. Don’t forget to check the constraint qualification conditions.Answer:The derivative of the constraints isdg=211113-100-1.A quick look at the constraints shows that at most two can bind simultaneously. Sinceany two rows ofdgare linearly independent, NDCQ is statisfied. Alternatively, you canuse the fact that all the constraints are linear.The Lagrangian isL=x+2y-λ1(2x+y-6)-λ2(x+y-4)-λ3(x+3y-9)+μxx+μyy.The first order conditions are0 = 1-2λ1-λ2-λ3+μx0 = 2-λ1-λ2-3λ3+μyIfx= 0, the first two constraints are slack (y≤6 andy≤4), soλ1=λ2= 0. Thenλ3>0, soy= 3. This impliesλ3= 2/3 andμx=-1/3<0, so it is not a solution.Now we knowx >0 andμx= 0. Supposey= 0. Then constraints (2) and (3) areslack (x≤4 andx≤9) since constraint (1) tells usx≤3. Nowλ2=λ3= 0. Hereλ1= 1/2 andμx=-3/2<0, so this is not a solution either.
MATHEMATICAL ECONOMICS EXAM #2, NOVEMBER 9, 2012Page 3We must have bothx >0 andy >0, soμx=μy= 0. If constraint (1) binds, constraint(2) impliesx≥2 while constraint 3 sayx≥9/5. Constraint (3) cannot bind, soλ3= 0.This implies 1 = 2λ1+λ2and 2 =λ1+λ2, soλ1=-1, which is impossible. This showsthat constraint (1) cannot bind.
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