A NOTE ON PROFIT MAXIMIZATION

Y. K . NG University of Sydney

I Scitovsky has shown, in a brilliant article,l that the assumption of profit

maximization by the entrepreneur implies that his indifference curves between income and inactivity are vertical displacements of .each other, the economic meaning of which is “that the entrepreneur’s choice between more and less activity-or between more income and more leisure-must be inde- pendent of his income.”2

The purpose of the present note is to show that Scitovsky‘s conclusion fol- lows only from a special definition of the entrepreneur’s wages, and, there- fore, that the assumption of profit maximization is not restricted to the case of “vertical displacements”. This note, however, is not concerned with other possible objections to profit maximization.

I1 Our Fig. 1 is a reproduction of Scitovsky’s Fig. .Z3 Money income m is

measured dong the vertical axis, and entrepreneurial inactivity i along the horizontal axis. Entrepreneurial activity is the negative of i and is measured from right to left. Money income is a function of entrepreneurial activity. The indifference curve Z, which goes through w , the point of total inactivity, represents the minimum satisfaction that will keep the entrepreneur in his profession. By this device, Scitovsky defines the entrepreneur’s wages thus : “That part of his income, therefore, which brings the entrepreneur to this

1 T. Scitovsky: “A Note on Profit Maximisation and its Implications”, Review of E c o m i c Stdies, XI, 1943. Reprinted in A.E.A.: Readings in Price Theory (Chicago: Richard D. Irwin, Inc., 1952). Review of Economic Studies, XI, 1943, p. 59.

3 I substitute “money income” for Scitovsky’s “net income”. Scitovsky uses “net income” to designate the difference between total receipts and total outlays (excluding entrepreneur’s wages). It is composed of profit and entrepreneur’s wages. But the word “net” has confused some economists. J. W. McGuire, for example, in his presentation of Scitovsky’s contribution, mistakes “net income” for profit. (Theories of Business Behmior, Prentice-Hall: Englewood Cliffs, N.J., 1964, pp. 75-6). Instead of contrasting the point of maximum satisfaction p, with the point of maximum profit (where the vertical distance between the net income curve and the indifference curve I is greatest, according to Scitovsky), he contrasts p with h, the point of maximum net income, which he regards as the “point of maximum profit”. Thus, the fundamental contribution of Scitovsky is totally missed; the crucial phrase “vertical displacements” is not mentioned at all.

1969 A NOTE ON PROFIT MAXIMIZATION 107

level of satisfaction is his minimum income, the wage of routine manage- ment. Only the part of his income that brings his satisfaction above the indifference curve Z can be regarded as profit proper. In Fig. 2 [our Fig. 13, therefore, of ps, the entrepreneur’s total net income, pr is profit, and rs is wage^."^

m

FIGURE 1

S w i

By this definition, the point of maximum profit, where the slope of the income curve is equal to that of indifference curve I , will coincide with the point of maximum satisfaction p onIy if the entrepreneur has a special type of indifference map. “For the net income curve to be tangential to an indifference curve at the same level of output at which its slope equals that of indifference curve Z, the tangents to the two indifference curves at that level of output must be parallel. In order that this condition may be satisfied for any and every kind of net income curve, all indifference curves must have the same slope for each abscissa. In other words-the several indiffer- ence curves must be vertical displacements of each other.’3

I11 Scitovsky‘s conclusion follows only by defining the entrepreneur’s wages to

be the income which brings the entrepreneur to the level of satisfaction which is equal to that of zero entrepreneurial activity. This definition is inap- propriate, at least in the analysis of profit maximization.

4Ibid., p. 58. 5 Ibid., p. 59.

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In the traditional analysis, the entrepreneur is said to maximize profit by adjusting ouput to a level where marginal cost equals marginal revenue. Cost, in this respect, includes interest on capital, and entrepreneur’s wages, “which he is supposed to pay to himself and include in his regular cost calculations.”a

In determining what amount should be the wages at each level of output (and hence each level of activity), the entrepreneur is comparing a certain level of activity and income with a slightly higher level of activity and income, in short, in marginal terms. Thus, in deciding whether to increase his activity from il to iz (see Fig. 2) ?, he will see whether the increase in money income is greater than bc. His initial position is at ml; c is the point on the indifference curve I’ passing through ml; b is on the same horizontal level with ml. Thus bc is the minimum money income necessary to induce the entrepreneur to increase his activity from i, to iz, assuming that there is no feasible intermediate level of activity. From m, to m2, therefore, bc is wages and cmz profit. Similarly for de and em3. What the entrepreneur t ies to maximize is h e n seen to be ( aml + cmz + em3) (profit), and his wages are ( i l a + b c + d e ) .

