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Assume that demand for services per period is P = 1000 − Q where Q is the stock of the durable consumed. Let the discount factor for both consumers and the firm be one. Assume that there are only two periods.

(a) What is the optimal rental rate for a monopolist that only leases? (b) What are the optimal prices for a monopolist that only sells, but is able to commit to its second-period price?

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Related questions

Consider a durable good monopolist whose product is usable for only 2 periods. However, each consumer can benefit from the product only for one period. The quality of the product does not diminish over time and thus those who bought the product in period 1 can sell it in period 2 (if they find buyers). Assume also that the production cost is zero. The inverse demand curve in each period is given by P = 16 - Q where Q denotes the quantity available at that period. Both the consumers and the monopolist use a discount factor d is an element of [0,1] to discount future income or profits.

a) Assuming that the monopolist can commit to future prices and quantities, determine the optimal quantities and price in each period. What is the monopoly's profit? Indicate whether your answer depends on the discount factor d

b) Assume now that the monopolist cannot commit to second-period prices and quantities. Determine the optimal quantities and price in each period. What is the monopoly's profit? Indicate whether your answer depends on the discount factor d

Dumping Assume that a firm is a monopolist at Home facing theinverse-demand curve, P = 10 - Q, but is one of many competitors inthe world market, where it can sell its output at a price Pw = 2.Furthermore, assume that the firms total cost is given by: T C (Q)= 10 + Q2 . Answer the following questions:

  1. (a) Find the optimal level of output that maximizes the firm's total profits. Is it optimal for the firm to export?

  2. (b) What happens if Pw increases to 4? Explain the economicintuition.

  3. (c) Now suppose that Pw = 6. Is this firm dumping?

Suppose that a fully automated production process can produce output according to the production function Q = 10K – 0.5K2 where Q = output and K = capital input. Assume further that the expression for the marginal product of capital, MPK, is given as MPK = 10 – K. In this production process capital costs $90 to rent (MFC = $90) and the output sells for $30 per unit (and therefore P = MR = $30).

a. Based on this information, what is the expression for the marginal revenue product of capital, MRPK in this production process? 

b. Given that the production function has a single variable input, what is the optimal quantity of capital the firm should rent (K*) given the existing rental cost of capital and price per unit of output?

c. Based on the optimal rental of capital, how much output (Q) would this company produce per production period? Total Output (Q) = ____________________.

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