Solving Models with Heterogeneous Agents Macro-Finance Lecture

Transcription

1 Solving Models with Heterogeneous Agents 2018 Macro-Finance Lecture Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan

2 Overview lecture Model Pure global Krusell-Smith Parameterized expectations Pure 1 st -order perturbation and still capturing nonlinearities Legrand-Ragot Combining global for idiosyncratic with 1 st -order perturbation for aggregate Bopart, Mitman, Krusell Reiter

3 Additional info more detailed notes and slides: summer school exercises and programs: ( me for password)

4 MODEL

5 Individual agent Subject to employment shocks (ε i,t {0, 1}) Incomplete markets only way to save is through holding capital borrowing constraint k i,t+1 0

6 Laws of motion z t can take on two values ε i,t can take on two values probability of being (un)employed depends on z t transition probabilities are such that unemployment rate only depends on current z t. Thus: u t = u b if z t = z b u t = u g if z t = z g with u b > u g.

7 Individual agent max E [ {c i,t,k i,t+1 } t=0 t=0 βt ln(c i,t ) ] s.t. c i,t + k i,t+1 = r t k i,t + (1 τ t )w t lε i,t + µw t (1 ε i,t ) + (1 δ)k i,t k i,t+1 0 for given processes of r t and w t, this is a relatively simple problem

8 Firm problem ( r t = z t α K t l(1 u(zt )) ) α 1 ( K t w t = z t (1 α) l(1 u(zt )) ) α

9 Government τ t w t l(1 u(zt ) = µw t u(z t ) τ t = µu(z t ) l(1 u(zt ))

10 Aggregate variables agents care about r t and w t They only depend on aggregate capital stock and z t!!! This is not true in general for equilibrium prices Agents are interested in all information that forecasts K t In principle that is the complete cross-sectional distribution of employment status and capital levels

11 Equilibrium - first part Individual policy functions solving agent s max problem A wage and a rental rate given by equations above.

12 Equilibrium - second part A transition law for the cross-sectional distribution of capital, that is consistent with the investment policy function. f t+1 = Υ(z t+1, z t, f t ) f t = beginning-of-period cross-sectional distribution of capital and the employment status after the employment status has been realized. z t+1 does not affect the cross-sectional distribution of capital but does affect the joint cross-sectional distribution of capital and employment status

13 Krusell-Smith Algorithm

14 Key approximating step 1 Approximate cross-sectional distribution with limited set of "characteristics" Proposed in Den Haan (1996), Krusell & Smith (1997,1998), Rios-Rull (1997) 2 Solve for aggregate policy rule 3 Solve individual policy rule for a given aggregate law of motion 4 Make the two consistent

15 Krusell-Smith (1997,1998) algorithm Assume the following approximating aggregate law of motion m t+1 = Γ(z t+1, z t, m t ; η Γ ). Start with an initial guess for its coeffi cients, η 0 Γ

16 Krusell-Smith (1997,1998) algorithm Use following iteration until η iter Γ has converged: Given η iter solve for the individual policy rule Γ Given individual policy rule simulate economy and generate a time series for m t Use a regression analysis to update values of η η iter+1 Γ = λ ˆη Γ + (1 λ)ηiter Γ, with 0 < λ 1

17 Parameterized Expectations

18 Paratemeterized expectations - Rep. agent Model: [ ( )] c γ t = Eβ c γ t+1 z t+1 αkt+1 α δ Approximation: k t+1 + c t = z t k α t + (1 δ) k t ln z t = ρ ln z t 1 + ε t, ε t N c γ t = P N (k t, z t ; ψ N ) k t+1 + c t = z t kt α + (1 δ) k t ( ) 0, σ 2 ε What to solve for? ψ N

