Midterm Review Questions

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1 Midterm Review Questions July 29, 2013 Consumer Theory 1. Parvez, a pharmacology student has allocated $120 per month to spend on paperback novels (N is the number of novels) and used CDs (C is the numbers of CDs). Novels cost $8 each; CDs cost $6 each. Write down his budget constraint and draw the budget line in the plane (C, N). 2. What happens when the price of a CD rises to $10? Answer graphically by drawing the new budget line. 3. What happens when the price of a CD rises to $10 AND Parvez s income rises to $2? Answer graphically by drawing the new budget line. 4. Now go back to the original assumptions of uestion 1. Suppose that Parvez is spending $120 monthly on paperback novels and used CDs. Suppose that at the uantities he consumes, the marginal utility of a novel is MU N = and the marginal utility of a CD is MU C = 24. Is he maximizing his utility? If not, should he consume (1) more novels and more CDs; (2) more CDs and fewer novels? Explain. 1. I = C + N so 120 = 6C +8N. Sincewewanttodrawthebudgetlineintheplane(C, N), N = C. 2. Since increases, the budget constraint rotates inward. 1

2 3. Overall, the budget set expands. 4. At the optimum, you know that it has to be that: = MU C. Here MU N = 8 =5and MU C = 24 6 =4. Thus MU C < MU N,meaningthatParvezshouldconsumemorenovelsandless CDs. As he does so, the marginal utility of novels will decrease and the marginal utility of CDs will increase until MU C = MU N. MU N Producer Theory Exercise 1 : Marginal Productivity of Accountants The following table shows total output (in tax returns completed per day) of the accounting firm of Hoodwink and Finagle: Number of Accountants Number of Returns per Day Assuming the uantity of capital (consumers, adding machines, desks, etc.) remains constant at all output levels: 1. Calculate the marginal product of each accountant. 2. Over what range of employment do you see increasing returns to labor? Diminishing returns? 3. Explain why MP L might behave this way in the context of an accounting firm. 2

3 Answer Key: 1. Accountants Number of Returns per Day Marginal Product There are increasing returns from 0 to 2 accountants and then for each additional accountant, returns become decreasing. 3. The crucial assumption here is that the uantity of other, non labor input (euipment like computers, storage facilities, printers...) remains fixed. Hence additional accountants might not have access to enough euipment and this would decrease the amount of services that each additional accountant could produce. Exercise 2 : Cost-Minimization in the Short and Long-Run You manage a plant that mass produces engines by teams of workers using assembly machines. The technology is summarized by the production function =4KL where is the number of engines per week, K is the number of assembly machines and L is the number of labor teams. Each assembly machine rents for r = $12, 000 per week and each team costs w = $3, 000 per week. Engine costs are given by the cost of labor teams and machines plus $2,000 per engine for raw materials. Your plant has a fixed installation of 10 assembly machines as part of its design. 1. What is the cost function C() for your plant- namely, how much would it cost to produce engines? What are average cost AC() and marginal cost MC() for producing engines? How do average costs vary with output? 2. How many teams are reuired to produce 80 engines? What is the average cost per engine? 3. You are asked to make recommendations for the design of a new production facility where K can be freely chosen (K is not fixed at 10 anymore). What would you suggest? In particular, what capital/labor ratio should the new plant accommodate? (Hint : What are the marginal product of labor and capital?) 1. You are asked to find a short-run cost function i.e. where the uantity of one input is fixed; here the uestion tells you that K = 10. This means that the short-run production function can be written as = L. This implies that for any level of output, thenumberoflaborteamshired will be L =. The total cost function is thus given by the sum of the costs of capital, labor and raw materials: TC() =rk + wl

4 , TC() = 12, , , TC() = 120, , 075 The average cost function is given by : AC() = TC() = 120,000 +2, 075 and the marginal cost function is just MC() =2, 075. Indeed, when increases by one unit, the total cost increases by 2,075 irrespective of the level of production; in this sense marginal costs are constant. On the other hand, average costs will decrease as uantity increases; this is due to the fixed cost of capital getting divided among more and more units. 2. To produce = 80 engines we need 2 labor teams as L = = 80 =2. Average costs are given by AC(80) = 120, , 075 = We no longer assume that K is fixed at 10. We need to find the combination of K and L which minimizes costs at any output level. Remember that the cost-minimization rule is given by MP L w = MP K r.tofindthemarginalproductofcapital,observethatincreasingk by one unit increases by 4L so PM K =4L. Similarly, increasing L by one unit increases by 4K so MP L =4K. Using these formulas in the cost-minimization rule, we obtain : 4K w = 4L r, K L = w r, K 3, 000 = L 12, 000 = 1 4. Thus the new plant should accommodate a capital/labor ratio of 1 to 4. The firm s capital-labor ratio is currently 10/2=5. To reduce average cost, the firm should either use more labor and less capital to produce the same output or it should hire more labor (keeping the same level of capital) and increase output. Exercise 3 : Returns to scale 1. Consider the production function F (K, L) =3LK 2. Is this production technology CRS, IRS or DRS? 2. Now consider the production function F (K, L) =min{k, L}. Is this production technology CRS, IRS or DRS? 1. F ( K, L) = 3( L)( K) 2 =3L 2 K 2 = 3 3LK 2 > 3LK 2 = F (K, L) for any > 1 ) IRS 2. F ( K, L) =min{ K, L} = min{k, L} = F (K, L) for any > 0 ) CRS 4

5 Exercise 4 : Profit maximization and producer surplus Suppose that a competitive firm has a total cost curve C() = The marginal cost of producing output is thus given by MC() =3+2. Assume that the market price of the firm s product is $9. 1. What level of output will the firm produce? (Hint: Don t forget the shut-down rule). 2. What is the firm s producer surplus? (Hint : Draw P =9and MC() =3+2 on a graph). Answer key : 1. To maximize profits, the firm should first set marginal revenue eual to marginal cost (output rule). Given that the firm is operating in a competitive market, marginal revenue is eual to market price. Thus the firm should set the market price eual to marginal cost : 9=3+2, 6=2 i.e. =3. Now, we need to check whether the candidate =3is better than =0. From your class notes, you know that this is the case if P > AV C( ) (shut-down rule). The average variable cost curve is simply AV C() = 2 +3 = +3. Thus AV C( )= +3=6. Clearly, since P =9,wehavethatP > AV C( ). We conclude that the optimal solution is indeed =3> Producer surplus is eual to the area below the market price of $9.00 and above the marginal cost curve MC() =3+2. Because MC is linear, producer surplus is simply a triangle with a base eual to $6 (9-3=6) and a height eual to 3 ( = 3 where P = MC). Therefore producer surplus is S = = $9. 5

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