Solving Heterogeneous Agent Models with Dynare

Transcription

1 Solving Heterogeneous Agent Models with Dynare Wouter J. Den Haan University of Amsterdam March 12, 2009

2 Individual agent Subject to employment, i.e., labor supply shocks: e i,t = ρ e e i,t 1 + ε i,t ε i,t N(µ ε, σ 2 ε ) For now, assume µ ε is such that aggregate labor supply equals 1, i.e., Z exp(e i,t )di = 1 Incomplete markets only way to save is through holding capital

3 Individual rm Competitive rm so agent faces competitive prices w t = (1 α) z t (K t 1 ) α r t = αz t (K t 1 ) α 1

4 Individual agent max fc i,t,k i,t g t=0 E t=0 βt ln(c i,t ) s.t. c i,t + k i,t = r t k i,t 1 + w t exp(e i,t ) + (1 δ)k i,t 1 P(k i,t )

5 Penalty term P(k i,t ) can capture many things: borrowing constraint: k i,t 0 individual return is lower with lower capital holdings P(k i,t ) cannot be too wild if you are going to solve the model with perturbation techniques

6 What aggregate variables do 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 levels and capital levels

7 Equilibrium Continuum of agents Individual policy functions that solve the agent s maximization problem A wage and a rental rate given by equations above. A transition law for the cross-sectional distribution of capital, that is consistent with the investment policy function. f t = beginning-of-period cross-sectional distribution of capital and the employment status after the employment status has been realized. f t+1 = Υ(z t, f t )

8 Key approximating step Approximate the cross-sectional distribution with a limited set of characteristics Proposed in Den Haan (1996), Krusell & Smith (1997,1998), Rios-Rull (1997) Explicitly or implicitly solve for aggregate policy rule Solve individual policy rule for given aggregate law of motion Make the two consistent

9 Krusell-Smith (1997,1998) algorithm Assume the following approximating aggregate law of motion m t+1 = Γ(z t, m t ; η Γ ), where z t are the exogenous shocks and m t is a vector with cross-sectional moments.

10 Krusell-Smith (1997,1998) algorithm Start with an initial guess for its coe cients, η 0 Γ Use following iteration until n iter has converged Γ Given η iter solve for the individual policy rule. Γ Given indivudual 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 Iterate until η iter+1 Γ is su ciently close to η iter Γ.

11 Solving for individual policy rules in KS algorithm Given aggregate law of motion you can solve for individual policy rules using your own favourite algorithm, including Dynare Note that the coe cients of the aggregate policy rules are simply parameters in the Dynare source le (the trick to rede ne parameters in the Dynare source le can again be used)

12 Explicit aggregation (Den Haan & Rendahl) The simulation part is an expensive part of the KS algorithm In theory simulating has bad properties Xpa avoids the simulation step Xpa works for any method to solve the individual policy rule and boils down to Preston-Roca if you solve individual policy rule with perturbation

13 XPA rst-order solution Individual policy rule: Aggregation gives k = a 0 + a k k 1 + a e e + a z z + a K K 1, K = a 0 + a k K 1 + a e + a z z + a K K 1 = a 0 + a e + (a k + a K ) K 1 + a z z = b 0 + b K K 1 + b z z

14 XPA rst-order solution Start with guess for fb 0, b K, b z g Use Dynare ( rst-order) to solve for fa 0, a k, a e, a z, a K g Update the law of motion for aggregate K: b 0 = a 0 + a e, b z = a z, b K = (a k + a K ) Iterate until convergence

15 XPA second-order solution k = a 0 + a k k 1 + a e e + a z z + a K K 1 +a k 2k a kek 1 e + a kz k 1 z + a kk kk 1 Aggregation gives +a e 2e 2 + a ez ez + a ek ek 1 + a z 2z 2 + a zk zk 1 + a K 2K 2 1 K = a 0 + a k K 1 + a e + a z z + a K K 1 +a k 2M(k 2 ) + a 1 kem(k 1 e) + a kz K 1 z + a kk K 2 1 +a e 2M(e 2 ) + a ez z + a ek K 1 + a z 2z 2 + a zk zk 1 + a K 2K 2 1 M(y) is the cross-sectional moment of variable y & M(e 2 ) is a known constant

16 XPA second-order solution Combining terms gives K = b 0 + b K K 1 + b z z + b zk zk 1 + b K 2K b z 2z2 +b M(k 2 ) M(k2 ) + b M(ke) M(ke) b 0 = a 0 + a e + a e 2M(e 2 ) b K = a k + a K + a ek b z = a z z + a ez b zk = +a kz + a zk b K 2 = a kk + a K 2 b z 2 = a z 2 b M(k 2 ) = a k 2 b M(ke) = a ke

17 XPA second-order solution There are now two new aggregate state variables M(k 2 1 ) and M(k 1e) If current K and r depend on these moments, then next period s values of K and r depend on next period s values of these moments. Thus, we need laws of motion to predict M(k 2 ) and M(ke +1 ) Note that M(ke +1 ) = ρ e M(ke)

18 First attempt Why not simply aggregate k 2 which is equal to (a 0 + a k k 1 + a e e + a z z + a K K 1 +a k 2k a kek 1 e + a kz k 1 z + a kk k 1 K 1 +a e 2e 2 + a ez ez + a ek ek 1 + a z 2z 2 + a zk zk 1 + a K 2K 2 1 )2 Problem: This introduces more cross-sectional moments (and it keeps on going)

19 XPA trick Get a second-order approximation for k 2 and ke In your Dynare source le you can simply add two equations like var1 var2 = kˆ2; = ke; Get laws of motion for M(k 2 ) and M(ke +1 ) by aggregating the second-order policy functions for var1 and var2.