Day 3. More insights & models with aggregate fluctuations. Gianluca Violante. New York University

Transcription

1 Day 3 More insights & models with aggregate fluctuations Gianluca Violante New York University Mini-Course on Policy in Models with Heterogeneous Agents Bank of Portugal, June 15-19, 2105 p. 1 /62

2 Question How progressive should labor income taxation be? p. 2 /62

3 Question How progressive should labor income taxation be? Argument in favor of progressivity: missing markets 1. Social insurance of privately-uninsurable lifecycle shocks 2. Redistribution with respect to initial conditions p. 2 /62

4 Question How progressive should labor income taxation be? Argument in favor of progressivity: missing markets 1. Social insurance of privately-uninsurable lifecycle shocks 2. Redistribution with respect to initial conditions Arguments against progressivity: distortions 1. Labor supply 2. Human capital investment 3. Redistribution from high to low diligence types 4. Financing of public good provision p. 2 /62

5 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (e.g., family) p. 3 /62

6 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (e.g., family) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. flexible labor supply 5. government expenditures valued by households p. 3 /62

7 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (e.g., family) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. flexible labor supply 5. government expenditures valued by households Tractable equilibrium framework closed-form SWF p. 3 /62

8 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (e.g., family) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. flexible labor supply 5. government expenditures valued by households Tractable equilibrium framework closed-form SWF Ramsey approach: mkt structure & tax instruments taken as given p. 3 /62

9 The tax/transfer function T(y i ) = y i λy 1 τ i The parameter τ measures the rate of progressivity: τ = 1 : 0 < τ < 1: τ = 0 : τ < 0 : full redistribution ỹ i = λ progressivity T (y) T(y)/y > 1 no redistribution flat tax rate (1 λ) regressivity T (y) T(y)/y < 1 Break-even income level: y 0 = λ 1 τ p. 4 /62

10 The tax/transfer function T(y i ) = y i λy 1 τ i The parameter τ measures the rate of progressivity: τ = 1 : 0 < τ < 1: τ = 0 : τ < 0 : full redistribution ỹ i = λ progressivity T (y) T(y)/y > 1 no redistribution flat tax rate (1 λ) regressivity T (y) T(y)/y < 1 Break-even income level: y 0 = λ 1 τ Restrictions: (i) no lump-sum transfer & (ii) T (y) monotone p. 4 /62

11 Empirical fit to US micro data PSID , age 25-60, N = 13,721: R 2 = 0.96: τ US = p. 5 /62

12 Empirical fit to US micro data PSID , age 25-60, N = 13,721: R 2 = 0.96: τ US = Log of post government income Log of pre government income p. 5 /62

13 Marginal and average tax rates Tax Rate Average Tax Rate Marginal Tax Rate Household Income x 10 5 At the estimated value of τ US = more p. 6 /62

14 Demographics and preferences Perpetual youth demographics with constant survival probability δ Preferences over consumption (c), hours (h), publicly-provided goods (G), and skill-investment (s) effort: U i = v i (s i )+E 0 (βδ) t u i (c it,h it,g) t=0 v i (s i ) = 1 κ i s 2 i 2µ u i (c it,h it,g) = logc it exp(ϕ i ) h1+σ it 1+σ +χlogg κ i Exp(η) ϕ i N ( vϕ 2,v ϕ ) p. 7 /62

15 Technology Output is a CES aggregator over continuum of skill types: Y = [ 0 N (s) θ 1 θ ] θ θ 1 ds, θ (1, ) Aggregate effective hours by skill type: N(s) = 1 0 I {si =s}z i h i di Aggregate resource constraint: Y = 1 0 c i di+g p. 8 /62

16 Individual efficiency units of labor logz it = α it +ε it α it = α i,t 1 +ω it with ω it N ( v ω 2,v ω ) α i0 = 0 i ε it i.i.d. over time with ε it N ( v ε 2,v ε ) ϕ κ ω ε cross-sectionally and longitudinally p. 9 /62

