Exam, Microeconomics A, August 1993
Unocial translation
Consider an economy with two individuals and one future period, in which two possible states may occur.
The two individuals agree that the states have probabilities 1 and 2 = 1 , 1, respectively. Individual
i will get an exogenous income y^is in state s, but may in advance obtain the possibility of consuming
yis 6= y^is by buying or selling state contingent claims in a market. Only the two individuals participate
in the trade, but we assume a competitive outcome is obtained. The individuals' budgets for buying state
contingent claims consist only of the market values of the exogenous incomes y^is .
(a) Write down the optimization problem of individual i for the general case in which the individual
has some preference ordering over the two state contingent consumption magnitudes. Derive the
conditions which characterize the behavior, and illustrate these in a diagram.
(b) Consider in particular the case with preferences determined according to von Neumann and Morgenstern's expected utility theorem, with the preference function of individual i given by
vi (y) ,e,i y ;
where i is a positive constant. Show that the optimal yi1, in the case of an interior solution, may
be written as
w^i , p2 ln 21 pp12
yi1 =
;
p +p
i
1
2
where ps is the price of a claim to one unit of income in state s, and w^i = p1y^i1 + p2y^i2 :
(c) Assume both individuals have preference functions of the given type, and dene
1 :
= 1
1
1 + 2
Show that in equilibrium the price ratio is given as
p=
p1 1 (^y2 ,y^1 )
= e
;
p2 2
where y^s is dened as y^1s + y^2s.
(d) Give an interpretation of how the price ratio in equilibrium depends on 1 =2, on y^2 , y^1 , and on
1 and 2.
(e) Assume y^1 = y^2 . Give an economic interpretation of this case. Illustrate the individual's behavior
in a diagram as in part (a), and calculate the optimal yi1 and yi2 under the assumptions of part (b).
(f) Assume that in addition to the future period discussed above, there is a present period which we call
\today." Consider two investment projects, both requiring reduced consumption today. One gives a
certain additional income x in the future period. The other gives an additional income z in state 1
and nothing in state 2. What do the equilibrium prices from part (c) tell us about
(i) the individuals' ranking of the two projects, and
(ii) the individuals' willingness to pay (today) for the two projects?
Suggested answer, Microeconomics A, August 1993
By D. Lund
(a) The maximization problem is a specialization of the general state-preference model, with only two
states, and (in part (b)) with a particular utility function, and (in part (c)) with only two individuals.
The formulation in part (a) is more general than the standard model, since this part does not assume
that preferences are derived from the von Neumann and Morgenstern expected utility theorem. One
may just work with a general utility function, Ui , which is increasing and quasi concave in (yi1 ; yi2).
The individual's optimization problem is then
max U (y ; y ) subject to
y 1 ;y 2 i i1 i2
i
i
2
X
s=1
psyis =
2
X
i=1
ps y^is :
(1)
(Prices are dened in the text of part (b).)
We may safely assume that the budget constraint is satised with an equality, since we assume
non-satiation.
The rst-order conditions for an interior solution consist of the budget constraint and
@Ui =@yi1 p1
= :
@Ui =@yi2 p2
(2)
The interpretation of this equation, and the diagram which illustrates it, is well-known from standard
microeconomics.
Assuming von Neumann and Morgenstern preferences, we have
U (yi1 ; yi2) 1 vi (yi1 ) + 2 vi (yi2):
(3)
Then the marginal rate of substitution gets the form
1vi0 (yi1 )
;
2vi0 (yi2 )
and the indierence curves have the special property that they all have the same slope at their
intersection with a 45 degree ray from the origin, equal to 1=2 in absolute value.
(b) Equation (2) may now be written
1i e, y 1 p1
= :
(4)
e, y 2 p
i i
2
i
i i
2
Together with the budget constraint this gives us two equations in yi1 ; yi2. The solution to this
system of equations gives that equation for yi1 which is shown in the text. A similar equation may
be derived for yi2 .
(c) Market equilibrium requires
y^1 y^11 + y^21 = y11 + y21;
(5)
and a similar equation, supply equals demand, for the claims on income in state 2. That second
equation is, however, superuous according to Walras' law.
By substituting in the expressions from part (b) for the demand for each of the individuals, we nd
p1 (^y11 + y^21 ) + p2 (^y12 + y^22 ) , p2 11 + 12 ln 21 pp12
:
(6)
y^1 =
p1 + p2
By dividing both numerator and denominator by p2, we get an equation in the price ratio p p1 =p2.
