E1)Kendall Square Inc makes and sells ramen to the large number of students around MIT. Kendall Square Inc uses the production function F(K,L)=(K+2L)1/3. Input prices are w=2 and r=3 for labor and capital respectively. It operates in a perfectly competitive environment facing a price p=$48, and uses an initial amount of capital K^=12.

Compute the firm's short-run cost function. To verify that you have the correct short-run cost function, calculate the short-run cost of producing 8 units of ramen.

C(8)=?

E2)What is the optimal output level for Kendall Square, Inc in the short run?

q=?

What are its profits?

=?

E3)Compute Kendall Square Inc's long-run cost function. To verify that you have the correct long-run cost function, calculate the long-run cost of producing 8 units of ramen.

C(8)=?

E4)Now suppose that, in addition to its existing capital and labor inputs, Kendall Square, Inc must hire a manager for a fixed amount of $54 in order to function properly. Suppose that all firms have the same long-run cost function as Kendall Square, Inc (they must also hire a similar manager), and the market demand is given by D(p)=9720/p. Suppose there are 59 other identical firms in the market (i.e. 60 identical firms in total) and no entry or exit due to government regulation.

What is the firm's profit?

=?

E5)For the rest of the problem, assume that we are in the long run with free entry in the market.

What will be the long-run equilibrium price in the market?

p=?

At that price, how much will each firm produce?

q=?

E6)What is the total number of firms in the market in the long run?

N=?

E7)If the manager of Kendall Square, Inc works hard, she can manage to transform the long-run cost function you derived in PS5.3.5 by cutting Kendall Square Inc's variable costs in half. That is, she can halve the part of the long-run cost function unrelated to her fixed wage of $54. Suppose that the rest of the market doesn't change: the other firms can only hire mediocre managers who cannot cut the cost functions and so the long-run market price remains fixed at the same level you calculated in E5.

What is the new optimal long-run production level for Kendall Square, Inc?

q=?

What are the new long-run profits for Kendall Square, Inc?

=?

E8)Instead of working hard, Kendall Square Inc's manager can shirk and not improve costs. In order to incentivize her hard work, Kendall Square Inc's shareholders want to give the manager a bonus if they see that variable costs are cut by half.

What is the maximum bonus that shareholders would be willing to give?