17 Individual efficiency units of labor logz it = α it +ε it α it = α i,t 1 +ω it with ω it N ( v ω 2,v ω ) α i0 = 0 i ε it i.i.d. over time with ε it N ( v ε 2,v ε ) ϕ κ ω ε cross-sectionally and longitudinally Pre-government earnings: y it = p(s i ) }{{} skill price exp(α it +ε it ) }{{} efficiency determined by skill, fortune, and diligence h it }{{} hours p. 9 /62

18 Government Disposable (post-government) earnings: ỹ i = λy 1 τ i Government budget constraint (no government debt): G = 1 0 [ yi λy 1 τ i ] di Government chooses (G, τ), and λ balances the budget residually WLOG: G g Y p. 10 /62

19 Markets Competitive good and labor markets Perfect annuity markets against survival risk p. 11 /62

20 Markets Competitive good and labor markets Perfect annuity markets against survival risk Competitive asset markets (all assets in zero net supply) Non state-contingent bond p. 11 /62

21 Markets Competitive good and labor markets Perfect annuity markets against survival risk Competitive asset markets (all assets in zero net supply) Non state-contingent bond Full set of insurance claims against ε shocks v ε = 0 bond economy v ω = 0 full insurance economy v ω = v ε = v ϕ = 0 & θ = RA economy p. 11 /62

22 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H p. 12 /62

23 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H Market clearing: C +G = Y p. 12 /62

24 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H Market clearing: C +G = Y Equilibrium allocations: logh RA (τ) = 1 1+σ log(1 τ) logc RA (g,τ) = log(1 g)+logh RA (τ) p. 12 /62

25 Optimal policy in the RA economy Welfare function: W RA (g,τ) = log(1 g)+χlogg+(1+χ) log(1 τ) 1+σ 1 τ 1+σ Welfare maximizing (g,τ) pair: g = χ 1+χ τ = χ Solution for g will extend to heterogeneous-agent setup Allocations are first best (same as with lump-sum taxes) p. 13 /62

26 Recursive stationary equilibrium Given (G,τ), a stationary RCE is a value λ, asset prices {Q( ),q}, skill prices p(s), decision rules s(ϕ,κ,0), c(α,ε,ϕ,s,b), h(α,ε,ϕ,s,b), and aggregate quantities N(s) such that: households optimize markets clear government budget constraint is balanced p. 14 /62

27 Recursive stationary equilibrium Given (G,τ), a stationary RCE is a value λ, asset prices {Q( ),q}, skill prices p(s), decision rules s(ϕ,κ,0), c(α,ε,ϕ,s,b), h(α,ε,ϕ,s,b), and aggregate quantities N(s) such that: households optimize markets clear government budget constraint is balanced In equilibrium, bonds are not traded b = 0 allocations depend only on exogenous states α shocks remain uninsured, ε shocks fully insured more p. 14 /62

28 No bond-trade equilibrium Micro-foundations for Constantinides and Duffie (1996) CRRA, unit root shocks to log disposable income In equilibrium, no bond-trade c t = ỹ t p. 15 /62

29 No bond-trade equilibrium Micro-foundations for Constantinides and Duffie (1996) CRRA, unit root shocks to log disposable income In equilibrium, no bond-trade c t = ỹ t Unit root disposable income micro-founded in our model: 1. Skill investment+shocks: wages 2. Labor supply choice: wages pre-tax earnings 3. Non-linear taxation: pre-tax earnings after-tax earnings 4. Private risk sharing: after-tax earnings disp. income 5. No bond trade: disposable income = consumption p. 15 /62

30 Equilibrium risk-free rater ρ r = (1 τ)((1 τ)+1) v ω 2 Intertemporal dis-saving motive = precautionary saving motive Key: precautionary saving motive common across all agents r τ > 0: more progressivity less precautionary saving higher risk-free rate p. 16 /62

31 Equilibrium skill choice and skill price p. 17 /62

32 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill) p. 17 /62

33 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill) Distribution of skill prices (in levels) is Pareto with parameter θ p. 17 /62

34 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill) Distribution of skill prices (in levels) is Pareto with parameter θ Offsetting effects of τ on s and p for pre-tax wage inequality var(logp(s;τ)) = 1 θ 2 p. 17 /62

