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### Me11e 08

1. 1. CHAPTER 5—PRODUCTION POLICY MULTIPLE CHOICE1. The production function Q = 0.25X0.5Y exhibits: a. constant returns to scale.b.increasing returns to scale.c.increasing and then diminishing returns to scale.d.diminishing returns to scale. ANS: B The law of diminishing returns: a.deals specifically with the diminishing marginal product of fixed input factors.b.states that the marginal product of a variable factor must eventually decline as increasingly more is employed.c.can be derived deductively.d.states that as the quantity of a variable input increases, with the quantities of all other factors being held constant, the resulting output must eventually diminish. ANS: B A new production function results following: a.a new wage agreement following collective bargaining.b.a surge in product demand.c.a decrease in the availability of needed inputs.d.the successful completion of a training program that enhances worker productivity. ANS: D The relation between output and the variation in all inputs taken together is the: a.factor productivity of a production system.b.law of diminishing returns.c.returns to scale characteristic of a production system.d.returns to factor characteristic of a production system. ANS: C When PX = \$60, MPX = 5 and MPY = 2, relative employment levels are optimal provided: a.PY = 16.7¢.b.PY = \$24.c.PY = \$60.d.PY = \$150. ANS: B When PX = \$100, MPX = 10 and MRQ = \$5, the marginal revenue product of X equals: a.\$100.b.\$50.c.\$10.d.\$5. ANS: B The returns to scale characteristic of a production system: a.is measured by the way in which inputs can be varied in an unbroken marginal fashion rather than incrementally.b.illustrates the distinct, or "lumpy," pattern of input combination.c.shows the relation between output and the variation in all inputs.d.is the relation between output and variation in only one of the inputs employed. ANS: C Returns to a factor denotes the relation between the quantity of an individual input employed and the: a.optimal scale of a firm.b.optimal size of production facilities.c.optimal length of production runs.d.level of output produced.
2. 2. ANS: DThe marginal product concept is:a.used to describe the relation between output and variation in all inputs in a productionfunction.b.the change in output associated with a one-unit change in an individualfactor.c.total product divided by the number input units employed.d.the complete outputfrom a production system.ANS: BA production function describes the relation between output and:a.technical progress.b.one input.c.total cost.d.all inputs.ANS: DTotal product divided by the number of units of variable input employed equals:a.average product.b.marginal revenue product.c.returns to scale.d.marginal product.ANS: AMarginal product is the change in output associated with a unit change in:a.all inputs.b.technology.c.scale.d.one input factor.ANS: DWhen the slope of the average product curve equals zero:a.total product is maximized.b.returns to the variable input are increasing.c.marginalproduct equals average product.d.marginal product equals zero.ANS: CTotal output is maximized when:a.average product equals zero.b.marginal product is maximized.c.average product ismaximized.d.marginal product equals zero.ANS: DAn isoquant represents:a.input combinations that can be employed at the same cost.b.input combinations that canefficiently produce the same output.c.output combinations that can be efficiently producedusing the same input combination.d.output combinations that can be produced for thesame cost.ANS: BRight-angle shaped isoquants reflect inputs that are:a.perfect complements.b.perfect substitutes.c.imperfect substitutes.d.inefficient.ANS: AThe marginal rate of technical substitution is:a.the slope of the marginal revenue product curve.b.the marginal product of eitherinput.c.minus one times the ratio of marginal products for each input.d.the slope of anisocost curve.ANS: CMarginal revenue product equals:
3. 3. a.marginal revenue multiplied by marginal product.b.marginal product multiplied by totalrevenue.c.total revenue multiplied by total product.d.marginal revenue multiplied by totalproduct.ANS: AA firm will maximize profits by employing the quantity of each input where the marginal:a.revenue product of each input equals its price.b.revenue equals the price of eachinput.c.product of each input is equal.d.product of each input equals its price.ANS: AIf tripling the quantities of all inputs employed doubles the quantity of output produced,the output elasticity:a.equals one.b.is greater than one.c.cannot be determined without further information.d.isless than one.ANS: DThe maximum output that can be produced for a given amount of input is called a:a.discrete production function.b.production function.c.continuous productionfunction.d.discontinuous production function.ANS: BThe output effect of a proportional increase in all inputs is called:a.returns to scale.b.returns to a factor.c.total product.d.marginal product.ANS: AAs the quantity of a variable input increases, the resulting rate of output increaseeventually:a.falls.b.rises.c.becomes constant.d.none of these.ANS: AEconomic efficiency is achieved when all firms equate the marginal:a.product and price for all inputs.b.cost of all inputs.c.revenue product and price for allinputs.d.product of all inputs.ANS: CWhen MRQ = \$25, PX = \$200, and MPX = 8, employment of X:a.is optimal.b.should expand.c.should contract.d.none of these.ANS: APROBLEMInput Combination. The following production table provides estimates of the maximumamounts of output possible with different combinations of two input factors, X and Y.(Assume that these are just illustrative points on a spectrum of continuous inputcombinations.) Units of Y
