Short & Long Run impact of volatility on the effect monetary shocks

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1 Short & Long Run impact of volatility on the effect monetary shocks Fernando Alvarez University of Chicago & NBER Inflation: Drivers & Dynamics Conference 218 Cleveland Fed Alvarez Volatility & Monetary Shocks April / 17

2 Output s impulse response Y (t; δ) to once-for all monetary shock δ, cumulative IRF M(δ). Sticky price models w/ idiosyncratic shocks w/ variance σ 2 & w/o strategic complementarity: Menu costs: deterministic and random Single and Multi-product firms Effect on Y (t; δ) and M(δ)) of permanent changes on idiosyncratic volatility σ: on impact and in the long run. Long vs Short run, difference across models. Help interpret state dependent or time varying VARs. Price setting (ours) vs investment responses (traditional). Alvarez Volatility & Monetary Shocks April / 17

3 percent deviation from steady state Once and for all monetary shock at t =. 1 / time t Alvarez Volatility & Monetary Shocks April / 17

4 Density function Aggregate shock hits x-section of desired adjustments p(x; ).5!(7x + /)!7x 7x! / 7x Price gap: x Alvarez Volatility & Monetary Shocks April / 17

5 percent deviation from steady state Impulse response of Price Level 1 /.8 P(t) time t Alvarez Volatility & Monetary Shocks April / 17

6 percent deviation from steady state Impulse response of Prices and Output 1 / Y t = /! P(t).8 P(t) time t Alvarez Volatility & Monetary Shocks April / 17

7 percent deviation from steady state Impulse response of Prices and Output 1 / Y t = /! P(t).8 P(t) time t Alvarez Volatility & Monetary Shocks April / 17

8 Price Setting Models Lifetime Utility : CES aggregate : c(t) = ( e r t log c(t) α l(t) + log M(t) P(t) ( 1 n ( Z ki (t) c ki (t) ) 1 1 η dk i=1 ) dt ) η η 1 Linear technology c ki (t) = l ki (t) / Z ki (t) and Z ki (t) = exp (σ W ki (t)). W ki (t) independent idiosyncratic Brownian motions. Equilibrium: constant nominal interest rate & wages W (t) M(t). Alvarez Volatility & Monetary Shocks April / 17

9 simultaneous adjustment of n products, subject to random menu cost: { ψ with probability 1 ζ dt or adjust n prices paying = with probability ζ dt. Examples: Golosov-Lucas, Calvo, Calvo +, Midrigan & extensions. Decision rule: change prices to ideal value when deviation of prices relative to (norm of) ideal price (vector) x first hit x or when cost is free. Kurtosis Kurt(Dp) & fraction free adjustment: Calvo-ness φ = ζ x 2 and n σ 2 ζ N(Dp) depend on Sufficient statistic": M(δ) = δ Kurt(Dp) 6 N(Dp) & shape of Y (t; δ) depend on Calvo-ness φ and n for given N(Dp). Kurt(Dp) & N(Dp) depends on structural parameters, among them σ 2. Alvarez Volatility & Monetary Shocks April / 17

10 Set up for σ and δ shocks (two MIT shocks) At t = in steady state for σ Once and for all permanent change to σ 1 > σ. After τ > unexpected permanent monetary shocks δ Shot run: monetary shock occurs at τ =, i.e. at the old steady state for σ, but new decisions rules corresponding to σ 1. Impact effect. Long Run: monetary shock occurs at τ =, i.e. at new steady state for σ 1, and new decision rules corresponding to σ 1. Comparative static. Average speed of convergence: from initial distribution with σ to final distribution with σ 1. Two forces: Effect of σ on decision rules (barriers x), and speed within barriers, which dominates.(same in SR and LR). Effect of σ on the initial distribution (different in SR and LR). Alvarez Volatility & Monetary Shocks April / 17

11 Long Run effect on IRF, Golosov-Lucas n = Alvarez Volatility & Monetary Shocks April / 17

12 Short Run effect on IRF, Golosov-Lucas n = Alvarez Volatility & Monetary Shocks April / 17

13 Impulse response Golosov-Lucas n = 1 Long Run: Y (t; σ 1 ) = Y M(σ 1 ) = σ σ 1 M(σ ) ( t σ ) 1 ; σ all t σ Short Run: Y (t; σ, σ 1 ) Y M(σ, σ 1 ) M(σ ) ( t σ ) 1 σ1 ; σ all t σ σ Expected time to converge: 2 π 2 N(Dp;σ 1 ) 1 5 N(Dp;σ 1 ) Alvarez Volatility & Monetary Shocks April / 17

14 Mechanics Higher σ increases barriers x, but less so that the increase in σ, so barriers are reached sooner. This decreases effect on output. The previous point means that N(Dp) increases with σ. Higher barriers makes invariant distribution wider (in the long run). In the short run, the initial distribution of gaps x is narrower than new stationary distribution, so that it has fewer price increases. This increases the effect on output. Short run, has two effect on opposite direction, twisting IRF Y, keeping same M. Long run has only one effect, shifting inwards IRF Y, lowering M. Alvarez Volatility & Monetary Shocks April / 17

15 Long run effect on cumulative IRF, different models Alvarez Volatility & Monetary Shocks April / 17

16 Long run effect on cumulative IRF, different models Alvarez Volatility & Monetary Shocks April / 17

17 Summary of Long Run Economies where, for given N(Dp), monetary shocks have larger effects, have smaller sensitivity to shocks to volatility σ. Kurtosis Kurt(Dp) is increasing in Calvo-ness" for each n, and increasing in n for each level of Calvo-ness". Main effect of increase in σ is to increase N(Dp). Price setting runs faster". Also it weakly decreases "Calvo-ness", but this effect is always dominated by the speed". Alvarez Volatility & Monetary Shocks April / 17