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Econ 201 Problem Set on Chapter 7 1. Complete the following table (round each answer to the nearest whole number): Total Variable Fixed Marginal Average Avg. Avg. Fixed Var. Output Cost Cost Cost Cost Cost Cost Cost 0 1 5 2 30 3 13 4 105 10 5 110 6 50 Solution: Total Variable Fixed Marginal Average Avg. Avg. Fixed Var. Output Cost Cost Cost Cost Cost Cost Cost 0 40 0 40 – – – – 1 45 5 40 5 45 5 40 2 60 20 40 15 30 10 20 3 79 40 40 19 26 13 13 4 105 65 40 26 26 16 10 5 150 110 40 45 30 22 8 6 200 160 40 50 33 27 7 2.Trisha believes the production of a dress requires 4 labor hours and 2 machine hours to produce. If Trisha decides to operate in the short run, she must spend $500 to lease her business space. Also, a labor hour costs $15 and a machine hours costs $35. What is Trisha's cost of production as a function of dresses produced? Solution: Since the production of a dress requires spending $60 for labor and $70 for machine hours, Trisha's cost function is: C q 130q 500. 3. A firm's total cost function is given by the equation: TC = 4000 + 5Q + 10Q2. (1) Write an expression for each of the following cost concepts: a. Total Fixed Cost b. Average Fixed Cost c. Total Variable Cost d. Average Variable Cost e. Average Total Cost f. Marginal Cost (2) Determine the quantity that minimizes average total cost. Demonstrate that the predicted relationship between marginal cost and average cost holds. Solution: PART (1) a. TFC 4000 b. 4000 AFC Q c. TVC TC TFC TVC 5Q 10Q2 d. TVC 5Q 10Q 2 AVC 5 10 Q Q Q e. TC 4000 5Q 10Q2 ATC Q Q f. MC 5 20Q PART (2) ATC is minimized where MC is equal to ATC. Equating MC to ATC 4000 5Q 10Q 2 5 20Q Q 4000 5Q 102 5Q 20Q 2 4000 10Q 2 Q 2 400 Q 20 ATC is minimized at 20 units of output. Up to 20, ATC falls, while beyond 20 ATC rises. MC should be less than ATC for any quantity less than 20. For example, let Q = 10: MC = 5 + 20(10) = 205 4000 510 10 10 2 ATC 505 10 MC is indeed less than ATC for quantities smaller than 20. MC should exceed ATC for any quantity greater than 20. For example, let Q = 25: MC = 5 + 20(25) = 505 4000 525 10 25 2 ATC 415 25 MC is indeed greater than ATC for quantities greater than 20. 4.Davy Metal Company produces brass fittings. Davy's engineers estimate the production function represented below as relevant for their long-run capital-labor decisions. Q = 500L0.6K0.8, where Q = annual output measured in pounds, L = labor measured in person hours, K = capital measured in machine hours. The marginal products of labor and capital are: MPL = 300L-0.4K0.8 MPK = 400L0.6K-0.2 Davy's employees are relatively highly skilled and earn $15 per hour. The firm estimates a rental charge of $50 per hour on capital. Davy forecasts annual costs of $500,000 per year, measured in real dollars. a. Determine the firm's optimal capital-labor ratio, given the information above. b. How much capital and labor should the firm employ, given the $500,000 budget? Calculate the firm's output Solution: a. K 0.8 MPL 300L0.4 K 0.8 300 L0.4 L0.6 MPK 400L0.6 K 0.2 400 0.2 K K 0.8 300 0.4 0.8 0.2 MRTS L 0.75 K K L0.6 L0.4 L0.6 400 0.2 K K MRTS 0.75 L w 15 Equate to . r 50 K 15 0.75 L 50 K 0.75 0.3 L K 0.4; K = 0.4L L b. C = 500,000 C = wL + rK 500,000 = 15L + 50K K = 0.4L from optimal ratio 500,000 = 15L + 50(0.4L) 500,000 = 15L + 20L 500,000 = 35L L = 14,285.71 or 14,286 hours Substitute to solve for K. 500,000 = 15(14286) + 50K 500,000 = 214,290 + 50K 285,710 = 50K K = 5714.20 or K = 5714 Q = 500(14,286)0.6(5,714)0.8 Q = 157,568,191 5.A paper company dumps nondegradable waste into a river that flows by the firm's plant. The firm estimates its production function to be: Q = 6KW, where Q = annual paper production measured in pounds, K = machine hours of capital, and W = gallons of polluted water dumped into the river per year. The marginal products of capital and labor are given as follows: MPK = 6W MPW = 6K The firm currently faces no environmental regulation in dumping waste into the river. Without regulation, it costs the firm $7.50 per gallon dumped. The firm estimates a $30 per hour rental rate on capital. The operating budget for capital and waste water is $300,000 per year. a. Determine the firm's optimal ratio of waste water to capital. b. Given the firm's $300,000 budget, how much capital and waste water should the firm employ? How much output will the firm produce? Solution: a. MPW = 6K MPK = 6W 6K K MRTS 6W W Rate of water charge to price of capital: PW 7.5 .25 PU 30 Equating MRTS to ratio of input prices K 0.25, K = 0.25W W b. C = PWW + PKK 300,000 = 7.50W + 30K recall K = 0.25W 300,000 = 7.5W + 30(0.25W) 300,000 = 7.5W + 7.5W W = 20,000 gallons K = 0.25W K = 0.25(20,000) K = 5000 Q = 6(5000)(20,000) Q = 600,000,000

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posted: | 1/16/2013 |

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