35 Equilibrium consumption allocation logc (α,ϕ,s;g,τ) = logc RA (g,τ)+(1 τ)logp(s;τ) }{{} skill price +(1 τ)α }{{} unins. shock (1 τ)ϕ }{{} pref. het. + M(v ε ) }{{} welfare gain from insurable variation Pass-through from (p,ϕ,α) to c mediated by progressivity Invariant to insurable shock ε p. 18 /62

36 Equilibrium hours allocation logh (ε,ϕ;τ) = logh RA (τ) ϕ }{{} pref. het. 1 σ(1 τ) M(v ε) }{{} welfare gain from ins. variation + 1 σ ε }{{} ins. shock Response to ε mediated by tax-modified Frisch elasticity 1ˆσ = 1 τ σ+τ Invariant to skill price p, uninsurable shock α, and size of g p. 19 /62

37 Social Welfare Function Economy is in steady-state with pair (g 1,τ 1 ) Planner chooses, once and for all, a new pair (g,τ) We make two assumptions: 1. Planner puts equal weight on all currently alive agents, discounts future cohorts at rate β SWF becomes average period-utility in the cross-section 2. Skill investments are fully reversible Optimal τ does not depend on the pre-existing skill distribution The transition to the new steady-state is instantaneous p. 20 /62

38 Exact expression for SWF(g, τ) log(1 τ) W(g,τ) = log(1+g)+χlogg +(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 1 ( ( )) ( )] 1 τ 1 τ [ log 2θ (1 τ) 1 θ θ (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ +(1+χ) 1ˆσ v ε (1+χ)σ 1ˆσ 2 v ε 2 p. 21 /62

39 Representative Agent component log(1 τ) W(g,τ) = log(1+g)+χlogg +(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) }{{} Representative Agent Welfare = W RA (g,τ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 1 ( ( )) ( )] 1 τ 1 τ [ log 2θ (1 τ) 1 θ θ (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ +(1+χ) 1ˆσ v ε (1+χ)σ 1ˆσ 2 v ε 2 p. 22 /62

40 Exact expression for SWF(τ) W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ +(1+χ) 1ˆσ v ε (1+χ)σ 1ˆσ 2 v ε 2 p. 23 /62

41 Skill investment component log(1 τ) W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 }{{} 1 2θ (1 τ) }{{} avg. education cost productivity gain = loge[(p(s))] = log(y/n) [ ( ( )) ( )] 1 τ 1 τ log 1 θ θ }{{} consumption dispersion across skills (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ +(1+χ) 1ˆσ v ε (1+χ)σ 1ˆσ 2 v ε 2 p. 24 /62

42 Skill investment component Optimal degree of progressivity (τ) Elasticity of substitution in production (θ) p. 25 /62

43 Uninsurable component W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ }{{ 2 } cons. disp. due to prefs ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ }{{} consumption dispersion due to uninsurable shocks (1 τ) 2 v α 2 +(1+χ) 1ˆσ v ε (1+χ)σ 1ˆσ 2 v ε 2 p. 26 /62

44 Insurable component W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ 1 +(1+χ) ˆσ v 1 v ε ε (1+χ)σ }{{}} ˆσ 2 {{ 2 } prod. gain from ins. shock=log(n/h) hours dispersion p. 27 /62

45 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } p. 28 /62

46 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 p. 28 /62

47 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 p. 28 /62

48 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 cov(logh,logw) = 1ˆσ v ε var(logh) = v ϕ + 1ˆσ 2v ε var 0 (logc) = (1 τ) 2 ( v ϕ + 1 θ 2 ) var(logw) = 1 θ 2 + δ 1 δ v ω p. 28 /62

49 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 cov(logh,logw) = 1ˆσ v ε v ε = 0.17 var(logh) = v ϕ + 1ˆσ 2v ε v ϕ = var 0 (logc) = (1 τ) 2 ( v ϕ + 1 θ 2 ) θ = 2.16 var(logw) = 1 θ 2 + δ 1 δ v ω v ω = p. 28 /62

