W1105 Principles: Recitation 2

Robert Ainsworth

September 24, 2015

1 The Algebra of Specific Taxes

1.1 Concepts

Definition: A specific tax is a fixed levy per unit of a particular good or service.1 Forinstance, in Texas there is a beer tax of $0.20 per can. In many locations, there are taxesper pack of cigarettes or per gallon of gas.

• The key characteristic of a specific tax is that the price paid by the buyer differs fromthat received by the seller. Formally, we say that a specific tax creates a wedge betweenthe gross price paid by the buyer, PB, and the net unit revenue received by the seller,PS. If the tax is t dollars per unit, then PB = PS + t.

• The market price is the per-unit amount exchanged between buyer and seller.

– If the tax is levied on (i.e., collected from) sellers, then the market price is PB =PS + t.

– If the tax is levied on buyers, then the market price is PS = PB − t.

– For instance, in the past airline taxes (and fees) were levied on buyers. Whenpurchasing an airline ticket, buyers would first see the market price PS and thenupon final purchase would be charged PB = PS + t. Now, taxes are levied onairlines: buyers see and pay the market price PB and airlines receive PS = PB− t.

• In the Texas example, t = $0.20 per can. Suppose it is levied on sellers, so that themarket price is equal to PB. Thus, if buyers face the market price of PB = $1.35 percan, then the price received by sellers is PS = PB − t = $1.35− $0.20 = $1.15 per can.

1.2 Algebra

Setup: Demand depends on the buyer’s price and supply depends on the seller’s price:QD = QD(PB) and QS = QS(PS), respectively. Due to the tax, PB = PS + t. Finally, therequirement for an equilibrium is QD = QS.

1Please let me know of any typos or errors by emailing ra2747@columbia.edu.

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How to solve: We can solve the system of equations by substitution.

1. QD = QS ⇒ QD(PB) = QS(PS).

2. Substitute to get either (a) QD(PS + t) = QS(PS) or (b) QD(PB) = QD(PB − t).

3. Solve either equation (a) to get P ∗S or equation (b) to get P ∗

B.

4. Calculate the other price according to P ∗B = P ∗

S + t.

5. Check that QD(P ∗B) = QS(P ∗

S).

1.3 Example

Suppose the demand for coffee is QD = 1, 000 − 100P while the supply of coffee is QS =300P − 200.

(a) Solve for equilibrium if there are no taxes.

Solution: As there are no taxes, PB = PS = P . We set QD = QS to get 1, 000 −100P = 300P − 200 ⇒ 1, 200 = 400P ⇒ P ∗ = 3. Substituting P ∗ into QD, we getQ∗ = 1, 000 − 100 · 3 = 700. Thus, the equilibrium is Q∗ = 700 cups and P ∗ = $3 percup.

(b) Now suppose the government imposes a tax of $1 per cup. Solve for equilibrium.

Solution: We have PB = PS +1, QD = 1, 000−100PB, and QS = 300PS−200. As in thealgorithm outlined above, we set QD(PB) = QS(PS). Following part (a), we substitutefor PB to get QD(PS + 1) = QS(PS) or 1, 000 − 100(PS + 1) = 300PS − 200. Thus,900− 100PS = 300PS − 200 ⇒ 1, 100 = 400PS ⇒ P ∗

S = 2.75. By step (4), P ∗B = P ∗

S + 1⇒ P ∗

B = 3.75. Finally, QD(P ∗B) = 1, 000−100 ·3.75 = 625, and QS(P ∗

S) = 300 ·2.75−200= 825− 200 = 625. Thus, Q∗ = 625 cups, P ∗

S = $2.75 per cup and P ∗B = $3.75 per cup.

If the tax is levied on sellers, the market price is P ∗B = $3.75. If the tax is levied on

buyers, the market price is P ∗S = $2.75. Note that relative to the case with no taxes,

the price paid by buyers increases by $0.75 and the price received by sellers decreasesby $0.25.

2 Public Goods

2.1 Excludability and Rivalry

Definition: A good is rival if one person consuming a unit means no one else can. Con-sumption reduces the amount of the good.

Definition: A good is excludable if people who don’t pay for the good can be excluded fromusing it.

The concepts of rivalry and excludability generate a matrix of four types of goods.

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• Private goods are excludable and rival. Examples include food, clothing, and haircuts.They are generally supplied by private firms.

• Club goods are excludable and non-rival. An example is cable television. People canbe excluded from access to cable channels if they do not pay (excludable). However,one person watching a cable channel does not prevent another person from watchingthe channel on another TV (non-rival).

