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PS4Answers - UCLA Economics 11 Fall 2009 Professor Mazzocco...

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1 UCLA Economics 11 – Fall 2009 Professor Mazzocco Problem Set 4 – Solutions 1) a) The Lagrangian is L = (1/3).ln(X) + (2/3).ln(Y) + λ(I – P X .X – P Y .Y), and its FOCs are: (1/3)X -1 – λ.P X = 0, and (2/3)Y -1 – λ.P Y = 0. Now using the budget constraint, i.e., I = P X .X + P Y .Y, it follows that d x (P X ,P Y ,I) = (1/3).(I/P X ), whereas d y (P X ,P Y ,I) = (2/3).(I/P Y ). b) Now we should minimize P X .X + P Y .Y subject to (1/3).ln(X) + (2/3).ln(Y) ≥ U for some U>0. Or equivalently, we can maximize –[P X .X + P Y .Y] subject to (1/3).ln(X) + (2/3).ln(Y) ≥ U for some U>0. Hence, the Lagrangian for this problem may be written as L = -[ P X .X + P Y .Y] + λ[(1/3).ln(X) + (2/3).ln(Y) – U]. The FOCs in this case are: -PX + λ (1/3)X -1 = 0, and -PY + λ (2/3)Y -1 = 0. Finally, combing the FOCs with the fact that the constraint should hold with equality, i.e., (1/3).ln(X) + (2/3).ln(Y) = U, we obtain the compensated demands. Specifically, after playing a bit with the algebra,
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PS4Answers - UCLA Economics 11 Fall 2009 Professor Mazzocco...

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