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Unformatted text preview: 130 BUYINGANDSELUNG (Ch. 9) .I'.‘ , .
(b) I]. Dudley can work as many hours a day as he Wishes for a wage rate of $10 an hour, how many hours of leisure will he choose? 7 . How many hours will he work? 9 . (Hint: Write down Dudley’s
budget constraint. Solve for the amount of ieisure that maximizes his
utility subject to this constraint. Remember that the amount of labor he
wishes to supply is 16 minus his demand for leisure.) (c) If Dudley’s nonlabor income decreased to $5 a day, while his wage rate remained at $10, how many hours would he choose to work? 9 . (a!) Suppose that Dudley has to pay an income tax of 20 percent on ali
of his income, and suppose that his bef'orr—ntax wage remained at $10 an
hour and his beforetax nonlabor income was $20 per day. How many hours would he choose to work? 8 . Chapter 10 NAME
lntertemporal Choice Introduction. The theory of consumer saving uses techniques that you
have already icarued. In order to focus attention on consumption over
time, we will usually consider examples where there is only one consumer
good, but this good can be consumed in either of two time periods. We
wili be using two “tricks.” One trick is to treat consumption in period I
and consumption in period 2 as two distinct commodities. If you make
period—1 consun‘iption the numeraire, then the “price” of period—2 con—
sumption is the amount of period1 consumption that you have to give
up to get an extra unit of period—2 consumption. This price turns out to
be 1/(1 + r), where r is the interest rate. The second trick is in the way you treat income in the two different
periods. Suppose that a consumer has an income of ml in period 1 and
mg in period 2 and that there is no inﬂation. The total amount of period
1 consumption that this consumer could buy, if be borrowed as much
money as he could possibly repay in period 2, is ml + As you
work the exercises and study the text, it should become clear that the
consumer’s budget equation for choosing consumption in the two periods
is always c2 m2
= m + mm.
1+r 1 l+r Ci. + This budget constraint looks just like the standard budget constraint that
you studied in previous chapters, where the price of “good 1” is 1, the
price of “good 2” is 1/(1 + r), and “income” is nil + Therefore
if you are given a consumer’s utility function, the interest rate, and the
consumer’s income in each period, you can ﬁnd his demand for consump—
tion in periods 1 and 2 using the methods you already know. Having
solved for consumption in each period, you can also ﬁnd saving, since the
consumer’s saving just the difference between his period1 income and his period—1 consumption. Example: A consumer has the utility function U ((31, c2) 1 c102. There is
no inﬂation, the interest rate is 10%, and the consumer has income 100
in period 1 and 123 in period 2. rEllen the consumer’s budget constraint
(:1 + (:2 / 1.1 = 100 + 121/11 3 210. The ratio of the price of good 1 to the
price of good 2 is 1 + r m 1.1. rFhe consumer will choose a consmnption
bundle so that [HUI/114072 = 1.1. But IVIUl = 02 and 1M U; ﬂ (:1, so the
consumer must choose a bundle such that (:2 /c1_ = 1.1. Take this equation
together with the budget equation to solve for c; and (:2. The solution is
c1 = 105 and c2 = 115.50. Since the consumer’s period—l income is only
100, he must borrow 5 in order to consume 105 in period 1. To pay back
principal and interest in period 2, he must pay 5.50 out of his periodu2
income of 121. This leaves him with 115.50 to consume. 132 INTERTEMPORAL CHOICE (Ch. 10) You wili also be asked to determine the effects of inﬂation on con—
sumer behavior. The key to understanding the effects of inflation is to
see what happens to the budget constraint. Example: Suppose that in the previous example, there happened to be
an inﬂation rate of 6%, and suppose that the price of period1 goods is
1. Then if you save $1 in period 1 and get it back with 10% interest,
you will get back $1.10 in period 2. But because of the ination, goods
in period 2 cost 1.06 dollars per unit. Therefore the amount of period—1
consumption that you have to give up to get a unit of period—2 consump
tion is 1.06 / 1.10 = .964 units of period2 consumption. If the consumer’s
money income in each period is unchanged, then his budget equation is
(:1 + .964c2 2 210. This budget constraint is the same as the budget
constraint would be if there were no inflation and the interest rate were
7', where .964 = 1 / (1 + 7'). rI‘he value of 7' that solves this equation is
known as the real rate of interest. In this case the real rate of interest
is about .038. When the interest rate and inﬂation rate are both smaii,
the real rate of interest is closely approximated by the difference between
the nominal interest rate, (10% in this case) and the inflation rate (6%
in this case), that is, .038 N .10 — .06. As you will see, this is not such a
good approximation if inflation rates and interest rates are large. 10.]. (0) Peregrine Pickle consumes (61,62) and earns (ml, pig) in periods
1 and 2 respectively. Suppose the interest rate is r‘. (0:) Write down Peregrine’s intertemporal budget constraint in present (:2 Tit2
value terms. C} ‘i— : ml ‘l’ (1+?) ' (i—H‘) (b) If Peregrine does not consume anything in period 1, what is the most
he can consun'le in period 2? m1 l 9") + m2 . rThis is the (future
value, present value) of his endowment. Future value . (c) if Peregrine does not consume anything in period 2, what is the most . . . m , . .
he can consume in period 1? 771.1 + (1 . I‘lns 1s the (future value, present value) of his endowment. Present value . What is the
slope of Peregrine’s budget line? ~i— 7") . 10.2 (0) Molly has a CobbDouglas utiiity function U {C}, (:2) = c‘f‘cya,
where 0 < a < l and where c; and c2 are her consumptions in periods 1
and 2 respectively. we saw earlier that if utility has the form n(.r1,m2) 2:
:r‘fnnlz‘i and the budget constraint is of the “standard” form plml +pgmg m, then the demand functions for the goods are 331 = (rm/I91 and 3:2
(1 elm/pg. El NAME _...____.__..___— 133 (a) Suppose that Molly’s income is m; in period 1 and 771.2 in period 2. Write down her hud et constraint in terms of aresent values. C +
g I 1 Cg/(1+T) mm1+m2/(1 +7"). (1)) \Ve want to compare this budget constraint to one of the standard form. In terms of Molly’s budget constraint, what is m? 1 . W'hat ispg? VVhatism? (c) If a = .2, solve for Moliy’s demand functions for consumption in
each period as a function of m1, mg, and 7". Her demand function for consumption in period 1 is C} = .2771: + .2m2/(1 § 7') . Her
demand function for consurnptior‘l in period 2 is (:2 = + r)m1 l— 87712 . (d) An increase in the interest rate will decrease her period—l
consumption. It will increase her period—2 consumption and increase her savings in period 1. 10.3 (0) Nickleby has an income of $2,000 this year, and he expects an
income of $1,300 next year. He can borrow and iend money at an interest
rate of 10%. Consumption goods cost $1 per unit this year and there is
no inﬂation. 134 INTERTEMPORAL CHOICE (Ch. 10) Consumption next year in 1,0003 . 1 z ~ ~. ‘ E i f i i i .2 l g i g i a 2, . ; i 2 3 3’ i r i i l 3 ..\._.;_Rec§ Ilnei E i 1 3 2 i e
i i i S E l i E % i i e i E s i 0 l 2 3 4 Consumption this year in LOOOS (a) What is the present value of Nicklehy’s endowment? $3 , 000 . What is the future value of his endowment? $3 , 300 . With blue
ink, show the combinations of consumption this year and consumption next year that he can afford. Labei Nickelby’s endowment with the letter
