Mean Field Games in Economics Part II

Transcription

1 Mean Field Games in Economics Part II Benjamin Moll Princeton GSSI Workshop, 31 August 217, L Aquila

2 Plan Lecture 1 1. A benchmark MFG for macroeconomics: the Aiyagari-Bewley-Huggett (ABH) heterogeneous agent model 2. The ABH model with common noise ( Krusell-Smith ) 3. If time: some interesting extensions of the ABH model Lecture 2 the wealthy hand-to-mouth and marginal propensities to consume (MPCs) present bias and self-control (economics meets psychology) 1. Numerical solution of MFGs with common noise based on When Inequality Matters for Macro Other stuff... 1

3 Recall Stationary MFG, Aiyagari s Variant Functions v and g on (a, ) (y, ȳ) and scalar r satisfy ρv =H( a v) + (wy + ra) a v + µ(y) y v + σ2 (y) yy v (HJB) 2 where H(p) := max {u(c) pc}, with state constraint a a c and = y v(a, y) = y v(a, ȳ) all a = a ((wy + ra + H ( a v))g) y (µ(y)g) yy (σ 2 (y)g) (FP) 1 = a gdady, g r =e Z K F (K, L) = 1 2 ez L/K, K = a agdady, L = w = e Z L F (K, L) = 1 2 ez K/L, a ygdady Coupling through scalars r and w (prices) determined by (EQ) (EQ) Algorithm: guess (r, w), solve (HJB), solve (FP), check (EQ) 2

4 Macroeconomic MFGs with Common Noise Thisiswherethemoneyis! Can fit 9% of macroeconomics into this apparatus so any progress would be extremely valuable To understand setup consider Aiyagari (1994) with stochastic aggregate productivity, Z, common to all firms First studied by Per Krusell and Tony Smith (1998), Income and Wealth Heterogeneity in the Macroeconomy, J of Political Economy Wouter Den Haan (1996), Heterogeneity, Aggregate Uncertainty, and the Short-Term Interest Rate, Journal of Business and Economic Statistics Language: instead of common noise economists say aggregate shocks or aggregate uncertainty 3

5 Macroeconomic MFGs with Common Noise Households: max {c t} t E e ρt u(c t )dt s.t. da t = (w t y t + r t a t c t )dt Firms: Equilibrium: L t = dy t = µ(y t )dt + σ(y t )dw t a t a { } max e Z t F (K t, L t ) r t K t w t L t K t,l t dz t = θz t dt + ηdb t, common B t for all firms r t = e Zt K F (K t, L t ), w t = e Zt L F (K t, L t ) a yg(a, y, t)dady, K t = a ag(a, y, t)dady 4

6 Macroeconomic MFGs with Common Noise Households: max {c t} t E e ρt u(c t )dt s.t. da t = (w t y t + r t a t c t )dt Firms: dy t = µ(y t )dt + σ(y t )dw t a t a { } max e Z t F (K t, L t ) r t K t w t L t K t,l t dz t = θz t dt + ηdb t, common B t for all firms r t = e Zt K F (K t, L t ), w t = e Zt L F (K t, L t ) Equilibrium if restrict to stationary y-process with 1st moment = 1: L t = 1, K t = a ag(a, y, t)dady 5

7 MFG System with Common Noise both g t and v t are now random variables dynamic programming notation w.r.t. individual states only E t is conditional expectation w.r.t. future (g t, Z t ) ρv t (a, y) =H( a v t (a, y)) + a v t (a, y)(w t y + r t a) (HJB) + µ(y) y v t (a, y) + σ2 (y) yy v t (a, y) dt E t [dv t (a, y)], t g t (a, y) = a [(w t y + r t a + H ( a v t (a, y)))g t (a, y)] y (µ(y)g t (a, y)) yy (σ 2 (y)g t (a, y)), w t = 1 2 ez t 1/Kt, r t = 1 2 ez t Kt, K t = (KF) ag t (a, y)dady dz t = θz t dt + ηdb t Note: 1 dt E t [dv t ] means lim s E t [v t+s v t ]/s sorry if weird notation 6

8 Analogous System for Textbook MFG $ See Cardialaguet-Delarue-Lasry-Lions Standard MFG with common noise W t dx i,t = dB i,t + 2βdW t See their equation (8) for MFG system with common pnoise: d t u t p1 `βq u t `Hpx,Du t q F px,m t q a2βdivpv t q ( dt `v t a2βdw t & in r,t s ˆT d, d t m t p1 `βq m t `div`m t D p Hpm t,du q a t dt divpm t 2βdWt, in r,t s ˆT % d, u T pxq Gpx,m T q, m m pq, in T d where the map v t is a random vector field that forces the solution u t of the backward equation to be adapted to the filtration generated by (W t ) t [,T ] q Previous slide is my sloppy version of this for my particular model 7

