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Financial Frictions and the Wealth Distribution

Jesús Fernández-Villaverde

University of Pennsylvania, NBER, and CEPR

Samuel Hurtado Galo Nuño

Banco de España Banco de España∗

March 2, 2020

Abstract

We postulate a nonlinear DSGE model with a financial sector and heterogeneous house-

holds. In our model, the interaction between the supply of bonds by the financial sector

and the precautionary demand for bonds by households produces significant endogenous

aggregate risk. This risk induces an endogenous regime-switching process for output, the

risk-free rate, excess returns, debt, and leverage. The regime-switching generates i) multi-

modal distributions of the variables above; ii) time-varying levels of volatility and skewness

for the same variables; and iii) supercycles of borrowing and deleveraging. All of these are

important properties of the data. In comparison, the representative household version of

the model cannot generate any of these features. Methodologically, we discuss how non-

linear DSGE models with heterogeneous agents can be efficiently computed using machine

learning and how they can be estimated with a likelihood function, using inference with

diffusions.

Keywords: Heterogeneous agents; wealth distribution; financial frictions; continuous-

time; machine learning; neural networks; structural estimation; likelihood function.

JEL codes: C45, C63, E32, E44, G01, G11.

∗A GitHub repository with all the codes for the paper and further examples is at https://github.com/ jesusfv/financial-frictions. We thank Manuel Arellano, Emmanuel Farhi, Xavier Gabaix, Lars Peter Hansen, Mark Gertler, Aubhik Khan, Davide Melcangi, Ben Moll, Chris Sims, Gianluca Violante, Ivan Wern- ing, and participants at numerous seminars and conferences for pointed comments. The views expressed in this manuscript are those of the authors and do not necessarily represent the views of the Eurosystem or the Bank of Spain.

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https://github.com/jesusfv/financial-frictions https://github.com/jesusfv/financial-frictions

1 Introduction

Recently, several researchers have documented the nonlinear relations between financial vari-

ables and aggregate fluctuations. For example, Jordà et al. (2016) have gathered data from 17

advanced economies over 150 years to show how output growth, volatility, skewness, and tail

events all seem to depend on the levels of leverage in an economy. Similarly, Adrian et al.

(2019b) have found how, in the U.S., sharply negative output growth follows worsening financial

conditions associated with leverage.

Consequently, we need to postulate fully nonlinear dynamic stochastic general equilibrium

(DSGE) models with a financial sector that can account for the empirical observations in these

papers. After suitable development, such DSGE models will serve as laboratories for policy

analysis and forecasting.

This paper takes a step in this direction by building, computing, and estimating a continuous-

time DSGE economy with a financial sector, modeled as a representative financial expert, and

households, subject to uninsurable idiosyncratic labor productivity shocks. The economy is hit

by aggregate shocks to the stock of capital, and it is subject to two financial frictions. First,

only the expert can hold the capital that is rented to a representative firm, and it cannot issue

state-contingent assets (i.e., outside equity) to finance it. Instead, the expert can issue a risk-

free bond to leverage its equity and accumulate more capital. Hence, the expert must absorb

all the capital-return risk. Second, the households can only save in the risk-free bond to self-

insure against idiosyncratic shocks and the variations in wages and the risk-free rate induced by

aggregate fluctuations. The bond, in addition, cannot be shorted.

We include a financial sector in our model to capture the evolution of debt and leverage.

We introduce household heterogeneity because, under the financial frictions we consider, such a

heterogeneity begets substantial endogenous aggregate risk. This aggregate risk derives from the

interaction between the supply of bonds by the expert, determined by the excess return of capital

over risk-free bonds, and the precautionary demand for bonds by the households, driven by their

income risk.

The endogenous aggregate risk induces an endogenous regime-switching process for output, the

risk-free rate, excess returns, debt, and leverage. The regime-switching generates i) multimodal

distributions of the variables above (as reported in Adrian et al., 2019a); ii) time-varying levels

of volatility and skewness for the same variables (as documented in Fernández-Villaverde and

Guerrón-Quintana, 2020); and iii) supercycles of borrowing and deleveraging (as evidenced in

Reinhart and Rogoff, 2009).

