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Additional Problems
Winter 2014
ECON 3328a
1. In the following diagram, a 2x2 pure exchange economy is illustrated.
XH
30
OH
YH
E
45
55
67.69
YG
OG
a.
b.
c.
d.
e.
XG
130
Indicate what the excess demand and excess supply is on the above diagram.
Calculate what the current market price is.
What is the MRSG and the MRSH at their current consumption points.
Explain what will happen to budget line in the above diagram.
Will the MRSG rise or fall as society moves to equilibrium.
2. Suppose that there is a pure exchange economy described by:
 Two commodities: automobiles (a) and boats (b)
 Two people: Mel (M) and Norm (N)
 Endowment = (a M , bM , a N , bN ) = (5, 5, 5 , 15)
a. Draw the Edgeworth box that represents this economy.
b. Suppose the Endowment is not Pareto optimal. Using indifference curves show the region
in which allocations offer a Pareto improvement over the endowment.
c. Show the region in which allocations offer a reduction in satisfaction for both individuals.
d. Would you recommend a perfectly equitable allocation?
e. Suppose that the market equilibrium price for boats is 1 automobile and Mel’s utility can
be represented by u  u(a M , bM )  10a 0.5b 0.5 . How much of each commodity does Mel
buy and sell.
f. Suppose that the market equilibrium price is 1 and Mel’s utility can be represented by
u  u(a M , bM )  10a 0.5b 0.5 . What is the competitive market allocation?
g. Suppose that the market equilibrium price is 1 and Mel’s utility can be represented by
u  u(a M , bM )  10a 0.5b 0.5 . What is Norm’s MRS at the market equilibrium price?
h. If the price rose above 1, what goods will be in excess supply?
3. Suppose a society’s preferred point on the contract curve is at point Z in the following
diagram, but the endowment is point E.
aH
F
OH
bH
●
G●
Z
●
●E
bG
OG
aG
a. Suppose that the government can only redistribute b and cannot redistribute a. Propose a
method for getting to Z.
b. Which allocation is socially more desirable, F or G? Be sure to provide an argument.
c. Suppose nature endowed H with all of the a and b, where would the endowment point be.
d. If nature endowed H with all of the a and b, what would be the market equilibrium price?
e. Suppose that a Hobbesian state of nature hits H and G. What will be the impact of this
event on the endowment point, the Edgeworth box, and the equilibrium price?
4. Suppose a utility function has the Stone-Geary form with constants x and y so that
u  u( x, y)  1 ln( x  x )   2 ln( y  y) . What is the total differential of the utility function
and what is the MRS at any value of x and y?
5. Suppose a utility function has the CES form with u  u( x, y)  (1 x     2 y   )1 /  . What is
the total differential of the utility function and what is the MRS at any value of x and y?
6. Suppose that a utility function has the log-linear utility function
u  u( x, y)  1 ln( x)   2 ln( y) and the individual has a budget line that depends on Px (the
money price of x), Py (the money price of y), and M (money income).
a. Write down the maximization problem that represents this person’s choice.
b. What are the first order conditions for a maximum?
c. Use the first order conditions to prove that the MRS must equal the price ratio when
the individual chooses their optimal choice.
d. Find the demand equation for x and y.
7. Should a new a new technology that reduces the cost of fishing cause regulators to increase
their target level of resource use?
8. Suppose that there is a pool of oil worth $1 billion. Suppose that there are N people who each
drill one well into this pool of oil, and each person has the expectation that their benefit from
drilling a well is ($1 billion / N). Suppose that each well costs $1 million.
a. What is the equilibrium number of wells?
b. What is the efficient number of wells?
c. Suppose that a new drilling technology arrives that allows more of the pool to be
extracted by drilling specialized wells. The more specialized wells drilled the more of
the oil can be extracted. Geologists re-estimate that the value of oil that can be
extracted is $1,000,000,000 + 5,000,000N – 10N2. What is the equilibrium number of
wells under this new technology?
d. What is the efficient number of wells under this new technology?
e. What price should government charge for a drilling permit?
f. What should the government charge for drilling if the cost of drilling wells increases
to $2 million.
9. Suppose that a fisheries catch (R) is a function of the number of fishers (L) in the industry,
and that this is given by the following equation: R = 500L - 0.01L2 . Suppose that each
fisher=s opportunity cost of participating in the fishery is 400 and that each fisher that
participates in the fishery can expect to get an equal share of the total catch.
a) What is the social equilibrium number of fishers.
b) What is the efficient number of fishers.
c) What should the cost of fishing license be if the government=s objective is to
maximize efficiency.
