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.

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 firms profit?

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

p=

q=

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

N=

If the manager of Kendall Square, Inc works hard, she can manage to transform the long-run cost function you derived by cutting Kendall Square Incs 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 doesnt 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.

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?

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Instead of working hard, Kendall Square Incs manager can shirk and not improve costs. In order to incentivize her hard work, Kendall Square Incs 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?