### How many firms will be operating in the long-run equilibrium

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Economics

Problem Set

Problem 1 (9.2 in the book)

Suppose a firm's production function is given by f (k, l) = kl - 0.8k2 - 0.2l2.

Problem 1.1 Fixing capital at k = 10, graph the marginal physical product of labor (MPl), as a function of l. At what l does MPl = 0?

Problem 1.2 Re-draw the MPl graph for k = 20.

Problem 1.3 Does this production function exhibit increasing returns to scale (IRS), decreasing returns to scale (DRS), or constant returns to scale (CRS)?

Problem 2 (9.4)

A single firm produces crayons in two locations. Location 1 has a production function given by q1 = 10√l1 and location 2 has a production function given by q2 = 50√l2, where l1 and l2 are the quantity of labor used in each location.

Problem 2.1 Given a fixed amount of total labor l, what allocation of labor between the two loca- tions maximizes total output? In other words, maximize q1 + q2 subject to l1 + l2 = l.

Problem 2.2 Derive a single production function f (l) that describes the firm's total output for a given total amount of labor l, assuming it always divides the labor optimally between locations.

Problem 3 (10.2)

Two partners, Mr. Jones and Mr. Smith, are producing a product. Let J be the hours of labor input from Jones, and S be the hours of labor input from Smith. The production function is simply f (J, S) = √(J · S). Smith's wage rate is \$3 per hour, while Jones' wage rate is \$12/hour. Suppose Smith has contributed S = 900 hours, but now refused to work any more.

Problem 3.1 How many hours will Jones have to work to achieve q = 150, q = 300, and q = 450?

Problem 3.2 What is the marginal cost of the 150th unit of output? Of the 300th unit? Of the 450th unit?

Problem 4 (11.2)

A firm has total cost function C(q) = 0.25q2. It sells its products in two countries. Demand in country A is given by qA = 100 - 2PA, and demand in country B is qB = 100 - 4PB (where PA is the price in country A and PB is the price in country B). Thus, total demand for this firm is qA + qB.

Problem 4.1 What is the firm's profit maximizing quantities to sell in each location? What are the resulting prices (PA and PB)?

Problem 5 (11.4)

A firm produces umbrellas at a cost of C(q) = 0.5q2 + 5q + 100. It chooses its output quantity at the beginning of the week. If it turns out to be a rainy week, umbrellas sell at a price of \$30. If it's a sunny week, they sell for \$20. The probability of it being a rainy week is 0.5 (meaning, 50%), and the probability of a sunny week is 0.5.

Problem 5.1 Write out the firm's expected profit function, denoted Eπ(q). (This is simply 0.5 times profit if sunny plus 0.5 times profit if rainy.)

Problem 5.2 What quantity q maximizes expected profits?

Problem 5.3 Suppose the owner is risk averse, where his utility for profits is given by u(π(q)) = √π(q). If he produces the expected-profit-maximizing quantity from the last problem, what will be his expected utility?

Problem 5.4 Give the first order condition for maximizing the owner's expected utility. (You don't have to solve for the maximum, but as an interesting math exercise you can try if you want.)

Problem 5.5 If the firm knew the weather in the coming week with certainty, what would be its optimal production plan?

Problem 6 (11.6)

Take a standard profit-maximizing competitive firm with profit function π(q) = R(q) - C(q, w, v).

Problem 6.1 Derive a the standard FOC for profit maximization. (This is simple; don't overthink it.)

Problem 6.2 Would a lump-sum tax of T (that doesn't depend on q) affect the profit-maximizing quantity of output?

Problem 6.3 Consider a corporate income tax, where the firm has to pay a fraction t of its profit to the government. Thus, it pays tπ(q) and keeps (1 - t)π(q). Would this tax affect the profit- maximizing quantity of output?

Problem 6.4 Would a tax of t dollars per unit of output affect the profit-maximizing quantity of output?

Problem 6.5 Would a tax on the labor input quantity (such as social security) affect the profit- maximizing quantity of output?

Problem 7 (Chapter 12)

Consider the smartphone market. There are many potential entrants in the long run. Economic costs for each firm are given by

C(q) = q3 - 20q2 + 200q + 1000,

where the \$1,000 fixed cost is an opportunity cost representing the accounting profit the firm would earn if they were building weapons guidance systems instead (which is the next-most profitable use of their production technology). Market demand for smartphones is given by

QD = 1060 - 5p.

Problem 7.1 Below what price will all firms shut-down in the short run?

Problem 7.2 What is the long-run price? (Notice: costs don't depend on number of firms here.) You will need to use a computer program (such as Wolfram Alpha at www.wolframalpha.com) to help solve a third-order polynomial. Your solution can be rounded to the nearest whole number.

Problem 7.3 With the current demand function, what will be the long-run total quantity produced in the market?

Problem 7.4 How many firms will be operating in the long-run equilibrium?

Problem 7.5 What are firms' accounting profits in the long run?

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