E. ([1) Suppose that Nickleby has the utility function U (01,02) = 0102.
Write an expression for Nickieby’s marginal rate of substitution between
consumption this year and consumption next year. (Your answer will be a function of the variables 01,02.) MRS : —02/ Cl .
(c) What is the slope of Nickleby’s budget line? M11. Write an equation that states that the slope of Nickleby’s indifference curve is equal to the slope of his budget line when the interest rate is 10%. 3
02/01. Also write down Nickleby’s budget equation. 01 l—
02/11: 3,000. (03) Solve these two equations. Nickieby wili consume 1 , 500 units in period 1 and 1 , 650 units in period 2. Label this point A on your
(.li;1.p;1‘a.nl. NAME —_._mm 135 ((3) Will he borrow or save in the ﬁrst period? Save . How much? 500. (f) On your graph use red ink to show what Nickleby’s budget line would
be if the interest rate rose to 2000. Knowing that Nicklehy chose the
point A at a 10% interest rate, even without knowing his utility function,
you can determine that his new choice cannot be on certain parts of his
new budget line. Draw a squiggly mark over the part of his new budget
line where that choice can not be. (Hint: Close your eyes and think of WARP.) {g} Solve for Nickleby’s optimal choice when the interest rate is 20%.
Nickleby will consume 1 , 458 . 3 units in period 1 and 1 , 750
units in period 2. (it) Will he borrow or save in the ﬁrst period? Save . How much? 541.7. 10.4 (0) Decide whether each of the following statements is true or
false. Then explain why your answer is correct, based on the Slutsky
decomposition into income and substitution eliects. (a) “If both current and future consumption are normai goods, an increase in the interest rate will necessarily make a saver save more.” False .
Substitution effect makes him consume less
in period 1 and save more. For a saver,
income effect works in opposite direction.
Either effect could dominate. (b) “If both current and future consumption are normal goods, an in—
crease in the interest rate will necessarily make a saver choose more consumption in the second period.” True . The income
and substitution effects both lead to more consumption in the second period. 10.5 (1) Laertes has an endowment of $20 each period. He can borrow
money at an interest rate of 200%, and he can lend money at a. rate of
0%. (Note: If the interest rate is 0%, for every dollar that you save, you
get back $1 in the next period. If the interest rate is 200%, then for every
dollar you borrow, you have to pay back $3 in the next period.) 136 INTERTEMPORALCHOICE (Ch. 10) (a) Use biue ink to illustrate his budget set in the graph below. (Hint:
The boundary of the budget set is not a single straight line.) C2
40 .é l l i Reilline‘ i l l r i 29 i I .3
l i 3 g ’ i l 3 ;B IO 5 .
l l 0 IO 20 30 40 C  (b) Laertes could invest in a project that would leave him with m1 m 30
and mg = 15. Besides investing in the project, he can still borrow at 200%
interest or lend at 0% interest. Use red ink to draw the new budget set
in the graph above. Would Laertes be better off or worse oil by investing
in this project given his possibilities for borrowing or lending? Or can’t one tell without knowing something about his preferences? Explain. Better off. If he invests in the project,
he can borrow or lend to get any bundle he
could afford without investing. (c) Consider an alternative project that would leave Laertes with the
endowment m1 3 15, m2 = 30. Again suppose he can borrow and lend
as above. But if he chooses this project, he can’t do the ﬁrst project.
Use pencil or black ink to draw the budget set available to Laertes if he
chooses this project. ls Laertes better off or worse off by choosing this
project than if he didn’t choose either project? Or can’t one tell without knowing more about his preferences? Explain. Can’t tell. He
can afford some things he couldn’t afford
originally. But some things he could afford
before, he can’t afford if he invests in
this project. 10.6 (0) The tabie below reports the inﬂation rate and the annual rate
of rcturu on treasury hilis in several countries for the years 1984 and 1985. NAME W 137 Inﬂation Rate and Interest Rate for Selected Countries % Interest % Inﬂation % Inﬂation 0/13 Interest Country R Rate, 1985 Rate, 1984 Rate, 1985
United States W. . .
Israel 48.1 217.3 210.1 Switzerland 3 .1 4.1
Ira1y 92 6722 NA 06 20 2
! (a) In the table below, use the formula that your textbook gives for the
exact real rate of interest to conn'nite the exact real rates of interest. (b) What would the nominal rate of return on a bond in Argentina have to be to give a real rate of return of 5% in 1985? 710.8% . What
would the nominal rate of return on a bond in Japan have to be to give . . . 0
a real rate of return of 5% in l985? 7 . 1 /o . (a) Subtracting the inflation rate from the nominai rate of return gives
a good approximation to the real rate for countries with a low rate of
inﬂation. For the United States in 1984;, the approximation gives you 6% while the more exact method suggested by the text gives you 5 . 79% . But for countries with very high inflation this is a poor approximation. The approximation gives you for Israel
. 0
in 1984, while the more exact formula gives you . For Argentina in 1985, the approximation would tell us that a bond yielding a nominal rate of 677 . 7% would yield a real interest rate of 5%. This contrasts with the answer 7 10 . 8% that you found above. 138 INTERTEMPORALCiiOICE (Ch. 10) Real Rates of Interest in 1984 and 1985 57 55
4157 1094 10.7 (0) We return to the planet Mungo. On Munge, macroeconomists
and bankers are jolly, clever creatures, and there are two kinds of money,
red money and blue money. Reeali that to buy something in Mungo you
have to pay for it twice, once with blue money and once with red money.
