Professor Scholz Posted November 27, 2006 Economics 441, Problem Set #4 Due in class, Monday, December 11, 2006

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1 Professor Scholz Posted November 27, 2006 Economics 44, Problem Set #4 Due in class, Monday, December, 2006 All problems are worth 40 points unless noted differently. (0 points). Suppose your preferences can be characterized by the simple utility function U = C. You enjoy rock climbing, where you have a 0 percent chance of getting injured and losing $50,000. Your income (and therefore consumption) in the uninjured state is $90,000. a) What is the most you are willing to pay for an insurance policy? Will insurance companies be willing to insure you? Without insurance, expected utility is.9 90, =290. The most someone would be willing to pay is X, where X solves X = 290, so X=5900. Insurance companies would be happy to offer insurance for prices between 5000, the expected payouts, and 5900, the most people would be willing to pay. b) Now, suppose there are a bunch of people who look just like you in their observable characteristics (they talk the same, they look the same, they make the same amount of income). But these people are passionate about their hobby of base jumping. Base jumpers make parachute jumps off immobile objects, like tall buildings, bridges, cliffs, etc. Suppose base jumpers have a 50 percent chance of getting injured and losing $50,000. What will this do to the market for insurance? The expected cost of insuring base jumpers, $25,000, is much higher than the expected cost of insuring rock climbers ($5,000). The problem is that insurance companies would lose money by offering policies attractive to rock climbers, since base jumpers would purchase them (since they are observationally equivalent to the insurance company). But policies attractive to base jumpers will be too expensive for rock climbers. The market would evolve in a way that rock climbers will not find attractive complete policies (that would cost between 5,000 and 5,900). c) This is an example of what phenomenon? Adverse selection. 2 (0 points): In the schools of Beat Town, students are educated to play music, which they then perform at home for their parents. When more money is spent on the schools, the students learn more songs, and their parents are more entertained. The family utility function for each Beat Town family is U = C S where S is Beat town s per-student expenditure on schooling and C is the amount of money the family has left over for other consumption after paying the school tax. All students attend the same school in Beat Town. a) Although all the families in Beat Town have the same utility function, they have different incomes. Suppose 50 families each earn $50,000; and 60 families each earn $80,000. If the town votes on the level of a lump-sum tax to finance schools, what level will win? Recall, since utility is invariant to monotonic transformations. So take the natural log of utility to find (note: you don t have to do this doing so just makes the algebra a little easier):

2 U = (.5)ln( C) + (.5)ln( S), and C = Y S, so U =.5ln( T S) +.5ln( S), du Y = + = 0. This implies S =. ds 2[ Y S] 2S 2 A simple majority vote will result in the $80,000 families winning the vote (since there are 60 of them and only 50 families with income of $50,000). From above, the $80,000 households prefer a lump-sum tax of $40,000. Consequently, all families will be assessed a lump sum tax of $40,000. b) Now suppose lump-sum taxes are found to be unconstitutional. Along with this court ruling, and the residents of Beat Town are ordered to substitute a flat percentage income tax levied at a rate of 50 percent. Explain how the welfare (utility) of Beat Town residents changes (if it changes). With a 50 percent tax, households with income of $80,000 will pay a tax of $40,000 and households with income of $50,000 will pay a tax of $25,000. So the total average tax payments are (50*(25,000)+60*(40,000))/0 = So the rich are worse off, since school spending is less than their optimal value (of 40,000), and private consumption remains unchanged. The poor are better off: consumption of 25,000 and school expenditures of 33,8.82 gives greater utility than consumption of 0,000 and school expenditures of 40,000. 3, 40 points total, 0 for each part) The superhero profession is a dangerous business. Every year, in fact, there s a probability p that a given superhero will be caught by a ruthless supervillain, who inflicts damage that requires $50 in medical costs to heal. Being a superhero doesn t pay well, but fortunately, superheroes receive an annual income of $00 from their (covert) civilian job. They first spend money on any medical costs, and use the rest for consumption. In one particular metropolis, there are two types of superheroes: clumsy and skillful. Clumsy superheroes have a 90% probability of being caught by a supervillian and suffering $50 in medical costs. Skillful superheroes, on the other hand, are only caught with probability 30%. Additionally, these two types have different utility functions. The utility of consumption for clumsy superheroes is.7.5 Uclumsy = ( Cclumsy ) while the utility of consumption for skillful superheroes is Uskillful = ( Cskillful ). Fortunately for this metropolis, there are eleven times as many skillful superheroes as clumsy superheroes. ACME Insurance Company has moved into the city, and is thinking about offering health insurance to superheroes. a) If ACME can perfectly identify whether each superhero is skillful or clumsy, then it can charge a different premium to each type. Suppose it charges an actuarially fair price for insurance. How much will ACME charge each type for $ in medical coverage? How much insurance will each type buy, and how much will each type have for consumption if they get caught, and how much will they have if they don t get caught? b) Now suppose that the type of the superhero is unobservable by the insurance company (although the superheroes themselves know), so ACME can only offer a single price for insurance. 2

