The Mathematics of Continuous Time Contract Theory

Transcription

1 The Mathematics of Continuous Time Contract Theory Ecole Polytechnique, France University of Michigan, April 3, 2018

2 Outline Introduction to moral hazard 1 Introduction to moral hazard 2 3 General formulation From Stackelberg game to control problem Proof through nonlinear representation

3 Moral hazard Adam Smith ( ), Scottish economist, philosopher and author, moral philosopher, a pioneer of political economy Moral hazard is a major risk in economics Moral hazard : Situation where an agent may benefit from an action whose cost is supported by others Should not count on agents morality Relationship between economic agents - should instead be based on mutual interest - each actor acts according to the incentives that are perceived

4 Moral hazard Adam Smith ( ), Scottish economist, philosopher and author, moral philosopher, a pioneer of political economy Moral hazard is a major risk in economics Moral hazard : Situation where an agent may benefit from an action whose cost is supported by others Should not count on agents morality Relationship between economic agents - should instead be based on mutual interest - each actor acts according to the incentives that are perceived

5 Moral hazard Adam Smith ( ), Scottish economist, philosopher and author, moral philosopher, a pioneer of political economy Moral hazard is a major risk in economics Moral hazard : Situation where an agent may benefit from an action whose cost is supported by others Should not count on agents morality Relationship between economic agents - should instead be based on mutual interest - each actor acts according to the incentives that are perceived

6 Moral hazard Adam Smith ( ), Scottish economist, philosopher and author, moral philosopher, a pioneer of political economy Moral hazard is a major risk in economics Moral hazard : Situation where an agent may benefit from an action whose cost is supported by others Should not count on agents morality Relationship between economic agents - should instead be based on mutual interest - each actor acts according to the incentives that are perceived

7 Moral hazard in real life

8 Moral hazard in real life

9 Moral hazard in real life

10 Moral hazard in real life

11 Resolution of moral hazard by contracting In all above examples, moral hazard addressed in our everyday life : Employee receives a salary indexed on performance Driver pays insurance premium indexed on sinistrality record Manager compensation indexed on performance AND risk Regulation introduces taxes indexed on the polluting externalities of firms, subsidies for green transition, carbon market thus introducing a price for pollution HOWEVER designing incentive regulation for some purpose can be misleading and subtle...

12 Resolution of moral hazard by contracting In all above examples, moral hazard addressed in our everyday life : Employee receives a salary indexed on performance Driver pays insurance premium indexed on sinistrality record Manager compensation indexed on performance AND risk Regulation introduces taxes indexed on the polluting externalities of firms, subsidies for green transition, carbon market thus introducing a price for pollution HOWEVER designing incentive regulation for some purpose can be misleading and subtle...

13 Moral hazard in insurance Fully insured drivers do not have incentive to increase caution Health insurance misleadingly lowers out-of-pocket expenses, and thus increases demand for medical services Un-employment insurance may give negative incentive to return to the job market

14 Balancing electricity grids systems by active demand Peak-Time Rebates (PTR) : paying customers to reduce their electricity demand with respect to past reference consumption = Make money out of nothing! Baltimore Baseball Field fined in 2013 by Federal Energy Regulation Commission (FERC)

15 Outline Introduction to moral hazard 1 Introduction to moral hazard 2 3 General formulation From Stackelberg game to control problem Proof through nonlinear representation

16 Delegation problem X : value of an output process owned by Principal Agent devotes effort a, thus impacting distribution of X = X a cost of effort c(a) compensation ξ : contract Choose ξ so that Agent devotes effort in the interest of Principal First best contract : Principal has full power, up to participation level R { sup E UA (ξ c(a)) : E U P ( ξ + X a ) R } a,ξ Centralized economy : Social Planner organizes Principal-Agent relation sup λ E U A (ξ c(a)) + (1 λ) E U P ( ξ + X a ) a,ξ

17 Delegation problem X : value of an output process owned by Principal Agent devotes effort a, thus impacting distribution of X = X a cost of effort c(a) compensation ξ : contract Choose ξ so that Agent devotes effort in the interest of Principal First best contract : Principal has full power, up to participation level R { sup E UA (ξ c(a)) : E U P ( ξ + X a ) R } a,ξ Centralized economy : Social Planner organizes Principal-Agent relation sup λ E U A (ξ c(a)) + (1 λ) E U P ( ξ + X a ) a,ξ

