Advanced Macroeconomcs II: Exchange economy Krzyszof Makarsk 1 Smple deermnsc dynamc model. 1.1 Inroducon Inroducon Smple deermnsc dynamc model. Defnons of equlbrum: Arrow-Debreu Sequenal Recursve Equvalence of defnons and effcency of decenralzed economy. Uly funcon Consder a sandard uly funcon U(c = β ln c I feaures: me separably, exponenal dscounng, Inada condons, belongs o he class of consan relave rsk averson (CRRA funcons u(c = (c 1 σ 1/(1 σ, neremporal elascy of subsuon s consan and equal o 1/σ, all CRRA funcons are homohec. 1.2 Assumpons Assumpons of he model There are J ypes of agens n our economy, each ype agen s denoed wh s ype {1, 2,..., J}. Tme s nfne = 1, 2, 3,... Le c denoe consumpon of agen n perod. Denoe he sequence of consumpons agen n all perods as c = {c }. Assume ha households maxmze lfecycle uly U(c = β u(c We resrc our aenon o exchange economy. Denoe endowmen of consumer n perod as e and denoe he sequence of endowmens for all perods as e = {e }. Endowmen for each agen s exogenous. 1
1.3 Effcen allocaon Pareo effcency Defnon 1.1. Allocaon {(c,2,...,j } s Pareo effcen f s feasble c = e, for all here does no exs anoher allocaon {( c,2,...,j } sasfyng c = e, for all where a leas one nequaly s src. u( c u(c, for all Fndng effcen allocaon To fnd Pareo effcen allocaons one needs o solve he socal planner problem (prove max (c,...,j sb. o ω U ( c c = e, for all where ω (usually J j=1 ω = 1 defnes wegh ha he socal planner assgns o agen. Dfferen weghs gve dfferen effcen allocaons. Solvng socal planner problem In order o solve he socal planner problem we consruc Lagrangan Frs order condons (FOCs Therefore for all, j we have L = ω [ ] β u(c [ ] λ (c e ω β u (c = λ ω u (c = ω j u (c j (1.1 Any allocaon { (c,2,...,j } s Pareo effcen f you can fnd weghs (ω,2,...,j such ha (1.1 s sasfed. Example Consder he followng wo perod economy endowmen economy. There are wo ypes of households, J = 2. The preferences of boh households are gven by U(c 1, c 2 = log c 1 + β log c 2, where β = 1. Endowmens are gven by (e 1 1, e 1 2 = (1, 0 and (e 2 1, e 2 2 = (0, 1. Defne Pareo allocaon. Wha condons do he se of Pareo-opma have o sasfy? Show n he Edgeworh box. Fnd Pareo opmal allocaons. 2
1.4 Arrow-Debreu equlbrum The concep of ADE In Arrow-Debreu equlbrum (ADE all decsons on exchange and paymens ake place n perod zero. Laer on only delvery of goods and servces (decded n perod zero akes place. Therefore we need prces for each perod and each sae of he world. Denoe (perod 0 prce of a good delvered n perod as p. In ADE consumpon good n each perod s reaed as a dfferen good. Preferences over consumpon goods n each perod are well defned. Why absrac ADE? As we show laer s equvalen o (more nuve sequenal equlbrum and n many cases smplfes analyss. Arrow-Debreu equlbrum (ADE Defnon 1.2. An Arrow-Debreu equlbrum (ADE s an allocaon {(c,2,...,j } and prces {p } such ha allocaon for consumer, {c }, solves, gven prces, he followng problem max {c } sb.o β u(c p c p e markes clear c = e, for all Fndng ADE Le u(c = ln c. To fnd ADE we need o: solve consumer problem subsue he soluon no feasbly condon To solve consumer problem we consruc Lagrangan: [ ] L = β u(c λ p c p e FOCs β u (c = λ p whch n perod 0 becomes elmnang λs β 0 u (c 0 = λ p 0 β u (c p = β 0 u (c 0 p 0 u (c 0 u (c = β p 0 p 3