10. Suppose that the total social benefits and total social costs of wells drilled into an
aquifer are as illustrated below.
a. What price gives the slope of the TSC curve?
b. Sketch the marginal social cost (MSC) and the marginal social benefit (MSB)
curves in the lower diagram.
c. Sketch the marginal private benefit (MPB) and the average social benefit
(ASB) curves in the lower diagram.
d. Show the equilibrium number of wells.
e. Show the efficient number of wells.
f. In the lower diagram, show the tax per well that would lead to the efficient
number of wells.
11. In the following diagram, the supply curve (S) and demand curve (D) for SUV’s is
illustrated. In addition, the marginal damage (MD) from carbon dioxide (CO2)
emissions is illustrated. On the same diagram show:
a. the market equilibrium number of SUV’s?
b. the efficient number of SUV’s?
c. the optimal Pigouvian Tax?
d. Suggest two more efficient policy instruments for reducing CO2 emissions.
P
25
S
P
10
MD
D
100
Q
12. The inverse demand and inverse supply curves are represented by the following
functions: P = 20 – 0.1Qd and P = 2 + 0.05Qs . What is the efficient level production
if there is an externality which imposes a marginal damage of $1 per unit?
13. In the following diagram, market demand and supply curve are illustrated. Suppose
the total external costs caused by this activity can be represented by C = 100 + 0.5Q2 .
a. What is the true social marginal cost of this activity?
b. What is the efficient level of output?
c. What is the deadweight loss at market equilibrium?
d. What Pigouvian tax on this activity would yield an efficient outcome?
e. What is the maximum that the polluter would pay to get pollution to the
optimal level?
f. What is the minimum that the polluter would accept to reduce pollution to the
optimal level?
P
25
10
S
D
100
Q
14. In class we examined how a negative externality and its associated marginal damage
led to overproduction. We also examined how there was room for a negotiated
solution that would “internalize” the negative externality.
a. Show how a positive externality introduces a marginal external benefit that
causes the market allocation to be inefficiently small.
b. Examine whether there are opportunities for a negotiated solution that would
internalize the positive externality.
c. Suggest some government interventions that might be alternative routes to
internalizing a positive externality.
d. Provide some examples of government interventions that seem to be intended
to internalize positive externalities.
15. Suppose that the long run marginal cost of steel production is $500 per tonne and the
demand for steel is defined by P = 1000 – 0.0001Qd.
a. What is the market equilibrium P and Q?
b. If steel has an associated negative externality that imposes pollution that
has total costs of TC = 5Q on the community, what will be the total
external costs?
c. What is the efficient level of steel production?
d. What is the deadweight loss at the market equilibrium?
e. What is the optimal Pigouvian tax?
f. What is the optimal Pigouvian tax if total external costs from pollution are
defined by C = 0.1Q2?
16. In class we examined how a negative externality and its associated marginal damage
led to overproduction. We also examined how there was room for a negotiated
solution that would “internalize” the negative externality.
a. Show how a positive externality introduces a marginal external benefit
that would cause the market allocation to be inefficiently small.
b. Examine whether there are opportunities for a negotiated solution that
would internalize the positive externality?
c. Suggest some government interventions that might be alternative routes to
internalizing the positive externality.
d. Provide some examples of government interventions that seem to be
intended to internalize positive externalities.
17. In the following diagram, the supply curve and demand curve are presented for beer.
The brewery makes a smell that creates a marginal damage of $1 per unit of beer.
a. Show the market equilibrium.
b. Suggest what the range of prices might be for a negotiated solution.
18. Suppose that the marginal private benefits from pollution for two firms (MPBa and
MPBb) are given in the following diagrams.
a. What is the total amount of pollution in an unregulated market?
b. If the government wants to eliminate pollution altogether what is the total
social cost of abatement?
c. If a uniform standard is proposed which cuts pollution in half, what is the total
social cost of abatement?
d. Why would plant specific pollution standards be more efficient way to cut
pollution in half than the uniform standard approach?
e. What is the total number of tradable emissions permits that would cut
pollution in half?
f. What is the market equilibrium price of tradable emissions permits that would
cut pollution in half?
g. How much would the firm a make by developing a new abatement technology
that cuts the marginal cost of abatement in half?
h. If the marginal social cost of pollution is $5 per tonne, what is the socially
efficient effluent fee?
i. If the marginal social cost of pollution is $5 per tonne, what is the socially
efficient number of tradable emissions permits?
