HANK. Heterogeneous Agent New Keynesian Models. Greg Kaplan Ben Moll Gianluca Violante

Transcription

1 HANK Heterogeneous Agent New Keynesian Models Greg Kaplan Ben Moll Gianluca Violante Princeton Initiative 2016

2 HANK: Heterogeneous Agent New Keynesian models Framework for quantitative analysis of aggregate shocks and macroeconomic policy Three building blocks 1. Uninsurable idiosyncratic income risk 2. Nominal price rigidities 3. Assets with different degrees of liquidity Today: Transmission mechanism for conventional monetary policy 1

3 How monetary policy works in RANK Total consumption response to a drop in real rates C response = direct response to r }{{} >95% + indirect effects due to Y }{{} <5% Direct response is everything, pure intertemporal substitution 2

4 How monetary policy works in RANK Total consumption response to a drop in real rates C response = direct response to r }{{} >95% + indirect effects due to Y }{{} <5% Direct response is everything, pure intertemporal substitution However, data suggest: 1. Low sensitivity of C to r 2. Sizable sensitivity of C to Y 3. Micro sensitivity vastly heterogeneous, depends crucially on household balance sheets 2

5 How monetary policy works in HANK Once matched to micro data, HANK delivers realistic: wealth distribution: small direct effect MPC distribution: large indirect effect (depending on Y ) 3

6 How monetary policy works in HANK Once matched to micro data, HANK delivers realistic: wealth distribution: small direct effect MPC distribution: large indirect effect (depending on Y ) C response = direct response to r }{{} + indirect effects due to Y }{{} RANK: >95% RANK: <5% HANK: <1/3 HANK: >2/3 3

7 How monetary policy works in HANK Once matched to micro data, HANK delivers realistic: wealth distribution: small direct effect MPC distribution: large indirect effect (depending on Y ) C response = direct response to r }{{} + indirect effects due to Y }{{} RANK: >95% RANK: <5% HANK: <1/3 HANK: >2/3 Overall effect depends crucially on fiscal response, unlike in RANK where Ricardian equivalence holds 3

8 Why does this difference matter? Suppose Central Bank wants to stimulate C RANK view: sufficient to influence the path for real rates {r t } household intertemporal substitution does the rest HANK view: must rely heavily on GE feedbacks to boost hh labor income through fiscal policy reaction and/or an investment boom responsiveness of C to i is, to a larger extent, out of CB s control 4

9 Literature and contribution Combine two workhorses of modern macroeconomics: New Keynesian models Gali, Gertler, Woodford Bewley models Aiyagari, Bewley, Huggett Closest existing work: 1. New Keynesian models with limited heterogeneity Campell-Mankiw, Gali-LopezSalido-Valles, Iacoviello, Bilbiie, Challe-Matheron-Ragot-Rubio-Ramirez, Broer-Hansen-Krusell-Öberg micro-foundation of spender-saver behavior 2. Bewley models with sticky prices Oh-Reis, Guerrieri-Lorenzoni, Ravn-Sterk, Gornemann-Kuester-Nakajima, DenHaan-Rendal-Riegler, Bayer-Luetticke-Pham-Tjaden, McKay-Reis, McKay-Nakamura-Steinsson, Huo-RiosRull, Werning, Luetticke assets with different liquidity Kaplan-Violante new view of individual earnings risk Guvenen-Karahan-Ozkan-Song Continuous time approach Achdou-Han-Lasry-Lions-Moll 5

10 HANK 6

11 Building blocks Households Face uninsured idiosyncratic labor income risk Consume and supply labor Hold two assets: liquid and illiquid Firms Monopolistically competitive intermediate-good producers Quadratic price adjustment costs à la Rotemberg (1982) Government Issues liquid debt, spends, taxes Monetary Authority Sets nominal rate on liquid assets based on a Taylor rule 7

12 Households max {c t,l t,d t} t 0 E 0 0 e (ρ+λ)t u(c t, l t )dt s.t. ḃ t = r b (b t )b t + wz t l t d t χ(d t, a t ) c t +Γ T (wz t l t + Γ) ȧ t = r a a t + d t z t = some Markov process b t b, a t 0 c t : non-durable consumption b t : liquid assets z t : individual productivity l t : hours worked a t : illiquid assets d t : illiquid deposits χ: transaction cost function T : labor income tax/transfer Γ: income from firm ownership no housing see working paper 8

