Public Economics — Problem Set #1
Due: October 3rd at 2:40pm (submit through the courseworks)
1. Consider a reform that changed welfare benefits in New Jersey and suppose that at the same time there was no
change in New York (note: even though there was no reform in NY, that does not mean that nothing in New
York has changed — for example, the composition of the group of welfare recipients may have changed ove
time). The data for welfare recipients (per month) in the two states looks as follows:
New Jersey New York
Average hours of work Average benefit Average hours of work Average benefit
Before the reform XXXXXXXXXX
After the reform XXXXXXXXXX
(a) Explain what assumption(s) you need to make to rely on this data in order to estimate the effect of the
eform
(b) What is the difference-in-difference estimate of the effect of the reform on hours of work of welfare recipients?
(c) What is the co
esponding estimate on the effect on welfare benefits?
(d) What is the implied elasticity of hours of work to the level of welfare benefits (ie., percentage change in
hours of work per a one percent change in the level of benefits)? Note: the previous parts tell you what
changes are, but the elasticity has to be evaluated at some reference point. It is obvious what the initial
point is when one evaluates the elasticity theoretically — that’s where you take the derivative — but with
eal-life data one could use a lot of different points — before or after the reform, in New Jersey or New
York or anything in between. One common choice is to take the mid-point between before and after reform
values for the treated group.
2. The demand for smartphones is given by D(p) = 400− p+ p
T
5 , where p is the price of a smartphone and p
T is
the price of a tablet (a substitute for smartphones). The supply is given by S(p) = 4 · p. The price of tablets is
fixed and set at pT = 500.
Suppose that the government imposes two taxes on phones: a $30 tax to be paid by the consumers and $70 tax
that producers have to pay.
(a) What is the economic incidence of this policy?
(b) What is the excess burden here?
(c) How would economic incidence change if government imposed instead a $70 tax on consumers and a $30
tax on producers
(d) Imagine that the tax on producers increases to $120, while the tax on consumers remains unchanged at
$30. How does the excess burden change? Divide the change in excess burden into components coming
from the surplus of each of the parties involved.
(e) Which component of the change in excess burden is the largest? Explain why.
3. The demand for food purchased in grocery stores is given by DG = 100 − PG + 12PT where PG is the price
(index) of food in supermarkets and PT is the price of take-outs. Co
espondingly, the demand for take-outs
is DT = 100 − PT + 12PG. The supply functions are given by SG =
1
2PG and ST =
1
2PT respectively. The
government imposes the tax of 40 on take-out food. Determine how the incidence of this tax is split between
consumers and producers of the two types of food. Note: you have to find prices for both goods that yield an
equili
ium in both markets at the same time.
4. The excess burden is an increasing function of a tax. True, false or uncertain: the excess burden of a (small?)
subsidy is therefore lower than the excess burden of a tax.1
5. Suppose that the marginal private cost of providing higher education for n students is given by MPC(n) = n
and that the marginal private benefit schedule is given by MPB(n) = 200 − n (ie, benefits decline with the
number of students, presumably because additional students are less qualified and derive lower return from being
educated). Imagine though that people with college education are more likely to vote and volunteer. Assume
(on faith) that these behaviors benefit everyone. The additional social benefit from these activities is valued at
20 per person with college education.
(a) Plot a graph showing private marginal benefit, private marginal cost and social marginal benefit.
(b) Find the price and quantity that co
espond to the private competitive equili
ium (i.e., with no intervention
of any kind).
1A topic for additional consideration (no extra credit): denote the total excess burden as EB(t) where t can be positive (tax) or negative
(subsidy). What happens at t = 0? Is EB(t) discontinuous or non-differentiable at t = 0? To be formal, this of course requires additional
structure (though not much of it really), so let’s just assume for simplicity that we are dealing with linear demand/supply curves in partial
equili
ium.
(c) Find the socially efficient quantity and the deadweight loss from being at the private competitive equili
ium
instead.
(d) What value of a monetary subsidy to education would implement the efficient solution?
6. There are 4 firms in the industry that have the total costs of eliminating pollution given by P 2/4, P 2/3, P 2/2
and P 2 respectively.
(a) Suppose that we want to reduce aggregate pollution in a way that minimizes the overall cost. Derive the
marginal cost of doing so as a function of the overall reduction in pollution P ∗
(b) Suppose we want to reduce the overall pollution by 100 units. How much should each of the firms reduce
pollution by in order to minimize the overall cost of doing so?
(c) Suppose that we require each firm to reduce pollution by 30 units. Firms are allowed to trade obligations
to lower their pollution reduction requirements. What will be the competitive market price of a unit of
pollution reduction and how many units will be traded?
