Incentives and Aggregate Shocks

The Review of Economic Studies Ltd.
Incentives and Aggregate Shocks
Author(s): Christopher Phelan
Source: The Review of Economic Studies, Vol. 61, No. 4 (Oct., 1994), pp. 681-700
Published by: The Review of Economic Studies Ltd.
Stable URL: http://www.jstor.org/stable/2297914
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Reviewof EconomicStudies(1994) 61, 681-700
? 1994The Reviewof EconomicStudiesLimited
Incentives
and
0034-6527/94/00330681$02.00
Aggregate
Shocks
CHRISTOPHERPHELAN
Northwestern University and University of Wisconsin-Madison
First version received August 1992; final version accepted May 1994 (Eds.)
theoryof thedynamicsof the distributionof consumpThispaperpresentsan incentive-based
tion in the presenceof aggregateshocks.Thepaperbuildson the modelsconcerningthedistribution
of incomeor consumptionand incentiveproblemsof Green(1987), Thomasand Worrall(1991),
PhelanandTownsend(1991),and Atkesonand Lucas(1992).By incorporatingaggregateproduction shocks,the modelallowsan examinationof the interactionsbetweenindividualand aggregate
consumptionseries given incompleteinsurance.Further,the methodologyoutlinedallows the
incorporationof incentiveconsiderationsto macroeconomicenvironmentssimilarto Rogerson
(1988) and Hansen(1985).
1. INTRODUCTION
This paper presents an incentive-basedtheory of the dynamicsof the distributionof
consumptionin the presenceof aggregateshocks.Its methodologicalaccomplishmentsare
two-fold: First, it displays a method for explicit considerationof aggregateshocks in
incentiveeconomiessuch as those of Green (1987), Phelan
dynamicgeneral-equilibrium
and Townsend(1991) and Atkeson and Lucas (1992). Second, it displaysa method for
explicitconsiderationof incentiveconditionsin macroeconomicenvironments,where,for
reasonsof tractability,incentiveconditionsare usuallyignored.One interpretationof the
limited-insuranceversion of the
environmentpresentedhere is an incentive-constrained,
unemploymentmodels of Hansen (1985) and Rogerson (1988) which, for reasons of
tractability,assumefull insuranceover unemploymentrisk.
In the model presented,each agent'sobservableoutcomesare a functionof his unobservablelevel of effort, an unobservableidiosyncraticshock, and a publicly observed
aggregateshock. While risk-averseagents would wish to pool (and thus insureagainst)
their idiosyncraticrisk, such completeinsuranceis not feasibledue to the unobservable
effort.Becauseof the non-incentivecompatibilityof full-insurance,highereffortlevelsare
achievedby makingeachagent'sconsumptiona functionof his currentandpast observable
outcomes.Agentswith favourablecurrentoutcomesreceiveboth highercurrentand future
consumption.This contingencyof individualconsumptionon individualoutcomescauses
the consumptionof ex ante identicalagents to differover time. Further,since individual
outcomesare also affectedby aggregateshocks, the aggregateshocks not only affect the
averagelevel of consumptionas in a single-agentmodel, but the efficientdistributionof
consumptionas well.
To this date, the only analyticallytractablemodel available to make predictions
regardingthe dynamicsof the distributionof consumptionin the presenceof aggregate
shocks is the full-informationor complete-marketsframework.With aggregablepreferences, such an economy reducesto an economywith a single representativeagent. After
economy, one can make predictionsregardingdistribusolving the representative-agent
economy relativelystraighttions in economieswhich reduceto the representative-agent
forwardly.Papersby Mace (1991), Cochrane(1991), and Townsend(1994) examinethe
681
682
REVIEWOF ECONOMICSTUDIES
relationshipbetweenindividualconsumptionseriesand aggregateconsumptionseriesin
light of the predictionsof full-information,complete-marketmodels. The generalresult
of these papers is that the full insuranceimplicationsof full informationmodels are
rejected.Thisimpliesthe needfor modelswhichallowricherinteractionsbetweenaggregate
and idiosyncraticshocks on consumptionlevels.
Solvingdynamic,stochasticmodels which make predictionsregardingconsumption
distributionsis difficultif one straysfrom assumptionswhich allow reductionto a representativeagent.With more than one type of agent at a given time, marketclearingprices
generallydependon the entiredistributionof agent types. In modelswherethe evolution
of the distributionof typesis endogenous,thiscausesintractability.Onewayof interpreting
the aggregationresultswhichallow the reductionof an economyto a singlerepresentative
agent is that they show one set of specialcircumstanceswhich cause distributionsnot to
affectprices.
An accomplishmentof this paper is the derivationof a differentset of assumptions
which allow predictions regardingdistributionswith the full-informationassumption
relaxed,but still allow the model to be solved. While the predictionsof this model differ
substantiallyfromthe testable(and rejected)full-insurancepredictionsof the full-information model, they are not without content. Specifically,I show that like the dynamic,
incentive-basedmodel of Green(1987) whichis void of productionand aggregateshocks,
individualconsumptionlevels follow a processwhich is the sum of an i.i.d. term and a
term which follows a randomwalk. Second, I show that while individualconsumption
changesare not perfectlycorrelatedas with full-insurance,they are positivelycorrelated
acrossindividuals,since they dependon a commonaggregateshock. Finally,I show that
the characteristicsof the cross-sectionaldistributionof consumptionitself will dependon
the historyof aggregateshocks.I show that in contrastto the full-informationframework
wherethe currentaggregateshock alone determinesthe distributionof consumption,the
frictionsassumedherecausethe predicteddistributionof consumptionto follow a Markov
process.
Thispaperbuildson the modelsconcerningthe distributionof incomeor consumption
and incentiveproblemsof Green (1987), Thomas and Worrall (1991), Phelan (1994),
Phelanand Townsend(1991), and Atkeson and Lucas(1992). Unlike each of the papers
except Phelan (1994), the long-rundistributionof consumptionis not degeneratedue to
the overlappinggenerationsframeworkadoptedhere. Banerjeeand Newman (1991) also
considerthe questionof consumptiondistributionand risk-bearingand developa model
with a non-degeneratelong-rundistribution.They do this by assumingfamily dynasties
with bequestmotiveswhereconsumptionis boundedfrom below by the fact that agents
arelimitedin the amountof debt theycan pass to theirchildrenand above by assumptions
causing the incentiveproblem to disappearfor agents with high enough wealth levels.
Taub (1990) considerslinear incentiveschemesin an economy with moral hazard and
aggregateshocks.
The model in this paper can also be consideredan incentive-constrained,
limitedinsurancemodel of unemploymentsimilarto the full-insuranceunemploymentmodels of
Hansen (1985) and Rogerson (1988). Thus while the model presentedin this paper is
simplerthat that of Hansen on dimensionsother than the considerationof incentives
(specifically,the modelin thispaperdoes not havecapital)themethodsin thispapershould
be seen as encouragingto macroeconomictheoristswho wish to incorporateincentivesor
dynamiccontractinginto models with both individualand aggregaterisk.
Section2 presentsthe technologyand preferencesassumedin this model. Section 3
presentsa fairly lengthy constructionof an equilibrium.The key of this section is an
PHELAN
INCENTIVESAND AGGREGATESHOCKS
683
ability to show that marketpricescan be representedby vectorswhich remainconstant
over time, or which are independentof the distributionof types in the population.The
aggregateshock affects realizedprices only by picking out elementsof these constant
vectors.(For example,therewill be a time-independent
vectorassociatingeach aggregate
shock with a one-periodinterestrate. The realizedinterestrate is determinedonly by the
currentrealizedaggregateshock.) Section4 discussesthe characteristics
of this equilibrium
as comparedto the characteristicsof the same model but given full information.Section
5 discussesoptima, using an extensionof the techniquesof Atkeson and Lucas (1992).
Here the constructedequilibriumof Section 3 is shown to be optimal, and a planning
problemis derivedto find the plan associatedwith the constructedequilibriumwithout
searchingfor market-clearingprices.Section6 concludes.
