Welfare, financial innovation and self insurance in dynamic incomplete markets models

Welfare, financial innovation and self insurance in dynamic incomplete markets models Paul Willen Department of Economics Princeton University First version: April 998 Tis version: July 999 Abstract We solve a dynamic general equilibrium model wit incomplete financial markets, consumers wit eterogeneous and time-varying income processes, discount factors and risk aversion. We sow tat under reasonable conditions, asset prices and allocations ave closed form solutions. Wile tis makes te model useful for analytical purposes, it also makes te model useful for empirical work see Davis and Willen 999 for an example. We ten sow: If socks are temporary, ten in many cases, a riskless asset alone can implement full risk-saring. Tis result parallels work of Levine and Zame998 but is more general in tat we sow tat full risk-saring can old exactly even wit subjective discount factors below one. If socks are temporary, computational evidence sows tat welfare losses due to incomplete markets are small even wen discount rates are significantly below one. Tanks to Laurent Calvet, Jon Geanakoplos, Sudipto Battacarya and Mattew Jackson for elpful comments and suggestions. Cris Telmer s discussion anoter of my papers at te Canadian Macro Study Group 997 meetings in Toronto stimulated my tinking on tis problem.welfareeffects8.tex Last compiled: Marc 29, 2000 Pone: 609 258-4032. E-mail: willen@princeton.edu

If socks to income are permanent, ten in many cases, one can analyze a dynamic model as a series of static models. If socks are combinations of temporary and permanent innovations, ten in many cases, one can solve for equilibrium as if all temporary socks were tradable. Tis is consistent wit empirical evidence in te consumption insurance tests of Cocrane99 and Attanasio and Davis996. Financial innovations tat allow for saring of permanent socks may, perversely, decrease individuals ability to sare temporary socks. In finite orizon settings, financial innovation can actually increase te variance of consumption for muc of te life-cycle. Deviations from complete markets equilibrium are non-linear in te persistence of socks. In an economy were income follows an AR process, welfare effects of incomplete markets are small even wit an AR coefficient of 0.5. However te welfare effects grow dramatically as te coefficient goes from 0.5 to. 2

Introduction In a recent paper, Levine and Zame998 ask in te title Do incomplete markets matter? Teir answer, more or less, is no. Specifically tey sow tat wit i socks tat are independent over time, ii a subjective discount rate close to one and iii infinite orizons, te equilibrium allocation of any incomplete markets economy converges to tat of a complete markets economy. In tis paper, we consider a specific incomplete markets model tat allows for analytical solutions and explore incomplete markets models in wic tese assumptions are relaxed. In general, we find tat wit tese assumptions relaxed, incomplete markets do matter. However, te extent to wic tey matter depends on many different tings. To carry te comparison wit Levine and Zame998 a little furter, one migt title tis paper, How muc do incomplete markets matter? Te field of incomplete markets as typically looked at two-period models. For many of te questions about risk-saring and asset pricing, tis approac is sensible. See Geanakoplos990 for a discussion. However, tere are at least two reasons wy one migt want to look at dynamic models. First, for applying tese models to data, one will typically need to look at time series data to estimate te parameters. Second, wile capturing te possibilities for risk saring troug risky assets, twoperiod models miss te substantial possibilities of risk saring tat one can acieve troug traditional consumption smooting: i.e. borrowing wen one is poor and saving wen one is ric. Researcers ave looked at dynamic incomplete markets models in some detail. Duffie, Geanakoplos, Mas-Collel and MacLennan994 sow existence for a general class of tese models. However, study in tis area as been limited by te complexity of te equilibria. Researcers ave typically ad to rely on computational solutions. Two well-known papers in tis area are Telmer994 and Heaton and Lucas996. More recently, Judd, Kubler and Scmedders998 ave developed a new tecnique for computing equilibria wic is considerably more powerful. However, te curse of dimensionality limits solution to small numbers of agents and assets. Te autors claim tat for more tan two agents and wit sensible parameters in particular discount rates, solutions are beyond te power of current computing tecnology. 2 However, as we sow ere, one class of dynamic incomplete markets, te exponen- Oter examples of dynamic incomplete markets models solved computationally include den Haan996,997, Marcet and Singleton 998, Storesletten, Telmer and Yaron 998, Krusell and Smit 997a,b. Imrooroglu989 and Hansen and Imrooroglu992 also consider related models. 2 Personal communication

tial normal as convenient, closed-form solutions for arbitrary numbers of individuals, assets and time periods. Te structure of te economy can be quite ric: individual income can follow any ARMA process; assets can be eiter one-period lived or longlived wit ARMA dividend streams. Yet we can still prove general results about te economy. In addition, we can calculate equilibria in virtually no time at all. Tis allows researcers as we do ere to simulate large numbers of economies wit different time orizons, individual parameters, market structures, etc. Finally, tis model, in its ability to andle ric parameter spaces, is well-suited to empirical work. Davis and Willen999, for example, use CPS data to calculate equilibria wit 44 different types of agents, 4 different assets, overlapping generations and age varying earnings processes and age-varying covariances between individual earnings and assets. Te caracterization of equilibrium is clear and intuitive, combining te insigts of te capital assets pricing model and te permanent income ypotesis into one straigtforward paradigm. In tis paper, we solve te model and sow te following: If socks are temporary, ten in many cases, a riskless asset alone can implement full risk-saring. Tis result parallels work of Levine and Zame998 and reflects an argument tat goes far back at least as far as Bewley980. It is more general in tat we sow tat full risk-saring can old exactly even wit subjective discount factors below one. If socks are temporary, computational evidence sows tat welfare losses due to incomplete markets are small even wen discount rates are significantly below one. If socks to income are permanent, ten in many cases, one can analyze a dynamic model as a series of static models. Tis suggests tat te results traditional two period incomplete markets researc can be generalized to a class of dynamic models wit permanent socks. Tis parallels work of Constantinides and Duffie 996 If socks are combinations of temporary and permanent innovations, ten in many cases, one can solve for equilibrium as if all temporary socks were tradeable. Tis is consistent wit empirical evidence in te consumption insurance tests of Cocrane99 wo sowed tat for example, sort illnesses tended not to affect consumption but tat long ones did, and Attanasio and Davis996 wo sowed tat sort run fluctuations in relative wages did not affect te distribution of consumption but tat long run fluctuations did. 2

Financial innovations tat allow for saring of permanent socks may, perversely, decrease individuals ability to sare temporary socks. In finite orizon settings, financial innovation can actually increase te variance of consumption for muc of te life-cycle. Deviations from complete markets equilibrium are non-linear in te persistence of socks. In an economy were income follows an AR process, welfare effects of incomplete markets are small even wit an AR coefficient of 0.5. However te welfare effects grow dramatically as te coefficient goes from 0.5 to. We develop a convenient measure of te welfare benefits of financial innovation. Te exponential-normal model as te convenient feature tat it is relatively easy to estimate. We argue tat tis metod is superior to consumption insurance tests as a way of looking for promising risk saring opportunities. Davis and Willen 990 estimates tis model using data from te CPS and calculates te welfare gains of introducing various assets 3. Tis paper builds on existing researc in te following ways. First, it sows tat te dynamic exponential-normal model discussed extensively in te literature see, for example, Stapleton and Subramanyan978 can be extended to a very ric incomplete markets setting. Second, it sows tat te partial equilibrium insigts of te consumption literature see Bewley980, for example, extend to a general equilibrium context in some cases and not oters. Tird, it sows tat te results of Levine and Zame 998, wile more general in terms of utilities, old exactly wit some more general specifications of te economy and approximately wit some oters. Te results described ere and in Levine and Zame998 sow tat in some cases, te riskless asset alone can implement a complete markets equilibrium. Tis result sould not be confused wit results in financial economics tat ave sown tat in dynamic economies, often only a small number of risky assets can be enoug to implement complete markets allocations. Various different versions of tis proposition ave been proved in static and dynamic contexts. 4 However, tese propostions are fundamentally different from te penomenon we describe ere. Risk-saring ere 3 Willen997 also uses tis model to calculate welfare gains. 4 See, for example, Sceinkman989 for a discussion. Kreps982 sows tat in a dynamic model, rougly if te number of assets is greater tan te maximum number of successor nodes for any give state, ten te incomplete markets allocation will be te same as te complete markets allocation. Duffie and Huang985 sowed someting similar in a continuous time context. Geanakoplos990 sowed tat in a two period incomplete markets model wit quadratic preferences, if endowments are spanned by assets and tere is a riskless asset, equilibrium is Pareto-optimal. 3

