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Econ 302

Homework #2 

1. (26 Total Points) Suppose a consumer’s utility function is given by U(X,Y) = X^1/2*Y^1/2.  Also, the consumer has $48 to spend, and the price of Good X, P_X = $1.  Let Good Y be a composite good whose price is P_Y = $1.  So on the Y-axis, we are graphing the amount of money that the consumer has available to spend on all other goods for any given value of X.

 

a) (2 points) How much X and Y should the consumer purchase in order to maximize her utility?

 

X* = __________________

 

Y* = __________________

 

 

b) (2 points) How much total utility does the consumer receive?

 

U(X*,Y*) = _________________

 

 

 c) (2 points) Now suppose P_X increases to $4.  What is the new bundle of X and Y that the consumer will demand?

 

X** = __________________

 

Y** = __________________

 

 

d) (2 points) How much additional money would the consumer need in order to have the same utility level after the price change as before the price change? (Note: this amount of additional money is called the Compensating Variation.)

 

Compensating Variation =  ________________________

 

 

e) (2 points) How much money would the consumer be willing to pay to avoid the increase in the price of Good X from PX = $1 to PX = $4?  (Note: the amount of money the consumer is willing to pay to avoid the price increase is called the Equivalent Variation.)

 

Equivalent Variation =  ________________________

 

 

f) (2 points) Of the total change in the quantity demanded of Good X, how much is due to the substitution effect and how much is due to the income effect? (Note: since there is an increase in the price of Good X, these values will be negative).

 

SE = __________________

 

IE = __________________

 

f) (14 points) In the space below, draw on a graph the

· original budget constraint (draw this in black)

· new budget constraint (draw this in green)

· compensated budget constraint (draw this in red)

Also, on your graph, indicate the optimal bundle on each budget constraint.  

· Label the optimal bundle on the original budget constraint X* and Y*

· Label the optimal bundle on the new budget constraint X** and Y**

· Label the optimal bundle on the compensated budget constraint X^C and Y^C

 

Do not scale your axes above 100.  In order to receive full credit, your graph must be neat, accurate, and fully labeled!!

 

2.  (12 total Points) Suppose a consumer’s utility function is given by U(X,Y) = MIN (6X, Y).  Also, the consumer has $56 to spend, and the price of Good X, P_X = $1.  Let Good Y be a composite good whose price is P_Y = $1.  So on the Y-axis, we are graphing the amount of money that the consumer has available to spend on all other goods for any given value of X.

a) (3 points) What is the Indirect Utility Function?

 

V(P_X, P_Y, M) =   _______________________________

 

 

b) (3 points) What is the Expenditure Function?

 

M(P_X, P_Y, V) =  ________________________________

 

Now suppose P_X increases to $2.  

 

 

c) (3 points) Calculate the Compensating Variation:

 

CV = _____________________

 

 

d) (3 points) Calculate the Equivalent Variation:

 

EV = _____________________

3.  (12 points) Suppose there are two consumers, A and B.  

 

The utility functions of each consumer are given by:

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

U_B(X,Y) = X + 4Y

 

The initial endowments are:  

A: X = 4; Y = 4

B: X = 12; Y = 4

 

a) (6 points) Using an Edgeworth Box, graph the initial allocation (label it “W”) and draw the indifference curve for each consumer that runs through the initial allocation.  Be sure to label your graph carefully and accurately.

b) (2 points) What is the marginal rate of substitution for consumer A at the initial allocation?

 

c) (2 points) What is the marginal rate of substitution for consumer B at the initial allocation?

 

d) (2 points) Is the initial allocation Pareto Efficient?