Persistent Private Information

Transcription

1 Persistent Private Information Noah Williams University of Wisconsin nwilliam Persistent Private Information p. 1

2 Overview Part of ongoing project on continuous time dynamic contracting models and applications. Many economic environments consider dynamic incentive provision with hidden actions or hidden information. Insurance, employment contracting, recent literature on economic policy. Incentive constraints may be difficult to handle in a dynamic setting. Even more difficult with hidden states (hidden savings, persistent hidden information). Little known thus far about quantitative implications in dynamic setting, especially with nontrivial dynamics. Continuous time methods simplify analysis of contracting problems, making possible quantitative analysis. Persistent Private Information p. 2

3 Why/How Does Continuous Time Help? Important to allow for history dependence in contract. Implies history dependence in income, consumption, etc. Formulate as agent s action or report changing the distribution of observed outcomes. Convenient methods for changes of measure (Girsanov) in continuous time. Similar to agency literature: change from state space to probability shift. Here build from state space structure. Variances are observed in continuous time diffusion setting, so information asymmetry in (local) means only. Easier to insure local concavity is preserved. Allows use of local optimality, first-order conditions. Leads directly to convenient state variables that capture history dependence. Not only analytic convenience: some results differ. Persistent Private Information p. 3

4 Particular Focus Problems with persistent private information. Set up as principal/agent model with hidden state variables. Version of Thomas & Worrall (1990) (like Green, 1987) in continuous time with persistent unobserved endowments. Most models with private info only characterize i.i.d. case (some exceptions). But idiosyncratic income shocks are highly persistent (cf Storesletten-Telmer-Yaron), likely to have sizeable private info component. Discrete time + i.i.d. optimal contract leads to immiserization. Inverse Euler equation governs consumption dynamics. Neither holds here: Examples where agent consumption increases over time. In i.i.d. limit, obtain efficiency. Persistent Private Information p. 4

5 Some Related Literature Hidden info models: Fernandes-Phelan (2000), Demarzo- Sannikov (2006), Sannikov (2006), Sung (2006), Zhang (2007), Battaglini (2005), Tchistyi (2006), Kapicka (2006) Relative to these: continuous time, continuous states, risk averse agent, persistent private information. I also use different methods (as in Williams, 2006). Stochastic max principle: Bismut (1973, 1978), Peng (1990), Yong-Zhou (1999). Sufficiency: Haussmann (1986), Zhou (1996). BSDEs: Pardoux and Peng (1990). Leads to 2 state variables: promised utility, promised marginal utility. Similar ideas: Abreu-Pearce-Stachetti (1986, 1990), Spear-Srivastrava (1987), Kydland-Prescott (1980), Werning (2001), Abraham-Pavoni (2002) Persistent Private Information p. 5

6 Persistent Private Information Continuous time version of Thomas and Worrall (1990) with persistent income process. Textbook (i.e. Ljungqvist-Sargent) dynamic private information model. Similar models underly optimal taxation literature, many others. Risk averse agent would like to borrow from risk neutral lender to stabilize income stream (unobservable to lender). Full stabilization (full info solution) not incentive compatible: agent always has incentive to lie and say income low. Thomas & Worrall: discrete time, discrete i.i.d. endowment Here: continuous time, continuous persistent endowment Persistent Private Information p. 6

7 Endowment Process db t = µ(b t )dt + σdw t = [µ 0 λb t ]dt + σdw t Focus on continuous time linear AR process, λ 0. Slightly more general in paper - allow for b to be a preference shock (like Atkeson-Lucas). Also allow b to be transformation (i.e. log) of privately observed process. Always persistent. λ = 0 permanent, i.e. b t = b 0 + µ 0 t + σw t Persistent Private Information p. 7

8 Endowment Process With λ > 0 have a stationary O-U process with statistics: E(b t b 0 ) = µ ( 0 λ + b 0 µ 0 λ Cov(b t, b s b 0 ) = σ2 2λ ) e λt, E(b t ) = µ 0 λ (e λ s t e λ(s+t)), V ar(b t ) = σ2 2λ Approximate i.i.d. process: λ with σ = σ λ, µ 0 = µλ. 1.3 Endowment, λ = Endowment, λ = Time Time Persistent Private Information p. 8

9 Conversion to Hidden Action Problem Solve using revelation principle. Convert reporting choice to a hidden action problem. Agent reports to principal income y t. Must be absolutely continuous w.r.t. b t (have nonzero prob) or else lie detected. Given Brownian information structure, report is endowment with drift perturbed by t : dy t = db t + t dt y t = b t + dy t dm t s 0 s ds b t + m t = [µ(y t m t ) + t ]dt + σdw t = t dt Hidden action ( t ) and hidden state (m t ): stock of lies. Persistent Private Information p. 9

