Lqudty and Asset Prcng (Prof. Pedersen) Short Sale Constrants Due to Lmted Commtment Stjn Van Neuwerburgh March 7, 2005 I. Introducton In the frst module of ths class we wll study asset prcng mplcatons of recursve contract theory. I wll start by revewng asset prcng n complete markets. Chapter 8 n Ljungqvst and Sargent (2004) provdes background readng. The man topc of ths module s asset prcng n an envronment wth lmted commtment. As n the complete markets envronment, agents can stll trade a complete set of contngent clams. However, we assume that they can walk away from ther debts. If they do so, they are excluded from tradng forever. 1 The nablty of agents to commt leads to endogenous restrctons on tradng. We wll start by characterzng Pareto-effcent allocatons. Then, we wll study the Kehoe and Levne (1993) decentralzaton, where all trade takes place at tme zero. Fnally we wll study a decentralzaton wth sequental trade due to Alvarez and Jermann (2000). Complete markets models mply perfect rsk-sharng: An agent s ndvdual consumpton growth does not depend on ts ndvdual ncome growth, only on aggregate consumpton growth. However, both consumpton data and asset prcng data reveal that households are unable to trade away all of ther dosyncratc rsk. 2 The lmted commtment model reproduces ths NYU Stern Fnance. svneuwe@stern.nyu.edu. 44 West Fourth Street, Tsch 9-14. I thank Hanno Lustg for provdng me wth hs teachng materals. 1 Ths s the standard punshment n the lterature, see Kehoe and Levne (1993), Kehoe and Perr (2002), Krueger and Perr (2003), etc. Lustg (2003) and Lustg and VanNeuwerburgh (2004b) propose a dfferent outsde opton, where agents retan access to credt markets but loose all collateral assets. 2 Papers fnd evdence at the household level (e.g. Cochrane (1991), Mace (1991), Nelson (1994),Krueger (2000), Blundell, Pstaferr and Preston (2002)), at the regonal level (e.g. Hess and Shn (1998) and Lustg and VanNeuwerburgh (2004a)), and at the nternatonal level (e.g. Backus, Kehoe and Kydland (1992)). 1
feature of the data. It brngs model and data closer together, both wth respect to quanttes and prces. II. Envronment A. Preferences and Endowments In each perod there s a realzaton of an event s t. s t = {s 0, s 1,, s t }. The hstory of events s denoted The condtonal probablty that a partcular event s t s realzed s denoted π t (s t s 0 ). There are I agents, = 1... I each agents owns a clam to a stochastc endowment y t(s t ) household purchase a consumpton plan { c t(s t ) } households rank consumpton streams accordng to U(c ) = β t ( u c t (s t ) ) π t (s t s 0 ) t=0 Inada condton: Assume lm c 0 u () = hstory: endowments possbly depend on the entre hstory of shocks s t s t III. Full Commtment In ths frst part, we assume that agents can fully commt to honorng ther promses. We start by characterzng Pareto-effcent allocatons. The frst and second welfare theorem allow us to make the connecton between the Pareto-effcent allocatons and the Arrow-Debreu equlbra. In the AD equlbra, agents trade a complete set of clams to consumpton whose delvery s contngent on a partcular realzaton of the state of the world. Completeness means that consumpton clams can be purchased that are contngent on any realzaton of the state of the world. One of the man results s that, n the case of addtve utlty, these Pareto-effcent allocatons do not depend on the hstory of the economy. There s an alternatve way of decentralzng 2
Pareto-effcent allocatons by allowng agents to trade Arrow securtes sequentally. decentralzaton, markets re-open each perod, and agents re-trade every perod. In ths Defnton 1. A feasble allocaton c satsfes c t(s t ) yt(s t ), for all t, s t A. Pareto Problem A benevolent planner produces a vector of weghts λ, one for each agent, to maxmze a weghted sum of utltes W = λ U(c ) subject to the feasblty condtons c t(s t ) y t(s t ), t, s t. Defnton 2. A feasble allocaton s effcent f t solves the planner problem for a strctly postve vector of weghts λ. B. Lagrangan We defne a Lagrangan L L = λ U(c ) + [ θ t (s t ) yt(s t ) t=0 s t c t(s t ) ] where θ t (s t ) s the Lagrange multpler resource constrant n node s t. Ths s a standard constraned optmzaton problem. The saddle pont problem conssts of maxmzng the value of the objectve functon w.r.t. c and mnmzng t w.r.t. θ: Max c Mn θ L The frst order condtons are necessary and suffcent. We derve the frst order condton for consumpton for agent n node s t : λ β t u ( c t(s t ) ) π t (s t s 0 ) = θ t (s t ). (1) 3