19 PEA algorithm 1 k 1 = k ss, z 1 = 1 2 Generate {z t } 10,000 t=2 3 Choose the initial value for ψ N, ψ 1 N 4 Find ψ N with following iterative process { [ 1 Generate c t, k t+1, y t+1 = β c γ t+1 using ψ i N 2 Find ψ N that is appropriate for this sample ψ i N = arg max ψ N 10,000 t=1001 ( zt+1 αk α 1 t δ)]} T t=1 (E t [y t+1 ] P N (k t, z t ; ψ N )) 2 3 ψ i+1 N = ζ ψ i N + (1 ζ) ψi N ζ is a dampening factor

20 PEA algorithm Global method Easy to incorporate occasionally binding constraints No curse of dimensionality as in grid-based methods Very useful implementation issues proposed in Judd, Maliar, Maliar (2010,2011) Computationally expensive for small/simple models relative to grid-based methods

21 Den Haan 1996: PEA and hetero Cross-sectional distribution characterized with finite set of moments No explicit approximate law of motion for aggregate variables = no additional inaccuracies introduced Full simulation method = easy

22 c 1 ν i,t = p N (s i,t ; ψ N ) solve k i,t+1 from budget constraint if k i,t+1 0 if k i,t+1 < 0 { done k i,t+1 = 0 solve c i,t from bc Conditional expectation of individual p N (s i,t ; ψ N ) s i,t = {k i,t, e i,t, z t,info about cross-sectional distribution}

23 Simulating a panel for given value psi generate aggregate productivity {z t } T t=1 start in t = 1 with cross-section of I agents Thus, k i,t and e i,t known at t = 1 use cross-section to calculate K t and other moments use K t and z t to calculate r t and w t for each agent calculate k i,t+1 for each agent draw new e i,t and go to the next period

24 Simulating a panel for given value psi To update individual problem: you only need variables for 1 agent But individual choices depend on aggregates = you need a panel

25 Updating individual law of motion If k t+1 > 0 c ν t = E t [ βc ν t+1 (r t δ) ] solve k t+1 from budget constraint collect observations with k t+1 > 0 regress E t [ βc ν t+1 (r t δ) ] on p N (s t ; ψ N ) = ψ N aggregate law of motion taken care of (implicitly) update ψ using weighted average of ψ N and old ψ N

26 Updating aggregate law of motion Not needed In the simulation, aggregate variables are constructed by explicitly aggregating the values across I individuals Thus, no approximation needed to describe law of motion of aggregate variables

27 Legrand-Ragot Algorithm

28 LeGrand-Ragot (LGR) environment Same as above, except exogenous aggregate risk does not affect employment risk (just for simplicity!) An unemployed worker works δ hours at home to produce δ goods (parameters are chosen such that agents do not prefer to work less than δ)

29 Key approximating assumption Key approximation step: All agents with the same employment history for the last N periods are identical If N = 2, then there are 4 types: uu, ue, eu, ee If N = 3, then there are 8 types: uuu,uue,ueu,uee,euu,eue,eeu,eee (in general, if there are E individual states then there are (E + 1) N types; here E = 2) Original model: N =, that is, an infinite number of different agents

30 Stories representing approximation LGR propose two "stories/models" so that the set of equations given to the computer looks like an actual economy and not just an approximation to the original model 1 quasi-planner 2 decentralized version with particular insurance mechanism This is useful, for example, to understand whether the set of equations of the approximation is well behaved

31 Quasi-planner "story" Agents with the same employment history for the last N periods have the same consumption and make the same savings choice independent of the wealth they bring into period t This savings choice is made by the quasi-planner The quasi-planner does take prices as given (in contrast to the conventional social planner)

32 Quasi-planner "story" Beginning of period t : all agents with the same N-period employment history go to the same "island" their savings are pooled quasi-planner chooses consumption and savings End of period t : All agents are entitled to an equal share of the savings Thus, quasi-planner cannot condition on next-period s unemployment status. This mimics market incompleteness

33 Quasi-planner model [ max E β t S t,e Nξ e NU ( ) ] c t,e N, l t,e N t=0 e N ε N s.t. a t,e N + c t,e N = w t l t,e Nn t,e N + δ1 e N =0 + (1 + r t) ã t,e N e N ε N a t,e N 0 e N ε N ã t,e N = S t 1,ẽN Π S t 1,(ẽ N,e N ) a t 1,ẽ N ẽ N ε N t,e N S t+1,e N = Π t S t,e N l t,e N 0 en ε N