2. THE MODEL
The economy consists of overlappinggenerationsof identical agents. Each generation
consists of a continuumof agents and is of equal size (unity) at birth. At the beginning
of each date, each agent takes an unobservableaction aEA where the set A is assumed
orderedand finite.This action resultsin a publiclyobservedoutput qe Q (Q orderedand
finite) accordingto the non-zeroprobabilityfunction P(qla, 0) where Ce0 (0 ordered
and finite) is a publicly observedaggregateshock drawn independentlyover time with
probabilityZ(0). It is importantto note that 0 is observedafterthe actionis taken.After
output q occurs,consumptionoccurs. Let Y-=R denote the set of possible consumption
amounts.After each period,agents are assumedto die with probability(1 -A), independent of age, and thus the size of the t-aged generationequals A''. Assume at the first
calendardate that thereexists the steady-statenumberof agents, 1/(1 -A). The calendar
dates are numberedr= 1, ..., ooc.
These variablescan be interpretedstraightforwardlyas a model where individuals
can "sleepon thejob" (take low efforta). In this interpretation,q representsthe physical,
stochasticallydeterminedwork outputof the agentmeasuredin units of the consumption
good and P(qla, 0) representsthe probabilityof each of these output levels for a given
level of effortand aggregateshock. Presumably,low effortlevels increasethe probability
of low outputs.An alternativeinterpretationis to considerq to be the qualityof a match
between a worker and a firm (again measuredin units of the consumptiongood and
representingthe deterministicproductivityof the workerin that match) and a to be the
unobservableefforttakenby an agent to find a productivematch.Thus, one can consider
an incentive-constrained
unemploymentmodel by letting q take on two levels: employment,with a correspondinghigh productivityq, and unemployment,with a corresponding
zero or low productivityq. The function P(qla, 0) is then the probabilityof a given
employmentstate for an agent given the effortlevel of the agent at findinga job, a, and
the aggregatestate 0.
At the beginningof his life, each agent entersinto a bindinglifetimecontractwith a
financialintermediaryor firm, in the case of equilibria,or the social planner,in the case
of optima.The contractspecifiesfor each date in the agent'slife his action level a and his
consumptionc. The contractmaymakeconsumptionor efforta functionof all information
availablein the economy at the appropriatetime. Agents care about their expecteddiscountedstreamof point-in-timeutilitiesover consumptionand effortwherethe common
point-in-timeutility function U(a, c) is the constant absolute risk aversionspecification
684
REVIEWOF ECONOMICSTUDIES
U: A x Y-+R_ such that U(a, c) = -exp (-y(c - v(a))), wherev(a) is any increasingfunction. Agents discountthe futureusing the constantdiscountparameter 1< 1. Define the
set of possibleex-anteutilities W_ R_.
3. EQUILIBRIA
This section constructsan equilibriumwhere all potential tradingis betweeninfinitelylived financialintermediariesor "firms"which exist from the first calendardate. When
agentsare born they are offeredlifetimecontractsby firmsand acceptwhatevercontract
gives them the highestexpecteddiscountedutility w from the perspectiveof birth.This is
the only markettransactionindividualsmay ever engage in. From then on they simply
follow the contract.In effect,agents "sell themselves"to a firmfor life. Enoughcommitment is assumedbetweenfirmsand individualsfor such contractsto be always binding.
While for now this is simply an assertedrestrictionof the interactionbetweenagents in
the economy, it will later be shown to non-restrictivein the sense that if the possibility
for such commitmentexists, it is optimal for society or the contractsthemselvesto bar
agentsfrom trade.
Firms, on the other hand, can trade the consumptiongood across dates and states
at prices they take to be given. For a given promisedutility w0, the firm'sproblemis to
maximizearbitrageprofits (or, as it will be stated later, to minimizearbitragelosses).
Firms do not have preferences.They take as given prices for date and state contingent
consumptionin termsof a numerairegood and use these pricesto transformprofitsand
losses at variousdates and states into a single number.Again, only firmsoperatein the
contingentclaimsand creditmarkets.Transactionsin these marketsare assumedobservable and thus barringindividualsfrom tradeis enforceable.'
Optimal contracts
I now considerthe questionof what kind of contractsfirmswill offer to agents. Instead
of assumingtime-zeromarketsallowingfirmsto tradedate-and state-contingentconsumption, I assume a sufficient set of spot markets. Let B,(0,lJ00...
,
O,'-1) be the spot price
at the beginningof date t of 0,-contingentconsumption,in termsof date t non-contingent
consumption(thus EZ.B,(Ol00, . . ., of- 1)= 1). That is, I assumea spot insurancemarket
at date t that allowsfirmsto buy and sell 0,-contingentconsumptionbefore 0, is realized.
For intertemporal trade, I assume a credit market. Let S,(0,l 0, . .
., O- 1)
be the spot
priceat the end of period t (after 0, is realized)of date t + 1 non-contingentconsumption
in termsof current(0,-contingent)consumption.Thesespot marketssufficientlyreplicate
the tradingopportunitiesof a completeset of time-zeromarkets.
The problemof findingefficientcontractscan be solvedusing recursivetechniquesif
these spot prices are independentof history and depend only on the currentaggregate
shock 0, and can thus be writtensimplyas vectorsB(0) and 3(0). This is trueunderany
framework(as does this one if not for
economywhich reducesto a representative-agent
unobservableeffort),but at this point I have given no reasonswhy this shouldbe the case
for this economy. Nevertheless,it is still a well definedquestion to ask how contracts
wouldlook if thesevectorsof spotpriceswereconstant.I will laterconstructan equilibrium
wherethese vectorsare, in fact, constant.
1. Note that unlike most overlapping-generations models the traders here, the firms, have access to all
markets.
PHELAN
INCENTIVESAND AGGREGATESHOCKS
685
Optimalcontractsare recursivein the sense that the expecteddiscountedutility of
an agent from any point time on (say w) sufficientlysummarizesall informationabout
that agent relevantfor constructinga continuationcontract between the firm and the
agent.2When an agent is born, the cost to a firm of contractingwith the agent is solely
a functionof the level of utility w the firmpromisesthe agent. (Here, cost is in termsof
presentnon-contingentconsumption.)An agent owed a high expecteddiscountedutility
(say wo)will cost more than an agent owed a low wo. After the agent'sfirst day of life,
he will have an expecteddiscountedutility (say w1)of followingthe contracthe signedat
birth. Since actions and transfersunder this contract can depend on the output of the
agent in his first period of life, this expecteddiscountedutility w1 will in generaldiffer
from the expecteddiscountedutilitywothe agenthad when he signedthe contract.Nevertheless,from the perspectiveof the firm,the one-period-oldagent looks no differentthan
a newbornagent who was given an initialutilitypromiseof wi. It costs the firmthe same
to providea lifetimecontractto an agent initiallypromisedw1as it does to continue a
contractthat providesthe agent with a continuationutility of w1.
Given that the firmconsidersthe expecteddiscountedutilityof the agentto be a state
variable,its choice variablesare as follows. The firmchoosesa functiona(w): W-+A,that
specifiesthe recommendedaction as a function of the agent's expectedutility from the
currentdate on. Likewise,the firmchoosesfor each we Wa consumptiontransferfunction
Y(q,01w):Q x 0-*R which again dependson the agent's expecteddiscountedutility at
the beginningof the period,and also the agent'scurrentoutputq and the currentaggregate
shock 0. The currentperiod output of the agent and the aggregateshock are known to
the firm at the time of the consumptiontransferbut not at the time actions are recommended. Finally, the firm chooses for each we Wa function W'(q, 01w): Q x 0-* Wwhich
specifiesthe utilitypromiseto the agent from the perspectiveof the next period.
The set of functions [a(w), Y(q, 01w), W'(q, 01w)] completely specifies a lifetime con-
tract for an agent with an initial utility of w0. His recommendedaction on his first day
of life is a(wo)and his consumptionis Y(qo,Oolwo),whereqois the agent'soutput on his
firstday of life and 0 is the aggregateshock for this date. The promisedutility function
W'(q,Olw) then generates the contract for the next period of life. For instance,
a( W'(qo, Oolwo)) is the agent's recommended action in his second period of life, and
Y(ql, 0OiW'(qo,Oolwo))is the agent'sconsumptionin his second periodof life.
