Production and Relative Consumption

Transcription

1 Mean Field Growth Modeling with Cobb-Douglas Production and Relative Consumption School of Mathematics and Statistics Carleton University Ottawa, Canada Mean Field Games and Related Topics III Henri Poincaré Institute, June 2015 Joint work with Son Luu Nguyen, Univ. Puerto Rico

2 Outline of talk Brief overview of the mean field game (MFG) equation system We present an application to continuous time growth problems Cobb-Douglas production HARA utility Absolute and relative consumption Product form coupling = simple solution

3 The game as an interacting particle system: review A game of N agents (players), 1 i N. Dynamics and costs: dx i = 1 N N f (x i, u i, x j )dt + 1 N j=1 J i (u i, u i ) = E T 0 1 N N σ(x i, u i, x j )dw i, j=1 N L(x i, u i, x j )dt, 1 i N. j=1 (u i : controls of all players other than player i) Write δ x (N) = 1 N N j=1 δ x j where δ is the dirac measure. For φ = f, σ, L, denote 1 N N φ(x i, u i, x j ) = φ(x i, u i, y)δ x (N) R n j=1 each x j R n (dy) := φ[x i, u i, δ (N) x ]

4 The twin equations: HJB McKV For the nonlinear diffusion model (CIS 06): HJB equation: V t = inf u U { f [x, u, µ t ] V } x + L[x, u, µ t] + σ2 2 V 2 x 2 V (T, x) = 0, (t, x) [0, T ) R. Optimal Response : u t = φ(t, x µ ), (t, x) [0, T ] R. Closed-loop McK-V equation (which can be written as a Fokker-Planck equation): dx t = f [x t, φ(t, x µ ), µ t ]dt + σdw t, 0 t T. The mean field game methodology amounts to finding a solution (x t, µ t ) in McK-V sense, i.e., Law(x t ) = µ t.

5 Recent applications Mean field control theory has very wide current and potential applications (see HN 15 (submitted to CDC 15) for references): Economic theory Finance Communication networks Social opinion formation Power systems Electric vehicle recharging control Public health (vaccination games)...

6 Two ends of modeling Model: nonlinear diffusions HJB + FP/McK-V-SDE (analytical complexity: high) Model: LQG/LEQG Riccati, closed-form solution linear fixed point operators; linear state feedback (analytical complexity: low) Remark: There is a jump of difficulty levels in analysis and computation We will describe a model lying between (medium difficulty).

7 Our goals Addressing mean field interactions in endogenous stochastic growth models with many firms Good mathematical tractability

8 Literature: growth modeling (discrete time) X i t : output (or wealth) of agent i, 1 i N u i t [0, X i t ]: capital stock (so no borrowing) c i t = X i t u i t: amount for consumption u (N) t = (1/N) N j=1 uj t: aggregate capital stock level The next stage output, measured by the unit of capital, is Xt+1 i = G(u (N) t, Wt i )ut, i t 0, (2.1) Regard u (N) t as being measured according to a macroscopic unit. See Olson and Roy (2006) for a survey on stochastic growth theory.

9 The utility functional The utility functional is J i (u i, u i ) = E T t=0 ρt v(x i t u i t), ρ (0, 1]: the discount factor c i t = X i t u i t: consumption, u i = (, u i 1, u i+1, ) We take the HARA utility v(z) = 1 γ zγ, z 0, γ (0, 1). Main results: (i) The mean field game equation system has a solution (proved by fixed point theorem); (ii) The set of decentralized strategies obtained is an ε-nash equilibrium. (for more detail, see Huang, DGAA 13)

10 The finite player model: general form Dynamics of capital stock for N players: dxt i = {F (Xt i, X (N) t ) δxt i }dt Ct i dt σxt i dwt i, 1 i N Individual utility: [ T J i (C 1,..., C N ) = E 0 e ρt U(C i t, C (N) t )dt + e ρt S(X i T ) ], F : gross growth rate Xt i : capital stock, C t i : consumption rate X (N) = 1 N N j=1 X j t, C (N) = 1 N N j=1 C j t δdt + dwt i : stochastic depreciation (see e.g. Wälde 11, Feicht and Stummer 10 for stochastic depreciation modeling)

11 The specific model to be analyzed Dynamics of capital stock for N players: dxt i = {A(X (N) t )[Xt i ] α δxt i }dt Ct i dt σxt i dwt i, Individual utility: [ T J i (C 1,..., C N ) = E 0 1 i N e ρt U(C i t, C (N) t )dt + e ρt S(X i T ) ], Below take γ = 1 α (see the note later; so α = 1 γ). U(Ct i, C (N) t ) = 1 γ [ α (0, 1), λ (0, 1) C i t 1 λ C i t C (N) t ] λ γ S(X i t ) = 1 γ X i t γ, Relative Performance: (Alonso-Carrera et al. 05), Turnovsky and Monteiro 07 (under the notion of consumption externality,...)

