Suppose you consider the savings decisions of three households - households 1, 2 and 3. Each household plans for this year€™s consumption and next year€™s consumption, and each household anticipates earning $100,000 this year and nothing next year. The real interest rate is 10%. Assume throughout that consumption is always a normal good.
A. Suppose the government does not impose any tax on interest income below $5,000 but taxes any interest income above $5,000 at 50%.
(a) On a graph with €œConsumption this period€ (c1) on the horizontal axis and €œConsumption next period€ (c2) on the vertical, illustrate the choice set faced by each of the three households.
(b) Suppose you observe that household 1 saves $25,000, household 2 saves $50,000 and household 3 saves $75,000. Illustrate indifference curves for each household that would make these rational choices.
(c) Now suppose the government changes the tax system by exempting the first $7,500 rather than the first $5,000 from taxation. Thus, under the new tax, the first $7,500 in interest income is not taxed, but any interest income above $7,500 is taxed at 50%. Given what you know about each household€™s savings decisions before the tax change, can you tell whether each of these households will now save more?
(d) Instead of the tax change in part (c), suppose the government had proposed to subsidize interest income at 100% for the first $2,500 in interest income while raising the tax on any interest income above $2,500 to 80%. (Thus, if someone earns $2,500 in interest, she would receive an additional $2,500 in cash from the government. If someone earns $3,500, on the other hand, she would receive the same $2,500 cash subsidy but would also have to pay $800 in a tax.) One of the three households is overheard saying:€œI actually don€™t care whether the old policy (i.e. the policy described in part A) or this new policy goes into effect.€ Which of the three households could have said this, and will that household save more or less (than under the old policy) if this new policy goes into effect?

Graph 8.7: Savings of 3 Households: Part II
B. Now suppose that our 3 households had tastes that can be represented by the utility function u(c1,c2) = c1α c2(1ˆ’α) , where c1 is consumption now and c2 is consumption a year from now.
(a) Suppose there were no tax on savings income. Write down the inter temporal budget constraint with the real interest rate denoted r and current income denoted I (and assume that consumer anticipate no income next period).
(b) Write down the constrained optimization problem and the accompanying Lagrange function. Then solve for c1, current consumption, as a function of α, and solve for the implied level of savings as a function of α, I and r. Do savings depend on the interest rate?
(c) Determine the α value for consumer 1 as described in part A.
(d) Now suppose the initial 50% tax described in part A is introduced. Write down the budget constraint (assuming current income I and before-tax interest rate r ) that is now relevant for consumers who end up saving more than $50,000. (Note: Don€™t write down the equation for the kinked budget€”write down the equation for the linear budget on which such a consumer would optimize.)
(e) Use this budget constraint to write down the constrained optimization problem that can be solved for the optimal choice given that households save more than $50,000. Solve for c1 and for the implied level of savings as a function of α, I and r .
(f) What value must α take for household 3 as described in part A?
(g) With the values of α that you have determined for households 1 and 3, determine the impact that the tax reform described in (c) of part A would have?
(h) What range of values cans α take for household 2 as described in part A?

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c original budget 3 neas bdet