Without loss of generality,I can split the consumptiontransferfunction Y into a
transferwhich does not depend on q or 0, z(w), and a transferwhich can depend on q
and 0, y(q, 01w), so that Y(q, 01w) = z(w) +y(q, 01w). Likewise, I can allow the promise
generatingfunction W' to be stated as a componentw'(q,Olw)which may dependon q
2. The proofthat the continuationutilityof the agentw sufficientlysummarizeshis history(moreformally
presentedin Spearand Srivastava(1987) and Phelanand Townsend(1991)) can be sketchedas follows.One
first formulatescontractsas specifyingthe work efforta and consumptionc of an agent at each point in time
as a functionof the agent'sentire output historyas well as the historyof aggregateshocks and consideran
efficientcontractin this (large) space. After any given historyh, the agent will have a utility w(h) associated
with continuingthe contract.One then considersgiving the firm a chance to re-optimizeor choose a new
continuationcontractsubjectto incentivecompatibilityconditionsand a constraintthat the agent'scontinuation
utilityremainconstant.However,one can showthatanynewcontinuationcontractcouldhavebeenincorporated
into the originalcontractwithoutupsettingthe incentivecompatibilityfor the originalcontractor the ex ante
utilitythe agent associatedwith the originalcontract.This impliesthat the firmcannot lowerits continuation
costs by reoptimizing.Any potentialgain from re-optimizationwould have alreadybeen incorporatedinto the
originalcontract.This impliesthat if afterhistoryh, the firmwereto "forget"the historyh, but rememberthe
agent'sutilityw(h)associatedwiththis history,the firmcouldrecoverthe continuationplanfromh by minimizing
its continuationcosts of providingan incentivecompatiblecontinuationcontractsubjectto the agentreceiving
Thusthe agent'sutilitycontainsall the informationnecessaryto recoverthe continuacontinuationutilitywv(h).
tion contractor sufficientlysummarizeshistoryh.
REVIEWOF ECONOMICSTUDIES
686
and 0 scaledby exp (-yz(w)), or W'(q, 0jw)=exp (-yz(w))w'(q, 01w). For now, simply
take this non-contingentpaymentz and the scalingof utility promisesas harmlessextra
degreesof freedomin the firm'schoiceproblem.The non-contingentpaymentz can always
be chosen to equal to zero and thus exp (- yz) =1. Further,for z equal to any non-zero
constant,the functionsy and w' can be chosen to "undo"the effectof the non-zeroz.
The firm is not free to pick just any contract. For an agent who was promiseda
specificexpecteddiscountedutilityof w, the firmmustpick policies[z, a, y(q, 0), w'(q,0)]
which actually deliver this expected utility. This can be expressedin the form of the
promise-keepingconstraint
W=Z EQ {-exp (-y(z+y(q,
0) - v(a))) + fA exp (-yz)w'(q, 0)}P(qla, 0)Z(0).
(1)
Equation(1) is the lifetimeexpecteddiscountedutilityof the agent from the beginningof
the periodgiven that the utilitypromiseexp (-yz)w'(q, 0) is actuallydeliveredgiven that
the agent lives to see the next period.
The firmmust also respectthe unobservabilityof actions,again for each we W.This
is expressedas incentiveconstraintsthat for all aeA,
E. EQ {-exp (-y(z+y(q,
>_,EQ
0)- v(a))) +f3A exp (-yz)w'(q, 0)}P(qla, 0)Z(0)
{-exp (-y(z +y(q, 0)- v(a))) + A exp (-yz)w'(q, 0)}P(qla, 0)Z(0).
(2)
The right-handside of equation(2) is the lifetimeexpecteddiscountedutilityof following
deviationaction a, again given that futureutilitypromisesare kept.
Let C*(w)denotethe cost in termsof currentnon-contingentconsumptionof providing an expecteddiscountedutilityw to an agent.Optimalcontractscan be foundby noting
that C*(w) is a fixed point of the operator T defined by the following programming
problem.The operatorT takes as given a cost functiongoverningutilitypromisesmade
today for tomorrowon, and generatesthe cost functiongoverningutility promisesfrom
the perspectiveof today. The objectiveof the programmeis to minimizethe value of net
consumptiontransfersin termsof currentnon-contingentconsumption,or,
(TC)(w)-
min
Y,
Q{z +y(q, 0)-q
z,a,y(q,0),Wv(q,f0)
+ S(0)AC(exp (-yz)w'(q, 0)))P(qla, 0)B(0)
(3)
subjectto the promise-keepingconstraint(1), and the incentiveconstraintsrepresented
by (2). It is importantto note that while q is a randomvariablefor a given agent, the
innersummationover q in the objectivefunction(3)-the expectedvalueof the expression
insidethe bracketsgiven 0-is knownwith certaintygiven 0. Firmsdo not face risk over
q realizationssince they can contractwith a positive mass of agents and thus P(qla, 0)
representsthe certainfractionof agents who realizeq. Thus the inner sum is the certain
cost, for a given 0 realization,of the plan [z, a, y(q, 0), w'(q,0)]. Summingover 0 and
weightingby the prices B(O) gives the total cost of the plan in terms in currentnoncontingentconsumption.
The exponentialutility functionallows a substantialsimplificationof this programming problem. The trick is to use the unconditionalpayment z to handle the utility
constraint,and the conditionalpaymenty(q, 0) and promisesw'(q,0) to handlethe incentive constraints.This is justifiedas follows.
PHELAN
INCENTIVESAND AGGREGATESHOCKS
687
In the incentiveconstraint(2), one can pull the expressionexp (-yz) outsideof both
summationson each side of the inequalityand cancel, giving
Z
ZQ {-exp (-y(y(q,
>,
0) - v(a)))
+
fAw'(q, 0)}P(qla, 0)Z(0)
(-y(y(q, 0) - v(&)))+ fAw'(q, 0)}P(qJ , 0)Z(0).
Q {-exp
I
(4)
This shows that if a contractis incentivecompatiblegiven one constantpaymentz, it is
incentivecompatiblefor all constantpaymentsz.
In the promise-keepingconstraint(1), one can again pull out from the right-hand
side the expressionexp (-yz). If this is set equalto -w, or z(w)= -log (-w)/y, this gives
(1) as
-1 =E. YQ{-exp (-y(y(q, 0)- v(a))) +/3Aw'(q,0)}P(qJa,O)Z(O),
(5)
thus removingz and w from the constraintset. The intuitionhere is that (5) insuresthat
if z = 0, the functionsy(q, 0) and w'(q,0) deliveran expectedutility of -1. If the policies
[a, y(q, 0), w'(q,0)] satisfy (5), this along with the assumptionthat z(w) -log (-w)/y
insures that the collection [z, a, y(q, 0), w'(q,0)] satisfies the original promise-keeping
constraint(1) for any w. This ability to state the constraintsindependentof the promise
w allows the followingresult.
Lemma1.
(1) Thefunction C*(w) takes theform
Q)- =
1I
1 -6A
-log (-w) + H~
Y
(6)
average
whereH is a constant(C(-1)) and -L S(0)B(0) is theprice-weighted
(over 0) of theprice of one-periodaheadconsumption,3(0).
(2) There exist efficientpolicies (a, y(q, 0), w'(q,0)) which do not dependon the
promisedutilityw.
Proof First, the true cost function C* can be shown to be a continuousfunction
which everywherelies betweentwo particularfunctionsof the form in (6). The first, a
lowerboundof the truecost function,is the cost functionassociatedwith the sameproblem
but withoutthe incentiveconstraint.The second,an upperboundof the truecost function,
is the cost functionwhere the action a is constrainedto equal the lowest elementof A.
Each of these problemsis static, and thus the true cost functionsfor these problemscan
be shown to take the form of (6), with (say), H equallingthe constant term for lower
bound, and ft equallingthe constanttermfor the upperbound. The space of continuous
functionslyingeverywherebetweenthe boundingfunctionsjust describedis completeand
can be normalizedby applyingthe supnorm.Callthis normedvectorspaceS. The operator
T can be shown to be a contractionon S by examiningthe objectivefunction (3) and
noting that Blackwell'ssufficientconditionshold. The right-handside is monotone in C
and T(c+ C) = T(C) + SAc for constantsc. This insuresthat thereis a uniquefixedpoint
of T in S. The set of functionsactuallyof the form in (6) constitutea completesubsetof
S, say S. Showingthat T maps S to itself completesthe proof.
