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Inequality and the Zero Lower Bound ∗
Jesús Fernández-Villaverde Joël Marbet
University of Pennsylvania, NBER, CEPR CEMFI
Galo Nuño Omar Rachedi
Banco de España ESADE Business School
April 28, 2022
Abstract This paper argues that the effects of the zero lower bound (ZLB) on aggregate dynamics cru- cially depend on household inequality. We establish this result within a heterogeneous agent New Keynesian (HANK) model that features an occasionally-binding ZLB. Importantly, we solve numerically for the fully non-linear stochastic equilibrium using a novel neural-network algorithm. We first highlight how the presence of the ZLB in a HANK economy alters the response of households’ decisions and macroeconomic aggregates to demand shocks. We then show that the interaction of the central bank’s inflation target and the amount of wealth in- equality is a key driver of the level of real interest rates and the frequency of ZLB events. This stands in contrast of standard macroeconomic models, in which the level of real rates is pinned down as an exogenous parameter. In our setting, a drop in the inflation target reduces the level of the real interest rate because households increase their precautionary savings against the higher risk of ZLB events. As a result, ZLB events become even more likely. This channel is further amplified at higher levels of wealth inequality. Keywords: Zero Lower Bound, Heterogeneous Agents, HANK, Non-linear Dynamics, Real Interest Rates, Inequality. JEL Classification Codes: D31, E12, E21, E31, E43, E52, E58.
∗The views expressed in this paper are those of the authors and do not necessarily represent the views of the Banco de España or the Eurosystem.
1 Introduction
The level of real interest rates in advanced economies is at a historic low. While three decades ago the natural interest rate was around 3%, nowadays the level is estimated to be well below 1% (Del Negro et al., 2017; Fiorentini et al., 2018). This fact has major implications for the conduct of monetary policy (Barsky, Justiniano and Melosi, 2014), as it implies a much higher likelihood of experiencing situations in which policy rates are constrained by the zero lower bound (ZLB). This explains why in the aftermath of the Great Recession virtually all central banks have been constrained by the lower bound on policy rates for several years.^1 This situation has been further exacerbated by the coronavirus crisis, such that nowadays the policy rate is expected to be stuck around zero for about three years (Lilley and Rogoff, 2020). Although interest rates play a major role in shaping the stance of monetary policy, standard macroeconomic models tend to feature a long-run neutral monetary policy, with the implication that the level of interest rates is pinned down by structural parameters unrelated to (monetary and fiscal) policy. In this paper we argue that the level of real interest rates crucially depends on the interaction between the stance of monetary policy – i.e., the central bank’s inflation target – and the amount of wealth inequality in the economy. To establish this result, we extend the workhorse representative-agent New Keynesian (RANK) model in two ways: ( i ) by allowing for household heterogeneity due to uninsurable idiosyncratic labor income risk, in the spirit of Bewley (1980), Huggett (1993), and Aiyagari (1994), and ( ii ) by explicitly incorporating the risk of hitting the ZLB in households’ expecta- tions, as in Fernández-Villaverde et al. (2015). This framework generalizes the heterogeneous agent New Keynesian (HANK) model of Gornemann, Kuester and Nakajima (2016), McKay, Nakamura and Steinsson (2016), and Kaplan, Moll and Violante (2018) by incorporating in the dynamics of the model the non-linearity implied by the existence of an occasionally-binding ZLB. To do so, we propose a novel neural-network algorithm that allows for the computation (^1) In few jurisdictions the policy rates have actually been set to negative values. However, policy rates are constrained by the effective lower bound (ELB), that is, a level of the interest rates under which the central bank cannot implement further rate reductions. Throughout this paper, we consider the ZLB as the ELB, but our results can be generalized simply by taking a stand of how negative policy rates can be.