34 Quasi-planner model Index to indicate a particular type: e N ε N S t,e N : population size island e N ã t,e N : per capita beginning-of-period wealth on island e N S t,e Nã t,e N equals sum of savings brought to island e N from different islands 1 e N =0 : indicator function if agents on this island are unemployed n t,e N : idiosyncratic productivity agents on island e N n t,e N = 0 if 1 e N =0 = 1) ξ e N : preference parameter agents with different employment histories have a different utility function Π t : transition matrix for the full N-period employment state examples below

35 Quasi-planner FOCs βe t [ ξ e NU c ( ct,e N, l t,e N) + νt,e N ê N ε N Π t,(e N,ê N ) ξ ê NU c = ( ) ct+1,e N, l t+1,e N (1 + rt+1 ) ν t,e Na t,e N = 0, a t,e N 0, ν t,e N 0 w t n e N t U c ( ct,e N, l t,e N) = Ul ( ct,e N, l t,e N) if nt,e N > 0 l t,e N = δ if n t,e N = 0 ]

36 Quasi-planner FOCs Note that the population sizes drop out! going to a large S t,e N island is bad because you have to share your wealth with more agents going to a large S t,e N island is good because the social planner gives it a larger weight these effecs exactly offset each other Note that the linearized Euler equation captures precautionary savings

37 Other model equations aggregate labor supply L t = S t,e Nn t,e Nl t,e N e N ε N aggregate savings K t = S t,e Na t,e N = S t+1,e Nã t+1,e N e N ε N e N ε N wage rate rental rate productivity ) α ( w t = (1 α) Kt 1 A t 1 L ) t α 1 depreciation r t = αa t 1 ( Kt 1 L t A t = 1 + u t u t = ρu t 1 + e t

38 Specific assumptions Greenwood, Hercowitz, Huffman preferences U c ( ct+1,e N, l t+1,e N) = ( c t+1,e N l1+1/φ t+1,e N 1+1/φ) 1 γ 1 1 γ = first-order condition for employed becomes w t n e N t = l 1/φ t,e N

39 Specific assumptions If n e N t (just as aggregate productivity) is known in period t, then L t is known in period t = r t is known in period t (risk-free r t means capital would be perfect substitute to risk-free government bonds) In fact, it is assume that n e N t = 1 for all employed agents Π t is constant = unemployment rate is constant N = 4

40 Constructing transition matrix 16 groups: unemployed employed 1. uuuu 9. euuu 2. uuue 10. euue 3. uueu 11. eueu 4. uuee 12. euee 5. ueuu 13. eeuu 6. ueue 14. eeue 7. ueeu 15.eeeu 8. ueee 16.eeee probability to become employed for unemployed equals 0.5 probability to become unemployed for employed equals 0.2

41 Π =

42 The tricky bit You have to figure out by trial and error (and some economic thinking) which group will be at the constraint Things would be problematic if that depends on the aggregate state (less likely to be problematic if aggregate fluctuations are small) Here, only group 1 turns out to be at the constraint

43 Some Dynare equations Budget constraint for group 1, uuuu, thus currently unemployed c1=delta+(1+r)*0.5*(s2*a2(-1)+s1*a1(-1))/s1-a1 this group gets members from groups 1 & 2 First-order condition for group 1 a1 = 0;

44 Some Dynare equations Budget constraint for group 2, uuue, thus currently unemployed c2=delta+(1+r)*0.5*(s4*a4(-1)+s3*a3(-1))/s2-a2 this group gets members from groups 3 & 4

45 Some Dynare equations First-order condition for group 2 weight2*(c2-delta^(1+1/phi)/(1+1/phi))^-sigma = beta*(1+r(+1))* ( 0.5*weight9*(c9(+1)-le(+1)^(1+1/phi)/(1+1/phi))^-sigma + 0.5*weight1*(c1(+1)-delta^(1+1/phi)/(1+1/phi))^-sigma ); Members of this group can go to group 1, uuuu, or group 9, euuu, with equal probability

