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Unformatted text preview: ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission University of Toronto, Department of Economics, ECO 204, 20082009. Ajaz Hussain Test 2 Solutions PLEASE FILL OUT THE INFORMATION BELOW Please write your name as it appears in ROSI: LAST NAME: FIRST NAME: MIDDLE NAME: UT ID #: SIGNATURE: PLEASE CIRCLE THE SECTION YOU'RE REGISTERED IN (NOT THE SECTION YOU'RE ATTENDING) M 122 M 46 T 122 T 46 W 68 PLEASE CIRCLE YOUR EXAM ROOM: EX 100 EX 310 EX 320 SCORES Question 1. 2. 3. 4. 5. 6. 7. Total Points 10 10 10 10 30 10 20 100 Score For your convenience there is a worksheet at the end of this test. This test has a total of 15 pages. Good luck! 1 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Imperfect Substitutes Formulas: q = L K L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/) + (/)]1/( + ) __________________________________________________________________________ Question 1 (10 points) Edison Chang has the production function q = L K + L + K. Suppose Edison Chang is in the long run. Calculate Edison Chang's MRTS show all calculations clearly. Answer: MRTS = MPL/MPK = (dq/dL)/(dq/dK) q = L K + L + K dq/dL = L1 K + dq/dK = L K1 + MRTS = ( L1 K + )/( L K1 + ) 2 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 2 (10 points) MazaMataHari has the production function q = L k. Suppose MazaMataHari is in the short run. What is MazaMataHari `s MRTS (in terms of q) at the fixed capital and optimal labor? Show all calculations clearly. Answer: Explanation: We need to derive the MRTS at k, q and the optimal short run labor. MRTS = MPL/MPK = (dq/dL)/(dq/dK) q = L k dq/dL = L1 k dq/dK = L k1 MRTS = (L1 k)/(L k1) MRTS = ( k)/( L) = (/)(k/L) The question asks for the MRTS at k and optimal L. Given q and k, the optimal short run labor is: L = (q/k )1/ = q1// k/ Substituting in MRTS yields: MRTS = (/)(k/L) MRTS = (/)(k/{q1// k/}) MRTS = (/) (k1 / /q1/ ) 3 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 3 (10 points) A company uses Labor (L), Capital (K) and Materials (M) as inputs to produce target output q with the production function q = L K M. (a) (5 points) For what values of , and will this company have constant returns to scale? Show your calculations below. Answer: Explanation: A production function has constant returns to scale (RTS = 1) if: f(L, K, M) = f(L, K, M). That is, when output with all inputs scaled up by a factor is equal to the output scaled up by . In lectures and HWs (see HW 5 for instance) we've used = 2. First, the output with L, K and M is: Initial q = f(L, K, M) = L K M. Next, the output with twice the L, K and M is: New q = f(2L, 2K, 2M) = (2L) (2K) (2M) New q = 2 2 2 L K M New q = 2 + + L K M New q = 2 + + (initial q) The new output (with all inputs doubled) will be double the initial output when + + = 1. 4 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission (b) (5 points) For what values of , and will this company have decreasing returns with respect to materials? Show your calculations below. Answer: Explanation: If the company is to have decreasing returns with respect to materials, it means that holding labor and capital fixed when materials are doubled output less than doubles. This will happen when < 1 regardless of and . To see this formally, the initial output is q = f(L, K, M) = L K M. Suppose we double materials, holding capital and labor constant: New q = f(L, K, 2M) = L K (2M) New q = 2 L K M New q = 2 Initial q With decreasing returns in materials, when materials is doubled, output less than doubles, which happens when < 1. 