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Unformatted text preview: Problem set 5  suggested solutions Question 1 (exercise 21.12 except e&g, add i&j)
• (a) Draw a graph with the aggregate demand curve D0 for the “greenies.”
Assume that green cars are competitively supplied at a market price p∗ —
and draw in a perfectly elastic supply curve for green cars at that price.
This is illustrated in panel (a) of Graph 21.12. • (b) There are two types of externalities in this problem. The ﬁrst arises
from the positive impact that green cars have on the environment. Suppose that the social marginal beneﬁt associated with this externality is
some amount k per green car and illustrate in your graph the eﬃcient
number of cars x1 that this implies for “greenies”. Then illustrate the
Pigouvian subsidy s that would eliminate the market ineﬃciency.
This is also illustrated in panel (a) of Graph 21.12. At price p∗, greenies buy x0
cars — but the marginal social beneﬁt curve lies k above D0 — which implies
the eﬃcient quantity of cars that greenies should buy is x1 . A subsidy of size
s = k will result in exactly that — because the entire subsidy will go toward
reducing the price of cars (given the elastic supply curve in the competitive
market.)
• (c) The second externality emerges in this case from the formation of social
norms — a form of network externality. Suppose that the more green cars
1 the “meanies” see on the road, the more of them become convinced that
it is “the right thing to do” to buy green cars even if they are somewhat
less convenient right now. Suppose that the “meanies’s” linear demand D1
for green cars when x1 green cars are on the road has vertical intercept
below (p∗ −k). In a separate graph, illustrate D1 — and then illustrate
a demand curve D2 that corresponds to the demand for green cars by
“meanies” when x2 (> x1 ) green cars are on the road. Might D2 have an
intercept above p∗?
This is done in panel (b) of Graph 21.12. As the number of green cars increases,
the demand for green cars by “meanies” increases — implying D2 lies above D1
and can certainly have intercept higher than p∗.
• (d) Does the subsidy in (b) have any impact on the behavior of the “meanies”? In the absence of the network externality, is this eﬃcient?
No, the subsidy in (b) has no eﬀect on the meanies because it does not reduce
price suﬃciently for any meanie to buy a green car given the number x1 of
green cars bought by the greenies. In the absence of the network externality, it
would indeed be eﬃcient for only the greenies to buy green cars — because the
meanies simply do not value them suﬃciently to buy them even when they get
the subsidy that takes the positive externality from green cars into account.
• (f ) Explain how the imposition of a larger initial subsidy has changed the
“social norm” — which can then replace the subsidy as the primary force
that leads people to drive green cars.
The social norm here is like peer pressure — the more green cars, the greater
the peer pressure felt by those who don’t drive green cars. As the social norm
changes, green cars are valued by meanies because others are driving them.
Whether meanies simply want to look good in front of others — or whether we
interpret the increased demand as an in crease in the perception that driving
green cars is “the right thing to do”, the subsidy might eventually be replaced
by the new social norm.
• (h) How could sin taxes like this be justiﬁed as means of maintaining social
taboos and norms through network externalities?
People who advocate such sin taxes might also be thinking of the impact that
indi vidual behavior has on social norms. The more people smoke, the more
socially acceptable it is to smoke — and thus the greater will be the demand
for cigarettes. The more available pornography is, the more socially acceptable
it might be for others to consume pornog raphy. To the extent to which consumption of pornography causes changes in behavior — such as extramarital
aﬀairs or casual relationships — that some might wish would not occur, we
have a network externality that reinforces other negative externalities. If these
network externalities are strong, and if you believe the social norms that might
form from less use of cigarettes or pornography to be important, you might
advocate sin taxes primarily on those grounds.
2 • (i) Consider an alternative policy to a subsidy on green cars to be a pollution tax on polluting cars. Illustrate how the markets for polluting &
green cars change with this new policy. (Assume all cars in each category
to produce the same amount of pollution.)
Graph:
Since we have negative externalities, our long run supply curve, incorporating
social pollution costs, or SMC, is upward sloping/exponential with pollution
costs. Private marginal cost curve (PMC) is horizontal, not incorporating social
pollution costs. Demand is downward sloping. Intersection of PMC and Demand
curve represents an ineﬃcient level of output supplied on the market at x∗ and
p∗ . Applying pollution tax will cause a decrease in output supplied down to
an optimal/desired level of output xopt and popt . The vertical distance between
SMC and PMC at xopt represents the size of the tax required so that ineﬃciencies
are eliminated.
