Dr. Donna Feir
Economic 313
Problem Set 7
Externalities
1. Suppose that the aggregate demand for good x is given by x = 100 – px, and aggregate supply is given
by x = px.11
a. Draw a diagram illustrating the competitive equilibrium price and quantity in the market for
good x. On this diagram, illustrate consumer surplus and producer surplus.
Equilibrium px = $50, equilibrium x = 50. CS = shaded triangle A, whose area is $1,250. PS = shaded
area, whose area is $1,250.
Now suppose that the production and consumption of good x results in marginal external costs of $20.
That, the marginal social cost curve will be parallel to the aggregate supply curve, but will lie $20 above
aggregate supply.
1
Note that these supply and demand curves can be derived from utility and production functions, as we did in
lecture, under the assumption that 𝑝𝑦 = $1.
Dr. Donna Feir
Economic 313
b. On your diagram from (a), draw the marginal social cost curve and identify the aggregate
external cost, given the equilibrium quantity of x in your answer to part (a).
We know MPC = MRT = x, so MSC = MPC + MEC = x + 20. Given this, at x = 50, total external costs =
$1,000, the shaded area.
Suppose that the government introduces a price floor in the market for good x.
Specifically, assume that the price of x cannot fall below $60.
c. Illustrate the effects on consumer and producer surplus resulting from the price floor, relative to
the unregulated equilibrium. By how much does consumer surplus plus producer surplus
decrease as a result of the regulation?
First diagram: just the change in equilibrium price and quantity. New price is $60, so x = 40.
Dr. Donna Feir
Economic 313
Second diagram: new CS = $800, shaded triangle below.
Dr. Donna Feir
Economic 313
Third diagram: decrease in CS = shaded area = $450.
Dr. Donna Feir
Economic 313
Fourth diagram: new PS= shaded area = $1,600.
Dr. Donna Feir
Economic 313
Fifth diagram: remember old PS from (a) = area B, redraw below.
Dr. Donna Feir
Economic 313
Sixth diagram: Change in PS has two components. First, a gain of the shaded rectangle: 40 units used
to be sold at a price of $50, now they sell at a price of $60,for a gain of $400. Second, a loss of the
shaded triangle. There used to be 50 units sold; now there are only 40. Those 10 units generated PS =
$50, and this is now lost.
Overall the change on PS is thus + $400 - $50 = +$350. Which adds up since we know new PS = $1,600
while old PS = $1,250.
Dr. Donna Feir
Economic 313
Overall effect? CS down by $450, PS up by $350. Net effect is thus - $100.
d. By how much do aggregate external costs decrease, as a result of the regulation?
First diagram: new aggregate external costs is shaded area. Each unit of x still results in $20 of
external costs, so 40 units in total result in $800 of aggregate external costs.
Dr. Donna Feir
Economic 313
Second diagram: decrease in aggregate external costs is shaded area. 10 fewer units of x are
produced, so the overall savings in external costs is $200 (each unit no longer produced lowers
external costs by $20).
Dr. Donna Feir
Economic 313
e. Use your answers to parts (c) and (d) to demonstrate that this policy is NOT a Pareto
improvement, but IS a potential Pareto improvement.
Market participants are worse off while those adversely affected by x are better off.
So this is not a PI. BUT, the gains in external costs ($200) more than offset the losses to market
participants ($100), so the winners from the policy could - in principle - compensate the losers from
the policy such that at least some people are better off and no-one is worse off. This is the definition
of a PPI: a change that results in gains and losses but where the gains are such that, with appropriate
compensation, we could turn the change into an actual PI. There are winners (who gain) and losers
(who lose) from the policy, but we can see that gains more than offset the losses. In some sense (in
the PPI sense), the policy is “worth it”, even though it is not a PI.
Now suppose that, instead of the price floor, the government introduces a $20 per unit tax on the
production of good x.
Dr. Donna Feir
Economic 313
f.
(Once again, demonstrate that this policy is not a Pareto improvement, but is a potential Pareto
improvement. Remember (from Econ 103) that your social welfare analysis in the presence of a
tax should account for the fact that tax revenues raised are a benefit to society.
The tax raises consumer price to 60, so CS decreases by areas a+b+c. The tax lowers producer price to
40, so PS decreases by areas d+e+f. Market participants, in aggregate are worse off by areas a+b+c+
d+e+f.