10 Hidden Action and Hidden State dy t dm t = [µ(y t m t ) + t ]dt + σdw t = t dt Assume agent cannot over-report income: t 0, m t 0. Idea: agent deposits y t in account monitored by principal Hidden State, m t m t Report y t Endowment b t Time Time Persistent Private Information p. 10

11 Contract and Change of Variables Principal observes only report y t. Entire history: ȳ = {y t : t [0, T]}. Contract specifies payment s t = s(t,ȳ) predictable, depends on history up to t. Given a contract, agent s consumption depends on whole history of past reports. Makes reporting decision difficult to handle directly: u(c t ) = u(b t + s t (ȳ)) = u(y t m t + s t (ȳ)) Consumption does not respond instantaneously to report. Current lie affects future consumption through m t and s t. For fixed report, principal tries to infer drift (including lie) vs. diffusion. Lying changes distribution of observed reports. Persistent Private Information p. 11

12 Distribution of Reports Endowment b t Time Persistent Private Information p. 12

13 Lying Changes the Distribution of Reports Report y t 0.75 Endowment b t Time Persistent Private Information p. 13

14 Change of Variables W 0 t : Wiener on C[0, T], i.e. report W 0 t. Other reports: dw y t = dw 0 t [µ(y t m t ) + t ]dt Under P 0, W 0 is a Brownian motion. Under P y, y t is. Analyze density of change of measure, Γ t = E 0 [ dp y dp 0 F t ], and its interaction with stock of lies z t = Γ t m t. They evolve as: dγ t = Γ t σ [µ(y t m t ) + t ]dw 0 t dz t = Γ t t dt + z t σ [µ(y t m t ) + t ]dw 0 t Persistent Private Information p. 14

15 Agent Reporting Problem Agent s preferences (note c t = b t + s t = y t m t + s t ): E y [ T 0 = E 0 [ T ] e ρt u (y t m t + s t (ȳ)) dt + e ρt U(y T m T + S T (ȳ)) 0 ] e ρt Γ t u (y t m t + s t )dt + e ρt Γ T U(y T m T + S T ) Agent chooses lying strategy { t } to maximize utility given contract s t (y), subject to evolution: dγ t = Γ t σ [µ(y t m t ) + t ]dw 0 t dz t = Γ t t dt + z t σ [µ(y t m t ) + t ]dw 0 t Persistent Private Information p. 15

16 Optimality Conditions Apply stochastic maximum principle. Hamiltonian for reporting (lying) problem: H(Γ, z) = Γu (y z/γ + s(y))+(γγ+qz) (µ(y z/γ) + )+pγ (q t, γ t ) co-states associated with Γ t, (p t, Q t ) co-states associated with z t. Agent maximizes H over t 0. Optimal to not lie currently when: γ + Qm + p 0 So truthful reporting (m = 0): γ p. Incentive constraint (typically) binds: γ = p. Persistent Private Information p. 16

17 Utility & Marginal Utility Processes Evolution of co-states by differentiating Hamiltonian (m t 0): dq t = dp t = [ ρq t [ ρp t ] H(Γ, z) dt + γ t σdwt 0, q T = U(y T s T ). Γ ] H(Γ, z) dt + Q t σdwt 0, p T = U (y T s T ). z Persistent Private Information p. 17

18 Utility & Marginal Utility Processes Evolution of co-states by differentiating Hamiltonian (m t 0): dq t = [ρq t u(c t )]dt + γ t σdw y t, q T = U(c T ). dp t = [ ρp t λγ t + u (c t ) ] dt + Q t σdw y t, p T = U (c T ). Persistent Private Information p. 18

19 Utility & Marginal Utility Processes Evolution of co-states by differentiating Hamiltonian (m t 0): dq t = [ρq t u(c t )]dt + γ t σdw y t, q T = U(c T ). dp t = [ ρp t λγ t + u (c t ) ] dt + Q t σdw y t, p T = U (c T ). Impose incentive constraint γ = p, solve: q t p t [ T ] = E y e ρ(s t) u(c s )ds + e ρ(t t) U(c T ) F t, t [ T ] = E y e (ρ+λ)(s t) u (c s )ds + e (ρ+λ)(t t) U (C T ) F t. t Persistent Private Information p. 18

20 Agent s Utility Process Idea of conditioning on utility process now well known. Form of the evolution in continuous time: dq t = [ρq t u(c t )]dt + γ t σdw y t γ t gives local volatility of agent s utility its loading on new information. It s the key for incentives. Discrete time analogue (interval ε): b t+ε = [b t + µ(b t )]ε + σ εw t+ε q t = u(c t )ε + e ρε E t q t+ε (W t+ε ) Discrete requires choice of continuation utility in each state. Analogue of continuous trading in assets. Persistent Private Information p. 19

