficients even though they may appear so at first sight, particularly to those not so familiar with literature using this approach to general equilibrium, developed by Amano (1964), Jones (1965, 1971), and many others since then.

To simplify the analysis, we assume that [B.sub.i] = 1, and thus [r.sub.E1] = [r.sub.E2] (=[r.sub.E], say), which means that in equity market rate of return across firms is identical. If we further assume that [A.sub.2] = 1, it is simply postulated that for the first firm cost of debt is higher (lower) if [A.sub.1] >1 (<1). Obviously, equations (1) and (2) describe the allocations of capital between firm 1and firm 2. Equations (3) and (4)--the dual to capital asset allocation--represent the price cost structure for the firms. More lucidly, equation (1), for instance, shows how many debt instruments are used to produce [X.sub.1] and [X.sub.2.] Thus, [a.sub.D1][X.sub.1] and [a.sub.D2][X.sub.2] measure the number of bonds issued by firm 1, and firm 2, and the sum of these numbers is equal to the total number of debt instrument (D) existing in the economy, Equation (2) can be interpreted exactly in the same way for equity capital allocation between the firms. Equation (3), as already noted in Baxter (1967), in competitive condition should read as follows:

[a.sub.D1][r.sub.D] + [a.sub.E1][r.sub.E] = [P.sub.1],

If market is perfectly competitive, the rate of return on each capital is identical across firms. That is, [r.sub.D] and [r.sub.E] and the rates of return, respectively, on debt capital and equity, and these rates are will then be firm-independent. Then, in above equation, [a.sub.D1][r.sub.D] is the cost of debt and [a.sub.E1][r.sub.E] is the cost of equity per unit of net output of firm 1. Since perfectly competitive equilibrium is a zero-profit situation, unit cost is equal to unit price, and that means: [a.sub.D1][r.sub.D] + [a.sub.E1][r.sub.E] = [P.sub.1]. Thus, it is the expression of zero profit cost allocation for firm 1. The relation (5), however, exhibits that rD1 is not equal to [r.sub.D2]; it is a situation of plurality of interest rates, as in the work of Modigliani and Miller (1958). If [A.sub.1] > 1, firm 2 has higher cost for debt, and if [A.sub.1] < 1, firm 2 has lower cost of debt than firm 1. Thus, the parameter Ai measures the extent of interest rate differential.

In this structure of capital market distortion, let us examine the effects of changes in debt ([D.sub.i]), equity ([E.sub.i]), prices ([P.sub.i]) of the firm products, and market distortion ([A.sub.i])--the parameters of the model--on the growth (decline) of firms, component cost of capital, and value. Here, let the asterisk as superscript denote the percentage change in a parameter or variable. That is, [D.sub.*/dD/D,] [a.sub.*.sub.D1] = [da.sub.D1]/[a.sub.D1] denote percentage change in debt (D), percentage change in [a.sub.D1], respectively, and so on. Total differentiation of equations (1) through (5) with some algebraic manipulations then results in the following expressions:

[[alpha].sub.D1] [X.sup.*.sub.1] + [[alpha].sub.D2] [X.sup.*.sub.2] = [D.sub.*] - ([[alpha].sub.D1][a.sup.*.sub.D1] + [[alpha].sup.*.sub.D2][a.sub.D2], (7)

[[alpha].sub.E1] [X.sup.*.sub.1] + [[alpha].sub.E2] [X.sup.*.sub.2] = [E.sub.*] - ([[alpha].sub.E1][a.sup.*.sub.E1] + [[alpha].sup.*.sub.E2][a.sub.E2], (8)

[[beta].sub.D1][r.sup.*.sub.D1] + [[beta].sub.E1][r.sup.*.sub.E] = [P.sup.*.sub.1] - ([[beta].sub.D1][a.sup.*.sub.D1] + [[beta].sub.E1][a.sup.*.sub.E1] (9)

[[beta].sub.D2][r.sup.*.sub.D2] + [[beta].sub.E2][r.sup.*.sub.E] = [P.sup.*.sub.2] - ([[beta].sub.D2][a.sup.*.sub.D2] + [[beta].sub.E2][a.sup.*.sub.E2] (10)