Ths mples that the rato of frst order condtons for two agents and j wth the same hstory s t s: u ( c t(s t ) ) ) = u (c λj j t (st ) λ (2) Also note that the complementary slackness condtons need to be satsfed: [ θ t (s t ) yt(s t ) c t(s t ) ] = 0, s t. Proposton 3. An effcent allocaton s a functon of the realzed aggregate endowment only. It depends nether on the specfc hstory s t, nor on the realzatons of the ndvdual endowments. ( Proof. To see ths, then equaton (2) for j = 1 mples that c t(s t ) = u 1 λ 1 u ( c 1 λ t (s t ) )). Because of feasblty and non-sataton, c t(s t ) = yt(s t ). Substtutng the expresson for c t(s t ) n the resource constrant, we see that c 1 t (s t ) only depends on the aggregate endowment n the current perod, yt(s t ). Because the weghts λ are constant, t follows that all agents consumpton only depends on the aggregate endowment n the current perod. In partcular, consumpton allocatons do not depend on the ndvdual ncome realzatons. They are not hstory dependent, n the sense that any other state s t that results n the same aggregate endowment gves rse to the same allocaton. C. Power Utlty Consder the smplest case of power utlty wth coeffcent of rsk averson γ, u(c) = c1 γ 1 γ. Verfy that the followng rsk sharng rule satsfes (1) the condton on the rato of margnal utltes and (2) feasblty: c t(s t ) = j λ 1/γ λ 1/γ j y j t (st ). (3) j In partcular, ths rule mples perfect correlaton between ndvdual consumpton and aggregate endowment! The consumpton share of agent, as a share of the aggregate, s fully pnned down by the ts ntal Pareto-weght λ, relatve to the 1/γ moment of the cross-sectonal dstrbuton of weghts. An mportant mplcaton s that changes n the consumpton share of an agent do not respond to changes n the ndvdual ncome share. Ths s the nature of emprcal tests n 4
the rsk-sharng lterature. Shadow Prces How does the planner value a unt of consumpton n dfferent states of the world? Well, that nformaton s embedded n { θ t(s t ) }. The Lagrangan multpler on the resource constrant s: ( θ t (s t ) = β t π t (s t j s 0 ) yj t (st ) j λ1/γ j The planner s wllng to trade unts of consumpton n s t for unts of consumpton at tme 0 at the rato: q 0 t (s t ) θ t(s t ) θ 0 (s 0 ) = βt π t (s t s 0 ) ( y j ) γ t (st ) y j 0 (s0 ) We defne q 0 t to be the tme-zero shadow prce of a unt of consumpton to be delvered at tme t n node s t. The consumpton trade-off between perods t 1 and t s analogous. The frst order condton for agent s consumpton (1) mples that the planner values resources n state s t (n unts of s t 1 consumpton) as follows: ) γ λ βu ( c t(s t ) ) λ u ( c t (st 1 ) )π t(s t s t 1 ) = θ t(s t ) θ t 1 (s t 1 ) qt 1 t (s t ). Usng the rsk sharng rule n (3), ths mples that, for any household : ( ) γ q t t 1 (s t ) = βπ t (s t j s t 1 ) yj t (st ). j yj t 1 (st 1 ) The shadow prce of aggregate consumpton q t t 1 ncreases n those states of the world n whch aggregate endowment growth between t 1 and t s low. Because allocatons are not hstory dependent, nether are shadow prces. D. Decentralzaton wth Tme Zero Tradng If we let agents trade clams to consumpton contngent on all states of the world s t, the equlbra that result are Pareto-effcent. Hence, ths means these equlbrum allocatons wll not feature any hstory dependence. Here s the setup. Households trade hstory-contngent clams to consumpton at tme 0 after s 0 has been realzed. Superscrpts refer to the dates at whch trades 5
occur, subscrpts refer to the dates at whch delveres are to be made. At tme t = 0 households can exchange clams on tme t consumpton at prces qt 0 (s t ). Households face a sngle budget constrant: Attach a multpler µ household s: qt 0 (s t )c t(s t ) qt 0 (s t )yt(s t ) s t t=0 s t t=0 to the household s budget constrant. The frst order condton for β t π(s t s 0 )u ( c t(s t ) ) = µ qt 0 (s t ). Defnton 4. A prce system s a sequence of functons { qt 0 (s t ) } and an allocaton s a lst of sequences of functons { c t(s t ) }, one for each. Defnton 5. A compettve AD equlbrum s a feasble allocaton and a prce system such that gven prces the household problem s solved for each and the markets clear n each s t. The frst order condton mples that the rato of margnal utltes satsfes: u ( c t(s t ) ) ) = u (c µ j t (st ) µ j, st. Note that f we choose µ = ( λ ) 1, ths s the same condton as the one that characterzed Pareto-effcency. Furthermore q t 0 = qt 0. Remark 6. A compettve AD equlbrum s a partcular Pareto-effcent allocaton, namely one that sets µ = ( λ ) 1 for all. Stochastc Dscount Factor as: The stochastc dscount factor or state prce deflator s defned m t (s t ) qt t 1 (s t ) π t (s t s t 1 ) u ( c = β t(s t ) ) u ( c t 1 (st 1 ) ), ( ) γ j = β yj t (st ). (4) j yj t 1 (st 1 ) Ths SDF s the one-perod ahead prcng kernel qt t 1, scaled by the correspondng state transton probablty. In complete markets, t s the ntertemporal margnal rate of substtuton of the 6
representatve household. Because of our rsk-sharng rule, t s a functon of the tme dscount factor β and the current growth rate of the aggregate endowment. In partcular, the SDF s not a functon of the hstory of aggregate endowment realzatons. An mportant property of the SDF s that t prces all clams n the economy. Consder an asset that pays a stream of dvdends {d t (s t )}. The prce at tme t 1 of that clam s p t 1 (s t ) = s t q t 1 t (p t (s t ) + d t (s t )) Defnng the one perod return as R t (s t ) p t(s t )+d t (s t ) p t 1, the prevous equaton mples that (s t 1 ) E t 1 [m t R t ] = 1 must hold. Ths s a no-arbtrage condton. If t ddn t hold, agents n the economy could make unbounded profts by reconstructng the dvdend sequence {d t (s t )} from the prmtve Arrow-Debreu securtes. Uncondtonal versons of ths no-arbtrage condton form the bass of emprcal tests n the asset prcng lterature. See chapters 8 