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Unformatted text preview: 2010231511; 13W] KIMlit 2%} Fill] as chap.9 4. Major League Baseball used to use what is known as a 53—1 system to vote for the
Most Valuable Player {MVP} in each league. Each yoter gets to vote fer three different
players they consider worthy of the award. Their firstplace candidate gets 5 points,
their secondplace candidate gets 3 points, and their thirdplace candidate gets 1
point. Points are then added up across all voters, and the player with the most total
points wins the award. Suppose there are three yoters—Neyer, Law, and Phillips—and
ﬁye potential candidates for the award—Alex, David, Raffy, Manny, and Mario. The
table below shows how each yoter ranks the candidates. Raffy is embroiled in a sub—
stance abuse scandal. The “guilty” or “innocent” yerdict will come out the day before
voting, and a guilty verdict will ban him from being voted on as an MVP. mum Rani Third Best salty Refiiy'
Fourth Best .‘vlanny .‘vIat'to
amen a. Who will win the MVP if Raffy is found innocent?
IfRaffy is found innocettt. David gets Hi points [5 frotn Meyer and S frotn Law]. Ales
gets 9' points [3 from each voter]. Raffy gets ?' poittts. and lvlanny gets 1 point. David
wins the MVP. b. Who will win the MVP if Raffy is found guilty?
IfRaffy is fouttd guilty. David still gets ll} points. But Ales now gets 1 1 points: 5
fmnt Phillips and 3 each ﬁatntbleyer and Law. So .ﬁtles wins the MVP. c. What problem with consistent aggregation does this illustrate?
This illustrates a violation of the independence of irrelevant alternatives. Raffy wasn‘t
going to win the competition either way. but the winner changes depending on whether he
is in the cotttpetition or not. T. A problem with the median yoter outcome is that it does not take into account inten
sity of preferences. Suppose that the government decided to give multiple votes to
people with strong preferences, pro or con. Does this solve the problem? 1I.I"~t'hy or why
not? This does not necessarily solve the problettt of preference intensities. For example. cott
sider a vote involving a sitnple "yes" or "no" decision [as opposed to one with multiple op
tions]. If individuals are multiple votes on this issue. they will not divide the votes between
"yes" and "no" in proportion to their preferences. Instead. they will sitttply cast all of their
votes in favor the position they prefer. regardless of their preference intensity. There will be
no change in the outcotne of the vote. Giving individuals tnultiple votes tnay help to solve the problettt of preference intensitie.t
when individuals can decide to split those votes across various different initiatives. .'‘l. person
with stmng opinions about education but only tnild preferences about building bike paths
could cast all of her votes for her favorite school councilor. for example. and none of her
votes for whether to build a bike path. This sort of tnechanistn has its own problems. how
ever: it provides strong incentives for strategic voting. for example. 11. Alfie, Bill, and Coco each value police protection differently. Alﬁe’s demand for the
public good is Cl = 55— 5F, Bill’s demand is Cl = El)— 4P, and Coco’s demand is Cl =
10:)— 10P. If the marginal cost of providing police protection is $13.5,what is the so
cially optimal level of police provision? Under Lindahl pricing,what share ofthe tax
burden would each of the three people pay? To answerthese questions. first rewrite each demand so that P is expressed as a function
of Q. Alfie: PA — ll— {1.29; Bill: PB— Eli — {1259 Coco: PC— ll} — ﬂ. [9. Adding each person ‘s willingness to pay yields PA I FBI PC — 41— {1559. The left
hand side gives the marginal social benefit of providing the ch unit of the good. Setting this
marginal benefit equal to the marginal cost gives the socially optimum level of provision: 41— {1559— 13.5.or Q— ﬁll 1.‘ifhen Q— 50. Alfie‘s marginal benefit is H — {lilﬁﬂl — 1. Similarly. Bill‘s marginal
benefit is El} — {125(501— IS. and Coco‘s is H) — B. [(50] — 5. Ilence. Alfie‘s share ofthe tax
burden under Lindahl pricing is [£115 a 14%. and Bill and Coco‘s shares are approximatel y
Sﬁﬁ‘lﬁ and 31%. respectively. 12. Carrboro has three equalsized groups of people: {1} type A. people consistently prefer
more police protection to less; {2} type B people prefer high levels of police protec
tion to low levels and they prefer low levels to medium levels; {3} type C people prefer
medium levels to low levels, which they in turn prefer by a modest amount to high
levels. a. Which types of people have singlepeaked preferences? Which have multipeaked
preferences?
Types Aand C have singlepeaked preferences. with peaks at "high" and "medium"
respectively. Type B has multiplepeaked preferences. with peaks at "high" and "low" and
a dip at "medium." b. Will majority voting generate consistent outcomes in this case? Why or why not?
Majority voting does not usually generate consistent outcomes when some voters have preferences that fail to be single peaked. But they do happen to generate consistent out
comes in this case. If "high" and "low" are the two options on the ballot. "high" will win.
since types A and B will vote for it. Similarly "high" wins when "high" and "medium" are
the two options on the ballot. 1.‘v‘hen "low" and "medium" are on the ballot. "medium"
wins. since types A and C will vote for it. Finally. when all three are on the ballot. types A.
and B will both vote for "high." which will therefore win. Notice that there are no cycles.
so the voting outcomes are. in fact. consistent. The decisions coincide with those that
would be made by a society that prefers "high" to “medium” and “medium” to "low." chap.12 12. There are two types of drivers on the road today. Speed Racers have a 5% chance of
causing an accident per year, while Low Riders have a 1% chance of causing an acci—
dent per year. There are the same number of Speed Racers as there are Low Riders.
