Household income risk, nominal frictions, and incomplete markets 1

Household income risk, nominal frictions, and incomplete markets 1 2013 North American Summer Meeting Ralph Lütticke 13.06.2013 1 Joint-work with Christian Bayer, Lien Pham, and Volker Tjaden 1 / 30

Research question What are quantitative implications of time-varying income uncertainty on the business cycles? Specifically: Uncertainty induced precautionary savings What role can monetary policy play? 2 / 30

Uncertainty Literature Firms Large literature that models cyclical variations in factor reallocation as transmission mechanism, e.g. Bachmann and Bayer (2013) or Bloom et al (2012) Households Rep. agent model Fernandez-Villaverde et al. (2011) measure fiscal policy risk, Born & Pfeiffer (2011) measure policy and TFP risk ZLB Basu & Bundick (2011) examine the effect of aggr. TFP and demand risk with particular focus on monetary policy constrained by the zero-lower bound 3 / 30

Uncertainty Literature Households cont d (Het. agent models with incomplete markets) Storesletten et al. (2001) with flexible prices (focus on asset prices) Guerrieri and Lorenzoni (2012) labor supply effect of uncertainty. Mericle (2012) looks at a setup with fixed wages (partial equilibrium). Raven and Sterk (2013) focus on unemployment-risk (long-vs. short-term) and search. 4 / 30

Empirical facts Why household income uncertainty? Magnitude up to two orders of magnitude larger than TFP risk Time-varying at business cycle frequency. In fact, countercyclical Storesletten et al (2001) find that STD of earnings is 126% higher in recessions 5 / 30

Model ingredients Incomplete Markets Household Problem Time-varying uncertainty σ t Adjust precautionary savings to reach optimal buffer stock Consumption/Savings decision 6 / 30

Model ingredients Time-varying uncertainty σ t Incomplete Markets Household Problem Adjust precautionary savings to reach optimal buffer stock Consumption/Savings decision New Keynesian Aggregate Supply Firm s optimal response to AD fluctuations Sticky Prices Price vs Quantity adjustment 7 / 30

Model overview Workers consumption / savings decision Entrepreneurs price setting Central Bank decision rule 1) Economy with fiat money nominal money m own capital competitive intermediate goods producers fixed nominal money supply M with zero interest 2) Economy with capital & bonds nominal bonds b & shares in fix capital stock k monopolistic resellers s.t. Calvopricing rent capital nominal interest rate on bonds B (=εsupply) according to Taylor rule 9 / 30

Worker-Households Supply labor, Endowment with human capital, h, is risky. Consume Save in money, no access to state-contingent claims. 10 / 30

Household s optimization problem Recursive formulation V ( m, h; Θ, s) = max c, m u (c) + β w EV ( m, h ; Θ, s ) s.t. : c + m = 1 1 + π(θ, s) m + w(θ, s)h N m 0 11 / 30

Entrepreneurs Final goods producers/resellers Differentiate goods Profit maximization s.t. demand schedule and price setting frictions à la Calvo(1983) Participate in a complete market, cannot trade assets with worker-households. Linearized Phillips curve logπ t = βe t (logπ t+1 ) + κ(log MC t + µ), 12 / 30

Entrepreneurs Intermediate goods producers: roundabout production Perfectly competitive Operate gross production function with constant returns to scale Employ (differentiated) pre-products X t max Π t = MC t Xt α Nt β Kt 1 α β X t w K t,n t,x t }{{} t N t (r t + δ)k t GDP 13 / 30

Entrepreneurs Intermediate goods producers: roundabout production Real wage and user costs of capital and output are a function of the marginal costs/markup w t = f N (MC t ) r t + δ = f K (MC t ) 14 / 30

Recursive competitive equilibrium A recursive competitive equilibrium in our basic model is a set of policy functions {c, m }, value functions V, pricing functions {w, π}, aggregate real money and labor supply functions { M, N}, distribution Θ over individual money holdings and productivity, and a perceived law of motion Γ, such that Given V, Γ, prices, and distributions, the policy functions {c, m } solve the household s problem and given the policy functions{c, m }, prices and distributions, the value function V is a solution to the Bellman equation. The labor, goods and money market clear M P t = m t ( m t 1, h t ; Θ t, s t )Θ t ( m t 1, h t )d m t 1 dh t =: A (Θ t, s t ) The actual law of motion and the perceived law of motion Γ coincide, i.e. Θ = Γ(Θ, s ). 15 / 30

Numerical implementation more Recursive equilibrium is not computable Krusell-Smith equilibrium (1998) Prices are a function of the endogenous distribution over wealth and idiosyncratic productivity Log-linear equilibrium forecasting rule for inflation depends on the uncertainty state, the real money stock and the variance of idiosyncratic productivity 16 / 30

Stochastic volatility process log h it = ρ h log h it 1 + ɛ it, ɛ it N (µ, σ ht ) σ 2 ht = σ2 s t, log s t = ρ s log s t 1 + ν t, ν t N (0, σ s ) Parameter Value Description ρ h 0.9873 Persistency of wage shocks σ 2 0.0864 mean short run variance of persistent shock ρ s 0.8 Persistency of variance σs 2 0.0140 Variance of innovations to the variance 17 / 30

