Heterogeneous agent models with DYNARE

Transcription

1 Heterogeneous agent models with DYNARE Xavier Ragot a a SciencesPo-CNRS and OFCE Dynare Summer School, June, 2017

2 The question Why do agents differ? Can have different characteristics before entering the economy : preferences, skills. They can grow older (overlapping generation models, Rios-Rull 1995 and Rios-Rull 1997) Ragot Heterogeneous Agents 1/62

3 The question Why do agents differ? Can have different characteristics before entering the economy : preferences, skills. They can grow older (overlapping generation models, Rios-Rull 1995 and Rios-Rull 1997) OR they can have different experiences/trajectories within the economy. Agents face some shocks (Heathcote, Storesletten, and Violante 2009 for a discussion of sources of risk) Workers find/loose their job firms find a new product/clients Individuals have health/family shocks Ragot Heterogeneous Agents 1/62

4 The question Why do agents differ? Can have different characteristics before entering the economy : preferences, skills. They can grow older (overlapping generation models, Rios-Rull 1995 and Rios-Rull 1997) OR they can have different experiences/trajectories within the economy. Agents face some shocks (Heathcote, Storesletten, and Violante 2009 for a discussion of sources of risk) Workers find/loose their job firms find a new product/clients Individuals have health/family shocks It is known that agents face much more risk than aggregate variables (Zeldes, Carroll, Rios-Rull...) Ragot Heterogeneous Agents 1/62

5 Example Ragot Heterogeneous Agents 2/62

6 Heterogeneous agents models Different names : Heterogeneous agents models : Incomplete insurance-market models, Bewley-Aiyagari-Huggett, Liquidity constrained, Standard Incomplete Market model (SIM) Start in economic theory (Bewley) rational expectation literature, but now solved with a form of bounded rationality (Krusell and Smith 1998) Frame Basic idea : agents understand well the risk they face and act accordingly/related to finance : self-insurance Ragot Heterogeneous Agents 3/62

7 Heterogeneous agents models Those models now used with a lots of frictions : Sticky prices (McKay and Reis, 2013; Nakamura et al. 2017) Unvoluntary unemplyment (search and matching) (Krusell, Mukuyama, Sahin 2010; ) Bounded rationality (Farhi, Werning 2017) Additional Financial frictions (Kaplan and Violante 2014;HANK, Kaplan, Moll, Violante, 2015; Den Haan, Pontus, Riegler, 2016;) New results : paradox of thrift : unemployment risk, precautionary saving, and aggregate demand (Ravn and Sterck 2015; Challe, Matheron, Ragot, Rubio-Ramirez, 2015) Ragot Heterogeneous Agents 4/62

8 Heterogeneous agents models : Different from Agent Based Models? Start in rational expectation literature but solved with forms of bounded rationality (Krusell and Smith 1998 : simple rules; Farhi and Werning 2017). Utility maximization to derive behavior and peform normative analysis Ragot Heterogeneous Agents 5/62

9 DSGE? DSGE : first models, representation agent, complete and perfect markets, rational expectations. Now : unemployment (search-and-matching), bounded rationality, learning, sticky prices... Many "frictions" DSGE models : Solved in DYNARE (using perturbation method around a steady- state): finite state-space representation, some econometric techniques (estimation) Ragot Heterogeneous Agents 6/62

10 DSGE? DSGE : first models, representation agent, complete and perfect markets, rational expectations. Now : unemployment (search-and-matching), bounded rationality, learning, sticky prices... Many "frictions" DSGE models : Solved in DYNARE (using perturbation method around a steady- state): finite state-space representation, some econometric techniques (estimation) In general, heterogeneous-agent models are not DSGE: Agents differ as time goes by Many (infinite) different agents. Ragot Heterogeneous Agents 6/62

11 Road map 1. The basic problem 2. No-trade equilibrium 3. Small-trade equilibrium 4. Truncation in the space of idiosyncratic histories Ragot Heterogeneous Agents 7/62

12 References 1. Ragot (2017) "Heterogeneous agents in DGSE models", contribution to Handbook of Computational Economics, vol 4. on my web page (References cited in the slides are there). 2. Challe and Ragot (2014) " Precautionary saving over the business cycle", Economic Journal. 3. Le Grand and Ragot (2017) " Optimal policies with incomplete insurance markets and aggregate shocks" Ragot Heterogeneous Agents 8/62