The cost of an accident is $12,000. a. Suppose an insurance company knows with certainty each driver’s type. What pre—
mium would the insurance company charge each type of driver? The insurance company expects to pay out $11000 in claims to 5% of the Speed Rac
ers it covers. so it must collect at least 0.05[$12.000] — $000 from each one. Similarly. it
must collect at least 0.01 [$ [2.000] — S [20 from each Low Rider. 0. Now suppose that there is asymmetric information so that the insurance company
does not know with certainty the driver’s type. Would insurance be sold if i. drivers self—reported their types to the insurance company? Every individual would claim to be a Low Rider. bth if the insurance company sold
insurance to everyone for $120. it would lose money because of the presence of Speed
Racers in the population. The insurance company would quickly increase premiums. bth if
it increased them by too much the Low Riders would leave the market. It cannot be deter
mined here exactly how much more than $120 the Low Riders would tolerate. as their risk
aversion is not speciﬁed. As more Low Riders chose not to purchase insurance. the pool
of covered drivers would include a higher and higher proportion of Speed Racers. requir
ing the insurance company to increase premiums again to coverthe claims. ii. no information at all is known about individual driver’s types? The insurance company could offer a premium that averages the expected claims. In a
population of half Low Riders and half Speed Racers. the pooling premium would be
[$600 ' $120131 — $3 I50. The Low Riders would have to be extremely risk averse to be
willing to pay $3 60 to cover an expected loss of $120. If they [the Lo w Riders] opted out
of the market. the insurance company would be back to the adverse selection problem dis
cussed above: an insured pool containing a high proportion of Speed Racers. 13. Your utility function is U: ln[2C',l where Cis the amount of consumption you have in
any given period. Your income is $40,000 per year and there is a 2% chance that you
will be involved in a catastrophic accident that will cost you $30,000 next year. a. What is your expected utility? Expected utility is the sum of the expected utility in each state. Assuming consump
tion of your entire income. when income is $40,000. utility is Infﬁﬂﬂﬂﬂj e [1.30; when
income is $0.000. utility is lnf20.000] a 0.00. Therefore. expected utility is 93(11301 ‘ 02(00012 [1.262. b. Calculate an actuarially fair insurance premium. What would your expected utility
he were you to purchase the actuarially fair insurance premium? The insurance company would expect to pay 330.000 each year to 2% of its cus
tomers. so over the population it would pay 002(530000] — $600 per person. So $600
would be an actuarially fair premium. A person who purchased actuarially fair insurance would have utility in both states of lniEHﬂﬂﬂﬂ — 60011211375. c. What is the most that you would be willing to pay for insurance, given your utility
function?
To answerthis question. first go back to the original utility to determine how much
money would yield the same utility as the expected utility of taking the risk. The wealth that would yield utility of l [.262 solves lnEEl’I’] — .Dﬁlntﬁﬂﬂﬂﬂl ‘ .02lni20.000]. so W —
explfﬂﬁlniﬁﬂﬂﬂﬂ] * .02lnt20.000]]."2 2 $3 FLU06.20. You are indifferent between the risky
situation and certain wealth of 533306.20 because they yield the same utility. Therefore. you should be willing to pay 340.000 — 33306.20 — $1003.30 for insurance. 14. Billy Joe has utility of U = Inlcl, while Bobby Sue has utility of U = Nl'EWhich person
is more risk averse? Which person would pay the higher insurance premium to
smooth consumption? It is not obvious from looking at the two utility functions which is more risk averse. A
numerical example is helpful. Consider starting from a wealth of [00 and having a 50% chance of losing 00. Billie .loe‘s expected utility is 13: lnflﬂﬂ] ‘ I": Infill] a 3.45. The amount
of [riskless] wealth Billie .Toe would need to have to be just as well off solves InEWHJ] — 1»? Inlflﬂﬂl ‘ 13: lnElﬂ]. or Wm e $3l.62. Bobby Sue‘s expected utility is Hilﬂﬂlj ‘ 1*?[1015 e
6.53. The amount of [riskless] wealth Bobby Sue would need to have to be just as well off
solves (WES? — 6.53. or Was a 43.3 1. This means that Billie Joe is more risk averse than Bobby Sue. To see this. note that the 50% risk of losing 00 is "like" losing 6333(100 — 31.62] for Billie Joe and like losing only 56.60 for Bobby Sue: the same risk hurts Billie .Toe
more. Equivalently. Billie .Toe would be willing to pay up to a 63.33 to fully insure himself.
while Bobby Sue would be willing to pay only up to 56.60. 15. Chimnesia has two equally sized groups of people: smokers and nonsmokers. Both
types of people have utility U = ln[C}, where C is the amount of consumption that peo—
ple have in any period. So long as they are healthy, individuals will consume their en— tire income of$15,000. lfthey need medical attention {and have no insurance}, they
will have to spend $1 0,000 to get healthy again, leaving them with only $5,000 to con—
sume. Smokers have a 12% chance of requiring major medical attention, while non—
smokers have a 2% chance. Insurance companies in Chimnesia can sell two types of policy. The “low de
ductible" {L—J policy covers all medical costs above $3,000, while the “high de
ductible" {Hj policy only covers medical costs above 00,000. a. What is the actuarially fair premium for each type of policy and for each group?