Snce u (c = c 1 and usng he fac ha he perod 0 consumpon s numerare, p 0 = 1 we ge c c 0 = β 1 p (1.2 Snce he rgh-hand sde s he same for all consumers we can defne he followng c c 0 = cj c j = β 1 α for all, j = 1, 2,..., J. p 0 Feasbly n perod 0 and n perod c = α c 0 (1.3 c 0 = e 0 (1.4 c = α c 0 = α c 0 = e e e Subsung from perod 0 feasbly condon, (1.4 α we ge he formula for α α = e 0 = e J e J e 0 (1.5 Usng α we can fnd p from (1.2 p = β 1 α (1.6 To fnd consumpon we subsue o he budge consran p c = p e β α α c 0 = c 0 p e β = p e c 0 = (1 β p e (1.7 and usng c 0 we can easly fnd c from (1.3. An ADE s an allocaon {(c,2,...,j } } and prces {p } gven by (1.7, (1.3 and (1.6, where α s gven by (1.5. 4
Effcency of an AD allocaon Theorem 1.1. [Frs Welfare Theorem, FWT] Suppose u(c s a srcly ncreasng funcon. An AD allocaon {(c,2,...,j } s Pareo effcen. Proof [by conradcon]. Suppose ha an AD allocaon s no effcen. Therefore here exss anoher feasble allocaon {( c,2,...,j } sasfyng (whou loss of generaly assume ha consumer 1 s srcly beer of u( c 1 > u(c 1 u( c u(c, for all 1 Snce u( c 1 > u(c 1 and he consumer 1 dd no pck herefore ( mus have been o expensve Nex we show ha for all oher consumers 1 he followng s rue If hs s no he case hen p c 1 > p e 1 p c 1. (1.8 hen here exs small enough ε > 0 such ha p c p c (1.9 p c < p c p e p c + ε < p e Whch means ha here exs he followng allocaon (p 0 s normalzed o 1 such ha ĉ 0 = c 0 + ε ĉ = c dla 1 p ĉ = p c + ε < p c < p c p e and u(ĉ 1 > u( c 1 u(c 1 (by monooncy of u(. And hs s mpossble (conradcon snce consumer would have chosen ĉ nsead of c. Therefore (1.9 s sasfed. From (1.8 and (1.9 we ge p c > p c and snce boh allocaons are feasble Conradcon. ( p ( p c e ( > p c ( > p e 5
Example, cn d Consder he followng wo perod economy endowmen economy. There are wo ypes of households, J = 2. The preferences of boh households are gven by U(c 1, c 2 = log c 1 + β log c 2, where β = 1. Endowmens are gven by (e 1 1, e 1 2 = (1, 0 and (e 2 1, e 2 2 = (0, 1. Defne A-D equlbrum. Fnd A-D equlbrum, s s opmal? (Hn: Fnd socal planner weghs Dscuss he SWT. Negsh mehod An ADE s effcen (FWT. Any effcen allocaon can be decenralzed (usng ransfers as an ADE (SWT proof omed. Snce FWT and SWT hold we can use he Negsh mehod n order o fnd equlbrum: fnd Pareo effcen allocaon as a funcon of weghs ω. consruc prces usng FOCs. compue ransfers T = p [e c ] necessary o acheve requred allocaon. fnd Pareo weghs ω, whch gve zero ransfers T = 0 for all. 1.5 Sequenal equlbrum. The concep of sequenal equlbrum In sequenal equlbrum exchange akes place n each perod, sequenally n me. We need o nroduce fnancal asses (one perod bonds, we wll call hem AD asses. An AD asse b +1 bough n perod by agen pays 1 n perod + 1. The prce of an asse b +1 n perod s equal o q. Addonally, we need a consran prevenng he households from playng he Ponz scheme (non- Ponz game condon, NPG. We are gong o nroduce n he smples form possble.e. we assume b +1 b, where b s large enough. We assume b 0 = 0. Sequenal markes equlbrum (SME. Defnon 1.3. A sequenal markes equlbrum (SME s an allocaon {(c, b +1,2,...,J } and prces of AD asses {q } such ha allocaon for consumer {(c, b }, gven prces, solves he consumer problem max {c,b } p.w. β u(c c + q b +1 e + b b +1 b markes clear c = e, for all b +1 = 0, for all 6