13 Households max {c t,l t,d t} t 0 E 0 0 e (ρ+λ)t u(c t, l t )dt s.t. ḃ t = r b (b t )b t + wz t l t d t χ(d t, a t ) c t +Γ T (wz t l t + Γ) ȧ t = r a a t + d t z t = some Markov process b t b, a t 0 c t : non-durable consumption d t : illiquid deposits ( 0) b t : liquid assets χ: transaction cost function z t : individual productivity T : labor income tax/transfer l t : hours worked Γ: income from firm ownership a t : illiquid assets no housing see working paper 8

14 Households max {c t,l t,d t} t 0 E 0 0 e (ρ+λ)t u(c t, l t )dt s.t. ḃ t = r b (b t )b t + wz t l t d t χ(d t, a t ) c t T (wz t l t + Γ) + Γ ȧ t = r a a t + d t z t = some Markov process b t b, a t 0 c t : non-durable consumption d t : illiquid deposits ( 0) b t : liquid assets χ: transaction cost function z t : individual productivity T : income tax/transfer l t : hours worked Γ: income from firm ownership a t : illiquid assets no housing see working paper 8

15 Households max {c t,l t,d t} t 0 E 0 0 e (ρ+λ)t u(c t, l t )dt s.t. ḃ t = r b (b t )b t + wz t l t d t χ(d t, a t ) c t T (wz t l t + Γ) + Γ ȧ t = r a a t + d t z t = some Markov process b t b, a t 0 c t : non-durable consumption d t : illiquid deposits ( 0) b t : liquid assets χ: transaction cost function z t : individual productivity T : income tax/transfer l t : hours worked Γ: income from firm ownership a t : illiquid assets no housing see working paper 8

16 Households Adjustment cost function d 2 χ(d, a) = χ 0 d + χ 1 max{a, a} χ max{a, a} Linear component implies inaction region Convex component implies finite deposit rates 9

17 Households Adjustment cost function d 2 χ(d, a) = χ 0 d + χ 1 max{a, a} χ max{a, a} Linear component implies inaction region Convex component implies finite deposit rates Recursive solution of hh problem consists of: 1. consumption policy function c(a, b, z; w, r a, r b ) 2. deposit policy function d(a, b, z; w, r a, r b ) 3. labor supply policy function l(a, b, z; w, r a, r b ) joint distribution of households µ(da, db, dz; w, r a, r b ) 9

18 Firms Representative competitive final goods producer: ( 1 ) ε Y = y ε 1 ε 1 ( ε pj ) ε j dj y j = Y P 0 Monopolistically competitive intermediate goods producers: Technology: y j = Zkj α n 1 α j m = 1 ( r ) α ( w Z α 1 α Set prices subject to quadratic adjustment costs: (ṗ ) Θ = θ (ṗ ) 2 Y p 2 p Exact NK Phillips curve: ( r a Ẏ ) π = ε (m m) + π, Y θ ε 1 m = ε ) 1 α 10

19 Determination of illiquid return, distribution of profits Illiquid assets = part capital, part equity a = k + qs k: capital, pays return r δ s: shares, price q, pay dividends ωπ = ω(1 m)y 11

20 Determination of illiquid return, distribution of profits Illiquid assets = part capital, part equity a = k + qs k: capital, pays return r δ s: shares, price q, pay dividends ωπ = ω(1 m)y Arbitrage: ωπ + q q = r δ := r a 11

21 Determination of illiquid return, distribution of profits Illiquid assets = part capital, part equity a = k + qs k: capital, pays return r δ s: shares, price q, pay dividends ωπ = ω(1 m)y Arbitrage: ωπ + q = r δ := r a q Remaining (1 ω)π? Scaled lump-sum transfer to hh s: Γ = (1 ω) z z Π 11

22 Determination of illiquid return, distribution of profits Illiquid assets = part capital, part equity a = k + qs k: capital, pays return r δ s: shares, price q, pay dividends ωπ = ω(1 m)y Arbitrage: ωπ + q = r δ := r a q Remaining (1 ω)π? Scaled lump-sum transfer to hh s: Γ = (1 ω) z z Π Set ω = α (capital share) neutralize countercyclical markups total illiquid flow = rk + ωπ = αmy + ω(1 m)y = αy total liquid flow = wl + (1 ω)π = (1 α)y 11