The actuarially fair premium for selling an Lpol icy to smokers is given by
{1. [Eff lﬂﬂﬂﬂ — $3.{}{}{}]— .1 BEHIND] — $3M}. since this is the expected amount the insur
ance company has to pay for medical care. Similarly. the actuarially fair premium for sell
ing an Lpolicy to nonsmokers is .ﬂEf $10M] — $l4ﬂ. The actuarially fair premiums for
selling an Ilpolicy to smokers and nonsmokers are $ﬂ.12[$lﬂ.ﬂﬂﬂ — seem] — $24ﬂ and
.ﬂEfﬂﬂﬂﬂ] — $4ﬂ. respectively. b. If insurance companies can tell who is a smoker and who is a nonsmoker and
charges the actuarially fair premiums for each policy and group, show that both
groups will purchase the L—policy. fine can compute the expected utility of each group with each policy to answer this.
bth there is an easier way. Since both policies are actuarially fairly priced for each group. both groups will prefer the policy that is closer to full insurance—the Lpolicy. Suppose th at smoking status represents asymmetric information: each individual
knows whether or not they are a smoker, but the insurance company doesn‘t. c. Explain why it is impossible, at any prices, for both groups to purchase L—policies
in this setting. Which groups, if any, do you expect to buy Lpolicies, and at what
price? With asymmetric information. companies can‘t offer one price to smokers and another
to nonsmokers: if. for example. the price offered to nonsmokers is lower. smokers will
simply pretend to be nonsmokers. Ilence. if both groups purchase Lpolicies. they must
both pay the same premium. and the premium must be at least [$R4ﬂ : $ Hill"2 — $490 or
the ﬁnn will lose money when it sells to both groups. E We use a to compute the average
cost of claims.] But at this high premium. nonsmokers won‘t find it worthwhile to buy in surance. To wit. their utility ﬁ"om no insurance is
.Dﬁlnfliﬂﬂﬂi : .ﬂElttHﬁ.ﬂﬂﬂ — [H.Wie 9.594
and their utility from an Lpolicy with premium $490 is:
.Dﬁlnflﬁﬁﬂﬂ —4E'ﬂ] : .ﬂElnHﬁﬂﬂﬂI— 490 — lﬂﬂﬂﬂl ‘ Willi] e 9.5m. The only group that would want Lpolicies at this [or higher] prices is smokers. And
if only smokers are buying insurance. it must have a premium of at least $340 for ﬁnns to l'lﬂt lﬂﬁ'ﬂ lllﬂllﬂy. d. Show that it is possible for both groups to purchase insurance, with one group
buying L—policies and one group buying H—policies. Imagine offering a menu of two insurance policies: an $R4ﬂ Lpolicy and a W Il
policy. We know ﬁ"om c that nonsmokers prefer no policy to the highpriced L policy.
Since a W Ilpolicy is actuarially fair for them. we know they prefer this policy to no
policy [they value insurance]. Ilence. when offered this menu. nonsmokers will choose
the cheap Ilpolicy. Firms will break even selling these policies to them. (in the other hand. smokers get expected utility .Eﬁlm’ liﬂﬂﬂ — R40] : .l Elnl.r l iﬂﬂﬂ — Hill} — lﬂﬂﬂﬂ ' 1000] 2 9.53171 from the Lpolicy and expected utility .33 lnIfliﬂﬂﬂ — 4D]
I .12 Iniliﬂﬂﬂ — 4ﬂ— lﬂﬂm I Eﬂﬂﬂ] — 9.521 from the Ilpolicy. Ilence. when offered
this menu of options. smokers will choose the hi ghpriced Lpolicy. Firms will break even selling these policies to them. too. chap.13 12. Lalaland is an extremely stable country with 200,000 residents, half of whom are
young workers and half of whom are retirees. At the end of each “year,” the 100,000
retirees die, the 100,000 young workers retire, and 100,000 new young workers are
born. Workers earn atotal of $5,000 for the year. Lalaland operates a “pay as you go"
social security system, where each current worker is taxed $2,500 and the revenue
collected is used to pay a $2,500 pension to each retiree. The neighboring country,
Gogovia, is larger and more dynamic. Gogovia has an active stock market that Lala
landians can invest in and earn a 10% rate of return. It also has an active banking
sector, which will gladly lend the Lalalandian government money, charging them 10%
interest per year. Lalaland is considering moving to a system of personal accounts, where each
Lalalander would take her $2,500 and invest it in Gogovian markets {and earn a much
higher rate of returnl]. The government would borrow $250 million {$2,500 x 100,000]
from Gogovian bankers to pay for current retirees. It would then tax retirees each year
by just enough to pay the interest on this debt. Would this new system be better or
worse for Lalaland? The new system would be neither better nor worse for Lalanland. In fact. it is an entirely
equivalent system. The interest due on the debt would be [0% x [$25 0m] — $25 m. so taxes would have to be ﬂint"100.000 — $250 per retiree. This is exactly enough to offset the
higher returns Lalalanders would earn in the stock market. (1}. Suppose that you had information about the amount of private savings during the .r" 1?. years before and after the introduction of the Social Security program. How might you
carry out a difference—in—difference analysis of the introduction of the Social Security
program on private savings? How to apply differencesindi fferences to examine private savings would depend on the
nature of the data. If one had data on only nggmgnie private savings. one could evaluate the
effects of introducing Social Security by comparing changes in the US. savings rate [the dif
ference between sav'ings before and savings after Social Security was introduced] to changes
in. say. the Canadian savings rate. In particular. one would compute the difference in the US.