FOCs In order o fnd a SME we: solve he consumer problem use equvalency of SME and ADE Frs, we consruc Lagrangan for he consumer problem L = {β u(c [ λ c + q b +1 e ] b } FOCs β u (c = λ λ q = λ +1 and he ransversaly condon (TVC lm λ +1b +1 = 0 Subsung from he frs equaon no he second or dfferenly β u (c q = β +1 u (c +1 q = β u (c +1 u (c u (c +1 u (c = q 1 β The res we can fnd usng equvalency of ADE wh SME. In ADE I can be also found drecly. u (c +1 u (c = p +1 p 1 β Example, cn d Consder he followng wo perod economy endowmen economy. There are wo ypes of households, J = 2. The preferences of boh households are gven by U(c 1, c 2 = log c 1 + β log c 2, where β = 1. Endowmens are gven by (e 1 1, e 1 2 = (1, 0 and (e 2 1, e 2 2 = (0, 1. Defne sequenal equlbrum. Fnd sequenal equlbrum, s s opmal? (Hn: Fnd socal planner weghs. Is he allocaon he same as n case of AD equlbrum? Equvalency of SME and ADE Nex, we show ha boh equlbrum conceps gve he same allocaon. Assume b 0 = 0 for all and endowmen of each consumer {e } s bounded from above. Theorem 1.2. Suppose ha allocaon {(c,2,...,j } and prces {p } are ADE sasfyng p +1 p γ < 1 (1.10 hen here exs allocaon {(b +1,2,...,J } and AD asses prces {q } such ha allocaon {(c, b +1,2,...,J } and AD asses prces {q } are SME. Suppose allocaon {(c, b +1,2,...,J } and AD asses prces {q } are SME sasfyng q 1 ε < 1 (1.11 (for any ε > 0 hen here exs AD asses prces {p } such ha an allocaon {(c,2,...,j } and prces {p } are ADE. 7
Proof. The followng relaonshp beween prces can be obaned from FOCs (we use he fac ha p 0 = 1 q = p +1 p p = p 0 q 0 q 1... q 1 = q 0 q 1... q 1 = Nex, we prove ha he budge consrans are equvalen. Sep 1: SME = ADE Take budge consran from SME c + q b +1 e + b and one perod earler subsung for b usng he fac ha b 0 = 0 we ge Le T hen Usng he consran (1.11 we ge Snce p = 1 q τ c 1 + q 1 b e 1 + b 1 c 1 + q 1 c + q 1 q b +1 e 1 + q e + b 1 T ( 1 ( 1 lm q τ c + q τ c + lm T T q τ b T +1 T T q τ b T +1 lm ( 1 q τ c T ( 1 T q τ b T +1 T q τ e ( 1 T q τ ( b = 0 ( 1 p c p e whch means ha any allocaon whch sasfes SME budge consran (wh he NPG condon sasfes he ADE budge consran. Sep 2: ADE = SME Defne bonds hold by consumers as b +1 = 1 p +1 q τ e 1 q τ e q τ p +τ (c +τ e +τ (1.12 hen hey sasfy he SME budge consran, because subsung for b +1 no he SME budge consran we ge c + q b +1 e + b c 1 + q p +τ (c +τ e +τ e + 1 p 1+τ (c 1+τ e 1+τ p +1 p c + q 1 p +1 p +τ (c +τ e +τ e + 1 p p +τ (c +τ e +τ 8
