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

CHAP. 14] GENERAL EQUILIBRIUM AND WELFARE ECONOMICS 329
a monetary equation, such as Fisher’s “equation of exchange.” This is introduced in a course in macroeconomics
and is not really needed in an introduction to general equilibrium and welfare economics. All that
we need here is the equilibrium output and input relative prices or price ratios—and those we have.
14.14 If we let P x ¼ $10 when the economy of Problem 14.11 is in general equilibrium of production and
exchange, (a) find P y and (b) findP L if the (MP L ) x ¼ 4 at perfectly competitive equilibrium; what is
the (MP L ) y ?(c) Find P K .(d) If we had set P x ¼ $20, what would P y , P L , and P K be?
(a) Since P x ¼ (1/2)/P y at equilibrium [see Problem 14.13(a)], if we let P x ¼ $10, P y ¼ $20.
(b)
With perfect competition, each profit-maximizing entrepreneur employs each factor up to the point where the
value of the marginal product of the factor in each use equals the factor price. Thus, P L ¼ (VMP L ) x ¼ (P x )
(MP L ) x ¼ ($10)(4) ¼ $40. At equilibrium P L ¼ (VMP L ) x ¼ (VMP L ) y ¼ $40. Since (VMP L ) y ¼ (P y )(MP L ) y
and P y ¼ $20, (MP L ) y ¼ $2.
(c) Since P L ¼ 2/3P K at equilibrium [see Problem 14.13(b)] and P L ¼ $40, P K ¼ $60.
(d ) If we had set (arbitrarily) P x ¼ $20, all other prices would have been double those found in parts (a), (b), and
(c). Thus, specifying an arbitrary absolute price for P x (the numéraire), we can find the corresponding price of
the other commodity and factors. Specifying a different P x will make all other prices proportionately different.
Note that we could have used one of the factor prices as the numéraire. In that case, our knowledge of the
equilibrium MP of the factor in the production of one of the commodities would have allowed us to find
all the other prices. Thus, we see how in a general equilibrium model all prices form an integrated
system—a change in the price of any commodity or factor affecting every other price (and quantity) in the
system (see also Problems 14.2 and 14.3).
WELFARE ECONOMICS
14.15 For an economy of many factors, many commodities and many individuals, state the condition for
Pareto optimum (a) in production, (b) in exchange, and (c) in production and exchange simultaneously.
(a)
(b)
(c)
The condition for Pareto optimum production in an economy of many factors and many commodities is that
the marginal rate of technical substitution between any pair of inputs be the same in the production of all commodities
that use both inputs. If this condition did not hold, the economy could increase its output of one or
more commodities without reducing the output of any other commodity. And a greater aggregate output is
better than a smaller output.
The condition for Pareto optimum in exchange in an economy of many commodities and many individuals is
that the marginal rate of substitution between any pair of commodities be the same for all individuals who
consume both commodities. If this condition did not hold, the satisfaction or welfare of one or more individuals
could be increased without reducing the satisfaction or welfare of any other individual. This represents an
unequivocal increase in social welfare.
The condition for Pareto optimum in both production and exchange simultaneously, in an economy of many
factors, many commodities, and many individuals, is that marginal rate of transformation in production be the
same as the marginal rate of substitution in consumption for every pair of commodities and for every individual
who consumes both commodities. If this condition did not hold, a reorganization of the productiondistribution
process until this Pareto optimality condition holds would represent an unequivocal increase in
social welfare. Once we reach Pareto optimum, no one can be made better off without causing someone
else to be made worse off at the same time. Note, however, that though the Pareto optimality conditions
carry us a long way toward defining policy recommendations for increasing social welfare, they do not
help us in deciding whether one particular distribution of income is better than another. In order to do
that, we must make some ethical or value judgment about the relative “deservedness” of different individuals
in the society.
14.16 If, in Fig. 14-13, A 1 refers to 150 utils, A 2 ¼ 300 utils, A 3 ¼ 450 utils and B 1 ¼ 300 utils, B 2 ¼ 600
utils, B 3 ¼ 750 utils, (a) derive the utility-possibility curve corresponding to the consumption contract
curve in Fig. 14-13. (b) What do points on, inside, and outside the utility-possibility curve stand for?
(c) At what point is this economy simultaneously at Pareto optimum in production and exchange?