deterministic steady state (DSS) of the HANK economy. Although this result traces back to the seminal work of Aiyagari (1994), our setting grants it a novel perspective, as the precautionary savings reduce the room of manoeuvre for the central bank’s policy rate. In this way, the precautionary savings in the DSS alter the ZLB frequency, and consequently affect also the behavior of the macroeconomic variables in the SSS. Importantly, this result does not emerge in the standard HANK literature, in which the drop in the real rate due to precautionary savings is immaterial for aggregate dynamics because of the lack of the ZLB. Second, the interaction of aggregate uncertainty with wealth heterogeneity also increases precautionary savings at the SSS, and thus exert a downward force in the real interest rate vis- à-vis the RANK economy. As ZLB events affect disproportionately more wealth-poor agents, the presence of aggregate uncertainty and the possibility of the occurrence of large recessions in which the policy rate is constrained leads households – and especially those at the bottom of the wealth distribution – to increase their buffer of precautionary savings (or equivalently, reduce their borrowing positions). These two channels then suggest that, as long as any variation in central bank’s inflation target may alter the frequency of ZLB events, then it would also alter households’ decisions about their optimal buffer of precautionary savings, and ultimately affect the level of the real interest rate in the SSS. In addition, the effect of changes in the inflation target on the level of real rates would be much larger in economies with more wealth inequality, as in this case households at the bottom of the distribution have a stronger motive for precautionary savings. This is the key mechanism that we put forward in this paper: the long-run level of the real interest rate crucially depends on the interaction between the stance of monetary policy and households’ inequality. Thus, our model generates a long-run Fisher equation in which the level of the real rate depends positively on the inflation target. That is, if we denote the steady-state nominal interest rate, real rate, inflation rate, and inflation target by i , r , π , and π ¯, respectively, then i (¯ π ) = r (˜ π ) + π (˜ π ), such that dr/dπ > ˜ 0. Crucially, the magnitude of this derivative increases with the amount of wealth inequality. We then leverage our model to evaluate the implications of the interaction of different
inflation targets and different amounts of wealth inequality on the level of the real interest rate. We show that reducing the inflation target from 4% – which corresponds to the average inflation rate between 1980 and 1999 – to 1.7% – which corresponds to the average inflation rates observed from the year 2000 on – reduces the real rate in the SSS by 21 bps. If in addition we consider a rise in wealth inequality – which in the model is achieved by increasing the volatility of the idiosyncratic labor earning process to replicate the same change in wealth inequality observed in the U.S. over the last three decades – then the drop in the level of the real rate at the SSS is amplified by an additional 29 bps. If we put these changes in the context of the decline in the level of the real rates observed over the last decades, the interaction between the drop in the inflation target and the rise in wealth inequality explains roughly a fourth of it. From this perspective, we propose a novel channel that is complementary to the factors proposed by the literature in accounting for the drop in the real rate, such as population aging (Carvalho, Ferrero and Nechio, 2016; Aksoy et al., 2019) and the differences in households’ marginal propensity to save out of permanent income (Mian, Straub and Sufi, 2021). The non-neutrality of the inflation target on the level of real rates due to the presence of the ZLB has already been studied in the context of RANK models, in which changes in the inflation target alter real rates through a deflationary spiral channel: when the inflation target is sufficiently low and the probability of hitting the ZLB is relatively high, agents correctly expect that the ZLB may be binding in the future and internalize that the central bank may not be able to fully stabilize inflation. As a result, nominal interest rates decrease, reducing real rates, and eventually increasing the likelihood of hitting the ZLB even more.^2 However, we show that the quantitative relevance of the deflationary spiral channel hinges on the presence of the precautionary savings channel generated by market incompleteness: if we reduce the inflation target from 4% to 1.7% in a RANK model under exactly the same parametrization of our HANK economy, the real rate drops just by 12 bps. Thus, our mechanism generates (^2) While this mechanism has been already uncovered by Adam and Billi (2007), Nakov (2008), Hills, Nakata and Schmidt (2019), and Bianchi, Melosi and Rottner (2020), these papers mainly highlight this mechanism to argue for variation in the systematic monetary policy stance. For instance, Bianchi, Melosi and Rottner (2020) show that the presence of deflationary spirals calls for an asymmetric Taylor rule that allows the central bank to overshoot more likely its inflation target.
paper looks at how inequality alters the ex-ante incentives to accumulate precautionary savings when households face an occasionally-binding ZLB, a mechanism that affects the levels of consumption, output, and the real rate in the stochastic steady state.