The government gains a+b+e+f in revenue and external costs fall by g+c+d. Gains from the tax are thus
a+b+e+f + g+c+d.
Gains exceed losses by g, so the policy is a PPI. The net gain to social surplus is g,the DWL due to the
uncorrected externality.
2. A soot-spewing factory that produces steel is next to a laundry. We will assume that the factory faces
a prevailing market price of P=$40. Its cost function is C=X2, where X is the steel output. The laundry
produces clean wash, which it hangs out to dry. Suppose each unit of steel produced produces one unit
of soot (S), that is, X = S. The soot from the steel factory smudges the wash, so that the laundry has to
protect the laundry from the soot of the factory and this increases its costs of producing clean clothes.
The cost function of the laundry is C = Y2 + ½ S, where Y is pounds of laundry washed. A pound of clean
laundry sells for $10. Both firms face a competitive market.
Dr. Donna Feir
Economic 313
a. What outputs X and Y would maximize the sum of the profits of these two firms? How big is the
joint profit?
Maximizing joint profits: Max{X,Y, S} 40 X +10 Y - X2 - Y2 - .5S but X=S
Or equivalently Max{X,Y} 40 X +10 Y - X2 - Y2 - .5X
Profit maximizing conditions: Price of output equals marginal cost of that output: with respect to X: 40
=2X + .5
…with respect to Y: 10 = 2Y
Hence X* = 39.5/2 = 19.75, Y* = 5.
Total profit: 50+790 – (25 + 19.752 + .5*19.75) = 415.06.
b. If each firm individually maximizes its own profit, what will be the output of each firm? How big
is each firm’s profit?
Steel Factory: Max{X} 40 X - X2
Profit maximizing conditions, Price of output equals marginal cost of that output: 40 = 2X. Hence X =
20.
Profit of steel factory: 800 – 400 = 400
Laundry:
Max{Y} 10 Y - Y2 - .5S
Profit maximizing conditions, Price of output equals marginal cost of that output: 10 = 2Y so Y = 5.
Profit of laundry: 50 – 25 - 10 = 15
Total profit: 50+800 – (25 + 400 + .5*20) = 415. This is less than in a).
c. What per-unit tax would we need to set on soot to obtain the outputs found in Part a) of this
problem? What is the government’s revenue from this tax? What is the profit of each firm?
Steel Factory: Max{X,S} 40 X - X2 – t*S but S=X
Or equivalently, max{X} 40 X - X2 – t*X
Profit maximizing conditions, Price of output equals marginal cost of that output: 40 =2X + t
Comparing this with the optimality conditions from a) with respect to X where 40 =2X + .5, we must
have t = .5. In this case X = 19.75 and therefore S = 19.75. Tax revenue is .5*19.75 = 9.875.
The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the profit of the laundry is: 10*5 –
52 -.5* 19.75 = 15.125.
Dr. Donna Feir
Economic 313
d. Draw the marginal benefit curve (marginal profit curve of steel factory) of soot and the marginal
external cost curve of soot in a diagram with soot on the x-axis and $ on the y-axis. Indicate the
socially optimal amount of soot and mark the government’s tax revenue from an optimal tax on
soot.
In order to find the steel factory’s marginal profit as a function of soot write down the profit of the
steel industry as a function of soot:
Profit = 40 X - X2
but we also know that S=X
so equivalently profit = 40 S - S2
Marginal profit as a function of soot is given by the derivative of the profit as a function of soot, that is
MP = 40 –2S.
In order to find the marginal external cost of soot, first write down the external cost of soot: e(S) =
.5S. Then marginal external cost of soot is given by e’(S) = .5.
The socially optimal amount of soot is where the marginal benefit of soot to society (given by the
marginal profit of the steel industry as a function of soot) equals the marginal external cost of soot.
That is, 40 S - S2 = .5, and therefore S = 19.75. The Pigouvian tax (optimal tax on soot) is equal to the
marginal external cost of soot at the socially optimal level of soot, that is t = . 5. Government revenues
from this tax is equal to the area of the rectangular with height .5 and length 19.75.
e. Suppose the laundry has the right to clean air and is willing to let the steel factory pollute for a
price of q per unit of soot. What is the equilibrium price of soot and how much revenue does the
laundry get from selling its rights to clean air? What is the profit of each firm?
