1
Problem Set 1 Solution Key
ECN 131
Public Finance
Prof. Farshid Mojaver
1. There are only two goods in the world: apples (A) and bananas (B). Your old budget constraint was
5A+10B=240. Your new budget constraint is 5A+12B=240.
a) What has changed? Be as specific as possible.
b) Sketch the old budget constraint, with A on the horizontal axis and B on the vertical. Label point
X as the consumer’s consumption choice. (Make sure it is an interior point.
c) Sketch the new budget constraint. Assuming A and B are substitutes, label point Z as the
consumer’s new choice.
d) Using a different color pen or pencil, sketch a third budget constraint to illustrate the substitution
effect separate from the income effect. Label one point on the graph Y such that the movement
from X to Y represents a substitution effect, and Y to Z represents the income effect.
a) The price of B has increased from $10 to $12.
b) See BC1 and point X on graph.
c) See BC3 and point Z on graph. Z must be below X, representing less consumption of good B
than before.
d) See BC2 (red dotted line) and point Y. This represents the new price ratio (hence its slope is
the same as BC3) but “gives back” enough income for the consumer to stay at his/her original
utility level. In this way, point Y on BC2 represents the consumer’s response to the change in
relative prices only (substitution effect), without the income effect. The income effect is
represented by the move from Y to Z, because this is a parallel shift of the budget constraint.
Note that as the consumer moves from X to Y (substitution effect), the demand for B decreases
but A increases. The income effect from Y to Z has demand for both goods decreasing. So
overall, demand for B must decrease, but demand for A could increase or decrease, depending on
which effect is stronger.
2. Assume Fifi has 2000 hours to allocate each year between leisure and work. Her wage is $8/hour.
a) Sketch Fifi’s budget constraint.
b) Suppose an income support program provides an income guarantee of $4000 and a benefit
reduction rate of 50%. Sketch Fifi’s budget constraint under this program. Label this budget
constraint BC2 and give coordinates for any endpoints or kink points.
c) Suppose a revised program provides $3000 and a benefit reduction rate of 25%. Using a different
color pen or pencil, sketch Fifi’s budget constraint under this program. Label this budget
constraint BC3 and give coordinates for any endpoints or kink points.
d) Does the reduced income guarantee in part (c) necessarily imply that Fifi will increase her labor
supply compared to her choice in part (b)? Use your graph to support your answer.
2
a) See BC1 on the graph.
b) See BC2 (the heavy black line) on the graph.
c) See BC3 (red line) on the graph.
d) The change from BC2 to BC3 will NOT necessarily imply that someone will increase labor
supply. Consider someone who was choosing point d on BC2.
The new budget constraint BC3 represents a higher income level (BC3 is above BC2 around
point d), which tends to increase demand for leisure and market goods.
But BC3 is steeper than BC2, indicating a higher price of leisure (relative to market goods) and
thus the substitution effect tends to make Fifi choose less leisure – and more labor. So if the
substitution effect dominates the income effect, Fifi chooses more labor supply rather than less.
3. Using examples and graphs
a) Show that a cut in TANF benefit can increase labor supply.
b) Show that a cut in TANF benefit does not necessarily increase labor supply
c) Show that a cut in benefit reduction rate can decrease labor supply (carefully illustrate substitution
and income effects).
Ans.a)
Wage
Income
Leisure
b)
3
Wage
Income
Leisure
c)
Wage
Income
SE
IE
TE
Leisure
4. Consider the case of a single mother with two children, who is trying to decide how many hours to work
each month. Suppose, due to constraints such as child care, she is limited to working a maximum of 40
hours per week, for a total labor supply endowment of 160 hours per month. This mother has no other
source of labor income, and if she works, would receive a wage of $6 for each hour worked.
Assume that the mother’s utility function is defined over money (C) and leisure (L), and takes the
following form:
U = 3 log C + 5 log L
a) Solve for the mother’s optimal choice of labor supply. How much money does the mother have for
consumption?
b) Suppose the state government introduces a welfare program for single mothers (similar in nature
to TANF). Under this new program, a mother with no labor income would receive $120 per month
from the government (i.e. $120 is the benefit guarantee). However, if the mother chooses to work,
this benefit amount is reduced by 20% of her labor income (i.e. the benefit reduction rate is 20%).
i.