and Canadian savings rates before and after the Social Security system was introduced and the
date the difference between these two differences. The validity of this estimate would rely on
there being no major contemporaneous changes that affected one country bth not the other. Better data would be on imfe'viifuni levels of private savings. Then one could use the fact
that some individuals E for example. certain government employees] were and still are outside
the Social Security system. ﬁne could thus compute the diﬁerence in savings rates before and
after the introduction of Social Security for individuals [ say. of a given age and income level]
affected by Social Security. CIne could then compute a similar difference for similar individu
als not affected by Social security. The difference between these differences would provide an
estimate of the change in private savings that was due to the introduction of Social Security. Consider an economy that is composed of identical individuals who live for two peri
ods. These individuals have preferences over consumption in periods 1 and 2 given
by U = Infci] + Infﬂz]. They receive an income of100 in period 1 and an income of 50
in period 2. They can save as mu ch of their income as they like in bank accounts,
earning an interest rate of 1 0% per period. They do not care about their children, so
they spend all their money before the end of period 2. Each individual’s lifetime budget constraint is given by It":1 + C2!“ + r} = Y1 + Y2!“
+ r]. Individuals choose consumption in each period by maximizing lifetime utility sub
ject to this lifetime budget constraint. a. What is the individual’s optimal consumption in each period? How much saving does he or she do in the first period?
Individuals solv'e
max U— ln{C'] I ln{C2] subject to C" I (TE"fill — lﬂﬂ I Sit"Nil. Rearrange the budget constraint C2 — l if) I if} — l. [CI and plug into the maximand:
max U— ln{C'] I Inflﬁﬂ — MCI]. Then take the derivative and set it equal to zero:
l."C' — l.l."{lﬁ{i — MCI]. or 2.2CI — lﬁﬂ.
So C" a Tl'f. and savings [DD — C" a 213. The optimal consumption in the second
period is then if} I l.l{l{}[}— CI] — Rf}. b. How the government decides to set up a social security system. This system will
take $10 from each individual in the ﬁrst period, put it in the bank, and transfer it to
him or her with interest in the second period. Write out the new lifetime budget
constraint. How does the system affect the amount of private savings? How does
the system affect national savings {total savings in society}? What is the name for
this type of social security system? Individuals now solv'e max [f—lnfC‘ll I In“:le subjectto C" I (2'2."{l.l]— lﬂﬂ— if} I {SDI lﬂfl.l]]."{l.l]. Rearranging the budget constraint giv'es C2 — lﬁﬂ — [.1le again—so the consumption levels are the same as in a. Since aftertax income is H) lower in period 1. however. this
means that private savings falls by Hi per individual. Total savings is unchanged. how
ever. since the increased savings through the government exactly offsets the decreased pri vate savings. This is an example of a funded social security system: the money needed
for second period benefits is saved in the first period. c. Suppose instead that the government uses the 31D contribution from each individ—
ual to start paying out beneﬁts to current retirees {who did not pay in to a social
security when they were working}. It still promises to pay current workers their 31D
{plus interest} back when they retire using contributions from future workers. Simi
larly, it will pay back future workers interest on their contributions using the contri
butions of the next generation of workers. An influential politician says: “This is a
free lunch: we help out current retirees, and current and future workers will still
make the same contributions and receive the same benefits, so it doesn’t harm
them, either." Do you buy this argument? If not, what is wrong with it? The Senator makes it seem as if we can pay benefits to current retirees without ever
paying for it. bth there ‘s {usually} no such thing as a free lunch. which makes us think
someone is harmed by this policy. The key observation is that this policy lowers the na
tional savings rate as compared with b. 1.‘v‘ith lower national savings. the economy will
grow less quickly. and future generations will. in fact. be worse off. 15. For each ofthe reforms listed, briefly discuss the pros and cons ofthe reform, paying
attention in particular to efficiency implications {through potential behavioral re—
spon ses to the change} and equity implications {who wins and who loses}. [N ote that
all reforms are intended to save the system money, so you do not need to list this as
a benefit.] a. In crease the number of years used to calculate beneﬁts from 35 to 40. Increasing the number of years used to calculate benefits could lower benefit s. be
cause more low or zeroearning years would be included in a retiree‘s average wage. To
avoid this reduction in benefits. workers might choose to delay retirement so that they had
40 highearning years included in the calculation. Wﬂfkﬂfﬁ' who spent many years in col lege and graduate schnnl might be mnst vulnerable. as they wnnld have had fewer full
time wnrking years by the time they reach retirement age. Similarly. wnrkers whn had
snme intermptinns in their emplnyment—tn raise a family nr tn retrain fnr a new career.
fnr example—alsn have tn delay retirement tn avnid inclusinn nf zernwage nr lnwwage
yea rs. . Reduce beneﬁts for beneﬁciaries with high asset levels {wealth}. Meanstesting by cnnsidering asset levels wnnld increase the redistributive nature nf
Sncial Security but wnnld induce snme perverse behavinr. Penple might be able tn in
crease their benefits by hiding assets by setting up trusts nr nther entities. fnr example.
They might alsn change the timing nf selling snme nf their assets tn retain Sncial Security
benefits. which distnrts resnnrce mnbility. an efficiency cnncem. 1.‘v‘hile this plan may ap
pear tn benefit the less wealthy at the expense nf the wealthy elderly. it seems vulnerable
tn lnnphnles and evasive behavinr. . Add new state and local government workers to the pool of covered workers {i.e.,
they pay payroll taxes now and receive beneﬁts when they are old}. Brnadening the tax base tn include these wnrkers wnnld yield a net increase tn the
system. Current Sncial Security participants wnnld. nvertheir lifetimes. pay in mnre than
they wnnld withdraw. Therefnre. increasing the number nf wnrkers cnvered wnnld prnvide
a net increase tn the cash flnw in the system. The new wnrkers wnnld stand tn lnse ﬁ"nm
this system relative tn a plan in which they had their nwn retirement accnnnts [because
with Sncial Security they wnnld pay in mnre than they receive]. bth the Sncial Security
system wnnld benefit. This new rule might induce snme tn exit these jnbs. bth since mnst
wnrkers are cnvered by the system. they wnnld have little chnice as tn where else tn wan
tn avnid this tax. . Gradually increase the normal retirement age {NRA} from 65 to TD {under current
laws, the NRA will gradually rise to E? by 2022; the proposal is to speed up this
process so that the NRA will be TD by 2022}. Gradually increasing the nnrmal retirement age will save the fund mnney by reducing
the nnmbernf years during which retirees can cnllect. Penple whn need tn retire earlier
fnr health nr physical limitatinn reasnns will be adversely affected. If they are able tn.