and 13 n Ljungqvst and Sargent (2004) and Duffe (2001) for more on the propertes of stochastc dscount factors. E. Decentralzaton wth Sequental Tradng There s another way of decentralzng Pareto-effcent allocatons by allowng markets to re-open n each perod. All we need s one-perod state contngent clams. In ths sequental tradng envronment, we have to resort to borrowng constrants to keep agents from runnng Ponz schemes. We chose to use natural borrowng constrants. These constrants requre the debt of a household to be smaller than what t could pay back f ts consumpton were zero from that perod onwards n all future states. If we mpose an Inada condton on the utlty functon, these borrowng constrants wll never bnd, because the margnal utlty of consumpton explodes as consumpton tends to zero! These natural borrowng constrants are the weakest possble debt lmts that suffce to mplement the AD equlbrum allocatons wth sequental tradng. Denote household net wealth condtonal on hstory s t by Υ t(s t ) = qτ t (s τ [ ) c τ (s τ ) yτ (s τ ) ], t=τ s τ s t where {q t τ } are prces that obtan when markets are re-opened at tme t. Ths s the value of all current and future net clams of household. The feasblty constrant at equalty mples that: Υ t(s t ) = 0 7
Debt Lmts We need some debt lmts n sequental tradng to prevent Ponz schemes. The natural debt lmt A t(s t ) = qτ t (s τ )yτ (s τ ) τ=t s τ s t s the value of clam to household s endowment; t s the maxmal amount that agent can repay. At tme t 1 household cannot promse to repay more n any state of the world tomorrow than the value of ts labor ncome stream A t(s t ). Household Problem Markets re-open each perod. Let Q t (s t+1 s t ) be the prce of one unt of consumpton delvered contngent on the realzaton of s t+1 after hstory s t. At tme t after hstory s t, household can purchase a full set of Arrow securtes, whose quanttes are denoted by a t+1 (s t+1, s t ). At tme t, the household chooses { c t(s t ), a t+1 (s t+1, s t ) } to maxmze expected lfetme utlty. The household faces a sequence of budget constrants: c t(s t ) + Q t (s t+1 s t )a t+1(s t+1, s t ) yt(s t ) + a t(s t ), s t+1 and a state-by-state borrowng constrant n node s t : a t+1(s t+1, s t ) A t+1(s t+1 ), s t+1. Because of the Inada condton, the natural debt lmt wll not be bndng. In the absence of bndng debt constrants, the frst order condton mples βu (c t+1 (st+1 )) u (c π(s t+1 s t ) = Q t (s t+1 s t ). (5) t (st )) Defnton 7. A dstrbuton of wealth s a vector a t (s t ) = { a t(s t ) } I =1, satsfyng a t(s t ) = 0. Defnton 8. A sequental tradng compettve equlbrum s an ntal dstrbuton of wealth a 0 (s 0 ), an allocaton { c t(s t ) } I =1 and prcng kernels Q t+1(s t+1 s t ) such that c solves the household problem, and { c t(s t ), a t+1 (s t+1, s t ) } I satsfy for all st =1 c t(s t ) = yt(s t ) and a t(s t+1, s t ) = 0, s t+1. 8
F. Equvalence of Allocatons The AD equlbrum and the sequental equlbrum are equvalent. To show ths, guess that for gven AD prces { q 0 t (s t ) }, we can recover Q t (s t+1 s t ) from the recurson q 0 t+1(s t+1 ) = q 0 t (s t )Q t (s t+1 s t ) If the prcng kernel satsfes ths recurson, the frst order condton for the AD problem βu (c t+1 (st+1 )) u (c t (st )) π(s t+1 s t ) = q0 t+1 (st+1 ) q 0 t (st ) = Q t (s t+1 s t ) concdes wth the frst order condtons for the sequental problem. Furthermore, by teratng forward on the sequental budget constrant and mposng a transversalty condton (see Sargent (1984), chapter 8), we can match up the wealth of agent n the sequental and n the AD economes: a t(s t ) = Υ t(s t ) We conjecture that the ntal wealth vector should be the null vector. Ljungqvst and Sargent (2004) show that ths portfolo strategy s affordable and allows the fnancng of the AD equlbrum level of consumpton. Furthermore, the household cannot ncrease consumpton beyond ths by lowerng a component of the asset portfolo, lest t jeopardzes beng able to fnance the AD consumpton n every state of the world tomorrow. Recursve Compettve Equlbrum structure on the transton probabltes π: A specal case arses when we mpose a Markovan π t (s t s 0 ) = π(s t s t 1 )π(s t 1 s t 2 )... π(s 1 s 0 ) and assume that the ndvdual endowment y t(s t ) s only a functon of the current state s t. Under these condtons equlbrum allocatons and prces are only a functon of the current state only: c t(s t ) = c (s t ) and Q t (s t+1 s t ) = Q(s t+1 s t ). 9
IV. Lmted Commtment We now relax the assumpton that agents can commt to honorng ther promses. In order to nduce contnued adherence to the contract, we requre that the consumpton allocaton satsfes a seres of partcpaton constrants: where U aut (s t ) = U(c )(s t ) U aut (s t ), s t,, β t ( u y τ (s τ ) ) π τ (s τ s 0 ) t=τ s τ s t The outsde opton s to revert permanently nto autarchy. Defnton 9. A feasble allocaton c satsfes c t(s t ) yt(s t ), for all t, s t and U(c )(s t ) U aut (s t ) n all s t, for all Theorem 10. A necessary and suffcent condton for perfect rsk sharng s that A. Pareto Problem U ( 1 I ) yt(s t ) U aut (s t ), s t,. A benevolent planner produces a vector of weghts λ, one for each agent, to maxmze W = λ U(c ), subject to the feasblty condtons c t(s t ) y t(s t ), t, s t, U(c )(s t ) U aut (s t ), s t,. 10