5 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 4 (10 points) A company uses Labor (L) and Capital (K) as inputs to produce target output q with the production function q = min(L, K) + . For what values of , and will this company have increasing returns to scale? Show your calculations below. Answer: Explanation: A production function has increasing returns to scale (RTS > 1) if: f( L, K, M) > f(L, K, M). That is, when the output with all inputs scaled up by a factor is greater than the output scaled up by . In lectures and HWs, we've used = 2. First, the initial output with L, K is: Initial q = f(L, K) = min(L, K) + Next, the output with twice the L and K is: New q = f(2L, 2K) = min(2 L, 2 K) + New q = 2 min(L, K) + We need to see when the output with twice the inputs is greater than double the initial output. Double the initial output is: Double Initial q = 2 min(L, K) + 2 New output > Double initial output when: 2 min(L, K) + > 2 min(L, K) + 2 Which can only happen if < 0. If you're having problems seeing this, you can use this technique from HW 5: 2 min(L, K) + > 2 min(L, K) + 2 6 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission 2 min(L, K) + + > 2 min(L, K) + 2 2 min(L, K) + 2 > 2 min(L, K) + 2 > 0 < 0 Thus: q = min(L, K) + has increasing returns to scale when < 0. 7 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 5 (30 points) The Colgate Company manufactures toothbrushes using labor and capital as inputs in a complements technology q = min(K, L). Every 1m toothbrushes requires 1 capital and 1 labor. Each unit of Colgate's capital (K) consists of: 1 Tufter machine which inserts the brush into the handle 1 Handle Mold machine which produces the handle 1 Packaging machine which packages the toothbrush These machines are specifically designed for Colgate toothbrushes and cannot be used by other companies to manufacture any product. Table 1 has the purchase price and depreciation time: Table 1 Purchase Price $500,000 $300,000 $150,000 Machine Tufter Handle Mold Packaging Depreciation Time 15 years 5 years 5 years Source: ColgatePalmolive Company: The Precision Toothbrush, HBS Case 9593064 Suppose Colgate hires workers in competitive markets at PL = $10. Currently, the average interest rate for bank deposits is 5%. Colgate's target output is 2m toothbrushes a year. (a) (5 points) Use the straight line depreciation method to calculate the annual depreciation (up to two decimal places) for the Tufter, Handle Mold and Packaging machines. Show your calculations below. Answer: Note: depreciation time is not the same as the actual life of a machine. Annual depreciation Tufter = $500,000/15 = $33,333.33 Annual depreciation Handle Mold = $300,000/5 = $60,000 Annual depreciation Packaging = $150,000/5 = $30,000 8 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission (b) (5 points) Use the answer in part (a) to compute the annual depreciation for Colgate's capital. Answer: Observe how the Handle Mold and Packaging machines have a depreciation time of 5 years while the Tufter has depreciation time of 15 years. Thus: For the first 5 years, the annual depreciation is = $33,333.33 + $60,000 + $30,000 = $123,333.33 For the next 10 years, the annual depreciation is = $33,333 For any year after, the annual depreciation is nil. (c) (5 points) What is the annual opportunity cost of Colgate's capital? Answer: Since the machines can only be used for Colgate products and cannot be used by other companies, there is no opportunity for the machines to be used elsewhere. Thus, the opportunity cost is zero. 9 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission (d) (5 points) Use your answers in parts (a), (b) and (c) to calculate Colgate's price of capital PK in Table 2 below: Answer: Colgate owns its capital. Thus it's PK is the: User cost of capital = Depreciation + Opportunity cost. Opportunity cost is zero for all years and depreciation is $123,333.33 for the first 5 years and $33,333 thereafter. Hence: Table 2 Year PK 1 123,333.33 2 123,333.33 3 123,333.33 4 123,333.33 5 123,333.33 6 33,333 7 33,333 8 33,333 9 33,333 10 33,333 11 33,333 12 33,333 13 33,333 14 33,333 15 33,333 Note: If you assumed and stated that the Handle Mold and Packaging machines were again purchased at the end of their depreciation time, you will be given credit. (e) (5 points) Assuming wages and the target output of toothbrushes are constant, calculate Colgate's