subsung q = p+1 p c + 1 p c + p +1 p 1 p +1 p +τ (c +τ e +τ e + 1 p p +τ (c +τ e +τ e + 1 p p (c e + 1 p c e + c e p +τ (c +τ e +τ p +τ (c +τ e +τ whch s rue. Nex, we show ha we can fnd hgh enough b such ha b +1 b where b +1 s gven by (1.12 b +1 = 1 p +1 p +τ (c +τ e +τ p +τ p +1 e +τ Subsung from (1.10 p+τ p γ τ (usng he fac ha {e } s bounded from above. b +1 so we ge he lower bound on deb b p +τ p +1 e +τ b = 1 + sup γ τ 1 e +τ > γ τ 1 e +τ < Whch means ha any allocaon whch sasfes he ADE budge consran sasfes he SME budge consran. Sep 3: If boh budge consrans are equvalen, he uly funcon s he same and he feasbly condons are he same he opmal choce s he same (equlbrum allocaons are he same. Effcency of he SME Allocaons under boh equlbra conceps are he same. As we shown earler he ADE s effcen. Therefore, he SME s effcen. Hence, can be found wh he Negsh mehod. 1.6 Recursve equlbrum The concep of recursve equlbrum We consder compeve economy wh only one ype of agen of measure 1. The endowmen mus saonary e = e for each. We drop he me ndces, b s replaced wh b and b +1 wh b. Denoe he aggregae bonds holdngs of all consumers as B. Snce each consumer s of measure 0 she has no conrol over aggregae varables and akes s evoluon n me as gven. Noaon, capal leers denoe aggregae and small ndvdual varables. Usng he concep of recursve equlbrum faclaes he use of numercal mehods for fndng compeve equlbrum (whch s necessary whenever FWT does no hold. 9
Recursve equlbrum Noe ha he consumer problem max {c,b } sub. o β u(c c + q b +1 e + b b +1 b can be expressed n recursve form v(b, B = max u(c + βv(b, B (c,b sub. o c + q(bb e + b B b = G(B b elmnang c gves ( e + b v(b, B = max u + βv(b, B (1.13 b b e+b q(bb q(b sub. o B = G(B where an aggregae polcy funcon G(B descrbes evoluon of B. The consumer mus know hs funcon n order o be able o derve fuure prces q(b. Denoe he polcy funcon ha solves he problem above as g(b, B. Defnon 1.4. A recursve equlbrum s a collecon of funcons: value funcon: v(b, B polcy funcons: g(b, B, G(B prce funcon: q(b; sasfyng funcons v(b, B and g(b, B solve he consumer problem (1.13. funcon g(b, B s conssen wh G(B Summary Defnons of equlbrum: Arrow-Debreu equlbrum Sequenal Markes equlbrum Effcency Equvalence Negsh mehod. Recursve equlbrum. G(B = g(b, B = 0, for all B Exercse: Defne equlbrum n he Ramsey model usng all defnon conceps learn here. 10
2 Smple dynamc sochasc general equlbrum (DSGE model 2.1 Inroducon Inroducon Expressng rsk n he sae space. The conceps of compeve equlbrum: Arrow-Debreu Sequenal Equvalency Effcency of decenralzed equlbrum. 2.2 Envronmen Represenng rsk n he sae space We use he noaon: Consder he economy wh J ypes of agens, where each ype s denoed as {1, 2,..., J}. Le S be a se of possble evens n perod (e.g. S = {1, 2} where 1 denoes ran and 2 no ran and le s S denoe an even n perod (elemen of he se S, n our example s = 1 denoes ran, and s = 2 no ran. Denoe he se of all possble hsores up o perod as S = S 0 S 1... S and le s S (elemen of he se S, s = (s 1, s 2,..., s, be a parcular hsory up o perod (n our example could be s = (1, 2, 1, 1, 2,..., 1. Le c (s denoe he consumpon of agen n perod afer hsory s (noe s jus he value of consumpon. Denoe he sequence of consumpons for agen for all possble perods and all possble hsores as c = {c (s } ;s S. Assume ha households maxmze he lfecycle expeced uly. Usng our noaon we can express expeced uly funcon (von Neumann-Morgensern as U(c = s S β π(s u(c (s where π(s denoes uncondonal probably of hsory s (whch s he same for all agens, herefore s no ndexed by. We resrc our aenon o he endowmen economy. Denoe he endowmen of agen n perod afer hsory s as e (s and he sequence of endowmens for all possble perods and all possble hsores as e = {e (s } ;s S. The endowmen for each consumer s gven (exogenous. Mos ofen s assumed ha here s no uncerany n perod 0. We assume as well. 2.3 Effcen allocaon Pareo effcency Defnon 2.1. Allocaon {(c (s,2,...,j },s S s Pareo effcen f s feasble c (s = e (s, for all s S 11