4. 4. Used Estimated Output perDay525836045554262042343324164965423206294372416455216824829433236011241 6820623425812345Units of X Used A.Do the two inputs exhibit the characteristics of constant, increasing, or decreasing marginal rates of technical substitution? How do you know?B.Assuming output sells for \$4 per unit, complete the following tables:X Fixed at 2 Units Units of Y Used Total Product of Y Marginal Product of Y Average Product of YMarginal Revenue Product of Y12345Fixed at 3 units Units of X Used Total Product of X Marginal Product of X Average Product of XMarginal Revenue Product of X12345C.Assume the quantity of X is fixed at 2 units. If the output of the production system sellsfor \$4 and the cost of Y is \$155 per day, how many units of Y will be employed?D.Assume that the company is currently producing 258 units of output per day using 1unit of X and 5 units of Y. The daily cost per unit of X is \$155 and that of Y is also \$155.Would you recommend a change in the present input combination? Why or why not?E.What is the nature of the returns to scale for this production system if the optimal inputcombination requires that X = Y?ANS:A.The inputs exhibit the characteristics of decreasing marginal rates of technicalsubstitution throughout. For decreasing MRTS, the slope of the production isoquantsdiminishes as one input is increasingly substituted for another. We can also see this pointalgebraically by holding X or Y constant in the input-output matrix and noting the declinein the relative marginal product of the other input as its usage level grows. B.X Fixed at 2 UnitsUnits of Y
5. 5. Used TPY MPY APY MRPY(1)(2)(3)(4) = \$4 × (2)1168168168\$672224880124320329446981844332388315253602872112Fixed at 3 unitsUnits of X Used TPX MPX APX MRPX(1)(2)(3)(4) = \$4 × (2)1206206206\$82422948814735233727812431244164410417654553991156C.Three units of Y will be employed. The marginal value of the first three units of Y isgreater than their marginal cost. The marginal value of the fourth unit is only \$152 or \$3less than its cost, and hence, the firm would employ no more than three units of Y.D.A change would be in order because the firm could produce 372 units at the same costusing 3 units of each output. That is, the marginal product to price ratios of the two inputsare not equal at the current input proportions. Relatively less Y and more X is needed toprovide an optimal combination. E.The system exhibits constant returns to scale. This is true because a given increase in both inputs causes an increases in output of the same proportion.XYOutput11124 × 1 = 12422124 × 2 = 24833124 × 3 = 37244124 × 4 = 49655124 × 5 = 620Input Combination. The following production table provides estimates of the maximumamounts of output possible with different combinations of two input factors, X and Y.(Assume that these are just illustrative points on a spectrum of continuous inputcombinations.) Units of Y Used Estimated Output perDay518426533439544041762483033523953164216264303334212817621624826518812 816417618412345Units of X UsedA.Do the two inputs exhibit the characteristics of constant, increasing, or decreasingmarginal rates of technical substitution? How do you know?B.Assuming output sells for\$3 per unit, complete the following tables: X Fixed at 4 Units Units of Y Used Total Product of Y Marginal Product of Y Average Product of YMarginal
6. 6. Revenue Product of Y12345 Fixed at 2 units Units of X Used Total Product of X Marginal Product of X Average Product of XMarginal Revenue Product of X12345C.Assume the quantity of X is fixed at 4 units. If the output of the production system sellsfor \$3 and the cost of Y is \$135 per day, how many units of Y will be employed?D.Assume that the company is currently producing 248 units of output per day using 2units of X and 4 units of Y. The daily cost per unit of X is \$135 and that of Y is also \$135.Would you recommend a change in the present input combination? Why or why not?E.What is the nature of the returns to scale for this production system if the optimal inputcombination requires that X = Y?ANS:A.The inputs exhibit the characteristics of decreasing marginal rates of technicalsubstitution throughout. For decreasing MRTS, the slope of the production isoquantsdiminishes as one input is increasingly substituted for another. We can also see this pointalgebraically by holding X or Y constant in the input-output matrix and noting the declinein the relative marginal product of the other input as its usage level grows. B.X Fixed at 4 UnitsUnits of Y employed TPY MPY APY MRPY(1)(2)(3)(4) = \$3 ×(2)1176176176\$5282248721242163303551011654352498814753954379129Y Fixed at 2 UnitsUnits of X employed TPX MPX APX MRPX(1)(2)(3)(4) = \$3 × (2)1128128128\$384217648881443216407212042483262965265175351C.Four units of Y will be employed. The marginal value of the first four units of Y isgreater than their marginal cost. The marginal value of the fifth unit is only \$129 or \$6 lessthan its cost, and hence, the firm would employ no more than four units of Y.D.A change would be in order because the firm could produce 303 units at the same cost