Steel Factory:
Max{X,S} 40 X - X2 – q*S but S=X and so can write profit in terms of S: 40 S - S2 – q*S
Set marginal benefit of soot equal to its marginal cost. Marginal benefit of soot to steel factory is how
much marginal profit is created: 40 –2S, marginal cost of soot is equal to q. Thus optimal amount of
soot is found by setting marginal benefit equal to marginal cost: 40 – 2S = q. The firm’s inverse
demand for pollution rights is q = 40 –2S.
Laundry:
Max{Y,S} 10Y – Y2 - .5*S + q*S
Profit maximizing with respect to Y: 10 =2Y, so Y = 5.
Dr. Donna Feir
Economic 313
with respect to S, marginal cost of soot given by.5 needs to equal marginal benefit of soot given by q
(now that steel factory has to pay $q to laundry for each unit of soot), so .5 = q. The firm’s inverse
supply of pollution rights to the steel factory is therefore q = .5.
Setting inverse demand equal to inverse supply, 40 –2S = .5, we find S = 19.75.
The laundry gets .5*19.75 = 9.875 from the steel factory.
The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the total profit of the laundry
(profit from laundering plus revenues from selling rights to pollute to steel factory) is: 15.125 + 9.875 =
25.
f.
Suppose the steel factory has the right to pollute the air up to S’ = 20. The steel factory is willing
to cut down its pollution for a price of q per unit of soot abated. What is the equilibrium price of
soot abated and how much revenue does the steel factory get from selling its rights to pollute?
What is the profit of each firm?
Steel Factory: Max{X,S} 40 X - X2 + q*(20 – S) but S=X
Or equivalently, max{S} 40 S - S2 + q*(20 – S)
Profit maximizing condition: marginal benefit of producing one more unit of soot needs to equal
marginal cost. Marginal benefit given by marginal profit, marginal cost given by lost opportunity to
collect q from laundry for reducing soot. Thus, 40 –2S = q.
Laundry:
Max{Y, S} 10Y – Y2 - .5*S - q*(20 – S)
Profit maximizing condition: marginal benefit of producing one more unit of laundry needs to equal
marginal cost. With respect to Y: Marginal benefit given by marginal revenue of Y, marginal cost given
by marginal production cost of Y. Thus 10 =2Yand Y=5.
With respect to S: Marginal benefit given by saving $q that would otherwise have to be paid to steel
factory and marginal cost given by increase in cost of producing laundry due to increase in soot: q = .5
From q = 40 –2S and q = .5, we find S = 19.75.
The steel factory gets .5*.25 = .125 from the laundry.
Total profit of the steel factory (profit from producing steel plus revenue from abating pollution) is:
790 – 19.752 + .125= 400.06; profit of the laundry is: 15.125 - .125 = 15.
In all the solutions to overcome the externality problem, we see that the efficient amount of pollution
is achieved. We also see that depending on the solution (i.e. who owns the rights) we have different
distributional effects.
Calculus Version of answer to question 2)
a. Maximizing joint profits: Max{X,Y, S} 40 X +10 Y - X2 - Y2 - .5S s.t. X=S
Dr. Donna Feir
Economic 313
Or equivalently Max{X,Y} 40 X +10 Y - X2 - Y2 - .5X
First order conditions: with respect to X: 40 –2X - .5 = 0
……with respect to Y: 10 – 2Y = 0
Hence X* = 39.5/2 = 19.75, Y* = 5.
Total profit: 50+790 – (25 + 19.752 + .5*19.75) = 415.06.
b. Steel Factory: Max{X} 40 X - X2
First order conditions: 40 –2X = 0
Hence X = 20
Profit of steel factory: 800 – 400 = 400
Max{Y} 10 Y - Y2 - .5S
Laundry:
First order conditions: 10 – 2Y = 0
Y = 5.
Profit of laundry: 50 – 25 - 10 = 15
Total profit: 50+800 – (25 + 400 + .5*20) = 415. This is less than in a).
c. Steel Factory: Max{X,S} 40 X - X2 – t*S s.t S=X
Or equivalently, max{X} 40 X - X2 – t*X
First order conditions: 40 –2X – t = 0
Comparing this with the optimality conditions from a) with respect to X where 40 –2X - .5 = 0,
we must have t = .5. In this case X = 19.75 and therefore S = 19.75. Tax revenue is .5*19.75 =
9.875.
