PARETO CONDITIONS, BEHAVIOURAL RULES, AND
THE THEORY OF SECOND BEST
YEW-KWANG N G
University of New England
Davis and Whinston [1965] have tried to prove that, despite the second-best theory,
even if one or more Pareto conditions are violated, the other Pareto conditions are
still valid, when there is separability in terms of the decision units in the system. As an
example, given separability, even with the existence of a monopolist firm, the Pareto
conditions for all the consumers as well as the competitive firms still remain unchanged.
In a recent paper, Sau [1971] argues that “this part of the Davis-Whinston analysis
is not necessarily correct” (p. 40), because if “a Pareto condition is violated by the
non-competitive firm, the consumers, who share the firm’s monopoly profit and
purchase the good(s) in which the firm has a monopoly would violate their respective
Pareto conditions for those specific goods in the second best equilibrium” (pp. 43-4).
It seems to me that Sau’s argument is questionable.
Sau arrives at his conclusion by considering first the utility maximization problem
of a consumer under perfect competition which yields the following conditions
..
...
,nz)
(k = 1,2,
,n)
where Ui = the utility function of individual i (= 1,2, , . . ,m).
Xik
= the quantity of good k (= 1,2, . . . , n) consumed by individual i.
Pk
= the price of good k .
li = Lagrange multiplier,
(i = 1,2,.
He then considers the utility maximization of consumer i in the presence of a
monopolistic firm r.
The corresponding condition is now
(i = 1,2, . . . ,rn)
where n is the good produced by the monopoly firm, a,, is the proportion of the
profit received by consumer i, and d,, is the degree of monopoly.
Sau regards both (1) and (2) as the Pareto optimality conditions under the respective
situations. This seems to me to be incorrect. Both (1) and (2) are the necessary conditions for utility maximization by each consumer under the respective situations. That
(1) happens to be the same as the Pareto conditions is only because of the special
assumption of perfect competition which, with some other assumptions, renders
the Pareto conditions identical with (1). The process of arriving at the Pareto condiWhich may be called the “classical assumptions”, such as no externality, indivisibility and increasing
returns.
1972
BEHAVIOURAL RULES, AND THE THEORY OF SECOND BEST
125
tions is quite different from the process which leads to (1). The former is by maximizing
the vector
(UI, u2, * . U,)
(3)
whilst the latter is by maximizing the scalar
(i = 1,2, . . . , rn)
ui
(4)
-
The Pareto conditions are necessary conditions for a situation where it is not
possible to make someone better off without making others worse off. In contrast,
the problem of individual utility maximization is maximizing one’s utility function
taking account only of the constraints, but not the utility functions of other consumers. Hence, in general, the two will yield different equilibrium conditions. For
example, if there is some monopolistic or monopsonistic power in the system, the
utility maximizing behaviour no longer satisfies the Pareto conditions, except by sheer
chance. Hence, (2) can only be regarded as the utility maximization conditions for
those consumers who have a share in the monopoly profit, but not as the Pareto
conditions.
Since Sau’s criticism of Davis and Whinston is based on contrasting (1) and (2)
which are both regarded as Pareto conditions, we must conclude that his criticism is
misleading. Specifically, Sau has not established that the Pareto conditions for the
second-best problem is different from the first-best;2 he has only shown that utility
maximizing behaviour of the second-best situation may be different from the first-best.
Though it is true that Davis and Whinston claimed the equivalence of first- and
second-best with respect to both Pareto conditions and behavioural rules, McManus
[1967, p. 3201 has already pointed out the mistake of their claim with respect to
behavioural rules.
In fact, the argument of Davis and Whinston with respect to Pareto conditions is
misleading in a different sense. Though their conclusion is valid with respect to
Pareto conditions, given the assumption of separability, this assumption is much
stronger than the mere absence of technological externality as claimed by Davis and
Whinston [1965, p. 31. This is so because the presence of substitutability and complementarity will make the relevant function inseparable even in the absence of noexternality. The assumption of separability is, therefore, very restrictive.
REFERENCES
Davis, Otta A. and Whinston, Andrew B. “Welfare Economics and the Theory of Second Best”,
Review of Economic Studies, Vol. 32, 1965, pp. 1-14.
McManus, M. “Private and Social Costs in the Theory of Second Best”, Review of Economic Studies,
V O ~34,1967,
.
pp. 318-21.
Sau, Ranjit K. “Monopoly, General Equilibrium, and the Theory of Second Best”, Australian
Economic Papers, Vol. 10,1971, pp. 40-44.
a Given separability I cannot see how
this part of Davis-Whinston’s argument can be invalidated.