12 Recent literature: relative performance From (Espinosa and Touzi, 2013) The performance of agent (manager) i (i = 1, 2,..., N): [ ] EU (1 λ)xt i + λ(xt i X ( i) T ), 0 < λ < 1 Coupling term X ( i) T = 1 N 1 j i X j T appears in the utility

13 A note on the choice of the HARA parameter Take the standard choice γ = 1 α This means equalizing the coefficient of the relative risk aversion to capital share

14 The limiting model and representative agent Write X t, C t instead of X i t, C i t. The dynamics: dx t = A(m t )X α t dt δx t dt C t dt σx t dw t = A(m t )Xt 1 γ dt δx t dt C t dt σx t dw t The utility functional (CARA, as a particular case of HARA): [ [ J = 1 T ( ) ] λ γ ] γ E e ρt Ct 1 λ Ct dt + e C ρt ηx γ T. t 0 A(m)x α is a mean field version of the Cobb-Douglas production function with capital x and a constant labor size. m t : population average state; C t : population average consumption; δdt + σdw t : stochastic depreciation; λ (0, 1).

15 The idea to solve the game problem Recall [ [ J = 1 T ( ) ] λ γ ] γ E e ρt Ct 1 λ Ct dt + e C ρt ηx γ T. t 0 Freeze the control mean field C t. Write [ C 1 λ t ( ) ] λ γ Ct = C t C λγ Ct γ t View the right hand side as HARA utility with a time-varying coefficient. Then try to exploit the nice properties of Cobb-Douglas and HARA Solve the optimal response Ĉt, and impose consistency C t = EĈ t

16 The HJB Write b t = C λγ γ 1 t ρv (t, x) = V t + σ2 x 2 V (T, x) = ηx γ. The HJB reads 2 V xx + (A(m t )x 1 γ δx)v x + 1 γ γ γ, x > 0 We look for a solution of the form V (t, x) = 1 γ [p(t)x γ + h(t)], x > 0, t 0 b tv By consistent mean field approximations, we will derive a fixed point equation for b t. γ γ 1 x.

17 HJB/McKV-SDE gives... 1 γ Denote X t = Zt. The HJB/McKV-SDE reduces to [ ṗ(t) = ρ + σ2 γ(1 γ) 2 + δγ ] p(t) (1 γ)b t p γ γ 1 (t), ḣ(t) = ρh(t) A(m t )γp(t), h(t ) = 0 { dz t = γa(m t ) γ [δ b t (p(t)) 1 γ 1 σ2 (1 γ) 2 p(t ) = η ] Z t } dt γσz t dw t, The consistency condition (replicating C t means replicating b t accordingly) 1 γ m t = EZt (= EX t ), b t = [b t p 1 γ 1 (t)ex t ] λγ 1 γ (RHS is (EĈ t ) λγ Existence = fixed point problem (Huang and Nguyen 15). Equilibrium strategy Ĉ t = b t p 1 γ 1 (t)x t 1 γ )

18 HJB/McKV-SDE gives... Special case A(m t ) A. Then the equation system reduces to [ ] ṗ(t) = ρ + σ2 γ(1 γ) 2 + δγ p(t) (1 γ)b t p γ γ 1 (t), p(t ) = η ḣ(t) = ρh(t) Aγp(t), h(t ) = 0 { dz t = γa γ [δ b t (p(t)) 1 γ 1 σ2 (1 γ) 2 where Z t is a GBM for given b t, p(t). The domino procedure: which defines b( ) p( ) h( ), Z( ) EX t [Γ(b )](t) = [b t p 1 γ 1 (t)ex t ] λγ 1 γ ] Z t } dt γσz t dw t,

19 Fixed point The solution of HJB/McKV-SDE reduces to analyzing the fixed equation b t = [Γ(b )](t) Existence result: There exists a solution b t (which further determines p(t), h(t), Z t ) for small λ. Proof. Schauder s fixed point theorem.

20 Numerical example dx t = AXt 1 γ dt δx t dt C t dt σx t dw t [ [ J = 1 T ( ) ] λ γ ] γ E e ρt Ct 1 λ Ct dt + e C ρt ηx γ T. t 0 T = 2, A = 1, δ = 0.05, γ = 0.6, η = 0.2, ρ = 0.04, σ = 0.08; i lambda=0.001 EX t i lambda=0.001 EC t i lambda=0.1 EX t i lambda=0.1 EC t i lambda=0.3 EX t i lambda=0.3 EC t i lambda=0.5 EX t i lambda=0.5 EC t

21 Observations from numerical solutions The existence proof via fixed point theorem needs small λ (so that the nonlinear mapping does not escape from a prior set). In practice, numerical convergence is seen for large λ. What difference the relative performance makes? The numerical solution suggests agents tend to be thriftier during the early phase if they care more about relative consumption (larger λ).

22 Back to the N player model? The proof of an ε-nash equilibrium theorem is of interest. Need very careful estimate of a denominator term (check decay, large deviations, etc) Recall the approximation 1 C t i 1 λ Ct i γ C (N) t λ γ 1 γ [ C i t 1 λ C i t C t λ ] γ

23 Some applications of MFGs to economic growth and finance. Guéant, Lasry and Lions (2011): human capital optimization Lucas and Moll (2011): Knowledge growth and allocation of time Carmona and Lacker (2013): Investment of n brokers Huang (2013): capital accumulation with congestion effect Lachapelle et al. (2013): price formation Espinosa and Touzi (2013): Optimal investment with relative 1 performance concern (depending on N 1 j X j ) Jaimungal and Nourian (2014): Optimal execution more...

24 Thank you!