does not exploit te covariance properties of assets wit endowments, as it does in any static model and in traditional complete markets models. Rougly we sow tat wen socks are temporary, te riskless asset provides for almost complete markets. Te complete markets paradigm underlies many strands of modern macroeconomics and finance. Tis paper suggests tat in spite of te fact tat we clearly don t observe te sort of ric markets imagined in te classical world of Arrow and Debreu, analysis wic presumes suc markets is not completely misguided. For example, Cocrane99 and Mace99 bot developed empirical metods for testing if an observed allocation was Pareto-optimal. Tis migt seem pointless it seems self-evident tat tey sould fail. We sow tat even wit te oldest and least sopisticated financial instrument a riskless bond we can potentially implement a Pareto-optimal allocation. Section 2 describes te model and solves te individual dynamic optimization problem and equilibrium. Section 3 develops a measure of te welfare effects of financial innovation. Section 4 considers models were agents only face only permanent socks. Section 5 considers models wit only temporary socks. Section 6 considers models wit permanent and temporary socks. Section 7 concludes. 2 A dynamic incomplete markets model We consider an exponential-normal economy ere. Agents ave exponential or constant absolute risk aversion CARA utility and socks to endowments and asset payoffs are normally distributed. Te convenience and simplicity of te exponentialnormal framework is very well-known in finance and in economics in market microstructure in particular, but in practically every oter area as well. Te two-period exponential normal model is te approac of coice in te financial innovation literature, wic is closely related to te topics discussed ere, 5 In general, researcers in te asset pricing literature ave used te more realistic constant relative risk aversion CRRA class of utility functions. 6 However, researcers ave made use of te exponential-normal framework. Stapleton and Subramayam978 consider a model similar to tis one i.e. a dynamic, discrete-time model wit exponential utility and normal returns. However, tis model is significantly more general in te following ways. First, individuals ave possibly non- 5 See te Journal of Economic Teory, 65, February 995, wic is devoted entirely to financial innovation. Almost every paper uses te exponential-normal framework. 6 See directly below and trougout te paper for discussions of ow CRRA preferences affect te issues discussed. 4

tradeable income. Second, we endogenize te interest rate i.e. tis is a full general equilibrium model. Huang and Wang997 consider asset pricing and risk-saring issues in a continuous time exponential-normal setting. Teir focus is different and tey also assume a fixed, exogenous interest rate. Endogenizing te interest rate is crucial to te analysis tat follows. Te idea tat one can smoot away almost all idisyncratic risk wen sock are temporary as been well-known in te partial equilibrium consumption literature for a long time. It is not obvious tat tese results will old wen te interest rate is allowed to equilibrate supply and demand. It is precisely tat issue tat we explore ere. Te main advantage of te exponential-normal framework is tat analytical solutions can generally be found even wit substantial incompleteness of markets or of information. Te disadvantages, owever, are also well understood, as follows: Negative consumption is possible. Wit CARA utility, te marginal utility of consumption at zero is less tan infinity. In addition, wit normal returns, any arbitrarily negative outcome is possible. In te computations tat follow, we will sow, owever, tat te probability of negative consumption is, in fact quite low. For reasonable parameterizations, te probability tat an individuals consumption will be negative one undred years in te future is less tan one percentsee figure 6. Absolute rater tan relative risk aversion is constant. Tis means tat te demand for precautionary savings and te demand for risky assets will not depend on te level of wealt te key state variable for individuals. Tis, in general, seems unrealistic. Normal distributions are unrealistic bot for income socks and for asset return socks. Te lognormal distribution is te distribution of coice for tese sorts of problems. 7 As we describe te solution to te model, we will point to places were te exponential-normal assumptions make a significant difference. 2. Te economy Tere is one consumption good. Tere are T + periods, t =0,..., T. Tere are H agents, =,..., H. A consumption pat is a random vector C = c t 7 Te problem wit te lognormal distribution is tat te sum of lognormals is not lognormal; te fact tat te sum of normals is normal is crucial to te results below. t T. 5

Let expectations be one period aead unless oterwise noted. All agents ave von Neumann-Morgenstern utilities of te type U [ T C =E t 0 A exp ] A c t t=0 Agents coose a portfolio of assets ω = ωt. Tere are no limits on ω, unlimited sort sales are possible. Let A = individual absolute risk aversion. t=,...,t H = A, te armonic mean of Condition Individual endowments yt T follow an ARMA process. Let t= η t be te income innovation for individual at time t. Te stocastic process for income as an MA representation. Let ψ i be te i t ỹ ỹ MA coefficient. I.e. E t t+i Et t+i = ψi η t. Define ỹ t H H = ỹ t. We now specify two alternate conditions on assets. In te appendix we sow tat if te payoffs on te one period assets described in condition 2 and te innovations on te long-lived assets described in 3 are te same, ten te equilibria of oterwise identical economies will be te same. For tis reason, we can witout any loss of generality, concentrate on te analytically simpler one-period asset case condition 2, keeping in mind tat all results can easily be generalized to te long-lived asset case. Condition 2 Tere are J + one-period assets J risky assets wic pay off x t J t =,..., T wic is iid over time. 2 Tere is a riskless asset wic always pays off one dollar wit certainty. I.e. x 0,t =t =,..., T Condition 3 Tere are J + assets J risky assets wit dividend stream d t J t =,..., T wic follows an ARMA process wit innovations t =,..., T wic are iid over time. 2 Tere is a riskless asset wic always pays off one dollar wit certainty. I.e. x 0,t =t =,..., T x t J Let te vector of errors be Φ= η T t=. η H T t= x. x J 6

Condition 4 Te error terms are distributed jointly normal Φ N 0, Σ We can assume witout loss of generality tat te assets are uncorrelated wit one anoter and tat tey ave an expected payoff of zero and a variance of one i.e. cov x i, x j =0, E x j =0, var x j =. βj,t is te covariance of individual s income innovation in period t wit asset j. So te covariance matrix is: 8 Σ= Σ β, β,t β, H β,t H...... βj, βj,t...... βj, βj,t βj, H βj,t H We define te covariance of an individuals income variations wit te a factor as beta. Note tat since var x j =, te coefficient of an OLS regression of income innovations on assets is βj,t. We can ten introduce R,t 2 = vareη t vareη t P J j= β j,t ex j I vareη t wic is analogous to te R 2 of a regression. It tells us ow muc of te variance of individual income socks an individual can edge away using risky assets. Consider a perfectly competitive economy in wic te consumption goods prices are normalized to be one and te asset prices are denoted by π R J T. 9 Te c budget set is B π ={ t,ω t t T c t + J j=0 ω j,t π j,t = y t + } J j=0 ω j,t x j An equilibrium π, C,ω is a price-vector and a consumption profile suc H tat: Eac individual optimally cooses in is budget set: C,ω B π and C,ω B π U C U C ; 2 Markets clear: for all t: H c t = H ỹ t and H ω t =0. Let Π t 2 t =Π t 2 i=t π 0,i be te price of a security at time t paying off a dollar wit certainty at time t 2 Let It = ỹt + J j= x jωj,t + ω0,t, wic we will call income. T Ten let Wt = It +E t s=t+ Πst+ỹ s wic we will call wealt. Finally, let Vt =var c t wic is te conditional variance of consumption one period aead of time. Table summarizes notation. 8 Te assumption tat te variance-covariance matrix for stocks is te identity matrix is made purely for expositional convenience and can be dispensed wit entirely. 9 We can define gross rates of return on long lived assets R j,t+ = xj+πj,t+ π j,t and do all te subsequent analysis in rates of return. However, since only te price and te expected return can vary, it seems more logical to work wit prices and exogenous payoffs rater tan rates of return. 7

earnings earnings innovations t t MA coefficient payoff of j t asset price of j t asset y t η t ψ t x j,t π j,t multi-period Π t 2 t =Π t 2 i=t π 0,i discount factor cov x j,t,ηt βj,t consumption asset oldings income wealt MPC out of wealt constant c t ω j,t It Wt a t 2.2 Individual optimization b t Table : Summary of notation In tis section, we will consider te individual optimization problem under te assumption tat asset prices are non-stocastic. In te following section, we sow tat in fact, under suitable conditions, equilibrium prices are in fact non-stocastic. Tis still may seem unduly restrictive or unrealistic. However, one must keep several tings in mind. In te one-period lived asset case wic is wat we explore in tis section, asset payoffs are iid over time. Tus, te outcomes of individual income and asset return draws can only affect te price of te asset troug its effect on te distribution of wealt wic CARA-utility effectively rules out. In te multi-period asset case, te asset payoff may contain information about future asset payoffs and tus affect te price of te asset. But as we sow in te appendix, te price of te asset will be normally distributed and in fact affine in te dividend innovation and tus equivalent to te one-period asset. 0 Wit tis in mind, we can now discuss te solution to te individual portfolio problem. Tere are two keys to te results tat follow. First, individual consumption is affine in wealt. I.e. c t = a tw t b t 0 In rates of return, we are essentially assuming tat te distribution of returns is deterministic, wic is generally assumed in partial equilibrium portfolio problems. 8