9 Today A computational method for MFGs with common noise, based on When Inequality Matters for Macro... Idea: linearize MFG with common noise Z t around MFG without common noise Z t = Works beautifully in practice and in many different applications But we have no idea about the underlying mathematics! Great problem for mathematicians Today: will do in terms of our specific example (Krusell-Smith) Good exercise for you: work this out for equation (8) in Cardialaguet-Delarue-Lasry-Lions 8

10 Warm-Up: Linearizing Economic Models Economists often solve dynamic economic models using linearization methods Explain in context of particularly basic macroeconomic model: neoclassical growth model for the moment: no heterogeneity, representative agent max {c t} t e ρt u(c t )dt s.t. k t = f (k t ) c t, k t, c t c t : consumption u: utility function, u >, u < ρ: discount rate k t : capital stock, k = k given f : production function, f >, f <, f ( ) < ρ < f () Interpretation: a fictitious social planner decides how to allocate production f (k t ) between consumption c t and investment k t 9

11 Warm-Up: Linearizing Economic Models You can obviously solve this problem numerically from the HJB equation: value function v satisfies ρv(k) = max c u(c) + v (k)(f (k) c) on (, ) But suppose you don t want to do this for some reason e.g. don t know finite difference methods or want to know more about optimal k t Can proceed as follows: differentiate HJB equation w.r.t. k v (k)(f (k) c(k)) = (ρ f (k))v (k) Define ν t = v (k t ), evaluate along characteristic k t = f (k t ) c t ν t = (ρ f (k t ))ν t k t = f (k t ) (u ) 1 (ν t ) (ν t, k t ) satisfy two ODEs with initial condition k = k, and can also derive terminal condition: lim t e ρt ν t k t = 1

12 Warm-Up: Linearizing Economic Models Recall (ν t, k t ) satisfy two ODEs ν t = (ρ f (k t ))ν t k t = f (k t ) (u ) 1 (ν t ) (ODEs) with k = k, lim e ρt ν t k t = t (BOUNDARY) Unique stationary (ν, k ) satisfying f (k ) = ρ, ν = u (f (k )) To understand dynamics: first-order expansion around (ν, k ) [ ] [ νt f (k )ν ] ] ] [ [ νt [ νt νt ν ] k t 1, := u (c ) ρ k t k t k t k }{{} B Easy to show: eigenvalues (λ 1, λ 2 ) of B are real, λ 1 < < λ 2 ] [ νt c 1 e k λ1t ϕ 1 + c 2 e λ2t ϕ 2, ϕ j R 2 = eigenvectors t constants (c 1, c 2 ) pinned down from (BOUNDARY) need c 2 = 11

13 Warm-Up: Linearizing Economic Models Linearization strategy also works with common noise. Consider max e ρt u(c t )dt s.t. {c t} t k t = e Zt f (k t ) c t, dz t = θz t dt + ηdb t = common noise Value function v(k, Z). Differentiate with respect to k: (ρ e Z f (k)) k v = (e Z f (k) c(k, Z)) kk v θz kz v + η2 2 kzzv Define ν t := k v(k t, Z t ). Then Ito s formula yields: dν t = b(k t, Z t )dt + η kz v(k t, Z t )db t b(k t, Z t ) := (e Zt f (k t ) c t ) kk v(k t, Z t ) θz t kz v(k t, Z t ) + η2 2 kzzv(k t, Z t ) t+s t+s ν t+s ν t = b(k u, Z u )du + η kz v(k u, Z u )db u t t expanding right-hand side terms 1 lim s s E t[ν t+s ν t ] = b(k t, Z t ) 12

14 Warm-Up: Linearizing Economic Models Recall (ρ e Z f (k)) k v = (e Z f (k) c(k, Z)) kk v θz kz v + η2 2 kzzv Evaluate along characteristic (k t, Z t ) using previous slide E t [dν t ] = (ρ e Z t f (k t ))dt dk t = e Z t f (k t ) (u ) 1 (ν t ) dz t = θz t dt + ηdb t with k = k, Z = Z and a terminal condition for ν t (in expect.) Expansion around stationary point w/o common noise (ν, k, ): E t [d ν t ] ν t ν t ν t ν d k t B k t dt + db t, k t = k t k dz t Z t η Z t Z t Can show: B R 3 3 has real eigenvalues λ 1 λ 2 < < λ 3 system of SDEs has unique sol n satisfying boundary conditions Impulse response functions (IRFs): ( ν t, k t, Z t ), t after db = 1 13 ( )

15 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 65 IRF to A Technological Shock Technology shock % dev Quarters

16 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 66 IRF to A Technological Shock Output Consumption Investment % dev. 1 % dev..4 % dev Quarters Quarters Quarters.8 Capital 1 Hours worked Labor productivity (Wages) % dev..4.2 % dev. % dev Quarters Quarters Quarters

