Lecture 10: Recursive Contracts and Endogenous Market Incompleteness

Transcription

1 Lecture 0: Recurive Contract and Endogenou Market Incompletene Florian Scheuer Why are the market for inuring againt idioyncratic rik imperfect/miing? Methodologically: recurive contract ( dynamic programing quared ) How can we look for evidence about thee mechanim? Frictionle benchmark One rik-neutral planner, one rik avere houehold Both dicount the future at rate β State of the world: {, 2,, S} Output : y() Effort e t Effort affect probability ditribution over tate: Pr( t e t ) Exogenou income i a pecial cae Social planner want to maximize houehold utility.t. budget contraint (Dual problem): planner want to maximize profit.t. utility of the houehold Planner chooe a equence of effort e and a equence of conumption c Timing:. Effort choen 2. State realized 3. Conumption choen

2 Way in which thi etup can be generalized: Pr( t e t ) if effort ha effect for more than on period Pr( t e t, t ) for the non iid cae Multiple houehold Rik-avere planner Different dicount rate between planner and houehold Planner problem: FOC:.t. V(w 0 ) max c,e Pr( t e)β t t Pr( t e)β t ( y( t ) c( t ) ) () t [ u ( c( t ) ) ] e( t ) = w 0 Pr ( t e ) β t + µ Pr( t e)β t u ( c( t ) ) = 0 u ( c( t ) ) = µ (2) Full inurance (becaue planner i rik-neutral) Conumption contant over time (becaue planner dicount at the ame rate a houehold) For many-houehold cae: u ( c i ( t ) ) u ( c j ( t ) ) = µ j µ i Ratio of marginal utilitie of two houehold contant over time Remember from the beginning of the coure that thi condition i what held when there were complete market 2 Recurive repreentation Promied utility a a tate variable on the planner problem V(w) = max Pr( e) [ (y() c()) + βv ( w () )] (3) e,c(),w ().t. Pr( e) [ u (c()) e + βw () ] = w (4) 2

3 Contraint (4) i ometime known a a promie keeping contraint: imagine that the planner ha promied the houehold a level of utility w and mut now keep the promie. FOC: Pr( e) + µ Pr ( e) u (c()) = 0 u (c()) = µ Pr( e)βv ( w () ) + µ Pr( e)β = 0 V ( w () ) = µ o continuation utilitie are alo equated acro tate Envelope condition: V (w) = µ (5) o continuation utility i the ame a current utility. We know thi from the equence problem too. Promied utilitie a tate variable are epecially ueful when we introduce variou contraint into program () 3 Moral hazard Now uppoe the houehold exert unobervable effort. V(w) =.t. max e,c(),w () Pr( e) [ (y() c()) + βv ( w () )] Pr( e) [ u (c()) e + βw () ] = w Pr( e) [ u (c()) e + βw () ] Pr( ẽ) [ u (c()) ẽ + βw () ] ẽ Special cae: e {0, } and the planner want to implement e = alway FOC: Pr( ) + µ Pr( )u (c()) + λ [Pr ( ) Pr ( 0)] u (c()) = 0 u (c()) = µ + λ [ Pr( 0) ] Pr ( ) (6) 3

4 Pr ( ) βv ( w () ) + µ Pr ( ) β + λβ [Pr ( ) Pr ( 0)] = 0 V ( w () ) = ( [ µ + λ Pr( 0) ]) Pr ( ) (7) Hence, Interpretation: u (c()) = V (w ()) (8) MRS between conumption and future promied utility: u (c()) β MRT between conumption and future promied utility: βv (w ()) no ditortion in the optimal tradeoff between incentive in the form of current conumption veru promied future utility The envelope condition (5) till hold, o uing (7): V ( w () ) [ = V (w) λ Pr( 0) ] Pr ( ) (9) Note that o V (w) i a Martingale ( E Pr( 0) ) Pr ( ) e = = ( Pr( ) Pr( 0) Pr ( ) ) = 0 Furthermore, uing (8) and (9): [ ] E u (c t+ ) = u (c t ) (0) Thi equation i known a the invere Euler equation, a oppoed to the regular Euler equation, which i E [u (c t+ )] = u (c t ) for the cae of βr =. It firt how up in Rogeron [985]. 3. The meaning of the invere Euler equation The uual Euler equation equate the expected marginal utility of a dollar over time. The invere Euler equate the expected marginal cot of a unit of utility over time. There i alway a way for the planner to change the timing of utility proviion, leaving the houehold indifferent and with the ame incentive 4

