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Unformatted text preview: & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 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 & ; f ( ax 1 ;ax 2 ) = af ( x 1 ;x 2 ) : Let a & : 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 2 if...
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This note was uploaded on 12/23/2009 for the course ECON 3130 at Cornell University (Engineering School).
 '06
 MASSON
 Microeconomics

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