Economics 742 Brief Answers, Homework #2

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1 Economics 742 Brief Answers, Homework #2 March 20, 2006 Professor Scholz ) Consider a person, Molly, living two periods. Her labor income is $ in period and $00 in period 2. She can save at a 5 percent interest rate or borrow at a 0% rate. a) Graph Molly s budget constraint with no taxation. Label the maximum consumption each period, the slopes of the budget constraint, income in each period, etc. b) Draw Molly s budget constraint when interest income is taxes at a 20 percent rate. Would this policy lead Molly to save more or less compared to part a? Y (+r s )+Y 2 /=520 If Molly was a saver in part (a), then we don t know whether she would save more or less. The tax Y (+r s )+Y 2 /=520 Y (+r s (-τ)) +Y 2 = Y +Y 2 /(+r b )=490.9 Y, C reduces the net interest rate, so the 00 Y +Y 2 /(+r b )=490.9 Y, C substitution effect implies she should save more, the tax reduces her second period income, so the income effect would imply she would save less. If Molly was a borrower in part a, she will not change her behavior. c) Draw Molly s budget constraint when interest income is taxes as in b), and interest paid is tax deductible. Would this policy lead Molly to save more or less, compared to your answer in part b? If Molly saved only a little or was at the kink in (b), either she will decide to borrow a little (because of the SE from the reduced cost of borrowing) of else she will exhibit no change in behavior (when the marginal rate of substitution of first and second period consumption is sufficiently low). If Molly was a borrower in (b), then she will unambiguously borrow more (save less). The deduction reduces the net interest rate, so the SE implies she will borrow more (save less). The deduction raises her income (a little) in period 2, so the income effect implies she will borrow more (and save less). Y (+r s )+Y 2 /=520 Y (+r s (-τ)) +Y 2 = Y +Y 2 /(+r b (-τ))=492.6 Y +Y 2 /(+r b )=490.9 Y, C

2 d) Draw Molly s new budget constraint when the tax system is like part c, but there is also a tax preferred saving option. Molly can put $00 in this account and the interest earned in the account is excluded from taxation. Would this policy cause Molly to save more of less compared to c? Would this policy raise or lower aggregate private saving (total annual saving divided by GDP)? What about public saving (total saving of the Government divided by GDP)? If Molly was borrowing a lot in (c), there would be no change in her behavior. If Molly was borrowing a little in (c) she may shift to saving a little ($00 or less) because of the substitution effect. If Molly was saving $00 of less in (c) the effect is ambiguous: the interest exclusion raises the net return from saving: the substitution effect implies more saving. The interest exclusion raises period 2 income, so the income effect implies less saving. If Molly was saving over $00, there is no substitution effect and the income effect of the interest exclusion implies there will be less saving. (Y 00)(+r s (-τ)) +00(+r)+Y 2 = 57 Y (+r s (-τ)) +Y 2 =56 00 Slope=-(+r s (-τ))= Slope=-(+r s )=-.05 Y +Y 2 /(+r b (-τ))=492.6 Y +Y 2 /(+r b )=490.9 Therefore, the effect of the new policy on aggregate private saving depends on the relative numbers and the initial net saving of borrowers and savers. Given te effect on private saving, the effect on total national (public plus private) saving will be smaller; the interest exclusion reduces government revenue and hence net public saving. e) Suppose that policy c) is in place and then the government introduces policy d). Suppose also that longitudinal wealth or saving data do not exist. From each of the following observations alone, would you conclude that the tax preferred accounts raise the aggregate private saving rate?! Over the period when the tax preferred saving rates are introduced, the aggregate private saving rate increases. This is consistent with the possibility that the ta preferred saving accounts increased saving, but other things might be changing over the same time period. It would be difficult to infer a causal effect if other tax factors or other things in the economy changed at the same time.! After the introduction of the policy, many tax-preferred saving accounts are opened. No, we don t know what is happening to saving outside the tax preferred accounts. Generally, those who save should first save in the tax preferred way first, so without additional information, the adoption of many new accounts could be consistent with the new incentives increasing or decreasing private saving.! People who have tax preferred saving accounts have greater total assets than people who do not have accounts. People who already were saving a lot would save in the tax preferred account, but would also reduce total saving; yet they may still have higher total assets than people who did not open taxpreferred accounts (a group that might include many borrowers). So this observation implies nothing about the effect of the tax preferred accounts on aggregate private saving. Y, C 2

