Economics 742 Homework #4

Transcription

1 Economics 742 Homework #4 May 4, 2009 Professor Scholz Please turn in your answers to the following questions in class on Monday, May 4. Each problem is worth 40 points, except where noted. You can work together on the problems, but please turn in your own work.. Write a final exam question (and provide an answer). Each student should provide a different question. t least one question will appear on the exam, so you have an incentive to write a good question (since you likely will get the problem correct if I choose your question!). I will distribute the questions (but not the answers) on Monday, December as an aid to preparing for the final exam (which is scheduled for 0:05 M on Tuesday, December 9). To meet this schedule, you must me your question (and answer) by class on Wednesday, December (Example final exam problem): Consider the following stylized model of an economy, which exists for two periods (period and period 2). There are two types of people in this economy: type s, who typically have long life spans, and type s, who typically have short life spans. There are 20 type s, and 0 type s. Each type cares only about consumption while alive, and each person has $00 in income, which he/she receives in period one (to make things simple, there are no work decision to be made in this economy, so we won t worry about labor supply and wages). Each person has an expected utility function of the following form: i = ln( C) + pln( C2), where p=.75 for type s, and p=.2 for the type s. In other words, no one wants to starve if they live into the second period, but no one is certain whether they will reach the second period and type s are more likely to reach the second period than type s are. Suppose that no annuity markets exist in this economy because potential annuity providers are unable to observe individual types (resulting in adverse selection problems).. lthough the annuity market doesn t exist, people can still invest money in period. In period 2, if the person is still alive, he or she receives his or her initial investment, plus a rate of return on the investment of 0% (i.e. the interest rate is 0%). How much will each type choose to save in the first period? What is first period consumption for each type, and how much will each type consume if they reach the second period? 2. Explain intuitively why an actuarially fair annuity market would or would not improve welfare. Would this market develop?. The federal government, recognizing a market failure in the annuity market, decides to correct the problem by implementing a social security program. The way this social security program will work is that the government sets a lump sum tax T that everyone must pay in the first period. The government takes this money, invests it at an interest rate of 0%, and uses the entire amount to pay in benefits to everyone who is alive in the second period. a. ssume that the government s social security budget must break even (in expectation). Write out the government s social security budget constraint. What multiple of taxes must benefits be for the budget to exactly break-even? (i.e., find the X such that =XT for the government to break even). b. Suppose the government is trying to choose the optimal lump sum tax T (and hence, the optimal social security benefit amount ). If the government cares only about the sum of utilities for everyone in the economy (i.e. it maximizes a utilitarian social welfare function), and if the budget must exactly break even, what level of T and will it choose? (note: you can safely assume that neither type will want to save privately in addition to social security). Is type better off with or without social security? How about type? In what sense is this program redistributive? c. The social security plan in part (b, just above) is never implemented because the administration that proposed the policy is voted out of office by the type s. Coincidentally (or perhaps not coincidentally), the new administration cares only about the type s. What level of T and will the

2 new administration choose, if the social security budget must still be balanced? (lthough the administration cares only about, each type must get charged the same lump-sum tax, and each type must receive the same amount in benefits if they live to period 2). ssume that everyone can supplement social security benefits with additional saving if they wish (i.e, they can save some of their post-tax income in period, as in part a). Is type better off with or without social security? What about type? Is either type better off than with the system in b (just above)? Is this new program redistributive, and if so, how does its redistributive nature differ from part b (just above). 4. Now, think about how this stylized economy and social security system relate to the merican economy and current social security system. In what sense is the social security system in the U.S. redistributive in ways similar to those illustrated above? Who are relevant type s and type s? nswer: If the person lives into the second period, her consumption is: C 2 = (l + r) * (l00 C ) =. * (00 - C ). Now just plug this in for C 2, and maximize the expected utility function with respect to C. maxu = ln C+ pln( C2) max U = ln C+ pln(.(00 C)) p 00 00p = 0 00 C = pc C =, C2 =. C 00 C + p + p So p=.75 for type s, so C = 80, saving = 20, C 2 = 22. p=.2 for type s, so C = 9.75, saving = 6.25, C 2 = Type s are saving more, and hence consuming less in period than s do but more in period 2, because the probability that they will live to see the second period is greater. b) Yes, people would be better off, because annuities provide a higher return on investment. Remember, the price of fair insurance is the probability of payout (i.e. the probability of having an accident). Here, the probability of payout is the probability of living into the second period. Type s only have a 20% chance of reaching the second period, so the price of one unit of annuity payout is.2. Provided the type reaches the second period, the return on his investment is huge- for every dollar he spends on insurance, he gets five dollars if he lives into the second period, for a net return of four dollars. This certainly is greater than his returns from saving. Type s must pay $.75 for $ of coverage, so s receive $. for every $ invested if they live into the second period. The return on their annuity investment is %, which is again higher than returns from saving. The reason that fair annuities can provide higher returns is because risks are pooled: everyone pays in, but only a fraction receive payment. nnuities are valuable because they assist in consumption smoothing throughout old age. In this simplified two period model, consumption smoothing can be accomplished through private savings, but doing so is inefficient, since each person needs to save on their own, and many people will die before they are able to eat their saving. n actuarially fair annuity market makes it much more efficient to smooth consumption (and hence increase utility). C, part a) The government receives 0T in revenue (since it taxes all 0 people equally), and invests it at an interest rate of 0%, so that it has 0T*. =T in funds to payout in period two. It pays out.75* in expectation for every type, and pays out.2* in expectation for every type - so in sum, it expects to pay out 20*.75 *+ 0*.2 *= 7. Since the budget must break even at the end of period two, this implies that T=7, or the benefits the government pays out to anyone who lives into period two is T/7. C, part b) The government is now trying to decide what T to choose to maximize the sum of utilities. In