Intuitively, how might you expect labor supply to differ under this program than if no welfare
program existed? Should the mother work more or less if welfare is provided, or is the answer
uncertain? Demonstrate graphically what happens to the budget set that this mother faces due
4
to the introduction of welfare (labeling the number of hours at which she stops receiving any
benefits from the government).
ii.
Now, solve for the optimal choice of labor supply. How many hours of work will the mother
now provide? What is her total labor income? What is her total income (including the welfare
benefit)? How has labor supply changed from in (a), and how has total income changed? Is
the woman better off?
c) The single mother moves to a neighboring state. This new state also has a welfare program,
similar in nature to the one considered in (b) – the difference is that the benefit guarantee in this
state is $300 and the benefit reduction rate is 50%.
i.
Intuitively, how might you expect labor supply to change relative to the TANF regime
represented in (b)? Demonstrate graphically how the budget constraint is different from that in
(b), and how the optimal choice of labor supply may change.
ii.
Now how many hours does the mother work? How is her labor income and total income
different from that in (b)?
d) Now, let’s apply these theoretical results. Consider a state that has an established welfare regime
as in (b). The Governor of the state is concerned that too many single mothers are out of the labor
force. To remedy this, he is trying to decide between implementing one of three different welfare
policy reforms:
1) lowering the benefit amount, but keeping the benefit reduction rate the same
2) keeping the benefit amount the same, but lowering the benefit reduction rate
3) lowering the benefit amount the same amount as (1) and lowering the benefit reduction rate
the same as (2)
i.
Theoretically, how should each proposal affect the labor supply of unemployed single
mothers, and why?
ii.
How should each proposal affect the labor supply of welfare mothers who are working?
iii.
Which proposal do you expect to result in the largest number of workers who were previously
not receiving benefits to now start receiving welfare benefits?
iv.
Based on these conclusions, which proposal would you suggest the Governor implement?
Why?
a) L = leisure
C = 6(wage) × hours of work = 6(160–L) = 960–6L
U = 3 log C + 5 log L = 3 log(960–6L) + 5 logL
U/L = 0  (–18)/(960–6L) + 5/L = 0  –18L + 4800 – 30L = 0
 4800 = 48L
 100 = L
So leisure = 100
hours of work = 60
C = 6 × 60 = 360
b i) LS(Labor supply) effects depend on individual’s previous choice of labor (i.e., whether
they’ll face the new budget constraint caused by the welfare program or not). If they’re working a
lot, there’s likely no effect. If they’re working a little, there might be a LS effect (probably
negative).
New BC meets old where C new = C old at same L
C new = 120 + 6 × .8(160–L)
C old = 6 × (160–L)
C new = C old  120 = 6 × .2(160–L)
120 = 192 – 1.2L
60 = L
5
C
960
Slope = – w = – 6
600
Slope = – w × (1–BRR)
= – 6 × .8 = – 4.8
120
G = 120
60
160
L
b ii) Assuming that the mother is on the lower (new) portion of the budget constraint,
then C = [ G – W(160–L)(.BRR)] + W(160–L) = 120 + .8 × 6(160–L) = 888 – 4.8L
U = 3 log(888 – 4.8L) + 5 log L  U/L = 0  (–14.4)/(888 – 4.8L) + 5/L = 0
 38.4 L = 4440
L = 115.625, C = 333
Labor income = 6 × (160 – 115.625) = 266.25.
Benefit = G – (0.BRR)W(160–L) = 120 – 0.2*266.25 = $66.25
Total Income = $66.25 + 266.25 = $332.5
The woman MUST be better off (you don’t even need to do a utility calculation for this), because
for any amount of work, she has at least as much income as before the program, so her utility is at
least as great as before (if she did not change her work behavior) or greater than before (if she did
change her behavior).
c i) Now, we’d expect labor supply to fall relative to the program in b. This is because the benefit
amount has increased (at any amount of work, she’ll have more total income than in b – i.e., a
negative income effect on LS) and the benefit reduction rate has also increased, such that returns
to an additional hour of work are less than before (negative substitution effect on LS).
