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E604
Problem Set #4.
Problems 4.7, 5.1, 5.2 and 5.4.
4.7 In Example, 4.3 we used a specific indirect utility function to illustrate the lump sum
principle that an income tax reduces utility to a lesser extent than a sales tax that garners
the same revenue. Here you are asked to:
a. Show this result graphically for a two-good case by showing the budget constraints that
must prevail under each tax (Hint: first draw the sales tax case. Then show that the
budget constraint for an income tax that collects the same revenue must pass through the
point chosen under the sales tax but will offer options preferable to the individual.)
4
Observe: The budget
constraint for the income tax
intersects the max indifference
attainable given the sales tax.
3
Y
Income Tax:
Y = 1.5 -.25X
2
1
U=2
U=1.4
1
0
0
2
4
Sales tax:
Y = 2 -.5X X
6
8
10
Y = 2 -.25X
b. Show that if an individual consumes the two goods in fixed proportions, the lump sum
principle does not hold because both taxes reduce utility by the same amount
4
3
Y
Income Tax:
Y = 1.5 -.25X
2
1
U=2
U=1.4
1
0
0
2
4
Sales tax:
Y = 2 -.5X X
6
8
10
Y = 2 -.25X
Notice that the sales tax rotates the budget constraint inward. However, the effect of the
income tax is identical, since the consumer enjoys only fixed combinations of the goods.
c. Discuss whether the lump sum principle holds for the many-good case too.
Yes. Intuitively, when goods are perfect complements, then an increase in the price of one
good is the same as an increase in the price of the bundle. Only when goods are
imperfect substitutes will (imperfect) substitution allow consumers to attempt to shift
away from the taxed good.
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 tow “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.
c. Graph the demand curve for X, holding I and PY constant.
Demand for X
5
Price
4
3
2
D
1
0
0
5
10
Quantity
d. How do changes in I and PY shift the demand curve for X?
Increase in I
Increase in Py
5
4
4
3
3
Price
Price
5
2
D
1
2
D
1
0
0
0
5
Quantity
10
0
5
Quantity
Demand for X moves directly with I. On
the other hand, increases in Py shift back the minimum point where the consumer will
purchase units of X
e. What would the compensated demand curve for X look like in this situation?
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.
10
P
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 changes 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
=
3
=
z= 12
.10z
.25z
+.15(z)
z = 24 ounces of peanut butter;
y = 12 ounces of jelly
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?
Y
In a two good world, one illustrates inferior good status when the expansion path
rotates toward the other good as income increases. Superior good status is just the
opposite. Thus, inferior good status for one good implies superior good status for the
other.
X
In the above chart, for example, good X is inferior. That implies that consumers
must be spending increasing portions of income on Y.
c. By how much should David’s allowance be increased to compensate for the rise
in the price of jelly in part (b)?
For David to return to his original utility level, he needs to consume z = 3 more
sandwiches. That requires 3 more ounces of jelly and 6 more ounces of peanut butter, or
6(.05) + 3(.15) = $.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
X - Peanut Butter Servings
50
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.