B. Lagrangan We defne a Lagrangan L L = λ U(c ) + µ t(s t ) β t ( u c τ (s τ ) ) π τ (s τ s t ) U aut (s t ) t=0 s t τ=t s τ s [ t + θ t (s t ) yt(s t ) ] c t(s t ) t=0 s t where µ t(s t ) s the multpler on the partcpaton constrant n node s t and θ t (s t ) s the multpler on the resource constrant. Ths s a standard optmzaton problem, except that we have an nfnte number of constrants. The saddle pont problem conssts of maxmzng the value of the objectve functon w.r.t. c and mnmzng t w.r.t. µ and θ: Max c Mn µ,θ L Marcet and Marmon (1999) noted that we can defne cumulatve multplers that make the problem recursve : ξ t(s t ) = λ + t τ=0 s τ s 0 µ τ (s τ ) Ths s a recursve formulaton, because t mples that : ξ t(s t ) = ξ t 1(s t 1 ) + µ t(s t ), ξ 0(s 0 ) = λ, The sequence { ξ t(s t ) } s a non-decreasng stochastc process. Ths follows from the nonnegatvty of µ t(s t ). Usng the recursve formulaton of ξ and Abel s partal summaton formula, we can restate the Lagrangan as: L = {[ ξ t(s t )β t ( u c t (s t ) ) ] π t (s t s 0 ) + µ t(s t ) t=0 s t t=0 s [ t + θ t (s t ) yt(s t ) ] c t(s t ). t=0 s t [ U aut (s t ) ]} The frst order condtons are necessary and suffcent. We derve the frst order condton 11
for consumpton for agent n node s t : ξ t(s t )β t u ( c t(s t ) ) π t (s t s 0 ) = θ t (s t ) (6) The rato of frst order condtons for two agents and j wth the same hstory s t s: u ( c t(s t ) ) ) = u (c ξj t (st ) j t (st ) ξ t(s t ) (7) As always, the complementary slackness condtons need to be satsfed: C. Power utlty µ t(s t ) β t ( u c τ (s τ ) ) π τ (s τ s t ) U aut (s t ) = 0 for all s t, all t=τ s τ s t [ θ t (s t ) yt(s t ) c t(s t ) ] = 0 for all s t As n the case wth full commtment, Pareto-effcent allocatons take on an elegant form wth power utlty. We conjecture the followng rsk sharng rule: c t(s t ) = ξ t(s t ) 1/γ j ξj t (st ) 1/γ y j t (st ) (8) It s easy to verfy that ths rsk sharng rule satsfes (7) and market clearng (by constructon). Proposton 11. Amnesa property: A household s consumpton share decreases as long as t does not swtch to a state wth a bndng constrant, but when t does, ts consumpton share ncreases to some cutoff level that does not depend on the hstory (s t ) f the endowment process s frst-order Markov. Proof. The frst part follows from the rsk sharng rule and the fact that { ξ t(s t ) } s a nondecreasng process for all. The second part follows from the complementary slackness condton, whch says that, when the constrant bnds: β t ( u c τ (s τ ) ) π τ (s τ s 0 ) U aut (s t ) = 0 t=τ s τ s t 12 j
Now, f y s frst-order Markov, then U aut (s t ) only depends on s t. Ths mples c t(s t cannot depend on s t, only on s t. Hstory dependence and tme-varyng Pareto-Negsh weghts are sgnature of lmted enforcement (and prvate nformaton) problems. Shadow Prces The frst order condton for agent s consumpton (6) mples that the planner values resources n state s t (n unts of s t 1 consumpton) as follows: ξ t(s t )βu ( c t(s t ) ) ξ t 1(s t 1 )u ( c t 1 (st 1 ) )π t(s t s t 1 ) = θ t(s t ) θ t 1 (s t 1 ) qt 1 t (s t ). Usng the rsk sharng rule n (8), ths mples that, for any household : q t 1 t (s t ) = βπ t (s t s t 1 ) ( j yj t (st ) j yj t 1 (st 1 ) ) γ ( ) j ξj t (st ) 1/γ γ j ξj t 1 (st 1 ) 1/γ The shadow prce of aggregate consumpton q t t 1 (s t ) ncreases n those states of the world n whch lots of agents are severely constraned, because n that case the aggregate weght shock g, g t (s t s t 1 ) ( j ξj t (st ) 1/γ j ξj t 1 (st 1 ) 1/γ ), would be large. In partcular, g t (s t s t 1 ) > 1. Note that f no agent s constraned, g t (s t s t 1 ) = 1, and the last part smply drops out. D. Smple Recursve Characterzaton Actually solvng that saddle pont problem s computatonally challengng. Instead we can try to use what we know about constraned effcent allocatons to ease the burden of computng these allocatons. We use consumpton weghts as state varables nstead of cumulatve multplers because we want statonary state varables. Defne an I 1 1 vector of consumpton shares ω where the th element s: ω t 1 = j c t 1 y j t 1, for = 2,..., I. The consumpton share of the frst household s just the resdual 1 1 ω t 1. A natural choce for the state varables s ( ω,s). In the smplest case of two agents, we would keep track only of (ω 1, s). 13
Cutoff Rule for Consumpton At the start of next perod, we compare the agent s consumpton weght n the prevous perod ω t 1 to the cutoff value for the current state of the world: ω ( ω, s). If the weght exceeds the cutoff weght, the consumpton weght ω t 1 s left unchanged and the agent s consumpton weght n t s: ω t = ω t 1 If the consumpton weght ω t 1 s smaller than the cutoff weght, the agent s consumpton weght n t s: Actual consumpton s gven by: ω t = ω ( ω, s) c t = ω t j ωj t for each agent. The cutoff rule s determned such that the constrant bnds exactly j y j t U(c )(s t ) = U aut (s t ) when the consumpton weght equals the cutoff weght ω t = ω t. At the end of each perod we store the vector ω, whch contans the consumpton shares ω t ω t These consumpton weghts ω are exactly lke the ξ 1/γ, but they are rescaled to make sure they sum to one at the end of each perod. The optmalty of ths cutoff rule follows mmedately from the frst order condtons n the saddle pont problem of the planner. Two Agent Example We consder the smplest example wth two agents, two states and no aggregate uncertanty. Suppose there are two ndvdual ncome states (y lo, y h ). In the frst state agent 1 draws a low endowment, n the second state agent 1 draws a hgh endowment. The endowments sum to one n each state. Suppose the events y are..d., then the only state varable s the consumpton share of the frst agent. How do we solve for the constraned effcent allocatons? 14