Graphically (dotted line is BC from b, solid line is BC from c)
C
960
Slope = – w = – 6
600
Slope = – 4.8
Slope = – 6 × (1–1/2) = – 3
300
120
60
160
L
6
c ii) The budget constraint is now: 300 + 3(160–L) = 780 – 3L
U = 3 log(780 – 3L) + 5 log L  U/L = 0  (–9)/(780 – 3L) + 5/L = 0
 24L = 3900  L = 162.5
However, the maximum amount of leisure a mother can take is 160, so the actual L = 160, and
the mother doesn’t work at all. She receives 300 in welfare benefit, and has no labor income.
Again, she must be better off than in b, because for any amount of work, she has more income
than before.
d) The following pictures represent the reforms:
C
C
w(1-BRR0)
BRR0
G0
G1
L
Baseline
C
L
① G↓
C
BRR1
BRR1
G0
② BRR↓
G1
L
③ G↓ BRR↓
L
i) All proposals should induce some unemployed women to enter the LF(Labor force). ① quite
clearly makes it more painful to remain unemployed. ② increases the returns to work (essentially
an increase in the effective wage). Normally, this would have apposing income & substitution
effects, but since the woman isn’t working, there is no income effect associated with the case in
the effective wage. Graphically, some women will have indifference curves that hit the corner of
the BC in the baseline, but are tangent to the BC with positive hours of work in ②. ③ is a mix of
① and ②, so LS also increases for some women.
ii) ① represents a negative income shock, should encourage LS. ② only represents a change in
the effective wage. Hence, there are apposing income & substitution effects: Income effect: for a
given L, wage and income are higher, so use some wealth to purchase leisure and reduce hours of
work. Substitution effect: returns to work increase, so increase hours of works. Since these effects
are apposing, total effects are ambiguous. ③ is also ambiguous since ② is ambiguous (although
technically it depends on the magnitude of the change in G and BRR)
iii) Proposal ② (because its “breakeven level” – where the new BC meets the old BC – is furthest
to the left).
7
iv) Anything reasonable here will do; depends how much you care about encouraging LS of the
unemployed & those who work little vs. discouraging LS for those who work a fair bit.
Empirical Economics (part 1)
1. According to a recent Op-Ed piece by Joshua Foer in the New York Times (2/14/06), “A study
conducted during the 1980's found that men who kiss their wives before leaving for work live longer, get
into fewer car accidents, and have a higher income than married men who don't.” Since reading this news, I
have made a point of kissing my wife before work every morning in order to (1) prolong my life, (2) avoid
car accidents, and (3) raise my income. Briefly explain (for one of these goals) why my logic is likely
mistaken.
The study does not appear to establish any causation between wife-kissing and any of the three
measured “benefits.” Even though there is apparently correlation between wife-kissing and
health/ safety/income, this doesn’t mean one causes the other. For example: avoiding car
accidents. Suppose there is a third variable that can’t be measured, a tendency to be late. And men
who tend to be late are often too much in a hurry to kiss their wives before work, and they are
also more likely to get into a car accident (because they are rushing). Thus the hidden “tendency
to be late” variable causes non-kissing and accidents. Nonkissing does not have any causal
relationship to car accidents.
2. A paper published in a medical journal found that children who said they dieted gained more weight
over a period of time than did children who said they did not diet. The authors concluded that dieting may
lead to binge eating and therefore cause weight gain, a conclusion that was widely reported in the press.
Can you think of any alternative explanations?
Ans) Although dieting may cause binge eating that in turn leads to increased weight gain,
dieting may not be the only explanation. For example, it could be that the children who
diet are the ones most prone to gaining weight. Consequently, dieting may not cause
weight gain.