Question: Question one A Consumer seeks to maximize a utility function U ( x , y ) subject to his income constraint given by : P

Question one

A Consumer seeks to maximize a utility function U(x,y) subject to his income constraint given by:P1x+P2y=M

a)What is meant by a duality problem in constrained optimization? Please provide examples. [6 marks]

b)Set up a Lagrange function for this optimization.[3 marks]

c)State the First order conditions and explain how you would solve for the critical values. [6 marks]

d)Explain the meaning of the Lagrange multiplier as it related to this optimization problem. [4 marks]

e) Given a utility function:U(x,y)=x(1/2)y(1/2) and an income constraint:50=3x+2y,maximize the utility function, subject to the constraint.[6 marks]

Question two

Differentiate the following:

a) 3x56x+1/3x [3 marks]

b) ln(3x2+8) [3 marks]

c) 4x3(2x25)3 [6 marks]

d) e2x/(2x1) [4 marks]

e) 4x2+3xy12y6 with respect to y [ 4 marks]

f) e3xlnx2 [ 5 marks]

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