(d) Suppose we do not allow firms to trade in part (c). What would be the deadweight loss compared to the
solution in part (c)
2
Economic Tools
▶ Utility function: a mathematical representation of preferences
▶ Assumption: individuals have well-defined “rational” preferences and
attempt to achieve the highest level of well-being
▶ Indifference curves
Consumption
Le
is
u
e
●
B
●
A
●
C
Utility
▶ Marginal utility
U(Z ,Y ) = 50 ln(Y XXXXXXXXXXln(Z )
(ln(X ))′ =
1
X
MUZ (Z ,Y ) = 0 + 30 ·
1
Z
=
30
Z
MUY (Z ,Y ) = 50 ·
1
Y
+ 0 =
50
Y
If one consumes (Z ,Y ) = (3, 2), the marginal utility of good Z is
MUZ (3, 2) =
30
3 = 10 while the marginal utility of good Y is
MUY (3, 2) =
50
2 = 25.
▶ The marginal rate of substitution (MRS) — the slope of the indifference
curve. MRS of good Z to good Y:
MRS = −MUZ
MUY
= − 30/Z
50/Y
= −3
5
Y
Z
Good Z
G
oo
d
Y
●
A
●
B
∆∆Z
∆∆Y
Slope:
∆∆Y
∆∆Z ≈≈ MRS
u((A)) == u((B))
u((B)) ≈≈ u((A)) ++ MUZ∆∆Z ++ MUY∆∆Y
MUZ∆∆Z ++ MUY∆∆Y == 0
∆∆Y
∆∆Z == −−
MUZ
MUY
Budget constraint
▶ Optimization is subject to (budget) constraints
Price of apples (A) is pA. Price of bananas (B) is pB . Income is Y .
The budget constraint is:
pAA+ pBB = Y
If price of apples was 5, price of bananas was 7 and income was 35, the
udget constraint would be
5A+ 7B = 35
▶ Equivalently:
pBB = Y − pAA ⇒ B =
Y
pB
− pA
pB
A
▶ The slope of the budget constraint is − pApB .
Characterization of the optimum
The budget constraint needs to be tangent to the indifference curve at the
optimum.
Two conditions:
1. The slopes of the budget constraint and the indifference curve need to
e the same:
−MUA
MUB
= MRS = −pA
pB
2. The optimum is on the budget constraint
pAA+ pBB = Y
Good Z
G
oo
d
Y
●
Example
▶ U(Y ,Z ) = 13 ln(Y ) +
2
3 ln(Z ). PY = 10, PZ = 20, Y = 120
▶ Method 1: MRS = −
1
3
1
Y
2
3
1
Z
= −12 ZY . The slope of the budget line is
−1020 = −12 . We need to solve:
−1
2
Z
Y
= −1
2
10Y + 20Z = 120
Solution: Z = Y = 4.
▶ Method 2: The budget constraint is 10Y + 20Z = 120 hence
Y = 12− 2Z . We want to pick the point with the highest utility on the
udget constraint, hence we want to maximize
1
3
ln(12− 2Z ) + 2
3
ln(Z )
That requires −23 112−2Z + 23 1Z = 0 ⇒ 12− 2Z = Z , hence Z = 4
and Y = 12− 2Z = 4.
Nonlinear budget constraints
▶ Why care? Because they are pervasive in the tax/welfare context.
▶ Examples: Earned Income Tax Credit (we’ll talk more about it) provides
a marginal subsidy if earnings are not too large and then slowly takes it
away. Many related provisions in welfare programs. Tax exemptions —
no tax (labor valuable, leisure costly) up to certain income level, tax
afterwards. Progressive taxation — price of labor depends on you
income. Health insurance subsidies — the amount depends on the level
of income.
Tax exemption
Leisure
C
on
su
m
pt
io
n
●
A
●
B
●
C
Income and substitution effects
Good Z
●
A
●
C
●
B
Total effect: A to C
Subsitution effect: A to B
Income effect: B to C
Elasticity (of demand)
▶ Demand at given price p is D(p)
▶ It could be individual demand or aggregate demand, we can derive it
ased on utility maximization or based on observation or assume
▶ Slope: D ′(p) — how much demand changes with a dollar change in price
▶ A common way is to instead measure the slope by the elasticity: the
percentage change in the demand in response to a 1% change in price
▶
ε =
% change in demand
% change in price
=
∆D(p)/D(p)
∆p/p
=
p
D(p)
D ′(p)
▶ Another (equivalent) definition:
ε =
∆ ln(D(p))
∆ ln(p)
▶ Aside: you can see it by substituting x = ln(p) so that
d ln(D(p))
d ln(p) =
d ln(D(e ln(x)))
dx and work through the derivative with respect to
x . Or, by simply noting that ∆ ln(x) is approximately a percentage
change in x .
Equili
ium and efficiency
Quantity
P
ic
e
Demand
Supply
P ●
Q
Quantity
P
ic
e
Demand
Supply
●
●
Q
Total surplus
Deadweight loss
Pareto efficiency
▶ An allocation at which the only way to make one person better off is to
make another person worse off is called Pareto efficient
▶ If an allocation is not Pareto efficient, there must exist a Pareto
improvement.
▶ At an (interior) Pareto efficient allocation MRSs for all individuals are
the same.
The First Theorem of Welfare Economics
▶ assume (1) perfect competition; (2) existence of markets for all