What could happen as a side eﬀect of taxing the cars that pollute is that due
to the network externalities, there would be less and less polluting cars on the
street, which could make people demand more green cars. Thus, on the green
car market, we may observe an upward shift in demand.
• (j) Finally consider a cap & trade policy for pollution, now with older
cars producing more pollution than younger cars. Green cars consume no
pollution. Explain what happens the “demographics” of cars on the road.
How will that change if greenies decide to purchase some of the pollution
rights?
Graph:
With cap and trade policy, new market of vouchers is created. Due to this
policy, we can then expect that there will be less older cars (that produce more
pollution) on the roads. If greenies decide to purchase some of the pollution
rights, demand on the market for vouchers will shift upwards, which will unambigiously increase the vouchers’ rental price. That means that even less people
will decide to purchase an old car, so we will be seeing more younger than
older cars on the streets. (Accepted alternative: Supply of vouchers for nongreenies/polluting cars decreases, which given their demand, increases rental
price of vouchers. Result follows as above.)
Question 2
1 1 2
The N consumers in the economy have the utilty function xn Y 2 where xn is the
amount individual n spends on private consumption and Y is the public good
national defense. Each consumer has an income I. The price of x is 1 and the
cost of y is y 2 . Of course, y1 + y2 + y3 + . . . + yn = Y • (a) What is the socially optimal level of provision of this public good?1
1 Your maximization problem setup looks something like this (check p.1060  ... for details) 3 Since our utility function is CobbDouglas, you know that the form of deI
mand function of u(x, y ) = xα y 1−α for y looks like y = (1 − α) P . Here, α = 1 .
2
1 I Thus, consumer i s demand is P = 2Y i . Since we have N consumers with
identical preferences and equal incomes, we obtain the market demand for Y
N
i
by vertically summing individual demands for the public good Y ( i=1 M BY ).
1 NI
Market demand for public good Y is thenP = 2 Y .
N
i
Optimal level of public good is provided when it holds i=1 M BY = M CY .
2
Since M C = Y , socially optiml level of provision of the public good Y equals
to 1 NI
2Y =Y2 Y = 3 NI
2 . Suppose, as in the book, MC=1. Then, Y = NI
2. • (b) If it not provided by the government, what is the individual contribution of each person toward purchasing the public good?
Optimization problem2 : 1 maxzi ,xi xi2 Y 1
2 st.
I = x i + zi
Y = zi + (N − 1)z, , where zi represents individual’s contribution towards the public good Y, where
z is the amount everyone but i contributes to the public good.
1
1
maxzi
ln(I − pi zi ) + ln(zi + (N − 1)z ) .
2
2
Take FOC wtr. to zi and equate it to zero. 1 (−1)
1
1
I
(N − 1)
+
= 0 ⇒ zi ( z ) = −
z,
2 (I − pi zi ) 2 (zi + (N − 1)z )
2
2 which represents the i s best response function of contribution to public good
Y given actions from all other than i .
In Nash equilibrium, each individual best responds to each other. Also, we
are in a symmetric case, where, repeating the same approach for “all other but
i ” consumers, will give us a symmetric best response function. To solve for how
much each individual contributes to a public good in an equilibrium , you can
maxx1 ,x2 ( 1 lnx1 + 1 lnY ) st. uN −1 (xN −1 , Y ) = u and Y = N I − x1 −
¯
2
2
maxx1, x2 ( 1 lnx1 + 1 ln(N I − x1 − N −1 xi )) st. uN −1 (xN −1 , Y ) = u
¯
i=1
2
2
N
N − 1
1
1
⇒ L = ( 2 lnx1 + 2 ln(N I − x1 − i=1 xi )) +
λi (¯ − ui (xi, Y ))
u
i=2
2 Assume p = p = 1.
x
i 4 N −1
i=1 xi (i) plug best reponse function z (zi ) into (zi (z ) and obtain z eq , or (ii), since you
know that each individual contributes the same amount towards the public good
Y, equate zi (z ) = z and solve for z eq . Due to simplicity, I will show (b).
I
(N − 1)
I
−
z = z ⇒ z eq =
2
2
N +1
• (c) What is the total amount of the public good provided? Is this society
more eﬃcient when it is large or small?
Total amount of the public good provided, assuming there are N individuals, is:
NI
.