Everything has a blue—money price and a redmoney price, and nobody
is ever allowed to trade one kind of money for the other. There is a blue
money bank where you can borrow and lend blue money at a 50% annual
interest rate. There is a red—money bank where you can borrow and lend
red money at a 25% annual interest rate. W. Germany A Mungoan named Jane consumes only one commodity, ambresia,
but it must decide how to allocate its consumption between this year and
next year. Jane’s income this year is 100 blue currency units and no red
currency units. Next year, its income will be 100 red currency units and
no blue currency units. The blue currency price of ambrosia is one b.c.u.
per Hagen this year and will be two b.c.u.’s per Hagen next year. The red
currency price of ambrosia is one 1‘.c.n. per Hagen this year and will he
the same next year. (a) if Jane spent all of its blue income in the ﬁrst period, it would be enough to pay the blue price for 100 ﬂagons of ambrosia. If Jane
saved all of this year’s blue income at the blue—money bank, it would have 150 b.c.u.’s next year. This would give it enough biue currency
19 to pay the blue price for 75 flagens of ambrosia. On the graph
below, draw J ane’s blue budget line, depicting all of those combinations
of current and next period’s consumption that it has enough blue income
to buy. NAME __—_m 139 Ambrosia next period iOO 75 50 25 0 25 50 75 i 00
Ambrosia this period (b) If Jane planned to spend no red income in the next period and to
borrow as much red currency as it can pay back with interest with next period’s red income, how much red currency could it borrow? 80 . (c) The (exact) real rate of interest on biue money is . The reai rate of interest on red money is . (d) On the axes below, draw J ane’s blue budget line and its red budget
line. Shade in all of those combinations of current and future anibrosia
consumption that Jane can afford given that it has to pay with both
currencies. 2W®KW§W§R§X¢W3$M$$EWQAmaﬁWWWm 140 INYERTEMPORALCIiOiCE (Ch. 10) Ambrosia next period 100 75 50 25 0 25 50 75 100
Ambrosia this period (6) It turns out that Jane ﬁnds it optimal to operate on its blue budget
line and beneath its red budget iine. Find such a. point on your graph and
mark it with a C. { f ) 0n the following graph, show what happens to Jane’s original budget
set if the blue interest rate rises and the red interest rate does not change.
On your graph, shade in the part of the new budget line where Jane’s
new demand could possibly be. (Hint: Apply the principle of revealed
preference. Think about what bundles were available but rejected when
.hrnc chose to consume at C before the change in blue interest rates.) NAME_ — 141 Ambrosia next period m0 f E _3 g
50
25
0 25 50 75 E00 Ambrosia this period 10.8 (0) Mr. O. B. Kendle will only iive for two periods. In the ﬁrst
period he wili earn $50,000. In the second period he wiil retire and live
on his savings. His utiiity function is U (61,02) 2: (1102, where (:1 is con—
sumption in period 1 and c2 is consumption in period 2. He can borrow
and lend at the interest rate r = .10. (a) If the interest rate rises, will his period—1 consumption increase, de
crease, or stay the same? Stay the same. His demand
for 01 is .5(nt;imrn2/(1—trﬁ) and TRQzZCL (1)) Would an increase in the interest rate make him consume more or less in the second period? More. He saves the same
amount, but with higher interest rates, he
gets more back next period. (c) if Mr. Kandle’s income is zero in period 1, and iii 55,000 in period 2,
would an increase in the interest rate make him consume more, less, or the same amount in period 1? Less . 10.9 (1) Harvey Hebit’s utility function is U (cl , Cg) = 1nin{cl , c2}, where
cl is his consumption of breed in period 1 and (:2 is his consumption of
bread in period 2. The price of bread is $1 per loaf in period 1. The
interest rate is 21%. Hervey earns $2,000 in period 1 and he will earn
$1,100 in period 2. 142 INTERTEMPORAL CHOICE (Ch. 10) (a) “trite Harvey’s budget constraint in terms of future value, assuming
no inflation. 1.210; § C2 = 3, . (b) How much bread does Harvey consume in the ﬁrst period and how much money does he save? (The answer is not necessarily an integer.)
He picks c12202. Substitute into the budget
to find c1 = 3,520/221 2 1,592.8. He saves
2, 000 — 3, 520/221 2 407.2. (c) Suppose that Harvey’s money income in both periods is the same as
before, the interest rate is still 21%, but there is a 10% inﬂation rate. Then in period 2, a loaf of bread will cost $ 1 . 10 . Write down Har
vey’s budget equation for period—1 and period2 bread, given this new information. 1.2101 + 1.102 = 3, 10.10 (2) In an isolated mountain village, the only crop is corn. Good
harvests alternate with bad harvests. This year the harvest will be 1,000
bushels. Next year it will be 150 bushels. There is no trade with the
outside world. Corn can be stored from one year to the next, but rats
will eat 25% of what is stored in a year. The villagers have CobbDouglas
utility functions, U(cl,c2) : c102 where (:1 is consumption this year, and
C2 is consumption next year. (or) Use red ink to draw a “budget line,” showing consumption possibilities
for the village, with this year’s consumption on the horizontal axis and
next year’s consumption on the vertical axis. Put numbers on your graph
to show where the budget line hits the axes. Next year's consumption l50 Illl
1000 l36 This year's consumption NAME ___.._—_.__..W 143 (b) How much corn will the villagers consume this year? 600
bushels . How much will the rats eat? 100 bushels . How much corn will the villagers consume next year? 450 bushels . (6) Suppose that a road is built to the village so that now the village is
able to trade with the rest of the world. Now the villagers are able to buy
and sell corn at the world price, which is $1 per bushel. They are also
able to bOrrOW and lend money at an interest rate of 1000. On your graph,
use blue ink to draw the new budget line for the villagers. Solve for the amount they would now consume in the first period 568 bushels and the second period 624: bushels . (d) Suppose that all is as in the last part of the question except that there
is a transportation cost of $.10 per bushel for every bushel of grain hauled
into or out of the village. On your graph, use black ink or pencil to draw
the budget line for the viliage under these circumstances. 10.11 (0) The table below records percentage interest rates and inflation
rates for the United States in some recent years. Complete this table. Inﬂation and Interest in the United States, 1965~1985 . . 1978 1980
CPI, Start of Year 38.3 ~ . 100.0 CPI, End of Year 49.2 . 133
4.2 . % Inﬂation Rate 4.‘
Nominal Int. Rate Real Int. Rate (a) People complained a great deal about the high interest rates in
the late 705. In fact, interest rates had never reached such heights in modern times. Explain why such complaints are misleading.
