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

CHAP. 4] CONSUMER DEMAND THEORY 101
Fig. 4-39 Fig. 4-40
(c) When only M is the binding constraint, the individual can exchange for work part of the 24 h available
for consumption. This will shift the individual’s T constraint down and the M constraint up. The individual
should keep on doing this until both the T and M constraints are binding, at a point analogous to
E, where the individual will reach the highest possible indifference curve of type I 00 . When T only is the
binding constraint, the individual should do the opposite, until both constraints become binding and it is
possible to reach the highest possible indifference curve of type I 0 .
CONSUMER DEMAND THEORY WITH CALCULUS
4.43 Starting with the utility function U ¼ U(X, Y), where X and Y refer, respectively, to the quantities of
commodities X and Y, derive the expression for the slope of the indifference curve using calculus.
Taking the total differential and setting it equal to zero (since utility remains unchanged along a given
indifference curve), we get
4.44
dU ¼ @U @U
dX þ
@X @Y dY ¼ 0
(Note: @ ¼ math symbol for partial derivative.)
Thus, the expression for the absolute slope of the indifference curve is
dY
dX ¼ @U=@X
@U=@Y ¼ MU X
¼ MRS XY
MU Y
Starting with utility function U ¼ U(X, Y) and budget constraint P X X þ P Y Y ¼ M, derive the equilibrium
condition using calculus.
We start by forming function V, which incorporates the utility function to be maximized subject to the budget
constraint set equal zero, and get
V ¼ U(X, Y) þ l(M P X X P Y Y)
where l is the Lagrangian multiplier. Taking the first partial derivative of V with respect to X and Y and setting them
equal zero, we get
@V
@X ¼ @U lP X ¼ 0
@X
Dividing the first by the second equation, we get
and
@V
@Y ¼ @U
@Y
@U=@X
@U=@Y ¼ MU X
MU Y
¼ MRS XY ¼ P X
P Y
lP Y ¼ 0