3 other words, its maximization problem is: T T max social welfare = 20* ln(00 T ) + *.75*ln 0* ln(00 ) *.2*ln 7 + T = 0 T = 5.89, = T T 00 T T 07 So each person pays 5.89 in taxes, and receives a benefit in period two of 0.84 (if they survive that long). noss = ln(80) + ln(22) noss = ln(9.75) + ln(6.875) SS = ln(84.) + ln(0.84) SS = ln(84.) + ln(0.84) is better off with social security than without - this is because it is highly probable that a type reaches the second period, and because the return from "saving" with social security is greater than the returns from private investment. Since =T/7.94T, one dollar "saved" (taxed) through social security returns.94 dollars if the person reaches the second period, for a return of 94% (which is certainly greater than the 0% return from the private market). From 's perspective, this isn't the optimal social security system -- the optimal system would in fact force more saving through an even higher tax - but because the high return results in high period two consumption, s find this system better than a world with no social security. s, however, are just slightly worse off than before. This is because the tax rate is much higher than what they would've chosen (see answers to the next part). So even though the rate of return is much higher than what private savings yields in the absence of social security, s are worse off because the tax rate is too high. One could view this system as redistributive since it taxes s more than they wish, and these tax revenues are partially redistributed to s since s are more likely to receive social security benefits. C, part c) Now the government chooses T to maximize 's utility: T 00 max ' s utility = ln(00 T ) + *.2*ln 0 T 6.25, = = = = 00 T 5T 6 which implies

4 SS 2 SS 2 = ln(9.75) + ln(2.) = ln(9.75) + ln(2.) Now is better off with this system than without, and is better than the first proposed system. is still better off than without social security (and if you assume that can supplement social security income with additional private savings, 's utility will be higher than 5.6), but is worse off than with the first social security system. This is because the forced savings in this system are lower than the other system - since this system doesn't account for 's preferences whatsoever, but the prior system partially weighted ' s preferences, must be better off with the first social security system. However, assuming can save in addition to social security, is not worse off than without social security - this is because can always save on top of social security at a 0% interest rate, but the returns on some investment (the 6.25 "invested" with taxes) is greater than the market rate of 0%. gain, this system redistributes from s to s - this is because the ratio of expected payout to payin is much higher for s than s. However, s are better off with this sort of redistribution than they are if the social security system didn't exist. D) There are many ways in which the merican social security system is redistributive. For example, since the PI is calculated as a progressive function of the IME, the social security system redistributes from the poor to the rich (the replacement rate for social security benefits is higher for the poor than the rich). What is most relevant from this example, however, is that the system also redistributes from the short lived to the long lived - because if two people are identical except for life expectancy, they'll pay the same amount in taxes, but the short lived person will receive less in benefits than the long lived person will. In merica, women tend to live longer than men, and whites tend to live longer than blacks, and richer people tend to live longer than poorer people - so due to differences in life expectancy, the system is redistributive from the first group to the second (of course, the progressivity of benefit calculations may make the system actually less redistributive from the first to second groups).. Consider a simple economy with two types of people, s and s, both of whom live and consume for two periods and are altruistic towards each other in the second period. Suppose each earns in the first period. In the second period, with probability of 0.5, earns and earns 0, and with probability 0.5, earns 0 and earns. Denote by a caret the second period state in which earns and earns 0, and denote by an asterisk the second period state where earns 0 and earns. lso assume an interest rate of zero. Then, assuming logarithmic utility, the expected utility of can be written where C Y is consumption when young, C O stands for consumption when old, and m is a parameter whose value lies between 0 and, and indicates the extent of altruistic caring by for and by for. n exactly analogous expression can be written for the expected utility of. a) Find C Y for and in a world without government intervention. Discuss the intuition for this result.