3 ! You have cross-sections of data before and after the introduction of tax-preferred saving accounts. Saving rates among people with low average income rose, while saving rates among people with high average income declined. This is somewhat more intriguing evidence that the tax preferred accounts are causally increasing saving. Low-income individuals tend to save less than high-income individuals. Therefore, when the tax incentives are introduced, low-income individuals are likely to be affected by the substitution effect, while high income individuals likely experience a pure income effect. The effect on aggregate private saving will depend on the relative numbers and initial net saving of low-income and high-income people. 2) In September 998 the U.S. Treasury Department began issuing Series I inflation-indexed saving bonds. These bonds were purchased using after-tax dollars. They accrued interest at a nominal interest rate of π percent per year, where π is the rate of inflation. The bonds mature in 30 years. After 30 years they no longer pay interest. The accruing interest on these bonds is not taxed until the bonds are cashed in, at which time the owner is liable for income tax on the full amount of accrued interest. a) Obtain an expression for the effective accrual tax rate on a Series I saving bond, assuming that the bond is held to maturity. This tax rate equals the interest income tax rate that, if it were applied to the annual of quarterly income generated by the bond, would lead to the same terminal wealth as the actual tax system. How does the effective accrual tax rate depend on the rate of inflation over the holding period, and the marginal tax rate that the saver faces when the bonds are redeemed? Call 2 the effective accrual tax rate. In continuous time, it satisfies π)( θ) T π) T π) T e = e τ ( e ), in which case the effective accrual tax rate is: ln(( τ ) e + τ ) θ =. The partial derivatives with respect to inflation and the marginal tax Tr ( + π ) rate are π) T τ) T ( τ) e ln(( τ) e + τ) = + 2 π (( τ) e + τ) π) T π) e = + τ Tr ( π) ( τ ) e τ, and The second derivative is positive, since all terms in the expression are positive. To show that τ τ < 0, note that its limit as T approaches 0 is + = 0. Show < 0, which then π r+ π r+ ρ T π implies < 0 for positive T. π The intuition for what you are doing is that as the rate of inflation increases, the effective accrual tax rate decreases. There are two effects of increasing inflation: () the end of holding period 3

4 capital gain is higher, because the nominal interest rate is now higher. The end of holding period tax burden will therefore be larger. This effect pushes theta up. (2) Because savings grow tax deferred, the higher nominal interest means there is a bigger opportunity to let interest compound without tax. This effect pushes theta down. Effect (2) dominates effect (), particularly as the holding period lengthens. As the marginal end-of-holding-period-tax-rate increases, so must the effectice accrual tax rate. The intuition is just that if the end of period tax burden is bigger, then an equvalent accrual tax must also be bigger. b) Now assume that the rate of inflation for the next 30 years is exactly 3.0% per year, and that an individual investor faces a marginal income tax rate of 33%. Assume further that the yield to maturity on a 30 year ordinary Treasury bond is 6.0 percent. Note that the interest on an ordinary Treasury bond is taxes on accrual, which means that it is taxed each year as it accrues. Obtain expressions for the after-tax wealth that an individual will have, after 30 years, if he or she invests $,000 in an ordinary Treasury bond and in a Series I bond. What is the effective after-tax yield on the ordinary Treasury bond, and how does it compare with the effective after-tax yield on the Series I bond? The after-tax wealth of an individual who invests $,000 in the ordinary bonds will be: (0.67)(0.06)(30) 000e = 3,340.. The after-tax wealth of an individual who invests $,000 in Series I ( )(30) bonds will be ((000 e )(0.67)) (000) = 3, The effective accrual tax rate on the Series I bond is: (.05)30 ln(( τ) e + τ) ln(0.67) e +.33) θ = = =.975. The effective after-tax yield on Tr ( + π ) 30(0.05) Series I bond is (-.975)(.05)=.040. The effective after-tax yield on the ordinary bond is 0.67(.06)= The investor does slightly better by buying the ordinary Treasury bonds over the Series I bonds. In fact, any investor with a marginal tax rate under 33% will do better with the ordinary Treasury bonds. Investors with a tax rate greater than percent will prefer the Series I bonds. For the bonds to exist, there must be taxpayers with rates higher than this threshold. In equilibrium, the supply of Series I bonds will be determined by the number of investors with MTRs greater than and by how much they wish to invest in Series I bonds. Investors will form strict clienteles for the two different saving vehicles. c) A high-ranking Treasury official asserts that Individuals can purchase up to $30,000 per year of Series I bonds. For most individuals, this is substantially more than their annual saving. Therefore, the availability of Series I bonds virtually eliminates the income tax burden on personal saving. Would you agree with this claim? How would you evaluate it? There are several reasons why the claim does not make sense. () For investors with MTRs less than.33293, Series I bonds will not eliminate the income tax burden on their saving, because these people prefer the ordinary bonds since the yield on these bonds is higher for them. There is an 4

