First and Second-Best Rawlsian Income Redistribution: Most governments raise tax revenues from higher income individuals and distribute them to lower income individuals in an attempt to achieve a more equal distribution of consumption. Such governments face a trade-off that emerges from the competing goals of achieving greater consumption equality while minimizing deadweight losses from taxation.
A: Consider in this exercise a government with Rawlsian goals — i.e. the goal of making the least well off individual the most well off. If the government does not have access to information about people’s utilities, it may choose instead to treat people’s consumption levels as if these represented utility values. Thus, instead of social indifference curves over utility allocations, the government would use social indifference curves over consumption allocations.
(a) Suppose individual 1 has income I1 and individual 2 has income I2, with I1 > I2. Draw the “consumption possibilities frontier” assuming that the government can costlessly redistribute income between the individuals. Indicate on your graph the consumption allocation that exists in the absence of government redistribution (and in the absence of any voluntary charitable efforts).
(b) What consumption allocation on this possibilities frontier would a Rawlsian government choose?
(c) Now illustrate how the consumption possibilities frontier changes if the government uses an inefficient tax system. Suppose the inefficiency takes the following form: As the tax rate on the rich increases, consumption of the rich decreases as if the tax system were efficient—but the deadweight loss increases at an increasing rate as the tax rate rises, with this loss reducing the amount available for distribution to the poor.
(d) Illustrate how a Rawlsian government might now not choose to equalize consumption levels between the rich and the poor.
(e) Suppose that income differences arise in part from “compensating differentials” in the labor market; i.e. suppose that higher income people make more money in part because they undertake unpleasant activities such as traveling a lot for their job and working long hours. If the government’s real goal is to apply its social welfare function to utility allocations instead of consumption allocations, how might the Rawlsian social welfare function applied to consumption allocations lead to excessive redistribution?
(f) Suppose instead that the marginal utility of consumption diminishes as consumption increases. Would the application of the Rawlsian goal to consumption distributions now lead to a tax rate that is too high or too low?
B: Suppose again that there are two individuals—one who makes an income I1 and the other who makes only I2, where I1 > I2. Assume that the government would like to redistribute income but does not have information on individual utilities. Thus, instead of applying a social welfare function to choose a utility allocation, the government applies a social welfare function to choose consumption allocations directly. The function it uses is the Rawlsian social welfare function W = min {c1, c2}.
(a) Give an example of a utility function u(c) that would make this equivalent to a social welfare function that chooses utility allocations. What has to be true about the marginal utility of consumption?
(c) Suppose next that the income tax used by the government is not a lump sum tax. For a tax rate t < 1, it is able to transfer the amount (tI1 − (ktI1)2) to individual 2. If the government wants to maximize the amount of transfer it can make, what tax rate will it choose?
(d) Suppose I1 = $200,000 and I2 = $10,000. What is the government’s “first best” income tax rate when it can tax individual 1’s income without any deadweight loss?
(e) Consider next the second-best case and suppose k = 0.0025. For the same two income levels as in (d), what is the government’s “second-best” income tax rate given that the tax system gives rise to the deadweight losses modeled in (c)?
(f) How much consumption do each of the two individuals undertake under the first-best outcome? How about under the second best outcome?