here does no exs anoher allocaon {( c (s,2,...,j },s S sasfyng c (s = e (s, for all s S and a leas one nequaly s src. u( c u(c, for all Fndng effcen allocaon In order o fnd Pareo effcen allocaon s enough o solve he socal planner problem (prove max (c,...,j p.w. ω U ( c c ( s = e ( s, dla każdego s S where ω (usually J j=1 ω = 1 defnes wegh ha he socal planner assgns o agen. Dfferen weghs gve dfferen effcen allocaons. Solvng socal planner problem In order o solve he socal planner problem we consruc Lagrangan L = ω [ ] [ ] β π(s u(c (s λ(s (c (s e (s s S s S Frs order condons (FOCs Therefore for all, j and for all s we have ω β π(s u (c (s = λ(s ω u (c (s = ω j u (c j (s (2.1 Any allocaon { {c (s s S,,2,...,J} s Pareo effcen f you can fnd weghs (ω,2,...,j, such ha (2.1 s sasfed. 2.4 Arrow-Debreu equlbrum The concep of ADE In Arrow-Debreu equlbrum (ADE all decsons on exchange and paymens ake place n perod zero, before he rsk s realzed. Laer on only delvery goods and servces (decded n perod zero akes place. Therefore we need prces for each perod and each sae of he world. Denoe (perod 0 prce of a good delvered n perod as p. Le p(s denoe he prce of one un of consumpon, quoed a perod 0; delvered n perod f and only f even hsory s has been realzed. In ADE consumpon good n each perod and n all hsory nodes s s reaed as a dfferen good. Preferences (von Neumann-Morgensern over consumpon goods n each perod and each hsory node s are well defned. Why absrac ADE? As we show laer s equvalen o (more nuve sequenal equlbrum and n many cases smplfes analyss. 12
Arrow-Debreu equlbrum (ADE Defnon 2.2. An Arrow-Debreu equlbrum (ADE s an allocaon {(c (s,2,...,j },s S and prces {p(s },s S such ha allocaon for consumer, {c (s },s S, solves, gven prces, he followng problem max {c (s },s S sb. o β π(s u(c (s s S p(s c (s p(s e (s s S s S markes clear c (s = e (s, for all s S Fndng ADE Le u(c = ln c. To fnd ADE we need o: solve consumer problem subsue he soluon no feasbly condon To solve consumer problem we consruc Lagrangan: [ ] L = β π(s u(c (s λ p(s c (s p(s e (s s S s S s S FOCs whch n perod 0 becomes elmnang λs β π(s u (c (s = λ p(s β 0 π(s 0 u (c (s 0 = λ p(s 0 β π(s u (c (s p(s = β 0 π(s 0 u (c (s 0 p(s 0 u (c (s 0 u (c (s = β π(s p(s 0 β 0 π(s 0 p(s snce p(s 0 = 1 (numerare, π(s 0 = 1 (no unverany n perod 0. Le u(c = ln c, solvng for c (s c (s c (s 0 = cj (s c j (s 0 = β π(s p(s α(s s 0 = α(s for all, j = 1, 2,..., J. (2.2 we ge The feasbly condon n perod 0 c (s = α(s c (s 0 (2.3 c (s 0 = e (s 0 (2.4 13
