Microeconomics Problems and Solutions

The population of country consists of workers and pensioners. The population share of workers is and the population share of pensioners is  (normalize the mass of the population to 1). There is only one consumption good, which is non-storable. Only workers can produce the consumption good, whereas pensioners depend on transfers from workers. Transfers are administered by the central government which can impose a tax rate of t 2 [0, 1] on each worker’s production output. All tax collections are evenly distributed among pensioners. Each person is endowed with one unit of time. If a worker spends an amount of time on working, his output is given by 2` and consumption by (1t)2`. Individual preferences are represented by utility function

where is the amount of consumption and is the amount of leisure. (a) In the space of consumption (c) and leisure (1`), draw the map of a worker’s indifference curves and the budget lines for tax rates, and 0.4. For each tax rate, locate the worker’s optimal choice of consumption and leisure. (b) Derive (i) workers’ optimal supply of labour at a given tax rate t and (ii) workers’ and pensioners’ resultant individual utility levels uw(t) and up(t), respectively. Explain how ntax rate t affects the supply of lab (c) The government’s preferences are represented by welfare function W where V ar(u) is the variance of the utility levels in the population. Plot the graph of welfare against tax rate given the optimal supply of labour.

There is a population of consumers, the mass of which is normalized to 1. 20% of consumers have an income of  consumers have an income of m = 50. There are two available consumption goods, 1 and 2, and a consumer’s preferences are given by where is an amount consumed of good.

(a) The government introduces a proportional sales tax t on the consumption of good 1 (i.e., the after-tax price of good 1 becomes p1(1 + t)). Find the value of t that enables the government to raise R = 2 in tax revenue.

(b) To support low-income consumers, the government introduces an income subsidy s for consumers with income m = 10. Find the subsidy s that fully compensates the lowincome consumers for the utility losses of the tax t from part (a). Solve for s in two different ways by applying (i) the indirect utility function and (ii) the Hicksian demand function.

(c) Redo (a) and (b) for the case when, instead of good 1, the government taxes the consumption of good 2 with a proportional sales tax t. Discuss welfare implications.

3. There is a population of students, the mass of which is normalized to 1. The cost of education is I = 1. A student’s return from education is his or her earnings y which are not known before the student starts education. It is known that 50% of graduates earn y1 = 4 and 50% earn y2 = 8. The government offers student loans of I = 1. The government can observe earnings and, therefore, can condition student loan repayments {R1, R2} on the student’s earnings y after graduation, where Ri  0 is the repayment when Student preferences are given by utility function U(x) = ln x where x is net (after repayment) income.

(a) In the space of repayment contracts, draw students’ indifference curves and the set of contracts that balance the government’s budget.

(b) Find graphically and analytically the optimal student loan repayment contract that maximizes students’ expected utility and balances the government’s budget.

(c) Now suppose that there are bad and good students and both types are equally likely in the population. Students know their type. A bad student’s return from education is fixed at y1 = 4, but a good student’s return depends on her learning effort. If she exerts no effort, then her return is y1 = 4, but if she exerts effort, then her return is y2 = 8. Exerting effort comes at a utility cost of 0.5, thus, yielding a (net) utility of ln(y2 R2)0.5. Assume that if utility from exerting effort and from exerting no effort is the same, the student chooses to exert effort. Find the optimal student loan contract {R1, R2} that maximizes students’ expected utility, balances the government’s budget, and induces good students to exert effort (i.e., a good student’s utility from exerting effort must at least be equal to her utility from exerting no effort)