were a t and b t are deterministic constants. a t is te marginal propensity to consume out of wealt. If te price of te riskless asset is at least one or te interest rate is weakly negative, ten as T, a t 0. Tis is a well-known result from te permanent income literature. Second, for any individual and asset j, cov c [ t+, x j = E x A j π ] j,t π 0,t Te rigt and side is te equity premium scaled up or down by te level of absolute risk aversion. Te covariance of individual consumption wit an asset will depend on te covariance of one s wealt wit tat asset, wic in turn depends on two tings: te covariance of income wit tat asset βj,t+ and 2 te position te individual takes in tat asset ωj,t. All else equal, tis means tat an individual wose income as ig covariance wit an asset will invest less tan someone wit lower covariance wit an asset. Te ratio of risky asset oldings of two individual wit different levels of absolute risk aversion but oterwise te same will be te inverse of te ratio of teir levels of absolute risk aversion. 2 Te following proposition formalizes tis intuition. Proposition Assume tat asset prices are nonstocastic. Under conditions, 2 and 4:. ω j,t = a t+ A π j,t π 0,t Ψ t+ β j,t+ were Ψ t+ = T t s=0 Π t++s t+ ψ s 2. ω 0,t = π 0,t +a t+ 3. c t = a tw t b t were a a t = b b t = π 0,t It J j= π j,tω j,t a t+ E + T t s=0 Π t++s t+ π 0,t +a t+ 2 A V T s=t+ Πst+ỹ s + 2 A V t+ A ln π 0,t + A ln t+ a t+ J j= c V t =var c t = R 2,t var at Ψ t η t b t+ π j,t π 0,t ωj,t ln π 0,t + b A t+ +var A J j= π j,t π 0,t x j E x j πj,t π 0,t is te difference between te expected payoff of buying a unit of te risky asset E x j and te expected payoff of investing te same amount of money in te riskless asset πj,t π 0,t. N.b. We ave assumed above tat E x j = 0. We put E x j in tese equations even toug it is zero for expositional purposes; in most cases we will leave it out. 2 Tis implies tat teir oldings of particular risky assets as a percentage of teir wole investment portfolio will be te same wic implies two fund separation. 9

Proof: See Appendix Letting φ t = π 0,t π 0,t + a t+ we can solve recursively for b t.i.e. T b t = φ s 2 A Vs+ a s+ s=t J j= π j,s ωj,s π 0,s A ln π 0,s We can get more insigt into individual risk bearing by looking at individual consumption variance. Vt = R,t 2 var at Ψ t η t πj,t +var E x A j x j π 0,t Te first term on te rigt and side R,t 2 var at Ψ t η t is idiosyncratic risk and te second term var J πj,t A j= π 0,t E x j x j is equity premium risk. 3 As we said before, two tings affect ow muc risk an individual olds in equilibrium. Te first is R,t 2. Te more assets are correlated wit an individual s income, te less individual risk an individual will old in equilibrium. However, te amount of risk is not equal to te sock itself: it is equal to j= a t Ψ t η t Ψ t is te marginal effect on wealt of an unit increase in income tis period. Terefore te marginal effect of an unit increase in income is te effect on wealt times te marginal propensity to consume. Te effect of an increase in income on consumption will be big all else equal if te effect on wealt is big. Te depends on te persistence of socks. Consider te two polar cases: if te sock is wite noise, ten Ψ t = and te marginal effect of a sock on income on consumption is a t = + T t s=0 Π t++s t+. If income follows a random walk, ten Ψ t = + T t s=0 Π t++s t+ and a t Ψ t =. Wen socks are permanent, ten te effect of a sock to income is unitary in consumption. Now, it is easy to understand te individual portfolio coice. In a two period model, individuals consume teir income in te second period, so te relevant idiosyncratic variable for edging is income. Here it clearly is wealt, so individuals will edge wealt. Te effect of a dollar increase in income is a Ψ t increase in wealt 3 Again we ave included E x j for expositional purposes even toug it is assumed to be zero. 0

so tey approac te edging problem as if tey were in a two period model in wic teyadanincomeofψ t in te next period. Te following results is similar to Proposition in Caballero990. Here we ave added in asset markets and a time-varying price of te riskless asset, but te result is essentially te same. Corollary Consumption follows a random walk wit drift c t+ = c t + 2 A Vt+ A ln π 0,t. + a tψ t R 2 2,t η t J π j,t x A j π j= 0,t Proof: By Lemma, E c = c t + A var c 2 t+ ln π 0,t. By te teorem A consumption equals c t+ =E c t+ + at Ψ t R 2,t η t π j,t A π 0,t x j. How would our results be different wit Constant Relative rater tan Constant Absolute Risk Aversion? Te principal difference is tat absolute risk aversion is wealt-dependent. In tis setup, note tat demand for risky assets is non-stocastic it doesn t depend on an individual s wealt level. Tis will not be true in a more general setup. Absolute risk aversion also affects te amount of precautionary savings people want to do. Here te expected slope of consumption is deterministic. In Corollary, te drift term depends on absolute risk aversion. In a more general setting, tis drift term would also be wealt-dependent. 2.3 Equilibrium In solving te individual optimization program, we assumed tat asset prices are non-stocastic. Tis assumption is essential for te straigtforward results we get. However, deterministic asset prices do not require deterministic individual or even aggregate income; tey require only tat te growt rate of aggregate income be deterministic. 4 Tus, we will need te following condition to calculate equilibrium prices. 5 Condition 5 Te expected growt rate of aggregate income is non-stocastic. E s ỹ t+ ỹ t =k t for all s<t I.e. Te processes tat meet condition 5 include obviously any non-stocastic income time-series but also more realistically a random walk aggregate income series wit a 4 Tis fact was noted by Battacarya98. 5 Condition 5 is also necessary for a non-stocastic interest rate.

possibly time varying growt term. Let, te covariance of a particular factor wit te market portfolio. Let = Π H = A H H. Let A = = A Wy is tis condition sufficient for a deterministic riskless rate? Intuitively, wen a single individual gets a goodbad temporary sock, e wants to saveborrow, because is income is fallingrising. By assuming tat aggregate income follows a random walk, we are implicitly assuming tat te good and bad temporary socks cancel eac oter out rougly, te number of people wo adjust teir expectations of future income up equals te number wo adjust teir expectations down. It is precisely tis problem tat Levine and Zame 998 confront wen trying to generalize teir model to te case of stocastic aggregate endowment in general te interest rate will go updown wen everyone wants to borrowsave. 6 As Battacarya98 points out, it is ard to tink of a tecnology tat will allow one to add production to tis framework wile preserving tis feature. However, te presence of unit roots in aggregate consumption and income time series as been a subject of intense debate in macroeconomicssee for example, Cristiano and Eicenbaum992 for a discussion, wic suggests tat tis assumption is not wolly unrealistic. Proposition 2 Under conditions, 2, 4 and 5.. Asset prices will be non-stocastic 2. Risky assets a π j,t π 0,t = A cov η t+, x j 7 b ω j,t = A a t+ cov η A t+, x j cov Ψ t+ η t+, x j 3. Riskless asset ] a π 0,t = exp A [ỹ t Eỹ t+ + H 2H = A Vt+ It J j= π j,tωj,t b ω 0,t = T π 0,t +a t+ a t+ E s=t+ Πst+ỹ s b t+ + 2 A Vt+ ln π A 0,t + ln A were V t =var c t = R 2,t var at Ψ t η t +var A A J j= cov η t+, x j x j 6 Levine and Zame sow tat by introducing a risky assets wose payoff is tied to aggregate income, tey can still get strong implications for risk saring. 7 Letting R j,t denote te return on long-lived asset j at time t i.e. Rj,t+ = xj+πj,t+ π j,t wecan re-write te risky asset pricing equation as E Rj,t+ R 0,t+ = A t cov η t+, R j,t+ wic is a dynamic version of te traditional CAPM. 2

Proof: See Appendix Te prices of te risky assets are determined by te covariance of te aggregate endowment wit te asset just as in te two-period CAPM-type model. Tis is a consequence of Stein s Lemma, wic tells us tat te covariance of a random variable and a function of anoter random variable is linear in te covariance of te random variables wen tey are bivariate normal. Tis allows us to sum across agents consumption. Te price of te riskless asset reflects te fact tat individual demand for te riskless asset is increasing in te variance of consumption, wic is commonly known as precautionary saving. So te price of riskless asset return on te riskless asset is increasing decreasing in te discount rate, decreasing increasing in te growt of income and increasing decreasing in te variance of consumption. Corollary 2 Te equilibrium values π, C,ω are continuous in te exogenous parameters of te economy i, Σ,, A H ψ =,...,H i=,...,i =,...,H =,...,H Proof: Follows trivially from Proposition 2. How would our results be different in a more general setting? Te key to te simplicity of tese results is tat te wealt distribution as no effect on te equilibrium. As we discussed at te end of te last section, canges in wealt affect consumption but not absolute risk aversion, and tus tey don t affect portfolio decisions. Tis means tat we can price assets using a combination of aggregate information and deterministic information about individuals. Tus our equilibrium parameters are deterministic. 3 Financial innovations and welfare A financial innovation is a new asset tat canges te span of te asset matrix. Evaluating te benefits of financial innovation may seem like an esoteric finance problem, but most welfare questions related to incomplete markets are equivalent to questions about financial innovation. Researcers ask, Wat is te welfare loss due to not aving complete markets? Tey migt well ask, Wat would te welfare cange be if we introduced financial innovations tat allowed individuals to trade all risks? We contend, owever, tat tinking about financial innovation wic need not imply market completion, more market disincompletion is different from tinking about market completion in a subtle and important way. 3

