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Moreover the positivity of du implies ? 0

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by complementary slackness. Moreover, the positivity of du implies λ > 0. Complementary slackness then implies px + y = I . The additional equation required is px + y = I , which gives us three equations in three unknowns ( x , y , λ ). c ) Are the second-order sufficient conditions satisfied? Answer: The Hessian of L is d 2 u , which is negative definite, so the second-order conditions for a maximum are satisfied. d ) Consider the system from parts (a) or (b) (as appropriate) as implicitly defining ( x * , y * , λ * ) as functions of ( p, I ). Is x * ( p, I ) a continuously differentiable function? Why? Answer: The relevant system is 0 = u x - λp 0 = u y - λ 0 = px + y - I
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MATHEMATICAL ECONOMICS FINAL, DECEMBER 10, 2002 Page 2 If we call the right hand side F ( x, y, λ, p, I ), we can apply the implicit function theorem provided d ( x,y,λ ) F is invertible. We have d ( x,y,λ ) F = 2 u x 2 2 u x y - p 2 u x y 2 u y 2 - 1 - p - 1 0 . The determinant is 2 p 2 u x y - p 2 2 u y 2 - 2 u x 2 = ( - 1 , - p ) d 2 u - 1 - p . Since d 2 u is negative definite, the determinant is negative, and so the matrix is invertible. The implicit function theorem then shows ( x, y, λ ) is a C 1 function of ( p, I ). 3. A consumer has the quasi-linear utility function u ( x, y ) = x + y 2 . The consumer consumes non-negative quantities of both goods, subject to the budget constraint: px + y 6. Find ( x * , y * ) that maximizes utility subject to the above constraints. Be sure to check the constraint qualification and second-order conditions. Answer: Note that dg = p 1 - 1 0 0 - 1 .
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