golden eggs and hyperbolic discounting - bill harbaugh homepage

GOLDEN EGGS AND HYPERBOLIC DISCOUNTING

DAVID LAIBSON

Hyperbolic discount functions induce dynamically inconsistent preferencesimplying a motive for consumers to constrain their own future choices This paperanalyzes the decisions of a hyperbolic consumer who has access to an imperfectcommitment technology an illiquid asset whose sale must be initiated one periodbefore the sale proceeds are received The model predicts that consumption tracksincome and the model explains why consumers have asset-specic marginal pro-pensities to consume The model suggests that nancial innovation may havecaused the ongoing decline in U S savings rates since nancial innovation in-creases liquidity eliminating commitment opportunities Finally the model im-plies that nancial market innovation may reduce welfare by providing ldquotoomuchrdquo liquidity

I INTRODUCTION

ldquoUse whatever means possible to remove a set amount of moneyfrom your bank account each month before you have a chance tospend itrdquo

mdashadvice in New York Times ldquoYour Moneyrdquo column [1993]

Many people place a premium on the attribute of self-controlIndividuals who have this capacity are able to stay on diets carrythrough exercise regimens show up to work on time and livewithin their means Self-control is so desirable that most of uscomplain that we do not have enough of it Fortunately there areways to compensate for this shortfall One of the most widelyused techniques is commitment For example signing up to givea seminar is an easy way to commit oneself to write a paper Suchcommitments matter since they create constraints (eg dead-lines) that generally end up being binding

Strotz [1956] was the rst economist to formalize a theory ofcommitment and to show that commitment mechanisms could bepotentially important determinants of economic outcomes He

This work has been supported nancially by the National Science Founda-tion (SBR-95-10985) and the Alfred P Sloan Foundation I am grateful to OlivierBlanchard for posing the questions that motivated this paper I have also bene-ted from the insights of Roland Benabou Ricardo Caballero Robert Hall Mat-thew Rabin an anonymous referee and seminar participants at the Universityof California at Berkeley Boston University the University of Chicago HarvardUniversity the Massachusetts Institute of Technology Northwestern UniversityPrinceton University Stanford University and the University of PennsylvaniaJoshua S White provided excellent research assistance All mistakes should beblamed on my t 2 1 period self

q 1997 by the President and Fellows of Harvard College and the Massachusetts Instituteof TechnologyThe Quarterly Journal of Economics May 1997

showed that when individualsrsquo discount functions are nonexpo-nential they will prefer to constrain their own future choicesStrotz noted that costly commitment decisions are commonlyobserved

we are often willing even to pay a price to precommit futureactions (and to avoid temptation) Evidence of this in economic andother social behaviour is not difcult to nd It varies from the gra-tuitous promise from the familiar phrase ldquoGive me a good kick if Idonrsquot do such and suchrdquo to savings plans such as insurance policiesand Christmas Clubs which may often be hard to justify in view ofthe low rates of return (I select the option of having my annualsalary dispersed to me on a twelve- rather than on a nine-monthbasis although I could use the interest) Personal nancial manage-ment rms such as are sometimes employed by high-income profes-sional people (eg actors) while having many other and perhapsmore important functions represent the logical conclusion of thedesire to precommit onersquos future economic activity Joining the armyis perhaps the supreme device open to most people unless it bemarriage for the sake of ldquosettling downrdquo The worker whose incomeis garnished chronically or who is continually harassed by creditorsand who when one oppressive debt is paid immediately incurs an-other is commonly precommiting There is nothing irrational aboutsuch behavior (quite the contrary) and attempts to default on debtsare simply the later consequences which are to be expected Inabil-ity to default is the force of the commitment

Strotzrsquos list is clearly not exhaustive In general all illiquidassets provide a form of commitment though there are some-times additional reasons that consumers might hold such assets(eg high expected returns and diversication) A pension or re-tirement plan is the clearest example of such an asset Many ofthese plans benet from favorable tax treatment and most ofthem effectively bar consumers from using their savings beforeretirement For IRAs Keogh plans and 401(K) plans consumerscan access their assets but they must pay an early withdrawalpenalty Moreover borrowing against some of these assets is le-gally treated as an early withdrawal and hence also subject topenalty A less transparent instrument for commitment is an in-vestment in an illiquid asset that generates a steady stream ofbenets but that is hard to sell due to substantial transactionscosts informational problems or incomplete markets Examples

QUARTERLY JOURNAL OF ECONOMICS444

include purchasing a home buying consumer durables andbuilding up equity in a personal business Finally there exists aclass of assets that provide a store of illiquid value like savingsbonds and certicates of deposit All of the illiquid assets dis-cussed above have the same property as the goose that laidgolden eggs The asset promises to generate substantial benetsin the long run but these benets are difcult if not impossibleto realize immediately Trying to do so will result in a substantialcapital loss

Instruments with these golden eggs properties make up theoverwhelming majority of assets held by the U S household sec-tor For example the Federal Reserve System publication Bal-ance Sheets for the U S Economy 1945ndash94 reports that thehousehold sector held domestic assets of $285 trillion at year-end 1994 Over two-thirds of these assets were illiquid including$55 trillion of pension fund and life insurance reserves $45 tril-lion of residential structures $30 trillion of land $25 trillion ofequity in noncorporate business $25 trillion of consumer dur-ables and at least $1 trillion of other miscellaneous categoriesFinally note that social security wealth and human capital tworelatively large components of illiquid wealth are not included inthe Federal Reserve Balance Sheets

Despite the abundance of commitment mechanisms andStrotzrsquos well-known theoretical work intrapersonal commitmentphenomena have generally received little attention from econo-mists This decit is probably explained by the fact that commit-ment will only be chosen by decision-makers whose preferencesare dynamically inconsistent and most economists have avoidedstudying such problematic preferences However there is a sub-stantial body of evidence that preferences are dynamically incon-sistent Research on animal and human behavior has ledpsychologists to conclude that discount functions are approxi-mately hyperbolic [Ainslie 1992]

Hyperbolic discount functions are characterized by a rela-tively high discount rate over short horizons and a relatively lowdiscount rate over long horizons This discount structure sets upa conict between todayrsquos preferences and the preferences thatwill be held in the future For example from todayrsquos perspectivethe discount rate between two far-off periods t and t 1 1 is thelong-term low discount rate However from the time t perspec-tive the discount rate between t and t 1 1 is the short-term high

GOLDEN EGGS AND HYPERBOLIC DISCOUNTING 445

discount rate This type of preference change is reected in manycommon experiences For example this year I may desire to startan aggressive savings plan next year but when next year actuallyrolls around my taste at that time will be to postpone any sacri-ces another year In the analysis that follows the decision-maker foresees these conicts and uses a stylized commitmenttechnology to partially limit the options available in the future

This framework predicts that consumption will track incomeSecond the model explains why consumers have a different pro-pensity to consume out of wealth than they do out of labor in-come Third the model explains why Ricardian equivalenceshould not hold even in an economy characterized by an innitelylived representative agent Fourth the model suggests that fi-nancial innovation may have caused the ongoing decline in U Ssavings rates since nancial innovation increases liquidity andeliminates implicit commitment opportunities Finally the modelprovides a formal framework for considering the proposition thatnancial market innovation reduces welfare by providing ldquotoomuchrdquo liquidity

The body of this essay formalizes these claims Section II laysout the model Equilibrium outcomes are characterized in SectionIII Section IV considers the implications of the model for themacroeconomic issues highlighted above Section V concludeswith a discussion of ongoing work

II THE CONSUMPTION DECISION

The large number of commitment devices discussed aboveis good news for consumers They have access to a wide arrayof assets that effectively enable them to achieve many forms ofcommitment However from the perspective of an economist theabundance poses a challenge It is hard to model the institutionalrichness in a realistic way without generating an extremely bur-densome number of state variables

I consider a highly stylized commitment technology that isamenable to an analytic treatment Specically I assume thatconsumers may invest in two instruments a liquid asset x andan illiquid asset z Instrument z is illiquid in the sense that asale of this asset has to be initiated one period before the actualproceeds are received So a current decision to liquidate part orall of an individualrsquos z holding will generate cash ow that can be


consumed no earlier than next period1 By contrast agents canalways immediately consume their x holdings

Consumers in this model may borrow against their holdingsof asset z Like asset sales such borrowing takes one period toimplement If a consumer applies for a loan at time period t theassociated cash ow will not be available for consumption untiltime period t 1 1

In later sections I embed consumers in a general equilibriummodel in which prices will be endogenous Now however I con-sider the consumer in isolation and assume that the consumerfaces a deterministic sequence of interest rates and wages Forsimplicity I assume that asset z and asset x have the same rateof return2

The consumer makes consumptionsavings decisions in dis-crete time t [ 1 2 T Every time period t is divided intofour subperiods In the rst subperiod production takes placeThe consumerrsquos liquid assets xt 2 1 and nonliquid assets zt 2 1mdashbothchosen at time period t 2 1mdashyield a gross return of Rt 5 1 1 rtand the consumer inelastically supplies one unit of labor In thesecond subperiod the consumer receives deterministic labor in-come yt and gets access to her liquid savings Rt times xt 2 1 In the thirdsubperiod the consumer chooses current consumption

c y R xt t t t + pound - 1

In the fourth subperiod the consumer chooses her new asset allo-cations xt and zt subject to the constraints

y R z x c z xx z

t t t t t t t

t t

+ + + 0

( )

- - - =sup3

1 1

The consumer begins life with exogenous endowments x0 z0 $ 0The consumer may borrow against her illiquid assets by giv-

ing a creditor a contingent control right over some of those assetsIn exchange the consumer receives liquidity that can be con-sumed Such a loan is formally represented as a reallocation ofassets from the illiquid account to the liquid account I assumethat a loan ie asset reallocation which generates consumableliquidity in period t 1 1 must be initiated in period t Specically

