Homework_7_Solution

Homework_7_Solution - 1. Suppose the production function...

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1. Suppose the production function for aluminum is given by 2 1 8 1 8 3 K E L Q = where L is the amount of labor hired, E is the amount of energy consumed and K is the amount of capital used. Suppose that the price of energy is 1, the wage rate is 3 and the price of capital is also 1. a) Suppose in the short run the number of units if capital is fixed at 16. Set up the factory’s cost minimization problem and solve for the optimal number of units of energy to use and labor to hire as a function of q. Using your answers for labor and energy used find the cost function and the marginal cost function. Illustrate the cost and marginal cost curves in a diagram. 2 1 8 1 8 3 K E L Q = short run: 8 1 8 3 2 1 8 1 8 3 4 ) 16 ( E L Q E L Q = = q E L wL E p E = + 8 1 8 3 4 s.t. min tangency : E E L p w MRTS = , L E L E E L E L E L p w f f E E L = = = = = - - 3 3 3 3 1 3 ) 8 / 1 ( 4 ) 8 / 3 ( 4 8 7 8 3 8 1 8 5 feasibility : 8 1 8 3 4 E L q = E q L q L q L L L q = = = = = 16 4 4 ) ( 4 2 2 1 2 1 8 1 8 3 demand for labor: 16 2 q L =
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demand for energy: 16 2 q E = The short run cost function is found by substituting these demands into p E E + wL + rK, with p E = 1, w = 3, r = 1, and K = 16 . 16 4 ) 16 )( 1 ( ) 16 )( 3 ( ) 16 )( 1 ( 2 2 2 + = + + = q q q C SR where SR stands for short run. The short-run marginal cost is the derivative of C SR with respect to q and is given by: 2 ) 2 ( 4 1 16 4 2 q q q dq d MC SR = = + = b) Suppose that in the long run the number of units of capital is also variable. Set up the cost minimization problem and solve for the number of units of energy, labor, and capital used. Using these answers find the cost and marginal cost curves. Illustrate your answer in a diagram. Now the cost minimization is: q K E L E p rK wL E K E L = + + 2 1 8 1 8 3 , , s.t. min which is solved again by some tangency condition and the feasibility condition. Here there are two tangency conditions reflecting the possibilities of substituting 3 factors of production among themselves. That is, one can trade some labor for some energy and the exchange rate is given by cost cost per unit q q 16 C SR MC SR
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the marginal rate of technical substitution between labor and energy (MRTS L,E as above). But one can also trade labor for capital, and the exchange rate is given by MRTS L,K , the marginal rate of technical substitution between labor and capital. (One can also trade K for E, but this is already captured by the two MRTS’s above). At an optimal combination of L and E as before, MRTS L,E has to equal E p w , which represents a feasible trade (it’s the opportunity cost of labor in terms of energy). Likewise, MRTS L,K has to equal the opportunity cost of labor in terms of capital, r w . In short, we have to solve the following three equations: q K E L r w MRTS p w MRTS K L E E L = = = 2 1 8 1 8 3 , , solving the first equation: L E L E E L E L E L p w f f p w MRTS E E L E E L = = = = = = - - 3 3 3 3 1 3 ) 8 / 1 ( 4 ) 8 / 3 ( 4 8 7 8 3 8 1 8 5 , solving the second equation:
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Homework_7_Solution - 1. Suppose the production function...

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