5 Provide util uing today conumption - reduce util uing tomorrow conumption (in every tate) Cot of providing util today: u (c t ) Expected aving from reducing util in every tate of the world tomorrow: [ ] β Pr( t+ ) t+ βu (c ( t+ )) = E u (c t+ ) Becaue thi cheme change utility by the ame amount in every tate, it doe not change incentive For there to be no that make thi a good idea, the IEE mut hold Note that the houehold i aving-contrained. Jenen inequality implie Therefore [ ] E u (c t+ ) > E [u (c t+ )] u (c t ) > E [u (c t+ )] E [ u (c t+ ) ] > u (c t ) Under the optimal allocation, if the houehold could ave at a rate R = β rate the planner can give it), it would want to do o. (which i the However, for incentive reaon, it i optimal not to let it ave Note that for thi to work it ha to be poible for the planner to oberve conumption/aving. Otherwie, we have a hidden aving problem. Exercie: where in the math do we ee that if incentive contraint don t bind the tandard Euler equation hold? 3.2 Immieration Propoition. Conumption converge to zero almot urely Proof. IEE implie that u follow a nonnegative Martingale (c t ) 5

6 By the Martingale Convergence Theorem, thi implie that u converge to a random (c t ) variable a.. (i.e. it converge to a contant but we don t know what contant) The contant cannot be poitive becaue that would mean full inurance, violating incentive contraint u (c t ) a.. 0 u (c t ) a.. c t a.. 0 (or to ubitence level under Stone-Geary preference) Note: thi i optimal! Note that commitment i required: the houehold would prefer to quit the cheme eventually Atkeon and Luca [992] how the immieration reult in general equilibrium for an economy a fixed aggregate endowment and hidden preference hock (rather than hidden effort). Special cae olved explicitly. Farhi and Werning [2007] characterize optimal allocation with OLG and Pareto weight on future generation. Immieration reult goe away. 3.3 Some tetable implication Conumption poitively correlated with current income (auming MLRP). Thi follow from (6) Everything about the pat that can help predict current income i ummarized by promied utility w Becaue of (8), u i a ufficient tatitic for today promied utility, o nothing (c t ) ele about the pat hould help predict current conumption 4 Limited commitment Suppoe income i exogenou (no effort choice) Suppoe houehold cannot commit to make payment to the planner and/or thee payment cannot be enforced. Recall that firt bet allocation will ometime dictate c() < y(). 6

7 Timing: Income realized If allocation dictate c() > y(), the planner tranfer c() y() to the houehold (aume for now that the planner can commit) If allocation dictate c() < y(), the houehold decide how much to give to the planner Conumption take place (aume the houehold cannot ave) Planner only mean of enforcement i to threaten the houehold with no longer inuring it if it mie a payment Let the value of autarky be v aut = t=0 ) β ( t Pr()u (y()) = Pr()u (y()) β At every hitory, the allocation ha to be better than autarky for the houehold Problem i V(w 0 ) max Pr( t )β t ( y( t ) c( t ) ) c t.t. Pr( t )β t u ( c( t ) ) = w 0 ( ) t ( ) u c( j ) + β Pr( j+t j )β t u c( j+t ) u ( y ( )) j + βvaut j+t j j In principle, thi could be very complicated: hitory-dependent allocation, S contraint for each hitory, etc. Recurive repreentation: V(w) = max Pr() [ (y() c()) + βv ( w () )] c(),w ().t. Pr() [ u (c()) + βw () ] = w () u (c()) + βw () u (y()) + βv aut (2) 7

