Download penr101 midterm

Fall 2007
Economics 431
Mid-Term Exam
Prof. Hamilton
Name:___________KEY________________
Question 1A. (15 points) Externalities and Monopoly Markets
Demonstrate on a diagram that the deadweight loss from a negative production externality is
larger under competition than under monopoly for sufficiently damaging pollutants. Carefully
label the axes and all relevant variables. Briefly explain.
$/Q
MSC
P*
Pm
MPC
MR
Q*
Qm
MPB
Qc
Quantity
In the diagram, the private market equilibrium occurs where MPB = MPC (at quantity
Qc) in a competitive industry and where MR = MPC (at quantity Qm) in a monopoly
industry. The social optimum occurs where MPB = MSC (at quantity Q*) must involve
smaller DWL (the red shaded triangle) than DWL in the competitive equilibrium (the
cross-hatched triangle) whenever Q* < Qm < Qc.
Question 1B. (10 points) According to the diagram you have drawn, would the optimal
externality policy for a monopolist involve a tax, a subsidy, or no policy at all?
If drawn (as above) such that Q* < Qm, the optimal policy for a monopoly market is a tax.
If drawn such that Qm < Q*, the optimal policy for a monopoly market is a subsidy.
Question 2. (25 points). Transferable Permits
Suppose two firms pollute a common water medium, and the water medium is well-mixed so that
the damage from water pollution does not depend on the identity of the polluter. Firm 1 receives
total benefit B1(X1) from polluting and firm 2 receives total benefit B2(X2) from polluting and
these benefits differ; i.e., B1(X) ≠ B2(X) for any (common) pollution level X. Pollution creates
external costs in the economy and marginal external cost is increasing in the pollution level.
Draw a diagram that shows the socially optimal level of pollution to be allocated between the
two firms and the efficiency gain from a transferable permit policy in the case of an equal
allocation of permits to each firm. Label the equilibrium permit price on your diagram.
$/X
MSC
P2
P*
MBT = Total pollution
demand
P1
MB1
X1*
χ*/2
X2*
X10 χ*
MB2
X20
χ0
Pollution
The permit price is P* and the shaded regions represent the efficiency gain from trading.
Question 3. (25 points) The Coase Theorem
A paper mill pollutes a river. The total benefit of pollution (X) to the mill is B(X) = 10X –
0.25X2. A few miles downstream from the paper mill is a popular fly fishing area. The
pollution from the paper mill damages fish and imposes total costs on fly fishers given by c(X) =
0.75X2. Suppose there are no other benefits and costs associated with water pollution in the
river.
A. Define the Coase Theorem, state its corollary and the requirements necessary for it to hold.
Coase Theorem: The private market achieves a Pareto optimal resource allocation whenever:
1) Property rights are clear and enforceable,
2) Property rights are transferable,
3) Economic agents have full information (no uncertainty),
4) Transaction costs are low.
Corollary: the ability of economic agents to achieve a Pareto optimal allocation does not depend
on which economic agent is given the property rights.
B. Suppose a local court order gives the paper mill the right to pollute the water. If the fly
fishers do not offer any compensation to reduce pollution, how much pollution will the firm
generate? Denote this quantity of pollution as Xc.
FP: Max. B(X) ≡ 10X – 0.25X2
Î
FOC: B′(X) = 10 – 0.5X = 0 Î
Xc = 20
C. Pollution abatement (A) refers to the amount of pollution reduced from the unregulated level
Xc; that is, actual pollution (X) is given by X = Xc – A. What is the marginal benefit to the
fly fishers from pollution abatement? What is marginal abatement cost for the firm? Provide
a graph that shows pollution abatement (A) on the horizontal axis, the marginal benefit of
abatement to fly fishers and marginal abatement cost (MAC).
MC(X) = c′(X) = 1.5X
Fly fishers incur significant damages when Xc = 20. The marginal cost fly fisher incur for the
last unit of pollution is MC(20) = 1.5(20) = 30. That is, the last unit of pollution by the paper
mill causes $30 of opportunity cost to be incurred in fly fishing costs. Because the demand
among fly fishers for pollution reduction must equal the opportunity cost of pollution:
Since X = 20 – A, MB(A) = 1.5(20 – A) = 30 – 1.5A is the marginal benefit of abatement
In words, the 20th unit of pollution causes $30 in damage to fly fishers, the 19th unit causes the
Troop MC(19) = 1.5(19) = $27.50 in damages, and so on, so that the marginal benefit of
pollution abatement has an intercept of $30 and a slope of 1.5.
Marginal abatement cost for the firm is the lost value incurred by the firm for each unit of
pollution abated.
Since X = 20 – A, MAC = 10 - 0.5(20 – A) = 0.5A is the marginal abatement cost
In words, with 20 units of pollution, the MB of the last unit is MB(20) = 10 – 0.5(20) = 0. (This
must be true, because this is how we determined where the firm stops polluting in part C.)
Similarly, the marginal abatement cost of the 19th unit is the foregone marginal benefit to the
firm, MB(19) = 10 – 0.5(19) = 0.5, and so on. The MAC starts at zero and rises at slope 0.5 as
pollution is abated: MAC = 0.5Q.
