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Problem Set 6 Solution
ECN 131
Public Finance Prof. Farshid Mojaver
I.Taxation and Savings
2. Consider a model where individuals live for two periods and have utility functions of the form: U = log (C1) + log (C2). They earn income of $100 in the first period. They save S to finance consumption in the second period. The interest rate, r, is 10%.
a)Set up the individual’s lifetime utility maximization problem. Solve for the optimal C1, C2, and S. [Hint: rewrite C2 in terms of income, C1, and r.] Draw a graph showing the opportunity set.
b)The government imposes a 20% tax on labor income. Solve for the new optimal levels of C1, C2, and S. Explain any differences between the new level of savings and the level in part a, paying attention to any income and substitution effects.
c)Instead of the labor income tax, the government imposes a 20% tax on interest income. Solve for the new optimal levels of C1, C2, and S. [Hint: what is the new after-tax interest rate?] Explain any differences between the new level of savings and the level in part a, paying attention to any income and substitution effects.
d)Suppose now that the government decides it wants to encourage savings for retirement and decides to subsidize savings instead of taxing it. Specifically, the government now offers a one-to-one match of interest income (for every dollar of interest income that you earn, the government gives you another dollar). Based on your answer to c, how do you think this will affect the level of savings? Why? You need not solve for the new level of S—just explain your answer intuitively.
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a) Rewrite C2 = (1+r)*S = (1.1)*(100-C1)
Max log C1 + log (1.1*(100-C1))
FOC: 1/C1 – 1.1/(1.1*(100-C1)) = 0, so C1=100-C1
C1=50, S=50, C2=55
b) same as a, except income = 100*(1-t) = 100*(1-.2) = 80
Max log C1 + log (1.1*(80-C1))
FOC: 1/C1 – 1.1/(1.1*(80-C1)) = 0, so C1=80-C1
C1=40, S=40, C2=44
Income effect: you are poorer, so you consume less in both periods; you still divide income evenly between C1 and S as in part a.
c) same as a, except after-tax interest rate = 1+r*(1-t) = 1+.10*(1-.2) = 1.08
Max log C1 + log (1.08*(100-C1))
FOC: 1/C1 – 1.08/(1.08*(100-C1)) = 0, so C1=100-C1
C1=50, S=50, C2=54
Same as a! Why?
Income effect: drop in interest rate has made you poorer; must save more to have same amount of consumption next year, so S increases.
Substitution effect: consumption in period 2 has become relatively more expensive, so shift towards more C1 and less C2; S decreases.
Net effect is ambiguous; with log utility, two effects exactly offset, so net effect on S is zero.
d) Part c showed that a decrease in r does not affect the optimal S.
The same is true if government increases r by subsidizing savings.
So S=50, as in part a.
Again there are income and substitution effects which offset exactly.
Income effect: rise in interest rate has made you richer; must save less to have same amount of consumption next year, so S decreases.
Substitution effect: consumption in period 2 has become relatively less expensive, so shift towards less C1 and more C2; S increases.
3. The government introduces a tax incentive program in which the first $5,000 of savings can be tax-deferred. Discuss the effects on saving.
In this budget constraint, the first $5,000 saved gains more future-period consumption than does other savings, so the slope of that part of the budget constraint is steeper. Any tax that would have been paid on that amount can be invested by the taxpayer, increasing future consumption. Savings in excess of $5,000 is taxed immediately, reducing future consumption. In the graph below, Y denotes current income. Tax1 denotes tax assessed on the undeferred income; Tax2 is the tax paid on the amount deferred. The bolded kinked line illustrates the tax-deferred budget constraint.
4. Gale and Scholz (1994) estimate that increasing the contribution limits for Individual
Retirement Accounts would have little effect on the overall rates of savings. Why do you think this might be the case?
Overall savings rates only increase when people take money from current consumption and invest it in savings. When people merely switch from non-IRA savings to IRA savings, the total savings rate is unaffected. One reason for Gale and Scholz’s finding may be that the current IRA limits are so high that most people could not afford to reduce their current consumption further to invest more in retirement savings. Perhaps the only people who could or would increase IRA savings are the wealthy, who already have substantial savings that could be moved to this less-liquid, but tax-favored, form of savings.
5. Suppose that the government proposes to eliminate unemployment insurance. Which model - the intertemporal choice model of saving or the precautionary saving model - would predict that this policy change would affect savings the most? Explain your answer and be sure to note any empirical evidence that might be relevant.