Thus, household heterogeneity allows us not only to replicate the nonlinear dynamics between

aggregate variables and leverage reported by Jordà et al. (2016) and Adrian et al. (2019b), but also

to offer a novel and simultaneous account of i) the recent heightened fragility of the advanced

economies to adverse shocks; ii) the rise in wealth inequality witnessed before the 2007-2009

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financial crisis (Alvaredo et al., 2017); iii) the increase in debt and leverage experienced during

the same period (Adrian and Shin, 2010, and Nuño and Thomas, 2017); and iv) the low risk-free

interest rates of the last two decades (Holston et al., 2017).

Let us unpack the previous three paragraphs. For parameter values that match U.S. data and

maximize the likelihood function, our economy has multiple stochastic steady states, or SSS(s),

despite the model having a unique deterministic steady state (DSS).1 In particular, we will have

a high-leverage SSS (HL-SSS) and a low-leverage SSS (LL-SSS), each with its basin of attraction

(there is a third, unstable SSS that we do not need to discuss).

The intuition for the existence of two stable SSS(s) is as follows. In the basin of attraction of

the HL-SSS, endogenous aggregate risk is high. After a negative aggregate shock, the economy

suffers deep and protracted recessions with persistently low wages (the next paragraph explains

why). Due to precautionary behavior, households accumulate more bonds when risk is high.

Higher savings have two consequences. First, higher savings increase wealth inequality among

households: households with good realizations of their idiosyncratic shock accumulate much

wealth. Second, higher savings lower the risk-free rate that clears the bond market and cause a

high expected excess return for capital, pushing the expert to leverage aggressively, the defining

feature of this basin of attraction. In comparison, aggregate risk is low in the basin of attraction

of the LL-SSS. Therefore, precautionary savings are lower, wealth inequality smaller, the risk-free

rate high, and the expected excess return reduced. These prices sustain the low leverage that

characterizes this basin.

Why are recessions more severe in the basin of attraction of the HL-SSS than around the

LL-SSS? When leverage is high, a negative aggregate shock to capital greatly erodes the expert’s

net wealth, since its relatively small equity must absorb all the capital losses. After a negative

shock, it takes a long time for the expert’s wealth and capital to recover. During this time,

wages are persistently low and the risk-free rate is high.2 Two forces cause the sluggish recovery

of capital. First, the supply of debt is low because the expert is reluctant to lever too quickly.

Second, the demand for debt is low because wealthy households, whose income is more dependent

on bond returns, have a weaker incentive to reduce their consumption when the risk-free rate is

persistently high. These two forces result in small debt growth and the slow recovery of capital

and output. By contrast, when leverage is low, the recessions after a negative aggregate shock

are mild. While the responses are similar on impact, the ensuing recovery is much faster because

the expert experiences less of a reduction in its net wealth.

Whether the economy has high or low leverage is a consequence of past aggregate shocks.

1An SSS, also known as a risky steady state, is a fixed point of the equilibrium conditions of the model when the realization of the aggregate shock is zero. A DSS is a fixed point of the equilibrium conditions of the model when the volatility of the aggregate shock (but not of idiosyncratic shocks) is zero. The multiplicity of SSS(s) is different from the multiplicity of equilibria: in our model, we find a unique equilibrium.

2In our model, wages are equal to the marginal productivity of labor (lower when capital is low) and the risk-free interest rate depends on the marginal productivity of capital (higher when capital is low).

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Sometimes, while the economy is traveling in the basin of attraction of the HL-SSS, a sequence

of aggregate shocks will move it to the basin of attraction of the LL-SSS (and vice versa). For

our baseline parameter values, the economy will spend more time, on average, in the basin of

attraction of the HL-

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