21 Interpretation Truthful reporting strategy leads to two endogenous state variables, promised utility q t and (minus) promised marginal utility p t. Full information: γ t = 0 and complete stabilization of utility. With private information, agent utility must respond to report: γ t p t 0. Incentive constraint ties volatility of promised utility to level of promised marginal utility. Mean reversion speed λ determines horizon of private information, thus discount in p t. As λ we approximate i.i.d. process and p t not relevant. Persistent Private Information p. 20

22 Truthful Revelation Contracts Participation constraint: contract must improve upon autarky: q 0 V a (y 0 ). Promise keeping: contract consistent with q t, p t (Instantaneous) Incentive constraint: γ t p t. Theorem: Assume that u and U are increasing and concave, and that λ 0. Then we have: 1. Any truthful revelation contract {s t, γ t, Q t } satisfies (i) the participation constraint, (ii) the promise keeping constraints, and (iii) the incentive constraint. 2. If u bbb 0 any contract satisfying (i)-(iii) and insuring: Q t E [ T t ] e ρ(τ t) [u bb (s τ, y τ ) + 2λQ τ ]dτ F t is a truthful revelation contract. Q t 0 sufficient. Persistent Private Information p. 21

23 Straightforward necessary conditions from above, sufficient condition easy to check but not necessary. Idea of proof: - Use representation of util, MU under contract - Evaluate potential gain from lying change distribution of reports - Concavity bounds difference by linear approximation - When µ(b) = µ 0, IC insures gain nonpositive. - More generally, need to bound term in λq t m 2 t, so Q t 0 is sufficient. - To allow Q t 0, optimize over bound. Persistent Private Information p. 22

24 Optimal Contracts on Infinite Horizon Principal risk neutral, wants to minimize payments: J = E y [ 0 ] e ρt s t dt, Minimizes J subject to y t, q t, p t, incentive constraint. Hamilton-Jacobi-Bellman equation for J(y, q, p): ρj = min {s, γ p, Q} {s + J yµ(y) + J q [ρq u(y + s)] +J p [ ρp γλ + u (y + s) ] + σ2 2 [ Jyy + J qq γ 2 + J pp Q (J yq γ + J yp Q + J pq γq) ]} Typically need to solve numerically. Consider examples where can do analytics. Note q 0 pinned down, p 0 free. Persistent Private Information p. 23

25 Exponential Utility Examples Now assume u(c) = exp( θc). Full information first. Ignore incentive constraint, condition on q t to enforce participation. Constant consumption consistent with q: c(q) = log( ρq) θ Optimal contract: s(y, q, p) = c(q) y, γ = 0, Q = 0. So q t = q 0, c t = c(q 0 ) t. Principal cost J (y, q) = µ 0 ρ(λ + ρ) y λ + ρ } {{ } gain from endowment log( ρq). ρθ }{{} cost of payment Persistent Private Information p. 24

26 Permanent Endowment Case Let λ = 0. Assume µ 0 > θσ 2, so endowment grows faster than risk compensation. Note u (c t ) = θu(c t ). With λ = 0 then have p t = θq t. So only need to condition on q t. Guess & verify principal value of same form as full info: J(y, q, p) = µ 0 ρ 2 y ρ + σ2 θ 2ρ 2 log( ρq) ρθ Additional constant term due to risk-compensation to agent. Same optimal policy function: s = c(q) y, but now:, dq t = θq t dw y t Persistent Private Information p. 25

27 Implications Optimal promised utility is a martingale. Optimal consumption can be written: c t = log( ρq t) θ = c(q 0 ) + σ2 θ 2 t + σw y t Consumption grows over time, due to risk compensation. Relative to autarky, contract only changes time profile of consumption, but has no risk sharing. Contrast with Thomas & Worrall where there is risk sharing, but promised utility tends to and consumption falls over time: immiserization. With permanent endowment (c t, q t ) can fan out over time to provide incentives and yet have c t still grow. Persistent Private Information p. 26

28 Evolution of the Consumption Distribution Mean, Autarky 1 SD, Autarky 2 SD, Autarky Distribution of Consumption under Autarky Consumption Time Persistent Private Information p. 27

29 Evolution of the Consumption Distribution Distributions of Consumption under the Optimal Contract and Autarky Mean, Contract Mean, Autarky 1 SD, Contract 2 SD, Contract 1 SD, Autarky 2 SD, Autarky 12 Consumption Time Persistent Private Information p. 28

30 Additional Notes Principal cost decreasing over time as drift greater than risk compensation. Can implement via deterministic transfer, independent of report: ( σ 2 ) θ s t = c(q 0 ) y µ 0 t. Insures participation constraint holds, trivially incentive compatible. Persistent Private Information p. 29