7. 7. using 3 units of each output. That is, the marginal product to price ratios of the two inputsare not equal at the current input proportions. Relatively less Y and more X is needed toprovide an optimal combination. E.The system exhibits constant returns to scale. This is true because a given increase in both inputs causes an increases in output of the same proportion.XYOutput1188 × 1 = 882288 × 2 = 1763388 × 3 = 2644488 × 4 = 3525588 × 5 = 440Production Relations. Indicate whether each of the following statements is true or false.A.L-shaped isoquants describe production systems where inputs are perfectcomplements.B.If the marginal product of capital increases as capital usage grows, thereturns to capital are decreasing.C.Marginal revenue product measures the output gainedthrough expanding input usage.D.The marginal rate of technical substitution will beaffected by a given percentage increase in the marginal productivity of allinputs.E.Increasing returns to scale and declining average costs are indicated when εQ > 1.ANS:A.True. L-shaped production isoquants reflect a perfect complementary relation amonginputs, i.e., no amount of input X can make up for the lack of input Y.B.False. Returns tothe capital input factor are increasing when the marginal product of capital increases ascapital usage grows.C.False. Marginal revenue product is the revenue generated byexpanding input usage, and represents the maximum that could be paid to expand usage.Marginal product measures the change in output given a change in an input.D.False. Themarginal rate of technical substitution is measured by the relative marginal productivity ofinput factors. This relation is unaffected by a commensurate increase in the marginalproductivity of all inputs.E.True. When εQ > 1, the percentage change in output is greaterthan a given percentage change in all inputs. Thus, increasing returns to scale anddecreasing average costs are indicated.Production Relations. Indicate whether each of the following statements is true or false.A.If the marginal product of capital decreases as capital usage grows, the returns to capitalare decreasing.B.The marginal rate of technical substitution will be affected by a givenpercentage increase in the marginal productivity of an input.C.Marginal revenue productrepresents the minimum revenue amount required to expand usage.D.Linear isoquantsdescribe production systems where inputs are perfect complements.E.Decreasing returnsto scale and declining average costs are indicated when εQ < 1.ANS:A.True. Returns to the capital input factor are decreasing when the marginal product ofcapital decreases as capital usage grows.B.True. The marginal rate of technicalsubstitution is measured by the relative marginal productivity of input factors. Thisrelation is affected by an increase in the marginal productivity of a single input.C.True.Marginal revenue product is the revenue generated by expanding input usage, andrepresents the minimum revenue required to expand usage.D.False. L-shaped productionisoquants reflect a perfect complementary relation among inputs, i.e., no amount of inputX can make up for the lack of input Y. Linear isoquants reflect perfect substitutability ofinput X for input Y, and vice versa.E.False. When ε Q < 1, the percentage change in outputis less than the percentage change in all inputs, implying decreasing returns to scale butincreasing average costs.
8. 8. Returns to Scale. Determine whether the following production functions exhibit constant,increasing, or decreasing returns to scale.A.Q = 0.25X + 5Y + 30ZB.Q = 4L + 15K + 600C.Q = 9A + 3B + 12ABD.Q = 4L2 + 6LK+ 3K2E.Q = 2L0.2K0.6ANS:A.Initially, let X = Y = Z = 100, so output is:Q1 = 0.25(100) + 5(100) + 30(100) =3,525Increasing all inputs by 4% leads to:Q2 = 0.25(104) + 5(104) + 30(104) =3,666Because a 4% increase in all inputs results in a 4% increase in output (Q 2/Q1 =3,666/3,525 = 1.04), the output elasticity is 1 and the production system exhibits constantreturns to scale.B.Initially, let L = K = 100, so output is:Q1 = 4(100) + 15(100) + 600 = 2,500Increasingboth inputs by 5% leads toQ2 = 4(105) + 15(105) + 600 = 2,595Because a 5% increase inboth inputs results in a 3.8% increase in output (Q2/Q1 = 2,595/2,500 = 1.038), the outputelasticity is less than 1 and the production system exhibits diminishing returns to scale.C.Initially, let A = B = 100, so output is:Q1 = 9(100) + 3(100) + 12(100)(100) =121,200Increasing both inputs by 1% leads to:Q2 = 9(101) + 3(101) + 12(101)(101) =123,642Because a 1% increase in both inputs results in a 2% increase in output (Q 2/Q1 =123,624/121,200 = 1.02), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.D.Initially, let L = K = 100, so output is:Q1 = 4(1002) + 6(100)(100) + 3(1002) =130,000Increasing both inputs by 2% leads to:Q2 = 4(1022) + 6(102)(102) + 3(1022) =135,252Because a 2% increase in both inputs results in a 4% increase in output (Q 2/Q1 =135,252/130,000 = 1.04), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.E.Initially, let L = K = 100, so output is:Q1 = 2(1000.2)(1000.6) = 80Increasing both inputsby 4% leads to:Q2 = 2(1040.2)(1040.6) = 82Because a 4% increase in both inputs results in aless than 4% increase in output (Q2/Q1 = 82/80 = 1.025), the output elasticity is less than 1and the production system exhibits decreasing returns to scale.Returns to Scale. Determine whether the following production functions exhibit constant,increasing, or decreasing returns to scale.A.Q = 25X + 0.5Y + 8ZB.Q = 9L + 5K - 400C.Q = 10A + 7B + 4ABD.Q = 6L 2 + 3LK +2K2E.Q = 2L0.4K0.6ANS:A.Initially, let X = Y = Z = 100, so output is:Q1 = 25(100) + 0.5(100) + 8(100) =3,350Increasing all inputs by 4% leads to:Q2 = 25(104) + 0.5(104) + 8(104) =3,484Because a 4% increase in all inputs results in a 4% increase in output (Q 2/Q1 =3,484/3,350 = 1.04), the output elasticity is 1 and the production system exhibits constantreturns to scale.B.Initially, let L = K = 100, so output is:Q1 = 9(100) + 5(100) - 400 = 1,000Increasing bothinputs by 5% leads toQ2 = 9(105) + 5(105) - 400 = 1,070Because a 5% increase in bothinputs results in a 7% increase in output (Q2/Q1 = 1,070/1,000 = 1.07), the output elasticityis greater than 1 and the production system exhibits increasing returns to scale.