1 One could alternatively assume that instantaneous access to asset z ispossible with a sufciently high transaction cost

2 The qualitative results do not hinge on the identical returns assumption


the asset reallocation occurs in subperiod 4 of period t therebyproviding consumable liquidity in period t 1 1 Such asset reallo-cations are subsumed in the consumerrsquos choice of xt and zt in sub-period 4

In the framework introduced above an uncollateralized loanhas occurred if an asset reallocation leaves the illiquid accountnegative Creditors are unwilling to make such loans because aconsumer who received such a loan would not have an incentiveto repay Hence I assume that zt $ 0

Finally the constraint xt $ 0 rules out forced savings con-tracts If the consumer could set xt to any negative value thenshe could perfectly commit her future savings behavior and henceher consumption level (or at least commit to any upper bound ontomorrowrsquos consumption level) For example if she foresaw ahigh level of labor income next period she could set xt negativeto force tomorrowrsquos self to save some of that income (recall thatct 1 1 yt 1 1 1 Rt 1 1 xt) A negative xt value would be interpreted as acontract with an outside agent requiring the consumer to transferfunds to the outside agent which the outside agent would thendeposit in an illiquid account of the consumer3 The constraint xt

$ 0 effectively rules out such contracts Two arguments supportthis implicit assumption against forced savings contracts

First such contracts are susceptible to renegotiation by to-morrowrsquos self and in any nite-horizon environment the contractwould unwind (In the second to last period renegotiation wouldoccur implying renegotiation in the third to last period etc) Sec-ond such contracts are generally unenforceable in the UnitedStates4 To make such a contract work tomorrowrsquos self must beforced to pay the specied funds to the outside agent or be penal-ized for not doing so (note that the transfer is not in the interestof tomorrowrsquos self) However U S courts will generally not en-force contracts with a penalty of this kind5

3 Mortgage payments are an example of a contract that xt $ 0 rules outHowever even though mortgage payments may be interpreted as forced savingscontracts they do not have the necessary exibility to achieve the full commit-ment solution Mortgage contracts generally do not make mortgage payments con-tingent on the level of labor income ows

4 I am indebted to Robert Hall for pointing out this fact to me5 U S contract law is based around the ldquofundamental principle that the

lawrsquos goal on breach of contract is not to deter breach by compelling the promisorto perform but rather to redress breach by compensating the promiseerdquo [Farns-worth 1990 p 935] Hence courts allow contracts to specify ldquoliquidated damagesrdquowhich reect losses likely to be experienced by the promisee but courts do notallow ldquopenaltiesrdquo which do not reect such losses


At time t the consumer has a time-additive utility functionUt with an instantaneous utility function characterized by con-stant relative risk aversion r Consumers are assumed to have adiscount function of the type proposed by Phelps and Pollak[1968] in a model of intergenerational altruism and which isused here to model intrapersonal dynamic conict6

(1) U E u c u ct t t t

T t

+ =eacute

eumlecirc

ugrave

ucircuacute+

=

-

aring( ) ( ) b d tt

t 1

I adopt equation (1) to capture the qualitative properties ofa generalized hyperbolic discount function events t periods awayare discounted with factor (1 1 a t ) 2 g a with a g 07 This classof discount functions was rst proposed by Chung and Herrnstein[1961] to characterize the results of animal behavior experi-ments8 Their conclusions were later shown to apply to humansubjects as well (see Ainslie [1992] for a survey)

Hyperbolic discount functions imply discount rates that de-cline as the discounted event is moved further away in time[Loewenstein and Prelec 1992] Events in the near future are dis-counted at a higher implicit discount rate than events in the dis-tant future

Given a discount function ƒ(t ) the instantaneous discountrate at time t is dened as

Applying the principle of lsquojust compensation for the loss or injury actuallysustainedrsquo to liquidated damage provisions courts have refused enforce-ment where the clause agreed upon is held to be in terroremmdasha sum xed asa deterrent to breach or as security for full performance by the promisor notas a realistic assessment of the provable damage Thus attempts to secureperformance through in terrorem clauses are currently declared unenforce-able even where the evidence shows a voluntary fairly bargained exchange[Goetz and Scott 1977 p 555]

In our case the promiseemdashthe outside agentmdashexperiences no loss if the con-sumer fails to make the payment Hence penalties or liquidated damages speci-ed in such contracts are not enforceable so the contract is incapable ofcompelling tomorrowrsquos self to make the payment For a more extensive discussionof these issues see Farnsworth [1990 pp 935ndash46] Goetz and Scott [1977] andRea [1984]

6 Zeckhauser and Fels [1968] provide an altruism-based microfoundationfor the Phelps and Pollak preferences Akerlof [1991] analyzes a special case ofthe Phelps and Pollak preferences ( d 5 1) Akerlof assumes consumer myopiawhile my analysis assumes that consumers foresee their future preferencereversals

7 See Loewenstein and Prelec [1992] for an axiomatic derivation of this dis-count function

8 Chung and Herrnstein claimed that the appropriate discount function isan exact hyperbola events t periods away are discounted with factor 1 t Thiscorresponds to the limiting case a 5 g reg yen


- f f9 ( ) ( )t t

Hence an exponential discount function d t is characterized by aconstant discount rate log(1 d ) while the generalized hyperbolicdiscount function is characterized by an instantaneous discountrate that falls as t rises

g a t + ( )1

Psychologists and economistsmdashnotably Ainslie [1975 19861992] Prelec [1989] and Loewenstein and Prelec [1992]mdashhaveargued that such declining discount rates play an important rolein generating problems of self-regulation

When 0 b 1 the discount structure in equation (1) mim-ics the qualitative property of the hyperbolic discount functionwhile maintaining most of the analytical tractibility of the expo-nential discount function I call the discount structure in equa-tion (1) ldquoquasi-hyperbolicrdquo Note that the quasi-hyperbolicdiscount function is a discrete time function with values 1 b d b d 2 b d 3 Figure I graphs the exponential discount function(assuming that d 5 097) the generalized hyperbolic discountfunction (assuming that a 5 105 and g 5 5 times 103) and the quasi-hyperbolic discount function (with b 5 06 and d 5 099) The

FIGURE IDiscount Functions


points of the discrete-time quasi-hyperbolic function have beenconnected to generate the curve in Figure I

The preferences given by equation (1) are dynamically incon-sistent in the sense that preferences at date t are inconsistentwith preferences at date t 1 1 To see this note that the marginalrate of substitution between periods t 1 1 and t 1 2 from theperspective of the decision-maker at time t is given byu9 (ct 1 1)( d u9 (ct 1 2)) which is not equal to the marginal rate of sub-stitution between those same periods from the perspective of thedecision-maker at t 1 1 u 9 (ct 1 1)( b d u9 (ct 1 2))

To analyze equilibrium behavior when preferences are dy-namically inconsistent it is standard practice to formally modela consumer as a sequence of temporal selves making choices in adynamic game (eg Pollak [1968] Peleg and Yaari [1973] andGoldman [1980]) Hence a T-period consumption problem trans-lates into a T-period game with T players or ldquoselvesrdquo indexed bytheir respective periods of control over the consumption decision(Note that self t is in control during all of the subperiods at timet) I look for subgame perfect equilibrium (SPE) strategies ofthis game

It is helpful to introduce some standard notation that will beused in the analysis which follows Let ht represent a (feasible)history at time t so ht represents all the moves that have beenmade from time 0 to time t 2 1 x0 z0 (c t x t z t ) t 2 1

t 5 1 LetSt represent the set of feasible strategies for self t Let S 5 P T

t 5 1 St

represent the joint strategy space of all selves If s [ S let s|ht

represent the path of consumption and asset allocation levelsfrom t to T which would arise if history ht were realized andselves t to T played the strategies given by s Finally let Ut(s|ht)represent the continuation payoff to self t if self t expects the con-sumption and asset allocation levels from t to T to be given bys|ht

III EQUILIBRIUM STRATEGIES

This section characterizes the equilibrium strategies of thegame described above Recall that the agent faces a deterministic(time-varying) sequence of interest rates and a deterministic(time-varying) labor income sequence Unfortunately for generalinterest rate and labor income sequences it is not possible to usemarginal conditions to characterize the equilibrium strategiesThis nonmarginality property is related to the fact that selveswho make choices at least two periods from the end of the game


face a nonconvex reduced-form choice set where the reduced-form choice set is dened as the consumption vectors which areattainable assuming that all future selves play equilibriumstrategies The nonconvexity in the reduced-form choice set of selfT 2 2 generates discontinuous equilibrium strategies for selfT 2 2 which in turn generate discontinuities in the equilibriumpayoff map of self T 2 3 This implies that marginal conditionscannot be used to characterize the equilibrium choices of selvesat least three periods from the end of the game9

I have found a restriction on the labor income process thateliminates these problems

(A1) u y R u y tt t ii

t9 9( ) ( ) 1sup3 Otildeaeligegrave

oumloslash sup3+

=+b d tt

t

t1

This restriction constrains the sequence yt t 5 Tt 5 1 to lie in a band

whose thickness is parameterized by the value of b the closer bis to zero the wider the band Calibration of the model revealsthat A1 allows for substantial exibility in the deterministic in-come process Ainslie [1992] reviews evidence that the one-yeardiscount rate is at least 13 This suggests that b should be cali-brated in the interval (0 23) (assuming that d is close to unity)To see what this implies consider the following example Assumethat Rt 5 R for all t d R 5 1 and u( times ) 5 ln( times ) Then A1 is satisedif for all t yt [ [y (1 b )y] If b 5 23 this interval becomes [y (32)y] and as b falls the interval grows even larger

Before characterizing the equilibria of the game it is helpfulto introduce the following denitions First we will say that ajoint strategy s is resource exhausting if s|hT 2 1 is characterizedby zT 5 xt 5 0 for all feasible hT 2 1 Second we will say that asequence of feasible consumptionsavings actions ct xt zt cT xT zT satises P1ndashP4 if t $ t

P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )

sup3 Otildeaeligegrave

oumloslash

gt Otildeaeligegrave

oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0

9 For an exposition of these problems see Laibson [1993] Related issues arealso discussed in Peleg and Yaari [1973] and Goldman [1980]


