Homework 2
Total points: 100
Problem 1 (15 points):
Graph Each of the following budget constraints. Be sure to label all intercepts and clearly label each constraint.
a) (5 points) px = 1; py = 2; I = 100
b) (5 points) px = 2; py = 2; I = 200
c) (5 points) px = 3; py = 6; I = 150
Problem 2 (10 points):
Suppose there are 2 goods, X and Y; and utility function of the consumer is
U (X; Y ) = 5X 2 Y 3 : M UX = 10XY 3 and M UY = 15X 2 Y 2
a) (5 points) derive demand functions for X and Y:
b) (5 points) Suppose I = 10; pX = 1; pY = 2: What is the optimal bundle?
Suppose there are 2 goods, X and Y; and utility function of the consumer is
U (X; Y ) = 5X 2 Y 3 : M UX = 10XY 3 and M UY = 15X 2 Y 2
a) (5 points) derive demand functions for X and Y:
b) (5 points) Suppose I = 10; pX = 1; pY = 2: What is the optimal bundle?
Problem 3 (15 points):
Suppose U (X; Y ) = minfaX; bY g where a > 0; b > 0: Also suppose that
pX > 0; pY > 0: Derive demand functions for X and Y: Show your work.
Suppose U (X; Y ) = minfaX; bY g where a > 0; b > 0: Also suppose that
pX > 0; pY > 0: Derive demand functions for X and Y: Show your work.
Problem 4 (20 points):
Suppose U (X; Y ) = aX + bY where a > 0; b > 0: M UX = a and M UY = b:
a) (10 points) Derive demand functions for X and Y: Show your work.
b) (10 points) Graph demand for X at I = 100; pY = 1, a = 2; b = 1:
Problem 5 (40 points):
Suppose U (X; Y ) = X Y where
> 0; > 0: M UX = X 1 Y and
1
M UY = X Y
: Assume also that pX = pY = p:
a) (5 points) Find the demand of X and Y as functions of p and I:
b) (5 points) Find the expression for the utility that the consumer is getting
from the optimal bundle (substitute demand functions that you derived in part
a in the utility function).
c) (5 points) suppose price of X becomes pX = p while price of Y remains
at pY = p: Solve for the new optimal bundle.
d) (10 points) What is the substitution e¤ect of the price change in terms of
X? Show your work.
e) (5 points) What is the condition on and under which the substitution
e¤ect derived in part d is positive. Interpret the condition.
f) (5 points) What is the income e¤ect of the price change in terms of X?
g) (5 points) What is the condition on
and
under which the income
e¤ect is positive. Interpret the condition. 1
Suppose U (X; Y ) = aX + bY where a > 0; b > 0: M UX = a and M UY = b:
a) (10 points) Derive demand functions for X and Y: Show your work.
b) (10 points) Graph demand for X at I = 100; pY = 1, a = 2; b = 1:
Problem 5 (40 points):
Suppose U (X; Y ) = X Y where
> 0; > 0: M UX = X 1 Y and
1
M UY = X Y
: Assume also that pX = pY = p:
a) (5 points) Find the demand of X and Y as functions of p and I:
b) (5 points) Find the expression for the utility that the consumer is getting
from the optimal bundle (substitute demand functions that you derived in part
a in the utility function).
c) (5 points) suppose price of X becomes pX = p while price of Y remains
at pY = p: Solve for the new optimal bundle.
d) (10 points) What is the substitution e¤ect of the price change in terms of
X? Show your work.
e) (5 points) What is the condition on and under which the substitution
e¤ect derived in part d is positive. Interpret the condition.
f) (5 points) What is the income e¤ect of the price change in terms of X?
g) (5 points) What is the condition on
and
under which the income
e¤ect is positive. Interpret the condition. 1
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