3 a. What is the maximum amount of money each type will pay in order to be fully insured against medical costs? Explain why this amount is the same or is different from the costs of actuarially fair insurance. b. Given this, what is the maximum amount that ACME can charge such that all types will fully insure? Will ACME stay in business if it does this? Is there a market failure in the insurance market why or why not? c) ACME instead considers offering two types of insurance coverage. The minimal coverage plan provides $20 in insurance coverage for total costs of $7. The extensive coverage plan offers $50 in insurance for total costs of $34. Which of these plans, if any, will each choose? Will ACME stay in business? Now is there a failure in the insurance market? Why or why not? d) Dr. Brain (a mad scientist) develops a new blood test that flawlessly identifies whether a superhero is clumsy or skillful, so that everyone (including insurance companies) will then be able to identify his or her type. However, this test is not free: the test costs $.50 to administer to a superhero. Suppose Dr. Brain offers to perform this procedure on any superhero who is willing to pay $.50. a. Assuming that there is perfect competition in the insurance market, and that once a test is undertaken all insurance companies know the superhero s type, will any type be willing to purchase this test? Why or why not? Note, in the absence of testing, the insurance plans from part c) above will be what is offered. To answer this, think about what will happen in the insurance market in response, i.e., will a single price be offered or will different policies be offered depending on who takes the test and what will be the equilibrium prices? b. Relative to c), which type is better off now that it is possible to reveal types or are both better off? c. Now is there a failure in the insurance market? Answer: Each superhero's utility exhibits diminishing marginal returns (you can see this by simply plotting the level of utility against consumption, or by finding that the second derivative is negative). This is equivalent to saying that the individual is risk averse. Since this is true, and since insurance is fair, the price that each type pays for a unit of insurance will be equal to the probability that they get caught - and further, we know from class that risk-adverse individuals (i.e. utility with diminishing marginal returns) will fully insure if insurance is fair. You could also get this result by solving for the optimal level of insurance that each type will choose. For clumsy superheros, the problem would be:.7.7 Max puaccident + ( p) U NoAccident Max.9( b.9 b) +.(00.9 b) du (.9)(.7)(.) (.)(.7)(.9) so = = db (50 +. b) (00.9 b) b = 50 Where b is the amount of insurance the clumsy superhero buys (which is equal to the amount he gets from the insurance company if he is caught). The procedure is similar for skilled superheros. 3