Since z can be set to any constant without harming the minimizedvalue of the
objectivefunction,set z equal to the constant -log (-w)/y. Second,considera function
REVIEWOF ECONOMICSTUDIES
688
CES (and thus of the form in (6)) with constanttermH1.Replacingz with -log (-w)/y
in the objectivefunction(3) and substitutingin for C allows us to derive
(T)(w)_
1
1-6A
-log (-w)+ 9
y
, P(qJa, O)B(O),(7
(7)
(0)A -log (-w'(q, '0))1Pa0B0
+ min
+
(G)(6)
0) -q +(?/I(
E.Z4Y~~
{Qy(q 0-q+
y
1-SAi
a,y(q,69),%'(q,f9)
wherethe minimizationis, again,subjectto (5) and (4). Sincethe solutionto the minimization problemin (7) subjectto (5) and (4) is independentof w, the sum of the second and
thirdtermson the right-handside of (7) is simplya constant.Thus we have shown that
the operatorT preservesthe guessedfrom of C* in the statementof the Lemma.The fact
that the minimizationproblemin (7) does not contain w impliesthat the loss-minimizing
policiesa*, y*(q, 0) and w'*(q,0) will not dependon w. 11
The true constant (say H*) can be found by using the fact that for the true cost
functionC*, T(C*)= C*, and thus the trueconstantH* equals 1/(1 - SA) times the solution to the minimizationproblem in (7). Given this H*, equation (6) is a fixed point
of T.
The fact that the loss-minimizingpoliciesa*, y*(q, 0) and w'*(q,0) will not depend
on w impliesthat all agents receivedthe same recommendedaction a and face the same
conditionalpaymentsy*(q, 0). Theirunconditionalpaymentsare determinedby the function z*(w)= -log (-w)/y. Future utilities are determined by the scaling factor
exp (-yz*(w))
=
-w, multiplied by the common updating function w'*(q, 0). This separa-
tion impliesthat one partof the cost-minimizingplanfor the representivefirmis to partially
pay the agents in termsof positive and negativeconstant annuities-fixed paymentsthe
agent receiveseveryperiod for the rest of his life. (This characteristicis in Green (1987)
and Atkeson and Lucas (1992) for optimalplans.)
To see this,consideran agentwithstartingutilitywoand thusan initialnon-contingent
payment z*(wo)= -log (-wo)/y. This agent's expected utility at the beginning of his
secondperiodof life can be writtenw, = -w0w'*(qOga,
00). Hereqois the agent'srealization
of q in his first period of life, and 0 is the economy-widerealizationof the aggregate
shock for that date. It quickly follows that z(w1) = -log (-wo)/y-log
(-w'*(qo, Oo))/y
An agent who "draws"qoand 00 in his firstperiod of life is deliveredthe unconditional
payment-log (-w'*(qo, Oo))/y consumptionunits in the next periodalong with his original unconditionalpayment-log (-wo)/y. In general,his unconditionalconsumptionpayment at any given period t is the sum of paymentsfor all previousperiods,includinghis
initial payment -log (-wo)/y, along with the latest addition -log (-w'*(qt,i,
Ot-1))/y.
Constructionof constant-price equilibrium
Everythingto this point relieson this assumptionthat the vectorsof spot pricesB(6) and
6(0) are actuallyconstant over time. I now turn to constructingan equilibriumwhere
this is true.Note that by constantpriceequilibriumI do not mean that realizedpricesare
constant,but only that the vectorsthat realizedpricesare drawnfrom are constant.
Assumea singleprice-takingrepresentativefirmwhichin everyperiodoffersall newbornagentsa contractwithutilitywo,makesnon-contingentpaymentsz*(w)= -log (-w)/
y to agentswhosecurrentexpectedutilitiesequalw, makescontingentpaymentsaccording
to y*(q, 0), and updatesutilitypromisesthroughthe functionw'*(q,0) as in the previous
INCENTIVES AND AGGREGATE SHOCKS
PHELAN
689
section. That is, assume that the representativefirm adopts policies which are optimal
given constantprices.I now look to see if marketscan clear this restriction.
For marketsto clear,this representativefirmneeds at all dates and statesto transfer
to agents (in aggregate)exactly as much consumptionas the agents produce.Put differently, the aggregatedeficit of the representativefirm (paymentsy+z minus outputs q)
must equal zero regardlessof the date or state. The strategyI use is to deriveconditions
firmto be constantacross
whichfor a givenperiod r cause the deficitof the representative
0 shocks and constant across the consecutivedates T and T + 1. I then show that these
conditionsimplythemselvesfor one periodahead,and thusmathematicalinductionimplies
the conditionsguaranteeconstantdeficitsfor all dates and states. Lastly,by constraining
this deficitto be zero, "marketclearing"conditionsare derivedfor the consumptiongood
over dates and states.
At the beginningof each period,everythingbut the distributionof promisedutilities
in the economyis the same.Let T' be the distributionof promisedutilitiesat date r. This
distributionis assumedto have total measure 1/(1 -A), the size of the population.Let
D*(Tr, 0) denotethe deficitof the representativefirm(or aggregatedeficit)at date r given
distributionT' and realization0 of the aggregateshock given policies [z*, a*, y*(q, 0)].
Lemma 2. Relative deficits across 0 shocks are independent of the distribution
of utilities T . That is, for all dates T and distributions Tr, the difference
(D*(TPr,0) - D*(Tr, O)) is a function only of the shocks 0Oe9 and Je0.
Proof. The aggregatedeficitcan be written
D*(Pr,
0)=
E (log (W) +y*(q, 0) -q)P(qIa*, o)dTr(w)}
={fJlo
>
) d'{PT(w)}+
1-A {Q {y*(q, 0)-q}P(qja*,
0)}.
(8)
The separationon the right-handside of (8) occurs because of the separationof
optimalpaymentsinto a non-contingentpayment-log (-w)/y whichdependson w and
a contingentpaymenty*(q, 0) whichdoes not dependon w. Thus total net paymentsare
the total of non-contingentpaymentswhich dependson the distributionof utilitiesT',
plus the total of net contingentpaymentswhich dependsonly on the expectedvalue of
the net contingentpaymentmultipliedby the numberof agents, 1/(1 -A). Differencing
(8) across 0 realizationscausesthe firsttermon the righthand side to cancel,provingthe
Lemma.
11
An immediatecorollaryis that if a set of polies [z*, a*, y*(q, 0)] causesthe aggregate
deficitD*(PT, 0) to be constantacross 0 realizationsfor one distributionT', then these
policieswill inducea constantdeficitacross 0 shocks for all distributionsVPT.
While the policies [z*, a*, y*(q, 0)] affect the deficit at date x, the policy w'*(q,0)
(along with z*) affectsthe date r + 1 distributionof utilitiesand thus the date r + 1 deficit.
Let G*(Tr, 0) denote the date r + 1 distributionof utilitiesgiven the date r distribution
T'Pand the date r shock 0 underthe policies [z*, w'*(q, 0)].
Lemma 3. If for a given distribution T'Pat date r, the deficit of the representative
firm D*(TQ, or) equals a constant d, regardless of the outcome of or, and iffor any given
REVIEW OF ECONOMIC STUDIES
690
j, 1, the T + I deficit D*(G*(Tr, Or), OFr+) equals this same constant d, regardless of Or,
then the deficit at all future dates (r + 2, T + 3,...)
will also equal d, regardless of the
outcomesof shocks(Or+I,Or+2,..
)-
Proof. This proof is based on mathematicalinduction.I show that if one assumes
constantdeficitsacross 0 at date r and this same equal deficit at date r + 1, then these
assumptionsimply themselvesone period ahead.
The conditionthat the date r deficitequal a constantd regardlessof the outcomeof
Or can be stated as, for all Ore-0
D*TPr, Or)
log w) +y*(q,
-{JWOQr)-q)P(qla*,
={{Jlo
-A {Q {y* (q, 0r)-q}P(qja*,
dPr(w)}+
$w)
Or)dTr(w)}
Or)} =d.
(9)
The requirementthat the date r + 1 deficitequal d regardlessof the outcome of or
can be writtenas, for all Or and any given Or+
D*(G*(TPr, Or), Ur?
i ))
-log (-w) dG*(Tr, Or)(w)}
=
+
I{Q{y*(q0,+1)
E,
- q}P(qja*,
O+)}
= d.
(10)
Condition(10) need hold only for one possible 0r+I becausecondition (9) holding
for all 0 impliesthat the expectedvalue of net conditionalpayments(the secondtermon
the right-handside of condition(9)) is constantacross 0 realizations.Thus condition(9)
holdingfor all 0 impliesthat if (10) holds for any particularvalue of 0r+I it holds for all
valuesof 0r+ I The next step in the argumentis to show that if the conditionsin equations
(9) and (10) hold for a distributionT', then this impliesthey also hold for each of the
possiblesuccessorsof 8T' G*(Tr, 0).
