Equlbrum wth Complete Markets Instructor: Dmytro Hryshko 1 / 33
Readngs Ljungqvst and Sargent. Recursve Macroeconomc Theory. MIT Press. Chapter 8. 2 / 33
Equlbrum n pure exchange, nfnte horzon economes, wth agents facng stochastc endowments. Two market structures: 1 Arrow-Debreu structure complete markets n dated contngent clams (tme τ, state vector s τ ), all traded at tme 0. 2 Sequental, complete, market structure: Arrow securtes traded one perod n advance; trades occur at each t 0. These two market structures result n dentcal consumpton allocatons. 3 / 33
Market Settng At t 0, a stochastc event s t s drawn from the state space S. Denote the hstory of events up to and nclusve the one at t as s t = (s 0, s 1,..., s t ), wth the uncondtonal probablty of observng s t, π t (s t ). The condtonal probablty of observng s t, gven s τ was realzed, s π t (s t s τ ), t τ. s t s publcly observable. Assume π 0 (s 0 ) = 1, s 0 gven. 4 / 33
of tme 2, are possble up to tme 3. Fg. 8.3.2 portrays a partcular hstory that t s known the economy has ndeed followed up to tme 2, together wth the set of two possble one-perod contnuatons nto perod 3 that can occur after that hstory. Example. State space S = {0, 1}. Arrow-Debreu settng t=0 t=1 t=2 t=3 (0,1,1,1) (0,1,1,0) (0,1,0,1) (0,1,0,0) (0,0,1,1) (0,0,1,0) (0,0,0,1) (0,0,0,0) Fgure 8.3.1: The Arrow-Debreu commodty space for a two-state Markov chan. At tme 0, there are trades n tme t = 3 goods for each of the eght nodes or hstores that can possbly be reached startng from the node at tme 0. 5 / 33
only on the current state s t. Remarkably, n the complete markets models o chapter, the consumpton allocaton at tme t wll depend only on s t. The m ncompleteness Example. of State chapter space 17 ands the = nformaton {0, 1}. Arrow and enforcement settngfrctons of ch 19 wll break that result and put hstory dependence nto equlbrum allocato t=0 t=1 t=2 t=3 (1 0,0,1) (0 0,0,1) Fgure 8.3.2: The commodty space wth Arrow securtes. At date t = 2, there are trades n tme 3 goods for only those tme t = 3 nodes that can be reached from the realzed tme t =2 hstory (0, 0, 1). 6 / 33
Market Settng Contd. Agents: = 1, 2,..., I. (Stochastc) Endowments of agent at t: y t(s t ). At tme 0, household purchases a hstory dependent consumpton plan, c = {c t(s t )} t=0. Ranks consumpton streams wth: U = β t [ u c t (s t ) ] π t (s t ). t=0 s t u C 2, strctly concave, c 0. A feasble allocaton must satsfy: c t(s t ) yt(s t ), all t, s t. 7 / 33
Equlbrum Outcomes and Hstory Dependence Indvdual s endowment hstory up to tme t can be descrbed as: y 0 (s 0), y 1 (s1 ), y 2 (s2 ),..., y t(s t ). (Note the general dependence of the tme t endowment realzaton on the entre hstory of stochastc process, precedng tme t.) In the complete markets models, consumpton allocaton does not depend on ndvdual hstory, and s only a functon of the aggregate endowment. 8 / 33
Pareto Problem Solve the planner s problem: s.t. I W = max λ U(c ) (8.4.1) =1 c t(s t ) y t(s t ), (8.2.2) where the Pareto weghts λ 0, = 1,..., I. 9 / 33
Pareto Problem Contd. Assgn non-negatve Lagrange multplers θ t (s t ) for each t and s t constrant (8.2.2), and form the planner s Lagrangan: [ [ y t (s t ) c t(s t ) ]] L = t=0 s t λ β t π t (s t )u [ c t(s t ) ] + θ t (s t ) For any, the FOC wth respect to c t(s t ) s: β t u [ c t(s t ) ] π t (s t ) = λ 1 θ t (s t ) 10 / 33
Effcent Allocaton Thus, the rato of the FOCs for and 1 s: u ( c t(s t ) ) u (c 1 t (st )) = λ 1, λ u (c t(s t )) = λ 1 λ 1 u (c 1 t (s t )) c t(s t ) = (u ) 1 [ λ 1 λ 1 u (c 1 t (s t )) ] (8.4.3) c t(s t ) = (u ) 1 [λ 1 λ 1 u (c 1 t (s t ))] = y t(s t ). 11 / 33
