Question

QUESTION ONE

A Consumer seeks to maximize a utility function Ux,y subject to his income constraint given by:   P1x+P2y=M.

1. What is meant by a duality problem in constrained optimization? Please provide examples.                                                                                                            [6 marks]
2. Set up a Lagrange function for this optimization.                                                  [3 marks]
3. State the First order conditions and explain how you would solve for the critical values.                                                                                                                                      [6 marks]
4. Explain the meaning of the Lagrange multiplier as it related to this optimization problem.                                                                                                                                       [4 marks]

Given a utility function: Ux,y=x12y12 and an income constraint: 50=3x+2y maximize the utility function, subject to the constraint.

Answer 1

Solution

Utility function =U(X,Y)

(Budget constraint) Xp1+YP2=M

(a) Here for the primal problem consumer want to maximize utility for a given level of the budget constraint

for this problem we have Lagrangian function

L=U(X,Y)+K(M-XP1-YP2)

for the dual problem, consumer want to minimize expenditure for a given level of utility for this we have Lagrange function

L=XP1+YP2+K[U-U(X,y)]

(b) Lagrange function for optimization is

L=U(X,Y)+K(M-XP1-YP2)

(c)

(d) Lagrange multiplier is the ratio of marginal benefit to marginal cost or we can say this is the benefit-cost ratio where numerator shows an increase in utility due to increase in consumption of one of good where denominator shows burden on budget function due to increase in one unit of consumption of a good.

K = MUX / P1= MUY/P2