50 Optimal progressivity 1 Social Welfare Function welfare change rel. to optimum (% of cons.) Welfare Gain = 0.5% τ * = τ US = Baseline Model Progressivity rate (τ) p. 29 /62

51 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) p. 30 /62

52 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) p. 30 /62

53 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) (3) + Pref. Het. τ = p. 30 /62

54 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) (3) + Pref. Het. τ = (4) + Unins. Shocks τ = p. 30 /62

55 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = (5) + Ins. Shocks τ * = Progressivity rate (τ) (3) + Pref. Het. τ = (4) + Unins. Shocks τ = p. 30 /62

56 Actual and optimal progressivity Average tax rate Actual US τ US = Utilitarian τ = Income (1 = average income) Income-weighted average marginal: from 33% to 24% p. 31 /62

57 Alternative SWF Utilitarian SWF embeds desire to redistribute wrt (κ, ϕ) Construct SWF for planner who is averse to redistribution p. 32 /62

58 Alternative SWF Utilitarian SWF embeds desire to redistribute wrt (κ, ϕ) Construct SWF for planner who is averse to redistribution Economy with heterogeneity in (κ,ϕ), χ = v ω = 0, and τ = 0 Compute CE allocations Compute Negishi weights s.t. planner s allocation = CE Use these weights in the SWF p. 32 /62

59 Alternative SWF Utilitarian κ-averse ϕ-averse (κ, ϕ)-averse Productive efficiency Y Y Y Y Insurance wrt ω Y Y Y Y Correction of χ-externality Y Y Y Y Redistribution wrt κ Y N Y N Redistriution wrt ϕ Y Y N N τ Welf. gain (pct of c) p. 33 /62

60 Optimal progressivity: alternative SWF Average tax rate Actual US τ US = Utilitarian τ = Ins. Only τ = Income (1 = average income) p. 34 /62

61 Alternative assumptions on G Cases G/Y τ Welf. gain g endogenous χ = % ḡ exogenous χ = % Ḡ exogenous χ = % With χ = 0, no externality and no push towards regressivity If G fixed in level, lower progressivity translates 1-1 into higher c p. 35 /62

62 Irreversible skill investment Assume skills are fixed after being chosen Planner has an additional incentive to increase progressivity: Can compress consumption inequality without distorting skill investment of existing generations (only concern is discouraging investment of future cohorts) Needed to preserve tractability: production is segregated by age Planner chooses τ = p. 36 /62

63 MODELS WITH AGGREGATE SHOCKS p. 37 /62

64 Aiyagari model with aggregate productivity shocks Aggregate and idiosyncratic risk. The aggregate productivity shock z only takes two values, z Z {z b,z g } with z b < z g. We also assume only two values for the individual productivity shock, ε E {ε b,ε g } with ε b < ε g. Let π(z,ε z,ε) = Pr(z t+1 = z,ε t+1 = ε z t = z,ε t = ε) be the Markov chain that describes the joint evolution of the exogenous shocks. Notation allows transition probabilities for ε to depend on z,z p. 38 /62

65 Aiyagari model with aggregate productivity shocks Production. From the firm s optimization problem we have: w = zf H (K,H) R = 1+zF K (K,H) δ Note: prices depend on the K/H ratio, not just on K, but the dynamics of H can be perfectly forecasted, conditional on (z,z ), through π because labor supply is exogenous. Rig π so that z is a sufficient statistic for H (KS, 1998) H is time varying, but we know how to compute it and forecast it: H(z) = ε EεΠ z (ε) p. 39 /62

66 Aiyagari model with aggregate productivity shocks State variables. The two individual states are (a,ε) S and the two aggregate states are (z,λ) Z Λ where λ( ) is the measure of households across states. λ represents the beginning-of-period distribution of wealth a and employment status ε, then, the new employment status is realized (while wealth is inherited from last period saving decision). The individual states (a, ε) are directly budget relevant The aggregate states z,λ are needed to compute prices The law of motion Ψ for λ is needed to forecast prices p. 40 /62