46 Some Dynare equations Budget constraint for group 9, euuu, thus currently employed c9=w*le+(1+r)*0.5*(s2*a2(-1)+s1*a1(-1))/s9-a9 this group gets members from groups 1 & 2

47 Some Dynare equations First-order condition for group 9 weight9*(c9-le^(1+1/phi)/(1+1/phi))^-sigma = beta*(1+r(+1))* ( 0.8*weight13*(c13(+1)-le(+1)^(1+1/phi)/(1+1/phi))^-sigma + 0.2*weight5*(c5(+1)-delta^(1+1/phi)/(1+1/phi))^-sigma ); Members of this group can go to group 5, ueuu, or group 13, eeuu, with 0.2 and 0.8 probability, respectively

48 Bopart-Mitman-Krusell Algorithm

49 Bopart-Krusell-Mitman Solve the no-aggregate uncertainty economy accurately Solve accurately the transition path for a one-time, positive, aggregate shock of fixed magnitude Assume linearity in aggregate responses Done

50 Solving for the transition path 1 Guess a time path for prices 2 Solve for the individual problem 1 Start at future date and assume economy has reached steady state 2 Policy rules known in this steady state 3 Solve policy rules in previous periods by going backwards 3 Update prices given individual policy rules 4 Iterate until convergence

51 Reiter Algotithm

52 Example environment for these slides Same as Krusell-Smith 1998 JPE paper except transition probabilities are assumed to constant (makes the exposition easier)

53 Example environment Recall that KS assume that aggregate productivity, z t, can take on only two values and transition probabilities vary such that employment can also take on only two values so that employment level is not needed as an additional state variable. When perturbation is used to deal with fluctuations in z t then it is implicitly assumed that z t can take on more than a finite number of values (even though one could restrict it to a finite number when simulating the model) = applying the Reiter method means that either n t becomes an additional state variable or one should apply a modification of KS "trick" to get z t = n t (see more detailed slides on my website)

54 Two key elements of Reiter procedure 1 A numerical solution to the model consists of: { 0 if P k i,t+1 = N (e i,t, k i,t, z t, m t ; ψ N ) < P N (e i,t, k i,t, z t, m t ; ψ N ) o.w. where m t is a characterization of the distribution 2 Knowing ψ N should be enough to write down a formula for Γ ψn ( ), where m t+1 = Γ ψn (z t+1, z t, m t ) "formula" means an exact algebraic expression (think: something that can be entered in the Dynare model block)

55 What does the second element require? An exact expression (i.e., formula) is required = Γ ψn cannot be such that it has to be determined with a simulation method or a subroutine Possibilities: 1 m t describes complete distribution = m t can be histogram values at a fine grid (as in Reiter 2008). Simulation slides give expression for Γ ψn ( ) m t could be limited set of moments if it is combined with a distributional assumption as in Winberry (2016) 2 m t are the moments of the levels of k i,t so that explicit aggregation is possible as in XPA (this will introduce additional policy functions if higher-order moments are used. See XPA slides)

56 Rewrite the policy function Rewrite the numerical solution to the model as ψ N,t = ψ N (z t, m t ) = ψ N (s t )

57 More on distribution Take KS environment. e i,t {0, 1} and z t { z b, z g} but transition probabilities constant Suppose m t contains the mean and uncentered variance of capital holdings for employed and unemployed and we make an assumption on the functional form of the distribution ) Thus, we know the density f (k i,t ; m [1],0,t, m [2],0,t for ) unemployed and density f (k i,t ; m [1],1,t, m [2],1,t for employed Notation: m [k],e,t is the k-order moment for capital of workers with employment status e in period t

58 More on distribution Expressions for end-of-period moments are easy to write down. E.g., for the second moment for the unemployed we get m [2],0,t = + (P N (0, k i,t ; ψ N,t ) 2 f (k i,t ; m [1],0,t, m [2],0,t ) dk i,t We use quadrature to turn this into a formula we can write down in say the Dynare model block ( becomes a sum) Getting a formula for beginning-of-next-period moments (which takes into account change in employment status), is just accounting (see simulation slides)

59 Notation & grid ε j and κ j : employment status and capital at grid point j Dimension of ψ N,t = n # ψ N If P N ( ) is 2 nd -order complete polynomial = n # ψ N = 6 number of grid points = n # grid n# ψ N no grid for s in the Reiter method!!!!!!!!!!!!!!!!!