23 Monetary authority and government Taylor rule i = r b + ϕπ + ϵ, ϕ > 1 with r b := i π (Fisher equation), ϵ = innovation ( MIT shock ) Progressive tax on labor income: T (wzl + Γ) = T + τ (wzl + Γ) Government budget constraint (in steady state) G r b B g = T dµ Transition? Ricardian equivalence fails this matters! 12

24 Summary of market clearing conditions Liquid asset market B h + B g = 0 Illiquid asset market A = K + q Labor market N = zl(a, b, z)dµ Goods market: Y = C + I + G + χ + Θ + borrowing costs 13

25 Solution Method 14

26 Solution Method (from Achdou-Han-Lasry-Lions-Moll) Solving het. agent model = solving PDEs 1. Hamilton-Jacobi-Bellman equation for individual choices 2. Kolmogorov Forward equation for evolution of distribution Many well-developed methods for analyzing and solving these simple but powerful: finite difference method codes: Apparatus is very general: applies to any heterogeneous agent model with continuum of atomistic agents 1. heterogeneous households (Aiyagari, Bewley, Huggett,...) 2. heterogeneous producers (Hopenhayn,...) can be extended to handle aggregate shocks (Krusell-Smith,...) 15

27 Computational Advantages relative to Discrete Time 1. Borrowing constraints only show up in boundary conditions FOCs always hold with = 2. Tomorrow is today FOCs are static, compute by hand: c γ = V b (a, b, y) 3. Sparsity solving Bellman, distribution = inverting matrix but matrices very sparse ( tridiagonal ) reason: continuous time one step left or one step right 4. Two birds with one stone tight link between solving (HJB) and (KF) for distribution matrix in discrete (KF) is transpose of matrix in discrete (HJB) reason: diff. operator in (KF) is adjoint of operator in (HJB) 16

28 Parameterization 17

29 Three key aspects of parameterization 1. Measurement and partition of asset categories into: 50 shades of K Liquid (cash, bank accounts + government/corporate bonds) Illiquid (equity, housing) 18

30 Three key aspects of parameterization 1. Measurement and partition of asset categories into: 50 shades of K Liquid (cash, bank accounts + government/corporate bonds) Illiquid (equity, housing) 2. Income process with leptokurtic income changes income process Nature of earnings risk affects household portfolio 18

31 Three key aspects of parameterization 1. Measurement and partition of asset categories into: 50 shades of K Liquid (cash, bank accounts + government/corporate bonds) Illiquid (equity, housing) 2. Income process with leptokurtic income changes income process Nature of earnings risk affects household portfolio 3. Adjustment cost function and discount rate adj cost function Match mean liquid/illiquid wealth and fraction HtM 18

32 Three key aspects of parameterization 1. Measurement and partition of asset categories into: 50 shades of K Liquid (cash, bank accounts + government/corporate bonds) Illiquid (equity, housing) 2. Income process with leptokurtic income changes income process Nature of earnings risk affects household portfolio 3. Adjustment cost function and discount rate adj cost function Match mean liquid/illiquid wealth and fraction HtM Production side: standard calibration of NK models Standard separable preferences: u(c, l) = log c 1 2 l2 18

33 Model matches key feature of U.S. wealth distribution Data Model Mean illiquid assets (rel to GDP) Mean liquid assets (rel to GDP) Poor hand-to-mouth 10% 9% Wealthy hand-to-mouth 20% 18% 19

34 Wealth distributions: Liquid wealth Model 2004 SCF Liquid wealth Lorenz curve Top 10% share: SCF 2004: 86%, Model: 73% Top 1% share: SCF 2004: 47%, Model: 16% Gini coefficient: SCF 2004: 0.98, Model:

35 Wealth distributions: Illiquid wealth Model 2004 SCF Illiquid wealth Lorenz curve Top 10% share: SCF 2004: 70%, Model: 87% Top 1% share: SCF 2004: 33%, Model: 40% Gini coefficient: SCF 2004: 0.81, Model:

36 Model generates high and heterogeneous MPCs Fraction of lump sum transfer consumed One quarter Two quarters One year Quarterly MPC $ Quarterly MPC out of $500 = 18% Amount of transfer ($) Liquid Wealth ($000) Illiquid Wealth ($000) 22