Nominal interest rates were high, but so
was inflation. Real interest rates were not high. (They were negative in 1978.) 130.0 144 INTERTEMPORAL CHOICE (Ch. 10) (b) If you gave up a unit of consumption goods at the beginning of 1985
and saved your money at interest, you could use the proceeds of your saving to buy 1 . 05 units of consumption goods at the beginning of
1986. If you gave up a unit of consumption goods at the beginning of
1978 and saved your money at interest, you would be able to use the proceeds of your saving to buy . 96 units of consumption goods at the
beginning of 1979. 10.12 (1) Marsha Mellow doesn’t care whether she consumes in period
1 or in period 2. Her utility function is simply U(C1,(12) = (:1 + c2. Her
initial endowment is $20 in period 1 and $40 in period 2. In an antique
shop, she discovers a cookie jar that is for sale for $12 in period 1 and that
she is certain she can sell for $20 in period 2. She derives no consumption
beneﬁts from the cookie jar, and it costs her nothing to store it for one
period. (a) On the graph below, label her initial endowment, E, and use blue ink
to draw the budget lii':e,showing combinations of period1 and period—2
consumption that she can aﬁ‘brd if she doesn’t buy the cookie jar. On the
same graph, label the consumption bundle, A, that she would have if she
did not borrow or lend any money but bought the cookie jar in period 1,
sold it in period 2, and used the proceeds to buy period—2 consumption. If she cannot borrow or lend, should Marsha invest in the cookie jar? Yes. (b) Suppose that Marsha can borrow and lend at an interest rate of 50%.
On the graph where you labelled her initial endowment, draw the budget
line showing all of the bundles she can afford it she invests in the cookie
jar and borrows or lends at the interest rate of 50%. On the same grapl‘i
use red ink to draw one or two of Marsha‘s indifference curves. Period2 consumption 80
60 40 20 0 20 40 60 80
PeriodI consumption NAME W 145 (c) Suppose that instead of consumption in the two periods being per—
fect substitutes, they are perfect con'lplements, so that Marsha’s utility
function is 1‘nii‘1{c],02}. If she cannot borrow or lend, should she buy the cookie jar? NO . If she can borrow and lend at an interest rate of 50%,
should she invest in the cookie jar? Yes . If she can borrow or lend as much at an interest rate of 100%, should she invest in the cookie gar? No. WVWRSWE‘MJFimx’mhmwéﬂﬂsﬁﬂzﬁﬂmvza NAU:W»‘:CL‘I£‘£1 160 ASSETMARKETS (Ch. 11) degree. Suppose that you are 22 years old and have just finished college. If
r = .05, ﬁnd the present value of lifetime earnings for a graduating senior
who will get an advanced degree in business and earn the average wage rate for someone with this degree until retiring at 65. $596 , 000 .
Make a similar calculation for medicine. $630 , 000 . 11.13 (0) On the planet Stinko, the principal industry is turnip growing.
For centuries the turnip fields have been fertilized by guano which was
deposited by the nowextinct giant scissorbiiled kikibird. It costs $5 per
ton to mine kiki—bird guano and deliver it to the ﬁelds. Unfortunately, the
country’s stock of kikibird guano is about to be exhausted. Fortunately
the scientists on Stinko have devised a way of synthesizing kiki—guano from
political science textbooks and swamp water. This method of production
makes it possible to produce a product indistinguishable from kikiguano
and to deliver it to the turnip ﬁelds at a cost of $30 per ton. T he interest
rate on Stinko is 10%. There are perfectly competitive markets for all
commodities. (a) Given the current price and the demand function for kiki—guano, the
last of the deposits on Stinko will be exhausted exactly one year from
now. Next year, the price of kiln—guano delivered to the ﬁelds will have
to be $30, so that the synthetic kiki—guano industry will just break even.
The owners of the guano deposits know that next year, they would get a
net return of $25 a ten for any guano they have left to sell. In equilibrium,
what must be the current price of kiki—guano delivered to the turnip fields? The price of guano delivered to the field
must be the $5 + the present value of $25.
This is 5 l 25/l.1 = . (Hint: In equilibrium, sellers must be indifferent between selling their kikiguano right now or at any
other time before the total supply is exhausted. But we know that they
must be willing to sell it right up until the day, one year from now, when
the supply will be exhausted and the price will be $30, the cost of synthetic guano.) (6) Suppose that everything is as we have said previously except that the
deposits of kiki—guano wiil be exhausted 10 years from 110w. What must
be the current price of kiki—guano? (Hint: 1.110 = 2.59.) 5 + 25/(1.1)10 214.65. Chapter 12 NAME
Uncertainty Introduction. In Chapter 1i, you learned some tricks that allow you to
use techniques you already know for studying interten'iporal choice. Here
you will learn some similar tricks, so that you can use the same methods
to study risk taking, insurance, and gambling. One of these new tricks is similar to the trick of treating commodi—
ties at different dates as different connnodities. This time, we invent
new commodities, which we call contingent commodities. If either of two
events A or B could happen, then we deﬁne one contingent commodity
as consmnption if A happens and another contingent commodity as con
sumption if B happens. The second trick is to ﬁnd a budget constraint
that correctly speciﬁes the set of contingent commodity bundles that a
consumer can afford. rThis chapter presents one other new idea, and that is the notion
of von Neumann—Morgenstern utility. A consumer‘s willingness to take
various gambles and his willingness to buy insurance will be determined
by how he feels about various combinations of contingent commodities.
Often it is reasonable to assume that these preferences can be expressed
by a utility function that takes the special form known as non N eumenn
Moryerrstern'utility. The assumption that utility takes this form is called
the ezrpected utility hypothesis. If there are two events, 1 and 2 with
probabilities in and ﬁg, and if the contingent consumptions are (:1 and
Cg, then the von Neumann—Morgenstern utility function has the special
functional form, U (c1, (32) = medal) + 7131111022). The consumer’s behavior
is determined by maximizing this utility function subject to his budget
constraint. Example: You are thinking of betting on whether the Cincinnati Reds
will make it to the World Series this year. A local gambler will bet with
you at odds of 10 to 1 against the Reds. You think the probability that
the Reds will n'iake it to the World Series is 7r = .2. If you don’t bet,
you are certain to have $1,000 to spend on consumption goods. Your
behavior satisfies the expected utility hypothesis and your von Neumann
l\rlorgenstern utility function is rrM/cT + Hg. The contingent commodities are dollars if the Reds make the World
Series and dollars if the Reds don’t make the World Series. Let cw be
your consumption contingent on the Reds making the World Series and
cNW be your consumption contingent on their not making the Series.