Introducing the denitions of , y^1 , and y^2, we arrive at the equation given in the text.
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It is possible to arrive at this solution by using only equation (2) for each individual, together with
the equations for market equilibrium. In general we should also require the budget constraint for one
individual, but for the particular utility function specied here, this is not necessary. Geometrically
this means that the equations for market equilibrium determine the Edgeworth box for the problem,
and (2) says that we should be on the contract curve through the box. With the given utility function
the price ratio is then determined, irrespective of the allocations y^is . This means that the marginal
rates of substitution are the same along the complete contract curve, which is not the case for all
utility functions.
(d) p is increasing in 1 =2: A higher payment is required in equilibrium for claims on income in a state
which is more likely. p is increasing in (^y2 , y^1 ): This is a measure of the scarcity of income in
state 1 compared to state 2. High scarcity requires a high equlibrium price. p is increasing in the
harmonic mean of the coecients of absolute risk aversion if income in state 1 is the most scarce,
but decreasing if income in state 2 is the most scarce. This is related to the willingness to pay to
avoid risk being higher when risk aversion is high. The special case of incomes in both states being
equal, is treated in part (e).
(e) Here the total incomes in the two states are equal. The Edgeworth box is quadratic, and the two
certainty lines (45 degree rays) coincide. There is no social (aggregate) risk. The price ratio will
be 1 =2 in equilibrium, and the market solution is found on the certainty line, since the demand
according to part (b) for each individual is the same for both states. The endowments may well be
outside the certainty line, but the market equilibrium implies that each individual ends up with full
certainty. Equilibrium demand for each individual will be
w^i
:
(7)
yi1 = yi2 =
p1 + p2
(f) We now assume that the projects are suciently small so that their eects on equilibrium prices can
be neglected. Using income in state 2 as the numeraire, the future cash ows may be valued as
px + x;
and
pz;
respectively.
Assuming that the cost (the \reduced present income") is the same for both projects, they may now
be ranked. The rst one is preferred if
px + x > pz;
while the second is preferred if the opposite holds.
There is no information telling us how large the willingness to pay today is for these incomes. The
only market described, is a market where claims to cash ows in two future states may be traded,
one for the other. There is nothing saying that the absolute price level may be interpreted so that
consumption today costs 1, and the abolute price level is not determined in the model. Thus we
cannot answer whether the net value of the projects is positive or negative.
2
Sensorveiledning, eksamen i Mikro A, august 1993
Ved D.L., 17.08.93, revidert 21.02.94
(a) Maksimeringsproblemet for individet, og frikonkurranselsningen, er et spesialtilfelle av modellen pa
s. 613 i Gravelle og Rees, 2. utg. Spesialiseringen bestar i at det bare er to tilstander, i punkt (b) ogsa
en spesiell nyttefunksjon, og i punkt (c) bare to individer. Dessuten forutsetter oppgaven i motsetning
til boka at vi -funksjonene er tilstandsuavhengige, og at sannsynlighetene s er individ-uavhengige.
Riktignok er formuleringen i (a) noe mer generell enn i boka, siden det i dette punktet ikke er
forutsatt von Neumann-Morgenstern preferanser. En generell nyttefunksjon, Ui , vil vre voksende
og kvasikonkav i (yi1 ; yi2), og individets problem er
max U (y ; y )
yi1 ;yi2 i i1 i2
gitt
2
X
s=1
ps yis =
2
X
i=1
ps y^is :
(1)
(Prisene er denert i teksten til punkt (b).)
Vi kan trygt forutsette at budsjettbetingelsen er oppfylt med likhet, siden vi forutsetter ikke-metning.
Frsteordensbetingelsene for en indre lsning bestar av budsjettbetingelsen og
@Ui =@yi1 p1
= :
@Ui =@yi2 p2
(2)
Tolkningen av denne likningen, og guren som illustrerer den, er velkjent fra grunnfag.
Om vi forutsetter von Neumann-Morgenstern preferanser som i resten av oppgaven, blir nyttefunksjonen
U (yi1 ; yi2) 1 vi (yi1 ) + 2 vi (yi2):
(3)
Da far den marginale substitusjonsbrken formen
1vi0 (yi1 )
;
2vi0 (yi2 )
og indierenskurvene har den spesielle egenskapen at alle har samme helning langs sikkerhets-linja
(45-graders-linja), i absoluttverdi lik 1=2.