optimal labor and capital usage over time. Explain your answer using algebra and/or graphs. Answer: Colgate uses a fixed proportions technology with labor and capital in a 1:1 ratio. Thus, it has a complements production function. The price of labor is constant while the price of capital is falling over time. This has no change in the amounts of labor or capital because with a complements technology, labor and capital 10 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission demands are independent of prices. Formally: q = min(L, K) Isoquant corners lie on the line K = L. The optimal labor and capital also lie on the K = L line. Thus given the target output q: L = q and K = q Colgate produces 2m toothbrushes so L = K = 2. Thus, despite stable wages and falling price of capital, there will be no change in the amounts of labor and capital. From lectures and HWs, note that even though amounts of labor and capital don't change, the cost of production will fall due to lower user cost of capital. (f) (5 points) Assuming that wages and the target output of toothbrushes are constant calculate Colgate's total costs in Table 3 below. Show all calculations below. Table 3 Year C(q) 1 246,686.66 3 246,686.66 5 246,686.66 7 66,686 10 66,686 Answer: The cost of producing output is: C(q) = PL L + PK K Now: L = K = q C(q) = PL q + PK q C(q) = (PL + PK)q Now: PL = $10 and q = 2 11 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission C(q) = (10 + PK)2 Now, PK changes over time. It was $123,333.33 for years 1 through 5 and $33,333 for years 6 through 12. Thus: For years 1 through 5: C(q) = (10 + PK)2 = (10 + 123,333.33)2 = $246,686.66 For years 6 through 15: C(q) = (10 + PK)2 = (10 + 33,333)2 = $66,686 Note: Credit will be given for correct cost calculations (i.e. provided you got C(q) = (PL + PK)q right) even if the price of capital was incorrect. 12 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 6 (10 points) ETorre Enterprises used labor (L) and capital (K) to produce its target output q = 80 according to the production function q = L + K where = 2 and = 8. ETorre procures labor and leases capital in competitive markets. Currently, PL = $100 and PK = $10. (a) (5 points) If ETorre is in the long run, how much labor and capital does it use? Show your calculations below. Answer: ETorre has a perfect substitute's production function. Optimal labor and capital depends on whether the isocost is steeper, tangent, or flatter than the isoquant. The isocost has slope PL/PK = 100/10 = 10 The isoquant has slope = MPL/MPK = / = 2/8 = 1/4. Given the isocost is steeper than the isoquant, ETorre will use zero labor. Thus: q = L + K q = K K = q/ K = 80/8 = 10 L = 0, K = 10. Incidentally, the y and x intercepts of the isoquant are 10 and 40 respectively. 13 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission (b) (5 points) Plot your answer to part (a) in Figure 1 below. Now suppose PL decreases: graph the resulting demand curve for labor in Figure 1 below: Figure 1 Answer: To see how the demand curve was derived note from part (a) that when PL = $100, there is no demand for labor. This implies L = 0 for PL 100. As PL falls, there will be no demand for labor so long as the isocost is steeper than the isoquant. See Figure 2 below. Figure 2 14 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission When the isocost becomes tangent to the isoquant, any amount of labor from 0 to 40 can be hired. This will happen when: PL/PK = / = PL = PK PL = 10 = $2.50 As PL falls further, the isocost will become flatter than the isoquant. This lets ETorre to produce his target output at a lower cost. See Figure 3. For any wage lower than $2.50, demand for labor will be 40. Figure 3 15 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 7 (20 points) The following page contains Exhibits 1 and 2 from the Prestige Telephone Company case, as well as Exhibit 3 from the solution. Recall Pcommercial = $800/hr and Pintercompany = $400/hr. (Note: In the case, all contribution