• Common resources are rival and non-excludable. An example is ocean fisheries. It isvery difficult to prevent someone from fishing in the ocean (non-excludable). However,excessive fishing depletes the fish population (rival).

• Public goods are non-excludable and non-rival. The canonical example is nationaldefense. It is impossible to exclude someone from benefiting from national defense(non-excludable). Also, one person benefiting from national defense does not preventother people from doing so (non-rival). However, there are many other examples.Public goods are often provided by governments.

Activity:

Identify whether each of the following is a private, club, common, or public good: (a)one-on-one tutoring, (b) clean air, (c) pay-per view movie, (d) city streets, (e) knowledgepool.

Solution:

(a) Private good. The tutor can exclude those who do not pay, and only one person can betutored by the tutor at a time.

(b) Common resource. It is difficult to prevent firms from polluting (non-excludable). Ifthey do, the amount of pollution that other firms can produce while maintaining cleanair is lower (rival).

(c) Club good. Viewers can be excluded if they don’t pay. However, one person watchingdoes not interfere with another person watching.

(d) Common resource. In the absence of a toll, people cannot be excluded from driving oncity streets. However, if the streets are congested, one person driving interferes withother peoples’ ability to drive.

(e) Public good. It is hard to exclude someone from access to knowledge, and consumingknowledge does not reduce the amount of it.

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2.2 Marginal Social Benefit for Non-Rival Goods

Recall that for private goods, marginal social benefit (or market demand) is calculated bysumming individual marginal benefit (or demand) curves horizontally. That is, to calculatemarginal social benefit at a given price, sum quantity demanded by each consumer for thatprice.

By contrast, for non-rivalrous goods, all (non-excluded) agents can consume the samequantity. Thus, quantity is defined at the collective or community level, not the individuallevel. Accordingly, we are unable to sum individual quantities demanded for a given price.Instead, we calculate marginal social benefit at a given quantity by summing users’ marginalwillingnesses to pay for that quantity. In other words, we sum individual marginal benefitcurves vertically. See the example below.

Activity:

A residential community has 100 residents who are concerned about security. The tablebelow summarizes the cost of hiring a 24-hour security guard as well as each individual totalbenefit.

Number ofGuards Total Cost Total Benefit to

Each Resident Marginal Cost MB to eachresident

Marginal SocialBenefit

0 $0 $0 $0 $0 $0

1 $100 $10 $100 $10 $1,000

2 $300 $16 $200 $6 $600

3 $450 $18 $150 $2 $200

4 $600 $19 $150 $1 $100

(a) Why is the security service a (local) public good for the residents?

Solution: The security service is non-excludable and non-rivalrous. Respectively: (i) if aresident does not pay, she will still benefit from the security (supposing she is not forcedto move); (ii) benefiting from the security service does not reduce the amount of securityprovided.

(b) What is the marginal cost of hiring security guards? What is the marginal benefit toeach resident? What is the marginal social benefit of hiring guards?

Solution: Answers shown in the table. The marginal cost of hiring guards is the changein total cost for each additional guard. The marginal benefit to each resident is thechange in total benefit to each resident for each additional guard. The marginal socialbenefit is the sum of the individual marginal benefits over all residents.

(c) How many security guards should the community hire?

Solution: Efficiency requires MSB = MC. Since the number of security guards hiredis discrete, there is no number for which MSB = MC exactly. Instead, we choose the

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last security guard for which MSB > MC, which in this case is 3 guards. Intuitively,we choose 3 guards because adding a 4th would increase costs by $150 but only increasesocial benefit by $100. It can be verified that 3 guards maximizes the difference betweenTotal Social Benefit and Total Cost.

(d) Do you think the community will choose the efficient number of guards?

Solution: We would expect the community to choose an inefficiently low number ofguards. We cannot observe marginal benefit for each resident. Since security is non-rivalrous, residents will gain the full benefit as long as they are not excluded. Thus, theywill have incentive to free-ride and under-report their marginal benefit, so as to pay less.Note that this assumes a voluntary contribution scheme.

2.3 Common Resources and the Tragedy of the Commons

If a good is non-excludable, agents will be able to consume it without paying. This causescommon resources to tend to be over-used. In selecting a quantity to consume, agents con-sider the marginal benefit from consumption but ignore the marginal social cost associatedwith depletion of the rivalrous good. Thus, consumers ignore the negative externality asso-ciated with their use. Examples include congestion (drivers over-use the common resourceof roads) and pollution (firms over-use the common resource of clean air).

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