67 Aiyagari model with aggregate productivity shocks Household problem in recursive form: v(a,ε;z,λ) = max c,a s.t. u(c)+β ε E,z Z c+a = w(z,λ)ε+r(z,λ)a a a λ = Ψ(λ,z,z ) v(a,ε ;z,λ )π(z,ε z,ε) where Ψ(λ,z,z ) is the law of motion of the endogenous aggregate state Dependence on z is inherited from π, but it does not apply to the marginal with respect to wealth (a is predetermined wrt to ε). Ψ(z,z,λ) is the new equilibrium object p. 41 /62

68 Equilibrium We focus on RCE, where (i) the individual saving decision rule is a time-invariant function g(a,ε;z,λ) and (ii) next period cross-sectional distribution is a time invariant function Ψ(λ,z,z ) of the current distribution and of current and next period aggregate shocks. Key complication: the decision rule depends on λ which is a distribution. Where is this dependence coming from? To solve their problem, households need to forecast prices next period. Prices depend on aggregate capital, and aggregate capital this period and next period, K and K, depends on how assets are distributed in the population. p. 42 /62

69 Why is the distribution a state variable? Consider the Euler equation associated to the problem above: u c (R(z,λ)a+w(z,λ)ε g(a,ε;z,λ)) βe[r(z,λ )u c (R(z,λ )g(a,ε;z,λ)+w(z,λ )ε g(g(a,ε;z,λ),ε ;z,λ ))] To solve for g, households need to forecast prices next period, and next period prices depend on λ Agents need to know the equilibrium law of motion Ψ to forecast λ given λ Ψ is a mapping from distributions into distributions: complicated Note: if prices were exogenous (SOE), the wealth distribution would not be a state variable in this example p. 43 /62

70 The Krusell and Smith approach Krusell and Smith (KS, 1998) propose to approximate the distribution through a finite set of moments Let m be a (M 1) vector of moments of the wealth distribution, i.e., the marginal of λ with respect to a Our new state vector has law of motion, in vector notation, m = Ψ(z, m) Note that we lost the dependence on z since we are only interested in dynamics of the wealth distribution. p. 44 /62

71 The Krusell and Smith approach To make this approach operational, one needs to: (i) choose M and (ii) specify a functional form for Ψ KS main finding: one obtains a very precise forecasting rule by simply setting M = 1 and by specifying a law of motion of the form: logk = b 0 z +b 1 zlogk, or similarly in levels instead of logs. Near-aggregation because (i) the wealth-rich have close-to-linear saving rules, (ii) they matter a lot more in the determination of aggregate wealth, and (iii) aggregate shocks do not induce significant wealth redistribution across agents p. 45 /62

72 The Krusell and Smith algorithm 1. Specify a functional form for the law of motion, for example, linear or loglinear. In the linear case: m = B 0 z +B 1 z m where B 0 z is M 1 and B 1 z is M M 2. Guess the matrices of coefficients { } Bz,B 0 z 1 3. Specify how prices depend on m. Since m 1 = K: ( ) m1 R(z, m) = 1+zF K δ H(z) ( ) m1 w(z, m) = zf H H(z) p. 46 /62

73 The Krusell and Smith algorithm 4. Solve the household problem and obtain the decision rule g(a,ε;z, m) with standard methods. Note you have additional state variables m on which you have to define a grid. At every point (a,ε,z, m) on the grid solve for the choice a = a that satisfies the following EE: u c (R(z, m)a+w(z, m)ε a ) β z,ε [R(z,Ψ 1 (z, m))u c (R(z,Ψ 1 (z, m))a +w(z,ψ 1 (z, m))ε g(a,ε ;z,ψ(z, m)))]π(z,ε z,ε) where Ψ 1 ( ) is the forecasting function of the mean (first moment) 5. Simulate the economy for N individuals and T periods. In each period compute m t from the cross-sectional distribution p. 47 /62

