Economics 440
Public Finance
Winter 2022
Allen Head
Answers to Assignment 2
1a. When firms are independent, the manufacturing firm chooses its output taking into
account the cost of pollution only to itself. Thus its profit maximization problem is
given by:
2
max Pm qm − 3qm
− (x − 2)2
(1)
qm ,x
|
{z
}
Cm (qm ,x)
subject to:
x = .5qm
(2)
2
2
max Pm qm − 3qm
− (.5qm − 2)2 = Pm qm − 3.25qm
+ 2qm − 4
(3)
or
qm
The first-order condition is given by:
Pm − 6.5qm + 2 = 0
(4)
Letting Pm = 13.5 and solving we have:
6.5qm = 13.5 + 2 = 15.5
15.5
qm =
≈ 2.38
6.5
x = .5qm ≈ 1.19.
1b. The merged firm maximizes total profits from both enterprises. Its profit maximization
problem is:
2
(5)
max Pm qm + Pf qf − 3.25qm
+ 2qm − 4 − 3qf2 − 1.5qm
qm ,qf
{z
} |
|
{z
}
Cm
Cf
Taking the first-order condition for choice of manufacturing output, qm , and solving
for the case in which Pm = 13.5 we have:
13.5 + .5 − 6.5qm = 0
6.5qm = 14
qm ≈ 2.15
x = .5qm ≈ 1.075.
1
The level of pollution in this case differs from that in part a. because the merged firm
takes into account the effect of pollution on the cost of fishing, whereas the independent
manufacturer does not. The merged firm thus produces a lower level of manuafacturing
output (qm = 2.15 rather than 2.38) and as a result pollutes less (x = 1.075 rather
than 1.19).
1c. If the government imposes a tax on the independent manufacturing firm for each unit
of output that it produces, then its profit maximization problem becomes:
2
max Pm qm − 3.25qm
+ 2qm − 4 − τ qm
qm
(6)
Taking the first-order condition and solving under the assumption that Pm remains
constant at 13.5, we have:
13.5 + 2 − τ − 6.5qm = 0
6.5qm = 15.5 − τ
τ = 15.5 − 6.5qm
We want to find the level of τ such that the independent manufacturer emits the same
pollution as it would if it were merged with the fishing firm, i.e. x = 1.075. Since
x = .5qm , it is sufficient to solve the equation above for the tax amount, τ that sets
qm = 2.15, the quantity of manufacturing output produced by the merged firm:
τ = 15.5 − (6.5 × 2.15)
τ = 1.525.
2. To derive the Samuelson condition consider the social welfare maximization problem,
where each household gets equal weight:
max αx1 + (1 − α) ln y + αx2 + (1 − α) ln y
x1 ,x2 ,y
(7)
subject to:
h y i γ1
= ω1 + ω2 = Ω
(8)
δ
Form the Lagrangian in the usual way, and derive the first-order conditions by differentiating with respect to x1 , x2 , y, and the Lagrange multiplier, λ, respectively:
x1 + x2 +
α = λ
α = λ
1−α 1−α
1 h y i γ1 −1
+
= λ
y
y
γδ δ
h y i γ1
x1 + x2 +
= Ω
δ
2
Now, divide the third first-order condition by λ and substitute using the first two:
1−α 1−α
1
+
=
αy
αy
γδ
| {z } | {z } |
1
M RSyx
h y i γ1 −1
δ
{z
M RT yx
2
M RSyx
.
(9)
}
Notice that this is the Samuelson condition. In this case, solving for y we can write
γ
2δ(1 − α)
y=γ
.
(10)
α
Thus, the optimal level of public good provision is determined by the parameters α, δ,
and γ, and does not depend on how the private good is distributed across households.
Notice also that when the planner weighs the two households the same, it does not
matter how much each consumes of the private good. If the planner assigns higher
weight to either of the households, then that household will get all of the private good;
and the other will get utility only from the public good.
3a. To show that drivers have incentive to drive an inefficient speed, we compare the first
order conditions from the individual drivers’ utility maximization problems to those
from a social welfare maximization problem. Driver h chooses his/her speed to solve:
max uh (sh ) − π(s1 , s2 )ch
sh
(11)
The first-order condition for this problem is:
∂π
∂uh
− h ch = 0
h
∂s
∂s
h = 1, 2.
(12)
So from the two drivers’ problems with have two equations:
∂u1
∂π
− 1 c1 = 0
1
∂s
∂s
2
∂u
∂π 2
−
c = 0
∂s2
∂s2
in the two unknown speeds, s1 and s2 .
We compare these equations with the first order conditions from the following social
welfare maximization problem:
max
ψ u1 (s1 ) − π(s1 , s2 )c1 + (1 − ψ) u2 (s2 ) − π(s1 , s2 )c2
(13)
1 2
s ,s
Choosing (without loss of generality) ψ =
max
1 2
s ,s
1
2
we have
1 1 1
u (s ) + u2 (s2 ) − π(s1 , s2 )(c1 + c2 )
2
3
(14)
The first order conditions for this problem are:
∂u1
∂π − 1 c1 + c2 = 0
1
∂s
∂s
2
∂u
∂π 1
2
= 0
c
+
c
−
∂s2
∂s2
These two equations are clearly different from the two above derived from the individual
drivers’ problems. In the social optimum, the planner recognizes that when an accident
occurs, costs accrue to both drivers. Individual Driver h, however, considers only the
cost to him or herself, ch , rather than the total cost, c1 + c2 .