Complete markets are someting of a knife-edge. Consider an economy tat were markets are incomplete. It is not, in general, te case, tat if we add a market for one type of risk, only beavior related to tat risk will be affected. Wen we add a market, prices of all assets may cange making it more or less difficult to insure oter risks. If we add yet anoter market, again beavior will cange wit respect to every risk. Only if we complete markets, will people perfectly insure every risk. Tus, if we consider making a market for a particular risk, assuming tat individuals will now be perfectly insured wit respect to tat risk is in general wrong. Consider te following examples. First, consider a case from international economics see van Wincoop994. Suppose we look at country-level data. We complete markets for country level risksl, and compute te allocation. We ten compare tis allocation wit te actual allocation. Our welfare conclusions from tis would almost certainly be wrong. Unless markets were complete witin countries to begin wit, completing markets for tese risks would in fact cange risk saring witin countries. Second, suppose individuals face two types of medical socks, temporarybroken legs and long-term eart diseasesee Gertler and Gruber997. 8 Suppose we find tat consumption data suggests tat temporary socks ave no effect on consumption and long-term socks do. Tis would suggest tat temporary socks were insurable and tat no insurance market for tem is needed. Tis inference may be wrong. As we sow in section 6.3, it is possible tat te existence of te long-term socks may actually make it possible to dynamically smoot away te temporary socks. By completing markets for te long-term socks, we may reveal te market incompleteness wit respect to te sort-term socks. Tus, we contend tat tis approac to analyzing consumption insurance is superior to te market completion approac, because it capture te complex general equilibrium effects of financial innovation. We can, witout loss of generality, assume tat any new financial instrument is uncorrelated wit existing financial instruments. It is not necessarily te case tat financial innovation is Pareto-improving even in tis relatively special case. Imagine, for example, an individual wo is endowed wit only riskless second period wealt. If financial innovation raises te interest rate, it will reduce is ability to borrow. However, we would still like to be able to make welfare comparisons. So we use consumer surplus analysis. As is common in te literature see, for example Attanasio and Davis996 and many oter papers we ask te question for te case of compensating variation ow muc more income in every date and state do we need to give an individual in te new equilibrium to give im te same utility as in te old 8 We discuss te example at lengt in section 6.3 4

equilibrium? Formally: Definition Te uniform equivalent variation is ow muc an individual s preinnovation consumption must be increased in every state so tat is utility is te same as it is after innovation. Define C to be consumption before innovation and C to be consumption after te innovation. Ten te uniform equivalent variation is θ suc tat U C + θ =U C Definition 2 Te uniform compensating variation is ow muc an individual s post innovation consumption must be increased in every state so tat is utility is te same before innovation. Define C to be consumption before innovation and C to be consumption after te innovation. Ten te uniform compensating variation is θ suc tat U C =U C + θ Because we are assuming CARA utility, in fact bot tese definitions are te same and we will call tem bot uniform variation. Tis proposition reflects te fact tat total utility is proportional to marginal utility wen utility is exponential. A reduction in te variance of second period consumption wic a financial innovation allows increases utility and decreases te marginal utility of second period consumption proportionally. Te interest rate reflects te marginal utility of an uniform increase in second period consumption wic is wat te riskless asset does, but since utility is proportional, it also reflects te cange in utility. Terefore, to determine te cange in consumer surplus from a financial innovation, it will suffice to look at te price of te riskless asset, bot before and after financial innovation. Proposition 3 Te average uniform variation for a T -period economy is θ = + T A ln t=0 Π t 0 + 2 T t=0 Πt 0 Proof: see Appendix Tis proposition reflects te fact tat total utility is proportional to marginal utility wen utility is exponential. A reduction in te variance of second period consumption wic a financial innovation allows increases utility and decreases te 5

marginal utility of second period consumption proportionally. Te interest rate reflects te marginal utility of an uniform increase in second period consumption wic is wat te riskless asset does, but since utility is proportional, it also reflects te cange in utility. 4 Permanent socks and two-period incomplete markets models As we noted above, wen socks are permanent, te effect of an increase in income on consumption is unitary. Tis makes a model wit permanent socks very similar to a two-period model: in a two period model individual consume teir income because tere is no tomorrow; ere tey consume te cange in income because tomorrow is expected to be exactly like today. Condition 6 Assume individual incomes follows a random walk. I.e. y t = k t + y t + η t Teorem Under conditions, 2, 4, 5 and 6 te asset prices, individual asset purcases and individual variances of consumption will be te same as in a two period model. Proof: Te marginal propensity to consume out of a sock to one s income is a t Ψ t. Wen income follows a random walk, Ψ t =+ T t s=0 Π t++s t+, so a t Ψ t =. Tus Vt =var η t J j= β j,t x A j +var J A j= cov η t, x j x j Teorem 3 as written, doesn t sound as strong as it in fact is. We can t say anyting about oldings of te riskless asset because tose depend on life-cycle factors an individual may borrow a lot tis period because e expects a big increase in labor income five periods in te future, wic obviously wouldn t appen in a two period model. So we can prove te following corollary. Corollary 3 Assume tat conditions, 2, 4, 5 and 6 old. k t Furter assume tat = k t. Ten all asset prices and portfolio oldings will be as in a two period model. In oter words, in tis situation dynamics more or less don t matter. Te permanent socks are not te only important assumption driving tese results. Exponential 6

utility plays a role too. In te general CRRA case, socks to wealt will affect absolute risk aversion and will cange te slope of te consumption pat. So in general, an innovation to income will not result in a one-for-one cange in consumption even wen socks are permanent. 9 As we said above, to determine te cange in consumer surplus from a financial innovation, it will suffice to look at te price of te riskless asset, bot before and after financial innovation. In general, te variance of tomorrow s consumption will depend on te interest rate tomorrow. Simply, if it is difficult to borrow in te future ten one anticipates tat one will not be able to smoot as muc. But ere, one does no intertemporal consumption smooting, so te price of te riskless asset [ ] π 0,t = exp A ỹ t Eỹ s + H A Vt+ 2H will only depend on te parameters tis period. If tose are constant ten te equilibrium interest rate will be constant and te value of te cange in social welfare will very straigtforward one will be able to use any one period riskless rate to carry out te calculation. Under tese circumstances, calculation of te welfare effects of financial innovation are trivial. Corollary 4 If te conditions of teorem 3 old and te growt of te economy is constant ten as T te uniform variation is of a financial innovation is θ A ln π 0,t π0,t Proof: By Teorem π 0,t ] = exp A [ỹ t Eỹ s + H 2H = A Vt+. By te permanent socks and stationarity, Vt+is constant and ỹ t Eỹ s is constant by assumption, so π 0,t is constant for any t. = 5 Temporary socks and consumption smooting As mentioned above, Levine and Zame998 sow tat wen income socks are iid, aggregate endowment is constant, discount rates are close to one and te time orizon is infinite, ten incomplete market allocations converge to complete markets allocations. Tis gives teoretical justification to te results of a large body of computation 9 Tis contrast was noted in Krusell and Smit 997a. Wit exponential utility, te expected slope of consumption does not depend on te level of wealt; tis is not, in general true. 7

evidence tat suggests tat te effects of incomplete markets on equilibrium allocations are marginal. Evidence of tis includes Telmer993, Lucas994, Heaton and Lucas996, Krusell and Smit 995, Krusell and Smit 997. Tis penomenon will old ere as well. Te sensitivity of individual consumption to individual socks depends on R,t 2 and on a tψ t. Assume tat R,t 2 is constant. As sown in section 2.2, wen socks are temporary Ψ t =. Ifpi 0,t t, a t must go to zero. We simply need to sow now tat in equilibrium pi 0,t t. Assume tat individual income is iid over time. I.e. y t = k t + y + ε t Let k t = H H = k t. Teorem 2 Suppose conditions, 2, 4, 5 old, individual income is iid and aggregate endowment is constant. As T and exp A k t+ k t, te incomplete markets economy allocation converges to te complete markets allocation. Since all te equilibrium values are continuous in te exogenous parameters, tis will be approximately true for any economy in a neigborood of exp A k t+ k t. Proof: π 0,t = exp A ỹ t Eỹ t+ exp A H 2H = A Vt+ and H 2H = A Vt+ 0, so π 0,t exp A ỹ t Eỹ t+. A ỹ t Eỹ t+ = A k t k t+. If exp A k t+ k t, ten π 0,t. Tis means tat te marginal propensity to consume out of one s own income is a t = + T t s=0 Π t++s t+, wic goes to 0 as T. Notice tat exp A k t+ k t implies tat te iger te growt rate is, te less agents must discount te future. If te growt rate is zero, te discount rate must be. If te overall economy is srinking, ten we can reduce te discount rate to a realistic level but in a very unrealistic way. Tis result will not old exactly if any of te conditions are not met. However, in te next section, we sow evidence from computations, wic sows tat te range of parameters for wic te effects of market incompleteness are small is quite large. 5. Evidence from computations Figures to 5 sow results from computations of te following economy. We consider an economy wit a continuum of ex ante identical agents. Eac faces an IID disturbance term wit variance defined as described on te plots. Since we ave a continuum of agents, we can assume tat tere is no aggregate variance. We coose a parameter of relative risk aversion and ten calculate te corresponding coefficient of 8