5 b) Find the Pareto efficient level of consumption when young. Compare this result to that in part a). c) In this simple model, show why compulsory saving (like social security) may be more efficient than subsidizing saving if the goal is to increase saving. To do this, include uncertain second-period earnings in the adverse state. In state ^ (which still occurs with probability of 0.5), will still earn, but earns 0 with probability 0.5 and.5 with probability 0.5. In state * (which still occurs with probability of 0.5), will still earn, but earns 0 with probability 0.5 and.5 with probability 0.5. If m is not too small, will still transfer to in state ^ and will transfer to in state *. Suppose further that and do not observe each other's earnings so each has an incentive to misstate earnings to the other. Discuss how social security might improve this situation. nswer: This problem is borrowed from Larry Kotlikoff s paper on the Contributions of Economics to the Evaluation and Formulation of Social Insurance Policy, May, 989, ER, pp If you want further elaboration of my comments, Kotlikoff s paper might be a good place to start. Part a, the individual s problem: You first need to figure out how transfers are determined in the model. In state ^, determines the consumption of the two at the margin. In ^, will choose to consume ˆ R where is the + m ˆR combined resources of and ( 2 m C Y ), and make a transfers so will consume R ˆ. + m Given this, the individual s optimization problem is to choose optimal consumption, given the 2 C Y m( 2C Y following log CY +.5 log + mlog. Solve and you will find C Y =. + m + m m + Part b), The Pareto Efficient problem Here you solve te planner s problem, maximizing the sum of and s expected utility. You ll find that C Y =. This problem demonstrates the Samaritan s Dilemma. While each person is 2m + 4 altruistic toward the others, those who fare better economically end up transferring resources to those who fare poorly. ecause each person can anticipate a transfer in the event of bad luck, each will free ride on the other when determining how much to save. In particular, when thinking about how much to save, each person will realize that another dollar of saving will not deliver, in their respective bad state, as much benefit as they would get if they were in control of the marginal allocation of consumption. s a consequence, both and have smaller saving incentive than optimal when they are young. Part c, can the Government, through something like social security, make things better? Since earnings are unobserved, the person to receive a transfer has an incentive to understate (as 0) their second period earnings. Thus, the person controlling the transfer will make the transfer subject to uncertainty. s you might have found out, there is not a convenient, neat closed for solution to the problem. Instead, you want to show that for any choice of (and therefore S), that s expected utility in period 2 state ^ is larger if s second period earnings are revealed prior to s choice of first period consumption. More surprisingly, you can also show that both people can be better off. The C Y C Y

6 intuition is the following: suppose that in s good outcome (when they both get 0.5), transfers exactly as much as they would in the case that s second period income is uncertain. Then allow to optimize again when earns 0 in period 2. will not consumer as much, which by revealed preference, means they are better off. will also clearly be better off (since their consumption is higher). Hence, observing s earnings can lead to Pareto improvement. If all saving of and is compulsory and done through Social Security, then there is no problem of observation. Hence, Social Security can resolve problems of deficient first-period saving and nonobservability of second period earnings. Social Security must, however, be able to link benefit payments to earnings. Question 4: Consider an economy where people live for 2 periods, with preferences of U = log( C) + log( C2). + Delta is the per period discount rate (between 0 and ). The person has first period income of Y and no second period income, and can save, earning interest of r. The government levies a capital tax at the rate tau on capital income received in the second period. The tax proceeds are paid as a lump-sum transfer to the following generation. The present generation does not care about the next one. a) Making use of indirect utility (as a function of Y and ( τ ) r ) evaluate the change in initial income that is required to compensate the individual for the welfare loss of the capital income tax (tau). b) What is the effect of a tax change on first period consumption? How about the second period? What conclusion about the welfare cost of capital income taxation can you draw from this? C2 nswer: a) The lifetime budget constraint is C + Y T so the lifetime consumption + r( τ ) = + + profile is C = ( Y+ T) and C2 = [ + r( τ )]( Y+ T). The use the balanced budget condition T = τ rs to eliminate T from consumption, so + T = τr( Y+ T C) = τr ( Y+ T) so Y+ T = Y. fter solving for optimal 2 + τr( + ) first and second period consumption, you can write indirect utility as ( + )[ + r( τ)] VY (, τ, r) = log Y + log Y. 2 + τr( + ) τr( + ) Now, without taxes and transfers and with initial income Y 0 the indirect utility function is ( + )( + R) VY ( 0,0, r) = log Y0 + log Y0. Equate the two indirect utility functions + and find the relationship between and Y and solve for the ratio of the two: Y0 Y0 + r( τ) = Y 2 + τr( + ) + r. The ratio is greater than one for any positive tax rate, r, and

7 discount rate (between 0 and ). Hence the capital income tax later in life with an equal transfer received earlier in life leads to a welfare improvement for an individual. b) When choosing the optimal consumption path, the consumers do not internalize the relationship between tax paid on capital income and the amount of transfer received. Hence, to find the consumers response on the change in the tax rate, we need to differentiate consumption with respect to tau, taking T C C2 + as given: = 0, = ry ( + T) < 0. Hence, an increase in the tax rate has no effect on τ τ 2 + first period consumption and it negatively affects second period consumption. Capital income taxation causes an individual welfare loss only for the first generation of the old, since they do not receive transfers when young. Similarly, a one-time increase in the tax rate causes an individual welfare loss only for the consumers who are old at the time of the change, whereas all future generations are better off. The change in aggregate welfare depends on how we define the aggregate welfare function. Question 5. Read one non-starred paper on social security and social insurance from the reading list. Write a one-pager on the paper, explaining the issue, how the paper fills a gap in the literature, what the author(s) do, and what they find.