AS.440.601 Microeconomic Theory

Assignment 1

1.(5 pts) Suppose that f(x,y)=xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: by substitution and by using the Lagrange multiplier method.

2.(5 pts) The dual problem to the one described in question 1 is

Minimize          x+y

subject to        xy=0.25.

Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrange multiplier with the value you got in question 1. Explain the relationship between the two solutions.

3.(10 pts) Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of

utility=U(t,s)=√ts.

a.   (5 pts) If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Paul spend the $1 his mother gives him to maximize his utility?

b.   (5 pts) If the school tries to discourage Twinkie consumption by increasing the price to $0.40, by how much will Paul’s mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?

4.(20 pts) Two of the simplest utility functions are:

1.   (10 pts) Fixed proportions: U (x, y) = min[x, y].

2.   (10 pts) Perfect substitutes: U (x, y) = x + y.

For each of these utility functions, compute the following:

Demand functions for x and y

Indirect utility function

Expenditure function

5.(30 pts) Consider the Cobb–Douglas utility function U(x, y) = x α y 1-α , where 0≤α≤1. This problem illustrates a few more attributes of that function.

a.   (10 pts) Calculate the indirect utility function for this Cobb–Douglas case.

b.   (10 pts) Calculate the expenditure function for this case.

c.   (10 pts) Show explicitly how the compensation required to offset the effect of an increase in the price of x is related to the size of the exponent α .

6.(30 pts). Suppose the utility function for goods x and y is given by

utility = U (x, y) = xy + y.

a.   (10 pts) Calculate the uncompensated (Marshallian) demand functions for x and y, and

describe how the demand curves for x and y are shifted by changes in I or the price of the other good.

b.   (10 pts) Calculate the expenditure function for x and y.

c.   (10 pts) Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good.