# sol_fin08 - S F E500 8:00 Question 1 Utility is quasilinear...

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S  : F  E  500, 8:00  December 18, 2008 Question 1 Utility is quasilinear, i.e., u ( x ) = v ( x 1 ) + x 2 . Thus, v ( x 1 ) = p 1 . Thus, 20 4 p 1 = x 1 implies p 1 = 5 (1 / 4) x 1 . Hence, v ( x 1 ) = 5 (1 / 4) x 1 , and therefore v ( x 1 ) = 5 x 1 (1 / 8) x 2 1 . Thus, u ( x 1 , x 2 ) = 5 x 1 (1 / 8) x 2 1 + x 2 Question 2 To produce q units of output, we must have z 1 + 2 z 2 = q 2 . Thus, costs are c ( w, q ) = min { w 1 , 0 . 5 w 2 } q 2 . The first order conditions for profit maximization is p = min { w 1 , 0 . 5 w 2 } 2 q . Thus, q = p min { 2 w 1 , w 2 } . Hence profit is p 2 min { 2 w 1 , w 2 } min { w 1 , 0 . 5 w 2 } p 2 4 min { w 1 , 0 . 5 w 2 } 2 = p 2 min { 2 w 1 , w 2 } p 2 2 min { 2 w 1 , w 2 } . Then the firm’s profit function is π ( w 1 , w 2 , p ) = p 2 2 min { 2 w 1 ,w 2 } Question 3 Suppose that a firm can operate at two locations, using one input at each location (the input also has the same price at the two locations). The production

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sol_fin08 - S F E500 8:00 Question 1 Utility is quasilinear...

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