18 Delegation problem X : value of an output process owned by Principal Agent devotes effort a, thus impacting distribution of X = X a cost of effort c(a) compensation ξ : contract Choose ξ so that Agent devotes effort in the interest of Principal First best contract : Principal has full power, up to participation level R { sup E UA (ξ c(a)) : E U P ( ξ + X a ) R } a,ξ Centralized economy : Social Planner organizes Principal-Agent relation sup λ E U A (ξ c(a)) + (1 λ) E U P ( ξ + X a ) a,ξ

19 Second best contract : Principal-Agent Problem Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) R ξ Non-zero sum Stackelberg game Difficult to solve, and so restrict to affine contracts...

20 (Static) Principal-Agent Problem Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) R ξ = Non-zero sum Stackelberg game Difficult to solve,... restrict to affine ξ(x) := a + bx

21 (Static) Principal-Agent Problem Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) R ξ = Non-zero sum Stackelberg game Difficult to solve,... restrict to affine ξ(x) := a + bx

22 (Static) Principal-Agent Problem Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) R ξ = Non-zero sum Stackelberg game Difficult to solve,... restrict to affine ξ(x) := a + bx

23 Contract theory at the heart of modern economic theory Nobel Prize 2014 winner : Jean Tirole Social interactions, incentives induced by well-designed contracts - Industrial organization - Organization theory - Regulation

24 From static to dynamic Principal-Agent problem Principal/Agent problem turns out to be More accessible in continuous time [Holmström & Milgrom 85] Nobel Prize 2016 winners : Oliver Hart and Bengt Holmström Cvitanić & Zhang ( 12, Book), Sannikov 08 : new approach Throughout literature : simple drift control and no diffusion control

25 Holmström & Milgrom 85 Agent solves the control problem : [ T V0 A (ξ) := sup E ξ(x ) α 0 1 ] 2 α2 t dt where α {F adapted valued in R}, and output process : dx t = α t dt + dw t Given solution α (ξ), Principal solves the optimization problem [ ] V0 P := sup E X α (ξ) T ξ(x ) ξ Ξ R where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

26 Holmström & Milgrom 85 Agent solves the control problem : [ T V0 A (ξ) := sup E ξ(x ) α 0 1 ] 2 α2 t dt where α {F adapted valued in R}, and output process : dx t = α t dt + dw t Given solution α (ξ), Principal solves the optimization problem [ ] V0 P := sup E X α (ξ) T ξ(x ) ξ Ξ R where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

27 Holmström & Milgrom 85 Agent solves the control problem : [ T V0 A (ξ) := sup E ξ(x ) α 0 1 ] 2 α2 t dt where α {F adapted valued in R}, and output process : dx t = α t dt + dw t Given solution α (ξ), Principal solves the optimization problem [ ] V0 P := sup E X α (ξ) T ξ(x ) ξ Ξ R where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

28 Application to Demand-Response programs 45 billions Euros for an objective of 200 millions smart meters London Demand response program : participation induced a decrease of 4% customers annual bill, i.e. 20 GBP out of 500.

29 Demand-Response programs in electricity tarification Output process : electricity demand under effort α dx t = α t dt + dw t Agent reduces consumption at high peak times by solving [ T V0 A ( (ξ) := sup E ξ(x ) + f (Xt ) 1 ) ] α 0 2 α2 t dt Given solution α (ξ), Principal solves the optimization problem [ T V0 P ( := sup E α ξ(x ) πxt g(x t ) ) dt] ξ Ξ R 0 where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

30 Demand-Response programmes with diffusion control Output process : electricity demand under effort α, β dx t = α t dt + β 1 t dw t, β = Agent responsiveness Agent reduces consumption and volatility by solving [ T V0 A ( (ξ) := sup E ξ(x ) + f (Xt ) 1 α,β 0 2 α2 t 1 ) ] 2 β2 t dt Given solution (α, β )(ξ), Principal solves [ T V0 P ( := sup E α,β ξ(x ) πxt g(x t ) ) dt] ξ Ξ R 0 where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

