Incomplete Markets, Heterogeneity and Macroeconomic Dynamics

Incomplete Markets, Heterogeneity and Macroeconomic Dynamics Bruce Preston and Mauro Roca Presented by Yuki Ikeda February 2009 Preston and Roca (presenter: Yuki Ikeda) 02/03 1 / 20

Introduction Stochastic general equilibrium models with incomplete markets and a continuum of heterogeneous agents Preston and Roca (presenter: Yuki Ikeda) 02/03 2 / 20

Introduction Stochastic general equilibrium models with incomplete markets and a continuum of heterogeneous agents The problem: the state vector includes the whole cross-sectional distribution of wealth (in nite-dimensional object) in the presence of aggregate uncertainty The pioneered method to solve this type of models: Krusell and Smith (1998) Summarize the in nite-dimensional cross-sectional distribution of asset by a nite set of moments Preston and Roca (presenter: Yuki Ikeda) 02/03 2 / 20

Introduction Stochastic general equilibrium models with incomplete markets and a continuum of heterogeneous agents The problem: the state vector includes the whole cross-sectional distribution of wealth (in nite-dimensional object) in the presence of aggregate uncertainty The pioneered method to solve this type of models: Krusell and Smith (1998) Summarize the in nite-dimensional cross-sectional distribution of asset by a nite set of moments The behaviour of future aggregate capital can be almost perfectly described using only the mean of the wealth distribution (Higher moments matter very little) Preston and Roca (presenter: Yuki Ikeda) 02/03 2 / 20

Introduction Contributions of the paper A new approach to solving this class of models based on perturbation methods Preston and Roca (presenter: Yuki Ikeda) 02/03 3 / 20

Introduction Contributions of the paper A new approach to solving this class of models based on perturbation methods An analytic characterization of the evolution of the wealth distribution (up to the order of the approximation) Continuously distrubuted random variables / not constrained by the number of state variables Preston and Roca (presenter: Yuki Ikeda) 02/03 3 / 20

Introduction Contributions of the paper A new approach to solving this class of models based on perturbation methods An analytic characterization of the evolution of the wealth distribution (up to the order of the approximation) Continuously distrubuted random variables / not constrained by the number of state variables Understanding of the role of heterogeneity in aggregate dynamics Preston and Roca (presenter: Yuki Ikeda) 02/03 3 / 20

The Model There are continuum of in nitely-lived agents. Total number of agents is normalized to one. Each household i has the following preference: E 0 t=0 β t u (c i,t ) u (c i,t ) = c1 γ i,t 1 1 γ Preston and Roca (presenter: Yuki Ikeda) 02/03 4 / 20

The Model There are continuum of in nitely-lived agents. Total number of agents is normalized to one. Each household i has the following preference: Leisure is not valued E 0 t=0 β t u (c i,t ) u (c i,t ) = c1 γ i,t 1 1 γ Preston and Roca (presenter: Yuki Ikeda) 02/03 4 / 20

The Model There are continuum of in nitely-lived agents. Total number of agents is normalized to one. Each household i has the following preference: E 0 t=0 β t u (c i,t ) u (c i,t ) = c1 γ i,t 1 1 γ Leisure is not valued The budget constraint for capital: a i,t+1 = (1 δ)a i,t + y i,t c i,t Preston and Roca (presenter: Yuki Ikeda) 02/03 4 / 20

The Model Agents face partially insurable labor market income risk Preston and Roca (presenter: Yuki Ikeda) 02/03 5 / 20

The Model Agents face partially insurable labor market income risk Labor input: l i,t = e i,t l Preston and Roca (presenter: Yuki Ikeda) 02/03 5 / 20

The Model Agents face partially insurable labor market income risk Labor input: l i,t = e i,t l e i,t : idiosyncratic employments shock e i,t = (1 ρ e )µ e e i,t + ε e i,t+1 where 0 < ρ e < 1, µ e > 0, ε e i,t+1 : i.i.d. with (0, σ2 e) Preston and Roca (presenter: Yuki Ikeda) 02/03 5 / 20

The Model The aggregate production function: y t = z t k α t l 1 t α Preston and Roca (presenter: Yuki Ikeda) 02/03 6 / 20

The Model The aggregate production function: y t = z t k α t l 1 t α The aggregate quantities of capital and labor: k t = l t = Z 1 0 Z 1 0 a i,t di l i,t di = µ e l Preston and Roca (presenter: Yuki Ikeda) 02/03 6 / 20

