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1) Assume a country consists of identical workers

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1) Assume a country consists of identical workers. Each worker earns a wage W when working and faces a probability a of losing the job. If the worker loses the job, earnings drop to zero (W = 0). However, the worker always has some non-labour income of 10 (even when working). The worker’s utility is U = log(C), where C is consumption. Consumption is equal to a worker’s total income.

(a) What is the expected utility of each worker? [Hint: you cannot get a numerical solution; instead write the mathematical expression for the expected utility as a function of W, a and non-labour income.]

Assume the government implements an employment insurance program. Under this programs, individuals pay a lump-sum tax t when they are employed, and get benefits B while they are unemployed. The system must break even at a point in time, i.e. benefits paid to unemployed workers must equal taxes collected from employed workers.

(b) What is the optimal employment insurance program, i.e. the program that, subject to the balanced-budget constraint, maximizes worker utility? Present both the tax rate and the benefit level for this program. (Hint: again, you cannot get a numerical solution; instead express the optimal t and B as functions of W and a.]

(c) Explain the intution behind your result in (b) (1-2 sentences). [1 mark]

Now assume that each worker who loses the job gets an amount kW from a friend to help out, where k is some constant such that 0 < k < (1 - a).

(d) What is the expected utility now if there is no employment insurance program? [Hint: you cannot get a numerical solution; instead write the mathematical expression for the expected utility as a function of W, a, k and non-labour income.] [1 mark]

(e) Now, reintroduce government unemployment insurance, which once again must break even. What is the optimal unemployment insurance system now (both optimal tax rate and benefit level)?

(a) How does this compare to your answer to (b)? Explain the intuition (1-2 sentences).

3. Consider a model in which individuals live only two periods. Individuala maximize the following utility function

  U = 3/5 In(C₁) + 2/5 In(C₂)

Where C₁ is the consumption in period 1 and C₂ is the consumption in period 2. In each period there are N young individuals and N old individuals. An individual receives an income of $300 in period 1 and no income in period 2. The market interest rate is 5 percent, and the person can borrow or lend money at this rate.

(a) Use a diagram of indifference curves and an intertemporal budget constraint to illustrate the individual’s optimal consumption in each period, if there is no public pension system. Make sure you label the axes and provide the numerical value of the slope of the budget constraint.

(b) Write down the individual’s lifetime budget constraint and solve to find C₁, C₂ and S, which is personal savings.

(c) Now assume that the government introduces a public pension program. The government takes $50 from each individual in period 1 and gives it directly to individuals in period 2.

i. What is the terminology for this public pension program?

ii. How much does an individual privately save now?

iil. How does an individual’s utility with the public pension system compare to an individual’s utility without the public pension system?

iv. Explain the intuition behind your findings in iii.