4 So, to answer the specific questions, ACME will charge the clumsy $0.90 for every dollar of insurance and will charge the skillful $0.30. Since they are risk averse and the insurance is priced fairly, they will both purchase $50 of insurance. The clumsy will consume $55 in both the accident and no accident state. The skillful will consume $85 in both states. B, part a) the maximum amount that each would be willing to pay for insurance is equal the amount that makes him or her exactly as well off with insurance as without. Without insurance, the expected utility of each type is clumsy skillful.7.7 =.9(00 50) +.(00) =.3(00 50) +.7(00) 9.2 Full insurance would mean that the superhero receives full compensation for injury if injured, resulting in complete consumption smoothing across the two possible states (caught or not caught). Hence, his income in each state is simply his full income less insurance costs. So to solve for the maximum each type would be willing to pay for full insurance, solve the following: = = (00 X ) X = clumsy, no insurance clumsy, FI = 9.2 = (00 X ) X = 6.80 skillful, NI skillful, FI.5.7 So the clumsy type is willing to spend for full insurance (for a per-unit price of.90) and the skillful type is willing to spend 6.80 (for a per-unit price of.336). Note that each is willing to pay more than the actuarially fair price - this is because their utility function exhibits diminishing marginal returns to consumption (risk aversion): B, part b) If ACME is looking for a single market price such that everyone will fully insure, it will have to offer the maximum price that the skillful are willing to pay: However, when it does this it will collect 2*6.80=20.6 in revenue for every *.3*50+.9*50=20 it expects to payout - since expected revenue is less than expected payout, ACME won't stay in business if it offers this plan. Hence, there's a market failure: we know that each type is willing to pay more than the actuarially fair price for full insurance, and ACME is certainly willing to offer insurance at a greater-than-fair price. The problem is that types are unobservable (that is, asymmetric information exists in this insurance market), so this arrangement cannot occur. Because there are transactions that would occur under full information that would make everyone better off, a market failure exists. c) Now, let's consider the expected utility that each type would receive under each plan: clumsy, PlanA skillful, PlanA.7.7 =.9( ) +.(00 7) =.3( ) +.7(00 7) 9.32 clumsy, PlanB skillful, PlanB.7.7 =.9( ) +.(00 34) =.3( ) +.7(00 34)

5 Comparing these expected utilities to those without insurance, as calculated in (b, part a), we see that clumsy types prefer plan B to plan A, and are better off by purchasing insurance than not purchasing insurance. The skillful types prefer plan A to plan B, and are also better off by purchasing insurance. So if these packages were offered, all the clumsy would purchase type B, and all the skillful would purchase plan A. ACME revenues are *7+ *34=. ACME expected payout is *.3*20+.9*50=. ACME would make zero economic profit from this package, but since zero economic profit allows the company to earn market rates of return on all factors of production it is willing to offer the insurance. Now everyone has some insurance However, there is still a market failure for the same reason as before - the skillful types still wish to purchase full insurance at actuarially fair prices, and the insurance company would offer insurance for those prices - but due to asymmetric information, this is impossible. The outcome may be more favorable than before, since now everyone gets some insurance, but a market failure still exists. D part a) There might be an incentive here for the skillful type to purchase the test. Consider what happens if the skillful pay for the test and reveal their type: given that there is perfect competition in the insurance market, the skillful will be offered actuarially fair insurance, so they would fully insure. The clumsy type would never purchase the test, because if they do their type will be revealed, and they will receive actuarially fair insurance - which is more expensive than the insurance plan they receive from c! To determine whether the skillful types will purchase the test, consider their expected utility if they do: skillful, NoIns skillful, PlanA skillful, test.5.5 =.3(00 50) +.7(00) =.3( ) +.7(00 7) 9.32 = (00.5.3*50) 9.38 Since the skillful are slightly better off by paying $.50 for the test, receiving fair insurance and fully insuring, they will be willing to pay for Dr. Brain's test. Now, ACME will offer two insurance packages. The first will be for those revealed to be skillful, who will be able to purchase insurance for a per-unit of coverage cost of.3. The second will be available only to those who haven't revealed themselves with a test (i.e. the clumsy), who will be able to purchase insurance for a per-unit of coverage cost of.9. Both types will fully insure! D, part b) The skillful are better off than in (c), because they are now fully insured. The clumsy are worse off, because although they are still fully insured, they have to pay more in order to be fully insured. D, part c) Now there is no longer a market failure. The presence of the market for blood tests has eliminated the asymmetric information problem, and everyone is fully insured at actuarially fair prices (although the skillful are technically paying more than actuarially fair prices to receive full insurance, since they first have to purchase the test). 5

