2 Certainty Equivalence and the Permanent Income Hypothesis(CEQ-PIH)

Transcription

1 1 Oveview Income Fluctuation poblem: Quadatic-CEQ Pemanent Income CARA pecuationay savings CRRA steady state inequality boowing constaints Geneal Equilibium: steady state capital and inteest ate 2 Cetainty Equivalence and the Pemanent Income Hypothesis(CEQ-PIH) 2.1 Cetainty assume βr = 1 T = fo simplicity no uncetainty: X max β t u (c t ) t=0 A t+1 =(1+)(A t + y t c t ) solution: " # X c t = A t + y t + R t y t+j 1+ j=1 1

2 2.2 Uncetainty: Cetainty Equivalence and PIH tempting... " # X µ t 1 c t = A t + y t + E t y t+j j=1 Pemanent Income Hypothesis (PIH) Cetainty Equivalence: x E (x) valid iff: pefeences: u (c) quadatic and c R main insight: given pemanent income X µ p 1 y t y t + E t 1+ j=1 t y t+j c t function of y p t and not independently of y t innovations X µ j 1 c t c t c t 1 = [E t y t+j E t 1 y t+j ] j=0 evisions in pemanent income implications: andom-walk: E t 1 [ c t ]=0 no insuance......consumption smoothing minimize c 2

3 maginal popensity to consume fom wealth: 1+ maginal popensity to consume fom innovation to cuent income depends on pesistence of income pocess example: {y t } is MA(2) c t = 1+ j=1 y t = ε t + β 1 ε t 1 = β (L) ε t X R j [E t y t+j E t 1 y t+j ] ª = y t E t 1 y t + R 1 (E t y t+1 E t 1 y t+1 ) 1+ = ε t + R 1 β 1 ε t = 1+R 1 β 1 ε t 1+ whee y t = ε t + β 1 ε t 1, E t 1 y t = β 1 ε t 1 and E t 1 y t+j =0fo j 1 and E t y t+1 = βε t ARMA pesistence c ε t > t 1+ with a unitootin y t α (L) y t = β (L) ε t β (R 1 ) c t = 1+α (R 1 ) ε t mg popensity to consume may be geate than 1 3 Estimation and Tests 3.1 CEQ-PIH andom walk (matingale): c t = u t E t 1 u t = 0 3

4 u t pefectly coelated with news aiving at t about the expected pesent value of futue income: X 1 c t = u t = 1+ j=0 (1 + ) [E ty t+j E t 1 y t+j ] j Two main tests: (geneally on aggegate data) andom walk unpedictability of consumption violations = excess sensitivity to pedictable cuent income popensity to consume too small given income peisistence = excess smoothness both tests ely on pesistence of income contovesial aggegation issues: acoss goods agents: Eule equation typically non-linea Attanasio and Webe leads to ejection on aggegate data time aggegation: data aveaged ove continuous time intoduces seial coelation 3.2 Eule Equations Hall: evolutionay idea: foget consumption function find popety it satisfies Eule equation! Attanasio et al u 0 (c t )= β (1 + ) E t [u 0 (c t+1 )] 4

5 4 Pecautionay Savings idea: beak CEQ 4.1 Two Peiods two peiod savings poblem: max u (c 0 )+βeu ( c 1 ) a 1 + c 0 = Ra 0 + y 0 = x c 1 = Ra 1 +ỹ 1 subsituting: max {u (x 0 a 1 )+βeu (Ra 1 +ỹ 1 )} a 1 f.o.c. (Eule equation) u 0 (x 0 a 1 )=βr EU 0 (Ra 1 +ỹ 1 ) U'[(1+)k 0 +z 0 -k 1 ] β(1+)eu'[z 1 +(1+)k 1 ] k 1 * k 1 Optimum: unique intesection k 1 * Figue 1: optimum: unique intesection k 1 5

6 mean peseving spead: second ode stochastic dominance eplace ỹ 1 with ỹ1 0 =ỹ 1 + ε with E ( ε y) =0 U'[(1+)k 0 +z 0 -k 1 ] β(1+)eu'[z 1 +E+(1+)k 1 ] β(1+)eu'[z 1 +(1+)k 1 ] k 1 * k 1 * ' k 1 Incease in uncetainty Compaative Static with u 000 > 0: Mean Peseving Spead thee possibilities: 0 U ( ) linea a 1 constant U 0 ( ) convex: RHS ises a inceases U 0 ( ) concave: RHS falls a deceases 1 1 intospection: a 1 inceases U 0 ( ) is convex U 000 > 0 CRRA: U 0 (c) =c σ fo σ >0 is convex somewhat unavoidable: U 0 (c) > 0 and c 0 U 0 (c) stictly convex nea 0 and 4.2 Longe Hoizon i.i.d. income shocks T = 6

