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April 10th, 2014, 08:49 AM   #1
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Kuhn-Tucker Conditions

I am using Matlab to solve a maximisation problem subject the the restriction that xi<=delta*TR(x1,x2). There are two lagrangians associated with my problem:

L1=U1(x1,x2)+lambda1*(delta*TR(x1,x2)-x1)

L2=U2(x1,x2)+lambda2*(delta*TR(x1,x2)-x2)

I need to solve for the vector (x1,x2)

I am investigating two cases:

Case 1: The restriction is binding in both lagrangians i.e. xi=delta*TR(x1,x2) for i=1,2. The Kuhn-Tucker conditions for this require that lambda1>0 and lambda2>0. Thus, I have four FOCS's:

(i) dL1/dx1 (ii) dL1/dlambda1 (iii) dL2/dx2 (iv) dL2/dlambda2

Case (ii)

The restriction xi<=delta*TR(x1,x2) is only binding in the first lagrangian (L1). The Kuhn-Tucker conditions for this require that lambda1>0 and lambda2=0.

My question is: How do I incorporate lambda1>0 and lambda2=0 into my system of equations? Do i need to drop one of the first order conditions? Or do I need to drop the lambda2(delta*TR(x1,x2)-x2) term in the second lagrangian.
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