Cost ATC AVC MC Output 2

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Vincent May
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1 Intermediate Microeconomics Answers to second midterm 1.a. The firm s short run supply curve is the portion of the marginal cost curve on and above the minimum of average variable cost, and zero for prices below the minimum of average variable cost. So we have to find the intersection of the marginal cost curve with the average variable cost curve. The marginal cost function is the derivative of the total cost function. Thus, the marginal cost function is MC(q) = q + 3q 2. The variable cost function is the portion of total cost that depends on q. This is V C(q) = 1200q 20q 2 + q 3. Average variable cost is this divided by q, or AV C(q) = q + q 2. Setting marginal cost equal to average variable cost, we have q + 3q 2 = q + q 2. Rearranging, we get 2q 2 = 20q. The solutions to this are q = 0 and q = 10. At q = 0, the average variable cost is decreasing, so it is not a minimum of AV C. The relevant solution is q = 10. At q = 10, MC(q) = (10) + 3(100) = So the firm s short run supply curve is q +3q 2 for q 10 and has q = 0 for P < b. A firm s average variable cost is increasing when marginal cost exceeds average variable cost, and is decreasing when marginal cost is lower than average variable cost. The marginal cost curve intersects the average variable cost curve at the minimum of average variable cost. The firm s long run average cost curve is the lower envelope of all its short run average total cost curves. Given a quantity of output, the firm s long run cost of producing that quantity is the short run cost of producing it using the optimal short run production process. For example, if the firm knows that in the long run it will be producing q units of a good, it will choose the firm size that leads to production of q units at minimum cost over all firm sizes. 1

2 Cost MC ATC AVC Output 2

3 Cost SAC1 SAC3 SAC2 LAC Quantity c. These hotels remain open because in the short run their fixed costs are sunk, and they cannot retrieve them no matter what they do. They may be making negative profits in the short run, but their prices are above average variable costs. In the longer run, considering the whole year, they expect to make positive or zero profits (otherwise they would go out of business). It would be costly to stop maintaining the hotels during the off season and then have to fix everything and rehire workers when the hotels reopen. d. When firms are earning zero economic profits, it means they are doing just as well as the next best activity they could be employing their resources in. The firms are covering their opportunity costs, which include the costs of not investing their capital in the next most productive area. So they are indifferent, and might as well stay in the market. 2.a. The manufacturer is not minimizing its costs. Labor and machinery are equally productive, but machinery costs more than labor. Therefore to minimize costs the manufacturer should use 10 hours of labor and no machinery to produce a rug. Formally, the slope of an isoquant is 1 everywhere, as its equation is L + K = 10. The slope of an isocost line is 1/2, as its equation is 20L + 40K = C. Therefore the lowest-cost isocost line touching the isoquant corresponding to 1 rug touches it at (10, 0). 3

4 Capital Isoquant Isocost line Optimal input combination Labor b. The condition is MP L /MP K = P L /P K. This is the condition that ensures that the slope of the isoquant ( MRT S = MP L /MP K ) equals the slope of the isocost curve ( P L /P K ). If the slopes were different so that MP L /MP K > P L /P K, costs could be decreased by increasing the amount of labor and decreasing the amount of capital, while continuing to produce the same amount. If MP L /MP K < P L /P K, costs could be decreased by increasing the amount of capital and decreasing the amount of labor, while keeping output the same. 3.a. If absolutely all inputs can be duplicated at every quantity, the firm cannot have decreasing returns, it must have either increasing returns or constant returns. By replicating the firm exactly, one would get at least constant returns. Decreasing returns functions such as F (L, K) = L + K come about because there are other inputs which are implicitly not being counted. Similarly, when a firm experiences decreasing returns due to an input getting more expensive when a lot of it has been bought, it is because there is some part of that input that is in fixed supply and cannot be replicated. b. A firm has increasing returns to scale if, when inputs double, output more than doubles. Indivisibility of some inputs can lead to increasing returns to 4

5 scale, because you might need an integer amount of some input to get any output at all. Thus, when you divide that integer amount by 2, output is divided by more than 2. c. Each firm is making zero profits, and is producing at the minimum of long run average cost. If price were raised above the minimum of long run average cost (for example due to a demand shock), firms would make positive profit, other firms would enter the market, short run supply would increase and price would decrease until profits were zero again and each firm producing at the minimum of long run average cost. 4.a. A monopolist finds the quantity to produce at the intersection of MC and MR curves, then finds price as the height of the demand curve at that quantity. Costs, Price P* MC AC MR D Q* Q b. The marginal costs in the two plants should be equal (quantities produced need not be equal). 5. The market s outcome is less efficient with the ceiling. Deadweight loss is as shown in the graph. Those consumers that are able to get apartments are better off, but those that do not get apartments, but would have been willing 5

6 to rent one at the original equilibrium price, are worse off. Producers are worse off. price Producer surplus Deadweight loss S P0 Pmax Gain to consumers D Q1 Q0 Q2 quantity 6

Eco 300 Intermediate Micro

Eco 300 Intermediate Micro Instructor: Amalia Jerison Office Hours: T 12:00-1:00, Th 12:00-1:00, and by appointment BA 127A, aj4575@albany.edu A. Jerison (BA 127A) Eco 300 Spring 2010 1 / 66 Page 261,