they may attempt tn find less physically demanding wan nr they may increase private
savings in nrder tn be able tn affnrd tn retire earlier. chap.19 2. The demand for rutabagas is C] = 2,000 —100P and the supply of rutabagas is C] = —100
+ 200P. Who bears the statutory incidence of a $2 per unit tax on the sale of rutaba gas? Who bears the economic incidence ofthis tax? If the tax is on the sale of rutabagas. the buyer bears the statutory incidence. since the
"sticker price“ of rutabagas does not include the tax. Economic incidence is determined by
relative elasticities. In this case. the quantity supplied is more responsive to a change in
price. so the less elastic consumers will bear most of the economic incidence. To calculate the relative burdens. solve the equilibrium condition with and without the
tax. Without the tax: 2.000 — 100P — — 100 I 200.”. Price — $71.00. With the tax. the price the
supplier receives is reduced by $2.00. The equilibrium condition is 2.000 — 100P — 2001? — 2] — 100
2.000 — 100P — 200? — 500
2.500 — 300.”. Price — $51.33. The consumers‘ tax burden — [posttax price — pretax price] I tax payments by con
sumers. here $51.33 — $71.00 I 0 a $1.33. The producers‘ tax burden — [pretax price — posttax price] I tax payments by producers.
here $71.00 — $51.33 I $2.00 a $01 In this case the consumer bears a larger share of the tax
burden than the producer. 4. The demand for football tickets is C] = 300 — 10F and the supply of football tickets is C]
= 20F. Calculate the gross price paid by consumers after a per—ticket tax of $4. Calcu—
late the aftertax price received by ticket sellers. For this answer. it does not matter whether the tax is added to the price paid by the con
sumers or subtracted from the price the sellers keep. Adding it to the consumers‘ price yields
demand of 360 — 101? I 4]. which is set equal to supply to yield the equilibrium aftertax
price: 360 — 10F — 40 — 20F. Simplified. 320 — 30?. so base price — $10.6]r and price I tax — $14.15]r paid by con
sumers. Producers keep only $14.15]r — $4 — $10.07*. Without the tax. 360 — 10F — 20F ; 360 — 30F; the price paid by consumers and kept by
sellers is $12. 13. Massive Products, Inc., is a monopolist whose cost of production is given by 10C] +
Gziso its marginal cost curve—equivalently, its inverse supply curve—is given by 10
+ 2G]. Demand for Massive Products’ massive products is C] = 200 — 2P. a. What price will the monopolist charge, and what profits will the monopolist earn?
What will consumer surplus be? First we calculate the profitmaximizing quantity by setting marginal cost equal to
marginal revenue. Marginal cost is 10 I 29. Marginal revenue can be found by solving
for the inverse demand curve. P — 100 — HQ and noting that the marginal revenue curve
has the same Paxis intercept and is twice as steeply sloped. Ilence. marginal revenue is
100 — Q. Setting MR — MC and solving for Q. 10I 2Q—100—Q,orEQ—00.orQ—30. Therefore. the profitmaximizing quantity is 30. and the profitmaximizing price can
be found from the inverse demand curve: P — 100 — ‘xf 130] — $515. Profits are computed as
the difference between total revenue and total cost. or $515 [30] — 10130] — 302 — 2.550 —
1.200 — 1.350. Consumer surplus can be computed as the area of the triangle with width (3— 30 and
height 100 — R5 — 15 [the diﬁ"erence between the Pintercept of demand and the price paid]. Computing. consumer surplus — 13.1 130 ]1 15] — 225. b. How will the monopolist’s price and profits change if atax of $15 per unit is im—
posed on the buyers of the product?1
Imposing a $15 tax on buyers will change their demand curve to Q — zoo — 2IP‘ l5 1.
or Q — lTﬂ — 2?. where P is the pretax [“sticker"] price. The new inverse demand is P —
R5 — 13: Q. and the new marginal revenue is P — R5 — Q. Setting equal to marginal cost and
solving gives ma 29—35 —Q,or3Q—T5.orQ—25. The profitmaximizing price is thus P— 35 — I": Q— $T2.5ﬂ_ Proﬁts are given by
$22.51?) [25 I — lﬂf25] — 252 — [.312 — 325 — 937.5. c. What is the deadweight burden of the tax? To compute the deadweight burden of the tax. we look at the change in total surplus
[including tax revenue as surplus]. The aftertax consumer surplus can be computed from
the new demand curve: I»? [25 ]I [2.510] — l5ﬁ_25. where 25 is the quantity purchased and
[2.50 — R5 — 22.5 is the difference between the Pintercept of demand and the price paid.