13 The basic problem : households The aggregate risk is represented by state variables h t R N in each period t, The history of aggregate shocks up to period t is denoted h t = {h 0,..., h t }. At the beginning of each period, agents face an idiosyncratic labor productivity shock e t E {e 1,.., e E } that follows a discrete first-order Markov process with transition matrix M(h t ), which is a E E Markov matrix. Aiyagari (1994) considers a a 7 states process. Heathcote (2005) uses a 3-state process. More generally, use Tachen (1986) procedure to find the transition matrix et the states. Ragot Heterogeneous Agents 9/62

14 The basic problem : households An household i can be either employed, e i t = 1 or unemployed e i t = 0 At period t, e t = {e 0,..., e t } E (t+1) denotes an history of the realization of idiosyncratic shock, up to time t. I consider a two-state process where agents can be either employed, when e t = 1 or unemployed, when e t = 0. In this latter case, the agent receives must supply a quantity of labor δ for home production to obtain a quantity of goods δ: The labor choice is constrained. The probability to stay employed is denoted α t M 1,1 (h t ) thus 1 α t is the job-separation rate. The probability to stay unemployed is denoted as ρ t M 0,0 (h t ), such that 1 ρ t is the job finding rate in period t. In the US quarterly, α = 0.95 and ρ = 0.2. Ragot Heterogeneous Agents 10/62

15 The basic problem : households Income fluctuation problem If employed, wage w t, unemployed, income δ She chooses how much to save, consume a i t+1, ci t face credit constraint a i t ā Derive utility from consumption and disutility from work U ( c i t, lt i ) Ragot Heterogeneous Agents 11/62

16 The basic problem : households ( ) max (a i t (.,.),ci(.,.)) E t=0 0 β t U c i t, lt i t t 0 a i t+1 + c i t = e i tw t lt i + (1 e i t)δ + (1 + r t )a i t c i t, lt i 0 a i t ā, a i 0 are given, Ragot Heterogeneous Agents 12/62

17 The basic problem : Production Markets are competitive and a representative firm produces with capital and labor. The production function is Y t = A t K λ t L 1 λ t + (1 µ)k t, µ is the capital depreciation rate, and A t is the technology level, which is affected by technology shocks. The first-order conditions of the firm imply that factor prices are and r t + µ = λa t Kt λ 1 L 1 λ t w t = (1 λ)a t K λ t L λ t Ragot Heterogeneous Agents 13/62

18 The basic problem : Production The technology shocks is defined as the standard AR(1) process A t e at, with a t = ρ a a t 1 + ɛ a t with ɛ a t N (0, (σ a ) 2) Ragot Heterogeneous Agents 14/62

19 The standard way to solve this problem Agents with different histories of shocks e i 0, ei 1, ei 2,... have different behavior Define the relevant distribution of agents F t across wealth Try to solve for the law of motion of this distribution, depending on the shocks h t hitting the economy. F t+1 = Υ (h t, F t ) This methodology is associated to Krusell and Smith (1998) who find a way to approximate the function Υ Ragot Heterogeneous Agents 15/62

20 The basic idea The idea is to truncate in the space of idiosyncratic histories : Only the last N shocks matter At each period t, relevant histories e i t N,..., ei t Thus 2 N+1 histories Ragot Heterogeneous Agents 16/62

21 1 - The N = 0, 2 agents case Assume that agents can not borrow ā = 0 There are no assets in the economy (no capital) Then the savings of all agents is a i t = 0 employed agents consume w t unemployed agent consume δ employed agents price the asset! Ragot Heterogeneous Agents 17/62

22 1 - The N = 0, 2 agents case Assume that U(c, l) = U(c, l) = c1 σ 1 1 σ χ l1+ 1 φ 1+ 1 φ log(c) χ l1+ 1 φ 1+ 1 φ if σ 1 if σ = 1 σ is the curvature of the utility function (not directly equal to risk aversion to the endogenous labor supply), and φ is the Frisch elasticity of Labor supply, ranging from 0.3 to 2 in applied work (see Chetty et al. 2011). Ragot Heterogeneous Agents 18/62

23 1 - The N = 0, Value functions V (a, h, {1}) = max c,l,a U(c, l) +βe (α V (a, h, {1}) + (1 α )V (a, h, {0})) a + c = Al + a(1 + r) a 0 and V (a, h, {0}) = max c,a U(c, δ) +βe (ρ V (a, h, {0}) + (1 ρ )V (a, h, {1})) a + c = δ + a(1 + r) a 0 Ragot Heterogeneous Agents 19/62

