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midterm1ans - MIDTERM#1 SUGGESTED ANSWERS Question 1 1 4...

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° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° MIDTERM #1 ° SUGGESTED ANSWERS ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° Question 1 1. [ 4 points ] A production function f ( x 1 ; x 2 ) exhibits constant returns to scale if for any a ° 0 ; f ( ax 1 ; ax 2 ) = af ( x 1 ; x 2 ) : Let a ° 0 : Then, f ( ax 1 ; ax 2 ) = min f 2 ax 1 ; ax 1 + ax 2 g = min f a (2 x 1 ) ; a ( x 1 + x 2 ) g = a min f 2 x 1 ; x 1 + x 2 g = af ( x 1 ; x 2 ) : 2. [ 6 points ] The cost minimization problem (CMP) of the ±rm is: min ( x 1 ;x 2 ) 2 R 2 + f w 1 x 1 + w 2 x 2 + ° g ; subject to min f 2 x 1 ; x 1 + x 2 g ° q Since the production function is not di/erentiable, we cannot use our usual calculus based method for ±nding a minimum. First, note that the objective function is strictly increasing in x 1 and x 2 ; the production constraint must bind. Hence min f 2 x 1 ; x 1 + x 2 g = q: However, unlike the usual Leontief production functions, since one of the arguments of the production functions involves both x 1 and x 2 ; the cost minimizing input combination does not necessarily lie on the kink of the isoquant. Moreover, we can write the production constraint as x 1 + min f x 1 ; x 2 g = q: From this expression it is easier to see that the isoquants will look like a combination of the usual Leontief case and that of a linear production function: Hence, the solution to the optimization problem is: 8 < : x ° 1 = x ° 2 = q
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