17 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 53 A good fit with estimated shocks.1.5 Output Data Model.4.2 Hours Worked Quarters Consumption Quarters Quarters Investment Quarters

18 Real Business Cycle (RBC) Model Aside: this model (neoclassical growth model + common noise in productivity Z t ) with addition of hours worked choice is called the Real Business Cycle (RBC) model fits aggregate data surprisingly well Finn Kydland and Ed Prescott got a Nobel prize for it what s a negative technology shock? Do we suddenly forget how to produce stuff? one example is oil price shock, but technology shocks probably a bit of a stretch 14

19 Summary of Linearization Method 1. Compute stationary point without common noise 2. Compute first-order Taylor expansion around stationary point without common noise 3. Solve linear stochastic differential equations 15

20 Key idea: same strategy in MFG with common noise 1. Compute stationary MFG without common noise 2. Compute first-order Taylor expansion around stationary MFG without common noise 3. Solve linear stochastic differential equations 16

21 MFG System with Common Noise Recall MFG System with Common Noise ρv t (a, y) =H( a v t (a, y)) + a v t (a, y)(w t y + r t a) (HJB) + µ(y) y v t (a, y) + σ2 (y) yy v t (a, y) dt E t [dv t (a, y)], t g t (a, y) = a [(w t y + r t a + H ( a v t (a, y)))g t (a, y)] y (µ(y)g t (a, y)) yy (σ 2 (y)g t (a, y)), w t = 1 2 ez t 1/Kt, r t = 1 2 ez t Kt, K t = (KF) ag t (a, y)dady dz t = θz t dt + ηdb t 17

22 Linearization and Discretization: Which Order? Numerical solution method has two components linearization (first-order Taylor expansion) around MFG without common noise discretization of (v, g) via finite difference method What we do: 1. discretization 2. linearization Reason: don t understand linearized infinite-dimensional system What one probably should do: 1. linearization 2. discretization i.e. analyze linearized infinite-dimensional system before discretizing and putting on computer 18

23 Interesting Exercise Start with equation (8) in Cardialaguet-Delarue-Lasry-Lions p q $ d t u t p1 `βq u t `Hpx,Du t q F px,m t q a2βdivpv t q ( dt `v t a2βdw t & in r,t s ˆT d, d t m t p1 `βq m t `div`m t D p Hpm t,du q a t dt divpm t 2βdWt, in r,t s ˆT % d, u T pxq Gpx,m T q, m m pq, in T d Linearize this system around stationary MFG with β = { = u + H(x, Du) in T d = m + div(md p H(x, Du)) in T d 19

24 Linearization: Three Steps 1. Compute stationary MFG without common noise 2. Compute first-order Taylor expansion around stationary MFG without common noise 3. Solve linear stochastic differential equations 2

25 Step 1: Compute stationary MFG w/o common noise ρv =H( a v) + (wy + ra) a v + µ(y) y v + σ2 (y) yy v (HJB ) 2 = a ((wy + ra + H ( a v))g) y (µ(y)g) yy (σ 2 (y)g) (FP ) r = /K, w = K, 2 K = a agdady (EQ ) 21

26 Step 1: Compute stationary MFG w/o common noise Compute using finite difference method, notation: a v(a i, y j ) a v i,j ρv i,j =H( a v i,j ) + (wy j + ra i ) a v i,j + µ(y j ) y v i,j + σ2 (y j ) yy v i,j (HJB ) 2 = a ((wy + ra + H ( a v))g) y (µ(y)g) yy (σ 2 (y)g) (FP ) r = /K, w = K, 2 K = a agdady (EQ ) 21

27 Step 1: Compute stationary MFG w/o common noise Compute using finite difference method, notation: v = (v 1,1,..., v I,J ) T ρv =u (v) + A (v; p) v, p := (r, w) (HJB ) = a ((wy + ra + H ( a v))g) y (µ(y)g) yy (σ 2 (y)g) (FP ) r = /K, w = K, 2 K = a agdady (EQ ) 21

28 Step 1: Compute stationary MFG w/o common noise Compute using finite difference method, notation: g = (g 1,1,..., g I,J ) T ρv =u (v) + A (v; p) v (HJB ) =A (v; p) T g r = /K, w = K, 2 K = a (FP ) agdady (EQ ) 21

29 Step 1: Compute stationary MFG w/o common noise Compute using finite difference method ρv =u (v) + A (v; p) v (HJB ) =A (v; p) T g (FP ) p =F (g) (EQ ) 21

30 Linearization: Three steps 1. Compute stationary MFG without common noise Yves finite difference method stationary MFG reduces to sparse matrix equations 2. Compute first-ordertaylorexpansion aroundstationary MFG withoutcommonnoise use automatic differentiation routine 3. Solve linear stochastic differential equation 22

