Assignment:
Assume that the rental price of labor in a firm which uses just input in production of its unique output good is fixed at 1. Assume also that when the price of output is 1/3 the firm chooses labor equal to ½ and output, y, equal to 2. Assume that over time the price changes and the output the firm produces changes as well. So that when price equals 1, labor equals 1 and output equals 4 and when the price equals 2, labor equals 3/2 and output equals 6.
a. Graph the isoprofit curves for all three prices.
b. Highlight the outer boundaries on the firm's potential technology curve.
2. Consider a two input firm which faces an aggregate technology for perfect compliments of y=min(3x1,x2).
a. Plot isoquants for y=3,6 and 9
b. What are the returns of scale for this production function?
c. For all possible prices on output, p, and on inputs, w1 and w2, are their price combinations for which a profit maximizing firm would not be able to select a price maximizing quantity (or at least one greater than 0)? Give a restriction on prices such that a profit maximizing firm will be able to solve their problem. What are the solution(s) and level of profits for such a firm.
d. Suppose we are looking to the solve the cost minimization for producing 3 units of output. Find x1 and x2 as a function of w1 and w2 as well as the level of cost for each price combination. Show graphically why your solution makes sense.
3. Repeat problem 2 for the production function of y=3x1+x2.
4. Suppose Farmer A has a technology for converting fertilizer, n, into corn with a marginal product of fertilizer given by MP(y)=1-N/200.
a. If the price of corn (p) is 3$ per unit and the price of fertilizer (w1) is 1 what is the profit maximizing amount of fertilizer? How does this work for a general price w1?
b. Can you find the amount of output as a function of the above prices if the production function is given by y=N-N2 /400? What are the levels of output and the average productivity as a function of w1 for this firm?
c. Suppose that these functions change so that the MP(y)=2-N/100 and y=2N-N2 /200? How do our solutions change?
5. Suppose a firm has a production function given by y=x10.5x20.2
a. Suppose that the output price is fixed at 2, the price of x1 is fixed at 1, and in the short run, x2 is fixed at some constant c. Find the solution to the short run problem as a function of c.
b. Find the price of x2 that would lead to a long run profit maximizing choice of x2 of c.
6. Suppose a production function is given by y=x10.5x20.5. Solve the cost minimization problem for an output target of 10 and generic prices w1 and w2 such that w2=(1-w1). For what value of w1 is output the cheapest to produce? [If you want to use the appendix for help with some of the math, you should reproduced the intermediate steps shown then.)