60 Model equation at grid points log utility and δ = 1 for simplicity = Euler equation becomes ( ) 1 r(s)κ j +w(s)ε j l PN (ε j, κ j ;ψ N (s)) E βr(s ) if P N (ε j, κ j ;ψ N (s)) > 0. = ( (r(s )) P N (ε j, κ j ; λ k (s)) + w(s)ε l P N (ε, P N (ε j, κ j ; λ k (s)); ψ N (s )) ) 1 ε j, s This equation is replaced by k i,t+1 = 0 if P N (ε j, κ j ;ψ N (s)) < 0.!!!! One must make a guess which part of the two-part Kuhn-Tucker conditions should be used at each grid point. (With perturbation we consider small changes in z so it is reasonable to assume that these characterizations will not change with z t )

61 Additional exact expressions 1 r(s) = αz ( K/ l ) α 1 2 w(s) = (1 α)z ( K/ l ) α 3 law of motion for z and ε 4 m = Γ ψn (z, z, m) m is histogram in Reiter = Γ ψn is fully known (see simulation slides) m can also be a limited set of moments if it is combined with a functional form assumption as in example above

62 Mental break Have I really done anything? Not much I constructed a grid I construct a system with individual choices substituted out using P N (e i,t, k i,t ; ψ N (s t ))

63 Perturbation system Suppose 2 nd -order polynomial is used: n # ψ N = 6 Suppose there are 6 grid points After substituting out r and w as well as taking care of E t [ ], we get the following type of system Thus, F(ψ N (s), s) = z = (1 ρ) z + ρz + ε z m = Γ ψn (z, z, m) F( ) known ψ N (s) consists of six elements that are unknown functions of s This is a standard perturbation system!!!!!!!!!

64 What is known and unknown? We can replace E [ ε j, s ] with a formula, either because variables have discrete support as in the KS environment or because we use quadrature approximaton The unknown in this system is ψ N (s) So these become variables in the perturbation system (for example, you can call them psia, psib, psic, psid, psie, psif)

65 Small comment If number of grid points exceeds n # ψ N, then you have to take a stand on how to weigh the elements of F ( ) to get a system of n # ψ N equations.

66 Perturbation system What are state variables in this perturbation system?

67 Perturbation system What are state variables in this perturbation system? state variables: s

68 Perturbation system What are state variables in this perturbation system? state variables: s What are not state variables in this perturbation system?

69 Perturbation system What are state variables in this perturbation system? state variables: s What are not state variables in this perturbation system? not state variables: ε and κ

70 Perturbation system What are state variables in this perturbation system? state variables: s What are not state variables in this perturbation system? not state variables: ε and κ What is this perturbation system solving for?

71 Perturbation system What are state variables in this perturbation system? state variables: s What are not state variables in this perturbation system? not state variables: ε and κ What is this perturbation system solving for? It will give you policy functions for the elements of ψ N (s). These describe how the coeffi cients of the individual policy rules fluctuate with s

72 A simple perturbation system? Reiter (2008) uses a fine histogram to characterize CDF = dimension of m typically high (> 1, 000 in Reiter (2008)) = λ k has many inputs = the perturbation system has to solve for many policy functions; the perturbation system solves how each element of λ k changes with each element of s t!!!!!! = higher-order perturbation becomes very tough (even first-order may be tricky) Winberry (2016) approach using moments makes problem more tractable