[R.sup.*.sub.Di] = [A.sup.*.sub.I] + [r.sup.*.sub.D], (11)

where [[alpha].sub.Di] = [a.sub.Di]Xi/D, [[alpha].sub.Ei] [equivalent to] [a.sub.Ei][X.sub.i]/E are the distributive allocations of debt and equity capital to the i-th firm, and [[beta].sub.Di] [equivalent to] [a.sub.D1] [r.sub.Di]/[P.sub.i] and [[beta].sub.Ei] [equivalent to] [a.sub.Ei][r.sub.E]/[P.sub.i] are the distributive shares of these capital assets to the value created in the i-th firm (i = 1, 2). Note a few interesting features now. The determinant of the coefficients of [X.sub.*.sub.I] in equations (7) and (8), given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines the relative leverage of the firms. If [absolute value of [alpha]] > 0, -(which is true if and only if ([a.sub.D1]/[a.sub.E1]) > ([a.sub.D2]/[a.sub.E2])),--obviously the first firm has more debt equity ratio in numerical terms, and that means it is relatively more levered in physical sense; similarly, [absolute value of [alpha]] > 0 signifies that the second firm is more levered in physical sense. Substituting equation (11) into equations (9) and (10), then the determinant of the coefficients of [r.sup.*.sub.D1] and [r.sup.*.sub.E], is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the absence of distortion in debt market (where [A.sub.i] = 1), the signs of [absolute value of [beta]] and [absolute value of [alpha]] are exactly the same (and in such a situation [absolute value of [alpha]] [absolute value of [beta]] > 0), which means the sign of [absolute value of [beta]] also defines the relative leverage of the first and the second firm. However, if [A.sub.1] > 1, the sign of [absolute value of [alpha]] may not necessarily be identical with that of [absolute value of [alpha]] [absolute value of [beta]] . Since ? is a stochastic matrix (that is, the sum of each row equals 1), one can easily find that: [absolute value of [beta]] = [[beta].sub.D1] - [[beta].sub.D2] = [[beta].sub.E2] - [[beta].sub.E1], from which then it follows that if [absolute value of [beta]] > 0, the first firm is more levered in financial sense (and if the sign of [absolute value of [beta]] is negative, the second firm is more levered in the same financial sense). This brings out the point that in the absence of distortion in the debt capital market (that is, for [A.sub.i] = 1), a firm is uniquely levered, no matter if numerical debt equity ratio or distributive share of debt capital relative to equity capital (that is, value sense of leverage measure) is considered. As already noted, if [A.sub.1] > 1, and the first firm is less levered in physical sense, it may nonetheless be more levered in financial sense compared with the second firm. Let us next introduce a few more basic ingredients of general equilibrium structure. First, the cost-minimization condition on the usage of debt and equity for firm i (i = 1, 2), defined by:

-[da.sub.D1]/[da.sub.Ei] = [r.sub.E]/[r.sub.Di],

yields upon some algebraic manipulations:

[[beta].sub.Di][a.sup.*.sub.Di] + [[beta].sub.Ei][a.sup.*.sub.E] (12)

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

is the elasticity of substitution between debt and equity capital in firm i, as shown in Ghosh and Sherman (1993), one may easily now find the following expressions by the solving the simultaneous equations (12) and (1 3) for i =1 first, and then for i = 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The substitutions of these [a.sup.*.sub.ji]'s (j = D, E; i = 1, 2) into (7) and (8) give rise to the relations (14) through (17):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[[beta].sub.D1][r.sup.*.sub.D] + [[beta].sub.E1][r.sup.*.sub.E] = [P.sup.*.sub.1] - [[beta].sub.D1][A.sup.*.sub.1] (16)

[[beta].sub.D2][r.sup.*.sub.D] + [[beta].sub.E2][r.sup.*.sub.E] = [P.sup.*.sub.2] - [[beta].sub.D2][A.sup.*.sub.2] (17)

From (16) and (17) one can immediately derive that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

