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Quantitative Microeconomics
Answers to Problem Set #2
1)
Slutsky equation:
∂
X
/
∂
P
X
= (
∂
h
X
/
∂
P
X
) [(
∂
X/
∂
M)*X]
a)
Lagrangian: L= XY
2
+
λ
(M P
Y
Y P
X
X )
i)
Firstorder conditions (FOC):
∂
L/
∂
Y = 2YX 
λ
P
Y
= 0
∂
L/
∂
X = Y
2

λ
P
X
= 0
∂
L/
∂λ
=M P
Y
Y P
X
X =0
ii)
Using the first and second FOC:
[Y/2X ]=P
X
/P
Y
i.e. MRS =ratio of the prices
This is equivalent to showing that the consumer is at his optimum when marginal
benefits=marginal costs. [Y/2X] is simply the marginal benefit of food in terms of clothing
since it is the slope of the indifference curve and is equivalent to the ratio of marginal
utilities.
Similarly, P
X
/P
Y
is simply the marginal cost of food in terms of clothing since it is
simply the slope of the budget line.
Rearranging this expression X=YP
Y
/2P
X
.
iii)
Substituting for X in the third FOC:
M P
Y
Y P
X
[(Y P
Y
)/2P
X
]=0
This condition is equivalent to saying that the constraint boundary condition principle is
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 Fall '07
 Loury
 Microeconomics

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