14.462 Lecture Notes: Aiyagari and Krusell-Smith

14.462 Lecture Notes Aiyagari and Krusell-Smith 1 The Economy • i ∈ [0, 1]. • Employment l(st ) = st i.i.d. across i (but not necessarily across t), with support S = {smin , ..., smax }, smin > 0. Let π(s0 |s) = Pr(st+1 = s0 |st = s0 ) and π(s) = P P Pr(st = s). Note that s0 π(s0 |s) = 1 for all s and π(s0 ) = s π(s0 |s)π(s). • Normalize Es = 1. • Preferences: E0 U = E0 ∞ X β t U (ct ) t=0 • Budget and borrowing constraint: ct + at+1 = wt st + (1 + rt )at − τ t 1 at = kt − bt ct ≥ 0 kt ≥ 0 bt ≤ bt at+1 ≥ −bt • The asset grid: at+1 ∈ A = {a1 , a2 , ...., aN } where a1 = −b, or at+1 ∈ A = [−b, ∞). • b is the borrowing limit. Either exogenous to the economy; or endogenous. E.g.: bt = inf {st+j }∞ j=1 ∞ X (qt+j /qt )[(wt+j st+j − τ t+j )] j=1 ∞ X [(qt+j /qt )(wt+j smin − rt+j D)] = j=0 qt ≡ qt−1 1 + rt Remark: If there is a steady state point (wt , rt ) → (w, r), then: τ t → τ = rD wsmin wsmin − rD = −D bt → r r 2 Equilibrium • Let Φt (a, s) = Pr(at = a and st = s) denote the joint probability of a and s in period t. 2 • The distribution of wealth in period t is given by X Φt (a, s) = Pr(at = a) ψt (a) = s∈S • Market clearing: Kt + D = X aψt (a) a∈A where D is (exogenous) government debt and Kt is aggregate (and per capita) capital. • Equilibrium prices: rt = f 0 (Kt ) − δ ≡ r(Kt ) ⇔ Kt = κ(rt ) 0 wt = f (Kt ) − f (Kt )Kt ≡ w(Kt ) ⇔ wt = ω(rt ) 2.1 Recursive Equilibrium • Suppose that, in equilibrium, the law of motion for the distribution of wealth is some functional Γ s.t.: Φt+1 = Γ(Φt ) This means that the evolution of Φt is deterministic. • Given Φt we can compute Kt by simply integrating: Kt = K(Φt ) It follows that wt = w(Φt ) and rt = r(Φt ), as well as bt = b(Φt ) Then, we can expresse the problem of the household in recursive form, provided we let Φt be a state variable. 3 • A recursive equilibrium is given by (V, A, Γ) such that: 1. V solves the Bellman equation; and A is the corresponding optimal choice: V (a, s, Φ) = max U(c) + β X s0 ∈S V (a0 , s0 , Φ0 )π(s0 |s) s.t. a0 = w(Φ0 )s0 + [1 + r(Φ0 )][a − c] − r(Φ0 )D 0 ≤ c ≤ a, a0 ∈ A(Φ), Φ0 = Γ(Φ) A(a, s, Φ) = arg max{...} 2. Γ is generated by A; that is, Γ maps Φ to Φ0 such that Φ0 (a0 , s0 ) = X Φ(a, s)1[A(a,s,Φ)=a0 ] π(s, s0 ) s∈S • The equilibrium path of the economy is then given by {Φt }∞ t=0 such that Φt+1 = Γ(Φt ), for given initial Φ0 . • Remark: I write Kt+1 + D = X aψt+1 (a) a∈A whereas SL write Kt+1 + D = X s∈S,a∈A 4 A(a, s, Φt )Φt (a, s) The two expressions are equivalent: X a0 ψt+1 (a0 ) = Kt+1 + D = a0 ∈A = XX a0 ∈A s0 ∈S = XX a0 Φt+1 (a0 , s0 ) a0 a0 ∈A s0 ∈S = X X s∈S,a∈A X Φt (a, s)1[A(a,s,Φt )=a0 ] π(s0 |s) = a0 1[A(a,s,Φt )=a0 ] Φt (a, s) s∈S,a∈A a0 ∈A = X X s0 ∈S π(s0 |s) A(a, s, Φt )Φt (a, s) s∈S,a∈A 2.2 Non-recursive Equilibrium ∞ • I could alternative define an equilibrium as sequences {Vt , At }∞ t=0 and {Kt , Rt , wt }t=0 such that ∞ 1. Given {Rt , wt }∞ t=0 , {Vt , At }t=0 solve X Vt+1 (a0 , s0 )π(s0 |s) Vt (a, s) = max U (c) + β s0 ∈S s.t. a0 = wt+1 s0 + [1 + rt+1 ][a − c] − rt+1 D At (a, s) = arg max[...] 