Econ 100A Solution Set for Homework #4, Due April 3 / 4. 1. A specific tax of $1 causes both the average and marginal cost curves of a competitive firm to shift up by $1 (assume that the firm starts out in long-run equilibrium): We know that the q* which each firm produces in the long run is the q minimizing AC, which occurs at the q where AC has zero slope and MC = AC. Since AC2 = AC1 + 1, the slope of AC2 is zero when: 0)(0)()1()(1112==+=+=dqACddqACdACdqddqACd. Thus the q* which minimizes AC2 is the same q* that minimizes AC1. The optimal scale of a competitive firm will NOT change. What will change is the number of firms in the industry: since total quantity sold in equilibrium decreases from Q1 to Q2 but the q* for each firm stays constant, the number of firms declines after the tax is imposed. 2. A usury law sets a maximum interest rate that lenders can charge to loan money. It acts like a price ceiling at i2: i p q Q D p1 S1 S2 p2 q* Q2 Q1 MC1 MC2 AC1 AC2 interest rate ceiling D S i1 i2 Q1Q2DQ2SD C A B E p Q(i) effects on equilibrium: i decreases from i1 to i2 the number of loans decreases from Q1 to Q2S borrowers demand Q2D, which implies an excess demand of Q2D – Q2S for loans after the usury law is enacted (ii) effects on welfare: no law with law change CS A+D A+B B-D PS B+C+E C -(B+E) W A+B+C+D+E A+B+C -(D+E) Thus some borrowers (“consumers”) gain since they get a loan at the lower rate of i2. Some borrowers lose because of the excess demand: they can’t get a loan at i2, even though they’d like one. Lenders (“producers”) lose since they make fewer loans in equilibrium and loan at a lower rate than before the law was enacted. Note: CS decreases if and only if B < D (this is the condition for the group of consumers as a whole to lose). 3. Demand facing monopoly: Q = 200 – 2p Þ p = 100 – ½*Q Monopoly’s cost curve: C(Q) = 40Q + 10 A. find equilibrium p & Q for monopoly: set MC = MR. QQQdQddQQQpddQdRMRdQQdCMC−=−=====100)21100()*)((40)(2 40 = 100 – Q Þ Qm = 60 pm = 100 – ½*Qm = 100 – 30 = 70 B. monopoly’s profit at profit-maximizing output: Πm = pmQm - C(Qm) = 60*70 – (40*60 + 10) = 4200 – 2410 = 1790 C. outcome if regulator forced firm to set p = MC: 100 – ½*Q = 40 ½*Q = 60 Þ Qc = 120 pc = 100 – ½*Qc = 100 – 60 = 40 Πc = pcQc - C(Qc) = 120*40 – (40*120 + 10) = 4800 – 4810 = -10 So if forced to follow the rule for a competitive firm, the monopoly would make a negative profit. 4. Monopoly can advertise at cost of A dollars and shift demand curve to Q = 400 – 2p. Þ p = 200 – ½*Q & C(Q, A) = 40Q + 10 + A A. new pm,A & Qm,A: set MC = MR 40 = 200 – QÞ Qm,A = 160 pm,A = 200 – 80 = 120 B. largest A such that the monopoly will advertise: need Πwith A ≥ Πno A Þ pm,AQm,A – C(Qm,A, A) ≥ pmQm – C(Qm) = 1790 120*160 – (40*160 +10 + A) ≥ 1790 19200 – 6410 – A ≥ 1790 A ≤ 11000 5. p = 20, MC = 5. What is elasticity of demand for profit-maximizing monopoly? For profit-maximizing monopoly, )11(ε+== pMRMC . Thus εε2020)11(205 +=+= ε2015 =−, or ε = -4/3. One could alternatively use the Lerner index formula: ε1−=−pMCp, and get the same result. 6. MC and AC increase for monopoly producing bomb-detecting equipment in airports. What is effect of this change on the monopoly equilibrium and on welfare? The monopoly equilibrium price increases and equilibrium quantity decreases. MC1 MC2 AC1 AC2 p Q MR D p2 p1 Q2 Q1To see the effects on welfare, we need to examine how the monopoly equilibrium shifts: Consumer surplus decreases from the triangle between p1 and D to the triangle between p2 and D, so consumers are unambiguously worse off. Producer surplus goes from the area below p1 and above S1 (and above the dotted horizontal line at the minimum price of S1) to the area below p2 and above S2 (and its corresponding horizontal dotted line at the minimum price of S2). The effect on producer surplus, however, is clear: producer surplus decreases. Informal proof: When costs are described by AC1, monopoly producer surplus is greater at p = p1 than at p = p2. When costs are described by AC2, monopoly producer surplus where p = p2, is now less than it would have been were p = p2 when costs were AC1. In other words, let PSP,C denote producer surplus given cost curve C and price P. PS2,2 < PS2,1 < PS1,1. Total welfare, the sum of consumer and producer surplus, decreases. It is originally everything between S1 and D, to the left of Q1; after the shift it is everything between S2 and D, to the left of Q2. 7. Video cassette manufacturers are engaging in price discrimination. The marginal cost of producing the cassettes is the same, whether or not they are popular. They know that both movie rental stores and individual families will buy popular movies, but only rental stores, whose demand is more inelastic, will buy less popular movies. Thus the demand for the unpopular movies is more inelastic. The manufacturers charge a lower price to induce more families to buy the popular movies, but charge the higher price for movies that only the group with more inelastic demand (the rental stores) will buy. 8. pSR = 5; pothers = 10; MC same for both groups. What can you say about demand elasticities of these groups? Using the formula for multimarket price discrimination when there are 2 groups: S1 S2 p Q MR D p2 p1 Q2 Q1SRDSRDothersDothersDothersDSRDothersDSRDothersothersDothersSRDSRSRMRpMCpMR,,,,,,,,,,12)2()11(10)11(5)11()11(εεεεεεεεεε−=+=+=+=+==+= Thus no matter what MC is, |εD,SR| > |εD, others|. If the elasticity of the senior citizens is –5, what’s the elasticity of the other group? εD,SR = -5 Þ )5(1)5(2,−−−=othersDε Thus εD, others =
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