73 Steady state What is the steady state of this system?

74 Steady state What is the steady state of this system? Simply set z = z = z and m = m = m

75 Steady state What is the steady state of this system? Simply set z = z = z and m = m = m Then this system of n # ψ N equations solves the "no aggregate uncertainty" version of the model (Intuitively: given m (and implied prices) the six equations associated with the capital Euler equation at the six grid points solve for the individual policy function. Given the policy function, one can solve for the steady-state distribution m (see the simulation slides that this is actually very easy if m is a histogram)

76 Important Remaining Issues

77 Important remaining issues Simulate accurately Importance of imposing equilibrium

78 How to simulate? Numerical procedure with a continuum of agents What if you really do like to simulate a panel with a finite number of agents? Impose truth as much as possible: if you have 10,000 agents have exactly 400 (1,000) agents unemployed in a boom (recession) Even then sampling noise is non-trivial

79 Simulation and sampling noise me1 Non random cross section me1 Monte Carlo simulation N=10, Time

80 mu1 Non random cross section mu1 monte Carlo simulation 10, Time

81 mu1 Non random cross section mu1 Monte Carlo simulation N=10,000 mu1 Monte Carlo simulation N=100, Time

82 0,75% muc Non random cross section muc monte Carlo simulation 10,000 muc monte Carlo simulation 100,000 0,50% 0,25% 0,00% Time

83 Imposing equilibrium In model above, equilibrium is automatically imposed in simulation Why?

84 Imposing equilibrium What if we add one-period bonds? Also solve for individual demand for bonds, b(s i,t ) bond price, q(s t ) Simulated aggregate demand for bonds not necessarily = 0 Why is this problematic?

85 Bonds and ensuring equlibrium I Add the bond price as a state variable in individual problem a bit weird (making endogenous variable a state variable) risky in terms of getting convergence

86 Bonds and ensuring equlibrium II Don t solve for but solve for b i (s i,t ) b i (q t, s i,t ) where dependence on q t comes from an equation Solve q t from ( ) I 0 = b i (q t, s i,t ) /I i

87 Bonds and ensuring equlibrium II How to get b i (q t, s i,t )? 1 Solve for d i (s i,t ) where d(s i,t ) = b(s i,t ) + q(s t ) this adds an equation to the model 2 Imposing equilibrium gives 0 = ( b i (q t, s i,t ) ) /I = q t = ( d i (s i,t ) ) /I b i,t+1 = d(s i,t ) q(s t )

88 Bonds and ensuring equlibrium II Does any b i (q t, s i,t ) work? For sure it needs to be a demand equation, that is b i (q t, s i,t ) q t < 0

89 Bonds and ensuring equlibrium II Many ways to implement above idea: d(s i,t ) = b(s i,t ) + q(s t ) is ad hoc (no economics) Alternative: solve for c(s i,t ) get b i,t from budget constraint which contains q t You get b i (q t, s i,t ) with b i (q t, s i,t ) q t < 0

90 References Algan, Y., O. Allais, W.J. Den Haan, P. Rendahl, 2010, Solving and simulating models with heterogeneous agents and aggregate uncertainty Den Haan, W. J., 1996, Heterogeneity, Aggregate uncertainty and the short-term interest rate, Journal of Business and Economic Statistics. Judd, K. L. Maliar, and S. Maliar, 2011, One-node quadrature beats Monte Carlo: A generlized stochastic simulation algorithm, NBER WP Judd, K. L. Maliar, and S. Maliar, 2010, Numerically stable stochastic methods for solving dynamics models, NBER WP 15296

91 Krusell, P. and A.A. Smith Jr., 1998, Income and wealth heterogeneity in the macroeconomy, Journal of Political Economy. Krusell, P. and A.A. Smith Jr., 2006, Quantitative macroeconomic models with heterogeneous agents, in Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Econometric Society Monographs, ed. by R. Blundell, W. Newey, and T. Persson Ríos-Rull, J. V., 1997, Computation of equilibria in heterogeneous agent models, Federal Reserve Bank of Minneapolis Staff Report.