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AgEc 301 Agricultural Economics I Homework Assignment 4. Due Wednesday, September 28, 2005 Please hand in via email to mcintosh@uidaho.edu as an attachment, or fax to 208-522-2954. Files received after 11:59 p.m. September 28 will be counted late. 1. U-Do-It Furniture, Inc., sells hardwood chairs, in both kits and fully assembled forms. Customers who assemble their own chairs benefit from the lower kit 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 are stable. The company has observed the following relation between the numbers of assembly workers employed per day and assembled chair output: Number of Finished Workers per day Chairs 0 0 1 5 2 9 3 12 4 14 5 15 a. Construct a table showing the net marginal revenue product derived from assembly worker employment. Number of Fully Assembled Marginal Product Net Marginal Value Workers Output of Labor Product of Labor 0 0 -- -- 1 5 5 125 2 9 4 100 3 12 3 75 4 14 2 50 5 15 1 25 b. How many assemblers would U-Do-It Furniture employ at a daily wage rate of $75? From the above table, we see that employment of three assemblers could be justified at a daily wage of $75. Employing a fourth assembler would drive the MVP of labor below $75. c. What is the highest daily wage rate U-Do-It Furniture would pay to hire four assemblers per day? From the table above, the daily wage of $50 per assembler would be the most the company would be willing to pay to hire a staff of 4. 2. Indicate whether each of the following statements is true or false, justify your answer. a. L-shaped isoquants describe production systems where inputs are perfect complements. True. L-shaped production isoquants reflect a perfect complementary relation among inputs, i.e., no amount of input X can make up for the lack of input Y b. If the marginal product of capital increases as capital usage grows, the returns to capital are decreasing. False. Returns to the capital input factor are increasing when the marginal product of capital increases as capital usage grows. c. Marginal revenue product measures the output gained through expanding input usage. False. Marginal revenue product is the revenue generated by expanding 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. The marginal rate of technical substitution will be affected by a given percentage increase in the marginal productivity of all inputs. False. The marginal rate of technical substitution is measured by the relative marginal productivity of input factors. This relation is unaffected by a commensurate increase in the marginal productivity of all inputs. e. Increasing returns to scale and declining average costs are indicated when εQ >1 True. When εQ> 1, the percentage change in output is greater than a given percentage change in all inputs. Thus, increasing returns to scale and decreasing average costs are indicated. 3. Determine whether the following production functions exhibit constant, increasing, or decreasing returns to scale. Q = 10X + 4Y + 0.25Z Initially, let X = Y = Z = 100, so output is: Q1 = 10(100) + 4(100) + 0.25(100) = 1,425 Increasing all inputs by 4% leads to: Q2 = 10(104) + 4(104) + 0.25(104) = 1,482 Because a 4% increase in all inputs results in a 4% increase in output (Q2/Q1 = 1,482/1,425 = 1.04), the output elasticity is 1 and the production system exhibits constant returns to scale. Q = 12L + 5K + 500 Initially, let L = K = 100, so output is: Q1 = 12(100) + 5(100) + 500 = 2,200 Increasing both inputs by 5% leads to Q2 = 12(105) + 5(105) + 500 = 2,285 Because a 5% increase in both inputs results in a 3.9% increase in output (Q2/Q1 = 2,285/2,200 = 1.039), the output elasticity is less than 1 and the production system exhibits diminishing returns to scale. Q = 4A + 14B + 3AB Initially, let A = B = 100, so output is: Q1 = 4(100) + 14(100) + 3(100)(100) = 31,800 Q2 = 4(101) + 14(101) + 3(101)(101) = 32,421 Because a 1% increase in both inputs results in a 2% increase in output (Q2/Q1 = 32,421/31,800 = 1.02), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale. Q = 5L2 + 5LK + 5K2 Initially, let L = K = 100, so output is: Q1 = 5(1002) + 5(100)(100) + 5(1002) = 150,000 Increasing both inputs by 2% leads to: Q2 = 5(1022) + 5(102)(102) + 5(1022) = 156,060 Because a 2% increase in both inputs results in a 4% increase in output (Q2/Q1 = 156,060/150,000 = 1.04), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale. Q = 3L0.3K0.4 Initially, let L = K = 100, so output is: Q1 = 3(1000.3)(1000.4) = 75 Increasing both inputs by 4% leads to: Q2 = 3(1040.3)(1040.4) = 77 Because a 4% increase in both inputs results in a 2.7% increase in output (Q2/Q1 = 77/75 = 1.027), the output elasticity is less than 1 and the production system exhibits decreasing returns to scale. Alternatively with this functional form you can simply sum the exponents, (0.3+0.4) = 0.7 which is less than 1 so this equation exhibits decreasing returns to scale. 