Principal-Agent Problems in Continuous Time

Transcription

1 Principal-Agent Problems in Continuous Time Jin Huang March 11, / 33

2 Outline Contract theory in continuous-time models Sannikov s model with infinite time horizon The optimal contract depends on the agent s continuation value and has a lower and upper boundary A procurement model in finite time horizon The optimal contract depends on the continuation value and the state process 2 / 33

3 Related literature Books: - Discrete-time contract theory: Salanie (1997), Laffont and Martinmort (21), Bolton and Dewatripont (24) - Continuous-time: Cvitanic and Zhang (213) Holmstrong and Milgrom (1987), Schatterler and Sung (1993), Williams (23), DeMarzo and Sannikov (27), Sannikov (28), He (28), Cvitanic and Wang and Zhang (29), He (28), Zhu (212), Cvitanic and Wan and Yang (212) 3 / 33

4 Modeling variations Time horizons: finite or infinite State processes: Brownian motion plus drift, geometric Brownian motion, or mean-reverting Payments: instantaneous payments C t, or lump-sum payment C T at the terminal time Information: moral hazard, adverse selection, and/or learning Modeling approaches: weak or strong formulation 4 / 33

5 The contracting environment of Sannikov s model t [, ) The output (state) process: dx t = A t dt + σdb t Effort: A t Hidden action (moral hazard) Per-period payment: C t Random fluctuations: B t 5 / 33

6 The contracting problem r is the common discounting factor The principal offers C to [ max E r C = max C max A E [ r ] e rt (dx t C t dt) ] e rt (A t C t )dt The agent best-responds with A to [ E r e rt( ) u(c t ) h(a t ) dt ] 6 / 33

7 Principal s constrained problem subject to: (P): E max A,C [ E r ] e rt (A t C t )dt [ r ) ] e (u(c rt t ) h(a t ) dt R (IC): A is agent s optimal response to C. 7 / 33

8 Representations of agent s continuation value Define W t E A [ t r t e rs( u(c s ) h(a s ) ) ds ] By the Martingale Representation Theorem, there is a unique Y such that W t = W +r t ( Ws u(c s )+h(a s ) ) t ds+r Y s dzs A The agent is indifferent between getting a promise of 1. {C s : t s }, or 2. W t 8 / 33

9 Incentive compatibility Proposition (Incentives) The following two conditions are equivalent. 1. A is an optimal response to the payment C. 2. A t (ω) arg max a {Y t (ω)a h(a)}, for almost every (t, ω). It is a continuous-time version of the one-shot deviation principle in that A is optimal if and only if A t is optimal at each moment t. 9 / 33

10 The main idea behind the IC proposition Let A be optimal, and assume that the agent follows A up to time t and switch to an alternative A after time t. V t = r ) t (u(c e rs s ) h(a s) ds + e rt W t Drift of V t : re rt( ) [Y t A t h(a t )] [Y t A t h(a t )] dt 1 / 33

11 Designing the optimal pair (A, C) Proposition (Transversality condition) If E A t [e rs Wt+s ] as s, then W t = W t. W t = W + r t ( Ws u(c s ) + h(a s ) ) ds + r t β sdzs A. Principal must pay the agent eventually. Let γ(a) = {y : a arg max a ya h(a )}. To enforce A, the principal promises W t = W +r t ( Ws u(c s )+h(a s ) ) t ds+r γ(a s )dzs A. 11 / 33

12 Converting the principal s problem into a stochastic control problem Let F (w) = u 1 (w). The principal s problem where [ τ ] max E r e rs (A s C s )ds + e rτ F (W τ ), A,C,τ W t = W + r t ( Ws u(c s ) + h(a s ) ) t ds + r γ(a s )db s. 12 / 33

13 The dynamic programming principal and the HJB equation Let F (w) = max A,C,τ E t [ τ ] r e rs (A s C s )ds + e rτ F (Wτ t,w ). t Following the standard arguments of DPP, F ( ) is a solution of the following ODE: ( ) rf = max a,c r(a c) + r w u(c) + h(a) F r2 γ(a) 2 σ 2 F F () = F (W gp ) = F (W gp ), and F (W gp ) = F (W gp) Let a (w) and c (w) be the maximizers. 13 / 33

14 Description of the optimal contract Theorem (Optimal contract) Let W t = W + r + r t t ( ) W s u(c (W s )) + h(a (W s )) ds γ(a (W s ))db s. The stopping rule τ is the first time W t hits lower boundary W t = or the upper boundary W t = W gp. The payment and requested efforts before τ is A t = a (W t ) and C t = c (W t ) and after τ, A t = and C t = F (W τ ). 14 / 33

15 Features of the contract W t summarizes the past history Lower boundary serves as a punishment scheme for incentives Upper boundary is due to income effect; too costly to compensate for the agent for his effort when W t is too high Probational period for low W t. 15 / 33

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17 Implementation 1. Find W that is the best starting point for the principal. It may be higher than the agent s reservation value R 2. At time t, calculate W t 3. Once W t is known, so is A t = a (W t ) and C t = c (W t ) 4. Retire the agent once W t hits the boundary 17 / 33

18 The contracting environment of a procurement problem t [, T ] The price (state) process is mean reverting: P t = P + t λ(a s P s )ds + t σdm s. Effort: A t Hidden action (moral hazard); the principal cannot contract on actions Lump-sum payment at terminal time: C T Random fluctuations: M t 18 / 33