Thus, c 1 t (s t ) s a functon of the aggregate endowment only, and not dependent on the specfc hstory s t leadng to that outcome, nor on the realzaton of ndvdual endowments. That s, c t(s t ) = c τ (s τ ) when y t(s t ) = y τ (s τ ). 12 / 33
Tme 0 Tradng: Arrow-Debreu Securtes Arrow-Debreu securtes: clams to consumpton for each possble date t, hstory s t. What s the optmal allocaton n a compettve economy wth Arrow-Debreu securtes? Assume: a complete set of state-contngent securtes traded at tme 0. The prce of a securty at tme 0 (now) for 1 unt of consumpton to be delvered at t f s t happens s q 0 t (s t ). 13 / 33
Household Problem Household chooses c (s t ) to s.t. max U(c ) = β t [ u c t (s t ) ] π t (s t ) (8.2.1) t=0 s t 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 ). (8.5.1) 14 / 33
Notes 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 ) The superscrpt 0 refers to the date of tradng, whle the subscrpt t refers to the date at whch the delveres are made. Multlateral trades are possble through a clearng operaton that keeps track of net clams; all trades happen at tme 0; after tme 0, trades agreed at tme 0 are executed but no more trades occur. No expectatons: the household s makng all of ts lfetme trades at tme 0 at a well-defned prce vector for all Arrow-Debreu commodtes. Household buys clams to dfferent contngent consumpton bundles: delvered f s t s realzed. 15 / 33
Lagrangan for household s: F.O.C.: L = β t [ u c t (s t ) ] π t (s t )+ t=0 s t [ ] µ qt 0 (s t )yt(s t ) qt 0 (s t )c t(s t ). t=0 s t t=0 s t U(c ) c t (st ) = µ q 0 t (s t ), or β t u [ c t(s t ) ] π t (s t ) = µ q 0 t (s t ) (8.5.4) 16 / 33
Compettve Equlbrum Defntons. A prce system s a sequence of functons {qt 0 (s t )} t=0. An allocaton s a lst of sequences of functons c = {c t(s t )} t=0, one sequence for each. Defnton. A compettve equlbrum s a feasble allocaton such that, gven the prce system, the allocaton solves each household s problem. 17 / 33
Equlbrum Allocatons Equaton (8.5.4) mples for any and j: u [c t (st )] u [c j t (st )] = µ µ j (8.5.5). That s, the ratos of margnal utltes for any consumers and j are constant across all dates and state hstores. An equlbrum allocaton solves (8.2.2), (8.5.1), and (8.5.5). (8.5.5) mples for any two consumers and 1: c t(s t ) = (u ) 1 [u ( c 1 t (s t ) ) µ µ 1 ]. Summng ths expresson over and utlzng (8.2.2) at equalty, we obtan: [ (u ) 1 u [ c 1 t (s t ) ] ] µ = c µ t(s t ) = yt(s t ). 1 18 / 33
Compettve Equlbrum The compettve equlbrum allocaton s a functon of the realzed aggregate endowment and depends nether on the specfc hstory leadng to that outcome nor on the realzatons of ndvdual endowments: c t(s t ) = c τ (s τ ) for all s t and s τ for whch y t(s t ) = y τ (s τ ). Unts of the prce system are arbtrary. Let q 0 0 (s0 ) = 1; the rest of the prces wll be quoted n the unts of tme 0 goods. A compettve equlbrum allocaton s a partcular Pareto optmal allocaton, the one for whch: λ = µ 1. The shadow prces θ t (s t ) = q 0 t (s t ). 19 / 33
Sequental Tradng of Arrow Securtes It can be shown that one-perod Arrow securtes are enough to make the markets complete f the markets are reopened each day, and the trades on one-perod ahead state-contngent consumpton clams occur. Ths sequental tradng attans the same allocaton as the compettve equlbrum n the economy wth tme zero tradng of state-contngent clams. We can prce dfferent (redundant) assets usng the prces for contngent clams from our compettve complete markets equlbrum. 20 / 33
Example 8.6.1. Rsk Sharng wth CRRA Preferences Utlty functon: U [ c t(s t ) ] = c t (st ) 1 γ 1 γ, γ > 0. Maxmzng the expected sum of dscounted utltes over all t, s t s.t. the ndvdual budget constrant, gves the followng equlbrum condton for agents and 1 (an applcaton of (8.5.5)): ( ) 1 c t(s t ) = c 1 t (s t µ γ ). Summng over, c t(s t ) = (µ 1 ) 1 γ c 1 t (s t ) (µ ) 1 γ = yt(s t ), and so [ ] 1 c 1 t (s t ) = (µ 1 ) 1 γ (µ ) 1 γ yt(s t ) = const yt(s t ). µ 1 21 / 33