The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the profit of the laundry is:
10*5 – 52 -.5* 19.75 = 15.125.
d. In order to find the steel factory’s marginal profit as a function of soot write down the profit
of the steel industry as a function of soot:
Profit = 40 X - X2
but we also know that S=X
so equivalently profit = 40 S - S2
Dr. Donna Feir
Economic 313
Marginal profit as a function of soot is given by the derivative of the profit as a
function of soot, that is MP = 40 –2S.
In order to find the marginal external cost of soot, first write down the external cost of soot:
e(S) = .5S. Then marginal external cost of soot is given by e’(S) = .5.
The socially optimal amount of soot is where the marginal benefit of soot to society (given by the
marginal profit of the steel industry as a function of soot) equals the marginal external cost of
soot. That is, 40 S - S2 = .5, and therefore S = 19.75. The Pigouvian tax (optimal tax on soot) is
equal to the marginal external cost of soot at the socially optimal level of soot, that is t = . 5.
Government revenues from this tax is equal to the area of the rectangular with height .5 and
length 19.75.
e. Steel Factory:
Max{X,S} 40 X - X2 – q*S
Or equivalently max{S} 40 S - S2 – q*S
s.t S=X
First order conditions: 40 –2S – q = 0
Or making q explicit q = 40 –2S
Laundry:
Max{Y,S} 10Y – Y2 - .5*S + q*S
First order conditions: with respect to Y: 10 –2Y = 0, with respect to S: -.5 + q = 0.
Therefore Y = 5, q = .5.
From q = 40 –2S, we find S = 19.75.
The laundry gets .5*19.75 = 9.875 from the steel factory.
The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the total profit of the laundry
(profit from laundering plus revenues from selling rights to pollute to steel factory) is: 15.125 + 9.875 =
25.
f.
Steel Factory: Max{X,S} 40 X - X2 + q*(20 – S) s.t S=X
Or equivalently, max{S} 40 S - S2 + q*( 20 – S)
First order conditions: 40 –2S – q = 0
Making q explicit: q = 40 –2S
Laundry:
Max{Y, S} 10Y – Y2 - .5*S - q*(20 – S)
First order conditions: with respect to Y: 10 –2Y = 0
Dr. Donna Feir
Economic 313
with respect to S: -.5 + q = 0
Y = 5, q = .5.
From q = 40 –2S, we find S = 19.75.
The steel factory gets .5*.25 = .125 from the laundry.
Total profit of the steel factory (profit from producing steel plus revenue from abating
pollution) is: 790 – 19.752 + .125= 400.06; profit of the laundry is: 15.125 - .125 = 15.
In all the two solutions to overcome the externality problem, we see that the efficient amount
of pollution is achieved. We also see that depending on the solution (i.e. who owns the rights)
we have different distributional effects.
3. Two firms, firm A and B, in a community pollute the environment. The government has decided that
12 units of pollution must be abated and that each firm must cut pollution by 6 units. The total cost of
pollution abatement is TCA =(1/6) qA2 for firm A, and TCB = (1/3) qB2 for firm B, where qA is the quantity
of abatement for firm A and qB is the quantity of abatement for firm B.
a. Is this solution cost efficient? Explain why or why not.
This solution is not cost efficient because the marginal cost of abatement is not the same for both
firms: MCA = (1/3)*qA and hence MCA(6) = 6/3 =2.
MCB = (2/3)*qB and hence MCB(6) = 12/3 =4.
If it is not the same, then we can show that we can save costs by having the firm with lower marginal
cost of abatement abate one more unit and the firm with higher marginal cost of abatement abate
one less unit, but still abate the same amount of pollution.
Total cost of abating with each firm abating 6 units of pollution: 36/6 +36/3 = 18. Now let firm A abate
one more unit and let firm B abate one less unit, then TCA = 49/6, TCB = 25/3, total cost of abating:
99/6 = 16.5. Clearly, there are cost savings if firm A abates 7 instead of 6 units and firm B abates 5
units instead of 6.
b. If the solution is not cost efficient, how much pollution should each firm abate at the cost
efficient outcome?
Both firms should abate a quantity that makes their marginal cost of abating equal. We also need to
ensure that 12 units of pollution are abated. MCA = (1/3)*qA and MCB = (2/3)*qB ; qA + qB = 12. This
means (1/3)*qA = (2/3)*qB and therefore qA = 2*qB and substituting into qA + qB = 12, we have 2qB + qB
= 12, and therefore qB = 12/3 = 4. The optimal amount for firm A is qA = 2*qB = 8.
