4. This problem deals with a very practical

question that concerns the inhabitants of this

planet. The question is "in a democracy, when

can we expect the majority of citizens to favor

a government that provides private goods in a

public way?". This problem also addresses

issues of efficiency arising from the public

provision of private goods.

On planet Jumpo, there are two goods,

aerobics lessons and bread. The citizens all

have a Cobb-Douglas utility function of the

form Ui(Ai, Pi) = A₁ P₁" where Ai and Pi

are citizen i's consumption of aerobics and

bread.

Although tastes are all the same, there are two

different income groups, the rich and the poor.

Every rich person on Jumpo has an income of

100 fondas and every poor person has an

income of 50 fondas (fonda is the currency of

planet Jumpo). There are two million poor

people and one million rich. Bread is sold in the

usual way and costs 1 fonda. Aerobics lessons

are provided by the state, in identical

quantities for each person, and the price to the

state for aerobics lessons is 2 fondas per/nlesson. The cost of state-provided lessons is

paid for by taxes collected from citizens. The

state has no other expenses, so the sum of the

taxes must equal the total cost of the aerobics

lessons. Jumpo is a democracy, and the

number of aerobics lessons to be provided is

decided by a vote of the citizens.

a. Assume that the cost of state-provided

aerobics lessons is paid for by requiring each

person to pay an equal amount of taxes (per

capita taxation). If each citizen receives 20

lessons, what will be the government's total

expenditure on lessons? How many taxes will

each citizen have to pay? If 20 lessons are

given, how much will a rich person have left to

eat bread after paying the tax? What about a

poor person?

b. Since aerobics lessons are provided

publicly, everyone receives the same amount,

and no one can have more lessons for that

matter, each person faces the same

optimization problem. Write down this

optimization program and explain it.

c. How many lessons will the rich want the

state to provide? How many lessons will the

poor want the state to provide? (Still assuming

per capita taxation and identical quantities for

each individual).

d. If the result is determined by a majority vote,

how many aerobics lessons will be provided?/nhow many aerobics lessons will be provided?

How many loaves of bread will the rich get?

How many loaves of bread will the poor get?

e. Assume that aerobics lessons are

"privatized" in such a way that no lessons are

provided publicly and no taxes are collected.

Each person can buy as many lessons as they

like and as many loaves of bread as they like.

Assume that the price of the bread remains 1

fonda per unit and the price of the lesson

remains 2 fondas per unit. How many aerobics

lessons will the rich receive? And the poor?

How many loaves of bread will the rich buy?

And the poor?

f. Suppose that aerobics lessons remain

publicly available, but are paid for by a tax

proportional to income. Suppose that if A

aerobics lessons are offered to every person in

Jumpo, the tax for the rich will be 3A fondas

and the tax for the poor will be 1.5A fondas.

With these tax rates, how many aerobics

lessons will the rich get? And the poor? How

many aerobics lessons per head will the

majority vote for? How many loaves of bread

will the rich get? (Hint: remember to rewrite

each group's budget constraint)./ng. Calculate the utility of a rich person and a

poor person

i. If we apply a per capita tax

ii. In case of privatization

iii. If a tax proportional to income is applied

h. Compare these three systems according to

the Pareto criterion. Is privatization

Pareto superior to the per capita tax? Is the tax

proportional to income superior to the per

capita tax in the Pareto sense? Is privatization

superior in a Pareto sense to a tax proportional

to income? Explain your answers.