An algebraic manipulation further gives rise to the following relationships:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Here one can note that at constant distortions,--that is, if [A.sup.*.sub.i] = 0, a change in [P.sub.1]/[P.sub.2] (in percentage terms) causes a more than equi-proportionate change in E Di r / r ; but the changes may not be uni-directional. If [absolute value of [alpha]] < 0, and [absolute value of [beta]] > 0, then in spite of the fact that the first firm is less levered in physical sense, higher cost of debt for the first firm may induce a situation in which an increase in the relative price of the physically levered firm serves to lower the return to the debt-holders relative to the returns to equity-holders in both firms. Next, consider the effects of the change in distortions at constant [P.sub.i]'s. As (18) and (19) show, an increased premium paid to debt-holders in the firm in which debt receives smaller distributive shares must raise cost of debt in both industries relative to the returns on equity capital. From (16) and (17) it is evident that an increase at [A.sup.*.sub.i] at constant [P.sub.i]'s works like a decline in [P.sub.i]'s in affecting the returns on debt and equity. It is instructive, however, to examine how the results just brought out will be influenced by the non-constancy of the [P.sub.i]'s. From (16) and (17) one can derive the following expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Equations (21) and (22) reveal that returns on debt capital in the first firm in real terms are increased (decreased) by the relative increase in the price of the product of the first firm ([([P.sub.1]/ [P.sub.2]).sup.*] [equivalent to] [P.sup.*.sub.1] - [P.sup.*.sub.2]) and/or decreased by the relative increase in the distortion in the first firm ([([A.sub.1]/ [A.sub.2]).sup.*] [equivalent to] [A.sup.*.sub.1] - [A.sup.*.sub.2]) if /[beta]/ > 0. The question immediately is to ascertain the relative strength of these two divergent pulls. A close examination of (19) and (20) shows that since 0 < [[beta].sub.Di] < 1, impact of relative price change is stronger than relative change in distortion on the change in the real rate of return on debt capital, In more simple terms, one may state that if the price of the first firm relative to that of the second firm rises by 5 percent while the debt market distortion for the first firm relative to that of the second firm also rises by the same 5 percent, bondholders will benefit in real terms in both firms if the first firm is financially more levered. Now, from (15) and (16), we can easily show:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At constant returns to debt and equity capital ([r.sup.*.sub.Di] = 0, [r.sup.*.sub.E] = 0) and at the invariant levels of distortion ([A.sup.*.sub.1] = 0). Obviously, if [D.sup.*] > [E.sup.*] then [X.sup.*.sub.1] > [D.sup.*] and [X.sup.*.sub.2] < [E.sup.*] provided first firm is more numerically (physically) levered. If [D.sup.*] = [E.sup.*] then [X.sup.*.sub.1] > [D.sup.*] and [X.sup.*.sub.2] > [E.sup.*] [D.sup.*] < [E.sup.*], and [X.sup.*.sub.1] < [D.sup.*] E, [X.sup.*.sub.1] < [D.sup.*] and [X.sup.*.sub.2] > [E.sup.*] provided first firm is more physically levered. Algebraically,

[X.sup.*.sub.1] > [D.sup.*] > [E.sup.*] > [X.sup.*.sub.2]

[X.sup.*.sub.1] = [D.sup.*] = [E.sup.*] = [X.sup.*.sub.2]

[X.sup.*.sub.1] < [D.sup.*] < [E.sup.*] < [X.sup.*.sub.2]

if [absolute value of [alpha]] > 0 (and the converse is true in the opposite physical leverage condition). Next, let us make use of (18) in (23) to get the expression defined by expression (24):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

A number of interesting observations can be made at this point. From (24), it is evident that at [P.sup.*.sub.i] = 0, and [A.sup.*.sub.i] = 0,--that is, at the constant prices and constant levels of distortion,

[X.sup.*.sub.1] > [D.sup.*] > [E.sup.*] > [X.sup.*.sub.2]

[X.sup.*.sub.1] = [D.sup.*] = [E.sup.*] = [X.sup.*.sub.2]

[X.sup.*.sub.1] < [D.sup.*] < [E.sup.*] < [X.sup.*.sub.2]

if [absolute value of [alpha]] > 0 (and the converse is true in the opposite physical leverage condition). All these mean that if debt capital increases relatively more that equity capital in the economy as a whole, the relatively levered firm (in physical sense) expands most and the less levered firm grows least (or may shrink). As a special case of this, one may conclude that if debt instruments are increased while equity instruments are held constant in the economy by some sort of pecking order policy, first firm will grow more than the rate of increase in debt, and the second firm will definitely shrink (provided the first firm is more levered in the physical sense, i.e., [absolute value of [alpha]] > 0). Technically, one finds the following scenario:

[X.sup.*.sub.1] > [D.sup.*] > [E.sup.*](= 0) > [X.sup.*.sub.2]

Since [D.sup.*] > 0, and [E.sup.*] = 0, [E.sup.*] > [X.sup.*.sub.2] means [X.sup.*.sub.2] < 0. In the opposite condition, that is, in the event of the economy-wide increase in equity instruments alone, less levered firm will expand more than the growth of equity, and the more levered firm will contract. In economic terms, growth in fixed income securities induce the physically levered firm to have more than proportional growth in EBIT, and that ultimately may increase earnings available to common stockholders of the physically more levered firm, and exactly the opposite fate will come to the stockholders of the other firm. Note that these results are predicated on the physical ranking of leverage. However, the results may be reversed in the presence of non-constancy of [P.sub.i]'s and [A.sub.i]'s. If leverage ranking of the firms are not the same in physical and financial senses, which means that [absolute value of [alpha]] [absolute value of [beta]] < 0, the unique price output response of the industries may not necessarily hold.