Finally we will say that a joint strategy s [ S satises P1ndashP4 iffor any feasible history ht s|ht satises P1ndashP4

It is now possible to state the main theorem of the paperThis theorem establishes that the consumption game has aunique equilibrium and the theorem characterizes thisequilibrium

THEOREM 1 Fix any T-period consumption game with exogenousvariables satisfying A1 There exists a unique resource-exhausting joint strategy s [ S that satises P1ndashP4 andthis strategy is the unique subgame perfect equilibriumstrategy of this game

(All proofs appear in the Appendix) Theorem 1 implies that theequilibrium consumption path is resource exhausting and satis-es P1ndashP4 It is straightforward to see why the equilibrium pathis resource exhausting the nal selfmdashself Tmdashconsumes all liq-uid resources in period T and self T 2 1 makes certain that allwealth is liquid in period T (ie zT 2 1 5 0) Hence no wealth goesunconsumed in equilibrium

Properties P1ndashP4 are also intuitive It is important to inter-pret them in light of the strategic self-control behavior that arisesin the intrapersonal consumption game In this game earlyselves prevent late selves from splurging Self t 2 1 uses theilliquid asset zt 2 1 to limit the liquidity available for consumptionin period t Note that self t cannot consume the illiquid asset dur-ing its period of control ct yt 1 Rt xt 2 1 On the equilibrium patheach self is endogenously liquidity constrained by the allocationchoices of earlier selves Property P1 is simply a standard Eulerequation relation for an environment in which liquidity con-straints exist The inequality arises because marginal utility canbe too high relative to future marginal utilities but it cannot betoo low since consumers always have the option to save PropertyP2 reects another standard Euler equation intuition when mar-ginal utility is strictly too high the liquidity constraint must bebinding Properties P3 and P4 reect the strategic decisions thatself t makes when it chooses asset allocation levels (xt and zt) P3implies that self t will limit self t 1 1rsquos liquidity as much as pos-sible (xt 5 0) if consumption at time t 1 1 is expected to be highrelative to what self t would prefer it to be P4 implies that self twill not limit self t 1 1rsquos liquidity at all (zt 5 0) if consumption attime t 1 1 is expected to be low relative to what self t would preferit to be Note that the equations associated with P3 and P4 donot contain the b term This omission arises because from the


perspective of self t utility trade-offs between period t 1 1 andany period after t 1 1 are independent of the value of b

IV ANALYSIS

In the following subsections I discuss several implications ofthe golden eggs model Some of the applications consider theinnite-horizon game that is analogous to the nite-horizon gamediscussed above When doing so I will focus consideration on theequilibrium that is the limit (as the horizon goes to innity) ofthe unique nite-horizon equilibrium10

A Comovement of Consumption and Income

There is a growing body of evidence that household consump-tion ows track corresponding household income ows ldquotoordquoclosely generating violations of the life-cyclepermanent-incomeconsumption model In particular household consumption is sen-sitive to expected movements in household income eg Hall andMishkin [1982] Zeldes [1989] Carroll and Summers [1991] Fla-vin [1991] Carroll [1992] Shea [1995] and Souleles [1995]11

Many of these authors nd that consumption tracks expected in-come changes even when consumers have large stocks of accumu-lated assets

Several models have been proposed to explain the consump-

10 For the innite horizon game a joint strategy s is resource exhausting ifthe continuation paths after all histories imply that the intertemporal budgetconstraint is exactly satised

z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig

egraveoumloslash Otildeaelig

egraveoumloslash

-yen -yen

aring aring=

For the innite-horizon game I will focus on the equilibrium that satises thefollowing innite-horizon analogs of P1ndashP4

P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1

11 Although Runkle [1989] is unable to reject the permanent income hy-pothesis there are reasons to believe his test lacks power (see Shea [1995])


tion-income comovement Carroll [1992] proposes a buffer-stocktheory of savings in which impatient consumers with a precau-tionary savings motive hold little wealth and choose optimal con-sumption policies in which consumption and income movetogether over the life-cycle Gourinchas and Parker [1995] simu-late an extended version of this model Attanasio and Weber[1993] argue that demographic dynamics explain much of theconsumption-income comovement

The golden eggs model provides a new explanation for theobserved comovement in consumption and income In the modelself t 2 1 chooses xt 2 1 to constrain the consumption of self t Inthis way ldquoearlyrdquo selves manipulate the cash ow process by keep-ing most assets in the illiquid instrument Hence at any givenmoment the consumer is effectively liquidity constrained thoughthe constraint is self-imposed In equilibrium consumption is ex-actly equal to the current level of cash ow ct 5 yt 1 Rt xt 2 1 (seeLemma 3 in the Appendix for a formal proof) However this doesnot imply by itself that consumption will track labor income Notethat xt 2 1 is endogenous and in equilibrium xt 2 1 covaries nega-tively with labor income Self t 2 1 varies xt 2 1 to try to offset thepredictable uctuations in yt When yt is large self t 2 1 willmake xt 2 1 small in an effort to prevent self t from overconsuming

However there are limits to the ways in which ldquoearlyrdquo selvescan constrain the choices of ldquolaterrdquo selves Self t 2 1 can only denyself t access to assets that have been accumulated in the pastSelf t 2 1 cannot deny self t access to yt labor income at time tSo when yt is particularly high (ie cash ow at time t is particu-larly high) consumption at time t will also be high This impliesthat on the equilibrium path predictable movements in incomewill tend to be reected in movements in consumption

An example may help to make this more concrete Let thehorizon be innite Assume that labor income follows a trendinghigh-low process yt 5 yegt when t is odd yt 5 yegt when t is evenAssume that the interest rate is constant and exp( r g) 5 d R (Thislast relationship is motivated by the steady state results below)Assume that yt

yent 5 1 satises A1 Finally assume that x0 5 0 z0 $

0 and z0 not be ldquotoordquo large relative to the labor income variabilitySpecically z0 must satisfy the relationship

u y Ru ye z e R eg g g9 9( ) (_ ( ) ) + pound -d 02 21

Then the equilibrium consumption path is


c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )

Figure II graphs the labor income path and equilibrium con-sumption path using parameter values b 5 06 R 5 104 g 5002 y 5 z03 5 1 and y 5 0812 Two properties stand out Firstthe illiquid asset is exclusively used to augment consumption inthe even periods ie in the periods with relatively low labor in-come However this increase is not sufcient to smooth consump-tion A regression of D ln ct on D ln yt yields a coefcient of 40Since the income process is completely deterministic this impliesthat predictable changes in income are associated with changesin consumption Hence consumption tracks income despite thefact that the consumer in this example controls a substantialasset stock (KY oslash 3)

B Aggregate Saving

In most intertemporal rational choice models high discountrates are a necessary condition for consumption-income comove-ment Such relatively high discount rates however tend to implyrelatively low levels of capital accumulation in general equilib-rium (see Aiyagari [1992]) The golden eggs model generates con-sumption-income comovement even when actors are wealthyThis is because in equilibrium decisions to dissave out of the illiq-uid asset stock do not depend on b Self t is not able to consumethe illiquid asset immediately so self t does not consider trade-offs between consumption today and consumption tomorrowwhen dissaving from the illiquid instrument Instead self t con-siders trade-offs between consumption at t 1 1 and consumptionat periods after t 1 1 The value of b is superuous for such adecisionmdashfrom self trsquos perspectivemdashand hence the steady statecapital stock is independent of b

The following general equilibrium analysis formalizes thisintuition Assume that there exists a continuum of individualagents indexed by the unit interval Individual decision and statevariables are represented with an i index (eg ct(i)) Consider astandard Cobb-Douglas production function with aggregate capi-tal Kt aggregate labor Lt and exogenous productivity At

12 The remaining variables d and r may take on any values that satisfythe steady state condition exp( r g) 5 d R


FIGURE IIConsumption and Labor Income

Y A K Lt t t t = -a a1

Aggregate capital is composed of the liquid and illiquid capitalholdings of individual agents in the economy

K x i z i dit t t + 0

1ordm - -ograve [ ( ) ( )] 1 1

Recall that labor is assumed to be supplied inelastically so L(i) 51 and

L L i dit t 10

1ordm =ograve ( )

In competitive equilibrium labor receives its marginal productso labor income of agent i at time period t is given by yt(i) 5(1 2 a )Yt Competitive equilibrium also implies that capital re-ceive its marginal product so Rt 5 1 1 a YtKt 2 d where d isthe rate of depreciation Liquid and illiquid gross asset returnsof agent i at time period t are respectively Rt xt 2 1 and Rt zt 2 1Finally At is assumed to grow exogenously at rate gA so in steadystate capital and output must grow at rate gA(1 2 a ) g

PROPOSITION 1 In the economy described above there exists aunique steady state that satises A1 In that steady state

(2) exp( )r dg R =


The important property of the steady state identied inProposition 1 is that the parameter b does not appear in the equa-tion relating the discount rate and the growth rate So b can becalibrated to generate excess sensitivity (ie consumption-income comovement) while d can be calibrated to match the his-torical capital-output ratio of three If b is in the interior of theunit interval then the equilibrium path will exhibit consump-tion-income tracking (eg see the example in previous subsec-tion) Meanwhile d can be chosen to satisfy the equation

(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1

which is a log-linearized version of equation (2) Setting d 5 098rationalizes KY 5 3 assuming that the other parameters in theequation take standard values r 5 1 g 5 002 a 5 036 d 5008

C Asset-Specic MPCs

Thaler [1990] argues that consumers have different marginalpropensities to consume for different categories of assets For ex-ample he presents evidence that an unexpected increase in thevalue of an equity portfolio will have a very small effect on con-sumption while an unexpected job-related bonus will be immedi-ately consumed Thaler divides consumer wealth into threecategories current income net assets and future income Hecites a wide body of evidence which suggests that ldquothe MPC from[current income] is close to unity the MPC from [future income]is close to zero and the MPC from [net assets] is somewhere inbetweenrdquo Thaler explains this behavior by postulating that con-sumers use a system of nonfungible mental accounts to guiderule-of-thumb decision-making By contrast the golden eggsmodel predicts that even fully rational consumers will exhibitasset-specic MPCs13

