tutorial 1 suggested solutions

8/9/2019 Tutorial 1 Suggested Solutions

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EC3351 Public Finance 2014/15 Semester 1

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Tutorial 1 Suggested Solutions

Q1. A negative externality: Deal, tax or pay-to-reduce?

The market demand for a good is given by the equation Q D = 120 – 2P, while the market

supply is given by the equation Q S = 2P, where P is the market price in dollars per unit.

Production of the good creates a negative externality to third parties, and the marginal

external cost associated with the Q th unit of the good is given by the equation MEC = Q.

A. Find the efficient quantity, the market equilibrium quantity and price. Compute the

value of the deadweight loss associated with the market outcome. Use a diagram to

illustrate your answers.

Refer to the diagram drawn on theleft. To find the deadweight loss,

you need to find the coordinates of

the shaded triangle ABC.

Point A is the market equilibrium.

To find its coordinates, equate

quantity demanded with quantity

supplied: 120 – 2PM = 2PM

PM = $30 and Q M = 60, where

subscript M denotes ‘market’. The

market quantity is thus 60 units.

Point B is the social optimum. Find

its coordinates by equating MSB

with MSC. The MSB equation is simply the demand equation with price (= $ willingness to

pay by consumers) as the subject1: MSB = 60 – ½Q. The MPC equation is simply the supply

equation with price (= $ willingness to accept by producers) as the subject: MPC = ½Q. The

MSC equation is given by MSC = MPC + MD = 3Q/2. Diagrammatically, the MSC curve isthe vertical summation of the MPC and MD curves.

Equate MSB with MSC: 60 – ½ Q = 3Q/2 Q* = 30 and P* = $45. The optimal quantity is

30 units.

The MSC corresponding to Point C is given by 3(60)/2 = $90.

Thus, the area of the shaded triangle is ½(60 – 30)($90 – $30) = ½ (30)($60) = $900.

1 Since there is no positive externality, MSB = MPB, so I will use MSB throughout.


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B. Assuming transaction costs are negligible, describe an agreement between consumers,

producers, and those affected by the externality, that would result in the efficient

quantity being produced.

Refer to the diagram from part A. If production shrinks from 60 to 30, consumer and

producer surplus will shrink by the area of triangle ABD = $450. But third parties would

gain $1,350, the area of trapezium ADBC2.

Third parties can offer to pay market participants an amount between $450 and $1,350 in

return for reducing production to 30 units, and market participants would agree to the

deal. This is a Pareto Improvement over the market outcome.

C. Suppose instead that the government imposes a $T-per-unit Pigouvian Tax on

producers. What is the value of T that would lead the market to produce the efficient

quantity? How is the Pigouvian solution different from the private solution in part B?

Since the vertical gap BD

between MSC and MPC at the

optimal quantity of 30 units is

$30, a $30/unit tax (placed on

producers in the diagram drawn

on the left) will shift the supply

curve up sufficiently for the

market to reach equilibrium

with 30 units produced. Thus, T

= 30.

As in part B, third parties gain

the area of trapezium ADBC =

$1,350.

The government gains tax

revenue = area of rectangle

DEFB = $30 x 30 = $900.

Market participants will lose surplus = area of triangle ABD + rectangle DEFB = $1,350.

Although gains outweigh losses by the same $900 as in part B, the result here is not a

Pareto Improvement over the market outcome, since market participants are worse off.

2 This trapezium is equal in size to the area under the MEC curve (not drawn) between Q = 30 and Q = 60.


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D. Suppose instead that the government pays producers $S-per-unit to reduce their

production. Is there a value of S that would lead the market to produce the efficient

quantity? How does this solution compare to the Pigouvian Tax solution?

A payment of $30 per unit to reduce production will effectively shift producers’ supply

curve up in exactly the same way as the Pigouvian tax of part C, and will thus also result in

30 units produced. This is because each unit produced now incurs an additional cost of

$30 in foregone government payment.

As before, third parties gain $1,350. Government loses 30 x $30 = $900 in subsidy

payments. Market participants gain $450. Once again, gains outweigh losses by $900, but

the distributional implications are very different.

In practice, it is probably not advisable to pay producers to reduce production, since this

may attract more producers to enter the market!


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Q2. Controlling Emissions in a cost-effective manner

This is a modified version of Question 12 in Chapter 5 of the textbook.

Suppose that two firms emit a certain pollutant. The marginal cost of reducing pollution for

each firm is as follows: MC 1 = 300e1 and MC 2 = 100e2, where e1 and e2 are the amounts (in

tons) of emissions reduced by the first and second firms, respectively. Assume that in the

absence of government intervention, Firm 1 generates 100 tons of emissions and Firm 2

generates 80 tons of emissions, and thus total emissions stands at 180 tons. Regulators aim

to reduce emissions by 40 tons i.e. to 140 tons.

A. Show that an emissions fee of $3,000 achieves the target at minimal cost. How many

tons of emissions will each firm reduce? How much would each firm pay in taxes?

With a fee, marginal costs are equalized, which is necessary and sufficient for cost

effectiveness.

