1Managerial EconomicsTutorial Solutions – Week 3Hirschey 15th Edition: Chapter 2: Problems P2.6, P2.10 and Chapter 3: Problems P3.3 and P3.4P2.6: Profit Maximization: Equations. 21st Century Insurance offers mail-order automobile insurance topreferred-risk drivers in the Los Angeles area. The company is the low-cost provider of insurance in thismarket but doesn’t believe its annual premium of $1,500 can be raised for competitive reasons. Rates areexpected to remain stable during coming periods; hence, P = MR = $1,500. Total cost for the company isas follows:TC = $41,000,000 + $500Q + $0.005Q2A. Calculate the profit-maximizing activity level.MC = ∂TC/∂Q = $500 + $0.01QSet MR = MC and solve for Q to find the profit-maximizing activity level: MR=MC$1,500=$500 + $0.01Q0.01Q=$1,000Q=100,000 This is a profit maximum because profits are decreasing for Q > 100,000.B. Calculate the company’s optimal profit, and optimal profit as a percentage of sales revenue (profitmargin).The total revenue function for 21st Century Insurance is:TR = P × Q = $1,500QThen, total profit is π=TR – TC=$1,500Q – $41,000,000 – $500Q – $0.005Q2=1,500(100,000) – 41,000,000 – 500(100,000) –0.005(100,0002)=$9,000,000TR=$1,500(100,000) 2 =$150,000,000 or $150 millionProfit Margin=π/TR=$9,000,000/$150,000,000=0.06 or 6 percent P2.10: Average Cost Minimization. Giant Screen TV, Inc., is a Miami-based importer and distributor of60-inch screen HDTVs for residential and commercial customers. Revenue and cost relations are asfollows:TR = $1,800Q – $0.006Q2TC = $12,100,000 + $800Q + $0.004Q2A. Calculate output, marginal cost, average cost, price, and profit at the average cost-minimizingactivity level.MR = ∂TR/∂Q = $1,800 – $0.012QMC = ∂TC/∂Q = $800 +$0.008QTo find the average cost-minimizing level of output, set MC = AC and solve for Q. AC=TC/Q=($12,100,000 + $800Q + $0.004Q2)/Q=$12,100,000/Q + $800 + $0.004QTherefore,MC=AC$800 + $0.008Q=$12,100,000/Q + $800 + $0.004Q0.004Q=12,100,000/Q Because,Q2 = 12,100,000/0.004Q = (12,100,000/0.004)1/2 = 55,000And, MC=$800 + $0.008(55,000) 3 =$1,240AC=$12,100,000/(55,000) + $800 + $0.004(55,000)=$1,240P==TR/Q($1,800Q – $0.006Q2)/Q=$1,800 – $0.006Q=$1,800 – $0.006(55,000)=$1,470π=P × Q – TC = $1,470(55,000) – $12,100,000 – $800(55,000) – $0.004(55,0002)= $12,650,000This is an average-cost minimum because average cost is rising for Q > 55,000.B. Calculate these values at the profit-maximizing activity level.To find the profit-maximizing level of output, set MR = MC and solve for Q(this is also where Mπ = 0): MR=MC$1,800 – $0.012Q=$800 + $0.008Q0.02Q=1,000Q=50,000AndMC=$800 + $0.008(50,000)=$1,200AC=$12,100,000/(50,000) + $800 + $0.004(50,000)=$1,242P=$1,800 – $0.006(50,000)=$1,500 4 π== = =TR – TC$1,800Q – $0.006Q2 – $12,100,000 – $800Q – $0.004Q2-$0.01Q2 + $1,000Q – $12,100,000-$0.01(50,0002) + $1,000(50,000) – $12,100,000=$12,900,000 This is a profit maximum because profit is falling for Q > 50,000.C. Compare and discuss your answers to parts A and B.Average cost is minimized when MC = AC = $1,240. Given P = $1,470, a $230 profit per unit of output isearned when Q = 55,000. Total profit π = $12.65 million.Profit is maximized when Q = 50,000 since MR = MC = $1,200 at that activity level. Since MC = $1,200 $10. To check your answer, calculate quantity at anindustry price of $12 and compare your answer with part B.When P > $10, both Cornell and Penn can profitably supply output. To derive the industry supply curvein this circumstance, simply sum the quantities supplied by each firm:P = $12: QC = -2,500 + 250($12) = 500P = $12: QP = -1,000 + 125($12) = 500Q (industry) = -3,500 + 375($12) = 1,000P3.4 Demand Function. The Creative Publishing Company (CPC) is a coupon book publisher with marketsin several southeastern states. CPC coupon books are sold directly to the public, sold through religiousand other charitable organizations, or given away as promotional items. Operating experience duringthe past year suggests the following demand function for CPC’s coupon books:Q = 5,000 – 4,000P + 0.02Pop + 0.25I + 1.5A,where Q is quantity, P is price ($), Pop is population, I is disposable income per household ($), and A isadvertising expenditures ($).A. Determine the demand faced by CPC in a typical market in which P = $10, Pop = 1,000,000 persons,I = $60,000, and A = $10,000.The demand faced by CPC in a typical market in which P = $10, Pop = 1,000,000persons, I = $60,000, and A = $10,000 is:Q = 5,000 – 4,000P + 0.02Pop + 0.25I + 1.5A= 5,000 – 4,000(10) + 0.02(1,000,000) + 0.25(60,000) + 1.5(10,000)7= 15,000B. Calculate the level of demand if CPC increases annual advertising expenditures from $10,000 to$15,000.If advertising rises from $10,000 to $15,000, CPC demand rises to:Q = 5,000 – 4,000P + 0.02Pop + 0.25I + 1.5A= 5,000 – 4,000(10) + 0.02(1,000,000) + 0.25(60,000) + 1.5(15,000)= 22,500C. Calculate the demand curves faced by CPC in parts A and B.The effect of an increase in advertising from $10,000 to $15,000 is to shift the demandcurve upward following a 7,500 unit increase in the intercept term. If advertising is$10,000, the CPC demand curve is:Q = 5,000 – 4,000P + 0.02(1,000,000) + 0.25(60,000) + 1.5(10,000)= 55,000 – 4,000PThen, price as a function of quantity is:Q = 55,000 – 4,000P4,000P = 55,000 – QP = $13.75 – $0.00025QIf advertising is $15,000, the CPC demand curve isQ = 5,000 – 4,000P + 0.02(1,000,000) + 0.25(60,000) + 1.5(15,000)= 62,500 – 4,000PThen, price as a function of quantity is:Q = 62,500 – 4,000P4,000P = 62,500 – QP = $15.625 – $0.00025Q

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