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101A_problemset4 - Econ 101A Problem Set 4 Due in class on...

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Econ 101A — Problem Set 4 Due in class on Th 2 April. No late Problem Sets accepted, sorry! This Problem set tests the knowledge that you accumulated mainly in lectures 14 to 17. The problem set is focused on production. General rules for problem sets: show your work, write down the steps that you use to get a solution (no credit for right solutions without explanation), write legibly. If you cannot solve a problem fully, write down a partial solution. We give partial credit for partial solutions that are correct. Do not forget to write your name on the problem set! Problem 1. Production: cost minimization and pro fi t maximization (11 points) In class we introduced two di ff erent characterizations of fi rm decisions: cost minimization and pro fi t maximization. In this exercise you are asked to show that the two optimization problems are equivalent ways of solving the same problem. Consider a mango farm in the Philippines. The farm has a production function y = f ( L, K, F ) , where L is labor, K is capital, and F is farm land. The price of labor is w, the price of capital is r, the price of land is s, with all of the prices positive. Assume also f 0 i > 0 and f 00 i,i < 0 , with i = L, K, F. 1. Write down the fi rst step of the cost minimization problem and the fi rst order conditions with respect to L, K, and F. (2 points) 2. The fi rst order conditions in step 1, together with the constraint y = f ( L, K, F ) , allow us to solve for L ( w, r, s, y ) , K ( w, r, s, y ) and F ( w, r, s, y ) . We can therefore write the cost function c ( w, r, s, y ) = wL ( w, r, s, y ) + rK ( w, r, s, y ) + sF ( w, r, s, y ) . In the second step of the cost minimization, the fi rm maximizes py c ( w, r, s, y ) . Write down the
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