Question

please help me the best you can

Part 1: Optimization with inequality constraints 1. A consumer lives on an island. Her utility function is U = (x²y)1/3. She produces two goods, x and y. She faces a production constraint and an environmental constraint: Her production possibility frontier is: x² + y2 s 300. She faces an environmental constraint given by x + y = 200. a) Set up the Lagrangian function. b) List all of the Kuhn-Tucker conditions. c) Interpret the Lagrange multipliers. (What do they tell you about the problem?) d) Explain the sign of the Kuhn-Tucker conditions relating to x. e) Solve for x, y and U. (Hint: only one of the constraints hold with equality.)

Answer 1

0.7 0.3 a) U= Coelys U= 28/3yl or u= x y constraint styl = 300 Lagrange function - U= se yo is a (300-ody) Use = ote 0.3 yo. 3_2082 0 - 0 Ug = 0.35 07 07 - ay a = 0 - 10 = 3000 y = 0 - 0.722 0.4 03 242 - 10.300 yo.7 - aya o Deviding by 0. Tocou yo. 3_ sch 01300 17 you ayn = 2.3 De' = 2.300 jo.4 = x/y - 2.327y -

b) Stationarity ,For maximizing

For minimizing

Primal feasibility

Dual feasibility

Complementary slackness

200 D 00- 03 OC sety= 200 U on yo.3 + 2C 200-x-4) Usc = 0.22 03 0.3 0 - Ug = 0.3 07 you ao to 2: 200 - -4 = 0 0.7220840132 0.387 4.72 - 0.73 € 0 3 Jo.3 - 1 10.30 e 7 yota = 2.3 ac yoots- C = 139.9 y = 601 a=0.54 /c = 1 1 y = 0.43 00 U2 = 200 - 0 - 014300 =0 ac +0.422 =200 1.480 = 200 ac = 200/1.48 = 139.9 200-139.9 - y = 0 , y = 60'|| 0.3007 yola 0 13X139.97 x (60-1707=2 9.5*0.06 = 0.54