The tax revenue is 25Lr l 5] — 3T5. Ilence. the deadweight burden of the tax is I,r l .3 5i} 1
225] — [932.5 I l5ﬁ_25 I 325 I — [5T5 — Hot3.25 — lﬂﬁ_25_ 15. The city of Malaise is considering a 10% tax on the revenues of all hotelsimotels in
side the city limits. Although not completely different from hotels and motels in the
nearby suburbs, the ones in Malaise have a distinct advantage in their proximity to
the interesting sights and convention centers. 5o individuals will pay some premium
to stay in Malaise rather than to stay nearby. Furthermore, all land is used equally well by hotelsimotels and other forms of
business; any Malaise land not talten by a hotelimotel is readily absorbed by other
forms of business. Mayor Maladroit calls you in to advise him on the incidence of such a tax. He is
particularly concerned with who will bear this tax in the short run {one month} and the
long run {ﬁve years}. a. What is the incidence of the tax in the short run? Answer intuitively, and use a dia
gram if possible. In the short run. the city hotels and motels will share the burden of the tax with their
guests. From the description of the town. it appears that demand for lodging near the city
center is relatively inelastic. so the businesses will be able to increase their rates without
losing many customers. Ilowever. the supply of hotel and motel space is likely to be fairly
inelastic too. particularly in the short run: it takes some time to convert hotels and motels
to other uses. and vice versa. Graphically. both demand and supply will be steep: Ron m rates
{FIFlW‘} Goneumer page ﬁlm P
llo1elJMotel Ioeepa Rooms pornlght In the short run. this tax yields very little quantity change but a significant price
change. b. What is the longrun incidence? Once again, use a diagram if possible.
In the long run. the number of hotel and motel rooms can be adjusted: thus. supply is more elastic. In the long run the tax incidence will fall primarily on the inelastically de
manding guests. The price kept by the firms is close to the original price. Room rates
{PFIW} CDI'IEIJI'HEI' paya Gunmen P
Hoteiila'lmei keeper c. How would your analysis in b change if hotelslmotels in the suburbs were perfect
substitutes for those in Malaise? What would happen to tax revenues? The availability of an untaxed substitute in the suburbs would make demand more
price elastic. This elasticity would have two effects on the market. First. the firms would
not be able to shift the lion‘s share of the tax burden to the consumers. because if they
did. visitors to the area would simply stay in the suburbs. Second. the number of rooms
rented would fall. so the revenue raised by this tax would decline relative to a. chap.20 1. The marltet demand for super—sticky glue is G = 24D — BF and the marltet supply is G =
—ED + 4F. a. Calculate the deadweight loss of a tax of $4 per unit levied on producers of super—
sticlty glue. Dead weight loss is calculated as the area of a triangle. the height of which is the dol
lar amount of the tax and the base of which is the change in quantity purchased resulting
from the tax. First. determine the change in quantity associated with this tax. 1.‘viithout the tax. equi
librium is 24H — ﬁP — —ﬁ{} I 4?. or 3TH} — lﬂP. Equilibrium price is El]. so equilibrium
quantity is —ﬁﬂ I [4 X 3’0] — [El] — Frﬂ— so. A tax levied on producers changes the supply function to Q — —ﬁr I 4E.” — 4] because
the price the producers can keep from any sale is reduced by $4. Recalculating equilib
rium. 24C! — ﬁP— —ﬁﬂ I 4P — lb. or 3 [Fr — lﬂP. Equilibrium price is $3 Lot}. so equilib
rium quantity is 240 — 6(31 .ﬁﬂ] — 5&4. The change in quantity is ﬁt} — 5&4 — as. so the area of the deadweight triangle is a
[9.ﬁil’4] — [9.2. b. How does deadweig ht loss change if the tax is levied on consumers of super—
sticlty glue? Intuitiver you would expect the deadweight loss to be exactly the same. The legal li
ability for the tax does not change the economic incidence of the tax. In this case. the height of the triangle is still the $4 tax. 1Mhen the tax increases the
price a consumer must pay. the new demand function is Q— 240 — ﬁt? I 4]. The new
equilibrium condition is 240 — 6P — 24 — —so I 4?, or 2T6 — lﬂP. Price is ﬂint}. and quantity is —ﬁﬂ[4 x 2160] — 5&4. exactly the quantity that re
sulted when the tax was imposed on the produce r. 1'. You are a consultant to the government of Buttony. The government has decided to
cut taxes on either apples, bananas, or cantaloupe, and it wants your input on which
fruit would be the best choice for a tax cut. It provides you with the following informa
tion. What is your recommendation, and why? Marginal lax .‘ﬂ arginal
revenue deadweight loss
(thousands of [thousands of
Sales dollars per $1 dollars per $1
("rood L'nit Price [thousands] L'nil lax additional tax} additional tax}
Apples SJ Jﬂﬂ HUD 2f] 5
Bananas 52 Hit] H125 3f] 2D
Cmtaloupc 54 5f] SH] 5 H] 2H Taxe s should be set so that the marginal dead weight burden per marginal dollar of re venue
generated is equal across goods. This is not true in Buttony. The marginal deadweightburden
per marginal dollar of revenue is much higher for cantaloupe [2W 1 {1 — 2 is Ell"Ell 'JI SEEM than
for the other goods. Cutting taxes on cantaloupe would be the most efficient: for a given rev
enue reduction, cutting cantaloupe tax es would reduce the deadweight loss the most. 10. The marltet demand for stuffed rabbits is Cl = 2,600 — 20F, and the government in
tends to place a $4 per bunny tax en stuffed rabbit purchases. Calculate the clearl
weightless of this tax when: a. Supply of stuffed rabbits is Cl = 400.