In the golden eggs model the current self is always endoge-nously liquidity constrained on the equilibrium path So the MPCout of current cash ow is one Proposition 2 formalizes thisclaim

13 Laibson [1994b] proposes another hyperbolic discounting model that gen-erates some mental accounting behavior In Laibson [1994b] rational consumersset up a system of self-rewards and self-punishments to motivate later selves toexert high effort Laibson [1994b] discusses effort-related mental accounts whilethe current paper discusses liquidity-related mental accounts


PROPOSITION 2 Fix a consumption game in which inequalityA1 is strictly satised Let ct 5 ct(Rt xt 2 1 Rt zt 2 1) represent theequilibrium (Markov) consumption strategy of self t Then

parapara

= sup3-

cR x

tt

t t( )1

1 2

when the partial derivative is evaluated on the equilibriumpath

In this subsection I contrast this MPC with its analog forilliquid assets At rst glance it is not clear how to best make thiscomparison I will consider two approaches

PROPOSITION 3 Fix a consumption game in which inequality A1is strictly satised Let ct(Rt xt 2 1 Rt zt 2 1) represent the equilib-rium (Markov) consumption strategy of self t Then

parapara

= sup3-

cR z

tt

t t( )1

0 2


This result is not surprising since on the equilibrium paththe individual always faces a self-imposed liquidity constraintSmall perturbations to the illiquid asset stock are not sufcientto stop the current self rsquos liquidity constraint from being bindingA more interesting question to ask is how a perturbation to zt 2 1

affects the choice of xt Recall that liquid assets set aside at timet will be completely consumed at time t 1 1 Unfortunately thevalue of para xt para (Rt zt 2 1) can take on any value between zero and oneFor example the partial derivative is equal to zero if the equilib-rium value of xt is equal to zero The partial derivative is equal tounity if t is the penultimate period of the game It would be help-ful to develop an MPC measure that provides a representativevalue of para xt para (Rt zt 2 1) The following proposition introduces such ameasure by considering the geometric average of MPCs over adeterministic business cycle of duration t

PROPOSITION 4 Fix any yen ndashhorizon consumption game with Rt 5R t Fix a particular value of t $ 1 Assume that yt

yent 5 1

satises A1 and yt 1 t 5 exp( t g)yt t $ 0 Assume thatexp( r g) 5 d R Let xt 5 xt(R times xt 2 1 R times zt 2 1) represent the equi-librium (Markov) consumption strategy of self t Let


1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt

evaluated on the equilibrium path Then

MPC MPC R ttz z 21 = ordm - sup3-1 1( )d r r

Note that Proposition 4 assumes that the growth rate of la-bor income is related to the return on capital by the steady stateequation in Proposition 2 exp( r g) 5 d R Note also that the re-sulting measure of the marginal propensity to consume 1 2( d R1 2 r )1 r is equivalent to the marginal propensity to consume inthe standard Ramsey model with no liquidity constraints and ex-ponential discount function d t For all reasonable parameter val-ues MPCz is close to zero Recall that the rst proposed measureof the MPC out of illiquid assets (ie the MPC measure intro-duced in Proposition 3) was exactly equal to zero Finally con-trast these proposed measures of the MPC out of illiquid assets(which take values close to or exactly equal to zero) with the unitymarginal propensity to consume out of liquid assets

D Ricardian Equivalence

In the economy analyzed in this paper the sequence of exoge-nous cash ows matters in a way that is independent of the pres-ent value of those cash ows This is immediately apparent fromFigure II Because taxation schemes affect these exogenous cashows Ricardian equivalence will be violated Moreover themodel generates such violations even when the consumer has alarge asset stock at all times Hence Ricardian equivalence isviolated for all agents whether or not they hold substantialwealth

E Declining Savings Rates in the 1980s

The golden eggs model may help to explain the decline inU S savings rates during the 1980s I pursue two approaches inthis subsection The rst explanation is driven by the fact thatduring the 1980s a relatively large proportion of national incomewas realized as cash ow to consumers However I am unsatis-ed with this rst story for reasons that I describe below HenceI focus most of my attention on a second explanation that isdriven by developments in the consumer credit market

Hatsopoulos Krugman and Poterba [1989] document the ob-


servation that cash ow to consumers (as a percentage of NNP)was high during the 1980s relative to the 1970s They report thatfrom 1970ndash1979 cash ow averaged 779 percent of NNP whilethe corresponding number for the 1980ndash1987 period was 808 per-cent They trace this increase to several sources notably higherinterest income (45 percentage points) higher transfers (22 per-centage points) and higher after-tax cash from takeovers (06percentage points)14

Using aggregate data Hatsopoulos Krugman and Poterba[1989] estimate a high marginal propensity to consume out ofcurrent cash ow Coupling this result with the higher cash owlevels they are able to explain most of the savings decline in the1980s However they do not explain why consumers should havesuch a high propensity to consume out of cash ow The goldeneggs model complements their analysis by providing a model thatexplains the high MPC However note that the golden eggs modelcan only explain the high MPC out of cash ow the model cannotexplain why the cash ow was high in the rst place Hence ap-plication of the golden eggs model may only relabel the puzzlechanging it from a consumption puzzle to a cash ow puzzle

The golden eggs model suggests a second explanation for thelow level of savings during the past decade The 1980s was a pe-riod of rapid expansion in the U S consumer credit marketIncreasing access to instantaneous credit has reduced the effec-tiveness of commitment devices like illiquid assets The goldeneggs model predicts that the elimination of commitment deviceswould lower the level of capital accumulation I will show that ifthe credit market were to become sufciently sophisticated thatconsumers could instantaneously borrow against their illiquidassets then the steady state capital-output ratio would fall Icalibrate this fall at the end of the subsection

The rapid expansion of the U S consumer credit market pro-vides the starting point for the argument summarized in the pre-vious paragraph One example of the expansion in instantaneouscredit has been the growth in credit cards15 In 1970 only 16 per-cent of all U S families had a third party credit card (eg prede-

14 Offsetting falls in cash ow occurred in labor income ( 2 03 percentagepoints) noninterest capital income in disposable income ( 2 20 percentage points)and taxes ( 2 21 percentage points)

15 Another important development in the U S credit market has been theexpanded use of home equity lines of credit Before the mid-1980s home equitylines of credit were almost unheard of By 1993ndash1994 83 percent of homeownershad a home equity line of credit See Canner and Luckett [1994] p 572


cessors of current cards like Visa and MasterCard) By 1989 54percent had one16 During this same period credit card acceptanceby retailers also increased dramatically Large retailers did notaccept credit cards during the 1970s In 1979 J C Penney brokeranks with its competitors and became the rst major retailer toaccept third party credit cards By the end of the 1980s almostall large retailers accepted third party cards The growth of ATMs(automatic teller machines) augmented the impact of the creditcard expansion by enabling credit cardholders to readily receivecash advances Regional and national ATM networks rst beganto form in the late 1970s and early 1980s17 Altogether thesedevelopments led to an explosion in revolving credit which isprincipally composed of credit card debt From 1970 to 1995 re-volving credit grew from 37 percent to 363 percent of total con-sumer credit18 No single year stands out as the date at whichmost consumers experienced a sharp increase in their personalaccess to instantaneous credit However it is safe to say that bythe mid-1980s most families had a third party credit card andthis card could be used in most large retail stores or could beused in ATMs to receive cash advances to make purchases in thestores that still did not accept credit cards Together these obser-vations suggest that the mid-1980s represents the rst time thata representative U S family had instantaneous access to con-sumer credit or could rapidly apply for such access

Introducing instantaneous credit into the golden eggs modeldramatically changes the equilibrium analysis (Recall that theoriginal model had credit which could be accessed with a one-period delay) In the original model consumption was boundedabove by cash on hand


With instantaneous access to credit consumption is now con-strained to lie below the sum of cash on hand and the value of allcredit lines that can be instantaneously set up or are already set

16 See Canner and Luckett [1992] p 65617 See Mandell [1990] for a short history of the credit card industry18 Consumer credit includes automobile loans revolving credit ldquootherrdquo in-

stallment credit and noninstallment credit ldquoOtherrdquo installment credit includesldquomobile home loans and all other installment loans not included in automobile orrevolving credit such as loans for education boats trailers or vacations Theseloans may be secured or unsecured Noninstallment credit is credit scheduled tobe repaid in a lump sum including single-payment loans charge accounts andservice creditrdquo [Economic Report of the President 1996 Table B-73]


up I assume that the value of these existing and potential creditlines is approximately equal to the value of the illiquid assetsheld by the consumer Hence consumption is now constrained by

c y R x R zt t t t t t + + pound - -1 1

In all other ways the model remains the same

PROPOSITION 5 Consider the general equilibrium economy ana-lyzed above but now assume that consumers can instantane-ously borrow against their illiquid asset This economy isequivalent to one in which there is no illiquid asset (ie x isthe only asset) In such an economy there exists a uniquesteady state and in that steady state

(4) exp( ) ( ) exp( )r b d b dg R g + = -1

COROLLARY In the steady state characterized in Proposition 5the capital-output ratio is less than the steady state capital-output ratio in the economy with the commitmenttechnology

Table I reports the magnitude of the reduction in steady statecapital that occurs when nancial innovation moves an economyfrom a golden eggs nancial technology to a new nancial tech-nology in which it is possible to instantaneously borrow againstthe illiquid asset (ie when nancial innovation eliminates theilliquidity that makes partial commitment possible) The entriesof Table I are derived in ve steps First I assume that the U Seconomy has historically been a golden eggs economy with a 5036 d 5 008 g 5 002 and KY 5 3 Second I calibrate prefer-ence parameters r and d based on equation (2) (the steady stateequation in golden eggs economies) and the competitive equilib-rium condition r 5 a YK 2 d 5 (036)1