24 1 - The N = 0 First order conditions A U c (c 1, l 1 ) = U l (c 1, l 1 ) U c (c 1, l 1 ) = βe (1 + r ) (α U c (c 1, l 1 ) + (1 α )U c (c 0, 0)) and the conditions for unemployed agents to be credit-constrained at the current interest rate is U c (c 0, 0) > βe (1 + r ) (ρ U c (c 0, 0) + (1 ρ )U c (c 1, l 1 )) Ragot Heterogeneous Agents 20/62

25 1 - The N = 0 one finds A t c σ 1 φ 1,t = ( χl1,t c σ 1,t = βe (1 + r t+1 ) α t+1 c σ 1,t+1 + (1 α t+1)δ σ) From the budget constraint of employed agents c 1,t = A t l 1,t, we obtain c 1+φσ 1,t = A 1+φ t /χ φ and c 0,t = δ Ragot Heterogeneous Agents 21/62

26 1 - The N = 0, Stochastic environment The labor market dynamics is exogenous : a t ρ a 0 0 a t 1 α t ᾱ = 0 ρ α 0 α t 1 ᾱ ρ t ρ 0 0 ρ ρ ρ t 1 ρ + ɛ a t ɛ α t ɛ ρ t with shocks N ( 0, (σ a ) 2) (, N 0, (σ α ) 2), N (0, (σ ρ ) 2) Ragot Heterogeneous Agents 22/62

27 1 - The N = 0, steady state The real interest rate is 1 + r = 1 ( ( ) ) δ σ 1 ᾱ + (1 ᾱ) β w Putting some numbers allows estimating the order of magnitude. Consider the period to be a quarter. Chodorow-Reich and Karabarbounis (2013) estimate a decrease in consumption of non-durable goods of households falling into unemployed between 10% and 20%. One can take the conservative value δ c 1 =.9 The quarterly job loss probability is roughly 5% (see Challe and Ragot, 2014), the discount factor is β =.99 One finds a real interest rate r = 0.45% for σ = 1, and r = when σ = 2 instead of r = 1% in the complete market case (where α = 1). Ragot Heterogeneous Agents 23/62

28 1 - The N = 0, linearizing One can express the real interest rate as a function of exogenous variables: E r t+1 = σµ 2 ρ α α t σ 1+φ 1+φσ (1 µ 1ρ a ) a t where µ 1 = αβ (1 + r) and µ 2 = β (1 + r) σ ( δ σ c σ ) 1 c σ ; 1 Ragot Heterogeneous Agents 24/62

29 1 - The N = 0, 2 agents case : Applications Krusell, Mukoyama and Smith (2011) apply this framework to price the assets, when idiosyncratic risk is time-varying. Ravn and Sterck (2016) investigate the unemployment risk and the hiring decision of firms (in the US during the great recession) McKay, Nakamura, Steinsson (2016) study "forward guidance" i.e. monetary policy in this setup. Challe (2017) studies optimal monetary policy with sticky prices and search-and-matching. Ragot Heterogeneous Agents 25/62

30 2 - The N = 1, 4 agents case For special utility functions Linear disutility of labor Lagos and Wright (2005). linear part in the utility function Challe and Ragot (2014). Then, all employed agents save the amount a e t and all uemployed agents are credit constraint a u t = ā Ragot Heterogeneous Agents 26/62

31 2 - The N = 1, example 1: linear utility two assumptions First, U(c, l) = u(c) l Scheinkman and Weiss (1986), Lagos and Wright (2005) and the credit limit is above the natural credit limit ā > δ/r Ragot Heterogeneous Agents 27/62

32 2 - The N = 1 Denote as V k (a t, X t ) the value function of agents in state k = 0, 1,... (0 is employed agents, here), where X t is the set of variables specified below which are necessary to form rational expectations). V 0 (a, X t ) = max c0,t,a 0,t+1 l t u(c 0,t ) l t + βe (α t+1 V 0 (a 0,t+1, X t+1 ) + (1 α t+1 )V 1 (a 0,t+1, X t+1 )) a 0,t+1 = w t l t + a(1 + r t ) c 0,t a 0,t+1 ā and for all unemployed people, k 1 V k (a, X t ) = max u(c t ) δ + βe (ρ t+1 V 0 (a k,t+1, X t+1 ) + (1 ρ t+1 )V k+1 (a k,t+1, X t+1 )) a k,t+1 = δ + a(1 + r t ) c k,t a k,t+1 ā Ragot Heterogeneous Agents 28/62