absolute risk aversion evaluated at average consumption. We also coose a standard deviation of earnings and report it on te plots as te ratio of standard deviation of earnings and expected income σ. In all te following plots, we sow effects at I various time orizons. Remember tat tere are no endogenous distributional effects so tere is no pat dependence. Terefore, any T -period model is nested in a T +- period model. Te standard deviation and interest rate calculations at time T are te first period of a T -period model and te second period of a T +periodmodel. Figure sows a contour plot of te welfare effects of incomplete markets wit various different time orizons and discount rates. Te welfare calculation is te uniform variation related to a financial innovation wic completes markets see section 3 for a full discussion te permanent cange in consumption required to bring individuals to te level of welfare tat te financial innovation provides. On practically all te plots, one odd feature emerges. In general, we would expect tat as time increases and increases, we would in general reduce te welfare benefits of market completion. However, for fixed time orizons, we see tat te effect of iger patience are somewat ambiguous. Tis is te backward bending portion of te contours in te upper left corner. Wen we raise te level of patience in te economy, welfare is affected in two ways: first, individual care more about te future wic means tat market completion is more valuable; second, te general equilibrium effects tat we ave discussed kick in, facilitating consumption smooting and increasing welfare. Clearly, for large T, te second effect dominates, as te plot sows. However, for large variances of income and ig risk aversion Figure 2 lower rigt panel, backward bending iso-welfare contours are present even for relatively large T. Te welfare effects of incomplete markets were not surprisingly muc more significant for ig risk aversion coefficients and for ig variances of consumption. As te lower-rigt panel of Figure 2 sows, tey converge for =.9 torouglytree percent. In addition, te welfare losses for sort time orizons are quite large. Figure 3 sows te standard deviation of consumption as a fraction of te standard deviation of income also equal to te marginal propensity to consume for various different parameters. In general tis fraction was very small. Only for very low discount rates, did it ever exceed 0%. For a reasonable discount rate =.95, it converged rougly five percent wic is not surprising te annuitized value of a security paying off a dollar every period wit a.95 discount rate is 20 and te MPC is just te inverse of tis see Proposition. One intriguing analytical issue ere is te interaction between precautionary saving, wic drives te interest rate down, and te effect of te interest rate on self- 9

insurance a lower interest rate facilitates self-insurance. Tis tensions leads to some odd results in tis finite-orizon setting. Figure 4 sows te effect on te interest rate for various different parameters. In general, tis effect was relatively well-beaved. It gets smaller as time gets larger or te discount rate gets smaller. However, for te ig risk aversion, ig variance case te lower left panel, te effect is ambiguous. For low s, te cange in te interest rate gets smaller as time gets sorter. Wat is driving tis? Wen te risk aversion and/or variance is sufficiently large, te last period of te model, in wic tere is no possibility of self-insurance begins to loom very large. Since people cannot self insure in te last period, tey demand uge amounts of precautionary saving in te second to last period, wic puses te interest rate down drastically. Tis in turn facilitates consumption smooting for a relatively large number of periods before. Figure 5 sows tis effect for RRA=0 and =.96. Notice, on te plot tat te standard deviation of consumption on te vertical axis actually drops as te standard deviation of income increases, starting rougly at a income standard deviation of 2000. In te last period, te plot sows tat te standard deviation of consumption is exactly equal to te standard deviation of income. 6 Models wit permanent and temporary socks Te results above are not entirely satisfying. Casual empiricism says tat consumption smooting plays an important role in risk-saring so a random walk model misses someting. But a model wit only temporary socks is clearly unrealistic as well. In addition, two ways of making te model realistic, putting in lower discount rates and aggregate income growt, bot pus up te interest rate and weaken te results. In tis section, we sow tat a model wit bot permanent and temporary socks captures te ideal properties of bot models: te permanent socks are consumed and te temporary socks are smooted away. Tis idea is not new; permanent income models ave exploited tis for quite some time. Hall and Miskin982 test weter consumption reacts more to permanent or temporary socks. Blundell and Preston 998 use tis property tat consumption sould react to permanent and not temporary socks to attempt to quantify canges in inequality. Te innovation ere is to place tis in te context of an equilibrium model and to explore te risk-saring implications of various market structures. As before, te key to consumption smooting is low interest rates or ig prices of te riskless asset. Levine and Zame998 use te convexity of marginal utility to guarantees tat te price of te riskless asset will always be iger tan te discount 20

rate. Tis is a consequence of Jensen s inequality wic tells us tat te expectation of marginal utility will be greater tan te marginal utility of expected consumption. EMU C >MUE C Since individuals equate te ratio of te marginal utilities to ratio of te prices, given a price equal to te discount rate, individuals will always want expected consumption to be iger in te second period. So to maintain equilibrium wit a fixed endowment, te price of te riskless asset will always ave to be iger tan te discount rate. Because of te convenient functional form we ave assumed, we can say someting stronger. Wen x is normal E[expx] = exp Ex+ 2 var x so we can go furter tan Jensen s inequality and precisely measure te cange in te interest rate, to te extent tat it depends on te variance or te growt in endowment someting we ave already taken advantage of in te previous result. Wile typically, tis result will be weakened as te discount rate drops below one or growt increases, ere te variance of permanent socks will pus te price of te riskless rate back up and reduce te marginal propensity to consume out of temporary socks. We will assume tat individual income as a specific form wit separate permanent and temporary socks. Tis is common in te literature on consumption. See Hall and Miskin982, Carroll and Samwick995, Gourincas and Parker996 and Blundell and Preston 998 and Viceira998 for examples. Condition 7 Suppose individual incomes follow a process wit permanent and temporary socks y t = k t + y t + η + ε t ε t We introduce te concept of temporarily complete market, wic formalizes te idea from Cocrane and oters tat individual can edge sort term but not long term risks. Definition 3 An economy is temporarily complete if tere exist φ j ε t = J j= φ j x j for all. =,...,H j=,...,j suc tat In oter words, a market is temporarily complete if all temporary socks are completely edgeable. 2

Proposition 4 Under conditions, 2, 4, 5 and 7 old and wen exp A k + R 2 2H var η as T, te equilibrium of te incomplete markets economy will converge to te equilibrium of te same economy wit temporarily complete markets. Since all te equilibrium values are continuous in te exogenous parameters, tis will be approximately true for any economy in a neigborood of exp A k t+ k t + 2H R 2 var η. Notice tat because cov ε t,η = 0 we can introduce te new assets to temporarily complete markets witout affecting te edgeability of te permanent socks. Proof: We just need to sow tat marginal propensity to consume out of one s temporary socks goes to zero. Te marginal propensity to consume of out te permanent socks is. So te condition exp A k t+ k t + 2H R 2 var η implies tat π 0,t, wic implies tat te marginal propensity to consume a t = + T t s=0 Π t++s t+ 0asT. 6. Evidence from computations Figures 7 troug 9 sow te effects of persistence on welfare calculations at various different levels of. Te assumptions for tese computations are identical to tose made in Section 5.. Persistence, not surprisingly, overturns te effective market-incompleteness results. However, te degree to wic it does tis is quite non-linear. Figure 7 sows te welfare benefits of market completion for various different parameters. As persistence increases, te welfare benefits increase. Tis is not so surprising. However, wat is interesting, altoug not surprising, is tat te effects are very non-linear. Te slope of te surface increases dramatically at around ρ =.6. Figure 8 sows te same plot in contour format. Notice tat te welfare benefits of market completion go from rougly tree percent of consumption at ρ =.6, =.97 to twelve percent of consumption at ρ =.8, =.97. Tis is particularly intriguing since empirical researc suggests tat AR coefficients are in precisely tis region. For example, Heaton and Lucas996 estimate an AR wit PSID data and give and estimate of ρ =.53. Davis and Willen999 estimate AR coefficients for 8 sex-education groups. For men, tey find coefficient rougly between.5 and.7. Te presence of measurement error would bias tese coefficients down. Te computations sow tat even a relatively small bias could ave a significant effect on te risk-saring properties of self-insurance. 22

6.2 Analysis using parameter estimates from te literature Table 2 sows te welfare effects of incomplete markets using estimates of income processes from te literature. For a relative risk aversion coefficient of 3, calculated welfare effects are consistent across studies and small between one and a alf and tree percent. Estimates diverge for ig risk aversion levels. Te Heaton and Lucas996 estimates give a welfare loss of tree percent wit RRA equal to ten, were using Carroll and Samwick996 estimates, we get a welfare loss almost tree times te size. 6.3 Tests of consumption insurance We argued above in section 3 tat ignoring te general equilibrium effects of introducing new markets migt lead to mistaken inference. In tis section, we consider te effects of financial innovation on an economy wit bot permanent and temporary socks. We sow ere tat allowing agents to sare te permanent socks will, by raising te interest rate, reduce teir ability to use consumption smooting to smoot te temporary socks. Figure 0 illustrates tis problem using computations. Te graps sows te variance of te permanent sock on te x-axis and te variance of consumption on te y-axis. Te area above te forty-five degree line is te effect of te temporary socks all variance due to permanent socks, as we ave seen, is passed directly troug to consumption. For large enoug permanent socks, te effect of te temporary socks goes away. Wy? Because as te permanent socks grow, precautionary savings puses down te riskless interest rate facilitating consumption smooting. Te problem is in te oter direction. As we srink te permanent socks, we see tat te variance of consumption goes down, but by significantly less tan te variance of te permanent socks. In some cases, te slope of te consumption variance curve is close to zero. Tis implies tat reducing te variance of income for an individual will ave practically no effect on te variance of is consumption. Gertler and Gruber997 is illustrates te possibility for mistaken inference. Tey argue tat te absence of evident consumption risk-saring for long duration illnesses suggests large welfare gains from te introduction of formal disability insurance. Tis analysis sows tat one must also take into account te potential reduction in risk saring due to introduction of tese markets. 23