32 008 (Recall that equa-

tion (2) is independent of b ) These equations jointly imply that

(5) exp( ( )) ( )r d0 02 1 04 =

Third I take the set of preference parameter values derived instep 2 (ie dened in equation (5)) and plug that set into equa-tion (4) the new steady state equation (ie the steady state equa-tion associated with the economy in which consumers caninstantaneously borrow against their illiquid assets) This yieldsthe following ldquoconstrainedrdquo steady state equation that holds inthe new economy


TABLE ISTEADY STATE INTEREST RATES AND CAPITAL-OUTPUT RATIOS IN ECONOMIES WITH

AND WITHOUT PARTIAL COMMITMENT

With commitment Without commitment(ie no instantaneous (ie instantaneous

credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR

Note that this constrained steady state relationship is indepen-dent of r and d and depends exclusively on b and R Fourth Ivary b the only free preference parameter in the constrained newsteady state equation and calculate the corresponding capital-output ratios (again using the competitive equilibrium relation-ship r 5 a (YK) 2 d) Fifth I compare these new capital-outputratios with the historical capital-output ratio

Note that when commitment is available the steady stateinterest rate and capital-output ratio are independent of thevalue of b (see Proposition 1) Now consider an example of a tran-sition from a commitment economy to an economy without com-mitment For a b value of 06 elimination of the commitmenttechnology raises the steady state real interest rate 13 percent-age points This corresponds to a reduction in the capital-outputratio of 03

These results should be compared with actual U S experi-ence during the 1980s and 1990s The model predicts that capitalaccumulation should have fallen at the same time that consum-ers gained access to instantaneous credit (approximately the mid-1980s) All measures of capital accumulation show a markeddownturn that starts in the 1980s and continues into the 1990sFor example U S personal savings as a percent of disposablepersonal income fell from an average of 73 percent from 1946ndash1984 to an average of 53 percent from 1985ndash1994 The 1985ndash1994 period had the lowest average saving rate of any ten-year


span in the postwar period19 The ratio of national net worth togross national product (ie the U S capital-output ratio) fellfrom an average of 32 from 1946ndash1984 to 28 in 1994 The 1994value is the low point for the series in the postwar period20

F Welfare Analysis of Financial Innovation

The introduction of instantaneous credit increases consum-ersrsquo choice sets Standard economic models imply that this devel-opment might lower levels of capital accumulation but wouldraise consumer welfare Yet in the United States policy-makersand pundits are concerned that instantaneous credit is somehowbad for consumers

The golden eggs framework provides a formal model of thecosts of nancial innovation By enabling the consumer to instan-taneously borrow against illiquid assets nancial innovationeliminates the possibility for partial commitment This has twoeffects on the welfare of the current self First the current self nolonger faces a self-imposed liquidity constraint and can thereforeconsume more in its period of control Second future selves arealso no longer liquidity constrained and may also consume at ahigher rate out of the wealth stock that they inherit The rsteffect makes the current self better off The second effect makesthe current self worse off (since the current self would like to con-strain the consumption of future selves) Under most parameter-izations the impact of the second effect dominates and thewelfare of the current self is reduced

Formally I measure the welfare loss by calculating the mini-mum one-time paymentmdashpaid to a representative consumermdashwhich would induce the representative consumer to switch froman innite horizon golden eggs economy to an innite horizon in-stantaneous credit economy (Using the notation of Section II thehypothetical payment that induces indifference is made duringsubperiod 2 of time t and the indifference is from the perspectiveof self t) I assume that the representative consumer starts in

19 National Income and Product Accounts Table 21 Bureau of EconomicAnalysis U S Department of Commerce

20 National net worth is calculated from Tables B11 and B109 in BalanceSheets for the U S Economy 1945ndash94 Board of Governors of the Federal ReserveSystem National net worth represents the sum of lines 1 and 30 from Table B11added to the difference between lines 43 and 42 from Table B109 Gross nationalproduct is calculated by the Bureau of Economic Analysis U S Department ofCommerce


the steady state of the golden eggs economy this steady state ischaracterized in Proposition 1 The representative agent remainsin that steady state if she remains in the golden eggs economyBy contrast if the representative agent switches to the instanta-neous credit economy (ie if she switches to the economy inwhich it is possible to instantaneously borrow against illiquidassets) then the new economy asymptotically converges to thesteady state characterized in Proposition 5 The starting pointfor this convergence is the golden eggs steady state capital stockaugmented (depleted) by a payment at time period one

I calibrate this exercise with a 5 036 d 5 008 g 5 002d 5 098 and r 5 1 and I assume that the golden eggs steadystate is characterized by the historical capital-output ratio KY 53 Note that these values are consistent with the steady stateequation for the golden eggs economy (see Proposition 1)

The convergence path for the economy in which instanta-neous borrowing is possible is characterized by a nonstandardEuler equation derived in Laibson [1996]

(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b

where l is given by21

(8) ld

d b

=

-- -

11 1( )

Note that when b 5 1 this Euler equation reduces to the standardcase To solve for the convergence path conditional on a startinglevel of nancial wealth it is necessary to search for the uniquesequence ct Rt

yent 5 1 that is 1) consistent with the nonstandard Eu-

ler equation given above 2) consistent with the dynamic budgetconstraint 3) consistent with the capital market competitiveequilibrium condition and 4) consistent with asymptotic conver-gence to the steady state characterized in Proposition 5 Identi-fying this sequence can be reduced to a one-dimensional searchover values of c1 start with a guess of the equilibrium value of c1combine this value of c1 with the dynamic budget constraint andthe competitive equilibrium conditions to generate R2 use thenonstandard Euler equation to calculate c2 as a function of c1 andR2 iterate the last two steps to generate a sequence ct Rt

yent 5 1

that can be checked for asymptotic steady state convergence ifthe sequence does not converge start with a new guess for c1

This algorithm provides a way of calculating the convergence

21 The derivation for l uses the calibration assumption r 5 1


path given any level of initial nancial wealth in the instanta-neous credit economy Once this has been done it is straightfor-ward to calculate the level of initial nancial wealth in theinstantaneous credit economy that induces indifference withthe level of initial nancial wealth in the golden eggs economyThe payment level is the difference between these two nancialwealth levels The payment level is reported in Table II where itis normalized by the level of output at time of payment22 Notethat a positive payment implies that the consumer needs compen-sation to induce her to willingly switch to the instantaneouscredit scenario Hence if payment were withheld the consumerwould be worse off in the instantaneous credit scenario Table IIreports these normalized payments for a range of b values

Note that when b 5 1 there is no welfare loss When b 5 1the consumerrsquos preferences are not dynamically inconsistent andthe consumer has no need to constrain her future selves By con-trast for the other cases ( b [ 02 04 06 08) the consumeris made worse off by nancial innovation Being able to borrowagainst illiquid assets is welfare reducing However note thatthis is not always the case For b values sufciently close to zerothe consumer is made better off by being able to splurge almostall of her nancial wealth immediately However for the range ofreasonable b values reported in Table II the consumer is alwaysmade worse off by nancial innovation

Of course the costs of nancial innovation explored abovemay be offset by unmodeled gains like being able to consume inunforeseen emergencies (which are ruled out in the deterministicframework of this paper) The point of this subsection is to dem-

TABLE IIPAYMENTS TO INDUCE INDIFFERENCE

BETWEEN GOLDEN EGGS ECONOMY

AND INSTANTANEOUS CREDIT ECONOMY

Payment aspercent of output

b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00

22 Note that output at time of payment is the same under the two scenariossince output at any given period is determined by capital put aside in the previ-ous period


onstrate that there are potentially important costs that accom-pany those other well-known benets of extra liquidity

V EVALUATION AND EXTENSIONS

I have analyzed the consumption problem of a dynamicallyinconsistent decision-maker who has access to a crude commit-ment mechanism The model helps to explain many of the empiri-cal puzzles in the consumption literature notably consumption-income tracking and asset-specic MPCs However the modelhas several drawbacks that suggest four important areas to pur-sue extensions

First the golden eggs model does not explain how consumersaccumulate assets in the rst place Note that consumption isalways greater than labor income on the equilibrium path How-ever this is less of a problem than it might rst appear Althoughthere is evidence that individuals often consume less than theyearn in labor income most of this saving is nondiscretionary (egpension contributions life-insurance payments mortgage pay-ments and other payments to creditors) Bringing such ldquonondis-cretionary savingsrdquo into the model can be done very simply Forexample the consumer could elect to take on a 30-period mort-gage obligation at time zero represented by a mortgage paymentof m for the next 30 periods Then the consumerrsquos cash ow attime t 30 would be yt 1 Rt xt 2 1 2 m which would be less thanyt if m were greater than Rt xt 2 1 A related way to model nondiscre-tionary savings would be to let the consumer set xt 2 1 itself lessthan zero (eg x xt 2 1 where x 0)

A second problem associated with the model is the anoma-lous prediction that consumers will always face a binding self-imposed liquidity constraint For example the golden eggs modelpredicts that after making their consumption choice consumersshould have no liquid funds left in their bank accounts This pre-diction contradicts many consumersrsquo experiences However thisproblem can be readily addressed by introducing a precautionarysavings motive for holding liquidity For example consider acontinuous-time analog of the golden eggs model and assumethat instantaneous liquidity needs arrive with some hazard rateThen in equilibrium the consumer will only rarely completely ex-haust her liquidity

A third problem with the golden eggs model is that some con-


sumers may not need to use external commitment devices (likeilliquid assets) to achieve self-control Consumers may have in-ternal self-control mechanisms like ldquowill powerrdquo and ldquopersonalrulesrdquo In Laibson [1994a] I analyze an innite-horizon consump-tionsavings game with no external commitment technology andnd a multiplicity of Pareto-rankable equilibria I interpret thismultiplicity as a potential model for self-control and willpowerHowever this approach raises problematic and as yet unresolvedequilibrium selection problems More work is needed to developtheoretically robust models of internal self-control mechanismsand to empirically validate such models