Betting on the Reds at odds of 10 to 1 means that if you bet $2: on the
Reds, then if the Reds make it to the Series, you make a. net gain of $1033,
but if they don‘t, you have a net loss of 33$. Since you had $1,000 before
betting, if you bet $11“. on the Reds and they made it to the Series, you
would have cw x 1,000 + 10:1: to spend on consumption. If you bet $3:
on the Reds and they didn’t make it to the Series, you would lose $.73, Yw‘ 162 UNCERTAINTY (Ch. 12) and you would have ch : 1, 000 — 2:. By increasing the amount its that
you bet, you can make cw larger and cNW smaller. (You could also bet
against the Reds at the same odds. If you bet $33 against the Reds and
they fail to make it to the Series, you make a net gain of 1'3 and if they
make it to the Series, you lose $3.1. If you work through the rest of this
discussion for the case where you bet against the Reds, you will see that
the same equations apply, with a: being a negative number.) We can use
the above two equations to solve for a. budget equation. From the second
equation, we have a: = 1,000 m CNw. Substitute this expression for :5 into
the ﬁrst equation and rearrange terms to ﬁnd cw + lOcN W = 11, 000, or
equivalently, .1cw + CNW = l, 100. (The same budget equation can be
written in many equivalent ways by multiplying both sides by a positive
constant.) Then you will choose your contingent consumption bundle (cw, CNW)
to maximize U(cw,cNW) 2: 2%? + .SW subject to the budget
constraint, .1cw + cNW m 1,100. Using techniques that are now familiar,
you can solve this consumer problem. From the budget constraint, you
see that consumption contingent on the Reds making the World Series
costs 1/10 as much as consumption contingent on their not making it. If
you set the marginal rate of substitution between cw and ch equal to
the price ratio and simplify the resulting expression, you will ﬁnd that
cNW 2 .16cW. This equation, together with the budget equation implies
that cw : $4, 230.77 and (WW = $676.92. You achieve this bundle by
betting $323.08 011 the Reds. If the Reds make it to the Series, you will
have $1,000 + 10 X 323.08 = $4,230.80. If not, you will have $076.92.
(We rounded the solutions to the nearest penny.) 12.1 (0) In the next few weeks, Congress is going to decide whether
or not to develop an expensive new weapons system. If the system is
approved, it will be very proﬁtable for the defense contractor, General
Statics. Indeed, if the new system is approved, the value of stock in
General Statics will rise from $10 per share to $15 a share, and if the
project is not approved, the value of the stock will fall to $5 a share. In
his capacity as a messenger for Congressman Kickback, Buzz Condor has
discovered that the weapons system is much more likely to be approved
than is generally thought. On the basis of what he knows, Condor has
decided that the probability that the system will be approved is 3/ 4 and
the probability that it will not be approved is 1/4. Let CA be Condor’s
consumption if the system is approved and CAM be his consumption if
the System is not approved. Condor’s von NeumannMorgenstern utility
function is U(cA,cN/1) = .75111cA + .25lncNA. Condor’s total wealth is
$50,000, all of which is invested in perfectly safe assets. Condor is about
to buy stock in General Statics. (a) If Condor buys as shares of stock, and if the weapons system is ap
proved, he will make a proﬁt of $5 per share. Thus the amount he can
consume, contingent on the system being approved, is (2,4 = $50, 000 + 5:13.
If Condor buys a: shares of stock, and if the weapons system is not ap proved, then he will make a loss of $ 5 per share. Thus the NAME —.__._ m 163 amount he can consume, contingent on the system not being approved, is CNA : 50, ‘“ 533. (b) You can solve for Condor’s budget constraint on contingent commod—
ity bundles (CA,CN,1) by elin'iinating a: from these two equations. His bud— get constraint can be written as . 5 cA+ . 5 cNA = 50,000. (c) Buzz Condor has no moral qualms about trading on inside infori'na—
tion, nor does he have any concern that he will be caught and punished.
To decide how much stock to buy, he simply maximizes his von Neurnanm
Morgenstern utiiity function subject to his budget, If he sets his marginal
rate of substitution between the two contingent corrunodities equal to
their relative prices and simplifies the equation, he ﬁnds that (1,4 /cNA = 3 . (Reminder: Where a is any constant, the derivative of alna:
with respect to m is a/ar.) (d) Condor finds that his optimal contingent commodity bundle is
(CA, cNA) x (75 , 000 , 25 , 000) . To acquire this contingent comw
modity bundle, he must buy 5 , 000 shares of stock in General Statics. 12.2 (0) 1Willy owns a small chocolate factory, located close to a river
that occasionally floods in the spring, with disastrous consequences. Next
summer, W lily plans to sell the factory and retire. The only income he
will have is the proceeds of the sale of his factory. If there is no flood,
the factory will be worth $500,000. If there is a ﬂood, then what is left
of the factory will be worth only $50,000. Willy can buy flood insurance
at a. cost of $.10 for each $1 worth of coverage. Willy thinks that the
probability that there will be a flood this spring is 1/10. Let op denote the
contingent comn‘rodity dollars if there is a flood and ch denote dollars
if there is no ﬂood. Willy’s von Neuman11Morgenstern utility function is U((31?‘:CNI«") w .1fc‘,¥+ ( a ) If he buys no insurance, then in each contingency, Willy’s consumption
will equal the value of his factory, so W illy’s contingent commodity bundle be (CI;1,C:NF) “n: . (b) To buy instn‘ance that pays him 85c in case of a ﬂood, Willy must
pay an insurance premium of 11:. (The insurance premium must be
paid whether or not there is a ﬂood.) If Willy insures for $.12, then if
there is a ﬂood, he gets $2: in insurance beneﬁts. Suppose that Willy has
contracted for insurance that pays him $.12 in the event of a ﬂood. Then
after paying his insurance premium, he will be able to consume CF = 50, l— . If Willy has this amount of insurance and there is no flood, then he will be able to consume ch = — . 164 UNCERTAINTY (Ch. 12} (c) You can eliminate :13 from the two equations for CF and CN 1» that
you found above. This gives you a budget equation for Willy. Of course
there are many equivalent ways of writing the same budget equation,
since multiplying both sides of a budget equation by a positive constant
yields an equivalent budget equation. The form of the budget equation in which the “price” of cm: is 1 can be written as .QCNF+ . 1 CF z 455,000. ((1) Willy’s marginal rate of substitution between the two contingent com~
modities, dollars if there is no ﬂood and dollars if there is a ﬂood, is M’RS(CF,CNF) :: To ﬁnd his optimal bundle of contingent commodities, you must set this marginal rate of substitution equal to the number 3 _9 . Solving this equation, you find that Willy will choose to consume the two contingent commodities in the ratio
CNF/CFIII. (e) Since you know the ratio in which he wiil consume CF and 6er, and
you know his budget equation, you can solve for his optimal consumption bundle, which is (ep~,ch)= . Willy will buy an insurance policy that will pay him $450 , 000 if there is a ﬂood. The amount of insurance premium that he wili have to pay is $45,000. 12.3 (D) Clarence Bunsen is an expected utility maximizer. His prof
erences among contingent commodity bundles are represented by the ex
pected utility function u(6150237r331r2) = ﬂlﬁI‘l KEV—(33‘ Clarence’s friend, Hjalmer Ingqvist, has offered to bet him $1,000 on the
outcome of the toss of a coin. That is, if the coin comes up heads, Clarence
must pay Hjaln‘ier $1,000 and if the coin comes up tails, IIjaimer must
pay Clarence $1,000. The coin is a fair coin, so that the probability of
heads and the probabiiity of tails are both 1 /2. If he doesn’t accept the
bet, Clarence will have $10,000 with certainty. In the privacy of his car
dealership office over at Bunsen Motors, Clarence is making his decision.