Dr. Donna Feir
Economic 313
Total cost of abating 64/6 + 16/3 = 96/6 = 16. Cost savings compared to a): 2 dollars.
c. Suppose the government tells firms to cut pollution by 6 units or trade with another firm so that
in total pollution is cut by the desired amount. If there is a competitive market for pollution
abatement, what would be the price for each unit a firm abates beyond its 6th unit on another
firm’s behalf?
(Let the price of abating on another firm’s behalf be denoted by p and the amount abated on another
firm’s behalf be denoted by X. Since firm A is able to abate 6 units of pollution at a lower marginal
cost than firm B, firm A will be the supplier of abatement and firm B will be the demander for
abatement. This means that firm A’s marginal benefit of abating on firm B’s behalf needs to be equal
to the cost of the last unit of abatement for firm A, i.e MCA(6+X) = p. For firm B the benefit of the last
unit not abated by firm B needs to be equal to the cost it has to pay firm A for having firm A abate this
unit on firm B’s behalf, i.e. MCB(6-X) = p. Thus we find the equilibrium amount of trade by setting
MCA(6+X)= MCB(6-X). Plugging in the functional forms, we have (6+X)/3 = 2*(6-X)/3. Solving for X, we
find X* = 2. The equilibrium price p* is equal to p* = MCA(6+X*) = MCB(6-X*) = 8/3. This means that
firm A receives (8/3)*2 from firm B for abating 2 units on its behalf. It gains (8/3)*2 – [64/6 –36/6] =
16/3 – 14/3 = 2/3. Firm B saves [36/3 – 16/3] - (8/3)*2 = 4/3. Overall we create 2 more dollars of social
welfare compared to a).
4. (Optional) Poldi and Gerald are married. Poldi likes to light scented candles in their house while
Gerald hates the smell of scented candles. They both value money and their utility function over money
(MP for Poldi’s money and MG for Gerald’s money) and smell of scented candles (S), measured in hours
of the day in which the candles are lit, is given by UP(M, S) = MPS for Poldi and UG (M,S) = MG(24-S) for
Gerald.
a. Find the marginal utility of smell of scented candles for Poldi and Gerald.
The marginal utility of smell is MP for Poldi and -MG for Gerald.
b. Is smell a positive or negative externality? Explain.
Smell is a negative externality because Gerald’s utility goes down with each unit of smell.
Suppose both Gerald and Poldi own initially $100 each and both people are price takers in all markets
(i.e. we have competitive markets for both goods). The price of one unit of money is, of course, equal to
$1.
c. If Gerald has the right to scent-free air and charges a price of q for every hour he lets Poldi
light her candles, how many hours will Poldi light her candles and how much money will she
have to pay Gerald to compensate him for the smell? What are Gerald’s and Poldi’s utilities
under this property right regime?
Dr. Donna Feir
Economic 313
If Gerald has the right to scent-free air, his maximizes his utility MG(24-S) subject to the following
budget constraint. His initial endowment is worth $100 + 24q, since he can charge a price of q for
every hour he tolerates the scent of Poldi’s candles and he can sell 24 hours of S at the most. This also
means that each hour of the day that he wants to enjoy scent-free he is forgoing the opportunity to
earn $q from Poldi. Therefore his expenditure on money and scent-free hours is given by MG +(24-S)q
and hence his budget constraint is MG +(24-S)q = 100 + 24q. To find Gerald’s optimal consumption
bundle, we must set Gerald’s marginal benefit of scent-free hours in terms of money equal to the
marginal cost of scent-free hours and we must be on Gerald’s budget constraint. The marginal cost is
just equal to q. The marginal benefit in terms of money is equal to the amount of money he is willing
to give up for an additional scent-free hour. It is equal to the marginal rate of substitution of money
for a scent-free hour: MRS = MG /(24-S).
Setting MRS equal to opportunity cost of (24-S), MG /(24-S) = q. From the budget constraint we can
make MG explicit: MG = 100+qS. Then substitute for MG in MG /(24-S) = q, we get (100+qS)/(24-S) = q.
Making S explicit, we find S = 12 –50/q. This equation can be interpreted as Gerald’s supply of hours in
which he lets Poldi light her scented candles.