Frst, solve for the value of autarchy: U aut 1 (y lo ) = u(y lo ) + β y π(y )U aut 1 (y ), U aut 1 (y h ) = u(y h ) + β y π(y )U aut 1 (y ). Usng vector notaton, the value of autarchy n each state y can be recovered from the followng equaton: U aut 1 = u(y)(i βπ) 1, where I s the dentty matrx. For the second household, we solve a smlar equaton: U aut 2 = u(1 y)(i βπ) 1 Remark 12. Necessary and suffcent condton for perfect rsk sharng: U(1/2) U aut 1 (y lo ) and U(1/2) U aut 1 (y h ) Remark 13. If the labor ncome process s too persstent or of agents are too mpatent, perfect rsk sharng s not feasble. As β 0 the nequalty u(y) + β y π(y y)u aut 1 (y ) < u(1/2) + β y π(y y)u(1/2) cannot be satsfed because u(1/2) < u(h). Smlarly, as the persstence ncreases, π(h h) 1, U aut 1 (h) u(h)/(1 β) > u(1/2)/(1 β). Second, we solve for the four cutoff values (ω 1 (y lo ), ω 1 (y h ), ω 1 (y lo ), ω 1 (y h )) 15
from the followng four non-lnear equatons: U 1 (ω 1 (y)) = u(ω 1 (y)) + β y π(y )U 1 (ω 1) = U aut 1 (y), y (y lo, y h ) U 2 (ω 1 (y)) = u(1 ω 1 (y)) + β y π(y )U 2 (ω 1) = U aut 2 (y), y (y lo, y h ), where ω 1 n the next perod s found by applyng the followng rule, the analogue of the more general rule we descrbed earler: f ω 1 (y) < ω 1 < ω 1 (y), ω 1 = ω 1 f ω 1 (y) > ω 1, ω 1 = ω 1 (y) f ω 1 (y) < ω 1, ω 1 = ω 1 (y) Perfect rsk sharng s when both ntervals [ω 1 (y lo ), ω 1 (y lo )] and [ω 1 (y h ), ω 1 (y h )] contan a consumpton share of 1/2. Shadow Prces Go back to the expresson for the shadow prces n ths economy, and recall that the aggregate endowment s normalzed to 1: q t 1 t (s t ) = θ t (s t ) θ t 1 (s t 1 ) = βπ t(s t s t 1 ) ( ) ( = βπ t (s t ξ t c s t 1 ) t ξ t 1 c t 1 ( 2 ) j=1 ξj t (st+1 ) 1/γ γ 2 j=1 ξj t 1 (st 1 ) 1/γ ) γ ( ) c = βπ t (s t γ s t 1 ) max t =1,2 c, (9) t 1 where the last lne follows from the fact that the unconstraned agent s the one wth the hghest IMRS. Remember that both agents cannot be constraned at the same tme n equlbrum, otherwse markets don t clear. We wll come back to ths later. 16
E. Decentralzaton wth Tme Zero Tradng: Kehoe-Levne Equlbrum Now, we actually let households trade. All tradng occurs at tme 0. Households trade hstorycontngent clams to consumpton at tme 0 after s 0 has been realzed. Superscrpts refer to the dates at whch trades occur, subscrpts refer to the dates at whch delveres are to be made. At tme t = 0 households can exchange clams on tme t consumpton at prce q 0 t (s t ). When tradng at tme 0, households face a sngle budget constrant: qt 0 (s t )c t(s t ) t=0 s t t=0 s t q 0 t (s t )y t(s t ) and a seres of partcpaton constrants, one for each node s t : U(c )(s t ) U aut (s t ) Defnton 14. A prce system s a sequence of functons { q 0 t (s t ) } and an allocaton s a lst of sequences of functons { c t(s t ) }, one for each. The Kehoe and Levne (1993) equlbrum concept constrans the feasble choce set usng the partcpaton constrants themselves. Households are not afforded the opton of walkng away from ther debts. Defnton 15. A Kehoe-Levne equlbrum s a feasble allocaton and a prce system such that, gven prces, the household problem s solved for each and the markets clear n each s t. Ths a standard saddle pont problem. Construct the Lagrangan for household. Attach a multpler µ to the household s budget constrant and attach multplers γ t(s t ) to the partcpaton constrant n each node s t. Derve the frst order condton for the household : ζ t(s t )β t u ( c t(s t ) ) π t (s t s 0 ) = µ q 0 t (s t ), (10) where the cumulatve multpler ζ s defned recursvely as follows ζ t = ζ t 1 + γ t, ζ 0 = 1. The sequence {ζ t (s t )} s a non-decreasng process because γ t (s t ) 0, t, s t. 17
The frst order condton mples that the rato of margnal utltes satsfes: q t 1 t (s t ) = = = qt 0 (s t ) qt 1 0 (st 1 ) = ζ t(s t )β t u ( c t (s t ) ) π t (s t s 0 ) ζ t 1(s t 1 )β t 1 u ( c t 1 (st 1 ) ) π t 1 (s t 1 s 0 ) ζ t(s t )βu ( c t(s t ) ) ζ t 1(s t 1 )u ( c t 1 (st 1 ) )π t(s t s t 1 ) ξ t(s t )βu ( c t(s t ) ) ξ t 1(s t 1 )u ( c t 1 (st 1 ) )π t(s t s t 1 ) for all hstores s t. In the last lne we recover the frst order condton of the Pareto problem for ξ t = ζ t (st ). There s a one-for-one mappng between the multplers n the KL equlbrum ζ and µ ξ the multplers n the Pareto problem ξ: t (st ) ξ t 1 (st 1 ) = ζ t (st ) ζ t 1 (st 1 ). Also, q0 t = qt 0. Remark 16. A Kehoe-Levne equlbrum s a partcular constraned Pareto-effcent allocaton, namely one that sets µ = ( λ ) 1 for all. Stochastc Dscount Factor strctly postve