As depicted in the diagram below, with a fee of $3,000, firm 1 cuts 10 tons while firm 2

cuts 30 tons. Firm 1’s abatement cost is ½ x $3,000 x 10 = $15,000 (triangle ABC), while

Firm 2’s is ½ x $3,000 x 30 = $45,000 (triangle AGH), so total abatement cost is $60,000.

Firm 1 pays a tax of 90 x $3,000 = $270,000 while Firm 2 pays 50 x $3,000 = $150,000.

B. If regulators instead

command each firm to reduceemissions by 20 tons, how large,

if any, is the resulting

deadweight loss?

Compared to part A, firms gain

by not having to pay tax, but

government loses by the same

amount. If each firm does 20

tons of abatement, Firm 1incurs a total abatement cost of

½ x $6,000 x 20 = $60,000

(triangle ADE), while Firm 2 incurs abatement cost of ½ x $2,000 x 20 = $20,000 (triangle

AFE). Total abatement cost is thus $80,000, which is $20,000 more than in part A. This

$20,000 is the deadweight loss of inefficient pollution reduction.3

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An equivalent way of explaining the issue is to see that by moving to 20 tons of reduction each, firm 1 incursadditional abatement cost of trapezium BCED, while firm 2 reduces its abatement cost by trapezium FEGH. The

former is larger than the latter by $20,000.


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C. Instead of an emissions fee, the regulatory agency introduces a tradable permit system

and issues 140 permits, each allowing one ton of emissions. Each firm is given 70

permits. Assume that the firms act as price-takers. Show that equilibrium permit price is

$3,000, and that the two firms will reduce emissions by the same amount as in Part A.

If both firms act as price-takers, each firm will reduce pollution until the marginal cost of

reducing a ton of pollution equals the permit price of $3,000. Thus, Firm 1 will choose to

cut emissions by 10 tons. Firm 2 will choose to cut emissions by 30 tons. This is identical to

the result in Part A.

The diagram to the right shows

the market for permits, with

individual firm demand curves

and the market demand curve

(which is the horizontal

summation).

Firm 1 will want to buy 20

permits, since it will emit 90

tons but it only has 70 permits.

Firm 2 will want to sell 20

permits, since it will emit 50

tons of emissions but has 70permits. Since 20 = 20, the permit market will be in equilibrium! Thus, $3,000 is the

equilibrium permit price.

D. Suppose the marginal damage of emissions is found to be constant at $2,000 per ton.

How should the government alter the number of permits, if at all, to obtain the optimal

quantity of emissions?

The marginal damage of emissions is the marginal benefit of reducing one ton of

emissions. The optimal amount of pollution is obtained when this is equal to the marginal

abatement cost. Thus, Firm 1 should reduce emissions by 2000/300 = 6.67 tons, while firm

2 should reduce emissions by 2000/100 = 20 tons. Total abatement should thus be 26.67

tons. The government should issue enough permits to allow for 153.33 tons of emissions.


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Q3. Applying the theory of externalities

Assess the following situations and answer the following questions: (1) Does an externality

exist? (2) Does the Coase Theorem apply? (3) what (e.g. legislation, regulation, tax, quota,

etc. or do nothing) is the appropriate government response?

A. Crying babies in a restaurant

1. Crying babies in a restaurant create a negative externality, because the noise disrupts

the dinners of other patrons but the babies’ parents do not need to take this into

account.

2. Rights on noise can be assigned, and negotiation costs may not be prohibitive given

the small number of parents with babies in a restaurant, with other patrons being

represented by the restaurant management, which has a strong stake in maintaining a

good dining environment. The Coase Theorem may thus apply, and a viable solution is

to grant the rights on noise to restaurant management. Restaurants are free to bancustomers with crying babies or to ask mothers to bring babies outside until they quiet

down.

3. Government should not do any further intervention.

B. Smoking in indoor indoor public areas

1. Smoking indoors in communal areas affects others negatively because of a number of

reasons such as damage to health form passive (second-hand) smoking, irritation from

the smoke, and increased risk of fire. Because these effects are not taken into account

by the smoker and are not transmitted by market prices, they qualify as externalities.2. As there are too many smokers and non-smokers, it is probably too costly to use the

Coase Theorem to allow for negotiations after assigning legal rights in public

communal areas. Yet the damage to non-smokers is substantial.

3. Some options for government response:

a. A tax placed on cigarette purchases would harm smokers even if they smoke

outdoors (where the external costs are much smaller).

b. A tax placed on smoking activity indoors, operating akin to an emissions fee,

would be more suitable, but it is extremely difficult to implement as it requires

tracking the number of cigarettes each person smokes in all indoor public

areas.

c. A third option is to do nothing (in effect giving non-negotiable rights to

smokers). This places the burden of adjustment on non-smokers – they have to

endure the smoke or walk away from it. Second hand smoking is extremely

harmful to health, and the added risk of fire is not trivial.

d. A fourth option is to impose a ban on indoor smoking in public areas. This

places the burden of adjustment on smokers, but it is not a heavy adjustment

because they can still step outdoors for smoke breaks. This is likely to be the

least costly option.

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