The quantity befe re the tax is 400; the quantity after the tax is 400. 1.‘fhen snppl y is al
ways 400 rabbits. the deadweight less ef the bunny tax is 13: [4 X 0]. er 0. There is ne
change in supply. se there is ne deadweight less. b. Supply of stuffed rabbits is Cl =12P. In this case. supply is net cetnpletely inelastic. se beferetax ancl aftertax quantities
must be calculated. Befere tax: 2.600 — 20P— 12?; P— $01.25; 9— [2 X 01.25 — 015. After tax: 2.600 — 20P— l2fP—4]: 2.600 — 20P— l2P — 40 2.640 — 32?; P— 02.15; 9— 2.600 — [20 X 02.?5]— 045. The quantity change is 025 — 045 — 30. se the area ef the cleaclweight less triangle is
1»? [30 X 4] — 60. c. Explain why the deadweight less calculatiens differ between a and b. Deadweight less is caused by changes in the equilibrium quantity. In a. because sup
ply was perfectly inelastic. there was ne change in quantity. 1I‘r'hen quantity clees net
change. the tax has caused ne clistertien. Thus. there is ne deadweight less. enly a transfer
ef tneney fretn the seller te the get'erntnent 15. The demand for snorkels in Eerhama is given by as = ﬁbb— 3P5 and the supply of
snorkels in Berhama is given by as = Ebb + 4P5. The demand for kayaks is given by
Gk = Ebb — BF]: and the supply of kayaks is given by Gk = Eb + 1.5Pk. Both goods are
currently untaxed, but the government of Eerhama needs to raise $5,0bb {to ﬁnance a
new lighthouse} by taxing snorkels and kayaks. What tax should it levy on each ofthe
two goods?
If it puts a tax of r3 on snorkels. the equilibrium price will solve ﬁﬂﬂ — EFF — Eﬂﬂ ‘ 4TB? — 1'3]. or [2P3 — 301'} 3 41'], or PR — 25 I 3.33. The quantity of snorkels sold will be QR — Sﬂﬂ —
REES I try"3] — 31313 — Sty"3. 1.‘v'ith no tax. 3th} snorkels are sold. Hence. the deadweight burden
of taxation is DWIR — 1.321} :19? — 1.321} [Eff"3].
The tax revenue from the snorkel tax is:
TR? _ r.st _ Eﬂﬂ rs _ RFE'IG' The marginal DWI is thus Eff}. and the marginal revenue is 31313 — [diff"3.
A similar exercise for kayaks yields D iﬂk — [3'2 q, ﬁg — [.3213 [r5 ff’ﬁi
and
TR}? — rd, Qk— 4 lﬂrk— erg'3. The marginal DWI. is thus ﬁt} I 5. and the marginal revenue is 4 ll} — [Err"5.
The optimal taxes must equate the ratio of the marginal D WI to the marginal revenue: Br]. 3 6ri :3 300 916:]. ..'3 ' 41:3— 12ri ﬁt"
[3;]. 3] 3 1 a — [3r]. 3] {1 3ri :3] = {my 3] (a:i 3]
ﬁt"
[3331333er = 333:] Hence, tf“tk— Inﬁll"325m — [Bi"4W for an efficient tax. Setting taxes 1", and r3 —
E [Si"illﬂirkand yields total revenues 4. or: 413:,r 43:; 33+ 300l135."410th _ $13334 may] Setting this equal to the $5,000 in revenue they need to raise [and solving numerically]
yields r,t =10.13 and r5. = 3.33. chap.21 0. Suppose that the government introduces an Earned Income Tax Credit such that for
the ﬁrst $0,000 in earnings, the government pays 50¢ per dollar on wages earned. For
the next $3,000 ofearnings, the credit is held constant at $4,000, and after that point
the credit is reduced at a rate of 20¢ per dollar earned. When the credit reaches zero,
there is no additional EITC. a. Draw the budget constraint that reﬂects this earned in come tax credit for a worker
who can work up to 4,000 hours per year at an hourly wage of $10 per hour. The 50;: cent subsidy applies to the first $0.000 of earnings. or the first 000 hours of
work. This corresponds to 3.200 hours of leisure and a consumption of $l2.000. The nest
$3.000 of earnings—300 hours of labor—is untaxed. Ilence. at 2.000 hours of leisure. the worker gets a consumption of $ [5.000. The $4.000 EITC benefit is phased out gradually.
disappearing after $20.000 in additional earnings. Ilence. at 000 hours ofleisure. the
worker gets to consume $31.000. Cons umption
$40,000 31 ,0 00 1 5,000
1 2,000 000 2,000 3,200 4,000 Leisure {hours} b. Illustrate on your graph the portions of the budget constraint where the labor sup—
ply effects ofthe policy are positive, negative, or ambiguous, relative to the “no
policy" status duo. Consumption
$40,000 31,000   D ism“ rages work Discourages work
15.naa ;   X 1 2‘ D ............... ............................................ .. . I Amhl guous lIm'ourngcs n or'k .1.
000 2.0003200 4,000 Leisure {hours} @ Suppose that you estimate the following female labor supply relation ship:
Labor supplyj— —32ﬂ I ﬁﬁtaftertax wagelj. ‘ 32ﬂtcollege graduate 11.—[2W marri edif.
where labor supply is measured in annual hours worked and wages are expressed in hourly
wages. a. Interpret the coefficient on aftertax wages. What does this coefﬁcient imply about
the effect of increasing wages from $6 to $10 per hour on labor supply? The coefficient on aftertax wages is positive. indicating that a higher aftertax wage
increases labor supply. The magnitude of the effect is 35: for each dollar increase in after
tax wage. all else equal. a female will work 35 more hours per year. For a $4 increase
[from $5 to f [H]. that translates to 4 X 35 — 340 hours. b. What can we learn from this estimate about the income and substitution effects of
wages on labor supply?