P
20
15
MPBb
60
MPBa
100
tonnes of pollution
19. In the following diagram, a production possibilities curve is given together with the
preferences of on individual named “b”. Find the optimal level of provision of the
public good.
P
20
PPF
Ub
60
100
Q of Public Good
20. Suppose that there are 100 identical individuals who’s preferences can be described
by utility function u i   ( X i , Gi )  10 X i0.2 Gi0.5 , where X is the quantity of the private
good consumed and G is the quantity of a pure public good consumed. What is the
efficient provision level of the public good if the PPF is described by 1,000,000 = X +
G, where X and G are the total quantity of the public good produced by society.
21. The Marginal Rate of Substitution between a pure public good (G) and pure private
good are transformed below into marginal benefit curves for three people (MBA,
MBB, MBC). Suppose the marginal cost of the public good is a constant $15.
a. What will the efficient provision level?
b. If the government decides to split the cost of the public good equally, how
much will each individual want?
c. If each individual faces an equal tax price (T=1/3P), what will be the
political equilibrium choice under the median voter theorem.
d. Describe how a Lindhal auction would be implemented and the
equilibrium prices and quantity of the public good if people honestly
revealed their preferences.
e. What would be the Lindhal equilibrium if people were strategic.
f. If the marginal cost of provision falls to $5, what will be the private
market provision level of the pure public good?
g. If the marginal cost of provision falls to $5, and exclusion becomes
possible, how much of the pure public good will be enjoyed by each
member of society under private supply.
h. If the marginal cost of provision falls to $5, exclusion becomes possible,
and it is possible to perfectly price discriminate. How much of the public
good will be supplied and what price will each person pay in the
marketplace?
$
20
15
10
5
MBA MBB
MBC
G
22. Suppose that there are people who like public radio and people who do not like public
radio. Suppose that the marginal private benefit function for those who like public
radio is given by MPBL = 4.8 - 0.2H, where H is the number of hours and MPBL is
the willingness to pay in dollars. Suppose that the marginal private benefit function
for those who do not like public radio is given by MPBN = 0 for all H. Suppose that
the marginal cost of producing 1 hour of public radio is $10,000,000 and that 50% of
a population of 20,000,000 likes public radio.
i. What is the efficient provision level?
j. What is the equilibrium provision level with voluntary provision if
everyone is honest about their preferences?
k. Suppose that one person who values public radio misrepresents their
preferences by pretending to be a person who does not value public radio.
What will be the impact on this individual’s consumption of public radio
in equilibrium? What will be the impact on this individual’s voluntary
contribution?
l. If exclusion is impossible, what would be the market equilibrium
provision level?
m. If exclusion becomes possible, describe how a profit maximizing firm
might get in the business of providing the public good?
23. In the following diagram, a utilities possibilities frontier (UPF) is given. Suppose that
society’s preferences can be represented by W = ∑ uiβ .
a. Identify the Utilitarian (β=1) social optimum as point A.
b. Identify the social optimum when β=0.5 as point B.
c. Identify the Rawlsian (β approaches 0) social optimum as point C.
U1
UPF
U2
24. Suppose that each person a natural disaster occurs in Victoria B.C. and families in
London Ontario would like to help. Assume that each family has identical preferences
given by u  f ( x, D)  x  D1 / 2 , where x = family’s consumption and D = total
amount of aid received by Victoria. Each family faces a budget constraint
of M  x  d , where d = the family’s contribution to the aid package.
25.
Essay questions: Argue whether the following statement is true, false or uncertain. Use
carefully constructed models and diagrammatic analysis if it helps your essay.
1. Human history records long periods with relatively little social change. The last
200 years is remarkable for the unprecedented level of social change. Without
government sponsorship of research and education, this would not have happened.
2. Only with an equal distribution of income will society be socially just.
3. A just society is only possible if society’s scarce resources are efficiently
allocated.
4. If government changed its priorities to be only concerned with efficiency, then
government would get out of the business of helping the poor and focus
exclusively on solving externalities and providing public goods.
5. In the absence of high transaction costs, government would be an un-necessary
social institution.
Coercive government is necessary for society to function.
6. The invisible hand of the market creates social harmony, and so government is not
necessary.
7. Good fences make good neighbors.
8. If the government regulated the fishery according to economic principals, catches
would be maximized.
9. The first priority of any government should be efficiency.