Model period is a quarter. Calibrated Parameters Parameter Value Target Households β W = β E 0.985 M2/Y = 3.5 (Worldbank, 2011) ξ 3 and r=0.007 Intermediate Goods α 0.450 Share of pre-products γ = β/(1 α) 0.666 Share of labor Final Goods κ 0.02 4 quarter avg. price duration 1/exp( µ) 39.56% Markup Capital K/Y 12 Capital to output ratio more 18 / 30

IRFs at inactive monetary policy One standard deviation increase in the variance of idiosyncratic productivity 0.1 GDP 3 Real Money Balances Percentage 0 0.1 0.2 0.3 0.4 0.5 0.6 Percentage 2.5 2 1.5 1 0.5 Percentage Points 0.7 0 10 20 30 40 50 60 70 80 0.05 0 0.05 0.1 0.15 0.2 0.25 Quarter Inflation 0.3 0 10 20 30 40 50 60 70 80 Quarter Percentage 0.2 1.2 1.4 0 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 Quarter Marginal Cost 1.6 0 10 20 30 40 50 60 70 80 Quarter 19 / 30

IRFs at inactive monetary policy One standard deviation increase in the variance of idiosyncratic productivity over 4 quarters. 0.1 GDP 4 Real Money Balances 0 3.5 Percentage 0.1 0.2 0.3 0.4 0.5 0.6 Percentage 3 2.5 2 1.5 1 0.5 Percentage Points 0.7 0 10 20 30 40 50 60 70 80 Quarter 0.05 0 0.05 0.1 0.15 0.2 0.25 Inflation 0.3 0 10 20 30 40 50 60 70 80 Quarter Percentage 0.2 0.2 0.4 0.6 0.8 1.2 1.4 0 0 10 20 30 40 50 60 70 80 Quarter 0 1 Marginal Cost 1.6 0 10 20 30 40 50 60 70 80 Quarter 20 / 30

Economy with active monetary policy portfolio choice problem: households self-insure by investing in bonds or trading shares of a fixed capital stock Central bank can conduct active monetary policy by offering ɛ-supply of nominal bonds and setting interest rates according to Taylor rule 21 / 30

Capital and Bonds Optimal savings and portfolio choice problem V (a, h; Θ, s) = max a u(c) + β max φ E { V ( R (φ)a, h ; Θ, s )} subject to c + a = a 0 a + w(θ, s)h N where a is total non-human wealth at the beginning of the period and a is next period s wealth before interest and dividends. 22 / 30

First order conditions Indifference equation: optimal portfolio choice φ (a, h; Θ, s) { q (Θ, s ) + r (Θ, s } { } ) I (Θ, s) E u c (c ) = E q(θ, s) 1 + π (Θ, s ) u c(c ) + Euler-equation 23 / 30

Central bank policy Central bank sets the gross nominal interest rate on bonds I according to the Taylor rule I = I ss (1 + π t ) φπ Bonds market clearing with B = ɛ B = (1 φ (a, h; Θ, s))a (a, h; Θ, s)θ(a, h)dadh Capital market clearing K = φ (a, h; Θ, s)a (a, h; Θ, s)θ(a, h)dadh 24 / 30

IRFs with Taylor Rule One standard deviation increase in the variance of idiosyncratic productivity 0 GDP 0.12 q 0.01 0.1 Percentage 0.02 0.03 0.04 0.05 0.06 Percentage 0.08 0.06 0.04 0.07 0.02 Percentage Points 0.08 0 10 20 30 40 50 60 70 80 0 0.01 0.02 0.03 0.04 0.05 Quarter PI 0.06 0 10 20 30 40 50 60 70 80 Quarter Percentage 0.05 0.1 0.15 0.2 0 0 10 20 30 40 50 60 70 80 0 Quarter MC 0.25 0 10 20 30 40 50 60 70 80 Quarter 25 / 30

Monetary Policy vs. Assets Combine Fisher equation and Taylor rule [ q E[P q + r (mc ] ) ] = (1 + φ π ) P E +i q 26 / 30

Monetary Policy vs. Assets Combine Fisher equation and Taylor rule [ q E[P q + r (mc ] ) ] = (1 + φ π ) P E +i q Model Monetary Policy Assets/Y Dividends C t I. Fiat Money fixed supply M 3.5-0.65 i. TR 3.5 const.? ii. TR 7.3 const.? II. Capital TR 7.3 r(mc t ) 0.0775 and Bonds 26 / 30

Conclusion Impose an empirically valid wage process with time varying uncertainty We show that an exogenous increase in wage uncertainty has depressing effects on output Output drop depends on the state of monetary policy 1 At the zero lower bound/inactive monetary policy: 0.65% 2 Active monetary policy: 0.08% 27 / 30

Thank you for your attention 28 / 30

Krusell-Smith forecasting rules I Economy with fiat money log M π =βm 1 (s) + β2 M (s) log M + βm 3 (s)var(h) ] = exp [(1 βm 2 (s)) log M βm 1 (s) β3 M (s)var(h) 1 II Economy with capital and bonds log q =β 1 q(s) + β 3 q(s)var(h) log π =β 1 π(s ) + β 3 π(s )var(h) back 29 / 30

Asset Distribution 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.5 1 1.5 2 2.5 3 @BC Quintiles Gini Data 10-0.2 1.1 4.5 11.2 83.4 0.82 Model 3.5 0.99 4.94 11.18 25.77 57.13 0.55 back 30 / 30