Equation (10) is simply a restatementof (9) where the distributionTr has been
replacedwith a successordistributionG*(Tr, or) and the currentshock has been set to a
particularvalue. Thus the assumptionof conditions(9) and (10) holdingfor T' immediately impliesthat condition (9) holds for each possiblesuccessordistributionG*(Tr, or).
To show that equation (10) will also hold for all possible ;Tr+' = G*(Tr, 0) requires
some more manipulation.Specifically,if I replacein for the successorfunction G*, the
sum of the unconditionalz paymentscan be writtenas
-log (-w) dG*e(r, Or)(w)
w
Y
-log (-wo)
-log (-wo) + A{{
(ww)*(q, or))
-log(-w)
dTr(w)}
~w
Y
+
-log
Y
-log (-W *(q, Or)) p(ql *, Or)}
+~~~EQ
0,'
P(qj a*,
(11)
PHELAN
INCENTIVESAND AGGREGATESHOCKS
691
Date r + 1 unconditionalpaymentswill equal the paymentsto newborns,plus A timesthe
unconditionaltime r payments(since only A fractionof the date r populationsurvives),
plus the new additionsto unconditionalpaymentsdeterminedby the functionw'*.Equation (11) allows the deficitat r + 1 (Equation(10)) to be writtenas
-log (-WO)+{{-log
7Y
+ 1A
+
(-W)dW()
w
IY
{E -log ( w *(q, or)) P(qja*, 0)}
I
1-A {EC {Y*(q, ?r+l) -q} P(q a* U,
?+ l)}
(12)
Since we know that (9) holds for both T' and each of its possible successors
G*(Tr, OT), the first term on the right-handside of (9) (the total of non-contingent
payments)must be equal betweenT' and each of its possiblesuccessors.This allows the
replacementof T' in (12) with any of its possiblesuccessorsT'+ l. Reversingthe derivations from (10) to (12) derivesfor all Or
-log (-W) dG(Tr+', Or)(w)}
D*(G*(Tr +l, Or), O + I))={J
7
w
+ 1!
{
*
y
{y(q,
Or+l)-q}P(qla*
r+,)}=d.
(13)
Thus the assumptionof conditions(9) and (10) for the distribution'r imply themselves for each possible successor distribution Tr+ l. Mathematical induction then implies
the constant deficit d at all dates x, r + 1, . . . regardless of the outcome of shocks
(Or, or+l - .).. ).
|
Lemma3 impliesthat if a set of constantprices {6(0), B(0)} inducespolicieswhich
satisfy(9) and (10) for any givendistributionat any givendate, this thenensuresconstant
deficitsacrossall statesat all futuredates.The pricesnecessaryto inducea constantdeficit
over all dates and states do not depend on the distributionT. Thus we can pick any
convenient distribution and find the prices which cause constant deficits for that
distribution.
In particular,considerequations(9) and (12) underthe assumptionthat the distribution of utilitiesis for all generationsdegenerateat a point w0-the date zero distribution
of utilities.Equation(9) becomesfor all 0
-log(W)
+ E (y(q, ) - q)P(ql a*, 0) = (1 - A)d.
(14)
Equation(12) becomesfor all 0 and any given 0
-log (-WO)+
E (y(q,
) - q)P(ql a*, 0)
-log (-w'*(q, 0))
+ AZQ
P(qja*, 0=(I-
A)d,
(15)
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REVIEW OF ECONOMIC STUDIES
or, from noticing that equation (15) is simply equation (14) plus an extra term, (15)
becomesfor all 0
E
-log ( wP*(q,0)) P(qja*, 0)=0.
(16)
Note thatequation(16) impliesthat the aggregateof new unconditionalz paymentsequals
zero. Further,since equation (16) is also the expectedvalue of additionalz payments
givena* and the realizationof 0, (and this holds for all 0), an individual'sunconditional
z paymentfollows a randomwork. Furtherstill, note that an agent's realizationof his
conditionalpaymenty(q, 0) is independentand identicallydistributedover time.Thus his
consumptionis the sum of a randomwalk componentand an i.i.d. componentand thus
has the samepropertyof every-increasing
cross-sectionalvarianceas a purerandomwalk.3
This impliesa step-by-stepmethod of constructingan equilibrium.Step 1 is to find
price vectors {a(0), B(0)}, and loss-minimizing policies given these prices
{a*, y*(q, 0), w'*(q,0)} such thatequation(16) holds and the aggregatedeficitis independent of the realizationof 0 or for any given 0 and all 0 # 0,
E {y*(q, O)-q}P(qja*, O)=Q {y (q, 0)-q}P(qja*, 0).
(17)
This condition,equation (17), is a direct implicationof (14) holding for all 0. If 0 can
take on n possiblevalues,equation(17) impliesn- I conditions.The conditionthat new
aggregatez paymentssum to zero for all 0 (equation (16)) impliesn conditions.These
are the correctnumberof conditions necessaryto pin down the 2n prices {a(0), B(0)}
given the restriction that E B(0) = 1.
Step 2 is to find the initialutilitypromisew0whichcausesmarketsto clearand which
resultsin zero losses for the representativefirm. Since the representativefirm is the sole
supplierand demanderof contingentconsumption,if pricesaresuchthatthe representative
firm chooses to run a zero deficit at all dates and states (which is impliedby equations
(17) and (16)) then marketsclear at all dates and states. Further,if the representative
firm runs a zero deficit at all dates and states, then its losses equal zero. Thus one can
find w0by simplysolving equation(14) for w0given a constantdeficitd=0.
The equilibriumis then definedas price vectors (6(0), B(0)}, the loss-minimizing
policies giving these prices {z*(w), a*, y*(q, 0), w'*(q,O)} and the initial utility w0constructedas above.
4. CHARACTERISTICSOF EQUILIBRIUM
Comparisontofull-information
framework
How do consumptionprofilesdifferin this economyrelativeto its full informationcounterpart-the same economy but with observableeffort?First, with full information,it is
possible there is a higher efficientlevel of effort, a*. I ignore this in this section and
focus only on consumptionprofilesholdinga* constant.Consumptionprofilesgiven full
informationare characterizedby full insuranceover idiosyncraticshocks. This implies
y(q,, 0) =y(q2, 0) for all (q,, q2, 0) and w'(q, 0) = -1 for all q and 0. That is individual
consumptionpaymentsy depend only on the aggregateshock 0 and agents start each
3. By the sequence{z,} being a randomwalk, it is meant that first differencesare i.i.d. over time and
E(z,+I-z,z,) = ).
PHELAN
INCENTIVESAND AGGREGATESHOCKS
693
period with the same expectedutility. Individualspecificvariablessuch as q cannot be
used to predictconsumptionor consumptionchanges.
This is not true givenunobservedeffort.In the equilibriumpresented,an individual's
consumptionat date t, c, can be expressed(wheres is the agent'sdate of birth) as
Ct=y(qt, 0t)log
and the first difference, ct+
-
(-wo)_
t-I log (-w'(qr, Or))
(18)
ct as
c,+,-c,=y(q,+,O,+,)_y(t
log
?(-
w'(qt, Ot)).(9
Underunobservedeffort,consumptionchangesdependon individualrealizationof q, and
q,+,, as well as 0, and 0,+ . This holds not only becausecontingentpaymentschange
dependingon q, and q,+,, but also because of the permanentconsumptionpayment
-log (-w'(q,, O,))/y.
With regardto the distributionof consumption,with identicalindividualseach causing zero profitsfor the hiringfirm,full informationimpliesthat each agent consumesthe
same as everyother. Consumptionvariesacrossdates only due to the aggregateshock. If
one assumesthat individualsstartlife with differentP(qja, 0) functions,then it is possible
to generate cross-sectionalinequalityin the full informationmodel. Nevertheless,full
insuranceimpliesthat the distributionof consumptionis simplya functionof the current
shock 0,. Underunobservedeffort,the distributionof consumptiondependson the entire
historyof aggregateshocks (0,1... , 0,). In fact, the evolutionof consumptiondistributions is a Markovchain.
Long-runproperties
Given that individualconsumptionlevels have a random walk component,one would
suspectthat the long-rundistributionof consumptionis degeneratesince as time goes on,
the varianceof consumptionacrossa cohortincreaseswithoutlimit. However,the assumption of a constantdeath rate and overlappinggenerationsallows more reasonablelongrun properties.Overlappinggenerationsand a constantdeath rate allows the distribution
of each generationto continuallyspreadwithoutcausinga degeneratelimitingdistribution
of the populationas a whole.
