Simulating models with heterogeneous agents

Transcription

1 Simulaing models wih heerogeneous agens Wouer J. Den Haan London School of Economics c by Wouer J. Den Haan

2 Individual agen Subjec o employmen shocks (ε i, {0, 1}) Incomplee markes only way o save is hrough holding capial borrowing consrain k i,+1 0 Compeiive firms, hus compeiive prices ( ) α w = (1 α) z K ll r = αz ( K ll ) α 1

3 Individual agen max E {c i,,k i,+1 } =0 =0 β ln(c i, ) s.. c i, + k i,+1 = r k i, + (1 τ )w lε i, + µw (1 ε i, ) + (1 δ)k i, k i,+1 0

4 Laws of moion z can ake on wo values ε i, can ake on wo values probabiliy of being (un)employed depends on z ransiion probabiliies are such ha unemploymen rae only depends on curren z Thus u = u b if z = z b and u = u g if z = z g wih u b > u g.

5 Complexiy of individual problem for given process of r and w his is a relaively simple problem sae variables? consrain?

6 Wha aggregae variables do agens care abou? r and w They only depend on aggregae capial sock and z!!! This is no rue in general for equilibrium prices Agens are ineresed in all informaion ha forecass K Thus, complee cross-secional disribuion of employmen saus and capial levels maers

7 Equilibrium Coninuum of agens Individual policy funcions ha solve he agen s maximizaion problem A wage and a renal rae given by equaions above. A ransiion law for he cross-secional disribuion of capial, ha is consisen wih he invesmen policy funcion. f = beginning-of-period cross-secional disribuion of capial and he employmen saus afer he employmen saus has been realized. f +1 = Υ(z +1, z, f )

8 Two differen ways o go Simulae a panel wih a large number of agens This uses Mone Carlo inegraion o calculae cross-secional momens Use ools from numerical lieraure grid mehod ha does no require he inverse of he policy funcion grid mehod ha requires he inverse of he policy funcion non-grid mehod

9 Wha is given? A policy funcion k (k i,, ε i,, s ) s : he aggregae sae variables iniial disribuion for = 1 characerizes he densiy of capial holdings of he employed and unemployed.

10 Grid mehod I Fine grid wih nodes: κ i, i = 0, 1,, χ Only mass AT grid poins p ε i, : mass of agens wih kε = κ i, i = 0, 1,, χ ε : employmen saus no mass in beween grid poins If k i 0 is binding = pε 0, > 0 (and CDF has some jumps a oher poins)

11 Grid mehod I Fix employmen saus remain wihin he period for now Nodes correspond wih beginning-of-period disribuion

12 Grid mehod I focus on node j wih mass p ε,j and capial value κ j find i such ha k (κ j, ε, ) saisfies if k (κ j, ε, ) > κ χ, i = χ κ i 1 < k (κ j, ε, ) κ i

13 Grid mehod I Se end-of-period fracions: f ε,i = 0 i Go hrough all nodes and allocae beginning-of-period p ε,j end-of-period f ε,i : o if k (κ j, ε, ) κ χ hen ω i,j f ε,i 1 f ε,i = k (κ j,ε, ) κ i 1 κ i κ i 1 = f ε,i 1 = f ε,i if k (κ j, ε, ) > κ χ hen f ε,χ + p ε,j + p ε,j ω i,j = f ε,χ ( + p ε,j 1 ω i,j )

14 0.6 Beginning of Period Disribuion 0.5 k'(0)= End of Period Disribuion 0.5 k'(0)=

15 0.6 Beginning of Period Disribuion k'(1)= k'(1)=0.8 End of Period Disribuion k'(0)= k'(1)=

16 0.6 Beginning of Period Disribuion k'(1)= k'(1)=0.8 End of Period Disribuion k'(0)= k'(2)= k'(1)=0.8 k'(2)=

17 Grid mehod I Use ransiion laws o go from end-of-period o beginning-of-period + 1 disribuion

18 Nex period s disribuion? g ε ε +1 z z +1 = mass of agens wih employmen saus ε ha have employmen saus ε +1, condiional on he values of z and z +1 For each combinaion of values of z and z +1 we have We hen have g 00z z +1 + g 01z z +1 + g 10z z +1 + g 11z z +1 = 1 p ε,i +1 = g 0εz z +1 f 0,i g 1εz z + +1 f 1,i g 0εz z +1 + g 1εz z +1 g 0εz z +1 + g 1εz z +1

19 Grid mehod II Disribuion uniformly disribued beween grid poins CDFs: wo piece-wise linear splines, P ε=0 (k) and P ε=1 (k)

