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.

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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 pr