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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)13. 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.
1. 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)=
2. What is the optimal output level for Kendall Square, Inc in the short run? (q∗) What are its profits? (π)
3. 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)=
4. 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? π=
5. For the rest of the problem, assume that we are in the long run with free entry in the market.
(a) What will be the long-run equilibrium price in the market? (p*)
(b) At that price, how much will each firm produce? (q*)
(c) What is the total number of firms in the market in the long run? (N)
6. If the manager of Kendall Square, Inc works hard, she can manage to transform the long-run cost function you derived in Q.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 Q.5.
(a) What is the new optimal long-run production level for Kendall Square, Inc? (q*)
(b) What are the new long-run profits for Kendall Square, Inc? (π
7. Instead of working hard, Kendall Square Inc’s manager can shirk and not improve costs. 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?