Econ 302 - how much money would the consumer need in order

Question

Instructions: Please Read Carefully

1. Suppose a consumer's utility function is given by U(X,Y) = X^1/2*Y^1/2. Also, the consumer has $576 to spend, and the price of X, P_X = 16, and the price of Y, P_Y = 4.

a) How much X and Y should the consumer purchase in order to maximize her utility?

b) How much total utility does the consumer receive?

c) Now suppose PX decreases to 9. What is the new bundle of X and Y that the consumer will demand?

d) How much money would the consumer need in order to have the same utility level after the price change as before the price change?

e) Of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect?

2. Suppose there are two consumers, A and B.

The utility functions of each consumer are given by:

UA(X,Y) = X^1/2*Y^1/2
UB(X,Y) = 2X + Y

The initial endowments are:

A: X = 4; Y = 4
B: X = 4; Y = 12

a) Using an Edgeworth Box, graph the initial allocation and draw the indifference curve for each consumer that runs through the initial allocation. Be sure to label your graph carefully and accurately.

b) What is the marginal rate of substitution for consumer A at the initial allocation?

c) What is the marginal rate of substitution for consumer B at the initial allocation?

d) Is the initial allocation Pareto Efficient?

3. Suppose there are two consumers, A and B, and two goods, X and Y. Consumer A is given an initial endowment of 6 units of good X and 1 units of good Y. Consumer B is given an initial endowment of 2 units of good X and 7 units of good Y.

Consumer A's utility function is given by:

UA(X,Y) = X^1/2*Y^1/2,

And consumer B's utility function is given by

UB(X,Y) = X^1/4*Y^3/4.

Therefore, consumer A's marginal utilities for each good are given by:

MUX = (1/2)X^-1/2Y^1/2
MUY = (1/2)X¹/2Y^-1/2

Also, consumer B's marginal utilities for each good are given by:

MUX = (1/4)X-3/4Y3/4
MUY = (3/4)X1/4Y-1/4

a)  Suppose the price of good Y is equal to one. Calculate the price of good X that will lead to a competitive equilibrium.

b) How much of each good does each consumer demand in equilibrium?

c) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

d) Illustrate the situation in an Edgeworth Box. Be sure to label your box carefully and accurately. Identify the initial endowment and label it W. Identify the competitive equilibrium and label it D. Draw the budget constraint that each consumer faces and identify the values where it intercepts the perimeter of the Edgeworth Box (there are two different intercepts to identify).