33 2 - The N = 1, example 1: linear utility We can derive the set of first order conditions. For employed agents; u (c 0,t ) = 1/w t u (c 0,t ) = βe(1 + r t+1 ) (α t+1 u (c 0,t+1 ) + (1 α t+1 )u (c 1,t+1 )) For unemployed agents, for k = 1...L 1 u (c k,t ) = βe(1 + r t+1 ) ( ρ t+1 u (c k+1,t+1 ) + (1 ρ t+1 )u (c 0,t+1 ) ) Ragot Heterogeneous Agents 29/62

34 2 - The N = 1 The conditions define a system of 2(L + 1) equations for 2(L + 1) variables (c k,t, a k,t ) k=0..l. 1/w t = βe(1 + r t+1 ) (α t+1 /w t+1 + (1 α t+1 )u (c 1,t+1 )), u (c k,t ) = βe(1 + r t+1 ) (ρ t+1 u (c k+1,t+1 ) + (1 ρ t+1 )/w t+1 ), a k,t+1 + c k,t = δ + a k 1,t (1 + r t ), u (c 0,t ) = 1/w t a L,t = ā Key question : How to find L? Guess and verify Ragot Heterogeneous Agents 30/62

35 2 - The N = 1, second example A second case, where the equilibrium structure is simple is { u ũ (c) if c < c, (c) = η if c c, See in the example this afternoon Ragot Heterogeneous Agents 31/62

36 2 - Applications Challe, Le Grand and Ragot (2013) study the effect of public debt on the yield curve Le Grand and Ragot, (2016) analyse the trading of derivative assets (options) in this setup. Challe, Matheron, Ragot, Rubio-Ramirez (2016) construct a model with many frictions and perform a Bayesian estimation of the model to investigate the effect of precautionary saving during the Great recession in the US Le Grand and Ragot (2016b) investigate the consumption risk of the rich and the poor in the US and the equity premium. Ragot Heterogeneous Agents 32/62

37 3 - The General Case, any N Recently, Le Grand and Ragot, (2016) present a general theory of truncation in the space of idiosyncratic histories. Based on a well-defined insurance structure, only the last N-periods matter. The model converges to Bewley when N tends toward Optimal Ramsey policy can be derived Ragot Heterogeneous Agents 33/62

38 Motivation Models with Incomplete insurance markets: A new benchmark ("Bewley-Aiyagari-Huggett", "Heterogeneous agents", "liquidity constrained"). Ragot Heterogeneous Agents 34/62

39 Motivation Models with Incomplete insurance markets: A new benchmark ("Bewley-Aiyagari-Huggett", "Heterogeneous agents", "liquidity constrained"). Empirically relevant : (Zeldes, 1989; Carroll 2001; Chodorow-Reich and Karabarbounis 2014; Heathcote, Storesletten, Violante 2009; Relative success for wealth Benhabib and Bisin 2016, among many others) Now solved with many additional frictions : New results (Krusell, Mukuyama, Sahin 2010; McKay and Reis, 2013; Ravn and Sterck 2015; Challe, Matheron, Ragot, Rubio-Ramirez, 2015;Heathcote 2005; Kaplan and Violante 2014;HANK, Kaplan, Moll, Violante, 2015; Den Haan, Pontus, Riegler, 2016; Farhi, Werning 2017) Ragot Heterogeneous Agents 34/62

40 Motivation Models with Incomplete insurance markets: A new benchmark ("Bewley-Aiyagari-Huggett", "Heterogeneous agents", "liquidity constrained"). Empirically relevant : (Zeldes, 1989; Carroll 2001; Chodorow-Reich and Karabarbounis 2014; Heathcote, Storesletten, Violante 2009; Relative success for wealth Benhabib and Bisin 2016, among many others) Now solved with many additional frictions : New results (Krusell, Mukuyama, Sahin 2010; McKay and Reis, 2013; Ravn and Sterck 2015; Challe, Matheron, Ragot, Rubio-Ramirez, 2015;Heathcote 2005; Kaplan and Violante 2014;HANK, Kaplan, Moll, Violante, 2015; Den Haan, Pontus, Riegler, 2016; Farhi, Werning 2017) But difficult to analyse/solve: Hard to solve with Aggregate Shocks (Krusell and Smith 1998; reiter 2009; Den Haan and coauthors 2010). Instead transitions, "MIT shocks". Hard/Impossible to solve for optimal policies with aggregate shocks ( Aiyagari McGrattan 1998; Shin 2006; Davila, Hong, Krusell, Rios-Rul 2012; Acikgoz 2013 ; Nuno and Moll 2015; Ragot 2015; Reis and McKay 2016; Bhandari, Evans, Golosov, Sargent 2015, 2016; Challe 2017; Bilbiie and Ragot, 2017) Ragot Heterogeneous Agents 34/62

