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Shock Propagation and Macroeconomic Fluctuations

Antoine Mandel, Vipin P. Veetil

Abstract

We study how idiosyncratic firm-level shocks generate aggregate volatility and tail risk when they propagate through a production network under overlapping adjustment: new productivity draws arrive before the economy reaches the static equilibrium associated with earlier draws. Each innovation generates a `productivity wave' that mixes and dissipates over time as it travels through the production network. Macroeconomic fluctuations emerge from the interference between these waves of different vintages. The interference between these waves is governed by the dominant transient eigenvalue of the production network, and therefore so is the macroeconomic fluctuations they generate. In such a dynamic regime, the tail of the degree distribution is a markedly weaker determinant of macro fluctuations than in the fully adjusted static benchmark. And the macroeconomic significance of the degree-heterogeneity of production networks cannot be known without knowing the rate at which the economy converges to equilibrium or equivalently the spectral properties of the production network. More concretely, once we permit the time-averaging of shocks, granular shocks may account for only a small fraction of the empirically observed aggregate volatility.

Shock Propagation and Macroeconomic Fluctuations

Abstract

We study how idiosyncratic firm-level shocks generate aggregate volatility and tail risk when they propagate through a production network under overlapping adjustment: new productivity draws arrive before the economy reaches the static equilibrium associated with earlier draws. Each innovation generates a `productivity wave' that mixes and dissipates over time as it travels through the production network. Macroeconomic fluctuations emerge from the interference between these waves of different vintages. The interference between these waves is governed by the dominant transient eigenvalue of the production network, and therefore so is the macroeconomic fluctuations they generate. In such a dynamic regime, the tail of the degree distribution is a markedly weaker determinant of macro fluctuations than in the fully adjusted static benchmark. And the macroeconomic significance of the degree-heterogeneity of production networks cannot be known without knowing the rate at which the economy converges to equilibrium or equivalently the spectral properties of the production network. More concretely, once we permit the time-averaging of shocks, granular shocks may account for only a small fraction of the empirically observed aggregate volatility.
Paper Structure (30 sections, 11 theorems, 149 equations)

This paper contains 30 sections, 11 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Organisation of the paper
  4. The Model
  5. Notation and preliminaries
  6. Economic environment
  7. Equilibrium, transient dynamics, and stability
  8. Existence and stability of equilibrium
  9. Transient dynamics and law of motion of quantity
  10. Aggregate output and its fluctuations
  11. Aggregate volatility and tail-risk
  12. Asymptotic Volatility Bounds
  13. The $\mathcal{L}$-economy
  14. Superposition of shocks of different vintages
  15. Growth rates and shock-interference
  16. ...and 15 more sections

Key Result

Proposition 1

Since $\rho(\mathbf A)<1$, $(\mathbf I-\mathbf A)^{-1}=\sum_{\ell=0}^\infty \mathbf A^\ell$. Then, for any $r\in(\rho(\mathbf A),1)$ there exist constants $C>0$ and $\ell_0$ such that $\|\mathbf A^\ell\|_1\le C r^\ell$ for all $\ell\ge\ell_0$. Consequently, for all $\mathcal{L}\ge\ell_0$, Hence $y^{[\mathcal{L}]}(\boldsymbol{\epsilon})\to y^\ast(\boldsymbol{\epsilon})$ geometrically as $\mathcal{

Theorems & Definitions (25)

  • Definition 1: Aggregate output
  • Definition 2: Aggregate volatility: static and dynamic
  • Definition 3: Aggregate tail risk: static and dynamic
  • Proposition 1: Neumann convergence and finite-depth error
  • proof
  • Lemma 1: Static bounds dynamic risk in the stationary infinite-history limit
  • proof
  • Corollary 1: Intertemporal overlap attenuates growth-rate volatility
  • proof
  • Proposition 2: Tail risk magnifies the timing wedge
  • ...and 15 more