{Ciarence uses the pocket calculator that his son, Elmer, gave him last
Christmas. You will ﬁnd that it will be helpful for you to use a calcrilator
too.) Let Event 1 be “doin comes up heads” and let Event 2 be “coin
comes up tails.” NAME 16’ (a) If Clarence accepts the bet, then in Event 1, he wiil have 9 , 000 doliars and in Event 2, he will have 1 1 , 000 dollars. ((1) Since the probability of each event is 1/2, Clarence’s expected utility
for a gai'nble in which he gets cl in Event 1 and Cg in Event 2 can be . 1 1
described by the formula §1/Cl i i1/C2 . Therefore Clarence’s expected utility if he accepts the bet with Hialrner will be 99 . 8746 .
(Use that calculator.) (c) If Clarence decides not to bet, then in Event 1, he wiil have
10 , 000 dollars and in Event 2, he will have 10 , 000 dollars. 'I‘herofore if he doesn’t bet, his expected utility will be 100 . (d) Having calculated his expected utility if he bets and if he does not bet,
Clarence determines which is higher and makes his decision accordingly. Does Clarence take the bet? NO . 12.4 (0) It is a slow day at Bunsen Motors, so since he has his calcu~
lator warmed up, Clarence Bunsen (whose preferences toward risk were described in the last problem) decides to study his expected utility func—
tion more closely. (at) Clarence first thinks about really big gambles. What if he bet his
entire $10,000 on the toss of a coin, where he loses if heads and wins if
tails? Then if the coin came up heads, he would have 0 dollars and if it
came up tails, he wouid have $20,000. His expected utility if he took the bet would be 70 . 71 , while his expected utility if he didn’t take the bet would be 100 . Therefore he concludes that he wouid not take
such a bet. {5) Clarence then thinks, “Well, of course, I wouldn’t want to take a,
chance on losing all of my money on just an ordinary bet. But, what
if somebody offered me a really good deai. Suppose I had a chance to
bet where if a fair coin came up heads, I lost my $10,000, but if it came
up tails, 1 would win $50,000. Would I take the bet? Ii’I took. tho. but, my expected utility would be 122 . 5 . If I didn’t take the bet, my expected utility would be 100 . Therefore I should take the bot.” 166 UNCERTAINTY (Ch. 12) (c) Clarence later asks himself, “If I make a bet where I lose my $10,000
if the coin comes up heads, what is the smallest amount that I would have
to win in the event of tails in order to make the bet a good one for me
to take?” After some trial and error, Clarence found the answer. You,
too, might want to ﬁnd the answer by trial and error, but it is easier to
find the answer by solving an equation. On the left side of your equation,
you would write down Clarence’s utility if he doesn’t bet. On the right
side of the equation, you write down an expression for Clarence’s utility
if he makes a bet such that he is left with zero consumption in Event 1
and a: in Event 2. Solve this equation for a). The answer to Clarence‘s question is where :L‘ m 10,000. rl‘he equation that you should write is = % £13 . The solution is 5:: = . (at) Your answer to the last part gives you two points on Clarence’s in
difference curve between the contingent commodities, money in Event 1
and money in Event 2. (Poor Clarence has never heard of indifference
curves or contingent commodities, so you will have to work this part for
him, while he heads over to the Chatterbox Cafe for morning coffee.) One
of these points is where money in both events is $10,000. On the graph
below, label this point A. The other is where money in Event 1 is zero and money in Event 2 is 40 , 000 . On the graph below, label this
point B. Money in Event 2 (x 1.000)
40 30
20 IO 0 10 20 30 40
Money in Event  (x 1,000) (8) You can quickly ﬁnd a third point on this indifference curve. The
coin is a fair coin, and Clarence cares whether heads or tails turn up only
because that determines his prize. Therefore Clarence will be indifferent
between two gambles that are the same except that the assignment of
prizes to outcomes are reversed. In this example, Clarence will be indif
ferent between point B on the graph and a point in which he gets zero if lzslvent 2 happens and 40 , 000 if Event 1 happens. Find this point
on the Figure above and label it C. NAME __. _. __.. 167 ( f ) Another gamble that is on the same indifference curve for Clarence
as not gan'ibhng at ali is the gai'nble where he loses $5,000 if heads turn up and where he wins 6 , 715 . 73 dollars if tails turn up. (Hint: To
solve this problem, put the utility of not betting on the left side of an
equation and on the right side of the equation, put the utility of having
$10, 000 w $5, 000 in Event 1 and $10, 000 + :5 in Event 2. Then solve the
resulting equation for On the axes above, plot this point and label it
D. New sketch in the entire indifference curve through the points that
you have labeled. 12.5 (0) Hjah‘ner Ingqvist’s sonin—law, Earl, has not worked out very
well. It turns out that Earl likes to gamble. His preferences over contin—
gent commodity bundles are represented by the expected utility function 2
u(e1,02,7r1,7r2) = me, + meg. (a) Just the other day, some of the boys were down at Skoog’s tavern
when Earl stopped in. They got to talking about just how bad a bet they
could get him to take. At the time, Earl had $100. Kenny Olson shuffled
a deck of cards and offered to bet Earl $20 that Earl would not cut a spade
from the deck. Assuming that Earl believed that Kenny wouldn’t cheat,
the probability that Earl would win the bet was 1/4 and the probability
that Ear} would lose the bet was 3/4. If he won the bet, Earl would have 120 dollars and if he lost the bet, he would have 80
dollars. Earl’s expected utility if he took the bet would be 8 , 4:00 , and his expected utility if he did not take the bet would be 10 , 000 .