41 What we do We provide a representation to derive optimal policies in incomplete insurance-market economies with aggregate shocks. Apply it to derive optimal fiscal policy over the business cycle in a model with capital (as Krusell-Smith), where the planner has various instruments (capital tax, labor tax, transfer, public debt) Time-varying labor and capital taxes : insurance vs incentives Time-varying public debt after TFP shock Ragot Heterogeneous Agents 35/62

42 The basic idea : Truncation of idiosyncratic histories Incomplete insurance markets : Agents with different histories of idiosyncratic risk have different consumption levels. State variable : large distribution of wealth. Infinite dimensional object (Huggett 1993, Hopenhayn and Prescott 1992 : Without aggregate shocks). Ragot Heterogeneous Agents 36/62

43 The basic idea : Truncation of idiosyncratic histories Incomplete insurance markets : Agents with different histories of idiosyncratic risk have different consumption levels. State variable : large distribution of wealth. Infinite dimensional object (Huggett 1993, Hopenhayn and Prescott 1992 : Without aggregate shocks). We construct an environment where : Finite number N of past shocks is a sufficient statistics Finite number of agents to follow : one representative agent for each history Simple structure with aggregate shocks Generalization of previous work, mostly N = 1 (Challe, Le Grand, Ragot 2013; Le Grand Ragot 2015; Challe Ragot 2014, Challe, Matheron, Ragot, Rubio-Ramirez 2016) Ragot Heterogeneous Agents 36/62

44 Why us this framework interesting? 1. The allocation of a family head : simple existence proof 2. Can be decentralized by a simple transfer scheme 3. Convergence toward the general model when N 4. Optimal Ramsey policies can be studied, "Lagrangian Approach" (Marcet and Marimon 2011) 5. Can be simulated with simple perturbation methods (DYNARE), (different from Reiter 2009; Preston and Roca 2007; Kim Kolman and Kom 2010) Ragot Heterogeneous Agents 37/62

45 The substantive contribution Optimal fiscal policy (capital tax, labor tax, transfer, public debt), with capital and TFP shocks Theoretical results: 1. Steady-state capital taxes directly determined by the severity of credit constraints (Aiyagari 1995). When credit constraints don t bind public debt is irrelevant ( Bhandari, Evans, Golosov, Sargent 2016) 2. Explicite formula for taxes, with a natural new concept of liquidity. Ragot Heterogeneous Agents 38/62

46 The substantive contribution Optimal fiscal policy (capital tax, labor tax, transfer, public debt), with capital and TFP shocks Theoretical results: 1. Steady-state capital taxes directly determined by the severity of credit constraints (Aiyagari 1995). When credit constraints don t bind public debt is irrelevant ( Bhandari, Evans, Golosov, Sargent 2016) 2. Explicite formula for taxes, with a natural new concept of liquidity. Quantitative results : US calibration 1. Average B/Y = 148%, τ L = 36%, τ K = 11%, (Aiyagari and McGrattan, 1998; Aickgoz 2013, Pedroni and Dyrda, 2016) 2. Capital taxes are much more volatile than labor taxes. 3. Saving increases too much after TFP shock, absorbed by public debt. Ragot Heterogeneous Agents 38/62

47 Other selected literature Optimal fiscal policy with incomplete markets for aggregate shocks (Aiyagari, Marcet, Sargent, and Seppala, 2002; Farhi 2012). Few papers on optimal policies : with incomplete insurance markets (Shin 2006). Bhandari, Evans, Golosov, and Sargent (2016) : Economy without capital where public debt is irrelevant : One instrument tax on total income. Ragot Heterogeneous Agents 39/62

48 Outline 1. The environment 2. The family-head allocation : The truncation 3. Decentralization and convergence properties : A theorem 4. Ramsey problem : Analytical results 5. Quantitative investigation Ragot Heterogeneous Agents 40/62