If Gerald has the right to scent-free air, Poldi maximizes her utility MPS subject to the following budget
constraint. Her initial endowment is worth $100. Gerald charges a price of q for every hour he
tolerates the scent of Poldi’s candles therefore Poldi’s expenditure on money and hours in which she
can light her scented candles is given by MP +Sq. Hence her budget constraint is MP+Sq = 100. To find
Poldi’s optimal consumption bundle, we must set Poldi’s marginal benefit of hours in which she lights
scented candles in terms of money equal to the marginal cost of hours in which she lights scented
candles and we must be on Poldi’s budget constraint. The marginal cost is just equal to q. The
marginal benefit in terms of money is equal to the amount of money she is willing to give up for an
additional hour with scented candles. It is equal to the marginal rate of substitution of money for an
hour with scented candles: MRS = MP /S.
Setting MRS equal to marginal cost of S, MP /S = q. From the budget constraint we can make MP
explicit: MP = 100-qS. Then substitute for MP in MP /S = q, we get (100-qS)/S = q. Making S explicit, we
find S = 50/q. This equation can be interpreted as Poldi’s demand of hours in which she can light her
scented candles.
Because both people are price takers, and with the assignment of the property right to Gerald we
have created a market for smell, we find the competitive price for smell, q, by setting demand equal
to supply. That is, 12 –50/q = 50/q and therefore q* = 100/12 = 25/3 and S* = 6. Gerald lets Poldi light
her candles for 6 hours and receives in return from her 6*25/3 = 50 dollars. This implies that Gerald’s
utility is equal to 150*(24-6) = 2700 and Poldi’s utility is equal to 50*6 = 300 under this property right
regime.
d. If Poldi has the right to light candles for 24 hours a day and charges a price of q for every
hour she does not light her candles, how many hours will Poldi light her candles and how
Dr. Donna Feir
Economic 313
much money will Gerald have to pay her to compensate her for not being able to enjoy the
scented candles the whole day? What are Gerald’s and Poldi’s utilities under this property
right regime?
If Poldi has the right to lighting her candles 24 hours a day, she maximizes her utility M G(24-S) subject
to the following budget constraint. Her initial endowment is worth $100 + 24q, since she can charge a
price of q to Gerald for every hour she does not light her scented candles and that’s at the most 24
hours. This also means that each hour of the day that she wants to enjoy scented candles she is
forgoing the opportunity to earn $q from Gerald. Therefore her expenditure on money and hours of
scented candles is given by MP +Sq and hence her budget constraint is MP +Sq = 100 + 24q. To find
Poldi’s optimal consumption bundle, we must set Poldi’s marginal benefit of hours of scented candles
in terms of money equal to the opportunity cost of hours of scented candles and we must be on
Poldi’s budget constraint. The opportunity cost is just equal to q. The marginal benefit in terms of
money is equal to the amount of money she is willing to give up for an additional hour of scented
candles. MRS = MP /S.
Setting MRS equal to opportunity cost of S, MP /S = q. From the budget constraint we can make MP
explicit: MP = 100+q(24-S). Then substitute for MP in MP /S = q, we get (100+q(24-S))/S = q. Making S
explicit, we find S = 12 +50/q. This equation can be interpreted as Poldi’s demand of hours in which
she lights her scented candles, given that she is the seller of scent-free hours to Gerald.
If Poldi has the right to scent-free air, Gerald maximizes his utility MG(24-S) subject to the following
budget constraint. His initial endowment is worth $100. Poldi charges a price of q for every scent-free
hour, therefore Gerald’s expenditure on money and scent-free hours is given by MG +(24-S)q. Hence
his budget constraint is MG +(24-S)q = 100. To find Gerald’s optimal consumption bundle, we must set
Gerald’s marginal benefit of scent-free hours in terms of money equal to the marginal cost of scentfree hours and we must be on Gerald’s budget constraint. The marginal cost is just equal to q.
Setting MRS equal to marginal cost of (24-S), MG /(24-S) = q. From the budget constraint we can make
MG explicit: MG = 100-q(24-S). Then substitute for MG in MG /(24-S) = q, we get (100-q(24-S))/(24-S) = q.
Making S explicit, we find S = 24 - 50/q. This equation can be interpreted as Gerald’s supply of scented
hours given that he is a demander of scent-free hours.