Ans) The precautionary saving model would predict larger effects of this policy change since that model accounts for the fact that saving is motivated in part by a person's desire to self-insure against risk as well as to smooth consumption over time. The government's proposal in this case would increase the risks people face; people would thus increase saving to compensate for the increased risk. Empirical evidence is consistent with the precautionary saving model in this regard; several studies have found that as social insurance programs are expanded (reducing risks), savings are reduced.
6. Suppose that you are going to save $1,000 of your income for one year, after which you will spend it along with any accumulated interest you earned. Assume that your marginal income tax rate is 50%. Consider the following two options:
Option 1: Invest in a regular savings account earning 10% interest.
Option 2: Invest in an IRA earning 10% interest.
Determine the after-tax value of your savings a year from now under both options. If the amounts are the same, explain why they're the same. If the amounts are different, explain why they're different.
Ans) Under Option 1, you pay taxes both on your contribution of $1,000 and on the interest earned on your savings. Consequently, you invest $500. It earns $50 in interest, of which you pay $25 in taxes. As a result, one year from now you will have $525.
Under Option 2, you do not pay taxes initially on your contribution of $1,000. Therefore, you earn $100 in interest. You pay 50% of the withdrawal in taxes, equal to $550. One year from now you will have $550.
The amounts are different for two reasons. The first is that in the IRA, you earn interest on income that would have been taxed away if you had invested in a regular savings account. Second, in the IRA you don't have to pay taxes on the interest earned, whereas you do have to pay such taxes if you invest in a regular savings account.
7. Suppose that Lilistan has two types of citizens: low-income citizens (income = $20,000) and high-income citizens (income = $80,000). Interest income is currently taxed and each type of citizen saves 10% of his or her income for retirement. The government is proposing to allow citizens to put money into an Individual Retirement Account where the contributions would be tax-deductible and interest would accumulate tax-free. Withdrawals would then be taxed as income. The government currently wants to set a $5,000 annual limit on the accounts. Discuss the effect of this policy on the saving choices of both the low-income and high-income citizens in terms of both income and substitution effects as well as the overall effect.
Ans) Currently, the low-income citizens are saving $2,000. Under the new policy, they would reshuffle and save in the IRAs. Since they would not yet hit the $5,000, a substitution effect would induce them to save more (a marginal effect) since the opportunity cost of consuming today is higher than it was without the tax subsidy. However, the policy also would create an income effect since people would have to save less to achieve a certain level of consumption in the future. This income effect would induce them to save less. Consequently, the net effect of this policy on low-income citizens would be ambiguous.
The high-income citizens are currently saving $8,000. Under the new policy, they would reshuffle $5,000 of this into IRAs. Since they meet the annual contribution limit, the creation of IRAs would not have any substitution effect on them. However, by moving $5,000 to an IRA, a high-income person would accumulate more wealth over time due to the tax subsidy and therefore would have to save less to achieve a given level of consumption in retirement. This income effect would induce high-income citizens to save less. Consequently, the effect of establishing the IRA would unambiguously reduce the savings of high-income citizens.
II. Taxes on Risk Taking and Wealth:Part A
1-What are the three ways in which capital gains are subsidized over interest-bearing savings accounts in the U.S. tax code?
Ans) The first is that while assets that earn interest are taxed on accrual, capital gains are taxed on realization. By paying taxes on sale of the asset rather than as value is accrued, you can earn interest on what you would have paid in taxes had you invested in an asset that was taxed on accrual.
The second is that the capital gains tax burden on an asset sold after the purchaser dies is based on the sales price minus the price of the asset at the time of death, not the purchase price.
The third is that capital gains on houses have typically been excluded from taxation. Consequently, there is a tax incentive to invest assets in one's home rather than in another form of asset on which the capital gains (or interest) would be taxed.
2-Suppose that a politician argues that capital gains should be taxed at a lower rate than other forms of income in order to protect investors against inflation. What do you think of this argument? Can you think of any inconsistencies in it? Is there an alternative approach to deal with the inflation problem? Explain.
Ans) The politician is referring to the fact that if inflation is 10% and the value of an asset (e.g., a share of stock) increases by 10%, then the investor pays capital gains taxes on the 10% despite the fact that the stock is worth no more in real terms. However, the same is true for other types of savings, so this argument does not justify taxing capital gains at a lower rate than earnings from other savings. In addition, the inflation problem is best addressed by indexing the tax system, not lowering the tax rate.
3-Suppose that Jack owns a warehouse in a rapidly expanding part of a town. Because of the location of the warehouse, it has increased in value from $300,000 (the price Jack paid) to $400,000 (the price Jack would get if he were to sell it). The doctor has recently told Jack that he has one year to live and the doctor is always right about such things. The capital gains tax rate is 25%. Jack's wealth will be distributed among his children when he dies. The warehouse is not currently being used. Jack must decide whether to sell the warehouse now (and pass the cash on to his children) or take it with him to his grave. Assume that there is no estate tax but that the capital gains tax code is the same as that currently in place in the United States.