The long-runpropertiesof this economy can be characterizedas follows: First, one
can note that set of continuationutilitiesan agent can have is countablesince thereare a
finitenumberhe can haveat any age. The distributionof promisedutilitiesfor the economy
at a point in time then is elementof the 1/(1 -A) simplexof R from the fact that the
assumedpopulationsize is 1/(1 - A). This is the state of the economy.(The distributions
for each generationdo not mattersince agents of all ages are identical.)
One can show that if the same aggregateshock is receivedat all dates, then the
distributionof utilities convergesto a non-degeneratedistribution.For an economy at
least r dates old, let uPTdenote the distributionof utilitiesfor agentsweaklyyoungerthan
r. (upT is an elementof theg= As-' simplexof R ', againbecausethe size of this segement
of the populationequalsE As-.) If the aggregateshock is the same at all dates, then
distribution TPris independent of the calandar date. The limTo
mass of agents older than r becomesarbitrarilysmall.
ur
exists because the total
694
REVIEWOF ECONOMICSTUDIES
For generalrealizationsof the sequenceof aggregateshocks,the probabilityof realizing an infinitesequenceof aggregateshockssuchthat the distributionof utilitiesconverges
is zero. Young generationsconstitute a positive percentageof the population and are
affectedonly by recentaggregateshocks,and thus the distributionof utilitiesis continually
buffeted.If an infinitesequenceof a particularshock 0 causes a relativelytight limiting
distributionof consumption,thena singlerealizationof this shockcausesthe consumption
distributionto tighten. If an infinitesequenceof a differentshock 0 causes a relatively
wide limitingdistributionof consumption,then a single realizationof this shock causes
the consumptiondistributionto spread.
Even though there is not a limitingdistributionof consumption,there is a limiting
probabilitydistributionover the distributionof consumption.This is provedby proving
the existenceof a limitingprobabilitydistributionover the distributionof utilities.For an
economy at least r periodsold, let p' denote the probabilitymeasureover the mass of
agentsweaklyyoungerthan r periodson utility point wi.4 That is if p' ([x, y]) =p, then
thereis a probabilityp that the mass of agentsweaklyyoungerthan r on utilitypoint w,
is in the interval[x, y]. This probabilitydoes not dependon the date sincethe distribution
of utilitiesfor agents weaklyyoungerthan r periodsdependsonly on the r most recent
shocks.
The collectionof probabilitymeasuresp,i over all possibleutilitypoints wi definesa
probabilitymeasureover the possibleutility distributionfor agentsweaklyyoungerthan
r. Again, the limpO j4, exists because the total mass of agents older than r becomes
arbitrarilysmall.This impliesa limitingmeasureon utilitydistributionsand thusconsumption distributions.
5. OPTIMA
This section considersthe social planner'sproblemfor this economy and whether the
equilibriumof the previoussection solves this problem.The first resultpresentedis that,
like the representativefirm'sproblem,the socialplanner'sproblemfor this economytakes
a recursiveform (Lemma4). This then allows proof that the equilibriumof the previous
sectiondoes indeedsolve the socialplanner'sproblem(Lemma5). Besidesbeingof interest
in and of itself, this is desirablefrom a purely practicalviewpoint since analyzingand
computingoptimatends to be easierthan analyzingand computingequilibria.Statingthe
socialplanner'sproblemin a formusefulfor computingequilibriumoutcomesand prices,
however,requiressomewhatfurtherargument.To this end, a functionalform for the cost
functionof the socialplanneris derived(Lemma6). Giventhis result,the sectionconcludes
by presentinga reformulationof the social planner'sproblemsuch that the equilibrium
prices fall out as functionsof the Lagrangemultiplierson the constraintsfaced by the
social planner.
The generalmethodologyused in this section (followingAtkeson and Lucas (1992))
is to considerthe social planneras a sort of "residualclaimant"on the resourcesof the
economy. Specifically,consider the social planner as having an ability to contributea
constantamountof the consumptiongood (positiveor negative)to the economyat each
date and state.The socialplanner'sproblemis then to minimizethis constantcontribution
to the economy (or maximizehis constantextractionfrom the economy) subjectto constraintsregardingthe ex ante utilities of the agents as well as the incentiveconditions
4. Measurepu maps the Borelsubsetsof [0,
AS-'] to [0, 1].
PHELAN
INCENTIVESAND AGGREGATESHOCKS
695
regardingunobservedeffort.If the representativefirm'splan from the previoussectionis
not an optimal social plan, then the social plannershould be able to extracta positive
constantamountfrom the economyand still give each agent the ex ante utilityhe expects
firm'splanfromthe previous
fromthe equilibrium.On the otherhand,if the representative
section is optimal,the maximumamountthe social plannercan extract(again subjectto
utility and incentiveconstraints)should be zero.
It is assumedthat the plannermust treatall agentsof all generationsequally,or that
each agent receivesthe same ex ante expecteddiscountedutility 1o.5In the course of
proving that the equilibriumplan is optimal subjectto this constraint,it is first shown
that the social planner'sproblemtakes a recursiveform wherethe state variablesof the
economy are the distributionof utilities owed by the social plannerto currentlyliving
agents, and (implicitly)the ex ante utility owed by the plannerto agents born in future
periods, i0. To this end, let B be the set of Borel-measurabledistributionswith total
measure 1/(1 -A), the size of the populationof living agents at any time, and let TPeB
representa distributionof continuationexpecteddiscountedutilitiesfor currentlyliving
agents.
Define X*(T): B-.R as the minimumconstant contributionby the social planner
which allows the provisionthroughan incentivecompatibleplan of P to currentlyliving
agents and the deliveryof voto agents born in futureperiods. I definean efficientplan
(or an optimum) as a specificationof contingentconsumptionand recommendedeffort
levelsthat achievesa givendistributionof utilities' with the minimumconstantcontribution by the social planner,X*(T ).6
The claim that the social planner'sproblemis recursiveis essentiallythat the cost
functionX*(T) can be stated as a function of itself in a Bellman-typeequation.Given
this, the distributionof promisedutilitiesto currentlylivingagents' completelysummarizes history,allowingthe plannerto choose one-periodplans conditionalon this distribution and a rule to generatenew promisedutilities.
In this recursiveformulation,for a given distributionT, let the choice variablesfor
the social plannerbe the same as for firms,except now let each of the functionsdepend
explicitlyon w. That is, the social plannerchooses functionsz(w), a(w), y(w, q, 0), and
firm'sproblem,let z(w) representan unconditional
w'(w, q, 0). As with the representative
paymentand let promisedfutureutilitiesequalexp (-yz)w'(w, q, 0). (Again, z can always
be set to zero.)
The following Lemma proves that X*(T) indeed solves a Bellman-typeequation.
Essentially,this Lemma argues that the minimumconstant contributionnecessaryto
support a plan [z(w), a(w), y(w, q, 0), w'(w, q, 0)] is the maximum of the contribution
necessarytoday for each possible 0 realization,and the constantcontributionnecessary
to support each of the possible utility distributionsfor tomorrow.Again, let G(UT,0)
denote the date r + 1 distributionof utilities if the date r distributionis T and shock
0 occurs, given the policies z(w) and w'(w,q, 0) for generating the r + 1 utility
distribution.
5. Thisassumptionthrowsout all trueoptimaif all of themhavethe propertythat generationsaretreated
unequally.Thatis, with overlappinggenerationsit mightbe possiblethroughintergenerational
transfersto raise
the utilityof latergenerationswithoutloweringthe utilityof earliergenerations.Sincethis modelhas an initial
date t = 0, discounting,and equalsize generations(afterthe first),I believeI have ruledout such schemes.
6. Atkesonand Lucas(1992)definethe cost of a givenpolicyas the supremum
overdatesof the necessary
gift, and the true cost of a distributionas the infimumof the cost of all possiblepolicies.Thus they do not
assumethat any cost can actuallybe attained,only arbitrarilycloselyattained.Thisdid not substantiallychange
theirresults,so here I simplyassumefor conveniencethat all maximaare attained.
REVIEWOF ECONOMICSTUDIES
696
Lemma 4.
as follows:
The costfunction X*(T) is afixedpoint of thefollowing operator Tdefined
CTX)(T)=
max max sX(G(T, 0)),
min
z(w),a(w),y(w,q,0),w'(w,q,0)
0
EC {z(w) +y(w, q, 0) - q}P(qja(w), 0)dT(w), X(G(T, 0))}
x
(20)
w
subject to thepromise-keepingconstraintsimpliedby (1) and the incentive constraints implied
by (2) holdingfor we W.
