Problem Set 2: Welfare, externalities and public goods

1. Suppose there are two voters and three candidates, x, y and z. Suppose that voter 1 ranks the

candidates x first, z second then y third. Suppose that voter 2 ranks the candidates y first, x second

and z third.

What is the total Borda count for each candidate? Which wins?

Suppose that candidate z once called a policeman a “pleb”. Voter 1 considers himself a man of the

people, and so is appalled by this comment. This leads him to change his ranking to x first, y second

and z third. Voter 2 is impressed as he hates everyone who wasn’t a member of the Bullingdon club.

He changes his ranking to y first, z second and x third.

Explain how this changes the Borda count outcome.

Can you see the connection between this problem and the Arrow Impossibility Theorem? Explain.

How reasonable are the ‘reasonable’ criteria that Arrow believes voting systems should satisfy?

2. Suppose the utility possibility frontier for two individuals is 𝑈𝐴 + 2𝑈𝐵 = 200. Graph this.

a) How would society maximise a ‘Nietzschean social welfare function’ 𝑊(𝑈𝐴, 𝑈𝐵) = max{𝑈𝐴, 𝑈𝐵}

given this utility possibility frontier? What would be the resulting values of utility for individual A and

B?

b) How would this change if we instead used a ‘Rawlsian social welfare function’ 𝑊(𝑈𝐴, 𝑈𝐵) =

min{𝑈𝐴, 𝑈𝐵}?

c) Suppose social welfare is instead given by 𝑊(𝑈𝐴, 𝑈𝐵) = 𝑈𝐴

1

2𝑈𝐵

1

2 , what is the social optimum in this

case?

Think about why this makes sense given the Cobb-Douglas formulation. (HINT: what do we know

about budget shares in this case?).

d) Show the three possible social indifference curves on the diagram.

3. A number of firms located in the western portion of a town after single-family residences took up

the eastern portion. Each firm produces the same product and, in the process, emits noxious fumes

that adversely affect the residents of the community. These noxious fumes are what we call negative

externalities (they are created by firms and enter the utility function of residents, although the

residents have no control over the quantity of the fumes). We can assume that the fumes decrease

the utility of the residents and lower property values.

a) From the social point of view, would it make sense to reduce to zero the negative externalities?

b) How might the community determine the efficient level of air quality?

4. In the (competitive) market for dry cleaning, the inverse demand function is given by 𝑃 = 100 − 𝑄

and the (private) marginal cost of production of each dry cleaning firm is given by 𝑐 = 10 + 𝑄. The

pollution generated by the dry cleaning process creates external damages given by the (negative)

externality 𝐸 = 𝑄.

a) Calculate the output and price of dry cleaning if it is produced under competitive conditions.

b) Calculate the socially efficient price and output of dry cleaning.

5. A beekeeper lives adjacent to an apple orchard. The orchard owner benefits from the bees because

each hive pollinates about one acre of apple trees. The orchard owner pays nothing for this service,

however, because the bees come to the orchard without his having to do anything. Because there are

not enough bees to pollinate the entire orchard, the orchard owner must complete the pollination by

artificial means, at a cost of £10 per acre of trees.

Beekeeping has a marginal cost of 10 + 2𝑄, where 𝑄 is the number of beehives. Each hive yields £20

worth of honey.

a) How many beehives will the beekeeper maintain?

b) Is this the socially efficient number of hives?

6. Bob and Ray are thinking about buying a sofa. Bob’s utility function is 𝑈𝐵(𝑆,𝑀𝐵) = (1 + 𝑆)𝑀𝐵 and

Ray’s utility function is 𝑈𝑅(𝑆,𝑀𝑅) = (4 + 𝑆)𝑀𝑅 where 𝑆 = 0 if they do not get the sofa and 𝑆 = 1 if

they do. 𝑀𝑖 is the amount of money individual 𝑖 has to spend on other stuff. Bob’s income is £800 and

Ray’s income is £2,000.

a) What is the most they could each spend on the sofa and still better off than without it?

b) A sofa is £600, explain why it may nonetheless be difficult for Bob and Ray to agree to buy it.

7. Lucy and Melvin share an apartment. They spend some of their money on private goods and some

on public goods like heating and furnishings. Lucy’s utility function is 𝑈𝐿 = 2√𝑋𝐿 + √𝐺 and Melvin’s

utility is 𝑈𝑀 = √𝑋𝑀𝐺, where 𝐺 is expenditure on public goods, and 𝑋𝐿 and 𝑋𝑀 are Lucy’s and Melvin’s

respective expenditures on private consumption. Lucy and Melvin’s combined income is £5,000.

a) What maximisation problem must we solve in order to find the Pareto efficient allocation, and why?

b) Write down a Lagrangian for this maximization problem.

c) Without performing any calculations, relate this problem to that faced by a social planner with social

welfare function given by: 𝑊 = 𝑈𝐿 + 𝜆𝑈𝑀.

d) Solve this maximisation problem to obtain a condition of the form 𝐴 + 𝐵 = 1 where 𝐴 and 𝐵 may

depend on 𝑋𝐿, 𝑋𝑀 and 𝐺, but do not depend on the Lagrange multipliers.

e) Interpret the condition you derived in d) in terms of marginal rates of substitution, and the marginal

cost of government spending.

f) Suppose that Lucy and Melvin each spend £1000 on public goods, but Lucy spends £2000 on private

goods for herself, whereas Melvin spends only £1000 on himself. Is this Pareto optimal?

8. Suppose people’s preferences are as given in the table below:

Voter Option A Option B Option C

1 70 40 20 2 10 80 30 3 40 20 80

Total 120 140 130

a) Under the VCG mechanism, which option will be chosen?

b) Which voter(s) are pivotal?

c) What fees would each agent have to pay under the VCG mechanism?