AEM 250 Spring 2007
Homework #3: Market Failure – Externalities:
SOLUTIONS
1)
a.
Step 1
:
Using the information in the table, find the MB and MC curves:
MB = -Qd + 220
MC = 1/2 Qs + 10
Set MB = MC to find Q* (private optimum quantity)
-Q + 220 = 1/2 Q + 10
Q* = 140
Plug Q* into either MB or MC to obtain P*
P* = MC = (1/2) * (140) + 10
P* = 80
Private equilibrium is at (Q*, P*) = (140, 80)
→
e* on graph
Step 2
:
Find the Social MC curve and solve for the social optimum:
SMC = MC + external costs (this is what we refer to as marginal
external social costs in the lectures)
SMC = 1/2 Qs + 10 + 60 = 1/2 Qs + 70
Set SMC = MB and follow the same process as above to obtain:
Social equilibrium at (Q*, P*) = (100, 120)
→
e’ on graph
$
220
120
80
70
10
100
140
Q (mil. tons)
SMC
MC
e’
MB
e*
$60 per ton
external cost

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b.
Net Economic Surplus at e*:
You can calculate CS and PS, and sum them, or simply calculate the entire
triangle:
CS = (1/2) * (220 – 80) * (140) = (1/2) * (140) * (140) = 9800
PS = (1/2) * (80 – 10) * (140) = (1/2) * (70) * (140) = 4900
NES = CS + PS = 14700
(note: times 1 million)
or
NES = (1/2) * (220 – 10) * (140) = (1/2) * (210) * (140) = 14700
You were not asked to calculate this, but remember that this NES does not
account for the loss from external costs, which we can subtract out:
External cost = 140 * 60 = 8400
Adjusted NES at market equilibrium= 14700 – 8400 = 6300
Net Economic Surplus at e’:
Using the same method as above we can find the NES at the social
optimum:
CS = (1/2) * (220 – 120) * (100) = (1/2) * (100) * (100) = 5000
PS = (1/2) * (120 – 70) * (100) = (1/2) * (50) * (100) = 2500
NES = CS + PS =7500
or
NES = ½(120-70)*100 = 7500
If we calculate the NES at the market optimum (without accounting for the
externality) we get an inflated measure because we are not taking into
account the external costs associated with pollution.
The NES at the
social optimum actually represents the maximum amount of benefit
society can achieve once all of the costs are taken into account.
c.
i.)
A Pigouvian tax is a tax levied on an agent who is creating a negative
externality with the intention of reducing the level of the negative
externality.
It is a per unit tax set equal to the Marginal External Social
Costs at the social optimum.
ii)
The optimal level of Pigouvian tax is equal to the marginal external
social cost at the social optimum.
In this case we have constant external
cost of $60 per unit (which is like our oil example discussed in lecture) for
all levels of output, including the socially optimal level. So the optimal tax
is $60 per unit.