N +1
The society is more eﬃcient when it is small. Why? Look at the best
response functions  when N increases, my individual contribution towards the
public good will decrease. For instance, if it is just two people in the world, each
of them contributes half of the whole contribution towards the public good. If
one extra person enters, it seems to you as if another person is actually giving
twice as much. That means that you can decrease your contribution to the public
good more and more as N increases  you have more and more people to free ride
on, as N increases. Why is this ineﬃcient? Each individual contributes up to
the level where his private marginal beneﬁts equal to marginal costs. However,
N
in the case of freeriding, our total marginal beneﬁts (M B1 + i=2 M Bi ) of the
public good’s consumtion are larger than MC. You can then see that the amount
contributed by an individual is less than eﬃcient, because an individual only
takes into account M B1 . Thus, as N increases, we are producing ineﬃciently
lower quantity of the public good, because we can freeride more and more.
Y = Nz = • (d) If the government supported individual contributions with a tax refund
of .5 for every unit of Y purchased by the individual, how would this change
the individual contribution?
Repeat the same optimization problem as in b), only add taxation and subsidy
into the story:
1
1
maxzi
ln((I + 0.5zi ) − pi zi ) + ln(zi + (N − 1)z ) .
2
2
You obtain the following best response function:
(N − 1)
z.
2
In equilibrium, each individual will contribute:
zi ( z ) = I − zi (z ) = z ⇒ z eq =
Y. 2I
.
(N + 1) Individual will increase his individual contribution towards the public good 5 • (e) Is it possible to arrive at the eﬃcient level of production with a subsidy?
If so, what should the subsidy be?
When N = 2, there are two of us, and each one therefore takes into consideration
half the overall beneﬁt from his private contribution to the public good. When
N = 3, the amount of the overall beneﬁt each of us takes into account falls to 1/3
— and when N = 4, it falls to 1/4. For a population of N , each person therefore
only takes into account 1/N of the overall beneﬁt from his private contribution —
−
leaving (NN 1) that he does not take into account. As N get large, the fraction of
the overall beneﬁt of a private contribution that each person takes into account
approaches zero. If I don’t take into account half the beneﬁt that I create by
giving to the public good, then a subsidy of s = 0.5 will cause me to internalize
the externality by having me pay for only half of the actual contributions I am
making. When I don’t take into account 2/3 of the beneﬁts I create (as when
N = 3), the subsidy must rise to s = 2/3 — and when N = 4, it must rise to s
= 3/4 because I now only take into account 1/4th of the beneﬁts I create. For
−
a population of N , the subsidy must therefore rise to s = (NN 1) because that is
the fraction of the beneﬁt I create with my contribution that I do not take into
account. This implies that the optimal subsidy will approach s = 1 as N gets
large.
In short  It is possible to arrive at the eﬃcient level of production wth a
−
subsidy equal to s = (NN 1) , since individuals internalize the externality they
are making with their own contribution towards the public good. When N is
getting large, optimal subsidy approaches 1.3
Question 3
Now there are only 2 consumers in the society, with demand curves y = −3P +10
and y = −2P + 10.
• (a) Suppose y is a public good like national defense. Draw the market
demand curve.
Sum both individual demand curves vertically. 4 • (b) The government has decided to protect only those who have passports.
What will be the level of national defense provided and what will be the
price of the passport that each citizen purchases? Use the Lindahl Pricing
concept.
We ﬁnd SMB (vertical summation):
10 y
y
1 5y
− ) + (5 − ) = 8 −
3
3
2
3
6
In order to ﬁnd socially optimal level of public good, we equate SMB=MC
P =( 3 Note that this case (when subsidy is approaching 1) is basically equivalent to the case of
government funding all the private contributions/providing the public good.
4 Graphs not drawn to scale. 6 1 5y
50
8−
= 2y ⇒ y =
3
6
17
Now the government uses price discrimination and has each individual pay
a diﬀerent amount such that each individual is paying the price that he or
she would have paid if he or she were demanding this amount. Thus, we use
individual’s demand curve to see what each individual would pay. This is then
called the Lindhal equilibrium.
• Type 1 consumer pays:
P= 50
2
10
− 13 = 2
3
3
39 • Type 2 consumer pays:
P =5− 50
13 2 =3 1
13 • (c) Now suppose y is a private good like electricity. Draw the market
demand curve.
Sum both individual −2P + 10 y = −5P + 20 0 demand curves horizontally.
, if 0 < y ≤ 3 1
3
, if 3 1 < y ≤ 20
3
, otherwise 5 • (d) What is the eﬃcient level of production if the cost of electricity is y 2 ? Optimality condition: M B = M C . We have TC = y 2 ⇒ M C = 2y , private
good.