It can readily be seen that the point of maximum profit will then neces- sarily coincide with the point of maximum satisfaction p . To the right of p ,

FIGURE 2

( 5 )

BIbid., p. 58. 7 We consider first the case of discontinuous increments to facilitate analysis. In this case, of course, the money income curve dissolves into discontinuous points like o, m,, m2, etc.

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the indifference curves passing through the several m’s necessarily lie below p . Movement to p will then bring additional money income greater than that necessary to compensate the increase in activity. To the left of p it does not pay to increase activity as the income curve lies below the indifference curve ZZ. The necessary minimum increment in income, fg, is greater than that obtainable, fm4.

If entrepreneurial activity can be varied continuously, we can draw (Fig. 3) a curve E passing through w. This curve is, in a sense, the supply curve of entrepreneurship (for that particular entrepreneur). It measures the sum of the minimum marginal increments in money income necessary to induce the entrepreneur to supply his activity continuously up to the respective level, given the income curve. The vertical distance between it and the income curve is residual profit which the entrepreneur seeks to maximize.

FIGURE 3

S w i At the risk of repetition, we may note ‘the difference between Scitovsky’s

and our definitions of the entrepreneur’s wages. Our entrepreneur’s wages q s is the sum of all minimum marginal increments in money income necessary to induce the entrepreneur to supply his activity continuously up to s. Sci- tovsky‘s rs is the money income that brings the entrepreneur, at the activity level of s, to the same level of satisfaction with w. If the entrepreneur’s choice is exclusively between activity level s and no activity at all, Scitov- sky‘s definition is appropriate.s Generally, however, an entrepreneur is free sIn this case, the money income curve dissolves into the points w and p . If p is higher/lower than r, p/o will be the point where maximum profit and maximum satisfaction coincide. Thus, the problem of divergence between maximum profit and maximum satisfaction does not arise.

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to choose between the intermediate levels. The assumption of profit maximi- zation is precisely that he is adjusting his output, hence activity level, SO as to equate marginal revenue with marginal cost. It is, therefore, more appro- priate to reckon the entrepreneur, in calculating his own wage element in his marginal cost, as behaving in our pattern.

The shape of the E curve is determined by the indifference curves passing through the various points on the income curve between o and p , assuming that this part of the income curve is continuous and monotonically increas- ing. More specifically, its slopes are equal to the slopes of the indifference curves at the income curve vertically above it, e.g., its slope at k is equal to that of the indifference curve I’ at i. To the left of 9 it is just the vertical displacement of the indifference curve ZZ. The reason is simple; to induce the entrepreneur to increase his activity level after he has attained the point p , additional income (wages) must be sufficient to keep him at least at the satisfaction level represented by the indifference curve ZZ.

It is now obvious that p is the point of maximum profit, as the slopes of E curve and income curve are equal at this point. To the right of this point, the slope (absolute value) of E curve (for example, the slope at k) is smaller than that of the income curve, because the slope of the indifference curve (Z’) at the respective point on the income curve ( i ) , is smaller than that of the income curve. To the left of g, the slope of the E curve is clearly greater than that of the income curve, since it is the vertical displacement of indifference curve ZZ.

In the case where the indifference curves are vertical displacements of each other, our E curve is identical with the indifference curve 1. But irres- pective of the shape of the indifference map, our point of maximum profit u~zuuys coincides with the point of maximum satisfaction. Therefore, the assumption of profit maximization is not restricted to the case where the indifference curves of the entrepreneur are vertical displacements of each other.

According to Scitovsky, if the entrepreneur’s choice between income and leisure is not independent of his income, he does not, while maximizing satis- faction, also maximize profit. For example, if the higher his income level, the greater he values leisure in terms of income, he will choose, according to Sci- tavsky, a level of activity less than that of maximum profit. According to our analysis, this simply means that more income (wages) will be necessary to induce him to increase his activity. And since entrepreneur’s wages are not to be regarded as a part of profit, the latter is necessarily a residue which will be maximized.