49 1 - Environment Close to Krusell and Smith 1998; Heathcote 2005 Unit mass of infinitely-living agents, discount factor β, (GHH) ( ) U(c, l) = u c l1+1/φ, 1 + 1/φ E idiosyncratic states, e t E {1,..., E}, Productivity θ e Aggregate state s t, first-order Markov structure Idiosyncratic risk : Markov, transition matrix M(s t ) Ragot Heterogeneous Agents 41/62

50 1 - Environment Close to Krusell and Smith 1998; Heathcote 2005 Unit mass of infinitely-living agents, discount factor β, (GHH) ( ) U(c, l) = u c l1+1/φ, 1 + 1/φ E idiosyncratic states, e t E {1,..., E}, Productivity θ e Aggregate state s t, first-order Markov structure Idiosyncratic risk : Markov, transition matrix M(s t ) The agents maximize t=0 β t U(c t, l t ), s.t. a t + c t = w t θ et l t + (1 + r t )a t 1 + T t a t ā Ragot Heterogeneous Agents 41/62

51 Environment : Production Production function (CRS), net of depreciation, TFP shocks : depends on s t F (K t 1, L t, s t ) Wage and real interest rate r t = F K (K t 1, L t, s t ) and w t = F L (K t 1, L t, s t ). After tax rates r t = (1 τt k ) r t, and w t = (1 τt) l w t Ragot Heterogeneous Agents 42/62

52 Environment: The Government Public spending G Taxes capital and labor, issues one period debt B t Public debt : same return as capital stock r t, (as in Heathcote 2005) Budget constraint G + (1 + r t )B t 1 + T t τ l t w t L t + τ k t r t (K t 1 + B t 1 ) + B t Market equilibria:, L t = 1 0 l i tλ(di) and K t + B t = 1 0 a i tλ(di) Ragot Heterogeneous Agents 43/62

53 The Question How should the government choose time-varying T t, τ l t, τ k t, B t as a function of the state of economy? Ragot Heterogeneous Agents 44/62

54 2. The Truncated Representation : Island Metaphor Denote the N-period history of any agent e N t = {e t N,...e t } E N, Agents with the same N-period history e N are on the same island. At the beginning-of each period, agents learn then productivity level e t and change islands accordingly. Agents take their wealth with them when they move to new islands Size of Island e N in period t is S t,e N, depends on M(s t ) Ragot Heterogeneous Agents 45/62

55 2. The Truncated Representation : Island Metaphor Denote the N-period history of any agent e N t = {e t N,...e t } E N, Agents with the same N-period history e N are on the same island. At the beginning-of each period, agents learn then productivity level e t and change islands accordingly. Agents take their wealth with them when they move to new islands Size of Island e N in period t is S t,e N, depends on M(s t ) For instance, N=3, two states (0,1), agent agent unemployed for three periods {0,0,0} agent in the island {0,0,0} in period t-1 becomes 1 in period t moves to island {0, 0, 1}. Ragot Heterogeneous Agents 45/62

56 The Truncated Representation : Island Metaphor The family head: cares equally for all agents on all islands allocates resources within islands can t transfer resources across islands choose consumption c(e N ), saving a(e N ) labor supply l(e N ) on each island before knowing next period status. is price-taker, tax-taker, face credit constraint ā on each island Ragot Heterogeneous Agents 46/62

57 Truncated Representation : Program of the Family Head ( max) E 0 β t S t,e N U at,e N,c t,e N,l t,e N t 0,e N E N e N E N a t,e N + c t,e N ( c t,e N, l t,e N ) = w t θ et l t,e N + (1 + r t )ã t,e N + T t, for all e N E N a t,e N ā, for all e N E N, with per capita beginning-of-period wealth ã t,e N ã t,e N = S t 1,ẽN S ẽ N E N t,e N Π t 1,ẽ N,e N a t 1,ẽ N. where Π t 1,ẽ N,e N is the probability to go from ẽn to e N in t 1 Ragot Heterogeneous Agents 47/62

58 Truncated Representation : First Order Conditions [ ] ( ) U c ct,e N, l t,e N + νt,e N = βe t M t (e 0, ê N ( ) )U c ct,ê N, l t,ê N (1 + rt+1 ), ê N E ( ) ( N ) w t θ et U c ct,e N, l t,e N = Ul ct,e N, l t,e N ν t,e N (a t,e N + ā) = 0 and ν t,e N 0, Same FOCs as in Bewley-Aiyagari-Huggett (simpler distributions). For instance, financial market equilibrium e N E N S t,e N a t,e N = B t + K t. Ragot Heterogeneous Agents 48/62