9. 9. C.Initially, let A = B = 100, so output is:Q1 = 10(100) + 7(100) + 4(100)(100) =41,700Increasing both inputs by 1% leads to:Q2 = 10(101) + 7(101) + 4(101)(101) =42,521Because a 1% increase in both inputs results in a 2% increase in output (Q 2/Q1 =42,521/41,700 = 1.02), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.D.Initially, let L = K = 100, so output is:Q1 = 6(1002) + 3(100)(100) + 2(1002) =110,000Increasing both inputs by 2% leads to:Q2 = 6(1022) + 3(102)(102) + 2(1022) =114,444Because a 2% increase in both inputs results in a 4% increase in output (Q 2/Q1 =114,444/110,000 = 1.04), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.E.Initially, let L = K = 100, so output is:Q1 = 2(1000.4)(1000.6) = 200Increasing both inputsby 4% leads to:Q2 = 2(1040.4)(1040.6) = 208Because a 4% increase in both inputs results ina 4.0% increase in output (Q2/Q1 = 208/200 = 1.04), the output elasticity is 1 and theproduction system exhibits constant returns to scale.Returns to Scale. Determine whether the following production functions exhibit constant,increasing, or decreasing returns to scale.A.Q = 2X + 25Y + 5ZB.Q = 3A + 5B - 200C.Q = 5A + 6B + 3ABD.Q = 4L2 - 3LK +2K2E.Q = 4L0.4K0.8ANS:A.Initially, let X = Y = Z = 100, so output is:Q1 = 2(100) + 25(100) + 5(100) =3,200Increasing all inputs by 4% leads to:Q2 = 2(104) + 25(104) + 5(104) = 3,328Becausea 4% increase in all inputs results in a 4% increase in output (Q 2/Q1 = 3,328/3,200 = 1.04),the output elasticity is 1 and the production system exhibits constant returns to scale.B.Initially, let L = K = 100, so output is:Q1 = 3(100) + 5(100) - 200 = 600Increasing bothinputs by 5% leads toQ2 = 3(105) + 5(105) - 200 = 640Because a 5% increase in bothinputs results in a 6.7% increase in output (Q2/Q1 = 640/600 = 1.067), the output elasticityis greater than 1 and the production system exhibits increasing returns to scale.C.Initially, let A = B = 100, so output is:Q1 = 5(100) + 6(100) + 3(100)(100) =31,100Increasing both inputs by 1% leads to:Q2 = 5(101) + 6(101) + 3(101)(101) =31,714Because a 1% increase in both inputs results in a 2% increase in output (Q 2/Q1 =31,714/31,300 = 1.02), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.D.Initially, let L = K = 100, so output is:Q1 = 4(1002) - 3(100)(100) + 2(1002) =30,000Increasing both inputs by 2% leads to:Q2 = 4(1022) - 3(102)(102) + 2(1022) =31,212Because a 2% increase in both inputs results in a 4% increase in output (Q 2/Q1 =31,212/30,000 = 1.04), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.E.Initially, let L = K = 100, so output is:Q1 = 4(1000.4)(1000.8) = 1,005Increasing bothinputs by 4% leads to:Q2 = 4(1040.4)(1040.8) = 1,053Because a 4% increase in both inputsresults in a 4.8% increase in output (Q2/Q1 = 1,053/1,005 = 1.048), the output elasticity isgreater than 1 and the production system exhibits increasing returns to scale.Returns to Scale. Determine whether the following production functions exhibit constant,increasing, or decreasing returns to scale.
10. 10. A.Q = 10X + 4Y + 0.25ZB.Q = 12L + 5K + 500C.Q = 4A + 14B + 3ABD.Q = 5L2 + 5LK+ 5K2E.Q = 3L0.3K0.4ANS:A.Initially, let X = Y = Z = 100, so output is:Q1 = 10(100) + 4(100) + 0.25(100) =1,425Increasing all inputs by 4% leads to:Q2 = 10(104) + 4(104) + 0.25(104) =1,482Because a 4% increase in all inputs results in a 4% increase in output (Q 2/Q1 =1,482/1,425 = 1.04), the output elasticity is 1 and the production system exhibits constantreturns to scale.B.Initially, let L = K = 100, so output is:Q1 = 12(100) + 5(100) + 500 = 2,200Increasingboth inputs by 5% leads toQ2 = 12(105) + 5(105) + 500 = 2,285Because a 5% increase inboth inputs results in a 3.9% increase in output (Q2/Q1 = 2,285/2,200 = 1.039), the outputelasticity is less than 1 and the production system exhibits diminishing returns to scale.C.Initially, let A = B = 100, so output is:Q1 = 4(100) + 14(100) + 3(100)(100) =31,800Increasing both inputs by 1% leads to:Q2 = 4(101) + 14(101) + 3(101)(101) =32,421Because a 1% increase in both inputs results in a 2% increase in output (Q 2/Q1 =32,421/31,800 = 1.02), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.D.Initially, let L = K = 100, so output is:Q1 = 5(1002) + 5(100)(100) + 5(1002) =150,000Increasing both inputs by 2% leads to:Q2 = 5(1022) + 5(102)(102) + 5(1022) =156,060Because a 2% increase in both inputs results in a 4% increase in output (Q 2/Q1 =156,060/150,000 = 1.04), the output elasticity is greater than 1 and the production systemexhibits increasing returns to scale.E.Initially, let L = K = 100, so output is:Q1 = 3(1000.3)(1000.4) = 75Increasing both inputsby 4% leads to:Q2 = 3(1040.3)(1040.4) = 77Because a 4% increase in both inputs results in a2.7% increase in output (Q2/Q1 = 77/75 = 1.027), the output elasticity is less than 1 and theproduction system exhibits decreasing returns to scale.Optimal Input Mix. Rachel Green, owner-manager of the Manhattan-based Central PerkCoffee Shop, is reviewing the companys compensation plan. Currently, the company paysits three experienced management staff members salaries based on the number of years ofservice. Chandler Bing, a new management trainee, is paid a more modest salary. Monthlysales and salary data for each employee are as follows: Sales StaffAverage Monthly Sales Monthly SalaryMonica Geller\$200,000\$12,000Phoebe Buffay 150,0009,750Joey Tribbian 120,0006,750Chandler Bing90,0004,500Bing in particular has shown great promise during the past year, and Green believes asubstantial raise is clearly justified. At the same time, some adjustment to thecompensation paid other sales personnel would also seem appropriate. Green isconsidering changing from the current compensation plan to one based on a 9%commission. Green sees such a plan as fairer to the parties involved and believes it wouldalso provide strong incentives for needed market expansion.A.Calculate Central Perks salary expense for each employee expressed as a percentage of