• Case 1: 0 < y ≤ 3 1
3
y = −2P = 10 ⇒ P = 5 − y
2 y
= 2y ⇒ y = 2
2
Since y = 2 falls in the Case 1 interval, this is a possible optimal solution.
5− • Case 2: 3 1 < y ≤ 20
3
y = −5P + 20 ⇒ P = 4 − y
5 y
20
= 2y ⇒ y =
5
11
does not fall in the Case 2 interval, this is not an optimal
4− Sine y = 20
11
solution.
We conclude that y = 2 is an eﬃcient level of production if the total cost of
electricity is y 2 .
5 Graphs not drawn to scale. 7 • (e) If a monopolist is providing the good, what is the proﬁt maximizing
price and quantity? How much proﬁts are made?
Optimality condition: MR=MC. We have TC = y 2 ⇒ M C = 2y , private good.
• Case 1: 0 < y ≤ 3 1
3
5 − y = 2y ⇒ y =
Since y =
solution. 5
3 5
3 falls in the Case 1 interval, this is a possible optimal monopolist’s • Case 2: 3 1 < y ≤ 20
3
4− 2y
20
= 2y ⇒ y =
5
12 Sine y = 20 does not fall in the Case 2 interval, this is not a possible optimal
12
monopolist’s solution.
We conclude that y = 5 is an optimal level of monopolist’s production if the
3
total cost of electricity is y 2 .
• (f) If the monopolist is able to discriminate, what is the price each consumer must pay?
In our case, there are only 2 types of consumers, therefore the monopolist can
charge a diﬀerent price to each of them (assuming he can recognize each type of
the consumer) and uses third degree price discrimination. Prices charged will
be:
• Consumer of type 1: M R1 = M C
10 2y
5
10 ( 5 )
11
= 2y ⇒ y = and P =
− 4 =2
3
3
4
3
3
12
Proﬁt:
π = yP − y 2 = 2 1
12 • Consumer of type 2: M R2 = M C
5 − y = 2y ⇒ y = (5)
5
1
and P = 5 − 3 = 4
3
2
6 Proﬁt:
π = yP − y 2 = 4
Together, he earns: 6 1 .
4
Proﬁt without discrimination: 4 1
6
8 1
6 • (g) Does discriminating monopolist have more proﬁts? Why? (answer
with logic, not math)
Yes, the monopolist has higher proﬁt, because he can, in our case, where he
knows willingness to pay for each type of the consumer, extract consumer surplus
from each of them by charging them diﬀerent price. Also, while discriminating,
he can reach customers with a lower demand that were previously exluded from
the market due to a very high price and a limited output.
• (h) Is the discriminating monopolist more eﬃcient than the nondiscriminating
monopolist? Justify by comparing monopolist outputs to the eﬃcient output.
Doing welfare analysis, one obtains the following results (see graphs for help):
• Discriminating monopolist:
– ydisc, type1 = y d1 = 5 , pd1 = 2 11
4
12
– ydisc, type2 = y d2 = 5 , pd2 = 4 1
3
6
∗ eﬃcient outcomes and prices on each market:
· y ef,1 = 10 , pef,1 = 2 6
7
7
· y ef,2 = 2, pef,1 = 4
∗ price at intersection M Ri = M C at y d,i
2( 5 ) · pM R1 = 10 − 34 = 2 1
3
2
· pM R 2 = 5 − 5 = 3 1
3
3 25
– DW L1 = (2 11 − 2 1 ) ∗ ( 10 − 5 ) ∗ 1 = 672
12
2
7
4
2
5
– DW L2 = (4 1 − 3 1 ) ∗ (2 − 5 ) ∗ 1 = 36
6
3
3
2
– Total DW L1+2 from both markets equals: 355
2016 • Nondiscriminating monopolist
– ynondisc = y nd = 5 , pnd = 4 1
3
6
∗ eﬃcient outcomes and prices on one market:
· y ef = 2, pef = 4
∗ price at intersection M R = M C at y nd
· pM R = 5 − 5 = 3 1
3
3 – DW L = (4 1 − 3 1 ) ∗ (2 − 5 ) ∗
6
3
3 1
2 = 5
36 DW L1+2 > DW L
We are dealing with the third degree price discrimination. We know that,
compared to an eﬃcient outcome, some DWL will arise, however it is ambigious
to say, whether it is more eﬃcient from the society’s point of view to allow the
monopolist to discriminate or not. As you can see from calculations above, third
degree price discrimination by the monopolist causes more DWL compared to
the case when monopolist is treating the market as one and is not discriminating
at all.
9 ...
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 Spring '08
 Woroch

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