31 Step 2: Linearize discretized system w common noise Discretized system with common noise ρv t = u (v t ) + A (v t ;p t ) v t + 1 dt E t[dv t ] dg t dt = A (v t;p t ) T g t p t = F (g t ; Z t ) dz t = θz t dt + ηdb t 23

32 Step 2: Linearize discretized system w common noise Discretized system with common noise ρv t = u (v t ) + A (v t ;p t ) v t + 1 dt E t[dv t ] dg t dt = A (v t;p t ) T g t p t = F (g t ; Z t ) dz t = θz t dt + ηdb t Structure basically the same as from warm-up exercise E t [dν t ] = (ρ e Z t f (k t ))dt dk t = e Z t f (k t ) (u ) 1 (ν t ) dz t = θz t dt + ηdb t 23

33 Step 2: Linearize discretized system w common noise Discretized system with common noise ρv t = u (v t ) + A (v t ;p t ) v t + 1 dt E t[dv t ] dg t dt = A (v t;p t ) T g t p t = F (g t ; Z t ) dz t = θz t dt + ηdb t... which we linearized as E t [d ν t ] ν t d k t B k t dt + db t, dz t Z t η ν t ν t ν k t = k t k Z t Z t 23

34 Step 2: Linearize discretized system w common noise Discretized system with common noise ρv t = u (v t ) + A (v t ;p t ) v t + 1 dt E t[dv t ] dg t dt = A (v t;p t ) T g t p t = F (g t ; Z t ) dz t = θz t dt + ηdb t Linearize in analogous fashion (using automatic differentiation) E t [d v t ] B vv B vp v t dĝ t = B gv B gg B gp ĝ t B pg I B pz p t dt + t dz t } {{ θ } Z t η B 23

35 Step 2: Linearize discretized system w common noise Discretized system with common noise ρv t = u (v t ) + A (v t ;p t ) v t + 1 dt E t[dv t ] dg t dt = A (v t;p t ) T g t p t = F (g t ; Z t ) dz t = θz t dt + ηdb t Can simplify further by eliminating p t E t [d v t ] B vv B vp B pg B vp B pz v t dĝ t = B gv B gg + B gp B pg B gp B pz ĝ t dt+ db t dz t θ Z t η Only difference to ( ν t, k t, Z t ) system: dimensionality rep agent model: dimension 3 MFG: 2 N + 1, N = I J, e.g. = 21 if I = 5, J = 2 23

36 Linearization: Three steps 1. Compute stationary MFG without common noise Yves finite difference method stationary MFG reduces to sparse matrix equations 2. Compute first-ordertaylorexpansion aroundstationary MFG withoutcommonnoise use automatic differentiation routine 3. Solve linear stochastic differential equation moderately-sized systems can diagonalize system, compute eigenvalues (typically N + 1 are < ) large systems, e.g. two-asset model from Lecture 1 = dimensionality reduction 24

37 Dimensionality Reduction in Step 3 Use tools from engineering literature: Model reduction Antoulas (25), Approximation of Large-Scale Dynamical Systems, available at Amsallem and Farhat (211), Lecture Notes for Stanford CME345 Model Reduction, available at Approximate N-dimensional distribution by projecting onto k-dimensional subspace of R N with k << N g t γ 1t x γ kt x k Adapt to problems with forward-looking decisions For details, see When Inequality Matters for Macro... 25

38 IRFs in Krusell & Smith Model Comparison of full distribution vs. k = 1 approximation = recovers Krusell & Smith s result: ok to work with 1D object 26

39 IRFs in Krusell & Smith Model Instead two-asset model from Lecture 1 requires k = 3 = not ok to work with 1D object 26

40 Our Method Is Fast, Accurate in Krusell & Smith Model Ourmethodisfast w/oreduction w/reduction Steady State.82 sec.82 sec Linearize.21 sec.21 sec Reduction.7 sec Solve.14 sec.2 sec Total.243 sec.112 sec JEDC comparison project (21): fastest alternative 7 minutes Ourmethodisaccurate Common noise η.1%.1%.7% 1% 5% Den Haan Error.%.2%.53%.135% 3.347% JEDC comparison project: most accurate alternative.16% 26

41 Linearizing MFGs with Common Noise: Summary Method works beautifully in practice and in many applications But we don t understand underlying mathematics Great problem for mathematicians! Again, from economists point of view, MFGs with common noise is where the money is Probably want to switch oder: 1. linearize then discretize and put on computer 27

42 Conclusion Mean field games extremely useful in economics lots of exciting questions involve mean field type interactions but mathematics often pretty challenging, at least for the average economist Potentially high payoff from mathematicians working on this! Questions? Come talk to me or shoot me an 28