74 The Krusell and Smith algorithm 6. Run an OLS regression m t+1 = ˆB 0 z + ˆB 1 z m t } and estimate the coefficients {ˆB z, 0 ˆB z 1. Since the law of motion is time-invariant, we can separate the dates t in the sample where the state is z b from those where the state is z g and run two distinct regressions. 7. If {ˆB0 z, ˆB 1 z } { B 0 z,b 1 z}, try a new guess and go back to step 1 8. Continue until convergence: fixed point algorithm in ( ) Bz,B 0 z 1 9. Assess whether the solution is accurate enough. If fit at the solution is not satisfactory add moments of the distribution or try a different functional form for the law of motion. p. 48 /62

75 Stochastic simulation This method approximates the continuum of agents with a large but finite number of agents and uses a random number generator to draw both the aggregate and the idiosyncratic shocks. 1. Simulate the economy for N individuals and T periods. For example, N = 10,000 and T = 1, Draw first a random sequence for the aggregate shocks of length T. Next, one for the individual productivity shocks for each i = 1,...,N, conditional on the time-path for the aggregate shocks. 3. Initialize at t = 0 from the stationary distribution, for example. 4. Drop the first 500 periods when computing statistics. p. 49 /62

76 Stochastic simulation There will be cross-sectional sampling variation in the simulated cross-sectional data, while conditional on the aggregate shock there should be none if the model has a continuum of agents. Algain-Allais-Den Haan-Rendhal document that moments of asset holdings of the unemployed (which are few) are subject to substantial sampling variation. Similarly for agents at the constraint It is pretty slow, but it can be parallelized easily across the N dimension p. 50 /62

77 Wealth per capita of the unemployed p. 51 /62

78 Fraction of agents on the borrowing constraint p. 52 /62

79 Non-stochastic simulation Same idea as approximation of pdf discussed ealier 1. Draw a long series of aggregate shocks of length T. 2. Construct a fine grid over capital [a min,a max ], say, of 1,000 points, i.e. J = 1, Initialize the distribution at t = 0 from the stationary distribution λ 0 (a,ε) = λ (a,ε) p. 53 /62

80 Non-stochastic simulation 4. Suppose we are at a given date t of the simulation with aggregate states ( m t,z t ) and next period aggregate shock is z t+1. Loop over the finer grid and for every ε and a j on the finer grid for wealth obtain g(a j,ε;z t, m t ) Identify the two adjacent grid points a k and a k+1 that contain g(a j, ). Then compute: λ t+1 (a k+1,ε ) = ε E λ t+1 (a k,ε ) = ε E π(ε,z t+1 ε,z t ) a k+1 g(a j,ε;z t, m t ) λ t (a j,ε) a k+1 a k π(ε,z t+1 ε,z t ) g(a j,ε;z t, m t ) a k a k+1 a k λ t (a j,ε) 5. Use these discretized distributions to compute the moments m t period by period. This method is called the histogram method. p. 54 /62

81 Assessing accuracy of the approximated law of motion KS suggest to compute the R 2 to measure the fit of the regression on the simulated data and use it to assess accuracy of the approximation for the law of motion. Unfortunately, it is not a good measure of fit: solutions with an R 2 in excess of sometimes can be inaccurate. Why? We want to assess the accuracy of the law of motion K t+1 = b 0 z t +b 1 z t K t Recall that this law of motion is based on the best linear fit of the time series for average capital {K t} obtained from a panel simulated via the decision rules, jointly with a sequence for {Z t }. Thus K t+1 and K t are only related through the decision rules, not directly through the previous law of motion p. 55 /62

82 Assessing accuracy of the approximated law of motion Define: u t+1 = K t+1 K t+1 where K t is the true value of the capital stock obtained from the simulation and K t+1 is the predicted one based on the law of motion. Then: u t+1 = K t+1 ( b 0 z t +b 1 z t K t since each period one starts with the true value and evaluates how the approximation performs starting from the truth. It is a one-step ahead forecast error of the law of motion Suppose the approximating law of motion is bad and would want to push the observations away from the truth each period. The error terms defined this way underestimate the problem, because the true dgp ( * ) is used each period to put the approximating law of motion back on track ) p. 56 /62

