This problem is set in a fanciful location, but it deals with a very practical issue that concerns residents of this earth. The question is, “In a Democracy, when can we expect that a majority of citizens will favor having the government supply pure private goods publicly?” This problem also deals with the efficiency issues raised by public provision of private goods. We leave it to you to see whether you can think of important examples of publicly supplied private goods in modern Western economies.
On the planet Jumpo there are two goods, aerobics lessons and bread. The citizens all have Cobb-Douglas utility functions of the form Ui(Ai,Bi) = A1/2i B1/2i, where Ai and Bi are i’s consumptions of aerobics lessons and bread. Although tastes are all the same, there are two different income groups, the rich and the poor. Each rich creature on Jumpo has an income of 100 fondas and every poor creature has an income of 50 fondas (the currency unit on Jumpo). There are two million poor creatures and one million rich creatures on Jumpo. Bread is sold in the usual way, but aerobics lessons are provided by the state despite the fact that they are private goods. The state gives the same amount of aerobics lessons to every creature on Jumpo. The price of bread is 1 fonda per loaf. The cost to the state of aerobics lessons is 2 fondas per lesson. This cost of the state-provided lessons is pAid for by taxes collected from the citizens of Jumpo. The government has no other expenses than providing aerobics lessons and collects no more or less taxes than the amount needed to pay for them. Jumpo is a democracy, and the amount of aerobics to be supplied will be determined by majority vote.
(a) Suppose that the cost of the aerobics lessons provided by the state is paid for by making every creature on Jumpo pay an equal amount of taxes. On planets, such as Jumpo, where every creature has exactly one head, such a tax is known as a “head tax.” If every citizen of Jumpo gets 20 lessons, how much will be total government expenditures on lessons? ___________ How much taxes will every citizen have to pay? ___________ If 20 lessons are given, how much will a rich creature have left to spend on bread after it has paid its taxes? ___________ How much will a poor creature have left to spend on bread after it has paid its taxes? ____________
(b) More generally, when everybody pays the same amount of taxes, if x lessons are provided by the government to each creature, the total cost to the government is ___________ times x and the taxes that one creature has to pay is ___________ times x.
(c) Since aerobics lessons are going to be publicly provided with everybody getting the same amount and nobody able to get more lessons from another source, each creature faces a choice problem that is formally the same as that faced by a consumer, i, who is trying to maximize a Cobb- Douglas utility function subject to the budget constraint 2A + B = I, where I is its income. Explain why this is the case.
(d) Suppose that the aerobics lessons are paid for by a head tax and all lessons are provided by the government in equal amounts to everyone.
How many lessons would the rich people prefer to have supplied? _____________
How many would the poor people prefer to have supplied? ___________
e) If the outcome is determined by majority rule, how many aerobics lessons will be provided? ______________ How much bread will the rich get? _________ How much bread will the poor get? __________
(f) Suppose that aerobics lessons are “privatized,” so that no lessons are supplied publicly and no taxes are collected. Every creature is allowed to buy as many lessons as it likes and as much bread as it likes. Suppose that the price of bread stays at 1 fonda per unit and the price of lessons stays at 2 fondas per unit. How many aerobics lessons will the rich get? _______ How many will the poor get? ___________ How much bread will the rich get? ___________ How much bread will the poor get? _____________
(g) Suppose that aerobics lessons remain publicly supplied but are paid for by a proportional income tax. The tax rate is set so that tax revenue pays for the lessons. If A aerobics lessons are offered to each creature on Jumpo, the tax bill for a rich person will be 3A fondas and the tax bill for a poor person will be 1.5A fondas. If A lessons are given to each creature, show that total tax revenue collected will be the total cost of A lessons.
(h) With the proportional income tax scheme discussed above, what budget constraint would a rich person consider in deciding how many aerobics lessons to vote for? _____________ What is the relevant budget constraint for a poor creature? _____________ With these tax rates, how many aerobics lessons per creature would the rich favor? _______________ What quantity of aerobics lessons per capita would be chosen under majority rule? ________ How much bread would the rich get? __________ How much bread would the poor get? __________
(i) Calculate the utility of a rich creature under a head tax. ____________ Under privatization. ___________ Under a proportional income tax.___________ Now calculate the utility of each poor creature under the head tax. ___________ .
(j) Is privatization Pareto superior to the head tax? _________ Is a proportional income tax Pareto superior to the head tax? ___________ Is privatization Pareto superior to the proportional income tax? ____________ Is privatization Pareto superior to the proportional income tax? __________