f06-answers-midterm2 - Economics 302 Intermediate...

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Economics 302 Silve Parviainen Intermediate Microeconomics Fall 2006 Answers to Midterm Examination II 1 Technology and Proft Maximization (a) The percentage by which the output grows when all inputs are increased by a given percentage. For a Cobb-Douglas function l α k β ,i f α + β< 1, the function exhibits decreasing returns to scale. 1 3 + 1 3 = 2 3 < 1= DRS. (b) max π =max l,k Pl 1 3 k 1 3 wl rk focs: 1 3 2 3 k 1 3 w =0 = l = P 3 w q 1 3 1 3 k 2 3 r = k = P 3 r q (c) Since the function exhibits decreasing returns to scale, a pro±t maximizing output level q can be found. q = ± P 3 w q ² 1 3 ± P 3 r q ² 1 3 = ± P 2 9 wr ² 1 3 q 2 3 = q 1 3 = ± P 2 9 wr ² 1 3 = q = P 2 9 wr . (d) If exactly same amounts of labor and capital are used in f 2 ( l, k )= l 1 2 k 1 2 as in the original f 1 ( l, k l 1 3 k 1 3 , the output is higher. That is, q 1 <q 2 l, k > 1 . Hence, technological progress takes place. (e) No. With α + β = 1, the function has constant returns to scale, and a pro±t maximizing output level is not de±ned.
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2 Cost Minimization (a) MRTS = w r = k l =1 = k = l. Using the production function, q = l 1 2 + l 1 2 = l = q 2 4 and k = q 2 4 . The total cost function is TC = q 2 4 + q 2 4 = q 2 2 . (b) If ¯ k = 100 the production function yields q =10+
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f06-answers-midterm2 - Economics 302 Intermediate...

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