7 Conclusion Tere are two main points to tis researc. First, te exponential-normal model is remarkably easy to use even in te ric framework described ere. In considering a model wit any sort of ric eterogeneity more tan two individuals, and more tan two assets, tis is essentially te only model one can use. A useful direction for future researc would be to see ow badly tis model does in comparison wit computational models. All dynamic incomplete markets involve some sort of approximation typically tis is in te computational algoritm. Here te approximation is in te utility function. Second, tis paper argues tat wit reasonable specifications of te time-series processes for individual income, we can make strong predictions about te beavior of individual consumption. Specifically, we sow tat if socks are completely temporary, ten allocations will be close to complete markets. More generally, if individual socks are a combination of permanent and temporary socks, ten individual allocations will beave as if all temporary socks were tradeable. We expose a ric and intriguing interaction between te interest rate, consumption smooting and precautionary saving, wic suggests tat evaluating new risk-saring instruments can be more complex tan is traditionally tougt. Tis model also makes strong predictions about te beavior of individual consumption in a general equilibrium model. Specifically, it allows us to consider te introduction of a new risk-saring opportunities, taking into account all te general equilibrium effects, wic can be considerable. Davis and Willen999 sows tat we can easily calculate equilibria in suc a model te properties of individual endowments and assets can be estimated using conventional tecniques. However, use of tis model to do empirical work is ampered by te unrealistic features described above. 24

8 Appendix 8. Proof of proposition: Dynamic Optimization Lemma Assume tat asset prices are nonstocastic. Under conditions, 2, 4. Suppose tat c t+ and x j J j= are jointly normally distributed. Ten for any : E c t+ π j,t =E x j A cov c π t+, x j j =,..., J 3 0,t c t = A var c 2 t+ ln π 0,t 4 A Proof: Te intertemporal Euler equation gives us for any π 0,t u C c t = E u C c t+ π j,t u C c t = E u C c t+ xj j =,..., J By definition of covariance and Stein s lemma: π j,t u C c t = E u C c t+ E xj + E u CC c t+ cov c t+, x j j =,..., J Dividing troug by π 0,t u C c t = E u C A we get π j,t =E x j A cov c π t+, x j 0,t c t+, and noting tat E u CCec t+ E u Cec t+ = j =,..., J Because x j and c t+ are normal π 0,t exp A c t = exp A E c A 2 t+ + 2 8.2 Solution at T ln π 0,t = c A t E c A t+ + 2 var c ln t+ + A 8.2. Solution to te portfolio problem var c t+ Consumption in te last period is trivially normally distributed it is te sum of income and asset returns wic are normal. By Lemma, we can solve for te portfolio oldingsremembering tat E x j =0andvarx j = by assumption ωj,t = π j cov η A T π, x j 0 25

ω0,t = +π 0,T IT E ỹt πj,t ωj,t + A var η 2 T + J j= x jωj,t ln π 0,T A 8.2.2 Caracterization of consumption as a function of wealt Wat we want to do is to caracterize consumption as an affine function of wealt. By definition, consumption is c T = I T J j= π j,t ω j,t π 0,T ω 0,T We substitute in our optimal solution for te olding of te riskless asset and we get: J = IT π j,t ωj,t We reorganize and get Let b T = π 0,T j= +π 0,T IT Eỹ T J j= π j,t ωj,t + A V 2 T ln π 0,T A = I T + π 0,T ỹ E +π 0,T +π T 0,T t π 0,T J π j,t ωj,t π + A 0,T 2 V T A ln π 0,T +π 0,T j= π 0,T +π 0,T J π j,t j= π 0,T ωj,t + V T ln π 0,T A But since W T = I T + π 0,T E t ỹ T = I +π T + π 0,T E ỹt b T 0,T = +π 0,T W T b T Let a T = b T = +π 0,T π 0,T +π 0,T J π j,t j= +vara η T + J j= x jωj,t π 0,T ω j,t A ln π 0,T Ten c T = a T W T b T 5 26

8.3 Solution at T 2 8.3. Solution to te portfolio problem Consumption is normal because it is affine in wealt, wic is normal. So by lemma π j,t 2 =E x j A cov x j, c T π 0,T 2 From 5 c T = a T WT b T π j,t 2 =E x j A cov x j,a T W π T b T 0,T 2 = E x j a T A x j, ỹ T + J j= cov x jωj,t 2 +ω0,t 2 + π ỹ 0,T E T T ỹ ỹ But since E T T ET 2 T = ψ η T, tis becomes J = E x j a T A cov x j, η T + x j ωj,t 2 + π 0,T ψ η T So j= at A cov η T, x π j j π 0 ω j,t 2 = E[ x j] +π 0,T ψ a T A t var x j E[ xj ] π j π = 0 +π a T A 0,T ψ var x j Let Ψ T = +π 0,T ψ so tis becomes E[ xj ] π j π = 0 Ψ cov η T, x j a T A T var x j var x j cov η T, x j var x j To calculate te demand for te riskless asset, lemma gives us: A ln π 0,T 2 = c T 2 E a T WT b T + 2 A var c T π 0,T 2 ω 0,T 2 + a T ω 0,T 2 = I T 2 J π j,t 2 ωj,t 2 j= at E ỹ T +E π0,t ỹ T b T Solving, we get ω 0,T 2 = π 0,T 2 + a T + 2 A var c T A ln π 0,T 2 I T 2 J j= π j,t 2ω j,t 2 at EỹT +E π 0,T ỹ T bt + 2 A var c T ln π 0,T 2 A 27

8.3.2 Caracterization of consumption as a function of wealt Substituting in to get consumption c T 2 = I T 2 = I T 2 J π j,t 2 ωj,t 2 π 0,T 2ω 0,T 2 j= J π j,t 2 ωj,t 2 j= I T 2 J j= π 0,T 2 π j,t 2ω j,t 2 at EỹT +E π 0,T ỹ T bt + π 0,T 2 + a T 2 A var c T ln π 0,T 2 A Re-organizing, we get a T a T c T 2 = I T 2 + π 0,T 2 EỹT +E π 0,T ỹt π 0,T 2 + a T π 0,T 2 + a T π 0,T 2 J π j,t 2 π 0,T 2 + a T 2 A var c T a T ωj,t 2 π 0,T 2 A ln π 0,T 2 + b T Noticing tat We get c T 2 = Let j= I T 2 + π 0,T 2 EỹT +E π 0,T ỹ T = WT 2 a T π 0,T 2 + a T W T 2 π 0,T 2 π 0,T 2 + a T 2 A var c T a T J j= ln π 0,T 2 A a T a T 2 = = π 0,T 2 + a T + π 0,T 2 a T π 0,T 2 2 b T 2 = A var c T a J T j= π 0,T 2 + a T ln π 0,T 2 + b A T = a T 2π 0,T 2 2 A var c T a J T j= a T ln π 0,T 2 + b A T Terefore we can generalize a t = + π 0,t b t = a tπ 0,t a t+ a t+ = + T Π s t+ s=t+ 2 A var c t+ a J t+ j= ln π 0,t + b A t+ 28 + b T π j,t 2 π 0,T 2 ω j,t 2 π j,t 2 π 0,T 2 ω j,t 2 π j,t π 0,t ω j,t π j,t 2 π 0,T 2 ω j,t 2

We now determine te variance of consumption: var =var a t J ỹt + ω j,t x j + a t Ψ t j= η t T s=t+ J βj,t x j j= V t =var c t Π s t+ E ỹs + a ta J j= π j,t π 0,t x j =var a t W t = η t J j= β j,t x j is a residual and so by construction uncorrelated wit x j, so we can re-write tis as J =var a t Ψ t η t j= β j,t x j +var = R,t 2 var at Ψ t η t +var A J j= 8.4 Proof of proposition 2: Equilibrium Proof: By Lemma A J j= π j,t π 0,t x j π j,t x j π 0,t A ln π 0,t = c t E c A t+ + 2 var c ln t+ + A Since in equilibrium, c t = yt, we take averages and get Or letting = A ln π 0,t = y t Eỹ t+ + Π H = A H H = π 0,t = exp A y t Eỹ t+ + A 2 var c H t+ + = H = A 2 var c t+ ln A To sow tat tese are non-stocastic, remember tat by condition 5 y t Eỹ t+ is non-stocastic. We can see tat var c t+ is non-stocastic by backward induction: in period T, tere will be no consumption smooting, so π 0,T will be nonstocastic wic means tat var c T will be non-stocastic; terefore π0,t 2 will be non-stocastic and so on. By Lemma again, we know tat π j,t =E x j A cov c π t+, x j 0,t 29