Proof. The proof is nearly identical to the proof in Atkeson and Lucas (1992),
Lemma4.1, and is in the Appendix. 11
An optimum can be defined as the policies [z(wjT), a(wj'P),
y(w, q, OJT),w'(w,q, 0JT)] whichsolve (20) subjectto (1) and (2) for all possibledistributions T, and an initial utility vo such that X(Wo)= 0, where'P0is the distributionwhere
all mass lies at the point vo. This result now allows the proof that equilibriumof the
previoussectionsis indeedan optimum.
Lemma 5. The equilibriumpolicies of the representativefirm [z*(w) = -log (-w)/
y, a*, y*(q, 0), w'*(q, 0)], together with the equilibriuminitial promised utility wo, are an
optimum.
Proof. The proof argues first that the representativefirm'spolicies are a feasible
solutionto the socialplanner'sproblemand implya cost X*(Wo)= 0 for the socialplanner.
If these policiesare not an optimumthen theremust exist anotherset of feasiblepolicies
which allow the social plannerto extracta constantpositiveamount from the economy.
This is shown to contradictthe assumedcost minimizationby the representativefirm.A
more detailedargumentis in the appendix. 11
At this point, the Bellman equation (20) is not in a particularlyuseful form for
calculatingor analyzingequilibriaby findingoptima.The followingLemmasolvesfor the
social planner'scost functionX* up to constant.
Lemma 6.
Thefunction X* takes the form
X(T)
- log (-w)
dT(w) +K,
(21)
where
K=
min
{Q
{y(q, O)-q}P(q a, 0)),
(22)
a,y(q,O),w'(q,O)
for any particular 0e0 and where the minimization is subject to the incentive condition (4),
the promise keeping condition (5), the constant payments conditionsfor all 0e0, (16), and
the constant current deficit condition, equation (17) for all 0$0.
Proof. See Appendix. 11
PHELAN
INCENTIVESAND AGGREGATESHOCKS
697
The formationallows us, with one qualification,to find the marketclearingprices
usingthe Lagrangemultipliersassociatedwith the constantdeficitconstraints
(17) (to get B(0)) and the constant paymentsconstraint(16) (to get the 6(0)). The
qualificationis that the constraintset for the minimizationproblemin (22) mustbe convex
and thus we.can invoke the Kuhn-Tuckertheorem.One way to avoid convexityproblems
is to formulatethe choiceproblemin termsof lotteriesas in Phelanand Townsend(1991).
This is avoidedhere solely for ease of exposition.
The Kuhn-Tuckertheoremallows us to state that thereexist multiplierslo for 0$ 0
and po for each 0e0 such that the optimalpolicies [a*,y*(q, 0), w'*(q, 0)] solve
{a(0), B(0)}
min
{Q {y(q, C)-q}P(qla,
0)) +Eo, {(A Q {y(q, 0) -q}P(qla,
{y(q, 0)-q}P(qja,
+ZQ
a,y(q,6),w'(q,6)
-(Q
0)))
{ZQ
0))
-log (-w'(q, 0)) P(qla, 0)}
(23)
subject to the incentivecondition (4) and the promise keeping condition (5), and, in
addition,that each of the marketclearingconstraints(equations(17) and (16)) holds for
all 0.
This Lagrangianis a particularlyusefulform of the social planner'sproblem.In this
form, the planner'sproblemhas the same constraintset as the firm'sproblem.Further,
the objectivefunctionsin the two problemsdifferonly in the coefficientson the moments
EQ {y(q, 0) -q}P(qla, 0)
and EQ
Pog(-w(q0))P(qla,
0).
(24)
Thus to give the firm the same objectivefunction as the planner,one simply needs to
choose pricesto equatethe coefficientson thesemomentsor (fromequations(7) and (23))
for 0, B(O) = I-E00
AO,
for
#0 , B(0)= AO,
and for all 0, ()A
-
1 SA p.
(25)
Since a =Eo B(0)3(0), the 6(0) are impliedby the system of equationsdefinedby the
last expressionin (25) for all 0e0.7
This derivationallowsus to easilyfindmarketclearingpricesby solvingthis planner's
problem.Further,the convexityassumptioncan be checkedex post. That is, if one derives
a solution to the optimum problem allowing for probabilisticchoice, but the optimal
policies are neverthelessdeterministicin the form above, then any non-convexityin the
constraintset given deterministicchoice is harmlessand one can then use the above rules
to deriveprices {B(0), 3(0)).
6. CONCLUSION
The purposeof this paperwas to createa tractableframeworkto makepredictionsregarding the dynamicsof consumptionand distributionsof consumption.Of course,it is possible
to think of other ways of generatinga non-trivialdistributiontheory. One would be to
simplyclose marketsover state-contingentconsumption(or insurancemarkets),but allow
tradingover dates (or creditmarkets).This would not necessarilybe easierto solve since
one would still face the problemthat withoutspecialassumptions,interestrateswill be a
7. If the 9 is broughtto the right-handside by multiplyingeachside of the expressionin (25) by (I - 9A)
then the expressionis linearin the 3(0) and thus the systemhas a uniquesolution.
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REVIEW OF ECONOMIC STUDIES
functionof the distributionof wealth. Further,normativeanalysiswould be hinderedby
the fact that one has left unexplainedwhy it is that insuranceis hindered.
In this model, limitedinsurancecomes directlyfrom the informationalassumptions
of the model.Giventhese,the equilibriumis efficient.The equilibriumheremost likelyhas
differentconsumptionseriesimplicationsthan those impliedby simplyclosing insurance
markets. The best insurance possible given information constraint-the equilibrium
presentedhere-is not only more insurancethanwould exist givenclosedinsurancemarkets, but should also be expectedto look differentin regardto how consumptionmoves
with aggregateshocks. The creationof models with specificdifferingpredictionsallows
these differencesto be evaluatedin relationto data.
APPENDIX
Proofof Lemma4. Again, let X*(T) be the trueminimumconstantgift whichallowsthe attainmentof
the distributionof utilities P and for the deliveryof utility voto agentsborn in futureperiods.The Lemma
statesthat T(X*)=X*. First I show that T(X*)>X*.
Suppose that for some T, (TX*)(T) <X*(T). This implies that there exist functions [z?(w),
ao(w), yo(w, q, 0), w'?(w, q, 0)J such that
X*(T) > max max
{J 2
Q {z0(w) +y?(w, q, O)- q}P(qla0(w),
O)dT(w), X*(GO(TP, ))}
(26)
whereG?(Q,0) is the distributionof utilitiesgeneratedby [z?(w),w'0(w)lgivenshock 0.
Let the possibly time-dependent plan [z,(w), a,(w), y,(w, q, 0), w'(w, q, 0)1,=O., denote the policy that
actuallyachievesX*(P). Definea new time-dependentplan [z,(w), ao(w), y,?(w,q, 0), w',(w, q, 0)1,=O.,, by followingplan [z?,ao,yo, w'0lat the firstdate and plan [z, a', y, w' I for each date thereafter.This is a feasible
planthatactuallydeliversthe requiredutilities,anddoes so at a cost (TX*)(T) <X*(T) whichis a contradiction.
Thus, (TX*)(TP) X*(TP).
Now suppose for some P that X*(T) <(TX*)(T).
Let [z,?(w),ao(w), yo(w, q, 0), w',(w, q, 0)1,=O,., denote
the policy that actuallyachievesX*(TQ).The firstdate of this plan, [zo,ao,yo, w'/o,satisfiesthe incentiveconstraints and promise keeping constraintsfor the recursiveformulation.Further,its date-zerodeficit must
be weakly less than X*(TQ) given distribution P and any 0. Further, since [z?,(w), ao(w),
y,(w, q, 0), w',(w, q, O)1,=O.<,has a deficit weakly less than X*(TQ)in every period, the cost of each of its possible
successorsunder[zoo,w'ol,G0(T, 0), must be weaklyless thanX*(T). This impliesthat
max max
{z?(w)+y?(w,q, O)-q}P(qla?(w),
{jE'Q
O)dT(w),X*(GO(T, O))} X*(T)<(TX*)(T).
(27)
Since[zoo,
is a feasiblerecursiveplan, this contradictsthe minimizationin the definitionof T. Thus,
ao,yo, w'OJ
(TX*)(TP)5X*(TP).
