2 answer them all

In public economics, we are often interested in transfer programs that maximize some social welfare function. For the current exercise, we will be interested in maximizing a utilitarian social welfare function, which is equivalent to maximizing the utility of a randomly chosen individual. Suppose a random process assigns individuals different characteristics that affect their ability to earn income. If these random characteristics were observable, the government could use these characteristics as a basis for lump sum transfers and achieve the first best outcome (see the discussion in the course textbook at the beginning of section 17.4). In practice, however, we can only observe rough indicators of these characteristics, such as income. The challenge is to design the best transfer program based only on these indicators, yielding what we call the "second best" solution. In this problem, we will consider a world where people choose to spend some of their time in leisure l and the remainder of their time (1 l) to earn an income Y (the maximum available leisure has been normalized to be 1). We will assume that all individuals in our society have a common utility function U(C, l) with consumption and leisure as arguments. If a person receives a wage w, income and leisure are related by Y = (1l)w. Consumption is equal to earned income Y plus any government transfers T, so that C = Y + T. Thus, the utility function can be written simply ( U(C, l) = U Y + T, 1 Y/w) The particular form of the utility function has intentionally been left unspecified, but in the following problems you should draw your plots for a well-behaved utility function that features a believable trade-off between consumption and leisure. It is sufficient to solve everything drawing graphs onlyno need for a mathematical representation of the utility function. 1. Plotting the utility function for a single individual.

(a) Suppose there are no government transfers (T = 0). Plot utility vs. income for a given individual (i.e. U on the y-axis, and Y on the x-axis). Discuss how the shape of the utility curve relates to decreasing returns and to the resulting trade-off between consumption and leisure. Mark the individual's optimal choice of earned income.

(b) Now suppose the government gives the individual in part (a) a transfer of size T. On the same graph as above, draw the individual's new utility curve under the transfer. Label the individual's new optimal choice of earned income, and explain why this does or does not differ from the optimal choice of earned income in (a).

(c) Finally, suppose the government levies a tax T on the individual in part (a). On the same graph as above, draw the individual's new utility curve under the tax.

2. Suppose we have two individuals A and B. Nature has given individuals A and B wages wA > wB, respectively. Plot the utility curves for A and B on the same graph (with taxes and transfers set to zero). Who chooses to earn more income? Why?

less than a specified limit Y , and taxes individuals a lump sum amount T if they earn more than Y . Suppose the two individuals A and B from part 2 above comprise our society. The government uses the transfer scheme just outlined, in the hope that A pays the tax while B receives the transfer. Of course, there is the concern that A may masquerade as a low ability individual by intentionally reducing his pre-tax income to qualify for the transfer. The government realizes that for a given cutoff Y , the transfer T must be limited so that A is not an impostor (i.e. does not choose to earn less than Y ). (a) Classify this program as Categorical vs. Means-tested, and Cash vs. In-kind.

(b) For a given tax/transfer T and cutoff Y , write out the condition that must be satisfied in order for A not to be an impostor.

(c) Since the government's goal is to transfer income to B, their intuition is that B's earnings should not be restricted. That is, if YB is the level of earned income chosen by B if he were given a transfer T, the income cutoff is set at Y = YB . Plot B's utility function when Y = YB , and mark B's choice of earned income.

(d) Continuing from 3(c), suppose the government has chosen the largest possible transfer T so that A is not willing to be an impostor, but that the constraint in part 3(b) binds. On the same graph from 3(c), plot two utility curves for A: the first utility curve in the case that A decides to earn more than Y , and the second utility curve in the case that A decides to be an impostor. Make sure that the two utility curves for A correspond to the fact that the constraint in 3(b) binds

(e) Using the graph from 3(d), decide whether it would be welfare-improving to restrict B's income. That is, can social welfare be increased by lowering Y slightly to Y ? Describe your reasoning carefully. Does your conclusion match the government's intuition expressed in 3(c)?