- What is the socially efficient thing for Jack to do? Explain your answer. Be sure to describe the tax implications of your answer as well.
- What would you advise Jack to do? Explain your answer. Be sure to describe the tax implications of your answer as well.
Ans) a. The efficient thing is to move assets to where they are most productive from a social perspective. Since Jack is not using the warehouse productively at all now, he should sell it for $400,000. By doing so, Jack would pay 25% of the $100,000 capital gain on the warehouse and would thus get to keep $375,000 from the sale.
b. Once Jack dies, the basis gets bumped up to the value of the asset at the time of death. Consequently, Jack's children would not have to pay any taxes on the capital gains and would get to keep the entire $400,000.
4-Suppose that you have a wealthy friend who is 60 years old and wants to know how to pass on her wealth to her children without paying more than necessary in taxes. What suggestions do you have?
Ans)
First, suggest that your friend make use of the annual gift that can be made to each child each year tax-free. Currently the limit is $11,000 per person per year. In this way your friend can give away some of her wealth to her children without being subject to transfer taxes.
You can also suggest that your friend look into setting up trusts, which is a legal way of ensuring that her children get all the benefits tax-free.
Finally, point out that taxes are levied only on transfers of property and cash, not on services. Consequently, your friend could pay for family trips, for example, without being subject to transfer taxes.
3. Consider three consumption tax systems:
(a)all goods are taxed according to the Ramsy rule
(b)all goods are taxed at the same rate, and
(c)the “necessities” are not taxed and “luxuries” are taxed at a higher rate.
Compare the equity and efficiency of these three systems.
Optimal tax theory would argue in favor of plan (a). This plan is a broad-based tax that is difficult to avoid, so it will distort behavior of consumers the least. Furthermore, given the tax’s broad base, the rate can be relatively low in order to raise the same amount of revenue.
Plan (c) violates most tenets of efficient taxation: it does not tax goods for which demand is inelastic (necessities), even though the Ramsey Rule indicates that taxes on necessities will generate the least deadweight loss.
Plan (b) is a broad-based tax like plan (a) but here consumers of the luxury goods reduce their consumptions and hence there the distortions are larger compared to plan (a).
Plan (c) does tax luxuries, for which demand is likely to be elastic. Thus, this tax will distort behavior and generate substantial deadweight loss. Plan (a) is clearly most efficient.
Plan (a) is most regressive: poorer taxpayers are hit the hardest. Here poorer taxpayers will pay a higher percentage of their income on taxes than will wealthier taxpayers. By consuming most of their income, poorer taxpayers are subjecting a high proportion of their income to this consumption tax.
Plan (b) is also regressive but less so compared to plan (a).Poorer taxpayers will pay a higher percentage of their income on taxes than will wealthier taxpayers.That is because poorer taxpayers cannot afford to save or invest large portions of their income; they spend it on the goods they need.
Plan (c) is not as regressive, because the kinds of goods that lower-income taxpayers purchase are not taxed but the kinds of goods purchased by higher-income taxpayers are taxed. Plan (c) is clearly more equitable.
4. Suppose that a nation is considering adopting a national sales tax of 25% on all goods in place of the progressive income tax currently in place.
- If the politicians want to use a sales tax but also want the tax to be progressive, what could they do? Evaluate this option in terms of what you know about optimal commodity taxation.
- Would all groups be affected by the transition to a consumption tax in the same way? Explain.
Ans)
a. To make the sales tax more progressive, the sales tax rate could be higher for some goods purchased disproportionately by the rich than for necessities such as food and basic clothing. However, this is exactly the opposite of what optimal commodity taxation theory indicates should be done, which is to tax goods with inelastic demands (like food and basic clothing) at higher rates than goods with elastic demands (like luxury goods).
b. For groups whose incomes and consumption will occur largely in the future (young and savers), a consumption tax would be similar to an income tax. However, some people have already paid taxes on their income and saved some of the posttax income for retirement. Those people (retirees) will be made worse off by a consumption tax since they will now pay taxes on their savings again when they spend it.
5. Suppose that the government is considering replacing the current income tax system with either a national sales tax or a value-added tax. Would one of these alternatives be more desirable than the other in terms of compliance? Explain.
Ans) To replace the income tax system with a national sales tax, the sales tax rate would have to be set quite high, 30-35% or so. Given a tax of that size, there would be incentive for people to make considerable efforts to avoid paying the tax. For example, retailers and buyers would have a significant incentive to make black market transactions.