ECON 302 HW 2 - Rebecca Chapman Econ 302-Spring 2008...

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Rebecca Chapman Econ 302-Spring 2008 Homework #2 Malcolm-Disc 303 Homework will be graded for both content and neatness. This homework does not require the use of Microsoft Excel, but you will find Excel speeds up the calculations greatly in this homework. 1) Consider an aggregate production function of the form Y=AK α L 1-α , where A represents the level of technology, L is the level of Labor, and K is the level of capital. Firms operate in a competitive market and therefore take factor prices as given, with W as the wage rate, R the rental rate of capital, and P the output price. a. Write down the firm’s profit function based on the information above, and find the necessary conditions for profit maximization (Hint: These conditions were discussed in class and are the same conditions you would get if you set the derivatives of profit with respect to capital and labor to zero using calculus.) Briefly explain why these conditions must hold if the firm is profit maximizing. Max π =P*Y-wL – Rk Y=AK^( α ) L^(1- α ) ∂π (K,L) =0 L 1- α AK^ ( α ) L^ (- α )=0 ∂π (K,L) =0 K α AK^( α -1) L^(1- α )=0 Max Y=(MPL x L) + (MPK x K) where, MPL= (1- α ) AK^( α ) L^(- α ) and MPK= α AK ^ ( α -1) L^ (1- α ) These conditions must hold if the firm is profit maximizing because a firm that is profit maximizing must be cost minimizing. You can attain this by setting the marginal products of capital and labor equal to zero. You can keep adding one unit of labor or capital until the marginal benefit you receive offsets the cost. b. If the labor income share in production is equal 0.75, find the value of α in the production function using one of the Cobb-Douglas properties discussed in class. Using one of the Cobb-Douglas production properties from class I was able to find out that α =0.25. This happens because if labor income share is 0.75 and the
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coefficients must be equal to 1, then it only makes sense to subtract the labor share from one. This also means that this Cobb-Douglas has constant returns to scale. c. What is the level of production if 10 units of capital are used in production, the rental rate of capital is 5, and the output price is 2? To find the level of production with what is given you first need to find the real rental price of capital or MPK, which is R/P. So you find that the MPK=5/2=2.5 and use that into the Capital Productivity of Y/K, where K=10.
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This note was uploaded on 10/03/2008 for the course ECON 302 taught by Professor Gold during the Spring '07 term at University of Wisconsin.

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ECON 302 HW 2 - Rebecca Chapman Econ 302-Spring 2008...

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