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 independent of his income.”2
The purpose of the present note is to show that Scitovsky‘s conclusion follows only from a special definition of the entrepreneur’s wages, and, therefore, 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. .Z3Money 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
1T.
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.
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A NOTE ON PROFIT MAXIMIZATION
level of satisfaction is his minimum income, the wage of routine management. 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^."^
FIGURE 1
m
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 indifference 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 inappropriate, 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 i e s
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 necessarily coincide with the point of maximum satisfaction p . To the right of p ,
+
+
FIGURE 2
(5)
BIbid., p. 58.
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.
7 We
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A NOTE ON PROFIT MAXIMIZATION
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 u p 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. Scitovsky‘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, Scitovsky‘s definition is appropriate.s Generally, however, an entrepreneur is free
s I n 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 maximization is precisely that he is adjusting his output, hence activity level, SO as
to equate marginal revenue with marginal cost. It is, therefore, more appropriate 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 increasing. 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 irrespective of the shape of the indifference map, our point of maximum profit
u~zuuyscoincides 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 satisfaction, 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 Scitavsky, 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.