2 Model
2.1 Environment
The economy is populated by a unit measure of ex-ante homogeneous households, which are ex-post heterogeneous due to the realizations of a persistent uninsurable labor earning risk. The idiosyncratic shock follows a Markov chain process. Households’ utility function over consumption and leisure is subject to an aggregate preference shock, which follows an AR(1) process. Households finance consumption expenditures with labor income as well as firms’ profits, and are allowed to borrow up to an exogenous limit. The production side of the economy consists of two levels. There is a continuum of in- termediate good producers, which produce different varieties of the intermediate good using labor, subject to a Rotemberg (1982) price-setting cost. Then, the different varieties of the intermediate goods are bundled into the final consumption good by the final good producer. Finally, the economy features a monetary authority that sets the nominal interest rate following a standard Taylor rule, subject to the ZLB constraint, as well as a fiscal authority, which levies progressive labor income taxes on the households to finance a fixed amount of outstanding public debt.
2.2 Households
There is a unit measure of ex-ante identical households indexed by i ∈ [0 , 1]. Households face idiosyncratic and aggregate risk. The idiosyncratic labor earning shock si,t ∈ { sm } Mm = determines the efficiency unit of hours supplied by each household, and follows a Markov chain with transition matrix Ω. We normalize the process such that the average realization of the idiosyncratic shock is ∫^ sitdi = 1. The aggregate shock ξt is a preference shifter, and follows an
AR(1) process in logs log ξt = ρξ log ξt − 1 + ζt , (1)
where ρξ ∈ (0 , 1) and ζt is normally distributed with mean 0 and variance σξ. The use of the demand shocks follow the work of Krugman, Dominquez and Rogoff (1998), Eggertsson et al. (2003), and Eggertsson and Krugman (2012), which show that shifts in households’ preferences
- a reduced form for any change that affect households’ leveraging capacity – is a powerful driver of ZLB events: Households choose consumption ci,t , bonds bi,t and labor services hi,t to maximize life-time expected discounted utility
{ ci,t,b^ max i,t,hi,t }∞ t =0^ E^0
∑^ ∞ t = βtξtu ( ci,t, hi,t ) (2) s.t. ci,t + bi,t = τt ( wtsi,thi,t ) 1 − γ
- Π tsi,t + R πt − t^1 bi,t − 1 , (3) bi,t ≥ b. (4)
where β denotes the time-discount parameter. Households’ optimization problem is subject to the budget constraint in Equation (3) and the borrowing constraint in Equation (4). The budget constraint posits that households finance consumption expenditures with firm profits Π t , which are rebated to households depending on their idiosyncratic labor productivity, and labor earnings, wtsi,thi,t , where wt denotes the real wage. Labor earnings are subject to a progressive taxation, where the parameter τt controls the level of the taxations and γ captures the degree of progressivity. When γ = 0, the tax rate is flat and does not depend on the level of households’ labor earnings.^4 Households also trade one-period non-contingent bonds, which yield the gross nominal return Rt − 1. We then refer to the gross real return on bonds as rt = R πt − t^1 , which divides the gross nominal return by gross inflation. The borrowing constraint posits that households bond position is limited by the exogenous limit b. (^4) Correia et al. (2013), McKay and Reis (2016), Wolf (2021) discuss the role of changes in the fiscal system in complementing monetary policy at the ZLB. In our framework, we abstract from this debate and consider a constant degree of progressivity over time.