7 Bellman equation FOC fom Bellman V (x) =max{u (x a 0 )+βev (Ra 0 +ỹ)} u 0 (c) =βr EV 0 (Ra +ỹ) again: V 0 convex pecautionay savings but V 000 endogneous! esult: u 000 > 0 then v 000 > 0 (Sibley, 1975) 4.3 CARA CARA pefeences u (c) = exp ( γc) V (x) =max{u (x a 0 )+βev (Ra 0 +ỹ)} a no boowing constaints (except No-Ponzi) no non-negativity fo consumption guess and veify: whee λ 1+ note with CARA V (x) =Au (λx) u (a + b) = u (a) u(b) veifying V (x) = max{u (x a 0 )+βaeu (λ (Ra 0 +ỹ))} µ ½ µ µ µ µ ¾ 1 1 V (x) = u 1+ x max u a0 x + βaeu a 0 x + ỹ R R R ³ n ³ o V (x) = u x max u ( α 0 )+βaeu α 0 + ỹ R R 7

8 whee α 0 = a 0 x/r o equivalently c = x α 0 1+ confims guess. Solving fo A : n ³ o A = max u ( α 0 )+βaeu α 0 + ỹ R ³ u 0 ( α 0 ) = βaeu 0 α 0 + ỹ ³ R u ( α 0 ) = βaeu α 0 + ỹ R wheeweused u 0 (c) = γu(c) ³ A = u ( α 0 )+βaeu α 0 + ỹ R = u ( α 0 )+ u ( α0 ) 1+ = u ( α 0 ) (note A > 0) coming back ³ u ( α 0 ) = β ( u ( α 0 )) Eu α 0 + ỹ ³ R u ( α 0 ) = β (1 + ) Eu ỹ R 1 ³ ³ α 0 = u 1 β (1 + ) Eu ỹ R 1 1 ³ = u 1 (β (1 + )) + Eu ỹ R veifying c(x) =λx + α using Eule... u 0 (c t ) = βr E t u 0 (c t+1 ) 1 = βr E t u 0 (c t+1 c t ) 1 = βr E t u 0 (c (x t+1 ) c(x t )) 1 = βr E t u 0 (λ (x t+1 x t )) since x t+1 = Ra t+1 + y t+1 and a t+1 = α 0 + x t /R x t+1 x t = R (α 0 + x t /R)+y t+1 x t = Rα 0 + y t+1 8

9 same as befoe ³ 1=βRE t u 0 α 0 + y t+1 R Veifying value function (again) note that u 0 (c) = γu(c) Then welfae Veifying u 0 (c t )=βr E t u 0 (c t+1 ) u (c t )=βre t u (c t+1 ) E t u (c t+1 )=(βr) t u (c t ) cuent consumption: X X 1+ V t β t E t u (c t+s )= β t (βr) t u (c t )= u (c t ) s=0 s=0 1+ c t = λx t V (x) = u (λx α 0 ) consumption function 1 1 log (β (1 + )) c (x) = λ x + y γ suppose βr = 1 no CEQ... y 1 λ u 1 [Eu (λy)]...but simple deviation: constant y CARA, the good: tactable useful benchmak helps undestand othe cases good fo aggegation (lineaity) CARA, the bad: negative consumption unbounded inequality 9

10 5 Income Fluctuation Poblem iid income y t c t 0 boowing constaints 5.1 Boowing Constaints: Natual and Ad Hoc natual boowing constaint maximize boowing given c t 0 c t 0 +No-Ponzi ad hoc boowing constaint: y min a t a t φ φ = min{y min /, b} Bellman V (x) = max {u ( x a 0 )+βev (Ra 0 +ỹ)} a 0 φ change of vaiables â t = a t + φ and â t+1 0 z t = Râ t + y t φ z t = â t + c t tansfomed poblem v (z) =max {u (z aˆ0 )+βev(râ 0 +ỹ φ)} â 0 0 (doppingbnotation) v (z) =max {u ( z a 0 )+βev(ra 0 +ỹ φ)} a