0 ≤ c ≤ a, a0 ∈ A(Φ) where rt+1 = f 0 (Kt+1 ) and wt+1 = f (Kt+1 ) − f 0 (Kt+1 )Kt+1 . ∞ 2. {Kt , Rt , wt }∞ t=0 is generated by Φ0 and {At }t=0 : for all t, X At (a, s)Φt (a, s), Kt+1 + D = s∈S,a∈A Φt+1 (a, s) = X Φt (a, s)1[At (a,s)=a0 ] π(s, s0 ) s∈S and rt = f 0 (Kt ) wt = f (Kt ) − f 0 (Kt )Kt 5 • In my work, this approach is much easier. But not in general. Note that there is no guaranty we could write Kt+1 = G(Kt ) where G is stationary. • Also, this approach proves useful in the characterization of the steady state of the economy. That’s what Aiyagari does. 2.3 Steady State • The steady-state distribution Φ is the fixed point of Γ : Φ = Γ(Φ) • The steady-state capital, interest rate, and wage are then computed as: Z K = a dΦ(a) − D r = r(K) w = w(K) 3 Aiyagari: Steady State 3.1 Individual Behavior • Let the economy be at the steady state, for all t: rt = r, wt = w = ω(r) ¾ ½ wlmin − D ≡ b(w, r, D) bt = b ≡ min b, r 6 • Define: xt ≡ at + b zt ≡ wlt + (1 + r)at + b − τ It follows that zt ≡ wlt + (1 + r)xt − ζ where zt are total resources available in t and xt+1 is investment in t and ζ ≡ rb + τ = r[b + D] = ζ(w, r, D) Remark: If ∆b = −∆D, as in the case of the natural borrowin limit, ζ is independent of D. Otherwise, an increase in D (an increase in τ ) is like a decrease in the labor income path. • Then, for individual i: ct = zt − xt+1 zt+1 = wst+1 + (1 + r)xt+1 − ζ Assume st+1 i.i.d. across t as well. • We can now write the value function in terms of z as: V (z) = max U(z − x) + β 0≤0≤z s.t. X V (z 0 ) π(s0 ) z 0 ≡ ws0 − ζ + (1 + r)x and the corresponding optimal investment as X(z) = arg max{...} x A(z) = X(z) − b 7 Remark: If ∆b = −∆D, then ζ and thus V (.) and X(.) are independent of D, implying A(z; D) = A(z; 0) + D. • In general, X need not be monotonic with either w or r. • If preferences are homothetic preference and if ζ is proportional to w, then X is proportional to w. • Also, X → ∞ as r → ρ and either X → −∞ as r → 0, if no ad hoc borrowing, or X = b for all r ≤ r, some r < ρ, if ad hoc b. Thus, X is “on average” increasing. 3.2 Individual Wealth Dynamics • We henceforth restrict to the case that st is i.i.d. across time and preferences are CEIS. • Suppose for a moment that market were complete. Then, the optimal consumption rule would be given by ct = m · [(1 + r)at + wt st + ht+1 ] = = m · [zt + (ht+1 − b)] where ht+1 is the present value of labor income and m is the marginal propensity to consume out of effective wealth. Note that m ∈ (0, 1) and ht+1 > (natural borrowing limit) ≥ b. Thus ct = c + m · zt where c > 0 and m ∈ (0, 1). • For zt ≤ c/m, ct > zt under compelete markets, but this is impossible under incomplete markets. Under incomplete markets, C(z) is bounded above by the 8 450 . In particular, there is zb ∈ [zmin , c/m) such that C(z) = z for all z ≤ zb and C(z) < z otherwise. Moreover, z > zb, 1 > C 0 (z) > m. But as z → ∞, C(z) − [c + m · zt ] → 0 and C 0 (z) → 0. Finally, C 00 < 0??? 