6 4, 40 points total, 5 points for each part, with a 5 point bonus for completing the entire problem) Consider a society of identical workers that each earns a wage W when they work. Each worker faces a probability of sustaining an injury. If they sustain an injury, they have no earnings (W=0). In any case, however, they always have some outside income of 5. Workers have utility of the form: U = log( C ), 2 where C=consumption=total income in the period (they do no saving). Assume there is no moral hazard. a) What is the expression for the expected utility of each worker? Now, suppose that the government introduces a worker s compensation program. Under this system, individuals pay some fraction of their wage when they are employed (i.e., their wage is taxed at a certain rate), and get a benefit when they are injured. The system must break even at a point in time; that is, the benefits paid to injured workers must be equal to the taxes collected from employed workers. b) What is the optimal workers compensation system? That is, what is the system that, subject to the constraint of breaking even, maximizes worker utility? Present both the tax rate and the benefit level for the system. c) Are there welfare gains or losses from introducing the Workers Compensation system (you don t actually have to measure the gains/losses just sign them)? Why? d) Would your answer to part c) change if utility was of the form: U=(/2)C? Now suppose that when workers get injured their spouses go to work. Each worker injured gets an amount kw from their spouse, where k is some constant and k < ( ). e) What is the expected utility now if there is no Workers Compensation program? f) Now, reintroduce Workers Compensation, which once again must break even. What is the optimal Workers Compensation system now (both tax rate and benefit level)? How does this compare to your answer to part (b)? Why? g) Are there welfare gains or losses now from introducing the system (once again, no precise measurement is necessary)? Intuitively (and not mathematically), are these gains or losses greater than in part c)? Why? Answer: a) ( ) = ( )log( w+ 5) + log(5) Remember, utility is invariant to monotonic transformations, so for ease of notation, I dropped the ½. b) The break-even condition dictates the benefit level, b, so w( ) b = 0 b= w. so solve max E( U) max[( )log( w( ) + 5) + log(5 + w)] The first-order conditions are ( ) ( ) w ( ) w ( w) + w= 0 + = 0 w( ) + 5 ( ) 5 5 w w w = w( ) + 5 = 5 + w = b= w( ) w( ) w c) Yes there are welfare gains. These occur because, due to the log utility, the workers are risk averse and therefore value the insurance provided by the workers compensation system. 6

7 d) If U=.5C then the workers are no longer risk averse and therefore they do not value insurance. There is no welfare gain to introducing workers compensation. e) ( ) = ( )log( w+ 5) + log( kw+ 5) f) max E( U) max[( )log( w( ) + 5) + log( kw+ 5 + w)] The first-order conditions are ( ) ( ) w ( ) w ( w) + w= 0 + = 0 w( ) + 5 ( ) 5 kw 5 w w kw + 5+ w = w( ) + 5 = kw+ 5 + w = ( k) b= w( k)( ) w( ) + 5 kw w g) Yes, there are gains, but they are smaller than in part c). They are smaller because spousal employment is already providing some insurance against injury. 5: 40 points total, 0 for each part) 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 As, who typically have long life spans, and type Bs, who typically have short life spans. There are 20 type As, and 0 type Bs. 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 As, and p=.2 for the type Bs. In other words, no one 3 wants to starve if they live into the second period, but no one is certain whether they will reach the second period and type As are more likely to reach the second period than type Bs 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). a) Although 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? b) Even though people in this economy can smooth consumption by saving and earning a return equal to the interest rate, would people be better off if an annuity market existed that provided actuarially fair annuities? (In answering this part of the question, there is no need to solve for anything. Just explain intuitively why you think an actuarially fair annuity market would or would not improve welfare). c) 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 7