On a more involved inspection of this analytical structure one can here note that since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[absolute value of [alpha]] [absolute value of [beta]] < 0 signifies that if the distortion in the less levered firm relative to the other firm increases, the growth of the firm will be perverse as well. The implications of all of these on the earnings available to stockholders are therefore quite nebulous.

Thus far we have discussed only the supply-side response to various parametric changes. To close the general equilibrium model, it is necessary that one has to bring out the demand-side in the structural set-up. To do so, let the relative demand be specified in the following functional form: (2)

[X.sub.1]/[X.sub.2] = f([P.sub.1]/[P.sub.2]), f(*) < 0. (25)

The functional relation (25) states that as [P.sub.1]/[P.sub.2] rises (falls), [X.sub.1]/[X.sub.2] falls (rises), as the normal demand structure should qualitatively suggest. A step further should lead one to the following expression of proportionate change:

[X.sup.*.sub.1] [X.sup.*.sub.2] = -[[xi].sub.D] ([P.sup.*.sub.1] [P.sup.*.sub.2], (26)

where [[xi].sub.D] (> 0) measures the elasticity on the demand side (as Haugen and Wichern use the concept extensively). Equating this expression (26) to the expression (24), one gets the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A little more algebraic manipulation can re-express (27) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

and this shows that under Anormal@ demand conditions, impact of distortion on price changes is less pronounced, and hence the relative change in expansion (or contraction) of the firms due to distortions is less significant provided the sign of [absolute value of [alpha]] [absolute value of [beta]] is positive.

Next, putting equation (27) into equations (21) and (22), we obtain the following expressions of the relative returns on debt capital under distortion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

From these expressions (29) and (30) one can recognize that if the first firm is physically less levered, real returns on debt capital in this firm gets better if prices do not change and/or demand elasticity is very low and/or percentage change in market distortion for firm 1 is higher than that of the second firm. All these conclusions are again valid if both of these firms are uniquely levered. This validity will be disturbed by different physical and financial leverage rankings of the firms.

III. CONCLUSION

In this paper we demonstrate that if the first firm is more levered in physical sense, and the debt capital is increased relative to equity capital in the economy, first firm expands most and the second firm least of all changes in percentage terms. A special case of this conclusion can be stated as follows: at constant returns to capital--debt and equity--and at constant distortion in the capital market, if debt alone increases (that is equity remains unchanged), relatively levered firm (in physical sense) expands and the less levered firm shrinks. If we go a step further we can conclude that at the constant level of debt and equity capitals with their respective returns remaining invariant as well, an increased premium paid to debt-holders in either industry will involve a substitution away from debt in that industry and affects the industry growth in the same way as the increase in debt and a decrease in equity. Also, the firm that is less levered in physical sense and yet pays a premium to the bondholders in that firm compared to the other firm, an increase in the price of the less levered firm product may not cause its output to grow, and thus the perversity can occur. This result is reminiscent of the result derived by Gup (1980), and yet a significantly different conclusion, as it involves leverage rankings of firms in two different ways in an integrated general equilibrium set-up as opposed to a partial equilibrium structure.

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Dilip K. Ghosh (a), Arun J. Prakash (b) and Dipasri Ghosh (c)

(a) Department of Finance/Management, The Institute of Policy Analysis

American University in Cairo

New York office of The American University in Cairo

420 Fifth Avenue, Third Floor

New York, NY 10018-2729

dghosh4@msn.com

(b) Florida International University

Department of Finance

Miami, FL 33199

prakasha@fiu.edu

(c) California State University-Fullerton

Department of Finance

Fullerton, CA 92834

dghosh@fullerton.edu

ENDNOTES

(1.) By standard normalization,--that is, by choosing the product units appropriately, one can make [P.sub.i] = 1, and hence [X.sub.i] can be called earnings before interest and taxes (EBIT) of firm i. Of course, here we do not have taxes in our paradigm. On a more critical analysis, it embraces the Cournot paradigm of production.

(2.) This type of demand function underlies the homothetic preference structure, often assumed in the general equilibrium literature.