31 Persistent Endowment Case Now let λ > 0. Contract must now condition on p t. Don t impose sufficient condition (it will fail), so must check incentives ex-post. First let p 0 be given, then maximize over it. Can show that solution of HJB can be written as: J(y, q, p) = j 0 + j 1y j 2 log( q) + h(p/q) where if we let k = p/q, h solves the 2nd order ODE: ρh(k) = 1 ( 1 θ log ρθ + h (k)(k θ) )+λh (k)k+ σ2 k 2 2 ( 1 ρθ h (k) 2 ) h (k) Persistent Private Information p. 30

32 Contract, Arbitrary Initial Condition Contract is of similar form as permanent case: s(y, q, k) = log θ + log(1/ρθ + h (k)(k θ)) log( q) θ θ ( h ) (k) Q(p, k) = p h (k) + k p Q(k) c(q, k) = log(1/ρ + θh (k)(k θ)) θ log( q) θ Dynamics depend on (q, p) or (q, k) equivalently: y, log( qĉ(k)) θ dk t = [ ĉ(k t )(k t θ) + λk t + σ 2 k 2 t (k t Q(k t )) ] dt +σk t (k t Q(k t ))dw t. Persistent Private Information p. 31

33 Optimal Initial Condition: Cost Function Component of Principal Cost, h(k), with λ = Optimal choice: k 0 = ρθ/(ρ + λ), makes k t = k 0 t. Persistent Private Information p. 32

34 Policy Functions 0.12 Agent Consumption Rate, ĉ(k) 2 Component of Volatility of p t, Q(k) k = p/q k = p/q c(q, k0) = log( ρq), ĉ(k) = ρ θ Q(p, k0) = pk0 Persistent Private Information p. 33

35 Implications Promised utility is a martingale, volatility depends on persistence of information: dq t = σρθ ρ + λ q tdw y t. λ = 0: previous case with no risk sharing. i.i.d. limit σ = σ λ, λ : volatility goes to zero efficiency (full info allocation) Cost: J(y 0, q 0, p 0) = J (y 0, q 0 ) + Agent consumption dynamics: σ 2 θ 2(ρ + λ) 2, c t = c(q 0 ) + σ2 θρ 2 2(ρ + λ) 2t + σρ ρ + λ W y t. Persistent Private Information p. 34

36 Consumption Distributions Autarky Consumption Distribution, λ = Autarky Consumption Distribution, λ = Consumption Consumption Time Contract Consumption Distribution, λ = Time Contract Consumption Distribution, λ = Consumption Consumption Time Time Persistent Private Information p. 35

37 Interpretations and Comparisons By tying consumption to shocks, able to achieve utility spread required for incentives but still allow consumption growth. As mean reversion increases, effective life of information is shorter, requiring less utility dispersion. Efficiency in limit due to information structure. By continuity here lie affects future increments of report, and hence future but not current consumption. Thomas & Worrall show that deviations from efficiency tied to cost of inducing efficient action for one period. As period length shrinks, this cost will vanish. DeMarzo & Sannikov study continuous time model where info is i.i.d. but distortions remain. They model info as increment of diffusion process. Problematic with risk averse agents. Persistent Private Information p. 36

38 Inverse Euler equation does not hold here, but does hold in moral hazard model of my earlier paper. There unobservable effort affects future increments of output but has instantaneous utility cost. Here lying has no consequence in current instant, only affects future transfers. So inverse Euler equation not sensitive to continuous or discrete time, but to source of information frictions & their costs. Persistent Private Information p. 37

39 Verifying Truthful Revelation Contract specifies s(y t, q t ) = log( ρq t )/θ y t, so agent consumption is c t = log( ρq t )/θ m t. Under agent s information, promised utility evolves as: dq t = σρθ ρ + λ q tdw y t = ρθ ρ + λ q t (σdw t + [µ(y t m t ) + t µ(y t )]dt) = ρθ ρ + λ q t(λm t + t )dt σρθ ρ + λ dw t, Persistent Private Information p. 38

40 Agent Problem Redux ρv = max V qq { exp( θ[ c(q) m]) V q ( ) } σρθ 2 ρ + λ ρθ ρ + λ q(λm + ) + V m Solution: V (q, m) = q exp(θm)(ρ + λ) ρ + λ + θλm. RHS of HJB increasing in for all m, so truthtelling optimal: V q And V (q, 0) = q. ρθ ρ + λ q + V m = q exp(θm)θ2 λ 2 m (ρ + λ + θλm) 2 0 Persistent Private Information p. 39

41 Conclusions Analysis of problems with persistent private information requires contract to condition on promised utility, promised marginal utility. Immiserization need not hold in my setting. Agent consumption increases over time. Obtain efficiency in the i.i.d. limit. Information structure and costs of deviating are key for properties of contracts. Continuous time approach provides simplifications in deriving optimality conditions, form of endogenous state variables, analytic solutions. In progress considering power utility cases, working toward quantitative versions. Persistent Private Information p. 40