Demand Shocks with Dispersed Information


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1 Demand Shocks with Dispersed Information Guido Lorenzoni (MIT) Class notes, 06 March 2007
2 Nominal rigidities: imperfect information How to model demand shocks in a baseline environment with imperfect info? Need consumer s decisions to be richer: Forward looking No fully revealing prices 1. Embed in something closer to neokeynesian benchmark 2. Add shocks to expected productivity
3 Ingredients Model of fundamental and sentiment shocks Fundamental information is dispersed across the economy Agents know potential output in their own sector, but not the aggregate Demand shocks: shifts in average beliefs about aggregate potential output
4 Model Households: consumer/producer on [0, 1]. Preferences: ( ) β t 1 logc it 1 + η N 1+η E it t=0 C it = ( ) σ σ 1 σ 1 dj J it C ijt σ random consumption basket: J it [0,1] Technology: Y it = A it N it
5 Shocks Individual productivity (private signal) is a it = loga it = a t 1 + θ it aggregate component and idiosyncratic component θ it = θ t + ε it Aggregate productivity is a t = a t 1 + θ t
6 Model Linear equilibrium Demand and supply shocks Monetary Policy Welfare Transparency Shocks (continued) Public signal about aggregate innovation s t = θ t + e t news aggregate statistics stock market... θ t = fundamental shock e t = sentiment shock
7 Trading Agents have nominal balances B it 1 with CB (cashless economy) Before observing current shocks: state contingent contracts CB sets nominal interest rate on balances R t Producer set price P it Consumer observes prices in consumption basket P jt for j J it Consumer buys goods All shocks publicly revealed, state contingent contracts settled
8 Budget constraint ( ) B it = R t B it 1 + (1 + τ)p it Y it P it C it + Z it (h t ) T t ( ) ( ) q t ht Zit ht dh t. P it price index for goods in J it Z state contingent contracts subsidy τ to correct for monopolistic distortion T t lump sum tax to finance subsidy
9 Random consumption baskets producers θ t
10 Random consumption baskets θ it producers θ t
11 Random consumption baskets producers consumers
12 Random consumption baskets producers 0.5 θ it consumers
13 Random consumption baskets (continued) θ it { } = θ jt : j J it additional idiosyncratic shock: sampling shock v it θ it = θ t + v it
14 Interest rate rule Price target Monetary policy rule r t ( ) = r + ξ p t 1 p t 1 p t = φ θ θ t + φ s s t no superior information only trying to keep nominal prices stable ξ > 1 active rule all lowercase = logs
15 Linear equilibrium Individual prices and consumption p it c it = = φ 0 + φ θ θ it + φ s s t ψ 0 + a t 1 + ψ ε θ it + ψ v θ it + ψ s s t in equilibrium p t = p t interest rate constant Proposition Linear equilibrium exists under given policy rule, determinate if ξ > 1
16 Potential output Linear equilibrium (continued) c t = ψ 0 + a t 1 + θ t aggregate output under first best allocation = aggregate output under full information (with right τ) = linear equilibrium iff ψ θ = 1 ψ s = 0
17 Linear equilibrium (continued) Mechanics and remark 1 full insurance + normal sampling shocks + isoelastic preferences closed form linear equilibrium e.g.: the price index for consumer i is P it = V p exp{p t + φ θ v i } where V p = exp{ 1 σ 2 φθ σˆε 2 } 2
18 Linear equilibrium (continued) Mechanics and remark 2 consumers observe whole distribution P jt for j J it a sufficient statistic is θ it this is like having two noisy signals of θ t : θ it = θ t + ε it θ it = θ t + v it
19 Linear equilibrium (continued) Mechanics and remark 2 consumers observe whole distribution P jt for j J it a sufficient statistic is θ it this is like having two noisy signals of θ t : θ it = θ t + ε it θ it = θ t + v it information structure is independent of monetary policy
20 Pricing Optimality condition ( ) I p it = η E it [c t + σ (p t p it )] a it + ( ) [ ] + E I it [p it + c it ] a it + η (ψ v + σφ θ )E I it v jt E I it expectation at pricing stage high demand relative to prod high price high consumption relative to prod high price
21 Consumption Euler equation c it II = E it a }{{} t+1 (r p t+1 + p it ) exp. income EII it expectation at consumption stage
22 Demand shocks Properties of monetary regime E t [p it+1 ] = 0 stable price level in expectation equilibrium r t constant Simple case σ ε σ θ agents disregard their private info E P [.] = E[. a t 1,s t ] t
23 Effects of e t and θ t 1 + η p t = 1 + ση (E t P [a t ] a t ]) Effects of e t > 0 only temporary effects raise c t, p t and n t y t = λ E P [a t ] + (1 λ )a t t Effects of θ t > 0 permanent effects raise c t lower p t and n t
24 y t n t p t a t t Dynamic responses to u t (solid) and e t (dash).