11. 11. sales generated by that individual.B.Calculate monthly income for each employee under a9% commission-based system.C.Will a commission-based plan result in efficient relativesalaries, efficient salary levels, or both?ANS:A. Sales StaffAverage Monthly Sales Monthly SalarySalary Percentage of Sales(1)(2)(3)(4) = (3) ÷ (2)Monica Geller\$200,000\$12,0006.00%Phoebe Buffay150,0009,7506.50%Joey Tribbian120,0006,7505.63%Chandler Bing90,0004,5005.00%B. Sales StaffAverage Monthly Sales 9% Commission(1)(2)(3) = (2) × 0.09Monica Geller\$200,000\$18,000Phoebe Buffay150,00013,500Joey Tribbian120,00010,800Chandler Bing90,0008,100C.The commission-based compensation plan will result in more efficient salaries for salespersonnel. Under this plan, Central Perks compensation costs average 9% of sales,irrespective of which member of the sales staff generates a given dollar of sales. Eachemployee is treated equally under this plan in the sense that all are paid the same rate forgenerating business.Although a commission-based plan will result in an efficient relativepay structure, a 9% commission may or may not result in an optimal level ofcompensation being paid to each employee. If 9% of sales represents the net marginalrevenue (marginal revenue minus all costs except sales expenses) generated by the salesstaff, then optimal levels of compensation would be generated under such a commission-based plan. However, if net marginal revenues are different than this rate, some adjustmentin the commission rate would be appropriate.Optimal Input Mix. Puerto Rico-based Chocolate Products, Inc., manufactures anddistributes a distinctive line of hand-packed candies. Lucy Ricardo president of Chocolateis reviewing the companys sales-force compensation plan. Currently, the company paysits three experienced sales staff members salaries based on the number of years of service.Matty Trumbull, a new sales trainee, is paid a more modest salary. Monthly sales andsalary data for each employee are as follows: Sales StaffAverage Monthly Sales Monthly SalaryEthel Mertz\$300,000\$9,000Caroline Appleby450,00014,000Ralph Ramsey520,00015,000Matty Trumbull390,0008,000Trumbull in particular has shown great promise during the past year, and Ricardo believes
12. 12. a substantial raise is clearly justified. At the same time, some adjustment to thecompensation paid other sales personnel would also seem appropriate. Ricardo isconsidering changing from the current compensation plan to one based on a 3.5%commission. Ricardo sees such a plan as fairer to the parties involved and believes itwould also provide strong incentives for needed market expansion.A.Calculate Chocolates salary expense for each employee expressed as a percentage ofthe sales generated by that individual.B.Calculate monthly income for each employeeunder a 3.5% commission-based system.C.Will a commission-based plan result in efficientrelative salaries, efficient salary levels, or both?ANS:A. Sales StaffAverage Monthly Sales Monthly SalarySalary Percentage of Sales(1)(2)(3)(4) = (3) ÷ (2)Ethel Mertz\$300,000\$9,0003.0%Caroline Appleby450,00014,0003.1%Ralph Ramsey520,00015,0002.9%Matty Trumbull390,0008,0002.1%B. Sales StaffAverage Monthly Sales 3.5% Commission(1)(2)(3) = (2) × 0.035Ethel Mertz\$300,000\$10,500Caroline Appleby450,00015,750Ralph Ramsey520,00018,200Matty Trumbull390,00013,650C.The commission-based compensation plan will result in more efficient salaries for salespersonnel. Under this plan, Chocolates compensation costs average 3.5% of sales,irrespective of which member of the sales staff generates a given dollar of sales. Eachemployee is treated equally under this plan in the sense that all are paid the same rate forgenerating business.Although a commission-based plan will result in an efficient relativepay structure, a 3.5% commission may or may not result in an optimal level ofcompensation being paid to each employee. If 3.5% of sales represents the net marginalrevenue (marginal revenue minus all costs except sales expenses) generated by the salesstaff, then optimal levels of compensation would be generated under such a commission-based plan. However, if net marginal revenues are different than this rate, some adjustmentin the commission rate would be appropriate.Optimal Input Mix. Salem-based Horton & Brady, Inc., is a small firm offering a widevariety of stock brokerage and financial services to high net worth individuals. MickeyHorton, president of Horton & Brady is reviewing the companys compensation plan.Currently, the company pays its three experienced financial advisors a salary based on thenumber of years of service. Nicole Walker, a new sales trainee, is paid a more modestsalary. Sales and salary data for each employee are as follows: Financial AdvisorsCommissions and