The Model The aggregate production function: y t = z t k α t l 1 t α The aggregate quantities of capital and labor: k t = l t = Z 1 0 Z 1 0 a i,t di l i,t di = µ e l z t, an aggregate technology shock, satis es z t+1 = (1 ρ z )µ z + ρ z z t + ε z t+1 where 0 < ρ z < 1, µ e > 0, ε z i,t+1 : i.i.d. with (0, σ2 z ) Preston and Roca (presenter: Yuki Ikeda) 02/03 6 / 20

The Model Rental rates and wage are determined by r(k t, l t, z t ) = αz t (k t /l t ) α 1 w(k t, l t, z t ) = (1 α)z t (k t /l t ) α Preston and Roca (presenter: Yuki Ikeda) 02/03 7 / 20

The Model Rental rates and wage are determined by r(k t, l t, z t ) = αz t (k t /l t ) α 1 w(k t, l t, z t ) = (1 α)z t (k t /l t ) α Household i s income is determined by y i,t = r(k t, l t, z t )a i,t + w(k t, l t, z t )e i,t l Preston and Roca (presenter: Yuki Ikeda) 02/03 7 / 20

Recursive Formulation The model is written as the dynamic programming problem: subject to v (a i,t, e i,t ; Γ t, z t ) = max c i,t,a i,t+1 [u(c i,t ) + βe t v (a i,t+1, e i,t+1, ; Γ t+1, z t+1 )] a i,t+1 = (1 δ)a i,t + r(k t, l t, z t )a i,t + w(k t, l t, z t )e i,t l c i,t Γ t+1 = H(Γ t, z t ) a i,t+1 + b 0 T t : the current distribution of consumers over asset holding and employment status Preston and Roca (presenter: Yuki Ikeda) 02/03 8 / 20

Recursive Formulation 1 De ne the interior function: I (a i,t+1 ) = (a i,t+1 + b) 2 Instead of imposing the borrowing constraint, a i,t+1 + b 0, solve the following dynamic programming problem: subject to v (a i,t, e i,t ; Γ t, z t ) = max c i,t,a i,t+1 [u(c i,t ) + βe t v (a i,t+1, e i,t+1, ; Γ t+1, z t+1 ) + φi (a i,t+1 )] a i,t+1 = (1 δ)a i,t + r(k t, l t, z t )a i,t + w(k t, l t, z t )e i,t l c i,t Γ t+1 = H(Γ t, z t ) φ > 0: penalty parameter Preston and Roca (presenter: Yuki Ikeda) 02/03 9 / 20

Perturbation Methods The Representative Agent Model The Representative Agent Model Assume there are no idiosyncratic labor shocks (All agents will be ex ante and ex post identical: a i,t = k t ) The optimality conditions can be summarized by E t F (c t+1, c t, x t+1, x t ) 2 ct σ βct+1 σ 6 (r(k 2φ t+1, l t+1, z t+1 ) + 1 δ) (k t+1 +b) 3 = E t 4 k t+1 (1 δ)k t r(k t, l t, z t )k t w(k t, l t, z t )l + c t z t+1 (1 ρ z )µ z ρ z z t ε z t+1 3 7 5 = 0 where x t = kt z t Preston and Roca (presenter: Yuki Ikeda) 02/03 10 / 20

Perturbation Methods The Representative Agent Model The solution is of the form c t = g(x t, σ) x t+1 = h(x t, σ) + ησε t+1 where σ > 0: the degree of uncertainty in ε t+1 (2 1) Perturbation methods approximate these functions g and h in the neighborhood of the model s steady state (c, x), c = g(x, 0) and x = h(x, 0). The second order approximation of g around (x, 0) yields g(x, σ) = g(x, 0) + g xm (x, 0)(x m x m ) + g σ (x, 0)σ m + 1 2 m,n g xm x n (x, 0)(x m x m )(x n x n ) + g xm σ(x, 0)(x m x m )σ Preston and Roca (presenter: Yuki Ikeda) m 02/03 11 / 20

Perturbation Methods The Representative Agent Model The second order approximation of h around (x, 0) yields h(x, σ) j = h(x, 0) j + h xm (x, 0) j (x m m x m ) + h σ (x, 0) j σ + 1 2 m,n h xm x n (x, 0) j (x m x m )(x n x n ) + h xm σ(x, 0) j (x m x m )σ m + 1 2 m h σxm (x, 0) j (x m x m )σ + 1 2 h σσ(x, 0) j σ 2 where j, m, n = 1, 2. Unknown terms can be solved by computing the rst and second order derivatives of F Preston and Roca (presenter: Yuki Ikeda) 02/03 12 / 20