8 FOC: Pr() + µ Pr()u (c()) + λ()u (c()) = 0 u (c()) = Pr() µ Pr() + λ() (3) Pr()βV ( w () ) + µ Pr()β + βλ() = 0 ( ) V (w µ Pr() + λ() ()) = Pr() (4) From (3) and (4): u (c()) = V (w ()) (5) Interpretation: a before, no ditortion in the tradeoff between tatic and dynamic incentive proviion (MRS=MRT) Envelope condition (5) till hold. Uing (5) in (4): V (w ()) = V (w) λ() Pr() V (w) never goe up w never goe down. (Concavity of the planner value function need to be hown) Given any w, divide tate into thoe where (2) bind and thoe where it doe not. When (2) bind: λ() > 0 V (w ()) < V (w) w () > w Binding contraint implie c() and w () olve: u (c()) + βw () = u (y()) + βv aut (6) u (c()) = V (w ()) Amneia : conumption and promied value do not depend on hitory! In particular, they do not depend on w 2. When (2) doe not bind: λ() = 0 V (w ()) = V (w): promied value doe not change 8

9 u (c()) = V (w : conumption i the ame acro tate ()) Note that (2) will bind for high tate (if at all) Implied dynamic:. A long a (2) doe not bind, conumption i contant 2. A oon a (2) bind, et conumption and promied utility to whatever level i conitent with the ytem of equation (6). Thi level will be higher than at the beginning 3. Keep conumption contant at the new level until (2) bind again (for higher income realization). Adjut conumption and promied utility to the level conitent with the ytem of equation (6) under that y(). 4. Eventually, tate S will be realized. After that, conumption i contant forever 4. Some tetable implication [Kocherlakota, 996] Conumption will have the following pattern: 9

10 Conumption poitively correlated with current income Everything about the pat that can help predict current income i ummarized by promied utility w Becaue of (5), u i a ufficient tatitic for today promied utility, o nothing (c t ) ele about the pat hould help predict current conumption Moreover, becaue of the amneia property for houehold that increae conumption, nothing about the pat hould predict current conumption, but current income hould predict conumption Intead, for the ret of the houehold, current income hould not predict conumption 4.2 Two-ided limited commitment [Thoma and Worrall, 988] In the one-ided limited commitment problem, the principal will be making profit at firt and loe later (auming, e.g. V(w 0 ) = 0) What if the principal can alo not commit? V(w) = max Pr() [ (y() c()) + βv ( w () )] c(),w ().t. Pr() [ u (c()) + βw () ] = w u (c()) + βw () u (y()) + βv aut y() c() + βv(w ()) 0 FOC: Pr() + µ Pr()u (c()) + λ()u (c()) η() = 0 u (c()) = Pr() + η() µ Pr() + λ() (7) Pr()βV ( w () ) + µ Pr()β + βλ() + η()βv ( w () ) = 0 V (w µ Pr() + λ() ()) = Pr() + η() (8) Equation (5) and the envelope condition (5) till hold Uing (5): V (w ()) = V (w) Pr() λ() Pr() + η() 0

11 Now V (w ()) may go up or down depending on which (if any) of the contraint bind General pattern If houehold contraint bind, promied utility (and conumption) go up If planner contraint bind, promied utility (and conumption) go down If no contraint bind, promied utility (and conumption) i contant (it eay to how that both contraint won t bind at once) Amneia: contract reet whenever any of the contraint bind Kehoe and Levine [993], Alvarez and Jermann [2000]: competitive equilibrium approach (rather than optimal contracting approach) to the limited commitment problem. Welfare theorem. Implication for aet price. 5 I the data conitent with optimal inurance under moral hazard? [Ligon, 998] Pure borrowing/aving implie tandard Euler equation. Under CRRA and βr = : [ (ct+ ) σ E t ] = 0 c t Optimal inurance with moral hazard implie Invere Euler Equation. Under CRRA and βr = : [ ( ) σ ct E t ] = 0 c t+ Etimate: [ ( ) b ct+ E t ] = 0 (9) c t Since we think σ > 0 b > 0 IEE optimal inurance with moral hazard b < 0 EE pure borrowing/aving Full inurance would imply both EE and IEE hould hold. Literally, thi require contant conumption