31 Another interpretation of volatility in electricity market Demand in excess of free energy exhibits high volatility High cost to match the supply to highly volatile excess demand Technological solution : acting on the supply side by using high capacity batteries for electricity storage Demand-Response : acting on the demand side by optimal contracting

32 Another interpretation of volatility in electricity market Demand in excess of free energy exhibits high volatility High cost to match the supply to highly volatile excess demand Technological solution : acting on the supply side by using high capacity batteries for electricity storage Demand-Response : acting on the demand side by optimal contracting

33 Another interpretation of volatility in electricity market Demand in excess of free energy exhibits high volatility High cost to match the supply to highly volatile excess demand Technological solution : acting on the supply side by using high capacity batteries for electricity storage Demand-Response : acting on the demand side by optimal contracting

34 Another example : Portfolio managers compensation Output process : portfolio value under effort π t dx t = π t ds t S t = π t (λdt + dw t ) Portfolio manager chooses portfolio allocation by solving [ T V0 A (ξ) := sup E ξ(x ) α 0 1 ] 2 π2 t dt Given solution π (ξ), investor solves the optimization problem V0 P ( ) := sup E π U P ξ(x ) + XT ξ Ξ R where Ξ R := { ξ : FT X measurable, such that V 0 A (ξ) R}

35 Yet, another example : Makers-Takers fees... towards Fintech

36 Modeling Makers-Takers fees Fundamental price {S t } t 0, Market Maker posts bid-ask prices p b t = S t δ b t and p a t = S t + δ a t MM inventory q t = Nt b Nt a, where Nt b, Nt a point process with unit jumps representing the arrivals of bid-ask order with intensities λ b t = λ(δ b t ) and λ a t = λ(δ a t ), with λ(x) = Ae kx MM chooses the departure from fundamental price : ( T ) V A (ξ) := sup E U A ξ + pt a dnt a pt b dnt b q t S t δ b,δ a 0 Given optimal response δ (ξ), Platform chooses optimal contract V P = sup ξ Ξ R E U P ( ξ + c(n a T + N b T ))

37 Modeling Makers-Takers fees Fundamental price {S t } t 0, Market Maker posts bid-ask prices p b t = S t δ b t and p a t = S t + δ a t MM inventory q t = Nt b Nt a, where Nt b, Nt a point process with unit jumps representing the arrivals of bid-ask order with intensities λ b t = λ(δ b t ) and λ a t = λ(δ a t ), with λ(x) = Ae kx MM chooses the departure from fundamental price : ( T ) V A (ξ) := sup E U A ξ + pt a dnt a pt b dnt b q t S t δ b,δ a 0 Given optimal response δ (ξ), Platform chooses optimal contract V P = sup ξ Ξ R E U P ( ξ + c(n a T + N b T ))

38 Modeling Makers-Takers fees Fundamental price {S t } t 0, Market Maker posts bid-ask prices p b t = S t δ b t and p a t = S t + δ a t MM inventory q t = Nt b Nt a, where Nt b, Nt a point process with unit jumps representing the arrivals of bid-ask order with intensities λ b t = λ(δ b t ) and λ a t = λ(δ a t ), with λ(x) = Ae kx MM chooses the departure from fundamental price : ( T ) V A (ξ) := sup E U A ξ + pt a dnt a pt b dnt b q t S t δ b,δ a 0 Given optimal response δ (ξ), Platform chooses optimal contract V P = sup ξ Ξ R E U P ( ξ + c(n a T + N b T ))

39 Modeling Makers-Takers fees Fundamental price {S t } t 0, Market Maker posts bid-ask prices p b t = S t δ b t and p a t = S t + δ a t MM inventory q t = Nt b Nt a, where Nt b, Nt a point process with unit jumps representing the arrivals of bid-ask order with intensities λ b t = λ(δ b t ) and λ a t = λ(δ a t ), with λ(x) = Ae kx MM chooses the departure from fundamental price : ( T ) V A (ξ) := sup E U A ξ + pt a dnt a pt b dnt b q t S t δ b,δ a 0 Given optimal response δ (ξ), Platform chooses optimal contract V P = sup ξ Ξ R E U P ( ξ + c(n a T + N b T ))

40 Outline Introduction to moral hazard General formulation From Stackelberg game to control problem Proof through nonlinear representation 1 Introduction to moral hazard 2 3 General formulation From Stackelberg game to control problem Proof through nonlinear representation

