100b_mt1_q147

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< Midterm Answer Keys - Bryan > #1. Please consider a °rm with the following cost function: C ( y ) = 1 2 y 3 ° 3 y 2 + y + 80 Please °nd (1) the short run minimum e¢ cient scale of production and (2) shut down price. (recall, in the SR, MC=AVC) If you recall the cost curves, you will see that the °rm will shut down in the short run when the price is equal to MC and AVC. That is, P ° = MC = AV C MC = C 0 ( y ) = 3 2 y 2 ° 6 y + 1 AV C = C ( y ) y = 1 2 y 2 ° 3 y + 1 Let MC = AV C: Then 3 2 y 2 ° 6 y + 1 = 1 2 y 2 ° 3 y + 1 3 2 y 3 ° 6 y 2 + y = 1 2 y 3 ° 3 y 2 + y y 3 ° 3 y 2 = 0 y = 0 or 3 The tagent line of MC at y = 0 (that is, take derivative of MC and evaluate at y = 0 ) is MC 0 (0) = 3 2 2(0) ° 6 = ° 6 < 0 So, at y = 0 ; MC will decrease if you increase y: That is, the °rm can minimize its cost by increasing y: The tagent line of MC at y = 3 (that is, take derivative of MC and evaluate at y = 3 ) is MC 0 (3) = 3 2 2(3) ° 6 = 3 > 0 So, at y = 3 ; MC will increase if you increase y: That is, y = 3 is the cost minimizing output level.

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