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final_ak - Final Exam Answer Key Econ 11 Summer Session C...

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Final Exam Answer Key Econ 11 °Summer Session C University of California, Los Angeles September 10, 2008 Write your name and ID number on your exam booklet. Exams without identi±cation will not be graded. Hand-held calculators are permitted. You have two hours to complete the exam. Each question is worth 20 points, for a total of 100. Good luck! 1. Let q = k 1 3 l 1 3 . Assume that both factors are variable, and the price of capital is v , while the price of labor is w . (a) Derive the cost function. Find the marginal and average cost func- tions. ANSWER: MRTS kl = l ° k ° = v w q = k ° 1 3 ° v w k ° ± 1 3 = ° v w ± 1 3 k ° 2 3 k ( q ) = ° w v ± 1 2 q 3 2 l ( q ) = ° v w ± 1 2 q 3 2 The cost function is C ( q ) = vk ( q ) + wl ( q ) = v ° w v ± 1 2 q 3 2 + w ° v w ± 1 2 q 3 2 = 2 ( vw ) 1 2 q 3 2 so the marginal and average cost functions are MC ( q ) = 3 ( vw ) 1 2 q 1 2 AC ( q ) = 2 ( vw ) 1 2 q 1 2 (b) Show that the marginal cost function intersects the minimum of the average cost function. ANSWER: The minimum of the average cost curve occurs where dAC ( q ) dq = ( vw ) 1 2 q ± 1 2 = 0 q = 0 1
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Marginal cost is equal to average cost where MC ( q ) = AC ( q ) 3 ( vw ) 1 2 q 1 2 = 2 ( vw ) 1 2 q 1 2 9 q = 4 q which only occurs where q = 0 . (c) Find the amount of k and l necessary to produce q = 400 when w = 4 and v = 1 . ANSWER: We use our demand functions from part ( a ) k (400) = (4) 1 2 (400) 3 2 = 16 ; 000 l (400) = ² 1 4 ³ 1 2 (400) 3 2 = 4 ; 000 (d) If the output price P = 3 , w = 4 and v = 1 , how much does a ±rm with this production technology choose to produce? ANSWER: The ±rm will maximize pro±t where P = MC ( q ) , which is P = 3 ( vw ) 1 2 q 1 2 3 = 3 (4) 1 2 q 1 2 q = ² 1 2 ³ 2 = 1 4 2. The tools you have learned in this class may help you plan your life. Let us split your life into two periods: period one, where you work, and period two, where you are retired. Suppose that in period one, you are able to earn an income from your labor of y 1 = $10400 . In period two, you retire ( y 0 = 0 ) and consume any savings or investments you made in period one. Your utility function is U ( c 1 ; c 2 ) = c 1 2 1 c 1 2 2 , where c 1 is your consumption in period one and c 2 is your consumption in period two. (a) Suppose that in period one, you must pay $400 in Social Security tax. In period two, the government pays you back a retirement bene±t of $400 . Assuming you make no additional savings of any kind , calculate your consumption in each period and your utility level. ANSWER: Since there is no possibility of savings, c 1 = y 1 ° t = 10400 ° 400 = 10 ; 000 and c 2 = y 2 + t = 400 U ( c 1 ; c 2 ) = c 1 2 1 c 1 2 2 U (10000 ; 4000) = (10000) 1 2 (400) 1 2 = 2000 2
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(b) Now assume there is no Social Security tax or bene±t. However, you want to make some savings s 1 in period one. The interest rate made on those savings, r = 100% . Compute your consumption in each period and the savings you make in period one.
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