(b) Likning (2) kan na skrives
1i e, y 1 p1
= :
(4)
e, y 2 p
i i
2
i
i i
2
Sammen med budsjettbetingelsen har vi to likninger i yi1 ; yi2, og lsningen av dette likningssytemet
gir den likningen for yi1 som er gitt i oppgaveteksten. Vi far en tilsvarende likning for yi2 .
(c) Markedslikevekt forutsetter
y^1 y^11 + y^21 = y11 + y21;
(5)
og en tilsvarende likning, tilbud lik ettersprsel, for krav pa inntekt i tilstand 2. Denne andre
likningen er imidlertid overdig etter Walras' lov.
Ved a sette inn uttrykkene fra punkt (b) for ettersprselen for hvert av individene, nner vi
p1 (^y11 + y^21 ) + p2 (^y12 + y^22 ) , p2 11 + 12 ln 21 pp12
y^1 =
:
p1 + p2
(6)
Ved a dividere over og under brkstreken med p2 , far vi en likning i prisforholdet p p1 =p2. Ved a
sette inn for denisjonen av , y^1 og y^2 , nner vi den likningen som star i oppgaveteksten.
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Det er mulig a komme fram til denne lsningen ved bare a basere seg pa likning (2) for hvert av
individene sammen med likningene for markedslikevekt. Generelt ville vi ogsa trenge budsjettbetingelsen for ett av individene, men det gjelder ikke for den nyttefunksjonen som er spesisert i denne
oppgaven. Geometrisk kan vi oppfatte det slik: Likningene for markedslikevekt fastlegger bytteboksen (allokeringen skal fordele det som er tilgjengelig), og (2) tilsier at vi skal vre pa kontraktkurven.
Med den gitte nyttefunksjonen er prisforholdet dermed fastlagt, uavhengig av de tildelte beholdningene y^is. Det betyr at de marginale substitusjonsbrkene er de samme langs hele kontraktkurven,
noe som apenbart ikke gjelder for alle nyttefunksjoner.
(d) p er voksende i 1=2: Det kreves hyere betaling i likevekt for krav pa inntekt i en mer sannsynlig
tilstand. p er voksende i (^y2 , y^1 ): Dette er et mal for knappheten pa inntekt i tilstand 1 i forhold
til tilstand 2, og stor knapphet gir hy pris. p er voksende i det harmoniske snittet av koesientene
for absolutt risikoaversjon dersom inntekt i tilstand 1 er knappest, og avtakende hvis inntekt i
tilstand 2 er knappest. Dette har a gjre med at betalingsvilligheten for a unnga risiko er hyere nar
risikoaversjonen er hy. Spesialtilfellet der samlede inntekter i de to tilstandene er like, er behandlet
i punkt (e).
(e) Her er samlet inntekt i de to tilstandene like. Situasjonen er drftet pa s. 606{607 i boka. Den
relevante guren er en Edgeworth bytteboks, som i boka, men med den spesielle egenskapen at den
er kvadratisk. Dermed faller de to sikkerhetslinjene sammen. Det er ingen sosial risiko (se andre
avsnitt s. 607). Prisforholdet blir i likevekt lik 1 =2, og markedslsningen havner pa sikkerhetslinja,
siden ettersprselen etter formelen fra punkt (b) for hvert individ blir den samme for begge tilstander.
Initialtildelingene ma antasa ligge utenfor sikkerhetslinja, men markedslsningen innebrer at begge
individer ender opp med full sikkerhet. Ettersprsel i likevekt er for hvert inivid gitt ved
w^i
:
(7)
yi1 = yi2 =
p1 + p2
(f) Vi antar na at prosjektene er sa sma at de ikke pavirker likevektsprisene. Om vi bruker inntekt i
tilstand 2 som numeraire, kan de framtidige inntektene verdsettes til henholdsvis
px + x
og
pz:
Hvis vi antar at kostnaden (\redusert inntekt i dag") er den samme for begge prosjektene, kan vi na
rangere dem. Det frste er foretrukket dersom
px + x > pz;
mens det andre er foretrukket ellers.
Det er derimot ingenting i markedet som forteller oss hvor stor betalingsvillighet vi har i dag for
disse inntektene. Det eneste markedet som er beskrevet, er et marked der inntekter i de to framtidige
tilstandene kan byttes mot hverandre. Det star ingenting om at det absolutte prisnivaet kan tolkes
slik at konsum i dag koster 1, og det absolutte prisnivaet er ikke bestemt i modellen. Derfor kan vi
ikke svare pa om nettoverdien av prosjektene er positiv eller negativ.
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