83 Meaningless of ther 2 Table1: MeaninglessnessoftheR 2 implied properties equation R 2 b u mean stand. dev. 3 =0:96404( ttedregression) 0: : :6723 0: =0: : : :6723 0: =0: : : :6723 0: =0: : : :6723 0:0113 Notes: The rst row corresponds to the tted regression equation. The subsequent rows arebasedonaggregatelawsofmotioninwhichthevalueof 3 ischangeduntiltheindicatedlevel ofther 2 isobtained. 1 isadjustedtokeepthe ttedmeancapitalstockequal. K t+1 = α 1 +α 2 z t +α 3 K t p. 57 /62

84 Assessing accuracy of the approximated law of motion Alternative procedure. Define the error as ũ t+1 = K t+1 ( b 0 z t +b 1 z t K t ), i.e., use K t instead of K t to predict the capital stock next period, thus we allow errors to propagate over time Then, report the maximal error in the simulation. One can even plot the true and approximated series, and by analyzing it one can figure out in which histories deviations are the largest. p. 58 /62

85 Trivial market clearing When prices are determined by marginal product of the aggregate state variable (e.g., physical or human capital), then iterating over the law of motion for the aggregate state yields a solution where prices are consistent with market clearing Given a simulated history of {z t }, and an initial condition K t : (i) prices at t depend only on K t which is known; (ii) individual saving decisions do not affect prices at t, but they aggregate into the same K t+1 predicted by the law of motion, and hence the asset market clears next period; (iii) next period prices equal again the marginal product of K t+1. What makes market clearing trivial is that prices (wages and rates of return) only depend on a pre-determined variable K t Suppose now decisions at time t affect prices at time t. Examples: (i) endogenous labor supply; (ii) risk-free bond market; (iii) housing market p. 59 /62

86 Endogenous labor supply Define first-step decision rules for assets and hours worked g a (ε,a,z,k;h) and g h (ε,a,z,k;h), a function of aggregate labor input H as well Define aggregate laws of motion: K = b 0 z +b 1 zk = Ψ K (z,k) H = d 0 z +d 1 zk = Ψ H (z,k) Guess initial decision rules and aggregate laws of motion At every point on the grid solve for (h,a ) using the EE: u c (R(z,K,H)a+w(z,K,H)εh a ) β z,ε [R(z,Ψ K (z,k),ψ H (z,ψ K (z,k))) u c (R(z,Ψ K (z,k),ψ H (z,ψ K (z,k)))a + w(z,ψ K (z,k),ψ H (z,ψ K (z,k)))ε g h (ε,a,z,ψ K (z,k),ψ H (z ;Ψ K (z,k))) g a (a,ε,z,ψ K (z,k);ψ H (z,ψ K (z,k))))]π(z,ε z,ε) p. 60 /62

87 Endogenous labor supply and the intratemporal first order condition: v h (h ) = w(z,k,h)ε u c (R(z,K,H)a+w(z,K,H)ε a ) Once these first-step decision rules are obtained from this system of equations, use the decision rules to simulate an artificial panel, but at every date t one must solve for H t that satisfies the labor market clearing condition at date t: g h (ε,a,z t,k t,h t)dλ t = F H ( w(zt,k t,h t) z t,k t ) 1 This time series {z t,h t,k t } is used to update the guess of the aggregate laws of motion. Once converged is achieved, we can solve for the second-step (and final) decision rules g a (ε,a,z,k) and g h (ε,a,z,k) only as a function of K p. 61 /62

88 Asset in zero net (or exogenous) supply Examples: household-supplied IOU, government bond, land Price of bond q. Cash-in-hand: ω = R(z,K)a+b Define the decision rule for capital and bonds g a (ε,ω,z,k;q) and g b (ε,ω,z,k;q) and the aggregate laws of motion: K = b 0 z +b 1 zk = Ψ K (z,k) q = d 0 z +d 1 zk = Ψ q (z,k) Obtained these first-step decision rules, the simulation step requires that, at each date t, one looks for the q t that satisfies the market clearing condition: g b (ε,ω,z t,k t ;q t)dλ t = 0 p. 62 /62