20 Grid mehod II Calculae he end-of-period disribuion as follows nodes correspond o he end-of-period disribuion go hrough he nodes, κ i, one by one calculae he beginning-of-period capial sock a which he agen would have chosen he value a he grid poin, x ε,i = k,inv (κ i, ε, s ) CDF a grid poin is equal o P ε(xε,i ) (Noe he wo ime subscrips) x ε,i F ε,i = dp(k) ε = 0 i ε p ε,i i=0 κ iε p ε,i ε+1 κ 1+iε κ iε + xε,i, where i ε = i(x ε,i ) is he larges value of i such ha κ i x ε,i Calculae nex period s beginning-of-period disribuion using he ransion laws

21 Nex period s disribuion? and P ε,i +1 = g 0εz z +1 F 0,i g 1εz z + +1 F 1,i g 0εz z +1 + g 1εz z +1 g 0εz z +1 + g 1εz z +1 p ε,0 +1 = P ε,0 +1 p ε,i +1 = P ε,i +1 Pε,i 1 +1

22 Parameerized cross-secional disribuion Grid mehods are likely o require a lo of nodes (1000 is ypical) For some procedures ha is cosly No clear you need such precise informaion

23 Parameerized cross-secional disribuion Grid mehods are likely o require a lo of nodes (1000 is ypical) For some procedures ha is cosly No clear you need such precise informaion Algan, Allais, and Den Haan (2006) propose o use polynomials P(k; ρ ) is a polynomial wih in period = 1 coeffi ciens equal o ρ 1 Using Simpson quadraure o calculae end-of-period momens Use ransiion laws o calculae nex period s beginning-of-period momens You need a way o find ρ 2 given values for momens in period 2

24 Fiing a disribuion given momens: Approach I Find N elemens of ρ such ha 0 [k m(1)] P(k; ρ)dk = 0 [ (k m(1)) 2 m(2) ] P(k; ρ)dk = 0 0 [ (k m(1)) N m(n) ] P(k; ρ)dk = P(k; ρ)dk = 1

25 Fiing a disribuion given momens: Approach II Use an alernaive funcional form ρ 1 [k m(1)] + ] P(k; ρ) = ρ 0 exp ρ 2 [(k m(1)) 2 m(2) + + ] ρ N [(k m(1)) N m(n)

26 Fiing a disribuion given momens: Approach II Find coeffi ciens ρ using min P(k, ρ)dk ρ 1,ρ 2,,ρ N 0 The firs-order condiions correspond exacly o he condiion ha he firs N momens of P(k, ρ) should correspond o he se of specified momens. ρ 0 is deermined by he condiion ha he densiy inegraes o one.

27 Fiing a disribuion given momens: Approach II The Hessian (imes ρ 0 ) is given by 0 X (m(1),, m(n)) X (m(1),, m(n)) P(k, ρ)dk, (1) where X is an (N 1) vecor and he i h elemen is given by (k m(1)) for i = 1 (k m(1)) i m(i) for i > 1 (2)

28 Does i make a difference? Numerical procedure wih a coninuum of agens Wha if you really do like o simulae a panel wih a finie number of agens? Impose ruh as much as possible: if you have 10,000 agens have exacly 400 (1,000) agens unemployed in a boom (recession) Even hen sampling noise is non-rivial This is done in he graphs below, bu sill he resuls are no accurae

29 Simulaion and sampling noise me1 Non random cross secion me1 Mone Carlo simulaion N=10, Time

30 mu1 Non random cross secion mu1 mone Carlo simulaion 10, Time

31 mu1 Non random cross secion mu1 Mone Carlo simulaion N=10,000 mu1 Mone Carlo simulaion N=100, Time

32 0,75% muc Non random cross secion muc mone Carlo simulaion 10,000 muc mone Carlo simulaion 100,000 0,50% 0,25% 0,00% Time

33 References Algan, Y., O. Allais, W.J. Den Haan, 2008, Solving heerogeneous-agen models wih parameerized cross-secional disribuions, Journal of Economic Dynamics and Conrol simulaes and solves model wih parameerized expecaion Den Haan, W. J., and P. Rendahl, 2010, Solving he incomplee markes model wih aggregae uncerainy using explici aggregaion, Journal of Economic Dynamics and Conrol aricle ha develops Xpa Young, E. R., 2010, Solving he incomplee markes model wih aggregae uncerainy using he Krusell-Smih algorihm and non-sochasic simulaions,journal of Economic Dynamics and Conrol inroduces he firs grid mehod