Again, averaging, we get πj,t H E x j π 0,t = H A = cov c t+, x j = π j,t π 0,t =E x j A cov ỹ t+, x j To get equilibrium oldings of assets, simply substitute te equilibrium prices into te individual optimization from te earlier teorem. 8.5 Proof of welfare cange measure Proof: Te Euler equation is π 0,t exp A c t = E exp A c t+ wic implies tat τ Eexp A c A τ = A So total utility is t π 0,s exp A c 0 s= [ U x = ] t A Eexp A c t t T = + t π 0,s exp A c 0 t T s= So consider equilibria before and after * innovation. We want to coose θ suc tat U C + θ =U C. I.e. + t π 0,s exp A c 0 + θ = + t π0,s exp A c 0 t T s= t T s= Taking logs, we get c 0 + θ c 0 = + t A ln t T s= π 0,s + t t T s= π 0,s Since bot c 0 and c 0 are bot equilibrium allocation, if we take averages, tey must be equal, wic gives us θ = H H = + t t T s= ln π 0,s A + t t T s= π 0,s = A ln + t T t s= π 0,s + t T t s= π 0,s 30

Proposition 3: Consider two economies: i An economy as described above; ii an economy tat is te same except tat te assets are long-lived and ave dividend streams d j,t t=,...,t j=,...,j were te d j,t s follows ARMA processes wit innovations x j,t. Ten te equilibrium consumption allocations are te same in te two economies. Proof: Te basic idea is tat given te long-lived assets, one can syntesize te sort-lived assets and vice versa. Te proof is by recursion. In te equilibrium of economy ii at period T, te conditional payoff on asset j is d j,t wic equals E T d j,t + x j,t. So, an individual can syntesize te one-period asset wit payoff x j,t by buying asset j and sorting te riskless asset by E T d j,t. Te price for te syntesized asset must be te same in equilibrium ii as in equilibrium i by proposition 2. By arbitrage, te price of te long-lived asset must be P j,t = π 0,T E d j,t +π j,t. T Clearly, conditional on te wealt distribution, te two equilibria must be te same. In period T 2, te conditional payoff of te long-lived security in period T is d j,t + P j,t. By te fact tat d j,t is an ARMA process, P j,t is linear in x j,t so te payoff on te long lived asset is affine in x j,t. Tus, one can syntesize a one-period asset paying off x j,t. Alternatively, wit just te one period asset, one can syntesize te payoff on te long-lived asset. So again, te two equilibria must be te same. References [] Atanasoulis, Stefano and Robert Siller, 995. World Income Components: Measuring and Exploiting International Risk Saring Opportunities. NBER working paper 5095. [2] Attanasio, Orazio and Steven J. Davis, 996. Relative wage movements and te distribution of consumption. Journal of Political Economy,046, pp. 227-262. [3] Bewley, Truman, 980. Te permanent income ypotesis and long-run economic stability. Journal of Economic Teory, 22, 252-292. [4] Battacarya, Sudipto, 98. Notes on multiperiod valuation and te pricing of options. Journal of Finance, 36, 63-80. 3

[5] Blundell, Ricard and Ian Preston, 998. Consumption inequality and income uncertainty. Quarterly Journal of Economics, May, 603-640. [6] Brainard, William C. and F. Trenery Dolbear. Social risk and financial markets. American Economic Review, 6, 360-370. [7] Caballero, Ricardo, 990. Consumption puzzles and precautionary saving. Journal of Monetary Economics, 25, 3-36. [8] Calvet, Laurent, 997. Incomplete markets and volatility. Yale University. [9] Carroll, Cristoper and Andrew Samwick, 997. Te nature of precautionary wealt. Journal of Monetary Economics 40, 4-7. [0] Cristiano, Lawrence and Martin Eicenbaum, 990. Unit roots in real GNP: Do we know and do we care? Carnegie-Rocester Conference Series on Public Policy, 7-6. [] Cocrane, Jon, 99. A simple test of consumption insurance. JPE 995, 957-976. [2] Constantinides, George and Darrell Duffie, 996. Asset princing wit eterogenous consumers. JPE 04, 29-240. [3] Davis, Stepen and Paul Willen, 999. Using financial assets to edge labor income risks: Estimating te benefits. Working paper. [4] Davis, Stepen and Paul Willen, 998. Feasible consumption insurance using financial assets. Work in progress. [5] den Haan, Wouter, 996. Understanding equilibrium models wit a small and a large number of agents. NBER working paper 5792. [6] den Haan, Wouter, 997. Solving dynamic models wit aggregate socks and eterogenous agents. Macroeconomic Dynamics 2, 355-86. [7] Duffie, J. Darrell, Jon Geanakoplos, Andreu Mas-Collel and Andy McLennan. Stationary Markov equilibria. Econometrica, 624, 745-78. [8] Duffie, J. Darrell and Ci-fu Huang, 985. Implementing Arrow-Debreu equilibria by continuous trading of a few long-lived securities. Econometrica, 53, 337-356. 32

[9] Elul, Ronel, 997. Financial innovation, precautionary saving and te riskless interest rate. Journal of Matematical Economics 27, 3-3. [20] Geanakoplos, Jon, 990. An introduction to general equilibrium wit incomplete markets. Journal of Matematical Economics, 9, -38. [2] Geanakoplos, Jon, 995. Lecture notes on general equilibrium wit incomplete markets. Yale University. [22] Gourincas, Pierre-Olivier and Jonatan Parker, 996. Consumption over te life cycle. Working paper. [23] Hall, Robert and Frederic Miskin, 982. Te sensitivity of consumption to transitory income: Estimates from panel data on ouseolds. Econometrica, 502, 46-48. [24] Imrooroglu, Ayse, 989. Cost of business cycles wit indivisibilities and liquidity constraints. JPE, 976, 364-383. [25] Hansen, Gary and Ayse Imrooroglu, 992. Te Role of Unemployment Insurance in an Economy wit Liquidity Constraints and Moral Hazard. JPE, 00, 8-42. [26] Heaton, Jon and Debora J. Lucas, 996. Evaluating te effects of incomplete markets on risk saring and asset pricing. JPE, 043, 443-467. [27] Huang, Jennifer and Jiang Wang, 997. Market structure, security prices and informational efficiency. Macroeconomic Dynamics,, 69-205. [28] Judd, Kennet, Felix Kubler and Karl Scmedders, 998. Incomplete asset markets wit eterogenous tastes and idiosyncratic income. Working paper. [29] Kreps, David, 982 Multiperiod securities and te efficient allocation of risk: A comment on te Black-Scoles Option Pricing Model. In McCall, J.J., ed. Te Economics of Information and Uncertainty. Cicago: University of Cicago Press. [30] Krusell, Per and Antony Smit, 997a. Income and wealt eterogeneity, portfolio coice and equilibrium asset returns. Macroeconomic Dynamics 2, 387-422. [3] Krusell, Per and Antony Smit, 997b. Income and welat eterogenity in te macroeconomy. JPE 065, 867-96. 33

[32] Levine, David and William Zame, 998. Does market incompleteness matter? Working paper. [33] Lewis, Karen, 996. Wat can explain te apparent lack of international consumption risk saring? JPE, 042, 267-297. [34] Mace, Barbara. 99. Full insurance in te presence of aggregate uncertainty. JPE, 995 928-956. [35] Marcet, Albert and Kennet Singleton, 998. Equilibrium asset prices and savings of eterogenous agents in te presence of incomplete markets and portfolio constraints. Fortcoming in Macroeconomic Dynamics. [36] Sceinkman, Jose, 989. Market incompleteness and equilibrium, in Battacarya, Sudipto and George Constantinides, eds., Teory of Valuation: Frontiers of Modern Financial Teory, Volume, Totowa: Rowman and Littlefield. [37] Stapleton, R.C. and M.G. Subramanyam, 978. A multiperiod equilibirum asset pricing model. Econometrica, 465, 077-096. [38] Storesletten, Kjetil, Cris Telmer and Amir Yaron, 998. Asset pricing wit idiosyncratic risk and overlapping generations. CMU working paper. [39] Telmer, Cris, 993. Asset pricing puzzles and incomplete markets. Journal of Finance, 48, 803-832. [40] van Wincoop, Eric, 994. Welfare gains from international risksaring. Journal of Monetary Economics, 34, 75-200. [4] Willen, Paul S. 996. Mean-variance equilibrium in incomplete markets models wit restricted participation and undiversifiable risk: A note Yale University mimeo. [42] Willen, Paul S. 997. Estimation of te benefits of financial innovation using micro level data. Princeton University. 34

Study RRA a 3 0 Carroll and Samwick996 b 0.56672.742 8.4603 Hall and Miskin982 c 0.5648.7759.975 Heaton and Lucas996 d 0.49595.42 3.785 MaCurdy982 e 07 2.9495 f Table 2: Welfare losses due to incomplete markets as a percentage of consumption. a We consider different risk aversion and assume te following: =.96, T = 000, kt =0, E yt =20, 000 b Carroll and Samwick s estimated an equation of te form described in Condition 7 in logs. We take a log-linear approximation and use teir estimates. Tey estimate te variance of permanent socks to be 0.027 and te variance of transitory socks to be 0.0440. c As reported in Caballero990, Hall and Miskin estimate te income process again in logs to be y t = y t + η t 0.360η t 0.080η t 2 0.060η t 3 were σ 2 =0.04. d Heaton and Lucas estimate a first-order AR process in logs. Te AR coefficient is 0.529 and te standard deviation of te error is 0.25 e MaCurdy estimates te earnings process in logs to be: y t = y t + η t 0.4η t 0.06η t 2 were σ 2 =0.054. f Tis specification implied negative interest rates; te comparison wit oter series is no longer informative. 35

