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Unformatted text preview: Economics 302 Spring 2008 Answers to Homework #2 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 L 1- , 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 firms 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. Profit Maximization with respect to K and L implies that the first order derivatives are set equal to zero. 1 1 1 Profit Revenue Cost * Profit * Profit *(1 ) P AK L WL RK P AK L R K R MPK P P AK L W L W MPL P ---- =- =--- = & = &-- = & = If labor (capital) first order condition did not hold, the firm could increase their profits by changing their level of labor (capital). If the cost of labor (capital) times the marginal product of labor (capital) exceed the wage (rental rate), the firm would increase the level of labor and profits as the added benefit of one more worker (unit of capital) exceeds the cost. If the cost of labor (capital) times the marginal product of labor (capital) is below the wage (rental rate), the firm would decrease the level of labor and increase profits as the additional cost of that last worker (unit of capital) exceeded the benefit. b. If the labor income share in production is equal 0.75, find the value of in the production function using one of the Cobb-Douglas properties discussed in class. 1 (1 ) (1 ) (1 ) (1 ) * (1 ) 0.75 0.25 MPL AK L L AK L L AK L Y L L L MPL Y --- =- =--- = = =- = = We can rearrange the Marginal Product of Labor as noted above, getting to the labor income as a share of total production. The above equations tells us that the labor income (L * MPL) share of income (Y) is equal to (1-). We then plug the specific value of 0.75 in to find that =0.25. 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? From the first order condition of profit maximization we have the following relationship. R MPK P = Rearranging the Marginal Product of Capital as we did above for MPL we have the following relationship....
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