SDF. It s defned as: As n the complete markets economy, there exsts a unque m t (s t ) qt t 1 (s t ) π t (s t s t 1 ) ξ = β t(s t )u ( c t(s t ) ) ξ t 1(s t 1 )u ( c t 1 (st 1 ) ), ( ) γ j = β yj t (st ) (gt (s t s t 1 ) ) γ. (11) j yj t 1 (st 1 ) In the lmted commtment economy, the SDF s a functon of the tme dscount factor β, the growth rate of the aggregate endowment, and the growth rate of the aggregate weght shock (thrd lne). Note that f no agent s constraned between t and t + 1, g t = 1 and the SDF m t collapses to the SDF n complete markets (Breeden (1979), Lucas (1978)). If many agents are severely constraned, the aggregate weght shock g t (s t ) >> 1. The second lne states that the SDF s the ntertemporal margnal rate of substtuton of each agent, weghted by the rato of cumulatve multplers. For all unconstraned households k, that rato s one, because when µ k t = 0, ξ k t = ξ k t 1. So, we have that the SDF equals the IMRS of the unconstraned households. Because the rato of cumulatve multplers s strctly greater than one for all constraned households (recall that { ξ t} s a non-decreasng stochastc process), 18
and because of the second lne, ther IMRS must be lower than the IMRS of the unconstraned households. Ths mples the SDF exceeds the IMRS of all agents, except for the unconstraned agents, more precsely those who dd not enter a state wth a bndng constrant n perod t, node s t : m t = max ( β u c ) t u ( ) c t 1 The SDF s maxmum IMRS (across all households). In sum, the unconstraned agents are the agents wth the hghest IMRS (or the lowest consumpton growth wth power utlty); ther IMRS prces all the assets n ths economy. The ntuton for ths result s that only the unconstraned agent can arbtrage when hs IMRS s smaller than the state prce of consumpton n a partcular state of the world. Furthermore, because of market clearng, there s always at least one unconstraned agent. However, ths does not mply that the prce of clam to a non-negatve dvdend stream equals the hghest margnal valuaton across all households. Before we move on, I want to stress ths pont. Consder an asset that s a clam to a stream of non-negatve dvdends {d}. Compute the margnal valuaton of an agent: MV 0 (s t ) = τ t max MV0 (s t ) = max s τ π(s τ t u (cτ ) s t )β u (c t ) d τ (s τ ) q t (s t ) [{d}] τ t s τ π(s τ t u (cτ ) s t )β u (c t ) d τ (s τ ) q t (s t ) [{d}] There s not a sngle agent who prces the payoffs n the economy. Rather, the dentty of the unconstraned household changes potentally n every node s t. The lesson s that we cannot just use the IMRS of any household to prce pay-outs, as we dd n the perfect enforcement model, even though markets are ex ante complete. Proposton 17. We can put bounds on the sze of the aggregate weght shocks: where ŷ denotes the labor ncome share. ( 1 g t (s t ŷt (s t ) 1 ) ) max ŷ t 1 (s t 1, ) Proof. The frst nequalty follows from the fact that the multplers do not change f nobody s constraned. The second nequalty follows from the fact that n autarchy the hghest IMRS s for the household who swtches from the hghest to the lowest ncome share. If the state prce 19
was hgher than ( ) γ ( j m t+1 = β yj t (st ) ŷ t (s t ) γ ) mn j yj t 1 (st 1 ) ŷ t 1 (s t 1 ) there could be no trade, because nobody would be wllng to buy contngent consumpton clams at ths hgh prce. Remark 18. AD prces wll be (weakly) hgher than n the correspondng economy wth perfect enforcement and nterest rates wll be (weakly) lower. Ths follows from the fact that the aggregate weght shock sequence {g t (s t )} 1. Hence the SDF s weakly hgher and nterest rates, r f t = E t[m t+1 ] 1, wll be weakly lower. F. Decentralzaton wth Sequental Tradng: Alvarez-Jermann Equlbrum Recently, Alvarez and Jermann (2000) devsed a more appealng decentralzaton whch uses solvency constrants n a sequental tradng envronment. These constrants are portfolo constrants as opposed to drect restrctons on the consumpton possblty set. They are judcously chosen so that they are not too tght : They are tght enough to make sure the Kehoe and Levne (1993) partcpaton constrants are always satsfed, but they do not bnd when the correspondng partcpaton constrants do not bnd. Ths way, we stll allow the maxmum amount of rsk sharng. The solvency constrants prevent default (revertng to autarchy) at the cost of reducng rsk sharng. We know from secton II that we always need borrowng constrants f we allow sequental tradng. Before we devsed natural borrowng constrants that were loose ; they were never bndng under the Inada condton. The AJ-borrowng constrants wll obvously be tghter and they may occasonally bnd. Envronment and Tradng Markets re-open each perod. Let Q t (s t+1 s t ) be the prce of one unt of consumpton delvered contngent on the realzaton of s t+1. At tme t, the household chooses { c t(s t ), a t+1 (s t+1, s t ) } to maxmze expected utlty. Agents face a sequence of budget constrants: c t(s t ) + Q t (s t+1 s t )a t+1(s t+1, s t ) yt(s t ) + a t(s t ) s t+1 Debt Lmts We need some debt lmts n sequental tradng to prevent Ponz schemes. Now we have an endogenous debt lmt that wll replace the natural debt lmt. 20