13. 13. Fees Generated SalaryHope Williams-Brady\$3.2 million\$185,000Austin Reed2.8 million112,000Sami Brady1.6 million56,000Nicole Walker1.2 million48,000Walker in particular has shown great promise during the past year, and Horton believes asubstantial raise is clearly justified. At the same time, some adjustment to thecompensation paid other sales personnel would also seem appropriate. Horton isconsidering changing from the current compensation plan to one based on a 5%commission. Horton sees such a plan as fairer to the parties involved and believes it wouldalso provide strong incentives for needed market expansion.A.Calculate Horton & Bradys salary expense for each employee expressed as apercentage of the commissions and fees generated by that individual.B.Calculate incomefor each employee under a 5% commission-based system.C.Will a commission-based planresult in efficient relative salaries, efficient salary levels, or both?ANS:A. Sales Staff Annual Sales SalarySalary Percentage of Sales(1)(2)(3)(4) = (3) ÷ (2)Hope Williams-Brady\$3,200,000\$185,0005.78%Austin Reed2,800,000112,0004.00%Sami Brady1,600,00056,0003.50%Nicole Walker1,200,00048,0003.33%B. Sales StaffAnnual Sales5% Commission(1)(2)(3) = (2) × 0.05Hope Williams-Brady\$3,200,000\$160,000Austin Reed2,800,000140,000Sami Brady1,600,00080,000Nicole Walker1,200,00060,000C.The commission-based compensation plan will result in more efficient compensation forsales personnel. Under this plan, Horton & Bradys sales costs average 5%, irrespective ofwhich member of the sales staff generates a given dollar of sales. Each employee is treatedequally under this plan in the sense that all are paid the same rate for generatingbusiness.Although a commission-based plan will result in an efficient relative paystructure, a 5% commission may or may not result in an optimal level of compensationbeing paid to each employee. If 5% of sales represents the net marginal revenue (marginalrevenue minus all costs except sales expenses) generated by the sales staff, then optimallevels of compensation would be generated under such a commission-based plan.However, if net marginal revenues are different than this rate, some adjustment in thecommission rate would be appropriate.Optimal Input Mix. Brisco, Van Buren & Associates is a New York City based law firm.Anita van Buren, managing partner of Brisco, Van Buren is reviewing the firmscompensation plan. Currently, the firm pays its staff attorneys salaries based upon thenumber of years of service. The value of billable hours generated by each staff attorneyduring the past year are as follows: Staff
14. 14. Billings SalaryEd Green\$5 million\$250,000Serena Southerlyn3.5 million157,500Jack McCoy2 million80,000Nora Lewin1.6 million72,000Van Buren believes some adjustment to the compensation paid all staff members would beappropriate. Van Buren is considering changing from the current compensation plan to onewhereby each staff member would be paid a salary equal to 10% of client billings (grossrevenue generated). Van Buren sees such a plan as fairer to the parties involved andbelieves it would also provide strong incentives for needed client development.A.Calculate Brisco, Van Burens salary expense for each employee expressed as apercentage of the client billings generated by that individual.B.Calculate income for eachemployee under a 10% commission-based system.C.Will a commission-based plan resultin efficient relative salaries, efficient salary levels, or both?ANS:A. Staff Billings SalarySalary Percentage of Sales(1)(2)(3)(4) = (3) ÷ (2)Ed Green\$5,000,000\$250,0005.0%Serena Southerlyn3,500,000157,5004.5%Jack McCoy2,000,00080,0004.0%Nora Lewin1,600,00072,0004.5%B. Staff Billings10% Commission(1)(2)(3) = (2) × 0.1Ed Green\$5,000,000\$500,000Serena Southerlyn3,500,000350,000Jack McCoy2,000,000200,000Nora Lewin1,600,000160,000C.The commission-based compensation plan will result in more efficient compensation forsales personnel. Under this plan, Brisco, Van Burens staff attorney salary costs average10%, irrespective of which member of the staff generates a given dollar of sales. Eachemployee is treated equally under this plan in the sense that all are paid the same rate forgenerating business.Although a commission-based plan will result in an efficient relativepay structure, a 10% commission may or may not result in an optimal level ofcompensation being paid to each employee. If 10% of sales represents the net marginalrevenue (marginal revenue minus all costs except sales expenses) generated by the staff,then optimal levels of compensation would be generated under such a commission-basedplan. However, if net marginal revenues are different than this rate, some adjustment in thecommission rate would be appropriate.Optimal Input Mix. Hydraulics Ltd. has designed a pipeline that provides a throughput of70,000 gallons of water per 24-hour period. If the diameter of the pipeline were increasedby 1 inch, throughput would increase by 4,000 gallons per day. Alternatively, throughputcould be increased by 6,000 gallons per day using the original pipe diameter with pumpsthat had 100 more horsepower.A.Estimate the marginal rate of technical substitution between pump horsepower and pipediameter.B.Assuming the cost of additional pump size is \$600 per horsepower and the costof larger diameter pipe is \$200,000 per inch, does the original design exhibit the property