37 Description Value Target / Source Preferences λ Death rate 1/180 Av. lifespan 45 years γ Risk aversion 1 φ Frisch elasticity (GHH) 0.5 ψ Disutility of labor 27 Av. hours worked equal to 1/3 ρ Discount rate (pa) 4.7% Internally calibrated Production ε Demand elasticity 10 Profit share 10 % α Capital share 0.33 δ Depreciation rate (p.a.) 10% θ Price adjustment cost 100 Slope of Phillips curve, ε/θ = 0.1 Government τ Proportional labor tax 0.25 T Lump sum transfer (rel GDP) % hh with net govt transfer ḡ Govt debt to annual GDP 0.26 government budget constraint Monetary Policy ϕ Taylor rule coefficient 1.25 r b Steady state real liquid return (pa) 2% Housing r h Net housing return (pa) 1.5% Kaplan and Violante (2014) Illiquid Assets r a Illiquid asset return (pa) 6.5% Equilibrium outcome Borrowing r borr Borrowing rate (pa) 8.4% Internally calibrated b Borrowing limit quarterly labor inc 23

38 Results 24

39 Transmission of monetary policy shock to C Innovation ϵ < 0 to the Taylor rule: i = r b + ϕπ + ϵ All experiments: ϵ 0 = , i.e. 1% annualized 25

40 Transmission of monetary policy shock to C Innovation ϵ < 0 to the Taylor rule: i = r b + ϕπ + ϵ All experiments: ϵ 0 = , i.e. 1% annualized Deviation (pp annual) Taylor rule innovation: ɛ Liquid return: r b Inflation: π Quarters Deviation (%) Output Consumption Investment Quarters 25

41 Transmission of monetary policy shock to C C 0 dc 0 = 0 rt b drt b dt + }{{} direct [ C0 0 rt a drt a + C 0 dw t + C 0 dt t w t T t }{{} indirect ] dt 26

42 Transmission of monetary policy shock to C C 0 dc 0 = 0 rt b drt b dt + }{{} direct [ C0 0 rt a drt a + C 0 dw t + C 0 dt t w t T t }{{} indirect ] dt 26

43 Transmission of monetary policy shock to C dc 0 = 0 C 0 drt b dt + r b t 0 [ C0 r a t drt a + C 0 dw t + C ] 0 dt t dt w t T t Intertemporal substitution and income effects from r b Total Response Direct: r b Deviation (%) Quarters 27

44 Transmission of monetary policy shock to C dc 0 = 0 C 0 drt b dt + r b t 0 [ C0 r a t drt a + C 0 dw t + C ] 0 dt t dt w t T t Portfolio reallocation effect from r a r b Deviation (%) Total Response Direct: r b Indirect: r a Quarters 28

45 Transmission of monetary policy shock to C dc 0 = 0 C 0 drt b dt + r b t Deviation (%) 0 [ C0 r a t drt a + C 0 dw t + C ] 0 dt t dt w t T t Labor demand channel from w Total Response Direct: r b Indirect: r a Indirect: w +Γ Quarters 29

46 Transmission of monetary policy shock to C dc 0 = 0 C 0 drt b dt + r b t 0 [ C0 r a t drt a + C 0 dw t + C ] 0 dt t dt w t T t Fiscal adjustment: T in response to in interest payments on B Deviation (%) Total Response Direct: r b Indirect: r a Indirect: w +Γ Indirect: T Quarters 30

47 Transmission of monetary policy shock to C C 0 dc 0 = 0 rt b drt b dt + }{{} 19% [ C0 0 rt a drt a + C 0 dw t + C 0 dt t w t T t }{{} 81% ] dt Deviation (%) Total Response Direct: r b Indirect: r a Indirect: w +Γ Indirect: T Quarters 31

48 Monetary transmission across liquid wealth distribution Total change = c-weighted sum of (direct + indirect) at each b 32

49 Why small direct effects? Intertemporal substitution: (+) for non-htm Income effect: (-) for rich households Portfolio reallocation: (-) for those with low but > 0 liquid wealth 33

50 Role of fiscal response in determining total effect T adjusts G adjusts B g adjusts (1) (2) (3) Elasticityof C 0 to r b Share of Direct effects: 19% 22% 46% Fiscal response to lower interest payments on debt: T adjusts: stimulates AD through MPC of HtM households G adjusts: translates 1-1 into AD B g adjusts: no initial stimulus to AD from fiscal side 34

