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

CHAP. 4] CONSUMER DEMAND THEORY 83
give up one unit of X. Thus, by moving up the budget line from point F toward point E, the consumer increases
her satisfaction. At point E, the MRS xy ¼ P x /P y .
Fig. 4-20
4.22 (a) Express mathematically the condition for consumer equilibrium given by the indifference-curve
approach. (b) Show that if a cardinal measure of utility exists, the condition in (a) reduces to
as given in Section 4.2.
MU x
P x
¼ MU y
P y
P x Q x þ P y Q y ¼ M
(a) As shown in Problem 4.18, the consumer’s budget line is given by the equation P x Q x þ P y Q y ¼ M.At
the point where this budget line is tangent to an indifference curve, the absolute slope of the curve,
MRS XY , equals the absolute slope of the budget line, P x /P y (see Problem 4.20). Thus,
MRS xy ¼ P x =P y
P x Q x þ P y Q y ¼ M
is the equilibrium condition under the indifference curve theory.
(b) Suppose that a consumer can measure utilities (and hence marginal utilities) numerically, and suppose
that for some Q x and Q y ,MU x ¼ 5 utils, MU y ¼ 1 util. Then the consumer would be willing to give up
5 units of Y for an additional unit of X, as the trade would not change that consumer’s net utility. Thus,
MRS xy ¼ 5 for the given quantities and, in general, MRS xy ¼ MU x /MU y . Substituting this expression
into the first equation of part (a), we obtain
MU x
MU y
¼ P x
P y
or
MU x
P x
¼ MU y
P y
4.23 Draw a diagram showing that (a) if the indifference curves are convex to the origin but are everywhere
flatter than the budget line, the consumer maximizes satisfaction by consuming only commodity Y,
(b) if the indifference curves are convex to the origin but are everywhere steeper than the budget
line, the consumer maximizes satisfaction by consuming only commodity X, and (c) if the indifference
curves are concave to the origin, the consumer maximizes satisfaction by consuming either only