4. Is use of the least-cost input combinations a necessary condition for profit maximization? Is it a sufficient condition? Explain. Employment of least-cost input combinations is a necessary but not sufficient condition for profit maximization. It is necessary because a failure to operate with a least-cost input combination means that costs could be lowered and profits increased at any given output level. It is not a sufficient condition because the cost-minimizing level does not incorporate any information concerning demand relations, and therefore provides no information about the optimal level at which to operate: that is, information concerning demand relations must be added to the analysis to determine how much to produce for profit maximization (an optimal level of output). In short, employment of a least-cost input combination will result in an optimal production of a target level of output. Conversely, employment of inputs such that MRPi = Pi for each input will result in an optimal production of an optimal level of output. 5. Medical Testing Labs, Inc., provides routine testing services for blood banks in the Los Angeles area. Tests are supervised by skilled technicians using equipment produced by two leading competitors in the medical equipment industry. Records for the current year show an average of 27 tests per hour being performed on the Testlogic-1 and 48 tests per hour on a new machine, the Accutest-3. The Testlogic-1 is leased for $18,000 per month, and the Accutest-3 is leased at $32,000 per month. On average, each machine is operated 25 eight-hour days per month. a) Describe the logic of the rule used to determine an optimal mix of input usage. The rule for an optimal combination of Testlogic-1 (T) and Accutest-3 (A) equipment is: MPT MPA PT PA This rule means that an identical amount of additional output would be produced with an additional dollar expenditure on each input. Alternatively, an equal marginal cost of output is incurred irrespective of which input is used to expand output. Of course, marginal products and equipment prices must both reflect the same relevant time frame, either hours or months. b) Does Medical Testing Lab usage reflect an optimal mix of testing equipment? (hint, examine the rule you describe in a) above on a per-hour or per-month basis using the information provided). On a per hour basis, the relevant question is, does: 27 48 18,000/(25 8) 32,000/(25 8) 0.3 0.3 On a per month basis, the relevant question is, does: 27 (25 8) 48 (25 8) 18,000 32,000 0.3 0.3 In both instances, the last dollar spent on each machine increased output by the same 0.3 units, indicating an optimal mix of testing machines. c) Describe the logic of the rule used to determine an optimal level of input usage. The rule for optimal input employment is: MVP = MP × PQ = Input Price This means that the level of input employment is optimal when the marginal revenue derived from added input usage is equal to input price, or the marginal input cost of employment. d) If tests are conducted at a price of $6 each while labor and all other costs are fixed, should the company lease more machines? (Again, examine on a per-hour or per- month basis). For each machine hour, the relevant question is, does: Testlogic-1 MVPT = MPT × PQ = PT 27 × $6 = $18,000/(25 × 8) $162 > $90 Accutest-3 MVPA = MPA × PQ = PA 48 × $6 = $32,000/(25 × 8) $288 > $160. Or, in per month terms: Testlogic-1 MVPT = MPT × PQ = PT 27 × (25 × 8) × $6 = $18,000 $32,400 > $18,000. Accutest-3 MVPA = MPA × MRQ = PA 48 × (25 × 8) × $6 = $32,000 $57,600 > $32,000. In both cases, each machine returns more than its marginal cost (price) of employment, and expansion would be profitable.