Therefore he refused the bet. (5) Just when they started to think Earl might have changed his ways,
Kenny oifered to make the same bet with Earl except that they would
bet $100 instead of 3520. What is Earl’s expected utility if he takes that hot? 10 , 000 . Would Earl be willing to take this bet? He is just indifferent about taking it or not. . :1 , . ' 
(a) Let Event 1 be the event that a card drawn from a fair deck of cards is
a spade. Let Event 2 be the event that the card is not a spade. Earl’s pref“
erences between income contingent on Event 1, (:1, and income contingent on Event 2, 02, can be represented by the equation “Lt 2 iii? ‘1“ . biue ink on the graph below to sketch Earl’s indifference curve passing
through the point (100, 100). 168 UNCERTAINTY (Ch. 12) Money in Event 2 mo _ .n, .m , w
E l i i 3
i.% E i i
5_§u gm z E. “E no §,g. g i g .:
t i i ' i i E mo 3 i g
i j
E E E 50 E ‘ s
i E
i 0 so 00 150 200 Money in Event I (d) On the same graph, let us draw Hjalmer’s son—in—law Earl’s indif—
ference curves between contingent commodities where the probabilities
are diiierent. Suppose that a card is drawn from a fair deck of cards.
Let Event 1 be the event that the card is black. Let event 2 he the event
that the card drawn is red. Suppose each event has probability 1 / 2. Then
Earl’s preferences between income contingent on Event 1 and income con— tingent on Event 2 are represented by the formula u = %C% "i‘ . On the graph, use red ink to show two of Earl’s indiiierence curves, in
cluding the one that passes through (100, 100), 12.6 (1) Sidewalk Sam makes his living selling sunglasses at the board—
walk in Atlantic City. If the sun shines Sam makes $30, and if it rains
Sam only makes $10. For simplicity, we will suppose that there are only
two kinds of days, sunny ones and rainy ones. (a) One of the casinos in Atlantic City has a new gimmick. It is accepting
bets on whether it will be sunny or rainy the next day. rl‘he casino sells
dated “rain coupons” for $1 each. If it rains the next day, the casino will
give you $2 for every rain coupon you bought on the previous day. If it
doesn’t rain, your rain coupon is worthless. In the graph below, mark
San'l’s “endowment” of contingent consumption if he makes no bets with
the casino, and label it E. NAME W 169 Cr
40 30 20 E0 O 10 20 3O 40 (b) 0n the same graph, mark the combination of consumption contingent
on rain and consumption contingent on sun that he could achieve by
buying 10 rain coupons from the casino. Label it A. (c) On the same graph, use blue ink to draw the budget line representing
all of the other patterns of consumption that Sam can achieve by buying
rain coupons. (Assume that he can buy fractional coupons, but not neg—
ative amounts of them.) W’ hat is the slope of Sam’s budget line at points above and to the left of his initial endowment? The slope is ~1. (0!) Suppose that the casino also sells sunshine coupons. These tickets
also cost $1. With these tickets, the casino gives you $2 if it doesn’t rain
and nothing if it does. On the graph above, use red ink to sketch in the
budget line of contingent consumption bundles that Sam can acl'iieve by
buying sunshine tickets. ((2) If the price of a dollar’s worth of consumption when it rains is set equal to 1, what is the price of a dollar’s worth of consumption if it shines? The price is 1. 12.7 (0) Sidewalk Sam, from the previous problem, has the utility func—
tion for consumption in the two states of nature u(cs, ans) 3 ciﬁrcfh where c, is the dollar value of his consumption if it shines, or is the dollar
value of his consun'iption if it rains, and if is the probability that it will
rain. The probability that it will rain is 7r 2 .5. wwnawmwmmWmmmnwmmwww_ bmrmvmwmmsymmwwmmwwwmwpm wwmmwwxw ., , Mn 170 UNCERTAINTY (Ch. 12) (a) How many units of consumption is it optimal for Sam to consume conditional on rain? 20 units . (b) How many rain coupons is it optimal for Sam to buy? 10 . 12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an ex»
pected utility maximizer. His von Neuniann~Morgenstern utility function
for wealth is 1.;(0) x in c. Sam’s brother also sells sunglasses on another
beach in Atlantic City and makes exactly the same income as Sam does.
He can make exactly the same deal with the casino as Sam can. (a) if Morgan believes that there is a 50% chance of rain and a 50% chance
of sun every day, what would his expected utility of consuming (cs,c,.) ta it::%lncsa—%lnc,. (b) How does Morgan’s utility function compare to Sam’s? Is one a monotonic transformation of the other? Morgan’ S utility
function is just the natural log of Sam’s, so the answer is yes. (c) What will hflorgan’s optimal pattern of consumption be? Answer: Morgan will consume 20 on the sunny days and 20 on
the rainy days. How does this compare to Sam’s consumption? This is the same as Sam’s consumption. 12.9 (0) Billy John Pigskin of Mule Shoe, rl‘exas, has a. von Neumann—
Morgenstern utility function of the form = Billy John also weighs
about 300 pounds and can outrun jackrabbits and pizza delivery trucks.
Billy John is beginning his senior year of college football. If he not
seriously injured, he will receive a $1,000,000 contract for playing pro~
fessional football. If an injury ends his football career, he will receive a
$10,000 contract as a refuse removal facilitator in his home town. There
is a 10% chance that Billy John will be injured badly enough to end his
career. (a) What is Billy John’s expected utility? We calculate .1./—10,“‘000 + .9,/—1,0‘0"0‘,W000 = 910. NAME m— 171 (b) If Billy John pays $1) for an insurance policy that would give him
$1,000,000 if he suffered a. careerending injury while in college, then he
would be sure to have an income of $1, 000, 000 — p no matter what hap
pened to him. Write an equation that can be solved to ﬁnd the largest
price that Billy John would be willing to pay for such an insurance policy. The equation is 910==I\/l,000,000—IL (c) Solve this equation for p. p = . 12.10 (1) You have $200 and are thinking about betting on the Big
Game next Saturday. Your team, the Golden Boar‘s, are scheduled to
play their traditional rivals the Robber Barons. It appears that the going
odds are 2 to 1 against the Golden Bears. That is to say if you want
to bet $10 on the Bears, you can ﬁnd someone who will agree to pay
you $20 if the Bears win in return for your promise to pay him $10 if
the Robber Barons win. Similarly if you want to bet $10 on the Robber
Barons, you can ﬁnd someone who will pay you $10 if the Robber Barons
win, in return for your promise to pay him $20 if the Robber Barons lose.
Suppose that you are able to make as large a bet as you iike, either on
the Bears or on the Robber Barons so long as your gambling losses do
not exceed $200. (To avoid tedium, let us ignore the possibility of ties.) (a) If you do not bet at all, you will have $200 whether or not the Bears
win. If you bet $50 on the Bears, then after all gambling obligations are settled, you will have a total of 300 dollars if the Bears win and 150 dollars if they lose. On the graph below, use blue ink to draw a
line that represents all of the con‘rbinations of “money if the Bears win”
and “money if the Robber Barons win” that you could have by betting
from your initial $200 at these odds. 172 UNCERTAINTY (Ch. 12) Money if the Bears lose 400 g 1: i i 1 l 5 ' 3. S . 5. . z . i
2 i
i
300 '2 .
l
i 200 28lueilinei. 00 0 Hit) 200 300 400
Money if the Bears win (5) Label the point on this graph where you would be if you did not bet
at all with an E. (c) After careful thought you decide to bet $50 on the Bears. Label the
point you have chosen on the graph with a C. Suppose that after you have
made this bet, it is announced that the star Robber Baron quarterback
suffered a sprained thumb during a tough economics midterm examination
and wili miss the game. The market odds shift from 2 to 1 against the
Bears to “even money” or 1 to 1. That is, you can now bet on either
team and the amount you would win if you bet on the winning team is
the same as the amount that you would lose if you bet on the losing team.
