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Firms, Prices & Markets Timothy Van Zandt © August 2012 Chapter 5 Short-Run Costs and Prices SOLUTIONS TO EXERCISES Exercise 5.1. This exercise asks you to provide a graphical illustration of overshooting. It will test your understanding of the distinction between short-run and long-run supply. Figure E5.1 shows the same initial scenario as Figure 5.5. The current equilibrium quantity and price are Qcurrent and Pcurrent . Then the demand curve shifts from d current (P ) to d new (P ), causing the price to rise in the short run to Pshort . Figure E5.1 P Short-run s(P ) d new (P ) Short-run s(P ; Q∗long ) d current (P ) Long-run s(P ) Pshort P long ∗ Pshort Pcurrent Qcurrent Qshort Qlong Q∗short Q∗long Q a. Suppose that all the firms believe the market price is going to stay at Pshort for the long term, so they initiate investments and adjustments to production processes and inputs in order to maximize profit given Pshort . Mark on the graph the intended long-run total output of the firms. Solution: Q∗long in Figure E5.1. b. After all these changes and investments are in place, the firms have new short-run cost curves that determine the new short-run supply curve. Draw in a plausible short-run supply curve. Solution: See s(P ; Q∗long ) in Figure E5.1. c. The new short-run equilibrium must be at the intersection of the new short-run supply curve and the new demand curve. Mark on the graph the short-run output and price levels. Solution: ∗ See Q∗short and Pshort in Figure E5.1. Firms, Prices & Markets • Solutions for Chapter 5 (Short-Run Costs and Prices) 2 d. Is the market now in a long-run equilibrium? Specifically, compare the resulting price to that which obtains in the long run when the firms do not overshoot. Do the firms regret their investments? ∗ Solution: The price is lower than if the firms had not overinvested (Pshort < P long ). ∗ ). The firms regret their investments because they presumed a higher price (Pshort > Pshort The market is not in long-run equilibrium. The firms will cut back their fixed inputs (less myopically, we would hope). Exercise 5.2. Consider the numerical example in this section. Suppose that the market starts out in long-run equilibrium with the demand curve d 0 , so that capacity is 700. Then a tax of Τ = €2 per unit is imposed. a. First model the new long-run equilibrium, treating the tax as a shift in the demand curve. What is the new long-run equilibrium price? Draw a graph showing the long-run supply curve, the initial and shifted demand curves, and the current and new long-run equilibria. Solution: Long-run equilibrium after the tax. The long-run supply curve is perfectly elastic. Therefore, there is no change in the price received by the sellers. The price paid by the buyers (including the tax) increases by the amount of the tax. The demand at this price b denotes what the is then the new equilibrium quantity. See Figure S1. In this figure, P long s buyer pays, including the tax; P long denotes what the seller receives, net of tax; d new is the shifted demand curve with the tax, when treating the tax as paid by the buyers. Figure S1 € b P long s Pcurrent , P long ⎧ ⎪ ⎪ ⎨ MCk ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ MCv ⎪ ⎪ ⎪ ⎪ ⎩ 11 Long-run 10 mc(Q), s(P ) 9 8 Τ 7 6 5 d current 4 3 d new 2 1 100 200 300 400 500 Qlong 600 700 Qcurrent 800 900 Q b. Next show what happens in the short run. What is the short-run price and quantity? Draw a graph showing the short-run supply curve and illustrate the equilibrium before and after the tax. Firms, Prices & Markets • Solutions for Chapter 5 (Short-Run Costs and Prices) 3 Solution: Short-run equilibrium after the tax. Supply is perfectly inelastic—with firms operating at full capacity—as long as the price does not drop below MCv . This does not happen because Τ < MCk . Therefore, output and demand do not change. The price paid by the consumers (including the tax) must stay the same; the price received by the sellers drops by Τ. See Figure S2. Figure S2 € Short-run mcs (Q), s s (P ) 11 Long-run mc(Q), s(P ) 10 9 8 s ⎧ Pcurrent , Pshort ⎪ ⎪ ⎨ MCk ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ MCv ⎪ ⎪ ⎪ ⎪ ⎩ 7 6 Τ s 5 Pshort d current 4 3 d new 2 1 100 200 300 400 500 600 700 800 900 Qshort , Qcurrent Q Exercise 5.3. This is a numerical example of the comparison between short-run and long-run costs. The production function is f (L, M) = L1/2 M 1/2 , where “M” = machines and “L” = labor. Suppose that PL = PM = 1. a. The cost-minimizing long-run input mix given this production function and these prices is to have equal amounts of L and M. Derive from this information the cost function. (See how many units of L and M you would need to produce Q units; then c (Q) is the cost of these inputs at the prices PL = PM = 1.) Solution: Need 1 unit of L and 1 of M for each unit of output. ⇒ c(Q) = 2Q. b. What are the AC and MC curves? Are there (dis)economies of scale? Solution: Constant AC = MC = 2. No economies of scale. c. Suppose that initially Qcurrent = 4; then there is a shift in demand (or some other change that makes the firm want to adjust its production). In the short run, M is a fixed input and L is a variable input. Determine how many units of M are employed initially. What is the short-run FC? Solution: To produce Qcurrent = 4, the firm used M = L = 4. Therefore, until the firm can adjust its machines, M is fixed at 4. These cost 1 per unit (“rental” cost); hence FC = 4. d. Given the fixed amount of M just determined, derive output as a function of L. € Firms, Prices & Markets • Solutions for Chapter 5 Solution: (Short-Run Costs and Prices) 4 Output can only be changed by adjusting L, such that Q = 2L1/2 . e. Invert this function in order to find the amount of L needed to produce Q units. Solution: Solving for L as a function of Q: L = Q2 /4 f. The variable cost is the cost of the labor. Derive variable cost as a function of Q. (This is so trivial you may wonder if you got the right answer.) Solution: Since L also costs just 1 per unit, vcs (Q) = Q2 /4. g. Add the short-run fixed cost and the variable cost to obtain the short-run total cost curve. Solution: c s (Q) = 4 + Q2 /4. h. Graph the short-run and long-run cost curves on the same axis. Solution: Figure S3 Short-run c s (Q) 20 Long-run c (Q) 10 PM Mcurrent Qcurrent i. What is the short-run marginal cost curve? Solution: mcs (Q) = Q/2. Q Firms, Prices & Markets • Solutions for Chapter 5 (Short-Run Costs and Prices) j. Graph the short-run and long-run marginal cost curves on the same axis. Solution: Figure S4 € 4.5 Short-run mcs (Q) 4.0 3.5 3.0 2.5 2.0 Long-run mc(Q) 1.5 1.0 0.5 Qcurrent k. Suppose, for example, that output is expanded from Q = 4 to Q = 8. What is the total cost in the short run? What is the total cost in the long run? Solution: Short run: Need L = 16 for a total cost of 20 = c s (8) Long run: Use L = 8 and M = 8 for total cost 16 = c(8). 5