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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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 Fall '07
 McDevitt
 Economics

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