Thus, ndvdual consumpton at t, s t s perfectly correlated wth the aggregate endowment and aggregate consumpton. 22 / 33
Rsk Sharng. Summary Consumpton of agents and j are constant fractons of one another, ndependent of t, s t. Condtonal on tme t, hstory s t, c t(s t ) s ndependent of the ndvdual endowment at t, s t, for all. There s an extensve cross-hstory, cross-tme consumpton smoothng. Constant-fracton consumpton allocaton comes from: 1) the complete markets assumpton; 2) a homothetc utlty functon. 23 / 33
Example (8.6.2): No Aggregate Uncertanty Assume s t [0, 1]; s t s a stochastc event; = 1, 2; y 1 t (s t ) = s t ; y 2 t (s t ) = 1 s t. Aggregate endowment at t, s t : yt(s t ) = s t + (1 s t ) = 1. Thus, t mmedately follows from (8.5.7) that c 1 t (s t ) s constant for each t, s t. c 2 t (s t ) = 1 c 1 t (s t ) s also constant t, s t. Equlbrum allocaton: c t(s t ) = c. 24 / 33
From (8.5.4) t follows β t π t (s t ) u ( c ) µ = q 0 t (s t ),, t, s t (8.6.3) Agent s b.c.: or c t t s t q 0 t (s t )[c y t(s t )] = 0, qt 0 (s t ) qt 0 (s t )yt(s t ) = 0. t s t s t Plug the defnton for q 0 t (s t ) from (8.5.4), to obtan: 25 / 33
c u (c ) µ c t t β t s t π t (s t )β t u (c ) π t (s t )β t y µ t(s t ) = 0 s t t s t π t (s t )β t yt(s t ) = 0 s t π t (s t )β t yt(s t ) (8.6.4) s t π t (s t ) t c = (1 β) t Is ths allocaton feasble? For each t, s t : c = (1 β) π t (s t )β t yt(s t ) t s t = (1 β) π t (s t )β t yt(s t ) t s t = (1 β) π t (s t )β t 1 t s t = (1 β) t β t π t (s t ) = (1 β)/(1 β) = 1. s t } {{ } =1 26 / 33
In ths example, consumer perfectly smoothes consumpton over tme and across hstores snce aggregate endowment does not fluctuate across tme. 27 / 33
Example: Perodc Endowment Processes s t [0, 1], s t determnstc. Economy starts at t = 0. Agent 1 s endowment sequence: 1, 0, 1, 0, 1,.... Agent 2 s endowment sequence: 0, 1, 0, 1, 0,.... π t (s t ) = 1 snce s t s determnstc. (8.5.4) mples: q 0 t (s t ) = β t π t (s t ) u (c ) µ = β t u (c ) µ. Normalze q0 0(s 0) = 1,.e., set the tme-0 good as a numerare. Thus, 1 = β 0 u (c ) µ = u (c ) µ. It follows that q t (s t ) = β t. 28 / 33
(8.6.3) mples: c = (1 β) t = (1 β) t π t (s t )β t yt(s t ) s t β t yt(s t ). s t c 1 = (1 β)(1+β 0+1 β 2 +0 β 3 +1 β 4 +...) = (1 β) 1 1 β 2 = 1 1+β. c 2 = 1 c 1 = β 1+β. Note that a lfetme rcher agent 1 consumes more. 29 / 33
Asset prcng n complete markets In complete markets, an asset wll offer a bundle of hstory-contngent clams, each component of whch has been already prced n the market. In ths case, the asset s vewed as redundant. 30 / 33
Prcng redundant assets An asset brngs {d t (s t )} t=0 a stream of clams on t, st consumpton (n physcal unts). (Thnk of dvdends here.) The tme 0 prce of ths asset n terms of tme 0, hstory s 0 consumpton must be: p 0 0(s 0 ) = t=0 s t d t (s t )q 0 t (s t ). Why? We can replcate ths asset by purchasng consumpton bundles {d t (s t )} t=0 at the total cost t=0 s t d t(s t )qt 0 (s t ). Hence, the asset must cost p 0 0 (s 0). 31 / 33
A rskless consol brngs 1 unt of consumpton n each t, regardless of s t. To replcate ts payoffs, we need to purchase q 0 t (s t ) unts of clams for each t, s t. The prce of ths asset, therefore, s: t=0 s t q 0 t (s t ). 32 / 33
A rskless strp pays d τ = 1 f τ = t 0, 0 otherwse. Thus, the strps pay off only at tme τ, regardless of the hstory realzaton s τ. To replcate the asset s payoffs, we need to purchase, at t = 0, q 0 τ (s τ ) unts of consumpton for tme τ, all hstores s τ. Thus, the prce of the strp must be s τ q 0 τ (s τ ). 33 / 33