Perturbation Methods Heterogeneous Agent Model Heterogeneous Agent Model Now describe the evolving distribution of wealth with the presence of heterogeneity Preston and Roca (presenter: Yuki Ikeda) 02/03 13 / 20

Perturbation Methods Heterogeneous Agent Model Heterogeneous Agent Model Now describe the evolving distribution of wealth with the presence of heterogeneity Approximate the wealth distribution around the model s deterministic steady state. (No aggregate shocks, no idiosyncratic shocks) State variables: The rst order terms: a i,t, e i,t, z t, k t The second order terms: bk t 2, bk t bz t, bz t 2, Φ t, Ψ t (bx t = (x t x)) Z 1 Z 1 Φ t = (a i,t a) 2 di and Ψ t = (a i,t a) (e i,t e) di 0 0 Preston and Roca (presenter: Yuki Ikeda) 02/03 13 / 20

Perturbation Methods Heterogeneous Agent Model Heterogeneous Agent Model Now describe the evolving distribution of wealth with the presence of heterogeneity Approximate the wealth distribution around the model s deterministic steady state. (No aggregate shocks, no idiosyncratic shocks) State variables: The rst order terms: a i,t, e i,t, z t, k t The second order terms: bk t 2, bk t bz t, bz t 2, Φ t, Ψ t (bx t = (x t x)) Z 1 Z 1 Φ t = (a i,t a) 2 di and Ψ t = (a i,t a) (e i,t e) di 0 0 In sum, x t = [a i,t, e i,t, z t, k t, Φ t, Ψ t ] 0 Preston and Roca (presenter: Yuki Ikeda) 02/03 13 / 20

Perturbation Methods Heterogeneous Agent Model Reformulate the optimality conditions in the representative agent model as E t F (c i,t+1, c i,t, x t+1, x t ) 2 βci,t+1 σ (r(k 2φ t+1, l t+1, z t+1 ) + 1 δ) (k t+1 c +b) 3 i,t σ (1 δ + r(k t, l t, z t ))a i,t + w(k t, l t, z t )le i,t c i,t a R i,t+1 = E 1 t 6 0 a i,t+1di k R t+1 4 1 0 (a i,t+1 a) 2 di Φ R t+1 1 0 (a i,t+1 a) (e i,t+1 e) di Ψ t+1 = 0 3 7 5 where x t = [a i,t, e i,t, z t, k t, Φ t, Ψ t ] 0 Preston and Roca (presenter: Yuki Ikeda) 02/03 14 / 20

Perturbation Methods Heterogeneous Agent Model Again, the solution takes the form c i,t = g(x t, σ) x t+1 = h(x t, σ) + ησε t+1 where h(x t, σ) is (6 1) vector. Preston and Roca (presenter: Yuki Ikeda) 02/03 15 / 20

The Solution Theorem All elasticities in the second order approximation to the solution of the model are independent of uncertainty. That is, g σ (x, 0) = h σ (x, 0) = g x σ (x, 0) = h x σ (x, 0) = 0 for all x 2 fa i,t, e i,t, z t, k t, Φ t, Ψ t g. At the rst order, uncertainty does not a ect any of the rst order elasticities. Preston and Roca (presenter: Yuki Ikeda) 02/03 16 / 20

The Solution Theorem All elasticities in the second order approximation to the solution of the model are independent of uncertainty. That is, g σ (x, 0) = h σ (x, 0) = g x σ (x, 0) = h x σ (x, 0) = 0 for all x 2 fa i,t, e i,t, z t, k t, Φ t, Ψ t g. At the rst order, uncertainty does not a ect any of the rst order elasticities. The direct impact of uncertainty is re ected in the solution via g σσ (x, 0) and h j σσ(x, 0). Preston and Roca (presenter: Yuki Ikeda) 02/03 16 / 20

The Solution Theorem All elasticities in the second order approximation to the solution of the model are independent of uncertainty. That is, g σ (x, 0) = h σ (x, 0) = g x σ (x, 0) = h x σ (x, 0) = 0 for all x 2 fa i,t, e i,t, z t, k t, Φ t, Ψ t g. At the rst order, uncertainty does not a ect any of the rst order elasticities. The direct impact of uncertainty is re ected in the solution via g σσ (x, 0) and h j σσ(x, 0). g σσ (x, 0) = 2 bc i,t(x t, σ) σ 2 Preston and Roca (presenter: Yuki Ikeda) 02/03 16 / 20