AJ mpose a dfferent state-by-state constrant on borrowng (or equvalently a lower bound on net wealth B), the so called solvency constrants: a t+1(s t+1, s t ) B t+1(s t+1 ), s t+1. Defnton 19. A dstrbuton of wealth s a vector a t (s t ) = { a t(s t ) } I =1 satsfyng a t(s t ) = 0. Defnton 20. A sequental tradng compettve equlbrum wth solvency constrants {B } I =1 s an ntal dstrbuton of wealth a 0 (s 0 ), an allocaton { c t(s t ), a t+1 (s t+1, s t ) } I, and prcng =1 kernels Q t+1 (s t+1 s t ) such that, for each, the allocaton solves the household problem: J t (a t, s t ) = max c t (st ),{a t+1 (s t+1,s t )} u(c ) + β π(s s)jt+1(a t+1, s t+1 ) s subject to : c t(s t ) + s t+1 Q t (s t+1 s t )a t+1(s t+1, s t ) y t(s t ) + a t(s t ) and a t+1(s t+1, s t ) B t+1(s t+1, s t ), s t+1, and markets clear for all s t c t(s t ) = y t(s t ) and a t(s t+1, s t ) = 0, s t+1. Defnton 21. An equlbrum has solvency constrants { Bt+1 (st+1 ) } I that are not too tght =1 f Jt+1(B t+1(s t+1 ), s t+1 ) = U aut (s t+1 ), t, s t+1. (12) J t+1 (B t+1 (st+1 ), s t+1 ) s the contnuaton utlty of a household startng wth assets B t+1 (st+1 ) n perod t + 1. Solvency constrants that satsfy ths condton prevent default by prohbtng agents to accumulate more state contngent debt than they are wllng to pay back. At the same tme, they maxmze the degree of rsk-sharng. The borrowng constrants are such that U(c )(s t ) U aut (s t ) and U(c )(s t ) = U aut (s t ) B t(s t ) = a t(s t ) 21
To enforce any generc solvency constrant, the agent s entre portfolo must be known. But here we need a lot more nformaton! Note that whoever s mposng these borrowng constrants needs nformaton about households preferences and endowments to determne whether constrants are not too tght. In ths sense, one could queston whether ths really s a decentralzaton. 3 Suffcent condtons for a maxmum are the Euler equaton and the transversalty condton: u (c t(s t ))Q t (s t+1 s t ) + βπ(s t+1 s t )u (c t+1(s t+1 )) 0 lm [ t βt a t(s t ) Bt(s t ) ] u (c t(s t )) The unconstraned agents between t and t + 1 have the hghest IMRS; they are the ones prcng all the assets n the economy n perod t: AJ call ths the hgh IMRS condton. { Q t (s t+1 s t ) = max βπ t+1 (s t+1 s t ) u (c } t+1 (st+1 )) u (c t (st )) Defnton 22. The mpled nterest rates are hgh f where { q 0 t (s t s 0 ) } s defned recursvely as: qt 0 (s t s 0 ) c t(s t ) < + t 0 s t q 0 t (s t s 0 ) = Q t (s t s t 1 )Q t (s t 1 s t 2 )... Q t (s 1 s 0 ). G. The Welfare Theorems and the Relaton between KL and AJ Equlbra We are now n good shape to nvoke the second welfare theorem: The Pareto optmal allocatons can be mplemented as an AJ equlbrum wth solvency constrants: Proposton 23. If an allocaton { c } satsfes the resource constrant, the partcpaton constrants, the hgh IMRS condton, and the hgh mpled nterest rate condton, then there exsts solvency constrants { B } and an ntal wealth dstrbuton a 0 (s 0 ), such that {a, c } are a 3 See Lustg (2003) and Lustg and VanNeuwerburgh (2004b) for a dfferent outsde opton that crcumvents the strngent nformatonal requrements. 22
compettve equlbrum. The solvency constrants can be chosen such that they are not too tght (equaton 12 holds for all ). Proof. See Alvarez and Jermann (2000), pages 791-792. Corollary 24. Any constraned effcent allocaton that has hgh mpled nterest rates can be decentralzed as a compettve equlbrum wth solvency constrants that are not too tght. Intutvely, these AJ equlbrum allocatons wll be the constraned effcent allocatons, because the solvency constrants n the sequental equlbrum problem serve the same purpose as the partcpaton constrants n the Pareto problem. Autarchy can always be decentralzed wth solvency constrants that are not too tght. Remark 25. The autarchc allocatons and prces are an equlbrum wth solvency constrants that are not too tght c t(s t ) = y t(s t ); a t(s t ) = 0 and Q aut t (s t+1 s t ) = max βπ t+1 (s t+1 s t ) u (y t+1 (st+1 )) u (y t (st )) In general these are not constraned effcent, even though they can be decentralzed! Also, {Q aut } may be so hgh (and mpled nterest rates so low) that the hgh mpled nterest rate condton may be volated n autarchy. It turns out there s a tght relaton between these AJ-equlbra n economes wth solvency constrants that are not too tght and the K-L equlbra. Proposton 26. Let {c, a, Q} be an equlbrum wth solvency constrants {B } and ntal wealth dstrbuton a 0 (s 0 ). If the solvency constrants are not too tght and the mpled nterest rates are hgh, then the consumpton allocatons and the mpled AD prces are a KL equlbrum. Proof. See Alvarez and Jermann (2000), pages 792-794. Snce KL equlbra are standard AD equlbra, ther allocatons are Pareto effcent. Ths s the frst welfare theorem. As a result of the prevous proposton, the AJ equlbra wth solvency constrants that are not too tght, are effcent. H. Asset Prcng n Economes wth Solvency Constrants In ths secton, we zoom n on the asset prcng mplcatons of the model wth lmted commtment. We start by allowng for stochastc growth n the aggregate endowment. 23