59 3. Truncated Representation : Recursive Decentralization More detail on the decentralization: Here Ragot Heterogeneous Agents 49/62

60 4. Solving the Ramsey Problem We can now use standard tools to solve the Ramsey problem with commitment (Sargent Ljungqvist, chapter 15; Chamley 1986; Marcet and Marimon 2011). As "finite" number of agents (in fact finite number of histories) BUT : Non-ricardian environment : Level of public debt is well-defined. Ragot Heterogeneous Agents 50/62

61 4. Solving the Ramsey Problem : 3 steps Write the problem in post-tax prices (Chamley 1986) Rewrite the Lagrangian with Euler Equations (Marcet and Marimon, 2011). Derive first-order conditions Ragot Heterogeneous Agents 51/62

62 The Ramsey Problem : Step 1 ( max ) E 0 r t,w t,b t,(a t,e N,c t,e N,l t,e N ) e N E N [ t=0 β t e N E N S t,en U en (c t,en, l t,en ) ], B t + F (K t 1, L t, s t) = G + T t + (1 + r t)b t 1 + r tk t + w tl t, a t,e N + c t,e N = w tθ et l t,e N + T + (1 + r t)ã t,e N, (all e N ) [ ] U c(c t,e N, l t,e N ) + ν t,e N = βe t Π U c,t(c en,ên t,ên, l t,ên )(1 + r t+1 ) ê N E N l t,e N = (w tθ et ) ϕ K t + B t = S t,en a t,en, e E N c t,e N, l t,e N 0,(a t,e N + a)ν t,e N = 0. Ragot Heterogeneous Agents 52/62

63 The Ramsey Problem : Step 1 ( max ) E 0 r t,w t,b t,(a t,e N,c t,e N,l t,e N ) e N E N [ t=0 β t e N E N S t,en U en (c t,en, l t,en ) ], (µ t) B t + F (K t 1, L t, s t) = G + T t + (1 + r t)b t 1 + r tk t + w tl t, a t,e N + c t,e N = w tθ et l t,e N + T + (1 + r t)ã t,e N, (all e N ) [ ] (λ t,e N ) U c(c t,e N, l t,e N ) + ν t,e N = βe t Π U c,t(c en,ên t,ên, l t,ên )(1 + r t+1 ) ê N E N l t,e N = (w tθ et ) ϕ K t + B t = S t,en a t,en, e E N c t,e N, l t,e N 0,(a t,e N + a)ν t,e N = 0. Ragot Heterogeneous Agents 52/62

64 The Ramsey Problem : Step 2 Denote λ t,e N Lagrange coefficient on the Euler equation of island e N (Marcet an Marimon 2011; Acikgoz 2013). max (.) β t S t,e N t=0 e N E N E 0 +U c (c t,e N, l t,e N ) ( U(c t,e N, l t,e N ) ( )) Λ t,e N (1 + r t ) λ t,e N where Λ t,e N ê N E N S t 1,ê N λ t 1,ê N Π t,e N,ê N, S t,e N Plus e N budget constraints, and budget of the State. Ragot Heterogeneous Agents 53/62

65 The relevant liquidity concept Social valuation of liquidity of agents e N ( ) ψ t,e N = U c (c t,e N, l t,e N ) U cc (c t,e N, l t,e N ) λ t,e N Λ t,e N (1 + r t ) Ragot Heterogeneous Agents 54/62

66 The relevant liquidity concept Social valuation of liquidity of agents e N ( ) ψ t,e N = U c (c t,e N, l t,e N ) U cc (c t,e N, l t,e N ) λ t,e N Λ t,e N (1 + r t ) All FOCs of the planner can be written as a function of liquidity-valuation gap for agents e N µ t ψ t,e N First Order Conditions: Here Ragot Heterogeneous Agents 54/62

67 Steady State Results Theorem 1. Marginal productivity of capital : 2. Capital tax is (where D > 0) 1 + F K (K, L) = 1 β τ K = D e N S e N ν e N 3. Labor tax is τ L ωe L (µ ψ N e N ) = µφ 1 τ L. e N E N Ragot Heterogeneous Agents 55/62

