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E604
Problem Set #5.
Problems 5.1, 5.2 and 5.4.
5.1. Thirsty Ed drinks only pure spring water, but he can purchase it in two differentsized containers - .75 liter and 2 liter. Because the water itself is identical, he regards
these two “goods” as perfect substitutes.
a. Assuming Ed’s utility depends only on the quantity of water consumed and that the
containers themselves yield no utility, express this utility function in terms of quantities
of .75L containers (X) and 2L containers (Y).
U
=
X
+
8/3Y
b. State Ed’s demand function for X in terms of PX, PY and I.
Since the products are perfect substitutes, Ed will spend all his income on X if
Py/Px> 8/3, and all Y otherwise. Thus, the demand function for X is
X
=
I/PX if Py/Px> 8/3, 0 otherwise
Symmetrically, the demand function for Y is
Y
=
I/PY if Py/Px< 8/3, 0 otherwise.
(Note: This doesn’t fit well into the standard Lagrangian set- up It’s easy to
solve for the price ratio Py/Px = 8/3. However, marginal utilities are constants, making it
impossible to recover a demand function from the first order conditions. This is because
the products are perfect substitutes)
c. Graph the demand curve for X, holding I and PY constant. Let I = 10, and Py
=2. Observe that when Px = .75, demand for X falls to zero.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Demand for X, Py = 2, I = 10
0
10
20
30
40
50
60
70
80
90 100 110
d. How do changes in I and PY shift the demand curve for X?
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Increase in Py
5
Demand for X, Py = 2, I = 10
Demand for X, Py = 2, I = 16
Price
4
3
2
D
1
0
0
0
10 20 30 40 50 60 70 80 90 100 110
5
10
Quantity
Demand for X moves directly with I. On the other hand, increases in Py shift
upward the switch point where consumers shift away from X to Y.
P
e. What would the compensated demand curve for X look like in this situation?
The products are perfect substitutes. So when X is being purchased, any further price
reduction results in an increase in X solely due to income effects. Abstracting out this
income effect would leave the quantity of X unchanged. On the other hand, beyond the
switch point, only Y is consumed and no changes in Px will have any income or
substitution effect.
Graphically, then, the demand curve would consist of a vertical line. At any price
ratio where X was consumed, every increase or decrease in Px would have enough
income added or subtracted to leave consumption at exactly the same point. The only
change occurs when Px rises enough to cause a shift to the consumption of Y. Any further
price increases would require income supplements that would keep consumption at the
same level of Y.
Compensated Demand for X
P1
X
5.2. David N. Gets $3 per week as an allowance to spend any way he pleases.
Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on
peanut butter (at $.05 per once) and jelly (at $.10 per ounce). Bread is provided free of
charge by a concerned neighbor. David is a particular eater and makes his sandwiches
with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will
never change these proportions.
a. How much peanut butter and jelly will David buy with his $3 allowance in a
week?
Let x = peanut butter, and y = jelly.
U = min{2x,y}, so he consumes in proportion z = 2x = y. That is, for each
sandwich, he needs 2x and y
The budget constraint is
I
=
Substituting
3
=
3
=
z = 15
.05x
+.10y
.10(z) +.10(z)
.20z
x = 30 ounces of peanut butter;
y = 15 ounces of jelly
b. Suppose the price of jelly were to rise to $.15 an ounce. How much of each
commodity would be bought?
3
=
.10z +.15(z)
3
=
.25z
z= 12
z = 24 ounces of peanut butter;
y = 12 ounces of jelly
c. By how much should David’s allowance be increased to compensate for the rise
in the price of jelly in part (b)?
In part (a), z = 3/.2 = 15. To consume 15 sandwiches when the price of z = .25,
Income must rise so that I/.25 = 15, or I’= $3.75, an increase of $.75.
d. Graph your results in parts (a) to (c)
Increase in the Price of
Jelly Servings
y -Jelly Servings
40
30
20
Compensating Income
U (Z=15)
10
U (Z=12)
0
0
10
20
30
40
50
X - Peanut Butter Servings
e. In what sense does this problem involve only a single commodity, peanut butter
and jelly sandwiches? Graph the demand curve for this single commodity.
David consumes z = 2 ounces of peanut butter and 1 ounce of jelly sandwiches
Price of PB&J Sandwiches
Thus, z = I/Pz
z = I/Pz
0
0
10
20
30
40
50
Z - PB&J Sandwiches
f. Discuss the results of this problem in terms of the income and substitution
effects involved in the demand for jelly.
Notice that in the case of perfect complements, an increase in the price of a good
prompts a pure income effect. David is simply unable to avoid the effects of an increase
in the price of jelly by consuming more relatively less expensive peanut butter.
5.4. Show that if there are only two goods (X and Y) to choose from, both cannot
be inferior goods. If X is inferior, how do changes in income affect the demand for Y?
Suppose not. Then when income increased, the consumption of both X and Y must
fall. But this implies an inward shift of the indifference curve, a contradiction of the
assumption that the products are ¨goods¨ e.g., that more is preferred to less.
Now suppose a good X is inferior. Then Y must be a normal good, causing an
increase in demand for Y.