19 An example: a bilateral contract to supply ancillary services A supplier can manipulate the market price without detection, for example 1. manipulate supplies 2. through virtual bids 3. by proxy The supplier manipulate the price to maximize payment The utility company seeks to pay as little as possible We want to know how to mollify the agent s incentives to manipulate prices through an adjustment C T 19 / 33

20 The contracting problem The agent s problem: max A [ T ( ) ] E U(C T ) + u(p s D s ) h(a s ) ds The principal s problem: min C T [ T ] E C T + P s D s ds 2 / 33

21 The same problem with the ABM state proces subject to: min A,C T [ T ] E C T + P s D s ds 1. Incentive compatibility constraint in that A is an optimal response to C T 2. The participation contraint [ T ( E U(C T ) + u(ps D s ) h(a s ) ) ] ds R 3. The state process is Arithmetic Brownian (not mean-reverting) P t = P + t A s ds + t σdm s 21 / 33

22 First-best (1) 1. Let F ( ) be the solution of the PDE t F (t, p) = min a pd t λu(pd t ) + λh(a) + a p F (t, p) σ2 pp F (t, p) F (T, p) =, for all p, and a,λ (t, p) be the solution of λh (a) + p F (t, p) =. 2. Let C,λ be the solution of 1 = λu (C T ). 22 / 33

23 First-best (2) 3. And λ is found by the participation constraint [ T ( R U(C,λ ) = E where P,λ t u(p,λ s ) ] D s ) h(a,λ (t, Ps,λ )) ds, = P + t a (t, Ps,λ )ds + t σdm s. Proposition (First-best contract) The first best contract is (C,λ, A ), where A = {a,λ (t, P t ) : t T }. 23 / 33

24 First-best (3): Steps to obtain the contract (numerically) 1. Solve the PDE and obtain a,λ ( ) 2. Obtain C,λ by inverting U( ) 3. Find λ by setting participation constraint at equality 4. Calculate (C,λ, A ). 24 / 33

25 Second-best: agent s problem The agent s continuation value is W t = E A t [ U(C T ) + T which has a representation: W A t = U(C T )+ T Proposition (Incentives) t t ] (u(p s D s ) h(a s ))ds, T (u(p s D s ) h(a s )+Z s A s )ds Z s σdb s. t The following two conditions are equivalent. 1. A is an optimal response to the payment C T. 2. A t (ω) arg max a {Z t (ω)a h(a)}, for almost every (t, ω). 25 / 33

26 The resolution of the problem of agency Let γ(a) (resp. η(z)) be the solution of z h (a) = in terms of z (resp. a). 1. Given C T, the response A t = η(z t ) is optimal, where (W, Z) is the unique solution of the BSDE W t = U(C T )+ T t T (u(p s D s ) h(η(z s ))+Z s η(z s ))ds Z s σdb s. t 2. Given (A, W ), the enforcing C T = J(WT A ) where J is the inverse of U( ), and W A t = W t ( ) u(p s D s ) h(a s ) ds + t γ(a s )σdb A s. 26 / 33

27 Second-best: principal s problem Let F be the solution to the PDE t F = min a pd t + a p F σ2 pp F + (h(a) u(pd t )) w F γ(a)2 σ 2 ww F + γ(a)σ 2 pw F F (T, p, w) = J(w), for all p and w, (1) and a (t, p, w) be the optimizer of min a a p F + h(a) w F γ(a)2 σ 2 ww F + γ(a)σ 2 pw F. 27 / 33

28 Second-best: the contract Theorem (Optimal Contract) Let a (t, p, w) be the minimizer in (1), and the agent is paid C T = J( W T ), where P t = P + t a (s, P s, W s )ds + t σdm s W t = R + [ t h(a (s, P s, W ] s )) u(p s D s ) ds + t γ(a (s, P s, W s ))σdm s. Then, the the contract (A, C T ) = ( {a (t, P t, W t ) : t T }, C T ) is incentive compatible for the agent, and optimal for the principal among all incentive compatible contracts that deliver an initial expected value of at least W to the agent. 28 / 33

29 Features of the contract The pair (P t, W t ) summarizes the past history; P t is needed because the restriction on payments There are no boundaries on which the agent is retired. This is due to finite time horizon The calculation of the contract relies on solving a PDE, as opposed to an ODE in infinite time horizon It is never optimal for the principal to gives the agent an initial value more than R 29 / 33

30 An implementation 1. The agent is asked to perform A 2. By time t, the agent is paid t P sd s ds 3. By time t, the agent is also paid R t ( ) u(p s D s ) h(a s) ds + t γ(a s)σdm s Note that P is observable and M is the standard Brownian motion, hence the payments can be calculated by time t. 3 / 33

31 Proof of the optimal contract theorem Verify four lemmas: 1. (C T, A) is optimal for the agent. 2. Any IC contract delivering R to the agent costs the principal at least F (, P, R). 3. (C T, A) costs the principal F (, P, R). 4. Any IC contract delivering more than R is not optimal for the principal. 31 / 33

32 First-best vs. second-best If U( ) and u( ) is linear, then first-best and second-best are identical Using a different approach, the condition for the second-best payment can be reduced to 1 = Γ T U (C T ), comparing to the first best 1 = λu (C T ), where Γ t λ + t σ 1 A sdm s 32 / 33

33 Thoughts Theories on continuous-time models are relatively developed Not too many completely solved models General frameworks have been developed but there are still limitations (Cvitanic and Zhang, 213) Limited work on applying the theories Computational complexities because general theories relie on solving PDEs and BSDEs (also FBSDEs) 33 / 33