0.08 8 6 4 0.32 0.6 0.04 0.02 σ/i=.,rra=3 0.02 0.02 0.04 0.04 0.08 0.08 0.08 2 Delta 8 0.6 0.6 0.6 6 4 2 200 400 600 800 000 200 400 Number of periods 8 6.6 3.2 0.4 0.2 σ/i=.3,rra=3 0.2 0.2 0.4 0.4 0.4 4 2 Delta 8 6 4.6.6.6.6 2 200 400 600 800 000 200 400 Number of periods Figure : IID socks: Welfare gain from completing markets as a percentage of consumption. Plot sows iso-welfare curves. Wen socks are IID over time, te welfare losses from incomplete markets are minimal. In te second panel, te standard deviation of te sock is 30 percent of income, yet even wit a low discount rate.92 and a sort time orizon 20 periods, te welfare loss is still less tan one percent of consumption. wfig.eps 36

0.2 5 0.064 0.032 0.06 σ/i=.,rra= 0.008 0.06 5.28 0.64 0.32 σ/i=.,rra=0 0.6 0.6 0.08 0.032 0.32 5 0.28 0.064 5 0.64 0.64 37 50 00 50 200 250 300 350 Number of periods σ/i=.3,rra= 0. 0. 50 00 50 200 250 300 350 Number of periods σ/i=.3,rra=0 6.4.6 3.2.6 5 0.2 5.6 0.4 0.4 5 5 25.6 2.8 3.2 3.2.6 6.4 50 00 50 200 250 300 350 Number of periods 50 00 50 200 250 300 350 Number of periods Figure 2: IID socks: Welfare gain from completing markets as a percentage of consumption. Plot sows iso-welfare curves. Wit iger risk aversion two rigt-and plots, te effects of incomplete markets are somewat more significant. For sorttime orizons, it can be quite large. Even for relatively long orizons, te effect is still significant. For a discount rate of.9, te welfare loss converges to about tree percent. wfig2.eps

0.52 0.64 0.32 0.64 0.32 0.08 0.02 0.04 0.6 5 0.28 0.064 σ/i=.,rra= 0.032 0.064 5 0.6 0.08 σ/i=.,rra=0 0.04 0.02 0.08 0.28 5 5 0.6 0.6 38 5 0 20 30 40 50 60 Number of periods σ/i=.3,rra= 0.08 0.04 0.02 5 0 20 30 40 50 60 Number of periods σ/i=.3,rra=0 0.0 0.02 0.0 0.04 0.32 0.6 0.08 0.02 0.04 0.08 5 0.6 5 0.08 0 20 30 40 50 60 Number of periods 0 20 30 40 50 60 Number of periods Figure 3: IID socks: Standard deviation of consumption divided by standard deviation of income. Plot sows iso-variance curves. Value is variance N periods before te end of time. In general, as te number of period gets larger and gets iger, te variance gets smaller. In te lower rigt panel te ig variance, ig risk-aversion case, as N gets larger, te variance gets larger as well. Tis is a curious finite-orizon effect wic is more clearly illustrated in Figure 5. wfig2b.eps

σ/i=.,rra= σ/i=.,rra=0 0.006 0.006 0.0032 0.0064 0.000 5 5 0.0032 0.0004 0.0002 0.000 0.0064 5 5 0.0008 0.0002 39 5 0 5 20 25 30 Number of periods σ/i=.3,rra= 5 0 5 20 25 30 Number of periods σ/i=.3,rra=0 0.0006 0.0007 0.0005 0.0006 0.00 0.008 0.004 0.00 5 5 0.00 0.002 0.032 0.002 0.0007 5 0.00 5 0.002 0.002 0.06 0.00 0.002 5 0 5 20 25 30 Number of periods 5 0 5 20 25 30 Number of periods Figure 4: IID socks: Difference in interest rates from complete markets. Plot sows iso-difference curves. Te presence of uninsurable risk always drives te interest rate down. In te last period, tis effect is dramatic because no self-insurance is possible and tis actually moderates te effect of market incompleteness in earlier periods. wfig2c.eps

6000 5000 40 std. dev. of consumption 4000 3000 2000 000 0 0 20 Number of periods 40 0 000 2000 3000 4000 5000 6000 std. dev. of income Figure 5: IID socks: Variance of consumption for differing variances of income for RRA=0, =.96. Tis plot sows near te end of time, te variance of consumption may become a decreasing function of te variance of income. Precautionary saving drives tis seeming paradox. In te last period of te model, no self-insurance is possible; agents demand large amounts of precautionary savings pusing te interest rate so far down tat consumption smooting actually becomes easier in earlier periods. Note tat tis effect persists for a considerable number of periods. wfigsta.eps

σ/i=.,rra= σ/i=.,rra=0 4 5 5 5 5 0.005 0.0 4e 009.6e 007 6.4e 006.6e 007 0.000256 4e 009 50 00 50 200 250 300 350 Number of periods σ/i=.3,rra= 0.005 0.02 0.04 0.08 0.0 0.6 0.005 0.02 0.04 0.08 50 00 50 200 250 300 350 Number of periods 5 5 5 5 4e 009.6e 007 6.4e 006 0.000256 4e 009.6e 007 50 00 50 200 250 300 350 Number of periods σ/i=.3,rra=0 0.004 0.00 0.002 0.008 0.06 0.00 0.002 0.004 0.032 0.06 0.064 0.008 50 00 50 200 250 300 350 Number of periods Figure 6: Te probability of negative consumption. Lines are iso-probability contours. Even wit low risk aversion and a ig variance of income, wit =.95, te probability of negative consumption does not exceed one percent even at a one undred year time orizon. wfig8d.eps

σ/i=.,rra=3 0 % of consumption 8 6 4 2 0 5 5 0 0.2 0.4 ρ 0.6 σ/i=.3,rra=3 80 % of consumption 60 40 20 0 5 5 0 0.2 0.4 ρ 0.6 Figure 7: Welfare gain from completing markets as a percentage of consumption, for socks of differing persistence. ρ is te coefficient 42 on a first-order AR-process is te discount factor. As persistence of socks increases, te benefits of completion grow non-linearly. At ρ =.5, te benefits are minimal, but at ρ =.8 tey ave become quite significant. wfig5b.eps

02.4 σ/i=.,rra=3 8 6 4 0.08 0.6 0.32 0.64.28 2.56 0.24 2 0.6 8 6 0.32 4 2 0.64.28 5.2 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 ρ 8 σ/i=.3,rra=3 6.4 6 4 2 8.6.6 3.2 25.6 5.2 6 4 6.4 2 3.2 2.8 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 ρ Figure 8: Welfare gain from completing markets as a percentage of consumption, for socks of differing persistence. Plots sow iso-welfare curves. ρ is te coefficient on a first-order AR-process. is te discount factor.wfig5.eps 43

6.4 5 0.04 σ=.,rra= 0.08 0.6 0.64 5.2 5 0.2 0.4 σ=.,rra=0.6 3.2 6.4.28 44 5 0 0.2 0.4 0.6 ρ σ=.3,rra= 5 0.2 0.08 0.6 0.32.6 3.2 6.4 2.8 5.6 0 0.2 0.4 0.6 ρ σ=.3,rra=0 5 6.4 3.2 6.4 2.8 3.2 25.6 5.2 02.4 25.6 0.4 3.2 5 5.6 0 0.2 0.4 0.6 ρ 3.2 6.4 2.8 0 0.2 0.4 0.6 ρ 25.6 Figure 9: Welfare gain from completing markets as a percentage of consumption, for socks of differing persistence. Plots sow iso-welfare curves. ρ is te coefficient on a first-order AR-process. is te discount factor. wfig6.eps

2500 RRA=3 σ=. 2000 RRA=0 σ=. std. dev. of consumption 2000 500 000 500 std. dev. of consumption 500 000 500 45 2500 0 0 500 000 500 2000 std. dev. of permanent sock RRA=3 σ=.3 2000 0 0 500 000 500 2000 std. dev. of permanent sock RRA=0 σ=.3 std. dev. of consumption 2000 500 000 500 std. dev. of consumption 500 000 500 0 0 500 000 500 2000 std. dev. of permanent sock 0 0 500 000 500 2000 std. dev. of permanent sock Figure 0: Variance of consumption as a function of variance of permanent sock. Different lines represent different rates of time-preference from =.8 at te top-left to = lower-rigt. As te variance of te permanent sock goes down, te variance of consumption goes down, but te interest rate goes up, tus reducing one s ability to smoot, counteracting te effect of te reduced variance. All te variance above te 45 degree line is due to temporary socks. wfig7v.eps