# Moreover the positivity of du implies ? 0

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business â€˜17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. Itâ€™s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania â€˜17, Course Hero Intern

• The ability to access any universityâ€™s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLAâ€™s materials to help me move forward and get everything together on time.

Jill Tulane University â€˜16, Course Hero Intern