3.3 Individual Wealth Dynamics • Given X(.), the low of motion for wealth zt of individual i is given by: zt+1 = wst+1 + (1 + r)X(zt ) − ζ or z 0 = G(z, s0 ). 3.4 Steady State: General Equilibrium • Let α(w, r, D) ≡ A(z; w, r, D) = EΦ X(z; w, r, D) − b. Remark: If ∆b = −∆D, then α(w, r, D) = EΦ X(z; w, r) + D − wlmin /r = = α(w, r, 0) + D and thus α(.) moves one-to-one with D. • If β(1 + r) ≥ 1, then U 0 (ct ) ≥ EU 0 (ct+1 ), which implies that xt , zt , at → ∞. Therefore, limr→ρ α(r) = +∞ and r is bounded above by ρ ≡ 1/(1 + β). If b = ∞, then limr→0 b(r) = −∞, implying limr→0 α(r) = −∞. In that case, r is bounded below by 0. If b < ∞, then ∃ r0 > 0 such that b(r) = b for all r < r0 , implying that ∃ r00 > 0 such that α(r) = −b for all r ≤ r00 and α(r) > −b for all r > r00 . In that case, α(r) is well defined for r < 0 as well. 9 • In equilibrium w = ω(r) and a(r, D) ≡ α(ω(r), r, D) That’s the steady-state supply of savings, as a function of r. • Remark: Even if αr > 0 and αw > 0, ω 0 < 0, and therefore ar is ambiguous. But we consider ar > 0. • Let κ(r) ≡ f 0−1 (r + δ) That’s the demand for capital, as a function of r. • General Equilibrium: Given D, r∗ solves a(r∗ , D) = κ(r∗ ) + D and K ∗ = κ(r∗ ) ≡ f 0−1 (r∗ + δ). • Complete vs Incomplete: rinco < 1/(1 + β) = rcompl ⇒ Kinco > Kcompl Saving rate δK/f (K) also higher under incomplete markets. • A higher b shifts a(r) left and therefore K ∗ falls. 3.5 The Effect of Government Debt • If ∆b = −∆D, then a(r, D) = a(r, 0) + D. In this case, r∗ is determined by a(r∗ , 0) = κ(r∗ ) and thus r∗ , K ∗ are independent of D. (Ricardian Equivalence) 10 • If b is independent of D, then ζ increases one-to-one with τ = rD. Because −ζ is like a deterministic income component, X(.) raises with −ζ/r but by less than one-to-one: ∂X(.)/∂ζ ≈ −s/r, where s ∈ (0, 1) is the saving rate. Therefore, an increase in D lowers X(z) but by less than one-to-one: ∂X(.)/∂D ≈ −s. Since a(r, D) = EΦ X(z; r, D) − b, we conclude ∂a(r, D)/∂D ≈ −s < 0. In this case, r∗ is determined by a(r∗ , D) = κ(r∗ ) + D and thus r∗ increases with D. It follows that K ∗ falls with D. (Crowding Out) 3.6 Simulations • Risk aversion • Volatility of idiosyncratic shocks l • Persistence in idiosyncratic shocks l 4 Krusell and Smith: Dynamics • An approximate or constrained equilibrium is given by 1. V solves the Bellman equation; and A is the corresponding optimal choice: V (a, s, m) = max U(c) + β X s0 ∈S V (a0 , s0 , m0 )π(s0 |s) s.t. a0 = w(Φ)s0 + [1 + r(Φ)][a − c] − r(Φ)D A(a, s, m) = arg max{...} 11 c ≥ a, a0 ∈ A(Φ), b m0 = G(m) 2. Given the initial Φ0 and the rule A, compute {mt , Φt }∞ t=0 by mt are the moments of Φt Φt+1 (a, s) = X s∈S The errors are very small. 0 Φt (a, s)1[A(a,s,m 0 π(s, s ). b t )=a ] b t) εt ≡ mt+1 − G(m • Simulations... • One moment (the mean) is enough... • Wealth distribution... not enough skewness • Introduce heterogeneity in discount factors (willingness to save) • Discuss Rios-Rul et al. 12