yj,t = lαj,t, (9)
where α captures the degree of diminishing returns to scale of the production function. The cost minimization problem of intermediate good firms imply the marginal costs mj,t are defined as mj,t = (^) αlwα j,tt − 1_._ (10)
Since all firms operate with the same labor-to-output ratio, both marginal costs and demanded labor services are equated across variety producers, that is, mj,t = mt and lj,t = lt , for all j ∈ [0 , 1]. Intermediate good firms cannot freely adjust their prices, but rather face price adjustment costs as in Rotemberg (1982). This feature introduces nominal rigidities in the model. We follow Bayer et al. (2019) in the specification of the adjustment costs, with the only difference that we extend their settings to the case of a trend inflation possibly different from zero. In this case, the adjustment costs, Θ j,t , take the following functional form
Θ j,t ≡ Θ
( (^) p j,t pj,t − 1
) = 2 θ
log
( (^) p j,t pj,t − 1 × π ˜
)
2 Yt, (11)
where π ˜ denotes the trend inflation which is targeted by the central bank, and θ crucially captures the degree of price stickiness. When θ = 0, the economy does not feature nominal rigidities, and therefore collapses to a standard neoclassical heterogeneous-agent model with imperfect competition on the supply side. Given the price adjustment costs, the problem of intermediate good producers is to choose a sequence of prices { pj,t } t ≥ 0 to maximize the expected discounted stream of profits net of the adjustment costs, E t ∑^ ∞ k = t βk
Π k ( pj,k ) − Θ
( (^) p j,k pj,k − 1
) (^) , (12)
where the profits of the intermediate good firm, Π k ( pj,k ), equal
Π k ( pj,k ) =
( (^) pj,k Pk^ −^ mk
) ( (^) pj,k Pk
)− ε Yk. (13)
In the pricing protocol, we follow Hagedorn, Manovskii and Mitman (2019) by assuming that the price adjustment costs are virtual. Basically, the adjustment costs do affect firms’ decision of setting optimal prices, but eventually do not result in any transfer of real resources. Accordingly, the profits of each firm, Π t ( pj,t ), describe the entire resources that are rebated back to the households. In this procedure, we further assume that profits are rebated to each household according to its idiosyncratic productivity level, such that we are consistent with the empirical observation that earnings-rich households tend to receive a disproportionately larger share of firm profits. Finally, the overall solution of the problem of the intermediate good firms yields the following New Keynesian Philips curve
log
( (^) πt π ˜
) = βEt
[ log
( (^) πt + ˜ π
) (^) Yt + Yt
]
( mt − ε^ − ε^1
) , (14)
where πt = (^) PPt − t 1 is gross inflation rate.
2.4 Government
The government consists of a monetary authority and a fiscal authority. The monetary authority consists of a central bank which sets the gross nominal interest rate Rt according to a standard Taylor rule Rt = max
^1 ,^ R ˜
( (^) πt π ˜
) φπ^ (^ Yt Y^ ˜
) φy^ ,
R^ ˜ is the gross nominal rate in the deterministic steady state, and Y ˜ denotes the level of output in the deterministic steady state. This specification of the Taylor rule implies two important considerations. First, the mone-
Finally, the aggregate resource constraint posits that total output equals the sums of the value added of the intermediate good firms, and also equals the overall consumption of the households, that is, Yt =
∫ (^1) 0^ l
αj,tdj =^ ∫^1 0^ citdi.^ (18)
3 Calibration
We calibrate the model to the U.S. economy by setting one time period of the model to cor- respond to a quarter. Table 1 shows the current parametrization of the model and the chosen targets. First, we set the gross inflation target of the monetary authority to π ˜ = exp ( 0_._ 02 / 4 ), so that the annual inflation target is 2% in the baseline economy. As far as the Taylor rule parameters are concerned, we set the sensitivity of the nominal rate to output deviations from the steady state to φy = 0_._ 1 , and that to changes in inflation to φπ = 2_._ 5. Finally, to discipline the level of the real rate in the deterministic steady state, we set the time discount factor to β = 0_._ 997. This choice implies that the real interest rate in the deterministic steady state equals 1%. Our choice binds the value of the real interest in the deterministic steady state to a value which is in the ballpark of the estimates provided by the literature (Del Negro et al., 2017; Fiorentini et al., 2018), and also consistent with the November 2018 FOMC’s Summary of Economic Projections. However, in the quantitative results we will show that different values of the inflation target imply a large variation in the level of real interest rates in the stochastic steady state. This happens even if all the economies share the same value of real interest rates in the deterministic steady state. Next, we calibrate the demand shock process, which is the only source of aggregate risk in our model. Since we want to understand how household heterogeneity and changes in the monetary policy stance interact alter households’ decision by modifying the risk of hitting the ZLB, we require our model to be consistent with the frequency of ZLB episodes observed in the U.S. economy. To do so, we first set the persistence of the autoregressive process to ρξ = 0_._ 6 , in line with the parametrization of Bianchi, Melosi and Rottner (2020), and then set the standard
deviation to ωξ = 0_._ 0105 , so that – under a 2% inflation target – the model reproduces a 10% ZLB frequency, in line with the frequency observed in the U.S. in the post-war period (Coibion et al., 2016).