51 Monetary transmission in RANK and HANK C = direct response to r + indirect GE response RANK: 90% RANK: 10% HANK: 2/3 HANK: 1/3 35

52 Monetary transmission in RANK and HANK C = direct response to r + indirect GE response RANK: 90% RANK: 10% HANK: 2/3 HANK: 1/3 RANK view: High sensitivity of C to r: intertemporal substitution Low sensitivity of C to Y : the RA is a PIH consumer HANK view: Low sensitivity to r: income effect of wealthy offsets int. subst. High sensitivity to Y : sizable share of hand-to-mouth agents Q: Is Fed less in control of C than we thought? 35

53 Monetary transmission in RANK and HANK C = direct response to r + indirect GE response RANK: 90% RANK: 10% HANK: 2/3 HANK: 1/3 RANK view: High sensitivity of C to r: intertemporal substitution Low sensitivity of C to Y : the RA is a PIH consumer HANK view: Low sensitivity to r: income effect of wealthy offsets int. subst. High sensitivity to Y : sizable share of hand-to-mouth agents Q: Is Fed less in control of C than we thought? Work in progress: perturbation methods estimation, inference 35

54 36

55 Numerical Solution of HJB Equations 37

56 Finite Difference Methods See Explain using neoclassical growth model, easily generalized to heterogeneous agent models Functional forms ρv(k) = max c u(c) + v (k)(f (k) δk c) Use finite difference method Two MATLAB codes u(c) = c1 σ, F (k) = kα 1 σ

57 Barles-Souganidis There is a well-developed theory for numerical solution of HJB equation using finite difference methods Key paper: Barles and Souganidis (1991), Convergence of approximation schemes for fully nonlinear second order equations Result: finite difference scheme converges to unique viscosity solution under three conditions 1. monotonicity 2. consistency 3. stability Good reference: Tourin (2013), An Introduction to Finite Difference Methods for PDEs in Finance. 39

58 Finite Difference Approximations to v (k i ) Approximate v(k) at I discrete points in the state space, k i, i = 1,..., I. Denote distance between grid points by k. Shorthand notation v i = v(k i ) Need to approximate v (k i ). Three different possibilities: v (k i ) v i v i 1 k v (k i ) v i+1 v i k v (k i ) v i+1 v i 1 2 k = v i,b = v i,f = v i,c backward difference forward difference central difference 40

59 Finite Difference Approximations to v (k i ) Backward Forward! #! #%$!! # $ Central # $ # #%$ " 41

60 Finite Difference Approximation FD approximation to HJB is ρv i = u(c i ) + v i [F (k i) δk i c i ] ( ) where c i = (u ) 1 (v i ), and v i is one of backward, forward, central FD approximations. Two complications: 1. which FD approximation to use? Upwind scheme 2. ( ) is extremely non-linear, need to solve iteratively: explicit vs. implicit method My strategy for next few slides: what works slides on my website: why it works (Barles-Souganidis) 42

61 Which FD Approximation? Which of these you use is extremely important Best solution: use so-called upwind scheme. Rough idea: forward difference whenever drift of state variable positive backward difference whenever drift of state variable negative 43

62 Which FD Approximation? Which of these you use is extremely important Best solution: use so-called upwind scheme. Rough idea: forward difference whenever drift of state variable positive backward difference whenever drift of state variable negative In our example: define s i,f = F (k i ) δk i (u ) 1 (v i,f ), s i,b = F (k i ) δk i (u ) 1 (v i,b ) Approximate derivative as follows v i = v i,f 1 {s i,f >0} + v i,b 1 {s i,b <0} + v i 1 {s i,f <0<s i,b } where 1 { } is indicator function, and v i = u (F (k i ) δk i ). 43