15. 15. required for optimal input combinations? If so, why? If not, why not?ANS:A.The marginal rate of technical substitution is calculated by comparing the marginalproducts of "diameter," MPD, and "horsepower," MPH:MPD = ∂Q/∂D = 4,000/1 = 4,000gal.MPH = ∂Q/∂H = 6,000/100 = 60 gal.So,MRTSDH = =- = -66.67 = -66.67This implies ∂H = -66.67 ∂D or ∂D = -0.015 ∂H. This means, for example,that output would remain constant following a one inch reduction in pipe diameterprovided that horsepower were increased by 66.67.B.No. The rule for optimal input proportions is: = In this instance the questionis: 0.02 0.10Here the additional throughput provided bythe last dollar spent on more horsepower (0.10 gallons/day) is five times the gain in outputresulting from the last dollar spent to increase the pipe diameter (0.02 gallons/day). Thus,horsepower and pipe diameter are not being employed in optimal proportions in thissituation.Optimal Input Mix. Electron Specialties, Inc. has designed an electric feeder cable thatprovides a throughput of 2,000 ampere hours (aH) per 24-hour period. If the diameter ofthe cable were increased by 1/2 inch, throughput would increase by 500 aH per day.Alternatively, throughput could be increased by 1,000 aH per day using the original cablediameter with an additional 100 mf of capacitance electronics designed by the firm.A.Estimate the marginal rate of technical substitution between capacitance electronics andcable diameter.B.Assuming the cost of additional capacitance electronics is \$50 per mfand the cost of larger diameter cable is \$20,000 per 1/2 inch, does the original designexhibit the property required for optimal input combinations? If so, why? If not, why not?ANS:A.The marginal rate of technical substitution is calculated by comparing the marginalproducts of "diameter," MPD, and "capacitance," MPC:MPD = ∂Q/∂D = 500/0.5 = 1,000aH.MPC = ∂Q/∂C = 1,000/100 = 10 aH.So,MRTSDF = =- = -100= -100This implies -C = -100 ∂D or ∂D = -0.01 ∂C. This means, for example, that outputwould remain constant following a one-half inch reduction in cable diameter provided thatcapacitance were increased by 100.B.No. The rule for optimal input proportions is: = In this instance the questionis: 0.05 0.125Here the additional throughput provided bythe last dollar spent on additional capacitance (0.125 aH) is two and one-half times thegain in power output resulting from the last dollar spent to increase the cable diameter
16. 16. (0.05 aH). Thus, capacitance electronics and cable diameter are not being employed inoptimal proportions in this situation.Optimal Input Mix. Boch, Ltd., has designed a fuel injector for oil fired generators thatprovides a throughput of 3 gallons of oil per minute. If the diameter of the industrialinjector nozzle were increased by 1 centimeter, throughput would increase 1 gallon perminute. Alternatively, throughput could be increased by 2 gallons per minute using theoriginal injector diameter with fuel pumps that had 50 more pounds of pressure (psi).A.Estimate the marginal rate of technical substitution between pump psi and injectordiameter.B.Assuming the cost of additional fuel pump size is \$10 per psi (due to changingof the pump and fuel supply peripheral), and the cost of larger diameter fuel injector is\$400 per centimeter (due to machining of combustion units), does the original designexhibit the property required for optimal input combinations? If so, why? If not, why not?ANS:A.The marginal rate of technical substitution is calculated by comparing the marginalproducts of "diameter," MPD, and "psi," MPP:MPD = Q/ D = 1/1 = 1 gal.MPP = Q/ P =2/50 = 0.04 gal.So,MRTSDP = =- = -25 = -25This implies ∂P =-25 ∂D or ∂D = -0.04 ∂P. This means, for example, that output would remain constantfollowing a one centimeter reduction in injector diameter provided that psi were increasedby 25.B.No. The rule for optimal input proportions is: In this instance the questionis: 0.0025 0.004Here the additional throughput provided bythe last dollar spent on more psi (0.004 gallons/minute) is about 1.6 times the gain inoutput resulting from the last dollar spent to increase the output diameter (0.02gallons/minute). Thus, psi and injector diameter are not being employed in optimalproportions in this situation.Optimal Input Mix. Third World Solutions, Inc., has designed a manual water pump thatattains a flow rate of 5 gallons per minute using 1 manpower. If the diameter of the pumpwere increased by 1 inch, throughput would increase 4 gallons per minute. Alternatively,throughput could be increased by an additional 8 gallons per minute using the originalpump diameter with one hydraulic chamber.A.Estimate the marginal rate of technical substitution between hydraulic chambers andpump diameter.B.Assuming the cost of additional hydraulic chamber size is \$5 perchamber and the cost of a larger pump diameter is \$2.50 per inch, does the original designexhibit the property required for optimal input combinations? If so, why? If not, why not?ANS:A.The marginal rate of technical substitution is calculated by comparing the marginalproducts of "diameter," MPD, and "hydraulic chambers," MPH:MPD = ∂Q/∂D = 4/1 = 4
