ECON1003 Quantitative Methods in Economics S1 2023 PROBLEM SET 10
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ECON1003 S1 2023
PROBLEM SET 10
1. A firm, selling two goods, X and Y, has a revenue function TR = 20xy−10x2 −50y2 + 800y.
(a) Find the stationary point (this is where the partial derivatives of TR with respect to x and y are 0)
(b) Solve for the optimal levels of x and y that maximize revenue (check second order conditions)
2. Consider a competitive firm with the production function Q = L1/3K1/2 and suppose the wage per unit of labor w = 8, the cost capital per unit is r = 3, and the price of output p = 1. Find the optimal levels of L and K that maximize profits.
3. Find and classify the stationary points for the following functions:
a. Q = 5ln(L) + 2ln(K) – 0.1L – 0.4K
b. F(x, y) = 2y2 + 2xy + x2 – 16x – 20y
4. A monopolist sells its products in two separate markets. The demand function for
each market is:
P1 = 80 – 2.5Q1
P2 = 125 – 10Q2
The total cost function for the monopolist is TC = 200 + 5Q where Q = Q1 + Q2.
a. Calculate the maximum profit with price discrimination.
b. Calculate the maximum profit with no price discrimination.
c. Compare the maximum profit in a) and b).
d. Calculate the price elasticity of demand in each market when profits are maximized with price discrimination. Comment.
2023-06-21