MANAGERIAL AND DECISION ECONOMICS, VOL. 7, 145-146 (1986)

Sales-optimization under an Unstable Profit-constraint

JACOB PAROUSH Department of Economics, Bar-llan University, Israel. Visiting in the Department of Economics and Finance and the Center for the Study of Business and Government at Baruch College of the City University of New York

This paper considers a firm’s manager who maximizes total sales under unstable profit-constraint. It is proved that, even if being risk-neutral, the manager is worse off than in the stable equivalent case and that his best policy is to produce less than in the stable case.

INTRODUCTION

The basic characteristic of managerial theories of the firm is the divorce of management from ownership. The owner’s power lies in appointing the board of directors which in turn appoints the top management. This separation of control from ownership allows the top management to deviate from profit-maximization and to maximize their own utility as far as they satisfy the shareholders with reasonable level of profits.

Baumol (1959) was the first to provide several reasons for sales revenue-maximization as an alterna- tive goal to profit-maximization. Since then, many managerial models of the firm’s behavior have intro- duced total sales as an argument in the manager’s utility function, and in this note we adopt Baumol’s target function as well.

The level of profits which has to be maintained by the manager in order to secure his own job depends obviously on the shareholders’ power, which in reality is not perfectly known and has to be estimated by the manager. Moreover, since this power depends on the size and composition of the body of shareholders, it is subject to fluctuations over time and has to be frequently re-evaluated. Actually, the top management operates under an unstable profit-constraint. Recently, a few works, such as Paroush and Venezia (1979), examine the effect of demand-uncertainty on the manager’s optimal policy. Here, we shall assume that the manager faces a known demand but is restricted by a wavering profit constraint and we shall study the effect of such a variability on the manager’s output policy as well as the effect on the level of his objective function.

The main result is that, even if he is risk-neutral, the manager produces less than in the stable equivalent case and, therefore, even under risk-neutrality the manager is worse off in the case of instability in comparison with the stable equivalent case. Compare

this result with Oi’s paradox (1961). Note that the effect investigated here is that of instability and not that of uncertainty, because we explicitly assume that output decision has to be taken ajier the profit-constraint is realized and set.

THE MODEL

Denote by R(x) and C(x) the revenue and the cost functions, respectively, where x is the firm’s output. Assume that over the relevant range of output dR/ dx > 0, dC/dx > 0, d2R/dx2 < 0 and d2C/dx2 > 0. These assumptions are quite standard.

Consider the problem: MaxR(x) with respect to x subject to the profit-constraint R(x) - C(x) b IT. De- note by IT , and IT2 the optimal values of R - C, where R is maximized with no constraint and where R - C is maximized, respectively. Note that if IT, i ll < 112 then the constraint R - C 3 IT is always binding at the optimum. We assume that internal unique regular solution always exists for every II such that IT, < II < II, and denote this solution by x*(II). Thus, x*(n) fulfils:

R(x*(ll)) - C(x*(IT)) = IT (1)

x*(II) 2 Z for every Iz such that R(2) - C(2) b IT ( 2 )

By total differentiation of Eqn. (1) with respect to II one finds that

(dR/dx - dC/dx)(dx*/dn) = 1

(d2R/dX2 - d2C/dx2)(dX*/dII)’ + (dR/dx - dC/dx)d2x*/dIT2 = 0

Since dR/dx - dC/dx < 0 and d2R/dx2 - d2C/dx2 < 0, one obtains:

dx*/dll< 0, d2x*/dI12 < 0 (3) Assume now that the restriction II fluctuates within

the limits IT, and II,, i.e. there is a random variable E

0143-6570/86/020145-02$05.00 0 1986 by John Wiley & Sons, Ltd.

I46 J . PAROUSH

such that E E = 0 and P , { l l , < ll + E < n,} = 1. De- note by xo the equivalent stable output, i.e.

R(xo) = ER(x*(H + E ) ) where x*(II + E ) is the so- lution to Max R(x) S.T. R - C > II + E.

Theorem

Note that risk-neutrality is explicitly assumed by taking R(x*(ll + E ) ) as objective function and not u(R(x*(II + E ) ) ) . It is easy to verify that the sign of xo - x*(n) depends on the sign of u"(dR/dII)2 + u'd2R/dx2, which is always negative for risk-averters but can also be negative for some risk-lovers.

The model of sales-maximization subject to profit- constraint is designed to explain why firms deviate from the fundamental equation between marginal

where marginal cost is larger than marginal revenue. This note does not come to refute such an observ-

ation but to reduce its significance and to indicate that such a deviation might be less important under the less restricted assumption of instability of the profit-

xo < x*(rI) By Eqn' (3)' dx*'dn < O' so that dR'dn =

by assumption d2R/dx2 <O, so that d2R/dnn2 =

(d2R/dx2)(dx/dn)' + (dR/dx)(d2x/dI12) < 0. Thus, R is a decreasing and convex function of n and therefore, by Jensen's inequality,

R(x*(II)) > ER(x*(II + E ) ) or x*(rI) > xo.

(dR/dx)(dx/dn.rc) < 0. By Eqn. ( 3 ) 3 d2x*/dnn2 < and revenue and marginal cost and produces at a point

QED constraint.

REFERENCES

W J Baumol ( 1 959) Business Behavior, Value and Growth, New York Macrnillan, revised edn Harcourt, Brace & World, Inc 1967

J Paroush and V lzhak (1 979) On the theory of the competitive

firm with utility defined on profits and regret European Economic Review 12, 193-202

W Y 01 (1 961 ) The desirability of price instability under perfect competition Econometrlca 29, 58-64