Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

332 GENERAL EQUILIBRIUM AND WELFARE ECONOMICS [CHAP. 14
14.19 For the economy of Problems 14.17 and 14.18, determine (a) the point of maximum social welfare and
(b) how much of X and Y is produced, how this X and Y is distributed between A and B (i.e., X A ,X B ,
Y A ,Y B ), the value of u A and u B , how much of L and K is used to produce X and Y (i.e., L x , L y , K x , K y ),
and the value of P x /P y and P L /P K when the economy reaches its maximum social welfare.
(a)
(b)
By superimposing the social welfare or indifference map of Fig. 14-16 on the grand utility frontier of
Fig. 14-15, we can determine the point of maximum social welfare, or “point of constrained bliss.” In
Fig. 14-17 this is given by point D 0 , where the grand utility frontier is tangent to W 2, the highest attainable
social welfare function. The choice of a point on the grand utility frontier is basically the choice of a particular
income distribution. A movement away from point D 0 along the grand utility frontier will increase the welfare
of one individual but reduce the total social welfare. Remember that the Pareto optimality conditions with
which we started our discussion of welfare economics are necessary but insufficient to determine the point
of maximum social welfare, since they simply define the grand utility frontier. This is as far as positive economics
will take us. To find the point of constrained bliss we need normative information on the values of the
society, so that we can construct a social welfare or indifference map.
Point D 0 (i.e., the point of maximum social welfare) on the grand utility frontier corresponds to point D on the
consumption contract curve and point M 0 on the transformation curve of Fig. 14-13. Thus, we now know how
much of X and Y this economy must produce in order to maximize its social welfare, and so we have removed
the indeterminancy that we talked about at the end of Problem 14.11. That is, having found the point of
maximum social welfare, we can now reverse the order of Problems 14.5 to 14.18 and find that this
society should produce 60X and 70Y [see Problem 14.12(a)]; X A ¼ 35, X B ¼ 25, Y A ¼ 35, Y B ¼ 35 [see
Problem 14.12(b)]. With X A ¼ 35 and Y A ¼ 35, u A ¼ 300 utils; with X B ¼ 25 and Y B ¼ 35,
u B ¼ 600 utils (see point D 0 in Fig. 14-17); L x ¼ 8, L y ¼ 10, K x ¼ 7, K y ¼ 5 [see Problem 14.12(c)], P x /
P y ¼ 1/2 and P L /P K ¼ 2/3 (see Problem 14.13).
Note that we have now obtained the complete solution to the simple general equilibrium model we
have set up, and in the process we have combined the theories of production, distribution, and consumption
and the value system of the society. Our simple model also shows that a change in one sector will
bring changes in every other sector of the economy, as indicated in our discussion of the circular flow in
Chapter 1.
14.20 Prove that when all markets in our simple economy are perfectly competitive, the following conditions
hold: (a) (MRTS LK ) x ¼ (MRTS LK ) y ,(b) (MRS xy ) A ¼ (MRS xy ) B ,(c) (MRS xy ) A ¼ (MRS xy ) B ¼ MRT xy .
(a)
(b)
(c)
We saw in Section 6.8 that under perfect competition, producers choose the quantity of L and K such that
MRTS LK ¼ P L /P K . Since P L and P K and thus P L /P K are the same in all uses under perfect competition,
(MRTS LK ) x ¼ (MRTS LK ) y .
We saw in Section 4.7 that under perfect competition, consumers choose the quantity of X and Y such that
MRS xy ¼ P x /P y . Since P x and P y and thus P x /P y are the same for all consumers under perfect competition,
(MRSx y ) A ¼ (MRS xy ) B .
The MRT xy ¼ Dy=Dx ¼ MC x =MC y . For example, if we must give up 2Y to produce 1X more, the
MC x ¼ 2MC y and the MRT xy ¼ 2. But in Chapter 10 we saw that under perfect competition, MC x ¼ P x
and MC y ¼ P y . Therefore, MC x /MC y ¼ P x /P y ¼ MRT xy . But since in the proof of part (b) we have seen
that the MRS xy for A and B also equals P x /P y , MRT xy ¼ MRS xy for A and B.
Similar results hold in a perfectly competitive economy of many factors, commodities, and individuals.
Thus perfect competition in every market in the economy guarantees (subject to the qualifications in Section
14.13) the attainment of Pareto optimum in production and distribution. This is the basic argument in favor of
perfect competition.
14.21 (a) Explain why with constant returns to scale and the absence of externalities, a Pareto optimum point
will not be attained if there is imperfect competition in some markets of the economy. (b) If the government
can make more but not all markets in the economy perfectly competitive, will social welfare
increase?
(a)
If industry X is imperfectly competitive, it will produce the output for which MC x ¼ MR x , P x . Thus P x is
higher, Q x is lower, and fewer resources are used than if industry X were perfectly competitive. If another