D. Graph
Fly Fishers
30
MC - Pollution
MB -Abatement
Paper
Mill
10
$7.50
$7.50
0
Pollution (X)
20
A* = 15 (X* = 5)
Pollution abatement (A)
E. What is the equilibrium level of pollution abatement, A*, after negotiation takes place?
What is the fair bargaining price for a unit of pollution abated by the paper mill? What cash
transfer does the paper mill receive from the fly fishers to abate water pollution?
The socially efficient level of abatement occurs where MB(A) = MAC for abatement units, or:
30 – 1.5A = 0.5A
=>
A* = 15.
That is, starting from 20 units of pollution, it is optimal to abate A* = 15 units, leaving X* = 5
units of pollution.
The fair bargaining price to obtain this outcome can be found from either MB or MAC at A* =
15 units of abatement. MAC(15) = 0.5(15) = $7.50. So, P* = $7.50. (This would be the same
bargaining price if the fly fishers received the property right and the paper mill paid the fly
fishers for pollution. Why?)
The cash transfer the paper mill receives from the fly fishers in the equilibrium negotiation is
C = $7.50(A*) = 15(7.50) = $112.50.
F. Suppose instead that the fly fishers received the property right to a clean river. What is the
equilibrium level of pollution, X*, after negotiation takes place and how does this relate to
A*? What is the fair bargaining price for a unit of pollution by the paper mill? What cash
transfer do the fly fishers receive from the mill in equilibrium for allowing water pollution?
If the fly fishers receive the property right, the equilibrium level of pollution will be
X* = 20 – A* = 20 – 15 = 5
The relationship between these values is X* + A* = Xc.
The cash value of the fly fishers receive for allowing these 5 units of pollution is:
C = $7.50(5) =$37.50
EXTRA CREDIT. In the context of this problem, what obstacle(s) might prevent bargaining
between fly fishers and the paper mill from achieving the socially optimal outcome?
The key is to pick one of the 4 requirements for the Coase Theorem to hold and explain why it
may break down. One major issue is transaction costs. There may be thousands of fly fishers
who use the river (or would wish to use it if it were less polluted) and organizing them together
to negotiate with the paper mill can easily preclude negotiation. One aspect of this transaction
cost is that a good fly fishing river is a public good, so that free riding can be a problem if fly
fishers try to collect money to negotiate with the paper mill for abatement. (Alternatively, if fly
fishers are to be given the property rights, some people who don’t fly fish would rush out to Wal
Mart and buy hats with fly fishing lures hanging off them to try to disguise themselves as fly
fishers in the hope of receiving a share of the property right.)
Question 4. (25 points) Uncertainty
More than half the electricity in the U.S. comes from burning coal. Suppose a regulator in the
U.S. wants to design a more efficient policy to control greenhouse gases (notably NOx, SOx,
CO2 and SO2), particulates and toxic compounds such as mercury. The regulator estimates the
total benefit of burning coal (X) at electric plants to be TB(X) = 100X – 2X2. Suppose the total
social cost of burning coal is TSC(X) = 10X + 0.5X2. The regulator wants an efficient policy
and can select between a tax and a standard.
A. Based on the information available on benefits and costs, what pollution tax (t^) would the
regulator set to obtain the social optimum? What pollution standard (X^) would the
regulator set to obtain the social optimum?
MB(X) = 100 – 4X
MSC(X) = 10 + X
Social optimal allocation (with available information): 100 – 4X = 10 + X Î
5X = 90
X^ = 18
Associated tax: t^ = 100 – 4X^ = 100 – 4(18)
Î
t^ = $28
B. Suppose the total benefit of burning coal (X) is actually TB(X) = 90X – 2X2. What is the
actual outcome that will occur under a tax of t^? Under a standard of X^?
MB(X) = 90 – 4X
MSC(X) = 10 + X
Social optimal allocation (with actual information): 90 – 4X = 10 + X Î 5X = 80
X* = 16
At a tax of t^ = $28, electric utilities burn coal until MB(X) = 90 – 4X = 28
Î 4X = 62
This implies an actual combustion level of: Xat = 15.5
Under a standard of X^ = 18, electric plants combust Xas = 18 units of coal (since MB(18) > 0)
C. Provide a diagram that shows the true social optimum, the outcome under a pollution tax and
the outcome under a pollution standard.
P
DWLstandard
DWLtax
MSC
t^=28
MBtrue
Qat=15.5
MB(X)est
Qas=18
X
Q*=16
D. See above. Taxes perform better, because MB of coal is inelastic.
E. The regulator makes the following public statement: “Because a transferable permit policy
operates much the same as a pollution tax, a transferable permit policy performs equally well
as a tax under uncertainty.” Evaluate this statement.
This statement is incorrect. Transferable permits are only like taxes in the sense that both obtain
allocative efficiency. At the aggregate level, a standard of X^ = 18 units will lead to 18 units of
coal being used. DWL under a standard without permit trading would be larger than the region
depicted above to the extent that pollution is not allocated efficiently among regulated electric
plants.