25 What restrictions does the theory impose? evidence on signals gives testable implications evidence on aggregate beliefs basic restrictions on joint behavior of error and actual series y t = λ E P [a t ] + (1 λ )a t t fraction of variance of y t due to demand shocks over total variance is bounded
26 e 2 / u e 2 / u e / 2 u 5
27 Changing σ e σ =.05 e σ =.1 e σ = σ = e e
28 Interest rate rule A richer policy rule r t ( ) = r + ξ p t 1 p t 1 Price target p t = µa t 1 + φ θ θ t + φ s s t use past information p t aggregate price index note the term µa t 1 inertial rule
29 Monetary Policy (continued) Consumption under µ 0 Euler equation c it II = E it a (r t + p it ) t+1 p t+1 }{{}}{{} exp. income future price
30 Interest rate rule A richer policy rule r t ( ) = r + ξ p t 1 p t 1 Price target p t = µa t 1 + φ θ θ t + φ s s t use past information p t aggregate price index note the term µa t 1 inertial rule
31 Power of policy rule Agents have different expectations about future output...but also different expectations about real interest rate 2 crucial ingredients: agents forward looking E II it [r µ θ θ t + p it ] in the future more information than now policy rule allows to manage expectations
32 Power of policy rule (continued) The choice of µ θ feeds back into optimal prices p it It also affects response to s t and response of relative prices An increase in µ θ increases ψ θ reduces φ θ increases φ s decreases ψ s
33 vector ψ θ,φ θ,φ s,ψ s s.t. Achievable linear equilibria ψ v = ψ ε δ v /δ ε φ θ (1 + ση)φ θ = η ((ψ θ + σφ θ )β θ 1) + ((ψ θ + φ θ )β θ /δ θ 1) + +η (ψ v + σφ θ )γ (1 β θ ), 0 = η (ψ θ + σφ θ )β s + (1 + η)ψ s + +(ψ θ + φ θ )(β s δ s )/δ θ η (ψ v + σφ θ )γβ s,
34 Another divine coincidence? Proposition There is a µ fs θ that achieves full stabilization: ψ θ = 1 ψ s = 0 here output is always equal to potential induce agents to respond more to private productivity
35 ψ θ 1 0 ψ s 0 μ θ μ θ fs
36 More on the relation between ψ θ and φ θ increase response of output to fundamental increase response of demand to local productivity reduce price adjustment (φ θ < 0)
37 Welfare 4 components: [ ] (1 + η)e (c t c t ) 2 a t 1 (1 + η)var (n it ) + Var ( ) ( ) c jt + σp jt j J it + σ (σ 1)Var pjt j J it 1. aggregate output gap () 2. labor supply cross sectional dispersion () 3. demand cross sectional dispersion () 4. relative price dispersion (+)
38 σ 7 η 2 σ 2 θ 1 σ 2 ε 1 σe 2 1/3 γ 0.5 Table: Parameters for the example
39 0 w w 2,w w μ fs θ μ θ
40 w μ * θ μ θ fs μ θ
41 Optimal monetary policy Proposition Full stabilization is typically not optimal Some accommodation of demand shocks is optimal It is optimal µ < µ fs It is optimal to partially accomodate ψ s > 0 Price dispersion is larger at optimal monetary policy than under full stabilization
42 Special case now it is optimal ψ θ = 1 η = 0 φ θ = 1 decreasing prices proportionally to productivity gives: 1. right relative prices 2. right response of consumption
43 Special case (continued) p it ( ) I = E i [p it + c it ] a it c it = E i II [a t+1 + p t+1 ] p it unit intertemporal elasticity of substitution proportional response is optimal
44 Transparency Is better public information good? (Morris and Shin (2002)) Effect on output gap may be bad Total effect always good
45 2 Effect on welfare w /σ e
46 0.02 Effect on output gap volatility w /σ e
47 Compare with Hellwig (2005) Lucas style model with unobserved money supply shocks more precision about monetary shocks is good: reduce output gap reduce price variance (spurious) Here uncertainty about real shocks more precision is good: ambiguous on output gap increase price variance (good) second effect dominates
48 Expectations shocks and business cycles Business cycles driven by news (Beaudry and Portier (2006), Jaimovich and Rebelo (2006)) Problem 1: in neoclassical setting demand disturbances have hard time generating right response of hours/consumption/investment Euler equation c t = E t at+1 (r }{{} t p t+1 + p t ) exp. income with flexible prices the real rate increases automatically
49 Expectations shocks and business cycles (continued) Nominal rigidity can help (Christiano, Motto and Rostagno (2006)) Problem 2: monetary policy accommodation of demand shocks is typically suboptimal Euler equation c t = E t at+1 (r }{{} t p t+1 + p t ) exp. income with full information optimal to increase r
50 Expectations shocks and business cycles (continued) Imperfect information + nominal rigidity can help Problem 3: policy rules still able to wipe out demand shocks...but this is not optimal a theory of demand shocks that survive optimal policy
51 Concluding Future superior information + forward looking consumers policy can induce efficient use of dispersed information Related themes: King (1982), Svensson and Woodford (2003), Aoki (2003) Efficient use of dispersed information = full stabilization output gap Still some offsetting of demand shocks is feasible and desirable Clearly this requires commitment, which may be tough (bubble example)
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