# Answer in part a we found that a is invertible it

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Answer: In part (a) we found that A is invertible. It follows that the system of equations has exactly one solution. For the record, the solution is (4 / 3 , - 2 , 3) t . 3. Suppose a firm’s production function is Q = K 1 / 3 L 2 / 3 and that K = 8000 and L = 64. a ) How much can the firm produce? b ) What are the marginal products of capital ( K ) and labor ( L )? c ) Suppose that the available capital falls by 2 units, while labor increases by 5 units. Without plugging the new numbers for K and L into the production func- tion, compute approximately how much the firm can now produce. 4. Let f ( x, y, z ) = x 2 + 3 y + z 3 - 5. a ) Find an ( x 0 , y 0 , z 0 ) satisfying f ( x 0 , y 0 , z 0 ) = 0. Answer: The point (1 , 1 , 1) works. b ) Can x be expressed as a function g ( y, z ) in some neighborhood of ( x 0 , y 0 , z 0 )? Answer: Since f/ x = 2 x , f/ x = 2 at ( x 0 , y 0 , z 0 ). The Implicit Function The- orem yields such a function g . Alternatively, note that g ( y, z ) = (5 - 3 y - z 3 ) 1 / 2

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MATHEMATICAL ECONOMICS FINAL, DECEMBER 11, 2003 Page 2 works. c ) Compute dg . Answer: By the Implicit Function Theorem, dg = ( - 1 / 2 x )( f/ y, f/ z ) = - (3 / 2)(1 , 3 z 2 ). At (1 , 1 , 1), this has the value ( - 3 / 2 , - 3 / 2). 5. Consider the problem of maximizing x 2 + y 2 subject to the constraints x 1, y 2, and x + 2 y 10. a ) Does this problem have a solution? b ) Is constraint qualification satisfied? c ) Assuming there is a solution, find it. Make sure to check the second-order con- ditions.
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