3b. If Driver h pays fine th in the event of an accident, then his/her maximization problem
becomes:
h = 1, 2.
(15)
max uh (sh ) − π(s1 , s2 ) ch + th
sh
The first-order condition for this problem is:
∂uh
∂π h
h
−
c
+
t
=0
∂sh
∂sh
h = 1, 2.
(16)
Now, if t1 = c2 and t2 = c1 , then the first-order conditions from the individual households’ problems are identical to those for the social welfare maximization problem. In
this case, the efficient fine involves each driver paying the other driver’s accident cost
as well as incurring their own.
3c. No, it is not. If drivers are fined in the event of an accident, then the cost to each driver
is effectively paid twice. First, Driver h incurs cost ch , because of the accident. Then,
he/she has to pay the other driver’s cost as a fine. In the end the government ends up
with t1 + t2 = c2 + c1 in revenue from fines, which has reduced drivers’ utility. Now
we might think that the government could spend this revenue in such a way that it
would compensate the drivers for the lost utility due to fines. If people know they are
going to get their fines back, however, then the fine will not induce them to drive any
slower! This policy does induce drivers to drive the correct speeds, but it lowers their
expected utility through fines.
4a. If households take G as given, then they will not voluntarily contribute anything for
provision of public good. Each household thinks that their individual contribution has
no effect on the stock of the public good. In particular, they all think that if they
contribute nothing, this will have no effect on the amount of public good available
for their consumption. So, each household maximizes utility by spending all of their
income on private consumption, and contributing nothing to public good provision:
xy = Y
gh = 0
4
h = 1, . . . , H
(17)
Now in this case, feasibility dictates that
G=
M
X
g h = 0.
(18)
h=1
Since no public good is provided in equilibrium, each household’s gets utility is
U (xh , G) = αY × 0 = 0.
(19)
This equilibrium allocation is inefficient. There are many ways to see this. Recall that
U (xh , G) = 0 for all households. Now consider the following feasible allocation that
is Pareto superior to the equilibrium. Suppose we take a small amount, , of private
good away from each household and use it to make public good. Now the total amount
of public good produced is
H
X
G=
= H > 0.
(20)
h=1
Utility in this case is
U (xh , G) = α(Y − )H > 0.
(21)
Since all households get higher utility in this feasible allocation than in the equilibrium
allocation, the equilibrium allocation is not Pareto efficient. Another way to see that
the equilibrium is not Pareto efficient is to derive the Samuelson condition and show
that it does not satisfy it. Note that with the given utility function,
h
M RSGx
xh
=
G
h = 1, . . . , H
(22)
and M RTGx = 1. So, the Samuelson condition can be written:
H
X
xh
h=1
G
=1
or
G=
H
X
xh .
(23)
h=1
Clearly this is not satisfied if G = 0 and xh = Y for all h.
4b. If the government levies lump-sum tax, τ , on each household, then the households all
have maximization problem:
max αxh G
(24)
xh ,G
subject to:
xh + g h = Y − τ
(25)
Households still take the stock of the public good as given, and so they continue
to consume their (now after-tax) income, and contribute nothing voluntarily to the
production of public good. So the equilibrium allocation is:
G = Hτ
xh = Y − τ
5
h = 1, . . . , H.
(26)
Now we want the τ so that Hτ is the efficient level of public good. Recall the Samuelson
condition:
H
X
G=
xh .
(27)
h=1
In the equilibrium here, this becomes:
Hτ = H(Y − τ )
solving,
τ=
Y
.
2
(28)
With these taxes we have
xh =
Y
2
G=
HY
.
2
(29)
Clearly, this allocation (the equilibrium with taxes) satisfies the Samuelson condition.
4c. In this case, households realize that when they change their level of contribution to
provision of the public good, g h , the stock, G, is affected. In this case, they solve:
"
#
H
X
(30)
max αxh g h +
gj
xh ,g h
j6=h
{z
|
}
G
subject to:
xj + g j = Y
(31)
Their Lagrangian is,
"
Lh = αxh g h +
H
X
#
g j − λ h xh + g h − Y .
(32)
j6=h
The first order conditions are:
α
H
X
gj = λ
j=1
αxh = λ
xh + g h = Y
Now, assume that all households behave symmetrically, that is g h = g, xh = x, etc..
Using the first two first-order conditions to eliminate the third we have:
H
X
gj + gh = Y
j=1
6
→
Hg + g = Y
(33)
Solving, we get
Y
H +1
The stock of the public good satisfies,
g=
x=
G = Hg =
HY
H +1
HY
= x.
H +1
(34)
(35)
Note: G 6= 0. With the number of households small enough so that they recognize the
influence that they have on the stock of the public good, they will realize that it is not
in their interest to contribute zero. Still, this is not the efficient outcome, as it does
not satisfy the Samuelson condition (assuming H > 1!). Households consider only the
benefits of their contribution to public goods provision that accrue to them directly,
not the total benefits to society.
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