11 5.2 Popeties of Solution βr= 1 CARA: E [a t+1 ] >a t and E [c t+1 ] >c t Matingale Convegence Theoem: If x t 0 and x t E [x t+1 ] then x t x (note: x< a.e.) Eule u 0 (c t )=βre [u 0 (c t+1 )] u 0 (c t ) conveges c t c if c < contadiction with budget constaint equality a t and c t βr< 1 Bellman equation v (z) =max{u (x a 0 )+βev (Ra 0 +ỹ φ)} a 0 v is inceasing, concave and diffeentiable Peview of Popeties monotonicity of c (z) and a 0 (z) boowing constaint is binding iff z z if then assets bounded u 00 (c) lim =0 c 0 u 0 (c) 11

12 if u HARA class c (z) is concave (Caol and Kimball) Figues emoved due to copyight estictions. See figues Ia and Ib on p. 667 in Aiyagai, S. Rao. "Uninsued Idiosyncatic Risk and Aggegate Savings." Quately Jounal of Economics 109, no. 3 (1994): boowing constaints cetainty: [0,z ] lage appoached monotonically uncetainty: [0,z ] elatively small not appoached monotonically concavity of v concavity of Φ (a 0 )=βev (Ra 0 +ỹ) standad consumption poblem with two nomal goods v (z) = max{u (c)+ (a 0 )} c,a 0 c + a 0 x a 0 0 c (z) and a 0 (z) ae inceasing in z 12

13 FOC (Eule) equality if a 0 > 0 define u 0 (x a 0 ) βrev 0 (Ra 0 +ỹ) u 0 (z )= βr Ev 0 (ỹ) z z c = z a 0 =0 Assets bounded above not a technicality......emembe CARA case idea: take a income uncetainty unelated to a (i.e. absolute isk) u 00 0 income uncetainty unimpotant u 0 βr bites a 0 <a falls Poof exist a z such that zmax 0 =(1+ ) a 0 (z)+ y max z fo z z Eule Eu 0 (c (z 0 )) u 0 (c (z)) = β (1 + ) u 0 ( c (z)) u 0 ( c (z)) whee c (z) = c (z 0 max (z)) = c (a 0 (z)+ y max φ) E [u 0 (c (z 0 ))] IF lim =1 DONE z u 0 ( c (z)) Eu 0 (c (z 0 )) u 0 (c (z)) u 0 ( c (z) ( c (z) c (z))) 1 u 0 ( c (z)) u 0 ( c (z)) u 0 ( c (z)) since a 0 is inceasing c (z) c (z) = c (Ra 0 (z)+ y max φ) c (Ra 0 (z)+ y min φ) < y max y min 13

14 Eu 0 (c (z 0 )) u 0 ( c (z) (y max y min )) 1 u 0 ( c (z)) u 0 ( c (z)) Since z a 0 (z),c(z) then c (z) = c (a 0 (z)+ y max φ). Apply Lemma below. Lemma. fo A > 0 u 0 (c A) 1 u 0 (c) Poof. 1 u 0 Z (c A) A u 00 (c s) = 1+ ds u 0 (c) 0 u 0 (c) since u0 (c s) > 1 fo all t > 0 u 0 (c) so u0 (c A) u 0 (c) 1. = 1 = 1 1 Z A 0 Z A 0 Z A 0 Z A 0 u 0 (c u 0 ( u 0 (c u 0 ( 00 s) u (c s) c) u0 ds (c s) s) γ (c s) ds c) γ (c s) ds γ (c s) ds 0 6 Lessons fom Simulations Fom Deaton s Saving and Liquidity Constaints (1991) pape: impotant boowing constaint may bind infequently (wealth endogenous) maginal popensity to consume highe than in PIH 14

15 Figue emoved due to copyight estictions. See Figue 1 on p in Deaton, Angus. Saving and Liquidity Constaints. Econometica 59, no. 5 (1991):

16 Figue emoved due to copyight estictions. See Figue 2 on p in Deaton, Angus. Saving and Liquidity Constaints. Econometica 59, no. 5 (1991):

17 Figue emoved due to copyight estictions. See Figue 4 on p in Deaton, Angus. Saving and Liquidity Constaints. Econometica 59, no. 5 (1991):

18 consumption smoothe tempoay shocks hade with pemanent shocks 7 Invaiant Distibutions initial distibution F 0 (z 0 ) laws of motion z 0 = Ra 0 (z)+ y 0 geneate F 0 (z 0 ) F 1 (z 1 ) F 1 (z 1 ) F 2 (z 2 ) steady state: invaiant distibution esult: 1. exists 2. unique 3. stable F (z) F (z) key: bound on assets and monotonicity A () E (a 0 (z)) continuous not necessaily monotonically inceasing in income vs. substitution; and w() effect typically: monotonically inceasing A () as R β 1. 18

