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WELLESLEY COLLEGE
DEPARTMENT OF ECONOMICS
ECONOMICS 20103
JOHNSON
Answers to Problem Set #2
1.
In
a Cobb/Douglas world, expenditure (price*quantity) is a constant fraction
ofincome for each good.
...so
x* = I
1
2Px in one
ofour examples implies
that the consumer always SPENDS half
ofhis/her income on the X good.
Looking at the table, therefore, I should expect that the share
ofincome spent
on food to be constant
if food preferences are Cobb/Douglas. Indeed, they are
not.
...as an example, using food expenditure as a fraction
ofincome  we go
from
4196/25,000
for the lower group to 5671145,000 for the upper group (we
used the average
ofthe income ranges for both). These denominators
represent an 8.0% increase and clearly the numerators do NOT represent such
a large
% increase. Thus, the expenditure
shaJ;'e on food clearly DROPS with
income (Engel's Law) andfood preferences are not Cobb/Douglas.
Entertainment is closer (likely because it's closer to being a luxury good), but
it still doesn't exhibit constant expenditure shares. Remember that
Cobb/Douglas preferences imply income elasticities
of 1, so neither food nor
entertainment exhibit that large an income elasticity.
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a)No
)
..rht.
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ki"m
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ba~.
MRS
=
MU
x
=
Y
+
1 . This shows that the MRS is not constant along a horizontal or
X,Y
MU
x
Y
vertical ray, which implies that the indifference curves are NOT parallel displacements
of
each other.
Remember that to be quasilinear it has to be linear in only one
of the variables. Actually,
perfect substitutes are also quasilinear but to me they're linear. For our purposes, quasi
linear means linear in only one
ofthe two arguments. The multiplicative term insures it
is not so.
b) The equation for a particular indifference arises from setting U(X,Y)= constant K.
So U(x,y)=X*Y
+
X=K implies
k
y=1
x
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°l.,;~~~
1

From the indifference curves we can see that preferences are strictly convex (MRS
diminishes). Also, ¢.ey are monotonic, strictly so
ifwe ignore the possibility ofX=O.
b)Since preferences are strictly convex but the indifference curves touch the xaxis, we
may have an interior solution but should also beware
ofcorners. Using the Lagrangean
gives us the potential interior solution (FOCS = First Order Conditions):
L
=
XY
+X
A(PxX
+
PyY
I)
FOeS
=>
y
+
1
=
P
x
x
P
y
y
=
G...otherwise
I
+P
y
x
=
.....
.for
.
.I>
P
y
;
2P
x
I
h
.
x
=
...ot erwzse
P
x
c) Plug in the prices and income: the optimal quantities consumed are
Y=5
and X=12.
Recall that the marginal utility
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 Fall '08
 JOHNSON
 Economics

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