Table 1: Baseline Parametrization Parameter Value Target/Source Panel A. Aggregate Risk ρξ AR coefficient of process for ξ 0_._ 6 Bianchi, Melosi and Rottner (2020) ωξ Standard deviation of ξ shock 0_._ 0105 10% ZLB frequency Panel B. Idiosyncratic Risk ρs AR coefficient of process for st 0_._ 8 10% Average MPC ωs Standard deviation of st shock 0_._ 05 30% Borrowers b Borrowing limit - 0_._ 29 Monthly average labor income Panel C. Preferences β Discount factor 0_._ 997 1% real interest rate in the DSS σ Risk aversion 1 Standard value 1 /ν Frisch elasticity of labor supply 1 Standard value χ Disutility of labor 0_._ 71 Labor supply equals 1 in the DSS Panel D. Production ε Demand elasticity 7_._ 67 15% price markup α Labor share 1 Constant returns to scale θ Rotemberg price adjustment cost 79_._ 41 Equivalent to 0.75 Calvo parameter Panel E. Monetary Authority ˜ π Inflation Target exp ( 0_._ 02 / 4 )^ 2% Annual inflation target φπ Taylor rule coefficient on inflation 2_._ 5 Standard value φy Taylor rule coefficient on 0_._ 1 Standard value deviations from steady state output Panel F. Fiscal Authority B Government outstanding bonds 0.25 Total liquid assets = 25% annual GDP γ Degree of tax progressivity 0_._ 18 Heathcote, Storesletten and Violante (2017)
As far as the calibration of the idiosyncratic risk is concerned, we want our economy to be consistent with the share of borrowers and savers observed in the data, and with the empirical estimates of the marginal propensity to consume (MPC). To do so, we first set the borrowing limit to b = − 0_._ 29 , so that it equals one monthly average wage. Given this tightness of the
of outstanding bonds which amount to 25% of annual GDP, in line with the estimate of liquid wealth in the U.S. economy derived by Kaplan, Moll and Violante (2018). Then, we set the degree of tax progressivity to γ = 0_._ 18 , in line with the estimates of Heathcote, Storesletten and Violante (2017) using PSID data. Finally, we set the tax shifter τt by clearing the government budget constraint, which implies a value of 0.975.
4 Algorithm
The computation of heterogeneous agent is complicated by the fact that an equilibrium requires the households to keep track of how the distribution of bonds evolves over time, so that it allows them to be able to properly form expectations. However, this feature makes the solution computationally intractable as one needs to keep track of an infinite-dimensional object. Thus, this class of model is often solved by assuming bounded rationality, as in Krusell and Smith (1998). Basically, the solution algorithm imposes that households need to keep track of just a moment of the bond distribution, and this moment tends to be the mean. Then, the Krusell and Smith (1998) hinges on linear law of motion approach, in which the individual policy function first solves for the updated individual policy function, and then updates the perceived law of motion derived within a stochastic simulation. The algorithm then repeats these steps until the convergence of the perceived law of motion. However, our economy is inherently non-linear, and therefore the linear approach of Krusell and Smith (1998) would not capture the full dynamics of the economy. An important innovation of this paper is to introduce the solution of a non-linear economy within the literature of HANK models. This allows us to evaluate the stochastic steady state in an economy that features price rigidities, heterogeneous agents, and a zero lower bound. In doing so, we follow the approach of Fernández-Villaverde, Hurtado and Nuño (2020), showing that neural networks can fully approximate in a non-parametric way the non-linearity of the law of motion of a heterogeneous agent economy. We report a thorough description of the algorithm in Appendix A. Figure 1 shows how our algorithm uncovers a marked non-linearity in both in both the
Figure 1: Non-Linearity due to the ZLB.
