Suppose a typical firm in a perfectly competitive market has the following short-run production function:

a. Use this information to solve for the technologically efficient quantity of labor (L) as a function of output (q). Assume that the price of a unit of labor is w = 10, and the price of a unit of the fixed capital is equal to r = 2.

b. Use your answer to part a and the above information to obtain the short-run cost function for this firm.

c. Write the equation for the marginal costs for this firm.

d. Use your answer in part c to solve for the firm's short-run supply function: q =f(P). Now assume there are n = 10 identical firms in the perfectly competitive market and that the market demand curve is

e. Use your answer in d with the above information to obtain the short-run market supply curve, Q = f(P).

f. Now use the short-run market demand and supply curves to solve for the equilibrium price (P), and quantity (Q).

g. Solve for the short-run quantity supplied by each firm in this market.

h. What is the typical firm's short-run cost of producing this level of output?

i. Find the short-run profit for a typical firm in this market. The long-run production function for the typical firm in this market is

j. Use the output level you determined in part g of this question to sketch the isoquant for that level of output; use the following values for L: 5, 10, 20, 25, 50,100.

k. On the same graph that you drew the isoquant, draw the relevant isocost that applies to the cost you calculated in part h being sure to completely label the horizontal and vertical intercepts.

l. How can you tell that this is not the long-run least cost of producing this level of output?

m. How would the firm adjust its labor and capital usage to produce this level of output at least cost in the long-run?