41 Continuous time Principal-Agent problem General formulation From Stackelberg game to control problem Proof through nonlinear representation Fix Ω := C 0 ([0, T ], R d ), paths space for output process X, and let V0 A (ξ) := sup E P[ T K T ξ K t c t (νt P )dt ], K t := e t 0 kν s ds P P 0 where P P weak solution of Output process : dx t = b t (X, ν t )dt + σ t (X, ν t )dw P t P a.s. Given solution P (ξ), Principal solves the optimization problem [ V0 P := sup E P (ξ) H T U(l(X ) ξ(x )) ], H t := e t 0 hsds ξ Ξ R where Ξ R := {ξ(x. ), such that V0 A (ξ) R}

42 General formulation From Stackelberg game to control problem Proof through nonlinear representation Principal-Agent problem : more natural formulation Fix Ω := C 0 ([0, T ], R d ), paths space for output process X, and let V0 A (ξ) := sup E P[ T K T ξ K t c t (νt P )dt ], Kt ν := e t 0 kν s ds P P 0 where P P weak solution of Output process : dx t = σ t (X, β t ) [ λ t (X, α t )dt + dwt P ] P a.s. Given solution P (ξ), Principal solves the optimization problem [ V0 P := sup E P (ξ) H T U(l(X ) ξ(x )) ], H t := e t 0 hsds ξ Ξ R where Ξ R := {ξ(x. ), such that V0 A (ξ) R}

43 General formulation From Stackelberg game to control problem Proof through nonlinear representation Principal-Agent problem : more natural formulation Fix Ω := C 0 ([0, T ], R d ), paths space for output process X, and let V0 A (ξ) := sup E P[ T K T ξ K t c t (νt P )dt ], Kt ν := e t 0 kν s ds P P 0 where P P weak solution of Output process : dx t = σ t (X,β t ) [ λ t (X,α t )dt + dwt P ], P a.s. Given solution P (ξ), Principal solves the optimization problem [ V0 P := sup E P (ξ) K T U(l(X ) ξ(x )) ], H t := e t 0 hsds ξ Ξ R where Ξ R := {ξ, such that V0 A (ξ) R}

44 Main result General formulation From Stackelberg game to control problem Proof through nonlinear representation Path-dependent Hamiltonian for the Agent problem : H t (ω, y, z, γ) := sup u { bt (ω, u) z σ tσ t (ω, u) : γ k t (ω, a, b)y c t (ω, a, b) } Theorem Under some technical conditions, the optimal contract is of the form ξ = Y Z,Γ T, with t Y t = Y0 + Zs dx s Γ s : d X s H s (Y Z,Γ s, Zs, Γ s )ds where (Y 0, Z, Γ ) is the solution of a standard stochastic control problem

45 The stochastic control problem General formulation From Stackelberg game to control problem Proof through nonlinear representation X := (X, Y ) satisfies d X t = µ( X t, Z t, Γ t )dt + σ( X t, Z t, Γ t )dw t : dx t = z H t (X, Y Z,Γ t dy Z,Γ t, Z t, Γ t )dt + { 2 γ H t (X, Y Z,Γ, Z t, Γ t ) } 1 2 dw t = Z t dx t Γ t : d X t H t (X, Yt Z,Γ, Z t, Γ t )dt V P = sup V 0 (X 0, Y 0 ), where Y 0 R t V 0 ( X [ ] 0 ) := sup Z,Γ E U(l(X T ) Y Z,Γ T ) V 0 ( X 0 ) = V (0, X 0 ) ; V solution of Hamilton-Jacobi-Bellman eq. t V 0 + sup z,γ { µ(., z, γ)dv σσ (., z, γ):d 2 V 0 ] } = 0 V 0 (T, x, y) = U(l(x) y)