The fourth problem with the golden eggs model is that someconsumers may have access to an array of ldquosocialrdquo commitmentdevices that are far richer than the simple illiquid asset proposedin this essay In Laibson [1994b] I analyze the problem of a con-sumer who can use social systems like marriage work andfriendship to achieve personal commitment Future work shouldtry to identify the most important mechanisms that consumersuse to overcome the self-control problems induced by hyperbolicpreferences

APPENDIX

Theorem 1 is proved with four intermediate lemmas Theselemmas apply to the game described in Theorem 1

LEMMA 1 Let s be a resource-exhausting element of the jointstrategy space S Assume that s satises P1ndashP4 Then for allhistories ht strategy s implies that ct $ yt

Proof of Lemma 1 Use induction to prove result Fix a periodt and feasible history ht Let s|ht 5 cA

t 1 t xAt 1 t zA

t 1 t T 2 tt 5 0 Assume

that ct 1 t $ yt 1 t t $ 1 By P1 u9 (ct) $ max t Icirc 1 T 2 t b d t

( P ti 5 1Rt 1 i) u 9 (ct 1 t ) If this inequality is strict then P2 implies that

ct $ yt So WLOG assume that u 9 (ct) 5 max t $ 1 b d t ( P ti 5 1Rt 1 i)u9 (ct 1 t )

u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash

pound Otildeaeligegrave

oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1


So u9 (ct) u9 (yt) and hence ct $ yt After conrming that cT $ yT

(by resource exhaustion) the proof is completed by applying astandard induction argument

LEMMA 2 Let sA and sB be resource-exhausting elements of thejoint strategy space S Assume that sA and sB satisfy P1ndashP4Let cA

t xAt zA

t Tt 5 1 and cB

t xBt zB

t Tt 5 1 be the respective paths of

actions generated by sA and sB Fix a particular value of tand assume c A

t 1 t $ cBt 1 t t $ 1 with cA

t 1 t cBt 1 t for at least

one t $ 1 Then c At $ cB

t

Proof of Lemma 2 By P1 u 9 (cAt ) $ maxt [ 1 T 2 1 b d t

( P ti 5 1 Rt 1 i) u9 (cA

t 1 t ) If this is satised with equality then

u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1

Hence u9 (cBt ) $ u9 (cA

t ) implying that c At $ cB

t So WLOG assumethat u9 (c A

t ) maxt [ 1 T 2 t 2 1 b d t ( P ti 5 1 Rt 1 i) u9 (cA

t 1 t ) By P2 c At 5 yt

1 Rt xt 2 1 If t 5 1 then cAt $ cB

t since cB1 y1 1 R1x0 5 cA

1 SoWLOG assume that t $ 2 If xB

t 2 1 $ xAt 2 1 then cB

t yt 1 Rt xBt 2 1

yt 1 Rt xAt 2 1 5 cA

t So WLOG assume xBt 2 1 xA

t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt

Hence u9 (cBt ) $ u 9 (cA

t ) implying that cAt $ cB

t


LEMMA 3 Let s be a resource-exhausting element of the jointstrategy space S Assume that s satises P1ndashP4 Let ct xtzt

Tt 5 1 represent the path of actions generated by s Then ct 5

yt 1 Rt xt 2 1 t $ 2

Proof of Lemma 3 Suppose that ct yt 1 Rt xt 2 1 for some t $ 2and look for a contradiction By P1 and P2 u9 (ct) 5max t [ 1 T 2 t 2 1 b d t ( P t

i 5 1 Rt 1 i)u 9 (ct 1 t ) so u9 (ct) max t 1 T 2 t 2 1 d t ( P t

i 5 1 Rt 1 i)u 9 (ct 1 t ) Hence by P3 xt 2 1 5 0 So ct yt which contradicts Lemma 1

LEMMA 4 Let ct xt ztTt 5 1 be a solution path to the following

problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c

x z tc y R x tx z R x z y c tx

( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-

Then satisfies P1 P4c x zt t t tT =

-1

Proof of Lemma 4 The rst step in the proof is to show thatthe solution set of the program above is a subset of the solutionset of the program below

max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to

x z tc y R x tx z R x z y c tx zx zc yc y R x t

t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3

Henceforth I will refer to these respectively as program I andprogram II Note that program II is a convex program with linear


constraints so the Kuhn-Tucker rst-order conditions are neces-sary and sufcient for a global optimum I will return to this factlater in the proof

The following notation will be used to prove the lemma LetV represent the set of all real vectors v 5 ct xt zt

Tt 5 1 Let CI

V (CII V ) represent the subset of vectors in V which satisfy theconstraints of program I (II) Let CI V (CII V ) represent thesubset of vectors in V that are solutions to program I (II)

The rst step in the proof is to show that CI CII Fix any v[ CI and let v 5 ct xt zt

Tt 5 1 Note that the rst ve constraints

of program I are identical to the rst ve constraints of programII Also note that if ct xt ztT

t 5 2 satises P1ndashP4 then by Lemma1 c2 $ y2 and by Lemma 3 ct 5 yt 1 Rt xt 2 1 t $ 3 Hence v [CII implying that CI CII

The next step is to show CI CII Fix any v [ CI Fix anyv 9 [ CII and let v 9 5 ct xt zt

Tt 5 1 Dene x1 such that c2 5 y2 1

R2 x1 Let v 0 be equivalent to v 9 except that x1 is replaced by x1and z1 is replaced by z1 5 z1 2 (x1 2 x1) Let U( v ) represent thevalue of the objective function evaluated at v Consider the fol-lowing two properties of v 0 v 0 [ CII U( v 9 ) 5 U( v 0 ) Recall thatv 9 [ CII Then v 0 must also be an element of CII Hence v 0 mustsatisfy the Kuhn-Tucker conditions of program II (since the con-ditions are necessary and sufcient) Using the Kuhn-Tucker con-ditions and the denition of v 0 it is straightforward to show thatv 0 [ CI Note that v 0 [ CII and v [ CI CI CII imply thatU( v 0 ) $ U( v ) Note that v [ CI and v 0 [ CI imply that U( v 0 ) U( v ) Hence U( v ) 5 U( v 0 ) which implies that U( v ) 5 U( v 9 ) Sov 9 [ CII and v [ CI CI CII imply that v [ CII HenceCI CII

I am now ready to complete the proof of the lemma Let v bea solution to program I and let v 5 ct xt zt

Tt 5 1 So ct xt zt

Tt 5 2

satises P1ndashP4 Since CI CII v must also satisfy the necessaryand sufcient Kuhn-Tucker conditions of program II Combiningthese constraints it is straightforward to show that ct xt ztT

t 5 1

satises P1ndashP4 N

Proof of Theorem 1 Suppose that there exist two resource-exhausting joint strategies sA sB [ S that satisfy P1ndashP4 Fix anyperiod t and any feasible history ht Let sA|ht cA

t 1 t xAt 1 t zA

t 1 t T 2 tt 5 0 sB|ht cB

t 1 t xBt 1 t zB

t 1 t T 2 tt 5 0 By resource exhaus-

tion and Lemma 2 cAt 1 t 5 cB

t 1 t t $ 0 Hence by Lemma 3 xAt 1 t 5 xB

t 1 t t $ 0 This in turn implies that zAt 1 t 5 zB

t 1 t t $ 0 as


a result of the savings constraints Because the proof started witharbitary ht we can conclude that sA 5 sB proving that there existsa unique resource-exhausting joint strategy s [ S that satisesP1ndashP4 The second part of the theorem follows from this unique-ness result Lemma 4 and a standard induction argument N

Proof of Proposition 1 If a steady state satises A1 thenexp( r g) $ Rd Moreover it is easy to construct a steady state atwhich exp( r g) 5 R d Suppose that there exists a steady stateat which exp( r g) Rd Then by P1ndashP4 limt reg yen zt 5 limt reg yen xt 5 0which implies that no such steady state could exist N

Proof of Proposition 2 By P1 u9 (ct) $ b d t (P ti 5 1 Rt 1 i) u 9 (ct 1 t )

t $ 2 t $ 0 Suppose that this inequality is satised exactly forsome t t pair Then xt 2 1 5 0 by P3 Hence

u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1

But u9 (yt) b d t ( P ti 5 1 Rt 1 i) u9 (yt 1 t ) violates A1 (since A1 is as-

sumed to hold strictly) So WLOG assume that u9 (ct) b d t

( P ti 5 1Rt 1 i) u 9 (ct 1 t ) t $ 2 t $ 0 Hence for sufciently small | e |

0 u9 (ct 1 e ) b d t ( P ti 5 1 Rt 1 i) u9 (ct 1 t ) t $ 2 t $ 0 So by P3 and

the uniqueness result of Theorem 1 in the subgame starting afterany sufciently small perturbation to the liquid asset stock theequality ct 5 yt 1 Rt xt 2 1 continues to hold and hence para ct(para (Rt

xt 2 1)) 5 1 N

Proof of Proposition 3 WLOG assume that u9 (ct) b d t

( P ti 5 1 Rt 1 i) u9 (ct 1 t ) t $ 2 t $ 0 (see Proof of Proposition 2) Hence

for sufciently small | e | 0 u9 (ct) b d t ( P ti 5 1 Rt 1 i)u 9 (ct 1 t 1

( P ti 5 1Ri) e ) t $ 2 t $ 0 So current consumption does not change

when zt 2 1 is perturbed N

Proposition 4 is proved with two intermediate lemmas

LEMMA 5 Fix the economy described in Proposition 4 On theequilibrium path of this game u9 (ct) 5 ( d R) t u9 (ct 1 t ) t $ 2

Proof of Lemma 5 Suppose that u9 (ct) ( d R) t u9 (ct 1 t ) for somet $ 2 Then P3 implies xt 2 1 5 0 implying that


u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt

Hence y 2 rt ( d R)t y2 r

t 1 t implying that exp( t g)yt yt 1 t which con-tradicts the assumptions of Proposition 4