II
Proof of Lemma5. Considerthe solution to the firm'sproblem.Therez*(w)= -log (-w)/y, and the
policies[a*,y*(q, 0), w'*(q,0)] solvedthe minimizationproblemin (7) subjectto (5) and (4). This impliesthat
the combination[z*(w),a*, y*(q, 0), w'*(q,0)1 satisfies(2) and (1), or that the solution to the equilibrium
problemis a feasiblesolutionto the optimumproblem.The plansofferedby the firmare incentivecompatible
and give each agent his requiredex-anteutility.
Further,recallthat the solutionto the equilibriumproblemdeliverszero aggregatedeficitsat all datesand
firmfaces on the firstcalendardate (all
states.Let TObe the distributionof utilitypromisesthe representative
mass on wo).Since the firm'splan is a feasibleplan for the social planner,the plannercan at least achievea
firm'splan
maximumdeficitover all datesand statesof zero, or X*(To):0. This impliesthat the representative
is not optimalonly if thereis anotherplan whichallows a negativeconstantpayment,or that X(TO)< 0.
Now suppose exactly that. This implies there exists a solution to (20) (say [z(wl1), al(wl'),
P(w,q, 01J), W'(w,q, 01T)Jwhichdeliversstrictlynegativedeficitsfor all datesand states.Couldthis planhave
been implementedby the representivefirm?Since I assumed(and confirmed)that the prices{6(0), B(0)} did
firmdid not conditionpolicieson T. Nevertheless,I did
not dependon the distributionT, the representative
firmto choose policiesindependentof T. It was simplynot in its interestto do
not constrainthe representative
INCENTIVES AND AGGREGATE SHOCKS
PHELAN
699
firmoptimallychoosespolicieswhichare independentof w, again,this
so. Further,althoughthe representative
was not arbitrarilyimposed,but derivedfrom the choice problem.Thus the representativefirm could have
chosento implement[A(wlIT),
a(wlT), P(w,q, 01IT),w'(w,q, 01IT)J.
Further,if thesepoliciesproducenegativedeficitsat everydateand everystate,thenif at leastone element
of {6(0), B(O)} is positiveand all othersare non-negative(a conditionwhichcan be checkedfor any candidate
firm.Thiscontradictsthe optimalequilibrium)thenthispolicywillproducepositiveprofitsfor the representative
firm.This contradictionimplies
ity of [z*(w),a*, y*(q, 0), w'*(q, 0)1 from the perspectiveof the representative
that the equilibriumconstructedin the previoussectionis optimalgiventhat it has non-negativepricesand at
least one positive price.8
11
Proof of Lemma 6. As in the proof of Lemma1, I exploita boundingfunctionof X* whichtakes the
form of (21). Let X(T) with constantK denotethe cost functionassociatedwith the removalof the incentive
conditions.
Imposingrestrictionson the minimizationproblemdefiningT (equation20) that z(w)= -log (-w)/y and
that policiesdo not dependon w or T definesa new operatorT(X). ApplyingT to a functionof the formin
(21) implies
- ~X)('P)
-log (-w) dP(w) +
-
W
min
maxmax -A
1
-log (-w'(q,
1_
G
(a.y(q.0).w'(q.0))
Y
E
(y(q, 0
-
q)P(qla, 0),
A
)) P(qla, 0) +
(28)
subjectto the promisekeepingconstraints(5) and incentiveconstraints(4) fromthe firm'sproblem.If the true
cost functionX* is of the formin (21) it must be a fixedpoint of P as well as T fromLemma5. Equation(28)
also showsthat the operatorT preservesthe guessedform (21), sincethe solutionto the minimizationproblem
does not dependon the distributionT.
Continuingin this line, one can imposecondition(16), that new-contingentpaymentsw'(q, 0) integrate
to zero for all 0. Again, this impliesa new operator(say T) for whichX* must again be a fixedpoint if it is
of the formin (21). This impliesthat (28) becomes
~~C
(TX)('P)iJ
~-log(-w)
W
dT'(w)+
minmax max {Q(y(q,
~~~~~(a.,y(q,0),w'(q,0))
Y
0)- q)P(qla, 0), K},
(29)
0
subjectto the promisekeepingconstraints(5), the incentiveconstraints(4), and condition(16). This makes
transparentthat for functionsof the formin (21), operatorT is non-decreasingin the constanttermK.
Now considerXi = T(X), or applyingoperatorT to the cost functiongivenno incentiveconstaints(which
is of the guessedform(21) withconstanttermK). SinceT(X) mustbe weaklygreaterthanX fromthe incentive
constraintsin the minimizationproblemdefiningT, one can discardthe secondpart of the innermaximization
and write
(TX)(v)f
w
-log (-w) dT(w)+
7
min maxE (y(q, 0)-q)P(qla,
(a.y(q6
))
0),
(30)
1
where,again,the minimizationis subjectto conditions(5), (4), and (16). Let K, be the constanttermassociated
withXi, equalto the minimizationin (30). Substitutingthis definitionfor K, into (29) impliesthatXI is a fixed
point of T.
At this point, it is clear that the true cost functionX* is a fixed point of T if it is of the guessedform
(21). The proof continuesby showingthat all fixedpoints of T otherthan Xi cannotbe the truecost function
and thus eitherXI=X*, or X* is not of the guess form.Furtherexaminationof (29) showsthat everyfunction
of the formin (21) with K> K, but less thanor equalto the upperboundis also a fixedpoint of T. Nevertheless,
sincethe policiesassociatedwith XI actuallydo achieveany distributionof utilitiesP with a constantgift less
than or equalto X,(T), only XI (the minimumof this set of fixed-pointcost functions)can be a candidatefor
8. Some readershavecommentedthat this resultis surprisinglyeasy to provegiventhe usualdifficultyof
provingthe FirstWelfareTheoremin infinite-goodsettings,especiallywithoverlappinggenerations.Whatmakes
this proof easy is that 1) the firmslive in everydate, and 2) that an optimumhere is definedvery differently
than usual. Here for one allocationto constitutean improvementover anotherallocation,it must use fewer
resourcesat every date and state. The specialcases that cause problemsin the usual First WelfareTheorem
proofswill probablycause the dualityapproachto breakdown here.
REVIEW OF ECONOMIC STUDIES
700
the truecost functionX* amongthe class of functionsof the form of (21). The functionXI is an upperbound
of the truecost function.
Thereare possiblyotherfixedpointsof T not of the formof (21). LetX be sucha fixedpoint. If for some
T,X(T) >X1(T), then X(T) cannot be the true cost function.Again, sinceX,(T) is an upperbound on the
true cost, any functionthat is anywheregreaterthan XI cannot be the true cost function.Lastly, since T is
for all P this implies (TX2)(T) ?(TX3)(T)
monotonic (if X2(T) <X3(T)
for all T), it is not possible that for
all T, X(TP)
<XI(TP). The statements(T(X)=X), (T(X1) =XI), and (for all T, XI(T') iX(TP))togetherviolate
monotonicity.Thusif thereareany fixedpointsof T not of the formin (21), they arenot the truecost function.
Thus X*=X,.
Giventhat we have derivedX* as T(X), the finalstep is to reconcilethe expressionfor the constantterm
in equation(30) with the constantterm in the statementof the Lemma.SinceX* is the true cost function,
imposinga constraintthat the currentperioddeficitbe constantacross0 is harmlessbecausethis a characteristic
This allowsus to specifythe socialplanner'sproblemas one of minimizing
of theconstructedmarketequilibrium.
the deficitgiven a specific0 realization,say 0, subjectto the constraintthat the deficitgiven other 0 values
equalsthis amount.Using equation(30) this implies
=
X*(T)
X
-log (-w) dT(w) +
min
{Q {y(q, O)- q}P(qla, O)},
(31)
a.y(q0).w'(q6)
wherethe minimizationis subjectto the incentivecondition(4), the promisekeepingcondition(5), the constant
paymentsconditionsfor all OeO, (16), and the constantcurrentdeficitcondition,equation(17) for all 0 #0. 11
Acknowledgements.Formerly,"Incentives,Inequality,and the BusinessCycle".The authorwould like
to thankAndrewAtkeson,John Cochrane,Robert Lucas,Lars Ljungqvist,Pete Streufert,LarrySamuelson,
RobertTownsendand participantsat seminarsat the Universityof Wisconsin,the FederalReserveBank of
Minneapolis,and the February1992 NBER EconomicFluctuationsMeetingsin Palo Alto, California.The
authorreceivedfinancialsupportfromNational ScienceFoundationgrantSES-9111-926.
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