63 Which FD Approximation? Which of these you use is extremely important Best solution: use so-called upwind scheme. Rough idea: forward difference whenever drift of state variable positive backward difference whenever drift of state variable negative In our example: define s i,f = F (k i ) δk i (u ) 1 (v i,f ), s i,b = F (k i ) δk i (u ) 1 (v i,b ) Approximate derivative as follows v i = v i,f 1 {s i,f >0} + v i,b 1 {s i,b <0} + v i 1 {s i,f <0<s i,b } where 1 { } is indicator function, and v i = u (F (k i ) δk i ). Where does v i term come from? Answer: since v is concave, v i,f < v i,b (see figure) s i,f < s i,b if s i,f < 0 < s i,b, set s i = 0 v (k i ) = u (F (k i ) δk i ), i.e. we re at a steady state. 43

64 Sparsity Discretized HJB equation is ρv i = u(c i ) + v i+1 v i k s + i,f + v i v i 1 s k i,b Notation: for any x, x + = max{x, 0} and x = min{x, 0} Can write this in matrix notation ρv = u + Av where A is I I (I= no of grid points) and looks like... 44

65 Visualization of A (output of spy(a) in Matlab) nz =

66 The matrix A FD method approximates process for k with discrete Poisson process, A summarizes Poisson intensities entries in row i: s i,b k }{{} inflow i 1 0 s i,b k s+ i,f k }{{} outflow i 0 s + i,f }{{} k inflow i+1 0 v i 1 v i v i+1 negative diagonals, positive off-diagonals, rows sum to zero: tridiagonal matrix, very sparse A depends on v (nonlinear problem). Next: iterative method... 46

67 Iterative Method Idea: Solve FOC for given v n, update v n+1 according to v n+1 i v n i + ρv n i = u(c n i ) + (v n ) (k i )(F (k i ) δk i c n i ) Algorithm: Guess v 0 i, i = 1,..., I and for n = 0, 1, 2,... follow 1. Compute (v n ) (k i ) using FD approx. on previous slide. 2. Compute c n from c n i = (u ) 1 [(v n ) (k i )] 3. Find v n+1 from ( ). 4. If v n+1 is close enough to v n : stop. Otherwise, go to step 1. ( ) See Important parameter: = step size, cannot be too large ( CFL condition ). Pretty inefficient: I need 5,990 iterations (though quite fast) 47

68 Efficiency: Implicit Method Efficiency can be improved by using an implicit method v n+1 i v n i + ρv n+1 i = u(ci n ) + (v n+1 i Each step n involves solving a linear system of the form 1 (v n+1 v n ) + ρv n+1 = u + A n v n+1 ( (ρ + 1 )I A ) n v n+1 = u + 1 v n but A n is super sparse super fast ) (k i )[F (k i ) δk i c n i ] See 48

69 Efficiency: Implicit Method Efficiency can be improved by using an implicit method v n+1 i v n i + ρv n+1 i = u(ci n ) + (v n+1 i Each step n involves solving a linear system of the form 1 (v n+1 v n ) + ρv n+1 = u + A n v n+1 ( (ρ + 1 )I A ) n v n+1 = u + 1 v n but A n is super sparse super fast ) (k i )[F (k i ) δk i c n i ] See In general: implicit method preferable over explicit method 1. stable regardless of step size 2. need much fewer iterations 3. can handle many more grid points 48

70 Implicit Method: Practical Consideration In Matlab, need to explicitly construct A as sparse to take advantage of speed gains Code has part that looks as follows X = -min(mub,0)/dk; Y = -max(muf,0)/dk + min(mub,0)/dk; Z = max(muf,0)/dk; Constructing full matrix slow for i=2:i-1 A(i,i-1) = X(i); A(i,i) = Y(i); A(i,i+1) = Z(i); end A(1,1)=Y(1); A(1,2) = Z(1); A(I,I)=Y(I); A(I,I-1) = X(I); Constructing sparse matrix fast A =spdiags(y,0,i,i)+spdiags(x(2:i),-1,i,i)+spdiags([0;z(1:i-1)],1,i,i); 49

71 Intermediate good firm pricing problem max {p t } t 0 e r a t 0 { )} (ṗt Π t (p t ) Θ t dt p t s.t. ( p ) ( p ) ε Π(p) = P m Y P m = 1 ( r ) α ( w ) 1 α Z α 1 α Θ(π)= θ 2 π2 Y Back to firms 50

72 Fifty shades of K Liquid Illiquid Total Household deposits Non-productive net of revolving debt Corp & Govt bonds B h = Indirectly held equity Productive Directly held equity Deposits at inv fund 2.92 B f Noncorp bus equity = 0.48 K net housing net durables Total B g = 0.26 A = Quantities are multiples of annual GDP Sources: Flow of Funds and SCF 2004 Working paper: part of housing, durables = unproductive illiquid assets back 51