12 Allowing for meaurement error, aggregate hock, etc, if we can forecat c t+ jut a well uing EE or IEE, thi i evidence of full inurance Equation (9) can be etimated by GMM, uing any information known at time t a intrument, i.e. etimate: [( ( ) ] b ct E ) x t = 0 c t+ Data from three village in India x : income, landholding, familiy ize, rainfall, village conumption 6 I the data conitent with other form of contrained optimal inurance? [Kinnan, 20] Data from Thai village Tet ome of the prediction of contrained-optimal inurance model (moral hazard, limited commitment and other, including hidden income or pure borrowing and aving with no inurance a in Aiyagari, 994) Who care? Example: India ha a National Rural Employment Guarantee Act. Doe thi crowd out exiting inurance? 2

13 Under moral hazard, perhap (lower penalty for low effort) Under limited commitment, maybe too (higher value of autarky) Under CRRA utility: o u (c t ) = c σ t log c t = σ log u (c t ) o prediction about u (c t ) can be teted by looking at log (c t) 6. I inurance imperfect? log c it = α log y it + β i + ɛ it Houehold bear idioyncratic rik: a % increae in income i aociated with 0.06% to 0.2% increae in conumption 3

14 But village do provide inurance (village-year fixed effect highly ignificant), conitent with the finding in Townend [994]. 6.2 I u (c t ) a ufficient tatitic for u (c t )? log c it = γ log c it + ζx it + ɛ it X i any other variable, for intance pat income Thi would ugget that moral hazard alone cannot account for the data Same ort of regreion reject both the Euler equation and the invere Euler equation But find evidence in favor of hidden income: planner cannot oberve realization of y(), mut provide incentive for agent to report it truthfully 6.3 I there amneia? Strictet tet: for anyone that ha conumption growth above village minimum, the pat hould not matter Slightly more lenient tet: etimate log c it = γ log c it + ζy it + ɛ it 4

15 allowing interaction coefficient with what quartile of the conumption-growth ditribution a houehold i in, i.e. log c it = γ log c it γ q log c it I (q) + ζy it + ζ q log Y it I (q) + ɛ it q=2 q=2 The trict amneia prediction ay that you hould find γ + γ 4 = 0 or at leat γ 4 < γ 3 < γ 2 < 0 (The argument i that high-conumption-growth houehold are thoe for whom the reneging contraint are more likely to be binding, o the pat hould matter le for them) Intead, we find γ 4 > γ 3 > γ 2 > 0 oppoite of what the limited-commitment model would ay 5

16 Reference Fernando Alvarez and Urban J. Jermann. Efficiency, equilibrium, and aet pricing with rik of default. Econometrica, 68(4): , July

17 Andrew Atkeon and Jr Luca, Robert E. On efficient ditribution with private information. Review of Economic Studie, 59(3):427 53, July 992. Emmanuel Farhi and Iván Werning. Inequality and ocial dicounting. Journal of Political Economy, 5: , Timothy J Kehoe and David K Levine. Debt-contrained aet market. Review of Economic Studie, 60(4):865 88, October 993. Cynthia Kinnan. Ditinguihing barrier to inurance in thai village. Northwetern Univerity Working Paper, 20. Narayana R Kocherlakota. Implication of efficient rik haring without commitment. Review of Economic Studie, 63(4): , October 996. Ethan Ligon. Rik haring and information in village economic. Review of Economic Studie, 65(4):847 64, October 998. William P Rogeron. The firt-order approach to principal-agent problem. Econometrica, 53 (6):357 67, November 985. Jonathan Thoma and Tim Worrall. Self-enforcing wage contract. Review of Economic Studie, 55(4):54 54, October 988. Robert M Townend. Rik and inurance in village india. Econometrica, 62(3):539 9, May