17. 17. gal.MPH = ∂Q/∂H = 8/1 = 8 gal.So,MRTSDH = =- = -0.50 =-0.50This implies ∂H = -0.5 ∂D or ∂D = -2 ∂H. This means, for example, that outputwould remain constant following a one-half inch reduction in pump diameter provided thatthe number of hydraulic chambers were increased by 1.B.No. The rule for optimal input proportions is: In this instance the questionis: 1.60 = 1.60Here the additional throughput provided by thelast dollar spent on more hydraulic chamber (1.60 gallons/minute) is the same as the gainin output resulting from the last dollar spent to increase the pump diameter (1.60gallons/minute). Thus, hydraulic chambers and pump diameter are being employed inoptimal proportions in this situation.Optimal Input Level. U-Do-It Furniture, Inc., sells hardwood chairs, in both kits andfully assembled forms. Customers who assemble their own chairs benefit from the lowerkit price of \$35 per chair. "Full-service" customers enjoy the luxury of an assembled chair,but pay a higher price of \$60 per chair. Both kit and fully assembled chair prices arestable. The company has observed the following relation between the number of assemblyworkers employed per day and assembled chair output: Number of Workers per day Finished Chairs001529312414515A.Construct a table showing the net marginal revenue product derived from assemblyworker employment.B.How many assemblers would U-Do-It Furniture employ at a dailywage rate of \$75?C.What is the highest daily wage rate U-Do-It Furniture would pay tohire four assemblers per day?ANS:A.Because the market for hardwood chairs is perfectly competitive, the \$25 price premium for fully assembled chairs versus kits is stable. Thus, the net marginal revenue product of assembler labor (sometimes referred to as the value of marginal product) is: Number of Assemblers per Day (1) Fully Assembled Output (2) Marginal Product of Labor (3)Net Marginal Revenue Product of Labor
18. 18. (4)=(3)×\$2500----155\$125294100312375414250515125B.From the table above, we see that employment of three assemblers could be justified at a daily wage of \$75 because MRPA=3 = \$75. Employment of a fourth assembler could not be justified because MRP A=4 = \$50 < \$75.C.From the table above, the MRPA=4 = \$50. Thus, a daily wage of \$50 per assembler is the most U-Do-It Furniture would be willing to pay to hire a staff of 4 assemblers.Optimal Input Level. Do-It-Yourself, Inc., sells budget-priced stereo receivers, in both kitand fully-assembled forms. Customers who assemble their own receivers benefit from thelower kit price of \$100 per receiver. "Full-service" customers enjoy the luxury of anassembled receiver, but pay a higher price of \$150 per receiver. Both kit and fullyassembled receiver prices are stable. The company has observed the following relationbetween the number of assembly workers employed per day and assembled receiveroutput: Number of Workers per day Finished Receivers0018214318420521A.Construct a table showing the net marginal revenue product derived from assemblyworker employment.B.How many assemblers would Do-It-Yourself employ at a dailywage rate of \$120?C.What is the highest daily wage rate Do-It-Yourself would pay to hirefour assemblers per day?ANS:A.Because the market for budget receivers is perfectly competitive, the \$50 price premium for fully assembled receivers versus kits is stable. Thus, the net marginal revenue product of assembler labor (sometimes referred to as the value of marginal product) is: Number of Assemblers per Day (1) Fully Assembled Output (2) Marginal Product of Labor (3)Net Marginal Revenue Product of Labor (4)=(3)×\$5000----188\$400214630031842004202100521150B.From the table above, wesee that employment of three employees could be justified at a daily wage of \$120 because MRPA=3 = \$200 > \$120. Employment of a fourth could not be justified because MRP A=4 = \$100 < \$120.C.From the table above, the MRPA=4 = \$100. Thus, \$100 is the most Do-It- Yourself would be willing to pay to hire a staff of 4 assemblers.Optimal Input Level. Just Bikes, Inc., sells tricycles, in partially-assembled and fullyassembled forms. Parents who assemble their own tricycles benefit from the lower price of
19. 19. \$40 per tricycle. "Full-service" customers enjoy the luxury of an assembled tricycle, butpay a higher price of \$60 per tricycle. Both partially and fully assembled tricycle pricesare stable. The company has observed the following relation between the number ofassembly workers employed per day and assembled tricycle output: Number of Workers per day Finished Tricycles0018214318421523A.Construct a table showing the net marginal revenue product derived from assemblyworker employment.B.How many assemblers would Just Bikes employ at a daily wagerate of \$100?C.What is the highest daily wage rate Just Bikes would pay to hire threeassemblers per day?ANS: A.Because the market for tricycles is perfectly competitive, the \$20 price premium for fully assembled versus partially assembled tricycles is stable. Thus, the net marginal revenue product of assembler labor (sometimes referred to as the value of marginal product) is:Number of Assemblers per Day (1)Fully Assembled Output (2)Marginal Product of Labor (3)Net Marginal Revenue Product of Labor (4)=(3)×\$2000----188\$160286120314480420360523240B.From the table above, we see that employment of two assemblers could be justified at a daily wage of \$100 because MRPA=2 = \$120 > \$100. Employment of a third assembler could not be justified becauseMRPA=3 = \$80 < \$100.C.From the table above, the MRPA=3 = \$80. Thus, \$80 is the highest daily wage Huffee Bikes would be willing to pay to hire a staff of 3 assemblers.Nonprice Competition. Tickets, Inc., uses mall intercept promotion services to promoteconcerts and sporting events. The St. Louis firm uses a team of ten students to hand-deliver flyers at shopping malls and other high traffic centers, where every hour incrementof flyer advertising costs \$130. Over the past year, the following relation betweenadvertising and ticket sales per event has been observed:Sales (units) = 7,000 + 200A - 0.6A2and∂ Sales/∂A = 200 - 1.2AHere A represents one hour of flyer distribution, and sales are measured in numbers oftickets.