Exercise 9. Individual Retirement Account (Intertemporal Choice Application)

Today you work, and tomorrow you retire. Today you are endowed with income ₁=100, tomorrow you don't work so that ₂=0. Your intertemporal utility is u=x₁²x₂, where x₁ is consumption today, and x₂ is consumption tomorrow. What you don't consume today, is saved to be consumption tomorrow.

a)Your budget constraint is: p₁₁ + p₂₂ = p₁x₁ + p₂x₂. Price today is normalized to p₁=1, and assuming no inflation, p₂=1 as well. For simplicity you accrue no interest rate on your savings. What portion of income do you consume today (x₁/₁)? How much do you save and consume tomorrow (x₂)? Show graphically the budget constraint and optimal indifference curve.

b)A more realistic model includes income tax brackets. Assume your average tax rate is t₁=20% on the income you made today, so that your budget constraint is (1-t₁)₁ = x₁ + x₂. What portion of disposable income do you consume today (x₁/((1-t₁)₁))? How much do you save and consume tomorrow (x₂)? Show graphically the budget constraint and optimal indifference curve.

c)Legislation in the 1970s introduced 401(k) and IRA plans. Savings placed in these types of retirement accounts allow for a tax break on deferred income. Retirement income is usually lower and falls under a lower tax bracket. Assume that for income below 50 an average tax rate t₂=10% applies. The budget in part b was: is ₁ = x₁/(1-t₁) + x₂/(1-t₁). For x₂<50 the new budget would be ₁ = x₁/(1-t₁) + x₂/(1-t₂). How do answers in part b change? Show graphically.

d)Aside for the advantage for individual savers, do you think there is also a long-term macroeconomic advantage?