46 The stochastic control problem General formulation From Stackelberg game to control problem Proof through nonlinear representation X := (X, Y ) satisfies d X t = µ( X t, Z t, Γ t )dt + σ( X t, Z t, Γ t )dw t : dx t = z H t (X, Y Z,Γ t dy Z,Γ t, Z t, Γ t )dt + { 2 γ H t (X, Y Z,Γ, Z t, Γ t ) } 1 2 dw t = Z t dx t Γ t : d X t H t (X, Yt Z,Γ, Z t, Γ t )dt V P = sup V 0 (X 0, Y 0 ), where Y 0 R t V 0 ( X [ ] 0 ) := sup Z,Γ E U(l(X T ) Y Z,Γ T ) V 0 ( X 0 ) = V (0, X 0 ) ; V solution of Hamilton-Jacobi-Bellman eq. t V 0 + sup z,γ { µ(., z, γ)dv σσ (., z, γ):d 2 V 0 ] } = 0 V 0 (T, x, y) = U(l(x) y)

47 Sufficient conditions General formulation From Stackelberg game to control problem Proof through nonlinear representation To prove the main result, it suffices that for all ξ?? (Y 0, Z, Γ) s.t. ξ = Y Z,Γ, P a.s. for all P P where we recall that Y t = Y 0 + t 0 OR, weaker sufficient condition : T Z s dx s Γ s : d X s H s (Ys Z,Γ, Z s, Γ s )ds for all ξ?? (Y n 0, Z n, Γ n ) s.t. Y Z n,γ n T ξ

48 Sufficient conditions General formulation From Stackelberg game to control problem Proof through nonlinear representation To prove the main result, it suffices that for all ξ?? (Y 0, Z, Γ) s.t. ξ = Y Z,Γ, P a.s. for all P P where we recall that Y t = Y 0 + t 0 OR, weaker sufficient condition : T Z s dx s Γ s : d X s H s (Ys Z,Γ, Z s, Γ s )ds for all ξ?? (Y n 0, Z n, Γ n ) s.t. Y Z n,γ n T ξ

49 Intuitions for the representation problem Solve ξ = Y Z,Γ, P a.s. for all P P, where Y t = Y 0 + T t 0 General formulation From Stackelberg game to control problem Proof through nonlinear representation Z s dx s Γ s : d X s H s (Ys Z,Γ, Z s, Γ s )ds H 0 : then Γ 0, P = {Wiener measure}, and the problem reduces to Path-dependent heat equation H linear in γ : then Γ is arbitrary, and P dominated family of measures... General case : Path-dependent semilinear parabolic equation Path-dependent fully nonlinear parabolic equation (HJB)

50 Extension to multiple agents General formulation From Stackelberg game to control problem Proof through nonlinear representation 1 Principal and N possibly interacting Agents, 1 Principal and continuum mean field interacting Agents, N possibly interacting Principals (or even continuum) and 1 Agent Another open problem in economics since Holström & Milgrom : Limited liability restriction = State constraint in the final stochastic control problem

51 Adverse selection General formulation From Stackelberg game to control problem Proof through nonlinear representation Agent preferences U and c depend on a parameter θ Θ called type, which is unknown to Principal Principal only knows distribution of types µ(dθ) on Θ = design tarification so as to push Agent to reveal its type Example : car dealers offer different qualities for the same model...

52 Adverse selection : work in progress... General formulation From Stackelberg game to control problem Proof through nonlinear representation Principal offers a menu of contracts {ξ θ, θ Θ}. Problem of Agent θ choosing contract ξ θ is : V A (θ, ξ θ ) := sup E ν[ T ] ξ θ c(θ, ν t )dt ν Principal s problem is sup {ξ θ } ICC Θ E ν (ξ θ ) [ U P (X T ξ θ ) ] µ(dθ) where {ξ θ } ICC if satisfies participation constraint and incentive compatibility : V A (θ, ξ θ ) V A (θ, ξ θ ) for all θ, θ Θ 0

53 Adverse selection : work in progress... General formulation From Stackelberg game to control problem Proof through nonlinear representation Principal offers a menu of contracts {ξ θ, θ Θ}. Problem of Agent θ choosing contract ξ θ is : V A (θ, ξ θ ) := sup E ν[ T ] ξ θ c(θ, ν t )dt ν Principal s problem is sup {ξ θ } ICC Θ E ν (ξ θ ) [ U P (X T ξ θ ) ] µ(dθ) where {ξ θ } ICC if satisfies participation constraint and incentive compatibility : V A (θ, ξ θ ) V A (θ, ξ θ ) for all θ, θ Θ 0

54 General formulation From Stackelberg game to control problem Proof through nonlinear representation Social interactions : a new challenge to mathematics Smart grids and smart cities