1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X, Y ) = X2Y and UB (X, Y ) = XY . You may find useful to know that MUAX = 2XY , MUAY = X2, MUBX = Y and MUBY = X. The initial endowments are: A: XeA = 9, YeA = 15, XeB = 4 and YeB = 20.
a) Suppose the price of good Y , PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium.
b) How much of each good does each consumer demand in equilibrium?
For 2, 3 and 4 make sure to label all your graphs accurately. You need to draw both isocosts and isoquants that go through each optimal cost minimizing labor - capital point.
2. Suppose firm's production function is given by Q = ALαKβ . Thus, the marginal product of labor is given by: MPL = αALα-1Kβ, and the marginal product of capital is given by: MPK = βALαKβ-1. Suppose that A = 2, α = 2/3 and β = 1/3. The wage w = $1 and the price of capital r = $4.
a) How much labor and capital should the firm hire if it wants to produce 8 units of output while minimizing its cost of production? What is the lowest cost firm incurs when producing 8 units of output?
b) What is the total cost of producing q = q‾ units of output?
c) On the same graph draw short-run and long-run Expansion paths for the level of outputs for the quantities q‾ = 8, q‾ = 16, q‾ = 32. For short-run Expansion path assume that capital is fixed at the optimal amount needed to produce q‾ = 8 units of output.
d) Does the economy exhibit increasing, decreasing or constant return to scale?
3. Suppose labor and capital are perfect substitutes. To produce 6 units of output the firm needs either 2 units of labor or 3 units of capital.
a) What is the functional form of the firm's production function?
b) Assume that w = $1 and the price of capital r = $4. How much labor and capital should the firm hire if it wants to produce 9 units of output while minimizing its cost of production? What is the lowest cost firm incurs when producing 9 units of output?
c) What is the total cost of producing q = q‾ units of output?
d) Does the economy exhibit increasing, decreasing or constant return to scale? Explain!
e) Illustrate your solution to part b) on a clearly labeled graph.
4. Suppose labor and capital are perfect complements. To produce 6 units of output the firm needs 2 units of labor and 3 units of capital.
a) What is the functional form of the firm's production function?
b) Assume that ω = $1 and the price of capital r = $4. How much labor and capital should the firm hire if it wants to produce 9 units of output while minimizing its cost of production? What is the lowest cost firm incurs when producing 9 units of output?
c) What is the total cost of producing q = q‾ units of output?
d) Does the economy exhibit increasing, decreasing or constant return to scale? Explain!
e) Illustrate your solution to part b) on a clearly labeled graph.