Statonary Economy Suppose we are n a growng economy, where the growth rate λ of the aggregate endowment e depends on the current state of the world: e t (s t ) = e t 1 (s t 1 )λ t (s t ) and the endowment share ŷ only depends on the current state: y t(s t ) = e t (s t )ŷ t(s t ). Assume a constant dscount factor β and restate the state transton probabltes as ˆπ(s s) = π(s s)λ(s ) 1 γ π(s s)λ(s ) 1 γ and ˆβ(s) = β π(s s)λ(s ) 1 γ. If the partcpaton constrants are satsfed n the growng economy, they are satsfed n the economy wth the unt aggregate endowment. Idosyncratc Rsk Independent of Aggregate State Denote the state s t = (x t, z t ). Suppose that we can wrte ŷt(x t ) and λ t (z t ) and suppose that aggregate shocks are..d. over tme: π(z, x z, x) = ϕ(x x)φ(z ) Ths mples we can state the value of autarchy only as a functon of the current state x: U aut (x) = u(ŷ (x)) + ˆβ ˆφ(z )ˆπ(x x)u aut (x ) Nether the resource constrant, nor the partcpaton constrants depend on the aggregate hstory z t. The constraned effcent allocatons mply consumpton shares that are only a functon of x t : ĉ t(x t ). There s another way of thnkng about ths. The cutoff rule ω (x) wll depend only on the current state x, but not on z t. Allocatons wll depend on y t, because your current consumpton weght depends on y t, but not on the hstory of aggregate shocks. In other words, we can wrte the second part of the SDF as a functon only of the dosyncratc hstory of shocks: (ĉ m t = βλ(z t ) γ φ(z t ) t+1 (x t+1 ) γ ) ĉ ϕ(x t x t 1) t (xt ) Proposton 27. The rsk premum on a one perod strp s dentcal to that n a standard representatve agent economy. 24
Proof. E t R t,t+1 [c t+1 ] E t R t,t+1 [1] c t+1 M t+1 c t+1 ] = E t[ 1 E t [ M t+1 ] = = = x t+1 z t+1 βλ(z t+1 ) γ φ(z t+1 ) 1 x t+1 z βλ(z t+1 ) γ φ(z t+1 ) t+1 z t+1 βφ(z t+1 )c t+1 (z t+1 ) ĉ t+1 (xt+1 ) ĉ t (xt+1 ) ĉ t+1 (xt+1 ) ĉ t (xt+1 ) z βφ(z t+1 )c t+1 (z t+1 ) t+1 γ ϕ(x t+1 x t )c t+1 (z t+1 ) γ ϕ(x t+1 x t ) z t+1 βλ(z t+1 ) γ φ(z t+1 )d t+1 (z t+1 ) x t+1 ĉ t+1 (xt+1 ) x t+1 ĉ t+1 (xt+1 ) ĉ t (xt+1 ) z βφ(z t+1 t+1 )d t+1 (z t+1 ) z t+1 βλ(z t+1 ) γ φ(z t+1 )d t+1 (z t+1 ) 1/ z t+1 βλ(z t+1 ) γ φ(z t+1 ) ĉ t (xt+1 ) 1 γ ϕ(x t+1 x t ) z βλ(z t+1 ) γ φ(z t+1 ) t+1 / γ / ϕ(x t+1 x t) But ths mples that the multplcatve rsk prema s unchanged from the complete markets economy, and all ths work was to no aval. For asset prcng purposes, t s not suffcent smply to ntroduce some dosyncratc rsk. The dosyncratc rsk has to nteract wth the aggregate rsk n some nterestng way! Idosyncratc Rsk Dependent of Aggregate State Suppose there are two aggregate states (recessons and expansons) z (re, ex) where λ(re) < λ(ex). Suppose there are two agents and two dosyncratc endowment states, then we have (y lo,re, y h,re, y lo,ex, y h,ex ). In the frst state agent 1 draws a low endowment, n the second state agent 1 draws a hgh endowment. The endowment sum to one n each state. Suppose the events y are..d. How do we solve for the constraned effcent allocatons? Frst, solve for the value of autarchy: U aut 1 (lo, re) = u(y lo,re ) + β z,x π(z, x lo, re)u aut 1 (x, z ) U aut 1 (h, re) = u(y h,re ) + β z,x π(z, x h, re)u aut 1 (x, z ) Usng vector notaton, the value of autarchy n each state can be recovered from the followng 25
equaton: U aut 1 = u(y)(i βπ) 1 For the second household, we solve the followng equaton: U aut 2 = u(1 y)(i βπ) 1 The only state varables are the consumpton share of the frst agent and the current aggregate state. Next, we solve for the cutoff values U 1 (ω 1 (s)) = u(ω 1 ) + ˆβ(s) y ˆπ(s s)u 1 (ω ) = U aut 1 (s) for all s (lo, re; lo, ex; h, re; h, ex) U 2 (ω 1 (s)) = u(1 ω 1 ) + ˆβ(s) y ˆπ(s s)u 2 (ω ) = U aut 2 (s) for all s (lo, re; lo, ex; h, re; h, ex) Clam 28. We get ω 1 (h, re) > ω 1 (h, ex) and ω 1 (h, re) < ω 1 (h, ex) f the cross-sectonal dsperson of labor ncome shares ncreases n recessons,.e. f ŷ(h, re) > ŷ(h, ex) Assume one of the constrants bnds n all four states of the world: that s agent one s always constraned n both of the hgh states and agent 2 s always constraned n both of the low states. So now, the ergodc set for the consumpton shares conssts only of (ω 1 (lo, re), ω 1 (lo, ex), ω 1 (h, re), ω 1 (h, re)). Clam 29. Ths mples that ( m(s, s ) = ˆβ(z)λ(z ω ) γ max ω, 1 ) γ ω 1 ω ( ) Verfy yourself that margnal utlty growth ncreases n recessons because max ω γ ω, 1 ω 1 ω s larger when z = re. Ths s a dfferent verson of an argument frst made by Mankw (1986) and elaborated upon by Constantndes and Duffe (1996). Idosyncratc rsk wll not affect rsk prema on stocks 26
unless the rsk tself s correlated wth the aggregate state of the economy. Storesletten, Telmer and Yaron (2004) actually provde evdence that the condtonal standard devaton dosyncratc labor ncome rsk more than doubles n recessons n the US. Ths fndng provdes an emprcal underpnnng for ths specfcaton of the labor ncome share process. [ ].75.25 Exercse 30. Take π = and take y(lo) =.35 and y(h) =.65. Set β =.65 and.25.75 γ = 4 and compute the equlbrum allocatons for the economy wthout aggregate uncertanty. Now assume that there s aggregate rsk and that λ(ex) = 1.04, λ(re) =.96, y(lo, re) =.20, y(h, re) =.80, y(lo, ex) =.34, and y(h, ex) =.65. Agan compute equlbrum allocatons. 27
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