Continuous Time Bewley Models

Transcription

1 1 / 18 Continuous Time Bewley Models DEEQA Quantitative Macro Sang Yoon (Tim) Lee Toulouse School of Economics October 24, 2016

2 2 / 18 Today Aiyagari with Poisson wage process : Based on so won t go through all the details Interest rate follows geometric Brownian motion : Based on Angeletos (2007); Benhabib et al. (2014), extended in above paper

3 3 / 18 Income Fluctuation Problem in Continuous Time Continuum of agents solve V (t, a, w) = E 0 e ρt u(c t )dt 0 da t = (w t + r t a t c t )dt, a t B, w t is CPP with jumps z w that arrive at rate λ w : z w = γ 2 γ 1 and λ w = λ 1 if w t = γ 1 z w = γ 1 γ 2 and λ w = λ 2 if w t = γ 2

4 4 / 18 Income Fluctuation Problem in Continuous Time With stationarity, r t = r. Since w takes only two values, let s instead write V (a, i) = E 0 e ρt u(c t )dt 0 da = (ra + γ i c) dt + (a a )dn it where i = 1, 2 and N it is Poisson with rate λ i. Thus we can take a t as the CPP instead of w t

5 5 / 18 Income Fluctuation Problem in Continuous Time With stationarity, V t = 0. Let j = 3 i. The HJB equation is ρv (a, i) = max {u(c) + V a [ra + γ i c]} c + λ i [V (a, j) V (a, i)] Lastly, need boundary condition u (γ i rb) V a ( B, i) which is just the Kuhn-Tucker condition at the boundary.

6 6 / 18 Some Theoretical Results Suppose r < ρ and γ 1 < γ At the constraint, s 1 ( B) = 0 but s 1 (a) < 0 for all a > B: Agents in bad state always borrow when they can 2. Suppose DARA. Then there exists a M s.t. s 2 (a) (a a M ) for all a a M. Stationary equilibrium is bounded when r < ρ 3. Easy to show savings explodes if r > ρ

7 7 / 18 Stationary Distribution in Continuous Time The solution to the Hamiltonian satisfies u (c(a, i)) = V a, so da = (ra + γ i c(a, i)) dt + (a a )dn it Let p(t, a) denote the density of a at time t. With stationarity, p t = 0 and we can ignore the time subscript. The KFE is thus a [ra + γ i c(a, i)] p(a) = λ }{{} j p(a; j i) λ i p(a) s(a,i)

8 8 / 18 Stationary Distribution in Continuous Time By definition, p(a) is the distribution of a when state is i. 1 Hence we have the set of KFE s d da s i(a)p i (a) = λ j p j (a) λ i p i (a), i = 1, 2 (1) d da [s 1(a)p 1 (a) + s 2 (a)p 2 (a)] = 0, For p(a) to have bounded support, the integrand s 1 (a)p 1 (a) + s 2 (a)p 2 (a) = 0. (2) Easy to solve the ODE resulting from plugging (2) in (1) 1 Does NOT mean that a was in state j.

9 9 / 18 More Theoretical Results If B <NBL, Dirac mass at B. If B is NBL, zero mass at B: Clustering at BL vs smooth wealth distribution For small λ 2, Dirac mass at a M : less wealth inequality (short tail) For large λ 2, zero mass at a M : more wealth inequality (long tail) We already knew this, but means for large wealth inequality λ 2 must be high But intuitively, γ 2 must also be high relative to γ 1 save a lot while rich in anticipation of becoming poor

10 10 / 18 Investment Risk in Continuous Time Continuum of agents solve V (t, k, b) = E 0 e ρt u(c t )dt 0 d(k t + b t ) = (w + R t k t + r t b t c t )dt where W t is BM. dr t = Rdt + σdw t, Assume no wage risk; easy to incorporate if Poisson For simplicity, assume k 0 but no borrowing limit Will think of constraints later * Assume u(c) = c 1 γ /(1 γ): Homotheticity! Makes policy rules linear

11 11 / 18 Income Fluctuation Problem in Continuous Time Total net worth A t w/r + k t + b t is stochastic with da t = [ra t + (R r)κa t c] dt + σκa t dw t where κ k/a is the share invested in risky assets. Thus A t is geometric BM HJB equation with Ito is { ρv (A) = max u(c) + V A [ra + (R r)κa c] c,κ + σ2 2 V AAκ 2 A 2} Note that HJB includes Ito correction term inside the max operator because of κ

12 12 / 18 Some Theoretical Results From Merton (1969), if κ allowed to be arbitrarily large we obtain: κ = R r γσ 2 S = 1 r + r ρ γ where (1 S) c/a. + (1 γ)(r r) κ 2γ Invest more in risky asset if return is high and risk is low Also save more when return is high and risk is low But note that investment and savings propensity is independent of wealth (κ, S are constants)

13 13 / 18 Stationary Distribution in Continuous Time Given the solution c(a) = (1 S)A, the A t process is [ ] r ρ (1 + γ)(r r)2 da t = + γ 2γ 2 σ 2 A t dt + σκa t dw t }{{} s A where s is a constant The KFE (for stationary distribution) is thus s d [Ap(A)] = σ2 κ [ 2 d2 A 2 p(a) ] da 2 da 2

14 14 / 18 More Theoretical Results The KFE (for stationary distribution) is thus [ ] r ρ (1 + γ)(r r) + d [Ap(A)] [ (R r) = γ 2γκ da 2γκ d2 A 2 p(a) ] da 2 Can easily guess p(a) = ζa η 1 : distribution is Pareto η = 2(ρ r) R r 1 κ γ Must be larger than 0 for Pareto to be defined (risk and discounting high compared to excess return)

15 15 / 18 Wealth Inequality η = 2(ρ r) (R r) 1 [ 2σ 2 ] κ γ = γ (ρ r) (R r) 2 1 More wealth inequality (thicker Pareto tail) when More investment; higher excess return lower risk and more patience...and less aversion? NO: higher EIS...ALL THIS ONLY VALID IN PARTIAL EQUILIBRIUM

16 16 / 18 Some Caveats Since Pareto only works with A > 0, only applies for large levels of wealth Need to add borrowing constraint for lower levels, but then need some modifications to the problem Top will still be approximately Pareto

17 17 / 18 General Equilibrium Since policy rules are linear, easily aggregated Assume L = 1 inelastic; representative firm with technology F (K, L) = f(k) and depreciation rate δ: R = f (K) δ, w = f(k) Kf (K) K and r determined by K = wκ, (1 κ)r + κr = 1 S r(1 κ)

18 18 / 18 Complete vs. Incomplete Markets With more risk, R down: tomorrow is more expensive If γ low (high EIS), substitution effect dominates invest less if κ high, income effect dominates invest more Angeletos (2007) shows there exists γ low enough so that SS capital is lower

19 18 / 18 References Angeletos, George-Marios, Uninsured idiosyncratic investment risk and aggregate saving, Review of Economic Dynamics, 2007, 10 (1), Benhabib, Jess, Alberto Bisin, and Shenghao Zhu, The Wealth Distribution in Bewley Models with Investment Risk, NBER Working Papers 20157, National Bureau of Economic Research, Inc May Merton, Robert C, Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case, The Review of Economics and Statistics, August 1969, 51 (3),