1Managerial EconomicsTutorial Solutions – Week 5Hirschey 15th Edition: Chapter 7: Problem P7.4 + Chapter 8: Question Q8.2 and Problem P8.8P7.4 Returns to Scale. Determine whether the following production functions exhibit constant,increasing, or decreasing returns to scale.A. Q = 0.5X + 2Y + 40ZInitially, let X = Y = Z = 100, so output is:Q = 0.5(100) + 2(100) + 40(100) = 4,250Increasing all inputs by an arbitrary percentage, say 2 percent, leads to:Q = 0.5(102) + 2(102) + 40(102) = 4,335Because a 2 percent increase in all inputs results in a 2 percent increase in output, the outputelasticity is 1 and the production system exhibits constant returns to scale.B. Q = 3L + 10K + 500Initially, let L = K = 100, so output is:Q = 3(100) + 10(100) + 500 = 1,800Increasing both inputs by an arbitrary percentage, say 3 percent, leads to:Q = 3(103) + 10(103) + 500 = 1,839Because a 3 percent increase in both inputs results in a 2.2 percent increase in output, the outputelasticity is less than 1 and the production system exhibits diminishing returns to scale.C. Q = 4A + 6B + 8ABInitially, let A = B = 100, so output is:Q = 4(100) + 6(100) + 8(100)(100) = 81,000Increasing both inputs by an arbitrary percentage, say, 1 percent, leads to:Q = 4(101) + 6(101) + 8(101)(101) = 82,618Because a 1 percent increase in both inputs results in a 2 percent increase in output, the outputelasticity is greater than 1 and the production system exhibits increasing returns to scale.2D. Q = 7L2 + 5LK + 2K2Initially, let L = K = 100, so output is:Q = 7(100)2 + 5(100)(100) + 2(100)2 = 140,000Increasing both inputs by an arbitrary percentage, say, 2 percent, leads to: Q =7(102)2 + 5(102)(102) + 2(102)2 = 145,656Because a 2 percent increase in both inputs results in a 4 percent increase in output, the outputelasticity is greater than 1 and the production system exhibits increasing returns to scale.E. Q = 10L0.5K0.3Initially, let L = K = 100, so output is:Q = 10(100)0.5(100)0.3 = 398Increasing both inputs by an arbitrary percentage, say, 4 percent, leads to:Q = 10(104)0.5(104)0.3 = 411Because a 4 percent increase in both inputs results in a 3.3 percent increase in output, the outputelasticity is less than 1 and the production system exhibits decreasing returns to scale.Q8.2 Assume that two years ago, you purchased a new Jeep Wrangler SE 4WD with a soft top for $16,500using five-year interest-free financing. Today, the remaining loan balance is $9,900 and your Jeep has atrade-in value of $9,500. What is your opportunity cost of continuing to drive the Jeep? Discuss thefinancing risk exposure of the lender.The opportunity cost of continuing to drive the Jeep is $9,500. If you sell the Jeep, $9,500 can begenerated to pay down your remaining loan balance. It is the current cost or replacement value of yourcurrent vehicle. It is the relevant economic cost of continuing to drive the Jeep. Historical cost of $16,500and the remaining loan balance of $9,900 are irrelevant for decision-making purposes. With a currentmarket value of only $9,500 against a remaining loan balance of $9,900, the lender faces the risk ofborrower default. Aggressive interest-free financing offered by the major automakers has the potentialto create big debt collection problems in the future.3P8.8 Multiplant Operation. Appalachia Beverage Company, Inc. is considering alternative proposals forexpansion into the Midwest.Alternative 1: Construct a single plant in Indianapolis, Indiana, with a monthly production capacity of300,000 cases, a monthly fixed cost of $262,500, and a variable cost of $3.25 per case.Alternative 2: Construct three plants, one each in Muncie, Indiana; Normal, Illinois; and Dayton, Ohio,with capacities of 120,000, 100,000, and 80,000, respectively, and monthly fixed costs of $120,000,$110,000, and $95,000 each. Variable costs would be only $3 per case because of lower distributioncosts. To achieve these cost savings, sales from each smaller plant would be limited to demand withinits home state. The total estimated monthly sales volume of 200,000 cases in these three Midwesternstates is distributed as follows: 80,000 cases in Indiana, 70,000 cases in Illinois, and 50,000 cases in Ohio.A. Assuming a wholesale price of $5 per case, calculate the breakeven output quantities foreach alternative. (NOTE: Use the Break-Even formula: Q = TFC / (P – AVC)The breakeven output quantity for the single plant alternative is:Q = TFC / (P – AVC)=$262,500 / ($5 – $3.25)= 150,000 cases per monthThe breakeven output quantities for the multiple plant alternative is:QMuncie =$120,000/ ($5 – $3)= 60,000 cases per monthQNormal =$110,000/($5 – $3)= 55,000 cases per monthQDayton =$95,000/($5 – $3)= 47,500 cases per monthThus, the firm-level breakeven quantity for the multiple plant alternative is:Q = 60,000 + 55,000 + 47,500= 162,500 cases per monthprovided that demand was distributed among the states in amounts equal to the breakeven quantitiesfor each individual plant.B. At a wholesale price of $5 per case in all states, and assuming sales at the projected levels, whichalternative expansion scheme provides Appalachia with the highest profit per month?4Single plant alternative:π = TR – TC= P × Q – TFC – AVC × Q= $5(200,000) – $262,500 – $3.25(200,000)= $87,500Multiple plant alternative:π = TR – TC= P X Q – TFCM – TFCN – TFCD – AVC X Q= $5(200,000) – $120,000 – $110,000 – $95,000 – $3(200,000)= $75,000Management would prefer the single plant alternative because of its greater profitability.C. If sales increase to production capacities, which alternative would prove to bemore profitable?Single plant at full capacity:π = TR – TC= P × Q – TFC – AVC × Q= $5(300,000) – $262,500 – $3.25(300,000)= $262,500Multiple plants at full capacity:π = TR – TC= P × Q – TFCM – TFCN – TFCD – AVC × Q= $5(300,000) – $120,000 – $110,000 – $95,000 – $3(300,000)= $275,000At peak capacity, management would prefer the multiple plant option because of its greaterprofitability.

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