This estimate does not explicitly include a measure of nonlabor income. The approach
described in the text subtracts the nonwage income effect from the wage effect to isolate the substitution effect of wages on labor supply. That cannot be done here. The most we can confidently state is that the total of income and substitution effects is positive. so any
negative effect on labor supply arising from the income effect is more than offset by the
substitution effect. The substitution effect. which induces more work as leisure becomes more expensive. must be greater than the income effect. which induces less work as in
come increases. c. How might this coefficient estimate be biased? Explain. This estimate holds marital status and having a college degree constant. Given those
controls. the coefﬁcient of interest is ‘ 35. indicating that women who earn a higher wage
work more hours. A number of other explanatory variables would have to be included to
avoid bias. There is no control for family size or presence of children. and it may be the
case that mothers are more likely to work parttime and to accept a lower hourly wage in
exchange for work hour flexibility. It is also possible that the women who are earning the
highest wages and working the longest hours are somehow different from other women.
notjust in presence of children bth in chosen careers. in attitudes aboth working. or in
ambition. Thus. there are a number of competing explanations for the observed correlation
between wages and hours that this crosssectional estimate cannot distinguish. 11. Why does the Earned In come Tax Credit exacerbate the marriage penalty for lowin—
come workers? Euggest an alternative method of calculating the EITC that reduces
this penalty. The EITC exacerbates the marriage penalty by combining both spouses‘ incomes to de
termine eligibility for the credit. Two fairly low incomes can combine to equal a total family
income high enough to place the family in the phaseout portion of the EITC. In that situa
tion. adding a second income to the first puts the second income in the range of a very high
marginal tax rate. This effect could result in a labor supply reduction for secondary earners
in these families. To counter this effect. the EITC.‘ could be amended so that the ave rage of
the two spouses‘ salaries detennined the family income. it could provide for a much longer
plateau before pha seout for twoeamer families. or it could be applied to :‘ndiw'tfmrl in
comes. regardless of marital status. ratherthan to family income. 13. You graduate from college and take a job at a consulting ﬁrm with a wage of $25 per
hour. Your job is extremely flexible: you can choose to work any number of hours
from 0 to 2,000 per year. a. Suppose there is an income tax of the following form:
Income up to $10,000: no tax
Income from 010,000—030,000: 20% tax rate
Income from $30,000 up: 30% tax rate
Draw a graph in hours workedlconsumption space, showing your opportunity set
with and without the tax system. With the tax system in place, are there any points
that you are particularly unlikely to ch oose? Why or why not? With an hourly wage of $25. the points of interest in the labor"leisure budget con
straint will be $10000 and 400 hours of labor [ [.600 hours of leisure] and $30000 and
[.200 hours of labor [000 hours ofleisure]. At leisure ofmore than [.600 hours. the slope
of the budget constraint is the wage of 25; between [.000 and 000 hours of leisure. the slope is 00% of the wage. or 20; at less than 000 hours of leisure. the slope is T0% of the
wage. or [15. The yintercept will be $10000 + .0[$20.000] + .T[$20.000] — $40000. Consu mptlon 550.000
540.000
526.0(1) $10.0ﬂ) 0 300 1.600 2.000 Lolsuro{l1:urs]l There are no points that you are particularly unlikely to choose because the re are no
sharp discontinuities or perfectly flat portions of the budget constraint. A marginal tax rate of l [or even greater!] would completely discourage work. bth there are no such tax rates
in this system. 0. Say you choose to work 1,500 hours per year. What is your marginal tax rate?
What is your average tax rate? Do these rates differ? Why or why not?
1.‘vhrking [.500 hours per year would yield an income of [.500 a $25 — $3 2.500 and
would put you in the highest tax bracket. with a marginal tax rate of 30%. To calculate the
average tax rate. divide total taxes paid by income: the first $l0.000 of income is untaxed;
the next $20000 of income is taxed at the rate of 20%. or $4.000; the remaining $2.500 is
taxed at the rate of 30%. or $2.250. Total taxes are $0,250. The average tax rate is 025031500 2 [0.00%. The marginal
tax rate is higher than the average tax rate because the progressive stmcture of' this tax
system taxes the last dollar earned at the highest rate; the average tax rate includes the
lower marginal rates paid on the ﬁrst $30000 of income. c. Suppose th at the two tax rates are increased to 25% end 50%. What is the likely ef—
fect on the labor supply of men? Wh at is the likely effect on the labor su pplyr of
married women? Explain how the responses might differ between these groups,
both in terms of underlying economic effects and in terms of the empirical evi—
dence on labor supply responses. The 25% rate is a slight increase o1.'er the current Eﬂ‘tﬁ rate; the ﬁﬂ‘ﬁ: rate is H} per
centage points higher than the original tax rate for the higher bracket. The labor supply of
men is generally thought to be inelastic: the empirical estimate of elasticity is approxi
mately —0. 1. This inelasticity suggests that the labor supply of men would be minimally
affected by this change. The labor supply of married women. though. has been estimated
to be much more elastic: a higher tax rate would tend to reduce their work hours by more.
These predictions based on empirical evidence are supported by economic theory. Sec
ondary earners (historically. married women] face high marginal tax rates e1.'en if they
earn low wages. since the primary earner‘s income pushes the family into a higher tax
bracket. This provides a strong di sincenti1.'e to work. particularly if the secondary worker
has home production alternatives such as child care. ...
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This note was uploaded on 12/10/2010 for the course ECONOMICS 432 taught by Professor Jannett during the Spring '10 term at Brown.
 Spring '10
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