MATH 125 Quiz: MATH 125 UWashington Quiz125W12 ans

• School: UW
• Department: Mathematics
• Course: MATH 125
• Professor: All

Document Summary

Mathematical economics final, december 9, 2014: a consumer has the unusual utility function u(x, y) = x2 + y2. The consumer consumes non-negative quantities of both goods, subject to the budget constraint: 2x + 3y 6. Find (x , y ) that maximizes utility subject to the above three constraints. Answer: there are three constraints: 2x + 3y 6 0, , x 0, and y 0. The matrix of derivatives of the constraints is: Constraint quali cation will be satis ed as long as at most 2 constraints bind. Exam- ining the constraints shows that at most 2 can bind (see the diagram). x2. The lagrangian is l = x2 + y2 (2x + 3y 6) + xx + yy. The rst-order conditions are 2x 2 + x = 0 and 2y 3 + y = 0. Since utility is increasing, there are 3 cases to consider: x, y > 0. Complementary slackness implies x = y = 0.