and n perod c (s = α(s c (s 0 = α(s c (s 0 = e (s e (s e (s Subsung from he perod 0 feasbly condon, (2.4 α(s e (s 0 = e (s we ge he formula for α(s Usng α(s we can fnd p(s from (2.2 J α(s = e (s J e (s 0 α(s = β π(s π(s 0 p(s 0 p (s (2.5 snce π(s 0 = 1 and p(s 0 = 1 p(s = β π(s α(s In order o fnd consumpon we subsue no he budge consran (2.6 Snce s S π(s = 1 s S p(s c (s = knowng c (s 0 we can easly fnd {( c (s from (2.3. An ADE s an allocaon c (s s S p(s e (s βπ(s α(s α(s c (s 0 = p(s e (s s S s S c (s 0 ( β π(s = p(s e (s s S s S c (s 0 β = p(s e (s s S c (s 0 = (1 β p(s e (s (2.7 s S,2,...,J } (2.7, (2.3 and (2.6, where α(s s gven by (2.5.,s S },s S and prces { } p(s,s S gven by 14
Example Consder he followng wo perod economy endowmen economy. There s no unereny n perod 1, n perod 2 wh probably 0.5, s 2 = 1 and wh probably 0.5 s 2 = 2. There are wo ypes of households, J = 2. The preferences of boh households are gven by U(c 1, c 2( = 2 =1 β 1 π(s log c (s, where β = 1. Endowmens are gven by (e 1 1, e 1 2(s 1, e 1 2(s 2 = (1, 0, 2 and (e 2 1, e 2 2(s 1, e 2 2(s 2 = (1, 2, 0. Defne Pareo allocaon. Wha condons do he se of Pareo-opma have o sasfy? Fnd Pareo opmal allocaons. Defne A-D equlbrum. Fnd. Is effcen? Effcency of an AD allocaon ADE s effcen (FWT. Proof s analogous o he one n case of deermnsc model. Any effcen allocaon can be decenralzed (usng ransfers as an ADE (SWT, proof omed. We can also use he Negsh mehod of fndng equlbrum: fnd Pareo effcen allocaon as a funcon of weghs ω. consruc prces usng he frs order condons. compue ransferst = s S p(s [e (s c (s ] necessary o acheve requred allocaon. fnd Pareo weghs ω, whch gve zero ransfers T = 0 for all. 2.5 Sequenal markes equlbrum. The concep of sequenal markes equlbrum (SME In sequenal markes equlbrum exchange akes place n each perod, sequenally n me. We need o nroduce fnancal asses (one perod bonds, we wll call hem AD asses. An AD asse b (s, s +1 bough n perod n node s by agen pays 1 n he sae s +1 and 0 oherwse. We assume ha AD asses may be raded one perod ahead. Prce of an asse b(s, s +1 n perod a node s s equal o q(s, s +1. A he begnnng of each perod afer he rsk s realzed he only hng ha maers s how much of an asse payng n a gven sae of he world one holds (s, s +1 whch a ha me can be denoed as (s +1. In perod a he node s agen can buy AD asses for each possble saes of he world whch can happen n + 1 afer hsory s, {b(s, s +1 } s+1 s. Assume b (s 0 = 0. Sequenal markes equlbrum (SME. Defnon 2.3. A sequenal markes equlbrum (SME s an allocaon {(c (s, {b (s, s +1 } s+1 s,2,...,j},s S and prces of AD asses {q(s, s +1 },s S,s +1 s such ha allocaon for consumer {(c (s, {b (s, s +1 } s+1 s },s S, gven prces, solves he consumer problem p.w. c (s + max {c (s,{b (s,s +1} s+1 s },s S s S β π(s u(c (s s +1 s q(s, s +1 b (s, s +1 e (s + b(s b (s, s +1 b 15
markes clear c (s = e (s, dla każdego s S FOCs In order o fnd a SME we: b (s, s +1 = 0, dla każdego s S, s +1 s solve he consumer problem use equvalency of SME and ADE Frs, we consruc Lagrangan for he consumer problem FOCs L = β π(s u(c (s s S λ(s [ c (s + q(s, s +1 b (s, s +1 e (s b(s ] s +1 s β π(s u (c (s = λ(s λ(s q(s, s +1 = λ(s +1 and he ransversaly condon (TVC. Subsung from he frs equaon no he second or dfferenly β π(s u (c (s q(s, s +1 = β +1 π(s +1 u (c (s +1 q(s, s +1 = β π(s+1 u (c (s +1 π(s u (c (s u (c (s +1 u (c (s = q(s π(s, s +1 βπ(s +1 The res we can fnd usng equvalency of ADE wh SME. In ADE I can be also found drecly. Equvalency of SME and ADE u (c (s +1 u (c (s = p(s+1 p(s π(s βπ(s +1 Analogously as n case of no uncerany we can prove he equvalency beween ADE an SME. The man argumen of he prof s he relaonshp beween prces n ADE and AD asse prces n SME gven by: q(s, s +1 = p(s+1 p(s p (s = p(s 0 q(s 0, s 1 q(s 1, s 2... q(s 1, s Beware! Equvalency requres ha he markes are complee. In our model he suffcency condon for he compleeness of markes (anyone can make any ransacon wh anyone s he exsence n each perod a each node s of a leas one AD asses for each possble sae of he world n + 1, (s, s +1 (afer hsory s. 16
Effcency of he SME Allocaons under boh equlbra conceps are he same. As we shown earler he ADE s effcen. Therefore, he SME s effcen. Hence, can be found wh he Negsh mehod. Example cn d Consder he followng wo perod economy endowmen economy. There s no unereny n perod 1, n perod 2 wh probably 0.5, s 2 = 1 and wh probably 0.5 s 2 = 2. There are wo ypes of households, J = 2. The preferences of boh households are gven by U(c 1, c 2( = 2 =1 β 1 π(s log c (s, where β = 1. Endowmens are gven by (e 1 1, e 1 2(s 1, e 1 2(s 2 = (1, 0, 2 and (e 2 1, e 2 2(s 1, e 2 2(s 2 = (1, 2, 0. Defne sequenal equlbrum. Fnd. Is effcen? Asses prces [o be done] 2.6 Recursve equlbrum (RCE The concep of recursve equlbrum [o be done] 2.7 Markov process Markov chans In many macro applcaons (RBC models, heerogeneous agen models s assumed ha he random varables x(s, s S follow he Markov process wh dscree me and fne sae spaces S. For example, any AR(1 process can be approxmaed wh he Markov process (Tauchen, 1986. The saonary (homogenous Markov process s a sochasc process such ha: he number of saes n each perod s fne and counable, s S (S s a fne se S = {e 1, e 2,..., e N } and does no change n me. ranson probably beween dfferen saes (oday and omorrow depends only on he sae oday and s ndependen of all hsory π(s +1 s = π(s +1 s, s 1,..., s 0 = π(s +1 s Le π(s +1 s denoe he condonal probably of endng up n he sae s +1 gven he oday sae s. Homogeney means ha he sae space does no change n me and ranson probables do no change n me. W can smplfy noaon and denoe ranson probably from sae s = k oday o he sae s +1 = l omorrow as π kl = π(s +1 = e k s = e l. Snce s S, s +1 S and S s fne and counable se π( can be expressed as N N ranson marx, where N denoes he number of elemens of he se S π 11 π 1l π 1N... π = π k1 π kl π kn... π N1 π Nl π NN Ths marx sasfes π kl 0 for all k, l and l π kl = 1 for all k. 17
Le he probably dsrbuon of he saes oday be an N elemen vecor P = (p 1, p 2,..., p N, noe k pk = 1. Then p l +1 = π kl p k k or P +1 = P π Saonary dsrbuon P of chan π sasfes P = P π Applcaon of Markov process Sandard RBC model s he sochasc dynamc general equlbrum (DSGE model. I has wo sae varables: capal (endogenous varables producvy (endogenous varable, follows an AR(1 process An AR(1 process can be approxmaed by (5-6 saes Markov process (Tauchen, 1986. Such a model can be relavely easly solved wh global mehods (value funcon eraons. Summary. Smple deermnsc dynamc model and noaon. Dfferen equlbrum concep. Equvalency of decenralzed and cenralzed allocaon. Smple sochasc dynamc model. Possbly of usng global mehods for solvng DSGE models. 18