Because both people are price takers, and with the assignment of the property right to Poldi we have
created a market for scent-free air, we find the competitive price for scent-free air, q, by setting
demand equal to supply. We can think of Poldi as the supplier of scent-free air and Gerald as the
demander of scent-free air. This means that the supply and demand of scent-free air (given by (24-S*)
must be the same in the competitive equilibrium, and therefore also that the amount of hours filled
with scent of candles, S* must be the same in the competitive equilibrium.
That is, 12 +50/q = 24 - 50/q and therefore q = 100/12 = 25/3 and S* = 18. Poldi forgoes to light her
candles for 6 hours and receives in return from Gerald (24- 18)*25/3 = 50 dollars. This implies that
Dr. Donna Feir
Economic 313
Gerald’s utility is equal to 50*(24-18) = 300 and Poldi’s utility is equal to 150*18 = 2700 under this
property right regime.
We can see here that the optimal amount of smell in this economy depends on who is assigned the
property right.
Alternative approach to answer questions 1d) and e):
d. If Gerald has the right to scent-free air, his utility maximization problem is
Max MG(24-S) s.t. MG = 100 + qS, because additionally to his $100 he is also receiving revenues from
letting Poldi light her candles S hours during the day. Substituting for MG in his utility function
we have:
MaxS (100 + qS) (24-S) and taking the derivative with respect to S and setting it equal to zero yields
q(24-S)-(100 + qS) = 0
From this optimality condition we can make S explicit to obtain S = 12 –50/q.
For Poldi: Max MPS s.t. MP + qS = 100, because she needs to decide how much of her $100 she wants
to keep and how much of it she wants to pay to Gerald in order to acquire the right to light
candles for S hours of the day. Substituting for MP in her utility function we have
MaxS (100 - qS) S and taking the derivative with respect to S and setting it equal to zero yields
-qS+(100 - qS) = 0
From this optimality condition we can make S explicit to obtain S = 50/q.
Because both people are price takers, and with the assignment of the property right to Gerald we
have created a market for smell, we find the competitive price for smell, q, by setting demand equal
to supply. We can think of Gerald as the supplier of smell and Poldi as the demander of smell.
That is, 12 –50/q = 50/q and therefore q* = 100/12 = 25/3 and S* = 6. Gerald lets Poldi light her
candles for 6 hours and receives in return from her 6*25/3 = 50 dollars. This implies that Gerald’s
utility is equal to 150*(24-6) = 2700 and Poldi’s utility is equal to 50*6 = 300 under this property right
regime.
e. If Poldi has the right to scented air, her utility maximization problem is
Max MPS s.t. MP = 100 +(24 – S)q, because additionally to her $100 she is also receiving revenues from
letting Gerald have 24- S hours of scent-free air during the day. Substituting for MP in her
utility function we have
Dr. Donna Feir
Economic 313
MaxS (100 +(24- S)q) S and taking the derivative with respect to S and setting it equal to zero yields qS+(100 +(24- S)q) = 0
From this optimality condition we can make S explicit to obtain S = 12 +50/q.
For Gerald: Max MG(24-S) s.t. MG + q(24-S) = 100, because he needs to decide how much of his $100 he
wants to keep and how much of it he wants to pay to Poldi in order to acquire the right to
scent-free air for (24-S) hours of the day. Substituting for MG in his utility function we have
MaxS (100 – q(24-S)) (24-S) and taking the derivative with respect to S and setting it equal to zero
yields
q(24-S)-(100 - q(24-S)) = 0
From this optimality condition we can make S explicit to obtain S = 24 - 50/q.
Because both people are price takers, and we with the assignment of the property right to Poldi we
have created a market for scent-free air, we find the competitive price for scent-free air, q, by setting
demand equal to supply. We can think of Poldi as the supplier of scent-free air and Gerald as the
demander of scent-free air. This means that the supply and demand of scent-free air (given by 24-S*)
must be the same in the competitive equilibrium, and therefore also the amount of smell, S*.
That is, 12 +50/q = 24 - 50/q and therefore q = 100/12 = 25/3 and S* = 18. Poldi forgoes to light her
candles for 6 hours and receives in return from Gerald (24- 18)*25/3 = 50 dollars. This implies that
Gerald’s utility is equal to 50*(24-18) = 300 and Poldi’s utility is equal to 150*18 = 2700 under this
property right regime.
We can see here that the optimal amount of smell in this economy depends on who is assigned the
property right.