0 2.^5 5.^0
5 10.^0
(^041). (^021). (^000). (^980). 96 −^ − 21
01
23
4
Preference Shock ξt Nominal Rate Rt− 1 (%)
Inflation
πt^ (%)
(a) Perceived Inflation ˆπ(ξt, Rt− 1 )
0 2.^5 5.^0
5 10.^0
(^041). (^021). (^000). (^980). 96 −^ − 21
01
23
4
Preference Shock ξt Nominal Rate Rt− 1 (%)
Inflation
πt^ (%)
(b) Simulated Inflation π(ξt, Rt− 1 )
perceived and simulated inflation rate, a non-linearity which is generated by the presence of the zero lower bound, which holds for sufficiently low levels of the realizations of the demand shock process ξt. To further corroborate the relevance of our approach, Figure 2 compares the nowcast errors on the dynamics of inflation generated by our neural network approach, with those implied by the naive application of the Krusell and Smith (1998) method, in which we predict the inflation rate as a log-linear function of the state variables. Panel (a) reports the errors for all simulated periods, and shows that the neural network approach increases the fit of households’ expectations by increasing the R^2 from 98.8% to 99.9%. This improvement, is concentrated in the left-hand part of the distribution of the errors, that is, cases in which inflation tend to be lower than expected due to the deflationary spirals generated amidst a ZLB event. Indeed, our approach is able to capture properly the non-linear dynamics at the ZLB. Panel (b) shows that the linear Krusell and Smith (1998) provides a very poor fit for inflation dynamics when the ZLB constraint binds, with an R^2 of 81.0%, whereas the neural network algorithm generates an R^2 of 99.2%. Hence, neural networks allow us to properly characterize the non-linear dynamics of the stochastic steady state at the ZLB, a feature that would not be present if we were to rely on standard computational algorithms.
The relevance of the non-linearity generated by the presence of the ZLB constraint can be appreciated by comparing the ergodic distribution of the key variables of our ZLB-HANK model with the implications of a counterfactual version of the baseline economy that abstracts from the the ZLB, that is, the standard HANK model.
Figure 3: Ergodic Distributions.
(^00) − 5 0 5 10
05
10
15
Inflation πt (%)
ZLB-HANK HANK
(^00) − 5 0 5 10
02
04
06
08
10
12
Nominal Rate Rt− 1 (%)
(^00) − 5 0 5 10
01
02
03
04
05
06
Real Rate rt (%)
(^0000). 90 0. 95 1. 00 1. 05
025
050
075
100
125
Aggregate Consumption Ct
Figure 3 reports the ergodic distribution of inflation, the nominal interest rate, the real interest rate, and aggregate consumption in both the baseline ZLB-HANK and the HANK economy. The graph shows how the presence of the ZLB skews the dynamics of the model to the left: it corresponds to cases in which the nominal interest rate is constrained by the ZLB, and the economy experiences a sharp drop in aggregate consumption amidst a deflationary spiral. All these dynamics are absent in the standard HANK model. The effect of the non-linearity of the ZLB constraint on the dynamics of inflation and output can be also evaluated by looking at the impulse-response function to small and large demand
shocks (i.e., a one standard deviation shock and a three standard deviation shock, respectively) in both the ZLB-HANK and the HANK economy, which we report in Figure 4.
Figure 4: Impulse Response Functions.
− 2. (^50 2 4 )
− 2. 0
− 1. 5
− 1. 0
− 0. 5
- 0
Quarter
Inflation
πt^
(pp) ZLB-HANK (1 std) ZLB-HANK (3 std) HANK (1 std) HANK (3 std) 0 2 4 6
− 3
− 2
− 1
0
1
2
Quarter
Real Rate
rt^
(pp)
0 2 4 6
− 4
− 3
− 2
− 1
0
Quarter
Nominal Rate
Rt −^1
(pp)
0 2 4 6
− 3
− 2
− 1
0
Quarter
Output
Yt^
(%)
0 2 4 6
− 3
− 2
− 1
0
Quarter
Wage
wt
(%)
0 2 4 6 − 3
− 2
− 1
0
Quarter
Preference Shock
ξt^
(%)
The plot shows that the impact response of the model to small demand shocks coincide both in the model with and without ZLB, as the shock is too small to generate a sufficiently large drop in the nominal interest rate that makes the ZLB binding. Instead, the responses to large demand shocks differ starkly depending on the presence or not of the ZLB. In the baseline model with the ZLB, a large shock brings the nominal interest rate down to zero, and it cannot go down further. This situation is accompanied by a much larger drop in both inflation and