73 Continuous time earnings dynamics Literature provides little guidance on statistical models of high frequency earnings dynamics Key challenge: inferring within-year dynamics from annual data Higher order moments of annual changes are informative Target key moments of one 1-year and 5-year labor earnings growth from SSA data Model generates a thick right tail for earnings levels 52

74 Leptokurtic earnings changes (Guvenen et al) 53

75 Two-component jump-drift process Flow earnings (y = wzl) modeled as sum of two components: log y t = y 1t + y 2t Each component is a jump-drift with: mean-reverting drift: βy it dt jumps with arrival rate: λ i, drawn from N (0, σ i ) Estimate using SMM aggregated to annual frequency Choose six parameters to match eight moments: 54

76 Model distribution of earnings changes Moment Data Model Moment Data Model Variance: annual log earns Frac 1yr change < 10% Variance: 1yr change Frac 1yr change < 20% Variance: 5yr change Frac 1yr change < 50% Kurtosis: 1yr change Kurtosis: 5yr change Transitory component: ˆλ 1 = 0.08, ˆβ 1 = 0.76, ˆσ 1 = 1.74 Persistent component: ˆλ 2 = 0.007, ˆβ 2 = 0.009, ˆσ 2 = 1.53 Density Year Log Earnings Changes Density Year Log Earnings Changes back 55

77 Adjustment cost function calibration d 2 χ(d, a) = χ 0 d + χ 1 max{a, a} χ max{a, a} Choose χ 0, χ 1, χ 2 together with: discount rate ρ borrowing intermediation wedge: κ b = r bor r b to match five features of the household wealth distribution Moment Data Model Mean illiquid assets (rel to GDP) Mean liquid assets (rel to GDP) Poor hand-to-mouth 10% 12% Wealthy hand-to-mouth 20% 17% Fraction borrowing 15% 14% 56

78 Monetary Policy in Benchmark NK Models 57

79 Monetary Policy in Benchmark NK Models Goal: Introduce decomposition of C response to r change Setup: Prices and wages perfectly rigid = 1, GDP=labor =Y t Households: CRRA(γ), income Y t, interest rate r t C t ({r s, Y s } s 0 ) Monetary policy: sets time path {r t } t 0, special case Equilibrium: C t ({r s, Y s } s 0 ) = Y t Overall effect of monetary policy r t = ρ + e ηt (r 0 ρ), η > 0 ( ) d log C 0 dr 0 = 1 γη 58

80 Monetary Policy in RANK Decompose C response by totally differentiating C 0 ({r t, Y t } t 0 ) dc 0 = C 0 dr t dt 0 r }{{ t } direct response to r + In special case ( ) d log C 0 = 1 [ η + dr 0 γη ρ + η }{{} direct response to r C 0 d Y t dt. 0 Y }{{ t } indirect effects due to Y ρ ρ + η }{{} ]. indirect effects due to Y Reasonable parameterizations very small indirect effects, e.g. ρ = 0.5% quarterly η = 0.5, i.e. quarterly autocorr e η = 0.61 η ρ + η = 99%, ρ ρ + η = 1% 59

81 What if some households are hand-to-mouth? Spender-saver or Two-Agent New Keynesian (TANK) model Fraction Λ are HtM spenders : C sp t Decomposition in special case ( ) = Y t d log C 0 dr 0 = 1 γη [ η (1 Λ) ρ + η }{{} direct response to r ] ρ + (1 Λ) ρ + η + Λ. }{{} indirect effects due to Y indirect effects Λ = 20-30% 60

82 What if there are assets in positive supply? Govt issues debt B to households sector Fall in r t implies a fall in interest payments of (r t ρ) B Fraction λ T of income gains transferred to spenders Initial consumption restponse in special case ( ) d log C 0 dr 0 = 1 γη + B. } 1 {{ λ Ȳ } fiscal redistribution channel λ T Interaction between non-ricardian households and debt in positive net supply matters for overall effect of monetary policy 61

83 Comparison to One-Asset Model Elasticity of Consumption Direct Effect: r b Wealth, $ Thousands (a) One-Asset Model (b) Two-Asset Model 62