You cannot cancel your original bet, but you can make new bets at the
new odds. Suppose that you keep your ﬁrst bet, but you now aiso bet
$50 on the Robber Barons at the new odds. If the Bears win, then after
you collect your winnings from one bet and your losses from the other, how much money will you have left? $250 . If the Robber Barons
win, how much money will you have left after collecting your winnings and paying off your losses? $200 . (of) Use red ink to draw a line on the diagram you made above, showing
the combinations of “money if the Bears win” and “money if the Robber
Barons win” that you could arrange for yourseif by adding possible bets
at the new odds to the bet you made before the news of the quarterbacks
misfortune. On this graph, label the point D that you reached by making
the two bets discussed above. 12.11 (2) The certainty equivalent of a lottery is the amount of money
you would have to be given with certainty to be just as wellmoff with that
lottery. Suppose that your von Neumann—Morgenstern utiiity function NAME —— 173 over lotteries that give you an amount .2: if Event 1 happens and y if
Event 1 does not happen is U (1‘, y, 7r) 2: + (1 — ark/r], where 7r is the
probability that Event 1 happens and 1 — 7r is the probability that Event
1 does not happen. (a) If 7r = .5, calculate the utility of a lottery that gives you $10,000 if Event 1 happens and $100 if Event 1 does not happen. Z .5 X 100+ .5 X 10. (b) If you were sure to receive $4,900, what would your utility be? 70 . (Hint: If you receive $4,900 with certainty, then you receive $4,900 in
both events.) (6) Given this utility function and 7r m .5, write a general formuia for the
certainty equivalent of a iottery that gives you $.13 if Event 1 happens and $3} if Event 1 does not happen. (.5931/2 l— .5y1/2)2 . (d) Calculate the certainty equivalent of receiving $10,000 if Event 1 hap pens and $100 if Event 1 does not happen. $3 , O25 . 12.12 Dan Partridge is a risk averter who tries to maximize the
expected value of ﬂ, where c is his wealth. Dan has $50,000 in safe
assets and he also owns a house that is located in an area where there
are lots of forest fires. If his house burns down, the remains of his house
and the lot it is buiit on would be worth only $40,000, giving him a total
wealth of $90,000. If his home doesn’t burn, it will be worth $200,000
and his totai wealth will be $250,000. The probability that his home wiil
burn down is .01. (a) Calculate his expected utility if he doesn’t buy fire insurance.
$498. (6) Calculate the certainty equivalent; of the lottery he faces if he doesn’t buy fire insurance. $248 , 004 . {c} Suppose that he can buy insurance at a price of $1 per $100 01" in—
surance. For example. if he buys $100,000 worth 01" irisurniute, he will pay
$1,000 to the company no matter what happens, but ll" his house burns,
he will also receive $100,000 from the company. ll" Dan buys $100,000
worth of insurance, he will be fully insured in the sense that no matter what happens his after—tax weaith will be $248 , 400 . “Aqws‘thMV’A‘WBW/ﬂ. ' .‘V " " "'3‘; m.wm:.ﬁmswwumwwmvmwwa 174 UNCERTAiNTY (Ch. 12} ((1) Therefore if he buys full insurance, the certainty equivalent of his wealth is $248 , 400 , and his expected utility is V . 12.13 (1) Portia has been waiting a long time for her ship to come in
and has conciuded that there is a 25% chance that it will arrive today.
If it does come in today, she will receive $1,600. If it does not come
in today, it will never come and her wealth will be zero. Portia. has a
von Neumann—Morgenstern utility such that she wants to maximize the
expected value of ﬂ, where c is total wealth. What is the minin'nin'i price at which she will seli the rights to her ship? $100 . Chapter 13 NAME
Risky Assets Introduction. Here you will solve the problems of consumers who wish
to divide their wealth optimally between a risky asset and a safe asset.
The expected rate of return on a portfolio is just a. weighted average of
the rate of return on the safe asset and the expected rate of return on
the risky asset, where the weights are the fractions of the consruner’s
wealth held in each. The standard deviation of the portfolio return just the standard deviation of the return on the risky asset times the
fraction of the consumer’s wealth held in the risky asset. Sometimes
you will look at the problem of a. consun'ier who has preferences over
the expected return and the risk of her portfoiio and who face; a budget
constraint. Since a consumer can always put all of her wealth in the
safe asset, one point on this budget constraint will be the combination
of the safe rate of return and no risk (zero standard deviation). Now
as the consun'ier puts a: percent of her wealth into the risky asset, she
gains on that amount the difference between the expected rate of return
for the risky asset and the rate of return on the safe. asset. But she also
absorbs some risk. So the slope of the budget line wiil be the difference
between the two returns divided by the standard deviation of the portfolio
that has :1: percent of the consui‘ner’s wealth invested in the risky asset.
You can then apply the usual indifference curveibudget line analysis to
find the consumer’s optimal choice of risk and expected return given her
preferences. (Remember that if the standard deviation is plotted on the
horizontal axis and if less risk is preferred to more, the hotter bundles will
lie to the northwest.) You will also be asked to apply the result from the
Capital Asset Pricing Model that the expected rate of return on any asset
is equai to the sum of the risk—free rate of return plus the risk adiustment.
Remember too that the expected rate of return on an asset is its expected
change in price divided by its current price. 13.1 (3) Ms. Lynch has a choice of two assets: The ﬁrst is a risk—free
asset that offers a rate of return of TI, and the second is a. risky asset (a
china shop that caters to large mammais) that has an expected rate of
return of rm and a standard deviation of am. (a) If 3: is the percent of wealth Ms. Lynch invests in the risky asset,
what is the equation for the expected rate of return on the portfolio? T3; 2 SET,” + — (13)?" f . What is the equation for the standard deviation of the portfolio? 0'3; 3 33 0' m . .' ' 'WWWWFW‘WWWWWWWMWVVWMWMWMW'hWA/PAJ ...
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This note was uploaded on 01/18/2012 for the course PADP 6950 taught by Professor Fergi during the Spring '11 term at UGA.
 Spring '11
 Fergi

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