analysis was conducted from March 2003 onwards.) (a) (10 points) Suppose Prestige Data Services raises commercial prices to $1,200/hr. Higher commercial prices will lead to lower commercial sales. What must the new number of commercial hours be for Prestige Data Services to be profitable? Show all calculations. Answer: From the case solution below, the AVC is constant and equal to $28. Suppose the new commercial hours are X. The change in the contribution from higher commercial prices is: = (1200 28)X (800 28)138 To bring the company into profitability, the increased contribution must be greater than the current net loss of $21,438 in March 2003. Thus, the new number of commercial hours needed is: = (1200 28)X (800 28)138 > 21,438 (1200 28)X > 21,438 + (800 28)138 X > {21,438 + (800 28)138}/(1200 28) X 109 hours. Hence, as long as the new commercial hours are greater than 109 hours, raising the price to $1200/hr will make the company profitable. 16 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission (b) (10 points) Suppose Prestige Data Services spends $7,720 on sales promotion (i.e. advertising). Assuming advertising raises sales of commercial hours, calculate the increase in commercial hours needed to make Prestige Data Services profitable. Show all calculations. Answer: From the case solution below, the AVC is constant and equal to $28. Suppose the new commercial hours are X. The change in the contribution from advertising is: = (800 28)X (800 28)138 7720 = (800 28)(X 138) 7720 To bring the company into profitability, the increased contribution must be greater than the current net loss of $21,438 in March 2003. Thus, the new number of commercial hours needed is: = (800 28)(X 138) 7720 > 21,438 = (800 28)(X 138) > 21,438 + 7,720 = (X 138) > {21,438 + 7,720}/(800 28) = X > {21,438 + 7,720}/(800 28) + 138 X 176 hours. If an advertising campaign budgeted at $7,7200 raising commercial hours to 176 hours, it will make the company profitable. The End 17 ECO 204, 20082009, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Exhibit 1 Prestige Data Services Summary of Computer Utilization, First Quarter 2003 Revenue Hours Intercompany Commercial Total revenue hours Service hours Available hours Total hours January 206 123 329 32 199 560 February 181 135 316 32 164 512 March 223 138 361 40 143 544 Exhibit 2 Prestige Data Services Summary Results of Operations, First Quarter 2003 January Revenues Intercompany sales Commercial sales Computer use Other Total Revenue Expenses Space costs Rent Custodial services February March $ 82,400 98,400 9,241 190,041 $ 72,400 108,000 9,184 189,584 $ 89,200 110,400 12,685 212,285 $ 8,000 1,240 9,240 $ 8,000 1,240 9,240 $ 8,000 1,240 9,240 Equipment costs Computer leases Maintenance Depreciation: Computer equipment Office equipment and fixtures Power 95,000 5,400 25,500 680 1,633 128,213 95,000 5,400 25,500 680 1,592 128,172 95,000 5,400 25,500 680 1,803 128,383 Wages and salaries Operations Systems development and maintenance Administration Sales 29,496 12,000 9,000 11,200 61,696 9,031 7,909 15,424 $ 231,513 $ (41,472) 29,184 12,000 9,000 11,200 61,384 8,731 7,039 15,359 $ 229,925 $ (40,341) 30,264 12,000 9,000 11,200 62,464 10,317 8,083 15,236 $ 233,723 $ (21,438) Materials Sales promotions Corporate services Total expenses Net income/(loss) Exhibit 3 Prestige Data Services Standard Costs  Average Month, First Quarter 2003 Variable Costs Power per hour Operations wages Total variable expenses per hour Fixed Costs Rent Custodial Computer lease Computer maintenance Depreciation Power Wages and Salaries Operations Systems development Administration Sales $4 $24 $28 $8,000 $1,240 $95,000 $5,400 $26,180 $200 $21,600 $12,000 $9,000 $11,200 Total Wages & Salaries Fixed Costs $53,800 Total nonvariable expenses Sales promotion $189,820 $8,000 $197,820 << Total Fixed Costs 18 ...
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 Fall '08
 HUSSEIN
 Economics, Microeconomics

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