Alternatively suppose that u9 (ct) (d R) t u9 (ct 1 t ) for some t $2 Then

u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1

Note that u9 (yt) 5 (d R) t u9 (yt 1 t ) follows from the assumptions Sothe previous inequalities imply that xt 1 t 2 1 0 which togetherwith P3 implies that u 9 (ct 1 t ) $ supn $ 1( d R)n u9 (ct 1 t 1 n) In additionxt 1 t 2 1 0 implies that zt 2 1 0 which together with P4 impliesthat u9 (ct) supn $ 1( d R)n u9 (ct 1 n) So there exists a nite t [ t 11 t 1 2 t 1 t 2 1 such that u9 (ct) supn $ 1( d R)n u9 (ct 1 n)Hence by P4 zt 2 1 5 0 contradicting the result that xt 1 t 2 1 0 N

LEMMA 6 Fix the economy described in Proposition 4 On theequilibrium path of this game xt 1 t 5 exp( t g)xt zt 1 t 5exp( t g)zt t $ 1

Proof of Lemma 6 By Lemma 5 u 9 (ct) 5 (d R) t u 9 (ct 1 t ) t $2 Combining this with Lemma 3 implies that u 9 (yt 1 Rxt 2 1)5 ( d R)t u9 (yt 1 t 1 Rxt 1 t 2 1) The assumptions exp( r g) 5 d R andyt 1 t 5 exp( t g)yt can be used to simplify the previous equationyielding xt 1 t 2 1 5 exp( t g)xt 2 1 t $ 2 Note that resource exhaus-tion and Lemma 3 together imply that zt 5 S yen

i 5 1 R2 ixt 1 i t $ 1So zt 1 t 5 S yen

i 5 1 R 2 ixt 1 t 1 i 5 S yeni 5 1 R 2 i exp( t g)xt 1 i 5 exp( t g)zt t $ 1N

Proof of Proposition 4 To prove this proposition I considertwo games an original game and a perturbed game The per-turbed game is identical to the original game except that in theperturbed game illiquid assets are higher at time zero Let D arepresent the difference between variable a in the perturbedgame and variable a in the original game Then Lemma 3 im-plies that

D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1

where the second to last equality follows from Lemma 6 N

Proof of Proposition 5 Laibson [1996] analyzes the economywithout the commitment technology I show that the innite hori-zon equilibrium which corresponds to the limit of the nite hori-zon equilibria is characterized by constant proportionalconsumption of the wealth stock where wealth is dened as thesum of nancial assets and the discounted value of future laborincome Let l represent the coefcient of proportionality I showthat l is given by

l d l br r + 1 = - --1 1 11[ ( ( ) )] R

With proportional consumption the steady state condition is

R g( ) exp( )1 - =l

Solving these equations to eliminate l yields equation (4) N

Proof of Corollary to Proposition 5 Let R represent thesteady state gross interest rate in the economy with commitmentRecall Proposition 1 exp( r g) 5 d R Using Proposition 5 it fol-lows that R 2 R 5 (r2 g)(1 2 b ) 0 as r g is required for theexistence of a steady state

HARVARD UNIVERSITY

REFERENCES

Ainslie George W ldquoSpecious Reward A Behavioral Theory of Impulsiveness andImpulsive Controlrdquo Psychological Bulletin LXXXII (1975) 463ndash96

mdashmdash ldquoBeyond Microeconomics Conict among Interests in a Multiple Self as aDeterminant of Valuerdquo in The Multiple Self Jon Elster ed (Cambridge UKCambridge University Press 1986)

mdashmdash Picoeconomics (Cambridge UK Cambridge University Press 1992)Aiyagari S Rao ldquoUninsured Idiosyncratic Risk and Aggregate Savingrdquo Working

Paper 502 Federal Reserve Bank of Minneapolis 1992


Akerlof George ldquoProcrastination and Obediencerdquo American Economic ReviewLXXXI (1991) 1ndash19

Attanasio Orazio P and Guglielmo Weber ldquoConsumption Growth the InterestRate and Aggregationrdquo Review of Economic Studies LX (1993) 631ndash49

Board of Governors of the Federal Reserve System Balance Sheets for the U SEconomy 1945ndash94 (Washington DC 1995)

Canner Glenn B and Charles A Luckett ldquoDevelopments in the Pricing of CreditCard Servicesrdquo Federal Reserve Bulletin LXXVIII (1992) 652ndash66

Canner Glenn B and Charles A Luckett ldquoHome Equity Lending Evidence fromRecent Surveysrdquo Federal Reserve Bulletin LXXX (1994) 571ndash83

Carroll Christopher D ldquoThe Buffer-Stock Theory of Saving Some Macroeco-nomic Evidencerdquo Brookings Papers on Economics Activity (19922) 61ndash156

mdashmdash ldquoHow Does Future Income Affect Current Consumptionrdquo Board of Gover-nors of the Federal Reserve System 1992

Carroll Christopher D and Lawrence H Summers ldquoConsumption Growth Paral-lels Income Growth Some New Evidencerdquo in National Saving and EconomicPerformance B Douglas Bernheim and John Shoven eds (Chicago IL Chi-cago University Press 1991)

Chung Shin-Ho and Richard J Herrnstein ldquoRelative and Absolute Strengthsof Response as a Function of Frequency of Reinforcementrdquo Journal of theExperimental Analysis of Animal Behavior IV (1961) 267ndash72

Farnsworth E Allan Contracts (Boston MA Little Brown and Company 1990)Flavin Marjorie ldquoThe Joint ConsumptionAsset Demand Decision A Case Study

in Robust Estimationrdquo NBER Working Paper No 3802 1991Goetz Charles J and Robert E Scott ldquoLiquidated Damages Penalties and the

Just Compensation Principle Some Notes on an Enforcement Model and aTheory of Efcient Breachrdquo Columbia Law Review LXXVII (1977) 554ndash94

Goldman Steven M ldquoConsistent Plansrdquo Review of Economic Studies XLVII(1980) 533ndash37

Gourinchas Pierre-Olivier and Jonathan Parker ldquoConsumption over the Life-Cyclerdquo MIT mimeo 1995

Hall Robert E and Frederic S Mishkin ldquoThe Sensitivity of Consumption toTransitory Income Estimates from Panel Data on Householdsrdquo Economet-rica L (1982) 461ndash81

Hatsopoulos George N Paul R Krugman James M Poterba ldquoOverconsumptionThe Challenge to U S Economic Policyrdquo American Business ConferenceWorking Paper 1989

Laibson David I ldquoNotes on a Commitment Problemrdquo MIT mimeo 1993mdashmdash ldquoSelf-Control and Savingrdquo MIT mimeo 1994amdashmdash ldquoAn Intrapersonal Principal-Agent Problemrdquo MIT mimeo 1994bmdashmdash ldquoHyperbolic Discounting Undersaving and Savings Policyrdquo NBER Working

Paper No 5635 1996Loewenstein George and Drazen Prelec ldquoAnomalies in Intertemporal Choice

Evidence and an Interpretationrdquo Quarterly Journal of Economics CVII(1992) 573ndash98

Mandell Lewis The Credit Card Industry (Boston MA Twayne Publishers1990)

Peleg Bezalel and Menahem E Yaari ldquoOn the Existence of a Consistent Courseof Action When Tastes Are Changingrdquo Review of Economic Studies XL(1973) 391ndash401

Phelps E S and R A Pollak ldquoOn Second-Best National Saving and Game-Equilibrium Growthrdquo Review of Economic Studies XXXV (1968) 185ndash99

Pollak R A ldquoConsistent Planningrdquo Review of Economic Studies XXXV (1968)201ndash08

Prelec Drazen ldquoDecreasing Impatience Denition and Consequencesrdquo HarvardBusiness School Working Paper 1989

Rankin Deborah M ldquoHow to Get Ready for Retirement Save Save Saverdquo NewYork Times March 13 1993 p 33

Rea Samuel A Jr ldquoEfciency Implications of Penalties and Liquidated Dam-agesrdquo Journal of Legal Studies XIII (1984) 147ndash67

Runkle David E ldquoLiquidity Constraints and the Permanent-Income Hypothe-sisrdquo Journal of Monetary Economics XXVII (1991) 73ndash98

Shea John ldquoUnion Contracts and the Life CyclePermanent-Income HypothesisrdquoAmerican Economic Review LXXXV (1995) 186ndash200


Souleles Nicholas ldquoThe Response of Household Consumption to Income Tax Re-fundsrdquo MIT mimeo 1995

Strotz Robert H ldquoMyopia and Inconsistency in Dynamic Utility MaximizationrdquoReview of Economic Studies XXIII (1956) 165ndash80

Thaler Richard H ldquoSaving Fungibility and Mental Accountsrdquo Journal of Eco-nomic Perspectives IV (1990) 193ndash205

Zeckhauser Richard and Stephen Fels ldquoDiscounting for Proximity with Perfectand Total Altruismrdquo Harvard Institute of Economic Research DiscussionPaper No 50 1968

Zeldes Stephen P ldquoConsumption and Liquidity Constraints An Empirical Inves-tigationrdquo Journal of Political Economy XCVII (1989) 305ndash46


showed that when individualsrsquo discount functions are nonexpo-nential they will prefer to constrain their own future choicesStrotz noted that costly commitment decisions are commonlyobserved

we are often willing even to pay a price to precommit futureactions (and to avoid temptation) Evidence of this in economic andother social behaviour is not difcult to nd It varies from the gra-tuitous promise from the familiar phrase ldquoGive me a good kick if Idonrsquot do such and suchrdquo to savings plans such as insurance policiesand Christmas Clubs which may often be hard to justify in view ofthe low rates of return (I select the option of having my annualsalary dispersed to me on a twelve- rather than on a nine-monthbasis although I could use the interest) Personal nancial manage-ment rms such as are sometimes employed by high-income profes-sional people (eg actors) while having many other and perhapsmore important functions represent the logical conclusion of thedesire to precommit onersquos future economic activity Joining the armyis perhaps the supreme device open to most people unless it bemarriage for the sake of ldquosettling downrdquo The worker whose incomeis garnished chronically or who is continually harassed by creditorsand who when one oppressive debt is paid immediately incurs an-other is commonly precommiting There is nothing irrational aboutsuch behavior (quite the contrary) and attempts to default on debtsare simply the later consequences which are to be expected Inabil-ity to default is the force of the commitment