5 implicit tax on Series I bonds for these investors. (2) For investors with MTRs greater than who hold Series I bonds, the income tax burden may be lessened but it is not eliminated. (3) A small fraction of the population holds a disproportionate share of wealth. For these individuals, the $30,000 limit will be binds and, even if they wanted, they could not put all their wealth in Series I bonds. 3) Consider an individual who lives for two periods and has an additively separable utility function of the form V( C, C ) = logc + logc + δ 2 2 Assume that the individual receives an endowment of W at the beginning of the first period, and then divides this endowment between consumption in the two periods. The pretax rate of return is r. The government has just announced that, for the first time, it will impose a capital tax at rate J on capital income in period two. The proceeds of this tax will be paid as a lump-sum transfer to the next generation. Individuals alive today do not care about the next generation. a) Find the lifetime indirect utility function, V(W,r(-J)), for an individual who is just beginning life. Using this expression evaluate the change in initial endowment (W) that would be required to make the individual as well off with the capital tax as without it. The indirect utility function is 2+ 2 VW (, r( )) log( ) δ + δ τ = + δ log(2 + δ) + log( + r( τ)) + log( W) + δ + δ + δ Then equate indirect utilities in the no-tax and capital tax worlds. VW (, r( τ )) = VW (, r) which, when you work through the algebra, will result in new Wnew W + r = + r( τ ) dc b) What is? What conclusions about the welfare cost of capital income taxation can you dτ draw from this finding? 2+ δ dc 0, since preferences over consumption in periods and 2 are Cobb-Douglas. Be clear, dτ = however. The capital tax still generates inefficiency, since the efficiency losses from taxes arise from substitution effects. If you calculate the compensated variation and the compensated revenue, you will see that the CV exceeds revenue from the tax. 5

6 4, 40 points) I want you to start understanding what maximum likelihood estimation is all about. The simplest way I know to do this is to start by estimating empirical models with a canned statistical package (like Stata, Limdep, or SAS) and then replicate those estimates after defining and maximizing the likelihood function that you specify, using the user-defined procedures in either Stata or Gauss. Gauss is, in my view, slightly easier for MLE problems but since you are investing in Stata this semester, it is fine if you would like to use Stata. This problem takes a small step toward become familiar with user-defined maximum likelihood estimation. a) Write down a simple empirical model of the probability of holding an IRA in 983. The purpose of this problem is to ask you to focus on computational methods, so do not write down an elaborate empirical model. Include a few sensible covariates and tell me why you chose them. Estimate this model as a linear probability model using Stata s regress command. b) Now estimate the same model by maximum likelihood (using either maxlik in Gauss, or ML in Stata read the documentation carefully). I want you to write down and maximize the correct likelihood function do not do this by computing ( X ' X) ( X ' y), but instead write down and maximize the appropriate likelihood function. If you did the problem correctly, you will get the same results (both point estimates and standard errors). Many minimized the sum of squared residuals (Y-Xb) 2, which will give you the correct answer (and indeed, I suggested this to one of your colleagues in my office hours). But this will make it very difficult to do the Tobit problem I asked you to do in part e (and which I will ask you to redo). Instead, use the normal density function to replicate the OLS regression estimates. If you do this correctly, it should be straightforward to estimate your Tobit model one branch (where the dependent variable is below the censoring point) will have the OLS likelihood function. The other will have the Probit likelihood function. If your starting values are not good, you may have an issue constraining sigma to be positive. If this is a concern, you can estimate exp(sigma), which ensures that it will always be positive in the likelihood routine. In Gauss, the likelihood function for the OLS replication can be written LIKELIHOOD FUNCTION */ retp(-ln(sig.*(2)^.5.*pi) - (/2).*((Y - Xbeta_hat)./sig)^2); c) Now estimate your empirical model using Stata s probit command. The probit coefficients are not directly interpretable. Show marginal effects using Stata s dprobit command. d) Using the user-defined maximum likelihood procedures, estimate the same model. If you did the problem correctly, you will get the same results. Calculate (directly) the marginal effects implies by the probit coefficients ( not via a canned problem). e) Estimate a Tobit model of IRA contributions in Stata. f) Generate the same estimates using a user-defined likelihood procedure. Compare the results. And explain the two branches of your likelihood function. 6