Utility functions and budget constraints

Question

Question 1: Evaluate the MU of x in the following utility functions at: 

u = (2x²y) (14x + y)

u = a/(x²y)

Question 2: Suppose you obtain utility from consuming x according to u = 40 x - 2x².

i) Each x costs $20 and you have a budget of $100. How many x should you buy?

ii) Now assume that you have a budget of $1000. How many x should you buy?

Question 3: Find the optimal amount of x and y to buy if you have the following utility functions and budget constraints.

i) u = x^1/2y + x and p_x = 4, p_y = 2 and I = 100

ii) u = 2x + y and p_x = 4, p_y = 2 and I = 100

Question 4: Find the optimal amount of x and y to buy if you have the following utility functions and budget constraints.

i) u = min(x, y) and p_x = 4, p_y = 2 and I = 100

ii) u = min(5x, y) and p_x = 4, p_y = 2 and I = 100

Question 5: Find the marginal rate of substitution between x and y (MUx/MUy) for the following utility functions:

i) u = x^1/2y + x evaluated at x = 15 and y = 10

ii) At what price ratio would the consumption in part above be optimal?

Question 6: Suppose that a firm produces a product with two inputs, capital K and labor L. Labor costs $7 per unit and capital costs $8. The price of the output is $20. The firm’s production function is y = Z_k^1/2 Z_L^1/4
i) Find the profit-maximizing set of inputs.

ii) Suppose that the production function were y = Z_k^1/2 Z_L^1/2. Describe why it is not possible in this case to find the profit-maximizing set of inputs.

iii) Now suppose, again for the production function y = Z_k^1/2 Z_L^1/2, that in the short run Z_k = 25. Find the short run input of L and the output y.

Question 7: Find the demand function for x for the following utility functions:

i) u = xy. y = 10, I = 100

ii) u = x^1/3y^1/3. y = 10, I = 100

iii) u = min[x, 2y]. y = 10, I = 100

Question 8: Assume a competitive industry. Ten perfectly competitive firms have production function y = Z_k^1/2 Z_L^1/2. P_k = 1 and P_L = 2. There are 1000 consumers, who each have identical utility function u = xy. The price of x is 3 and each consumer has income 100. Find theprice, the total number of units of y consumed, and the total amount of each type of input used.

Question 9: Two types of consumers live on an island. They all consume only xylophones. The consumers live only two periods, because xylophones are not actually very nutritious. The two types of consumers are the perfect substituters (PS) and the CDs. The PS utility function is u = x₁ + x₂. The CD utility function is u = x₁²x₂. There are 10 PS individuals and 15 CD individuals. Each PS person has endowment of 100 in the first period and 100 in the second period. Each CD person has endowment of 100 in the first period and 0 in the second period.

i) Find the equilibrium interest rate.
ii) Find the consumption in period 2 of a PS individual.