Strotzrsquos list is clearly not exhaustive In general all illiquidassets provide a form of commitment though there are some-times additional reasons that consumers might hold such assets(eg high expected returns and diversication) A pension or re-tirement plan is the clearest example of such an asset Many ofthese plans benet from favorable tax treatment and most ofthem effectively bar consumers from using their savings beforeretirement For IRAs Keogh plans and 401(K) plans consumerscan access their assets but they must pay an early withdrawalpenalty Moreover borrowing against some of these assets is le-gally treated as an early withdrawal and hence also subject topenalty A less transparent instrument for commitment is an in-vestment in an illiquid asset that generates a steady stream ofbenets but that is hard to sell due to substantial transactionscosts informational problems or incomplete markets Examples




















y R z x c z xx z

t t t t t t t

t t

+ + + 0

( )

- - - =sup3

1 1

















T t

+ =eacute

eumlecirc

ugrave

ucircuacute+

=

-

aring( ) ( ) b d tt

t 1










- f f9 ( ) ( )t t


g a t + ( )1










t 5 1 St










oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES



























































y R z x c z xx z

t t t t t t t

t t

+ + + 0

( )

- - - =sup3

1 1

















T t

+ =eacute

eumlecirc

ugrave

ucircuacute+

=

-

aring( ) ( ) b d tt

t 1










- f f9 ( ) ( )t t


g a t + ( )1










t 5 1 St










oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES




















































T t

+ =eacute

eumlecirc

ugrave

ucircuacute+

=

-

aring( ) ( ) b d tt

t 1










- f f9 ( ) ( )t t


g a t + ( )1










t 5 1 St










oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES









































- f f9 ( ) ( )t t


g a t + ( )1










t 5 1 St










oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES














































t 5 1 St










oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES













































oumloslash sup3+

=+b d tt

t

t1




P max

P max +

P

1

2

3

1

11

u c R u c

u c R u c c y R x

t T t t ii

t

t T t t ii

t t t t t

9 9

9 9

( ) ( )

( ) ( )


oumloslash


oumloslash THORN =

- +=

+

- +=

+ -

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

uu c R u c x

u c R u c

t T t t ii

t t

t T t t ii

t

9 9

9 9

( ) ( )

( ) ( )

+ - - +=

+ +

+ - - +=

+ +

Otildeaeligegrave

oumloslash THORN =


oumloslash THORN

1 1 11

1 1 11

0

4

lt max

P max

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1zzt = 0










IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES
















































IV ANALYSIS







z x R y R cit

t it

ti

t

i

t

0 0

1

1

1

11 1 + +

= =

= =Otildeaelig


egraveoumloslash

-yen -yen

aring aring=


P sup

P sup +

P lt sup

1

1

1

1

2

3

1

11

1

u c R u c

u c R u c c y R x

u c R

t t ii

t

t t ii

t t t t t

t t

9 9

9 9

9

( ) ( )

( ) ( )

( )

sup3 Otilde

gt Otilde THORN =

sup3

sup3

sup3

+=

+

+=

+ -

+ +

( )( )

tt

tt

tt

b d

b d

d

t

t

t

t

iii

t t

t t ii

t t

u c x

u c R u c z

=+ +

+ +=

+ +

Otilde THORN =

gt Otilde THORN =

( )( )sup3

11

11

1

0

4 0

t

t

t

tttd

9

9 9

( )

( ) ( )

P sup 1












c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES


















































c

y

y z e R e

t

tt

t

tg gt

+

if odd

if even =

igraveiacuteiuml

icirciuml -02 21( )


B Aggregate Saving









1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES













































1ordm - -ograve [ ( ) ( )] 1 1


L L i dit t 10

1ordm =ograve ( )



(2) exp( )r dg R =



(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES










































(3) r d a dg r Y K d raquo - - = - - -( ) ( ) ( )1 1


C Asset-Specic MPCs






parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES










































parapara

= sup3-

cR x

tt

t t( )1

1 2




parapara

= sup3-

cR z

tt

t t( )1

0 2





yent 5 1



1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES









































1 11

0

1

1

- ordm -para

para

aelig

egraveccedil

ouml

oslashdivideOtilde

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

+

+ -=

-

MPCx

Rztz t i

t ii ( )

tt




























(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES



























































(4) exp( ) ( ) exp( )r b d b dg R g + = -1





(5) exp( ( )) ( )r d0 02 1 04 =






credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES












































credit) credit)

rKY

rKY

b 5 02 0040 300 0119 181b 5 04 0040 300 0070 240b 5 06 0040 300 0053 270b 5 08 0040 300 0045 288b 5 10 0040 300 0040 300

(6) 1 04 02 )exp( ) + (1 = -b bR
















(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES




















































(7) u c R u ct t t9 9( ) ( )[ ( ) ] + = -+ +1 1 1 1d l b


(8) ld

d b

=

-- -

11 1( )




yent 5 1












b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES
















































b 5 02 696b 5 04 295b 5 06 90b 5 08 16b 5 10 00












APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES


















































APPENDIX




t 1 t xAt 1 t zA





u c R u c

R u y

u y

t T t t ii

t

T t t ii

t

t

9 9

9

9

( ( )

( )

(

) max by assumption

max by assumption

) by A1

= Otildeaeligegrave

oumloslash


oumloslash

pound

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1





t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES












































t xAt zA

t Tt 5 1 and cB

t xBt zB






t


( P ti 5 1 Rt 1 i) u9 (cA


u c R u c

R u c

u c

tB

T t t ii

tB

T t t ii

tA

tA

9 9

9

9

( ( )

( )

(

) max by P1

max by assumption

) by assumption


oumloslash


oumloslash

=

- - +=

+

- - +=

+

t

tt

t

t

tt

t

b d

b d

Icirc 1

Icirc 1

1 1

1 1










t yt 1 Rt xBt 2 1



t 2 1 $ 0

0 1

11

1

11

1 1

lt by assumption

by Lemma 1

+ + by res exhaust

R c c

R c y

R x z c y

R

t ii

T t

tA

tB

t ii

T t

tA

t

t tA

tA

tA

tA

t

+-

==

-

+ +

+-

==

-

+ +

- -

Otildeaeligegrave

oumloslash -

Otildeaeligegrave

oumloslash -

= -

=

aring

aringpound

t

tt t

t

tt t

( )

( )

( )

zz c y R xtA

tA

t t tA

- -=1 1as +

So zAt 2 1 0 and

u c R u c x

R u c

u c z

tB

T t t ii

tB

tB

T t t ii

tA

tA

t

9 9

9

9

( ( )

( )

(

) max by P3 and 0

max by assumption

) by P4 and


oumloslash gt


oumloslash

sup3

- - +=

+ -

- - +=

+

-

t

tt

t

t

tt

t

d

d

Icirc 1

Icirc 1

1 11

1 1

1AA 0gt



t




yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES











































yt 1 Rt xt 2 1 t $ 2





problem

max ) +

subject to 0 1 + 1 + + + 1

c x z

T

t t

t t t t

t t t t t t t

t t t tT

u c u c


( ( )

( )

= =

-

+

-

- -

aring

sup3 sup3pound sup3

= - sup3

11

1

1

1

1

1 1

0

b d t

tt

zx zc x zT T

t t t tT

0

2

0 fixed

satisfies P1 P4

= =

=-


-1


max ) + c x z

T

t t t tT

u c u c

( ( )= =

-

+aring1

11

1

1b d t

tt

subject to


t t

t t t t

t t t t t t t

T T

t t t t

0 1 + 1 + + + 1

fixed +

sup3 sup3pound sup3

= - sup3

= =sup3=

-

- -

-

1

1 1

0 0

2 2

1

0

( )

sup3sup3 3





Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES











































Tt 5 1 Let CI










Tt 5 1 So ct xt zt

Tt 5 2


t 5 1

satises P1ndashP4 N


t 1 t xAt 1 t zA


t 1 t xBt 1 t zB





t 1 t t $ 0 as






u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES













































u y u c

R u c

R u y

t t

t ii

t

t ii

t

9 9

9

9

( (

( )

(

) ) by Lemma 3

by assumption

) by Lemma 3

=

= Otildeaeligegrave

oumloslash


oumloslash

+=

+

+=

+

b d

b d

tt

t

tt

t

1

1






xt 2 1)) 5 1 N










u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES









































u y u c

R u c

R u y

t t

t

t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 3

by assumption

by Lemma 3

=

lt

pound

+

+

d

d

tt

tt




u y u c

R u c

R u y Rx

t t

t

t t

9 9

9

9

( (

( ) ( )

( ) ( )

) ) by Lemma 1

by assumption

+ by Lemma 3

sup3

gt

=

+

+ + -

d

d

tt

tt t 1







D DD

DD

Dz R z

xRz

xRzt t

t

t

t

t+ - -

-

+ -

+ -

= -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdividet

t t

t1 1

1

1

2

1 1 ( )( ) ( )

L


for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES









































for all t $ 2 Hence

1 1 1

1

0

0

01

1

2

01

1

1

1

- = -aelig

egraveccedil

ouml

oslashdivide -

aelig

egraveccedil

ouml

oslashdivide

eacute

eumlecircecirc

ugrave

ucircuacuteuacute

=aelig

egraveccedil

ouml

oslashdivide

=

reg-

+ -

+ -

reg+ -

-

MPCx

Rzx

Rz

zz R

gR

tz

zt

t

t

t

zt

t

lim( ) ( )

lim

exp( )

D

D

DD

DD

DD

L t

t

t

t

t

== -( ) d r rR1 1



l d l br r + 1 = - --1 1 11[ ( ( ) )] R


R g( ) exp( )1 - =l



HARVARD UNIVERSITY

REFERENCES









































golden eggs and hyperbolic discounting - bill harbaugh homepage

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