Macroeconomics
1. Let r = 0 and β = 1. Households live for two periods and have utility log(c1) + log(c2) and pre-tax income $3000 each period. The government needs to finance a project in the first period that costs $300 each household.
Policy 1: tax $300 in the first period.
Policy 2: issue debt in the first period and pay back in the second period.
Let T1 and T2 be the lump-sum taxes in period 1 and period 2.
(a) Write down the consolidated life time budget constraint for the households using T1 and T2
(b) Write down the household’s maximization problem. (3 pts)
(c) What are T1 and T2 under policy 1? (3 pts)
(d) What are T1 and T2 under policy 2? (3 pts)
(e) What are c∗
1 and c∗
2 under policy 1? (3 pts) (Hint: c∗
1 = I
1+β and c∗
2 = β(1+r)I
1+β )
(f) What are c∗ 1 and c∗
2 under policy 2?
(g) Does Richardian Equivalence hold?
(h) What are the savings s under policy 1?
(i) What are the savings s under policy 2?
(j) If the household can not borrow at all, does the solution under policy 1 change? If yes, what are the new c∗ 1 and c∗ 2?
(k) If the household can not borrow at all, does the solution under policy 2 change? If
yes, what are the new c∗ 1 and c∗ 2?
(l) If the household can not borrow at all, Does Richardian Equivalence hold?
2. Let T (y) be a tax function, T ′(y) be the marginal tax function and t(y) be the average tax function. Prove that T (y) is Regressive if and only if T ′(y) < t(y).
3. Hardy is single. His salary income is $96,000 and interest income is $4,000. He puts $10,000 in his IRA accounts. He choose to take the standard deduction.Answer the following questions use the 2018 tax table provided.
(a) What is his gross income
(b) What is his AGI?
(c) What is his taxable income?
(d) What is Hardy’s federal income tax amount?
(e) If the tax withholdings in his paycheck is $16,000, does he qualify for a refund? If
yes, how much? If not, how much tax does he own the IRS?
4. In 2022, the income threshold for social security tax is $147,000. The social security tax rate is 6.2%. Hardy makes 180, 000 in 2022, how much social security tax will he pay in 2022?
5. There are a continuum of households, who work in the first period to earn income y and receive social security benefit b in the second period. The period utility function is log(c). The discount factor β = 1. Households die at the end of the first period with probability p. The size of the population is normalized to one in the first period. The population grows at rate n and income grows at rate g. The social security system is PAYGO.
Assume there is only social security tax in the economy and the tax rate is τ. Government balances the social security budget. Let s be households’ savings in period
1 and let r be the interest rate. Please answer the following questions:
(a) Write down households’ maximization problem (6 pts)
(b) Write down the government budget constraint (3 pts)
(c) Derive the consolidated budget constraint of the household. The resulting budget onstraint should NOT contain any unknowns.
2
(d) Let I(τ) be the present value of the household’s life-time income, solve c∗
1, c∗
2, and
s∗. (9 pts)
(e) Is s decreasing or increasing in τ?
(f) What is the implicit return of social security?
(g) Derive the condition such that I(τ) is increasing in τ.
(h) What is the interpretation of the above condition?
6. Answer the following:
(a) What is the difference in hours worker per person and hours worked per worker?
(b) Why unemployment rate can only partially reflect the labor market conditions?