68 5. Quantitative investigation 1. Labor process, 3 states Domeij and heathcote (2005) M = Productivity e 1 = 0.213, e 2 = 0.848, e 3 = N = 4 : thus 3 4 = 81 agents (solve for N = 6 and 729 agents and similar results). U(c, l) = 1 ( ) 1 σ c l1+1/φ 1 σ 1 + 1/φ F (K, L, s) = A(s)K α L 1 α δk with A(s) = exp(s) and s t = ρ s s t 1 + ε s t, ε s t N (0, σ 2 s) Ragot Heterogeneous Agents 56/62

69 Parameter Values A period is a year. N β φ σ χ δ α G ρ s σ s Table: Other Parameter Values 1. G/Y = 19% (US average). 2. φ =.5 (Chetty et al. 2011) 3. α = 0.36, share of capital in GDP 4. Standard value for techno process Ragot Heterogeneous Agents 57/62

70 Steady State r(%) w τ K (%) τ L (%) B/Y (%) K/Y (%) L Y Table: Steady State Values Ragot Heterogeneous Agents 58/62

71 Sensitivity Parameters Output φ σ χ N τ K (%) τ L (%) B/Y (%) Benchmark calibration Change in φ Change in σ Table: Sensitivity Analysis Ragot Heterogeneous Agents 59/62

72 Taxes over the Business Cycle Figure: Aggregate IRF after a TFP shock Ragot Heterogeneous Agents 60/62

73 Second Moment Standard deviations sd(y ) sd(τ K ) sd(τ L ) sd(b) Autocorrelations corr(y, Y 1 ) corr(τ K, τ K 1 ) corr(τ L, τ L 1 ) corr(b, B 1) Table: Second Moment Ragot Heterogeneous Agents 61/62

74 Conclusion Volatile capital taxes : a substitute for monetary policy? Introduce more frictions to obtain richer trade-off : search and matching, nominal frictions, and so on. Ragot Heterogeneous Agents 62/62

75 Appendix

76 Heterogeneous Agents Appendix 1 Truncated Representation: Recursive Decentralization (go back) Assume now that agents no aggregate shock and 1 + r < 1/β agents maximize inter-temporal welfare (as in Bewley) get a lump sum transfer Γ N+1 (e N+1 ), which depends on last (N + 1)-period history. V (a, e N+1 ) = max U(c, l) + βe M e,e V (a, e N+1 ) a,c,l e E a + c = wθ e l + (1 + r)a + T + Γ N+1 (e N+1 ) c, l 0,a ā Solution g N+1 a (a, e N+1 )

77 Heterogeneous Agents Appendix 2 Truncated Representation: Recursive Decentralization (go back) Theorem There exists a balanced transfer Γ N+1 (en+1 ) such that the equilibria of the family head and decentralized economy are the same. If g N+1 a (a 1, e N+1 ) g N+1 a (a 2, e N+1 ) < κ a 1 a 2, with κ < 1, then max Γ N+1 (e N+1 ) 0 and lim V N+1 (a, (e) = V Bewley (a, e). N Intuition : when the saving rate is less than 1, the effect of initial wealth level disappears as time goes by. True for all Bewley models in the literature.

78 Heterogeneous Agents Appendix 3 Truncated Representation: Sequential Decentralization (go back) How can we think about this allocation? If agents: are identical in period 0 can share risk at the beginning of each period t only if they have the same e N t history. recall each agent is measure 0 The family head allocation is the maximum ex ante welfare they can get: Market outcome back to main text

79 Heterogeneous Agents Appendix 4 First Order Conditions : Transfers (go back) Transfers : e N E N S t,e N ( ) µ t ψ t,e N 0, with equality when T t > 0.

80 Heterogeneous Agents Appendix 5 First Order Conditions : Debt (go back) Choice of Public debt µ t = βe t [µ t+1 (1 + r t+1 )].

81 Heterogeneous Agents Appendix 6 First Order Conditions : Labor Taxes(go back) e N E N ω L e N ( ) µ t ψ t,e N = µ t ϕ τ t L 1 τt L where e N E N ωl e N = 1 is the labor share of island e N.

82 Heterogeneous Agents Appendix 7 First Order Conditions : Capital Taxes(go back) e N E N ω K e N ( ) µ t ψ t,e N = S t,e N U c (c t,e N, l t,e N )Λ t,e N. e N E N where e N E N ωk e N = 1 is the capital share of island e N. back to main text