Results Benchmark parameter values β = 0.98, δ = 0.025, γ = 2, α = 0.36 l = 0.32, µ z = 1, ρ z = 0.75, σ z = 0.0132 µ e = 0.93, ρ e = 0.70, σ e = 0.05, ρ ze = 0 Preston and Roca (presenter: Yuki Ikeda) 02/03 17 / 20

Results Optimal decision rules Preston and Roca (presenter: Yuki Ikeda) 02/03 18 / 20

Results Optimal decision rules Law of motion for ba i,t+1 ba i,t+1 = 0.0003 + 0.9993ba i,t + 0.6288be i,t + 0.8574bz t 0.0278bk t +0.0002ba 2 i,t + 0.0006ba i,tbe i,t + 0.0458ba i,t bz t 0.0031ba i,t bk t +0.0006be 2 i,t 0.6465be i,t bz t + 0.0300be i,t bk t + 0.0036bz 2 t 0.0010bz t bk t + 0.0025bk 2 t 0.0009 bφ t 0.00005bΨ t Preston and Roca (presenter: Yuki Ikeda) 02/03 18 / 20

Results Optimal decision rules Law of motion for ba i,t+1 ba i,t+1 = 0.0003 + 0.9993ba i,t + 0.6288be i,t + 0.8574bz t 0.0278bk t +0.0002ba 2 i,t + 0.0006ba i,tbe i,t + 0.0458ba i,t bz t 0.0031ba i,t bk t +0.0006be 2 i,t 0.6465be i,t bz t + 0.0300be i,t bk t + 0.0036bz 2 t Law of motion for bk t+1 0.0010bz t bk t + 0.0025bk 2 t 0.0009 bφ t 0.00005bΨ t bk t+1 = 0.0003 + 0.8573bz t + 0.9714bk t + 0.0036bz 2 t + 0.0449bz t bk t 0.0006bk 2 t 0.0007 bφ t + 0.0006bΨ t Preston and Roca (presenter: Yuki Ikeda) 02/03 18 / 20

Results Optimal consumption and saving decisions depend on all variables Preston and Roca (presenter: Yuki Ikeda) 02/03 19 / 20

Results Optimal consumption and saving decisions depend on all variables ba i,t+1 and bk t+1 are negative. Φ bφ t bφ t = R 1 t 0 (a i,t a) 2 di! More capital is held by individuals with a lower marginal propensity to save bk t+1 is positive. Ψ bψ t = R 1 t 0 (a i,t a) (e i,t e) di! Individuals with lower capital have worse employment outcomes Preston and Roca (presenter: Yuki Ikeda) 02/03 19 / 20

Results Optimal consumption and saving decisions depend on all variables ba i,t+1 and bk t+1 are negative. Φ bφ t bφ t = R 1 t 0 (a i,t a) 2 di! More capital is held by individuals with a lower marginal propensity to save bk t+1 is positive. Ψ bψ t = R 1 t 0 (a i,t a) (e i,t e) di! Individuals with lower capital have worse employment outcomes ba i,t+1 ba i,t = 0.9993 + 0.0004ba i,t + 0.0006be i,t + 0.0458bz t 0.0031bk t! It varies across individuals according to ba i,t, but other e ects are small. Preston and Roca (presenter: Yuki Ikeda) 02/03 19 / 20

Comments Advantages of Perturbation methods Preston and Roca (presenter: Yuki Ikeda) 02/03 20 / 20

Comments Advantages of Perturbation methods It analytically determines individual decision rules which are optimal to the second order Preston and Roca (presenter: Yuki Ikeda) 02/03 20 / 20

Comments Advantages of Perturbation methods It analytically determines individual decision rules which are optimal to the second order It can handle high dimension state spaces and exible speci cations of the disturbance processes It is robust; value function iteration are often sensitive to approximation methods Preston and Roca (presenter: Yuki Ikeda) 02/03 20 / 20

Comments Advantages of Perturbation methods It analytically determines individual decision rules which are optimal to the second order It can handle high dimension state spaces and exible speci cations of the disturbance processes It is robust; value function iteration are often sensitive to approximation methods Disadvantages of Perturbation methods Preston and Roca (presenter: Yuki Ikeda) 02/03 20 / 20

Comments Advantages of Perturbation methods It analytically determines individual decision rules which are optimal to the second order It can handle high dimension state spaces and exible speci cations of the disturbance processes It is robust; value function iteration are often sensitive to approximation methods Disadvantages of Perturbation methods It cannot be used in the presence of occasionally binding constraints (the use of interior function) Preston and Roca (presenter: Yuki Ikeda) 02/03 20 / 20