19 8 Geneal Equilibium GE effects of pecuationay savings? moe k, lowe how much? 8.1 Huggett: Endowment endowment economy no govenment zeo net supply of assets idea: any pecuationay saving tanslates to lowe equilibium inteest ate computational GE execise: CRRA pefeences boowing constaints 8.2 Aiygai adds capital y t = wl t and l t is andom; w is economy-wide wage P N is given by N = l i p i define steady state equilibium: 3equations /3unknowns: (K,, w) Z A (z,,w) df (z;, w) φ = K = F k (K, N) δ w = F N (K, N) 19

20 solve w () and substitute: Z A GE () = a (z,,w ()) dμ (z;, w ()) = K intesect with = F k (K, N) δ A GE () continuous not necessaily monotonically inceasing in (a) income vs. substitution; (b) w() effect typically: monotonically inceasing A () as R β 1 Figues emoved due to copyight estictions. See Figues IIa and IIb on p. 668 in Aiyagai, S. Rao. "Uninsued Idiosyncatic Risk and Aggegate Savings." Quately Jounal of Economics 109, no. 3 (1994): compaative statics b A (0,b) > 0 typically: A (, b) > 0 b 20

21 σ 2 y A Table emoved due to copyight estictions. See Table II on p. 678 in Aiyagai, S. Rao. "Uninsued Idiosyncatic Risk and Aggegate Savings." Quately Jounal of Economics 109, no. 3 (1994): wealth distibution: not as skewed tansition? monotonic? 9 Inequality CEQ-PIH and CARA inequality inceases linealy unbound inequality CRRA inequality inceases initially bounded inequality 21

22 Figue emoved due to copyight estictions. See Figue 2 on p. 444 in Deaton, Angus, and Chistina Paxson. "Intetempoal Choice and Inequality." Jounal of Political Economy 102, no. 3 (1994):

23 Figue emoved due to copyight estictions. See Figue 4 on p. 445 in Deaton, Angus, and Chistina Paxson. "Intetempoal Choice and Inequality." Jounal of Political Economy 102, no. 3 (1994):

24 Figue emoved due to copyight estictions. See Figue 6 on p. 450 in Deaton, Angus, and Chistina Paxson. "Intetempoal Choice and Inequality." Jounal of Political Economy 102, no. 3 (1994):

25 Figue emoved due to copyight estictions. See Figue 1d) on p. 769 in Heathcote, Jonathan, Kjetil Stoesletten, and Giovanni L. Violante. "Two Views of Inequality Ove the Life-Cycle." Jounal of the Euopean Economic Association 3, nos. 2-3 (2005): Deaton and Paxson Revisionisist (Heathcoate, Stoesletten, Violante) Guvenen Stoesletten, Telme and Yaon: 10 Life Cycle: Consumption tacks Income Caoll and Summes: 11 Othe Featues and Extensions Social Secuity: Hubbad-Skinnne-Zeldes (1995): Pecautionay Savings and Social Secuity Scholz, Seshadi, and Khitatakun (2006): Ae Ameicans Saving "Optimally" fo Retiement? Medical Shocks: Palumbo (1999) 25

26 Figue emoved due to copyight estictions. See Figue 1 in Guvenen, Fatih. "Leaning You Eaning: Ae Labo Income Shocks Really Vey Pesistent?" Ameican Economic Review. (Fothcoming) Figue 9 Figue emoved due to copyight estictions. See Figue 1 on p. 613 in Stoesletten, Kjetil, Chis Telme, and Ami Yaon. "Consumption and Risk Shaing ove the Life Cycle." Jounal of Monetay Economics 51, no. 3 (2004): Figue 10 26

27 Figue emoved due to copyight estictions. See Figue 5 on p. 624 in Stoesletten, Kjetil, Chis Telme, and Ami Yaon. "Consumption and Risk Shaing ove the Life Cycle." Jounal of Monetay Economics 51, no. 3 (2004): Figue 11 Figue emoved due to copyight estictions. Figue 12 27

28 Figue emoved due to copyight estictions. Figue 13 Figue emoved due to copyight estictions. Figue 14 28

29 Figue emoved due to copyight estictions. Figue 15 Leaning Income Gowth: Guvenen (2006) Hypebolic pefeences: Hais-Laibson Leisue Complementaity Aguia-Hust (2006): Consumption vs. Expenditue Attanasio-Webe: Demogaphics and Taste Shocks 29