production cost sol

production cost sol - COMM/FRE 295 Solution to Questions...

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COMM/FRE 295 Solution to Questions Set #3 Production and Costs Question #1. Wheat is produced according to the production function Q = 100(K 0.8 L 0.2 ) where Q denotes output, K denotes capital and L denotes labor. (a) Beginning with K = 4 and L = 49, show that the MP of K and L are both decreasing. Show that this production function exhibits a diminishing MRTS. (b) Does this production function exhibit increasing, decreasing or constant returns to scale? Show the mathematics you used to arrive at your answer. (c) With capital fixed at 4 units, derive an expression for the MP of labor and AP of labor. Graph your expressions for MP L and AP L (L on the horizontal axis). (d) If w =10 and r = $5, find cost minimizing combination of K and L to produce 5000 units of Q. Draw a graph to illustrate your result. a) MP L = Q/ L = 0.2*100(K .8 L -.8 ) and MP K = Q/ K = 0.8*100(K -.2 L .2 ). Next calculate: (MP L )/ (L) = -0.8*0.2*100(K .8 L -1.8 ) and (MPK)/ (K) = -0.2*0.8*100(K -1.2 L .2 ). Both of these values are negative, so they are clearly diminishing for all values of L and K (you may be able to tell it by simply looking at the MP functions). With diminishing MP, the slope of the isoquant becomes flatter as we move down the isoquant, which is consistent with diminishing MRTS. b) Suppose both the inputs increase by t ( t > 1) times, then old new tQ KL t tL tK Q = = = 100 ) ( ) ( 100 2 . 0 8 . 0 This means when all the inputs are increased by a factor of
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production cost sol - COMM/FRE 295 Solution to Questions...

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