Problem Key: Ch. 5. 5.6, 5.7, 5.8, 5.9
Suggested Answers
5.6 Suppose than an individual’s utility for X and Y is represented by the CES function
(for =-1):
utility =
U(X,Y)
=
-1/X
-
1/Y
a. Use the Lagrangian multiplier method to calculate the uncompensated demand
functions for X and Y for this function.
L
=
-1/X
FONC imply
1/X2 =
-
1/Y + (I – PxX – PyY)
PX
;
=
PX/ PY or
PY
1/Y2
=
=
(PY/ PX).5Y
Taking the ratio
Y2/X2
Y
=
(PX/ PY).5X
and
X
Insert Y or X into the budget constraint to recover uncompensated demand functions for X
or Y, respectively. First demand for X is
I
+
PY(PX/ PY).5X
=
PXX
=
I/[PX + (PYPX).5]
or
X
Similarly, for Y,
I
=
PX(PY/ PX).5Y +
Y
=
=
PYY
I/[PY + (PYPX).5]
b. Show that the demand functions calculated in part (a) are homogeneous of degree zero
in PX, PY and I.
This is obvious. Consider, for example, the demand for X. Multiply I, PY and PX by 1+,
X
=
I(1+)/[PX(1+) + (1+)(PYPX).5]
Similarly for Y.
1
=
I/[PX + (PYPX).5]
c. How do changes in I or in PY shift the demand curve for good X?
Income effect: X is a normal good, since X and I move directly. Increases in I will shift
out the demand for X.
Related good effect: Y is a complement for X. An increase in PY reduces X, shifting in the
demand curve.
5.7 As in Example 5.1 assume that utility is given by
utility =
U(X,Y)
=
X.3Y.7.
a. Use the uncompensated demand functions given in Example 5.1 to compute the
indirect utility function and the expenditure function for this case.
From problem 5.1
X
=
.3I/PX ;
Y
=
.7I/PY
The Indirect utility function is generated by inserting the demand functions for X and Y
into the utility function:
V
=
(.3I/PX).3(.7I/PY).7
=
IK/[ PX.3 PY.7] where K = .3.3.7.7
To get the expenditure function, solve indirect utility for I
E
=
V PX.3 PY.7/K
b. Use the expenditure function calculated in part (a) together with Shephard’s Lemma
(footnote 5, ch. 5) to compute the compensated demand function for good X.
Shephard’s Lemma simply applies the envelope theorem to the expenditure function.
Specifically, we can take the derivative of the expenditure function with respect to a
parameter in the system (here Px) as we would take the derivative w.r.t. any variable, if
the variables are at their optimal values. Here
E/PX
=
hX
=
.3V PX-.7 PY.7/K
c. Use the results from part (b) together with the uncompensated demand function for
good X to show that the Slutsky equation holds for this case.
The Slutsky Equation states
dX/PX
=
hX/PX
-
dX/E [E/PX]
This problem requires showing that the LHS of the above equality equals the RHS.
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First, dX/PX =
-.3I/PX-2
Now consider compensated demand
hX/PX
=
-(.3)(.7)V PX-1.7 PY.7/K =
=
.3/Px
=
hx
=
hX/PX
-
=
=
=
-.7X/PX-X/PX
-.3I/PX-2,
.3 X/Px
-.7X/PX
Finally,
dX/E
and
E/PX
=
(.3) V PX-.7 PY.7/K
Thus,
dX/PX
dX/E [E/PX]
implies
-.3I/PX-2
since X =
.3I/PX
5.8 Suppose the utility function for goods X and Y is given by
utility =
U(X,Y)
=
XY + Y
a. Calculate the uncompensated (Marshallian) demand functions for X and Y and
describe how the demand curves for X and Y are shifted by changes in I or in the price of
the other good.
L
=
XY + Y + (I – PxX – PyY)
FONC imply
Y
=
PX
Taking the ratio, Y/(X+1)
Y = (PX/PY)(X+1)
;
;
X+1
=
PY
=
PX/PY . Thus
X= PYY/PX – 1
Inserting this expressions into the budget constraint yields demand functions for X and Y.
I
=
=
PxX
PxX
+
+
PY(PX/PY)(X+1)
PxX + Px
3
Thus,
X
=
(I - PX)/(2PX)
Solving for Y,
I
=
=
PX(PYY/PX – 1)
PYY PX
Y
=
(I + PX)/(2PY)
+
+
PYY
PYY
Notice that both X and Y are normal goods. The demand for both X and Y will shift out as
income increases.
Consider finally cross price effects. Observe that in the demand for X the price of Y does
not appear. On the other hand notice that the demand for Y is directly related to PX. An
increase in the price of X will shift out the demand for Y as people substitute away from
X.
(By the way, consider how such a result can arise. When PY increases, demand
for X will fall only through an income effect. On the other hand, an increase in the price
of X both decreases income, and causes consumers to substitute away from X. We study
this in Chapter 6.)
b. Calculate the expenditure function for X and Y.
V
=
=
=
(I - PX)/(2PX) [(I + PX)/(2PY)]
(I - PX) (I + PX) /[4PXPY]
(I + PX)2/[4PXPY]
=
2V.5(PXPY).5 - PX
+
+
(I + PX)/(2PY)
2PX(I + PX)/(4PXPY)
Thus
E
c. Use the expenditure function calculated in part (b) to compute the compensated
demand functions for goods X and Y. Describe how the compensated demand curves for
X and Y are shifted by changes in income or by changes in the prices for the other good.
By Shephard’s Lemma
E/PX
=
X
=
=
V.5(PY/PX).5- 1
V.5(PY.5 - PX.5) /PX.5
E/PY
=
Y
=
V.5(PX/PY).5
By definition, income changes in compensated demand functions only to leave V constant.
Thus, the effect of an income increase on an (income compensated) demand curve is
definitionally zero. On the other hand, observe that an increase in the price of Y
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increases the compensated demand for X (implying that Y is a net substitute. Similarly, an
increase in the price of X increases the compensated demand for Y, again implying that
X is a net substitute for Y.
The difference in demand in the two cases is that compensated own price effects
are stronger for X than for Y. For a given change in PX, a consumer will shift away from
X to a greater extent than the customer would shift away from Y in response to a change
in PY ..
5.9 Over a three-year period, an individual exhibits the following consumption behavior:
Px
Py
X
Year 1
3
3
7
Year 2
4
2
6
Year 3
5
1
7
Is this behavior consistent with the strong axiom of revealed preference?
Y
4
6
3
No. “Reveal preferred” implies that the quantity -weighted price of a selected bundle is
less than for any nonselected bundle. But in the above this clearly does not hold. Fr
example, bundle 3 could be purchased for less than the bundle 1 bought at year 1 prices.
(Since less of Y is purchase in year 3)
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