midterm_question - kept the same changes in Q due to...

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Production function is KL Q = . Price of capital is $40, price of labor is $20. What is the cost minimizing choice of capital and labor if we want to produce Q=4000? Optimal choice of inputs is the point of tangency of isoquant and isocost line. That is, slopes of the two curves will be the same at the cost minimizing point. K L p p RTS = Slope of the isocost curve is equal to K L p p L K - = . If we give up L units of labor we save L p L dollars. With this money we can buy K L p L p units of capital. Now we have to find the slope of the isoquant. Consider again giving up L units of labor for K units of capital. The amount of the production falls by L K L L K Q Q Q L = - = - = ) ( 0 1 0 1 If labor decreases by L . Quantity produced rises by K L K K L Q Q Q K = - = - = ) ( 0 1 0 1 If capital increases by K . Since along the isoquant quantity of the good produced is
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Unformatted text preview: kept the same changes in Q due to increased capital and due to decreased labor should cancel out, i.e. = + = + = L K K L Q Q Q L K . In this way we get RTS equal to L K L K-= Using the numbers given in the problem we have 2 1 40 $ 20 $ = = = K L p p L K Therefore, K L 2 = . Intuitively, since labor is cheaper than capital it is optimal to choose more labor than capital. Plug this relationship in the production function to get 4000 2 * = K K 72 . 44 2000 = K 44 . 89 2000 * 2 = L Now compute total costs 71 . 3577 * 40 $ * 20 $ + = K L TC In case labor an capital is indivisible take K=45, L=89 and TC=3580....
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