Short run and long run expansion path

Homework 3 Instructor: Sergii Golovko Econ 302 - Summer 2016 Due: Thursday, June 9, IN CLASS

PLEASE STEPLE ALL YOUR PAGES!

Exercise 1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y ) = X

2Y and UB(X,Y ) = XY . You may find useful to know that MUAX = 2XY , MUAY = X

2, MUBX = Y and MUBY = X. The initial endowments are: A: XeA = 9, Y

e A = 15, X

e B = 4 and Y

e B = 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 exercises 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.

Exercise 2. Suppose firm’s production function is given by Q = ALαKβ. Thus, the marginal product of labor is given by: MPL = αAL

α−1Kβ, and the marginal product of capital is given by: MPK = βAL

αKβ−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?

Exercise 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!

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e) Illustrate your solution to part b) on a clearly labeled graph.

Exercise 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 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.

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