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On the macroeconomic effects of heterogeneous productivity shocks

  • Christian Jensen EMAIL logo
Published/Copyright: September 15, 2015
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 16 Issue 1

Abstract

The conventional wisdom that producer heterogeneity washes out, and is therefore irrelevant for the aggregate economy, does not apply when producers compete monopolistically. Despite this, the effects of such heterogeneity can be reproduced with an appropriately redefined representative-agent framework where the equilibrium values of aggregates are expressed in terms of the moment generating function of the distribution of heterogeneity, or its asymptotic distribution. Increased heterogeneity raises aggregate productivity and production, more so the fiercer competition is. We propose a framework where the entire distribution of heterogeneity matters, yet computationally requires no more than a representative-agent model.

Keywords: aggregate total factor productivity; aggregation; idiosyncratic shocks; monopolistic competition; producer heterogeneity
JEL: E20; E23
Corresponding author: Christian Jensen, Department of Economics, University of South Carolina, 1014 Greene Street, SC 29208, Columbia, Tel.: +1-803-777-2786, e-mail:

Appendix: Proof proposition 4

It is first necessary to establish that if a random variable converges in distribution toward 𝒟t with moment generating function Γ𝒟t, then in the limit, its moment generating function will equal Γ𝒟t.

Proposition 7Let Γait (θt–1) be the moment generating function of the random variable ait∀i. If assumption 2 holds and Γ𝒟texists, then

(64)limMt→∞Γait(θt−1)=ΓDt. (64)

Proof. Since the characteristic function φait of any random variable ait always exists and is given by

(65)φait(τ)=E(eτait−1) (65)

where

(66)eτait−1=cos(τait)+−1sin(τait) (66)

is a bounded continuous function, and ait→dDt as Mt→∞, it follows from the Helly-Bray theorem, see Loève (1963, 182), that φait converges toward the characteristic function of the distribution 𝒟t as Mt→∞. Hence, in the limit, the two characteristic functions are identical. When the moment generating function of ait exists, it is related to the characteristic function of ait through the relation

(67)Γait(τ)=φait(τ−1) (67)

so when Mt→∞, and the characteristic functions of ait and 𝒟t become identical, their moment generating functions become the same too, so Γait exists and equals Γ𝒟t.

This result can now be combined with that on convergence in probability (proposition 2) to compute aggregate total factor productivity.

Proposition 4

Proof. Proposition 2 implies that conditional on θt, for any ε>0,

(68)Pr(|A⌣t−(Γait(θt−1))1θt−1|≥ε)→0 (68)

when It→∞. Proposition 7 implies that

(69)limMt→∞(Γait(θt−1))1θt−1=ΓDt1θt−1 (69)

since the function f(ω)=ω1θt−1 is continuous. Combining the two conditions, (68) and (69), we have

(70)Pr(|A⌣t−ΓDt1θt−1|≥ε)→0 (70)

when It→∞ and Mt→∞, which is equivalent to the statement (47) in the proposition.      □

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Published Online: 2015-9-15
Published in Print: 2016-1-1

©2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. On the macroeconomic effects of heterogeneous productivity shocks
  4. Fiscal policy in an open economy
  5. Understanding entry and exit: a business cycle accounting approach
  6. Contributions
  7. Predicting US recessions with stock market illiquidity
  8. The corruption-inflation nexus: evidence from developed and developing countries
  9. Credit channel and capital flows: a macroprudential policy tool? Evidence from Turkey
  10. Optimistic about the future? How uncertainty and expectations about future consumption prospects affect optimal consumer behavior
  11. Forecasting exchange rates using multivariate threshold models
  12. Firms’ operational costs, market entry and growth
  13. Commonalities and cross-country spillovers in macroeconomic-financial linkages
  14. Identifying conventional and unconventional monetary policy shocks: a latent threshold approach
https://doi.org/10.1515/bejm-2013-0160
Keywords for this article
aggregate total factor productivity; aggregation; idiosyncratic shocks; monopolistic competition; producer heterogeneity

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. On the macroeconomic effects of heterogeneous productivity shocks
  4. Fiscal policy in an open economy
  5. Understanding entry and exit: a business cycle accounting approach
  6. Contributions
  7. Predicting US recessions with stock market illiquidity
  8. The corruption-inflation nexus: evidence from developed and developing countries
  9. Credit channel and capital flows: a macroprudential policy tool? Evidence from Turkey
  10. Optimistic about the future? How uncertainty and expectations about future consumption prospects affect optimal consumer behavior
  11. Forecasting exchange rates using multivariate threshold models
  12. Firms’ operational costs, market entry and growth
  13. Commonalities and cross-country spillovers in macroeconomic-financial linkages
  14. Identifying conventional and unconventional monetary policy shocks: a latent threshold approach
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