8 government takes this money, invests it at an interest rate of 0%, and uses the entire amount to pay B in benefits to everyone who is alive in the second period. a. Assume 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 B=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 B). 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 B will it choose? (note: you can safely assume that neither type will want to save privately in addition to social security). Is type A better off with or without social security? How about type B? 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 Bs. Coincidentally (or perhaps not coincidentally), the new administration cares only about the type Bs. What level of T and B will the new administration choose, if the social security budget must still be balanced? (Although the administration cares only about B, 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). Assume 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 A better off with or without social security? What about type B? 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). d) Now, think about how this stylized economy and social security system relate to the American 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 As and type Bs? Answer: Don't get confused because we're thinking about consumption in different periods. Just treat the problem as deciding how much to spend on two goods: consumption today and. consumption tomorrow. The easiest way to solve intertemporal utility maximization is to consider how much the person can consume in the second period, given how much he chose to consume in the first period. So in this example, if the person lives into the second period, his 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) maxu = ln C+ pln(.(00 C)) 3 3 p p = C = pc C =, C2 =. C 3 00 C 3 + p 3 + p A A B So p=.75 for type As, so C = 80, saving = 20, C 2 = 22. p=.2 for type Bs, so C = 93.75, saving B = 6.25, C 2 = Type As are saving more, and hence consuming less in period than Bs do 8

9 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 Bs only have a 20% chance of reaching the second period, so the price of one unit of annuity payout is.2. Provided the type B 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 As must pay $.75 for $ of coverage, so As receive $.33 for every $ invested if they live into the second period. The return on their annuity investment is 33%, 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. Annuities 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. An actuarially fair annuity market makes it much more efficient to smooth consumption (and hence increase utility). C, part a) The government receives 30T in revenue (since it taxes all 30 people equally), and invests it at an interest rate of 0%, so that it has 30T*. =33T in funds to payout in period two. It pays out.75*b in expectation for every type A, and pays out.2*b in expectation for every type B - so in sum, it expects to pay out 20*.75 *B+ 0*.2 *B= 7B. Since the budget must break even at the end of period two, this implies that 33T=7B, or the benefits the government pays out to anyone who lives into period two is 33T/7. C, part b) The government is now trying to decide what T to choose to maximize the sum of utilities. In other words, its maximization problem is: 33T 33T max social welfare = 20* ln(00 T ) + *.75*ln + 0* ln(00 T ) + *.2*ln = 0 T = 5.89, B= T T 00 T 3T 07 So each person pays 5.89 in taxes, and receives a benefit in period two of (if they survive that long). 9

10 A noss B noss A SS B SS = ln(80) + ln(22) = ln(93.75) + ln(6.875) = ln(84.) + ln(30.84) = ln(84.) + ln(30.84) A is better off with social security than without - this is because it is highly probable that a type A reaches the second period, and because the return from "saving" with social security is greater than the returns from private investment. Since B=33T/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 A'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, As find this system better than a world with no social security. Bs, 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, Bs are worse off because the tax rate is too high. One could view this system as redistributive since it taxes Bs more than they wish, and these tax revenues are partially redistributed to As since As are more likely to receive social security benefits. C, part c) Now the government chooses T to maximize B's utility: 33T 00 max B' s utility = ln(00 T ) + *.2*ln 0 T 6.25, B = = = = 00 T 5T 6 which implies A SS 2 B SS 2 = ln(93.75) + ln(2.3) = ln(93.75) + ln(2.3) Now B is better off with this system than without, and is better than the first proposed system. A is still better off than without social security (and if you assume that A can supplement social security income with additional private savings, A's utility will be higher than 5.6), but A 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 A's preferences whatsoever, but the prior system partially weighted A' s preferences, A must be better off with the first social security system. However, assuming A can save in addition to 0

11 social security, A is not worse off than without social security - this is because A 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%. Again, this system redistributes from Bs to As - this is because the ratio of expected payout to payin is much higher for As than Bs. However, Bs 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 American social security system is redistributive. For example, since the PIA is calculated as a progressive function of the AIME, 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 America, 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).