A TANK Model of Fiscal Policy Uncertainty∗

Filipe Stona Marcelo S. Portugal

January 2020

Abstract

To demonstrate the transmission channels of fiscal policy uncertainty, we build aTwo-Agent New Keynesian (TANK) model with stochastic volatility shocks. First, aTANK model highlights the importance of the labor market on the transmission of un-certainty to households, reinforcing real consequences that uncertainty shocks have onthe economy. Next, we show that these shocks can be amplified giving the combinationof fraction of hand-to-mouth agents on the economy and their risk-aversion character-istics. Extending our analysis to a developing country, comparing results for the USand Brazil, we are able to understand what attributes of a country lead to differentresults, which is manly driven by the hand-to-mouth agent wealth characteristics. Themodel also allows us to understand how government spending can be manged to easethe negative impact on households, demonstrating how a policy that generates similaroutcomes on output can harm agents constitution level.JEL: E12, E23, E32, E52, E62.Keywords: fiscal policy, volatility, uncertainty, policy risk, New Keynesian model,Two-Agent model.

∗Stona: Department of Economics, Universidade Federal do Rio Grande do Sul, Brazil (e-mail: fs-tona@live.com); Portugal: Department of Economics and Department of Business Administration, Uni-versidade Federal do Rio Grande do Sul, and National Council for Scientific and Technological Development(CNPq), Brazil (e-mail: msp@ufrgs.br). We thank Jesús Fernández-Villaverde and Oren Levintal for helpfulcomments and suggestions. All errors are our own.

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mailto:fstona@live.commailto:fstona@live.commailto:msp@ufrgs.br

1 IntroductionThe significant decrease of natural rates of interest on both advanced and emerging economieshas pressured monetary policy, which lost much room to stimulate the economy during adownturn. On a scenario of low growth, inflation and interest rates, the government maybe compelled to take action using its fiscal capacity to stimulate the economy. The usage ofdiscretionary fiscal policy to reactivate the economic performance when aggregate demandand interest rates are low has been aired by the economic literature at least since Feldstein(2002), and every time these combination hits a country and the central bank seems to runout of ammunition to fight a downturn, it makes a comeback. However, fiscal policy decisionsare not as easy to be taken as monetary policy. Fiscal adjustments are under the influence ofpoliticians, that may have political interest on its usage and a different interpretation of whatan active fiscal policy implies. Even with increased levels of public indebtedness, governmentscan be tempted to use fiscal instruments to fight back an economic slowdown, increasing theconcern on its debt sustainability. Furthermore, depending on the system of governance,some fiscal action have to be approved by the congress, leading into a long waiting period ofuncertainty.

Among the aggregate effects of uncertainty described by Bloom (2009), an increase inuncertainty may depress investment due to the “wait-and-see” approach. This behavior offirms would harm households, who would have to use precautionary savings to hedge againstunexpected decrease on their income. At the same time, Kaplan, Violante and Weidner(2014) highlight that a great fraction of individuals in the US are borrowing constraint,with little liquid asset on their portfolio. Kuhn, Schularick and Steins (2017) also show theevolution of portfolio composition of households according to their income level, showing thatbottom half of the distribution have a high degree of indebtedness and some iliquid assets,but little room for unexpected expenses. These are some evidences that demonstrate whypeople without liquid capital are more exposed to uncertainty shocks.

This paper aims to indicate how uncertainty about fiscal policy affects the borrowingconstraint and unconstrained household. We consider a Two-Agent New Keynesian economywith Rotemberg pricing and Epstein–Zin preferences This framework shows that with sim-ple modifications on a otherwise standard model with fiscal policy stochastic volatility byFernández-Villaverde et al. (2015) we are able to observer responses neglected on the fullyaggregated case. For instance, a TANK model highlights indirect channels of uncertaintytransmission, such as the labor market. Furthermore, we not only reinforce real effects thatuncertainty shocks can have on the economy, demonstrating that these can be bigger thanfiscal policy decisions, but also demonstrate that the fraction of hand-to-mouth agents on theeconomy risk-aversion characteristics can be a key factor to the outcome in the real economy.Also, we also provide an alternative fiscal policy rule refined to a developing economy, withfiscal revenue estimations that lack on the literature due to data constraints. As highlightedby Gavin and Perotti (1997), fiscal policy outcomes present sharp differences between LatinAmerican and industrial country. We identify these divergences and redefine a rule properto an emerging economy. Finally, the model also allows us to understand how governmentspending can be manged to ease the negative impact on households, demonstrating how apolicy that generates similar outcomes on output can harm agents constitution level.

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Kaplan, Moll and Violante (2018) show that models with a representative agent mutateindirect channels of monetary policy transmission witch are pivotal to understand its effectson consumption changes. In this way, we also add up to the discussion about the relevance ofheterogeneous agents New Keynesian models (HANK) on the analysis of policy transmissions,illustrating that Two-Agent New Keynesian model (TANK) can account for aggregate resultsand indirect policy channels highlighted by Kaplan, Moll and Violante (2018), as argued byDebortoli and Galí (2017). Since Fernández-Villaverde et al. (2015) demonstrate the relevanceof monetary policy on fiscal volatility shocks, once the central banks reacting to uncertaintycan deviate from its standard reaction rule, a Two-Agent framework plays a meaningful roleon the analysis of their transmission channels of these shocks.

This study does not focus on wealth distribution and its reaction to uncertainty shocks,such as Bayer et al. (2019), but on borrowed constraint households, yet we understand thatportfolio composition data presented in Kuhn, Schularick and Steins (2017) are a relevantevidence of borrowing constraint households on the US economy. According to Kuhn, Schu-larick and Steins (2017), houses and other non-financial assets make up more than 80 percentof the asset side of the balance sheet of the bottom 50 percent, while the middle class andthe top 10 percent have higher levels of bonds and financial assets on their portfolio. At thesame time, Kaplan, Violante and Weidner (2014) show that hand-to-mouth agents representroughly 20 percent of total U.S. income, but three-quarters of these are wealthy households,who owns sizable amounts of iliquid assets, which carry a transaction cost, such as housing.The authors also give evidences that the share of hand-to-mouth agents can vary roughlyfrom 20 to 50 percent depending on the definition.

We also aim to understand in what extend the results presented in this paper are relatedto country specific characteristics, such as fiscal structure and political uncertainty. For thisreason, the analysis is extended to a developing economy. Emerging markets are known forits political instability, which make them more prone to fiscal policy volatility shocks. Brazilis facing fiscal uncertainty about its fiscal policy path since 2011, when a fall in extraordinaryrevenues observed in previous years unleashed an acceleration of increased levels of publicindebtedness. At the same time, Brazil has a complete different fiscal policy structure andhigher income inequality, which can represents a higher fraction of HtM on the economy. Asan example, while the US tax system focus on the of capital income, the Brazilian system haslower taxes on capital and a higher focus on income charges. Studies like Kaplan, Violante andWeidner (2014) are scarce for Brazil, but according to Brazil’s national survey of householdbudgets1, around 75 percent of households reported difficulty to make it to the end of theirpay period with current income. Also, 80 percent the portfolio composition is committed toconsumption, 2.1 percent to settle liabilities, such as payment of loans, and 5.8 percent toincrease asset levels.

Related Literature. This paper is an intersection of three branches of the literature:two-agent models, uncertainty shocks and fiscal policy. The growing literature on New-Keynesian models with heterogeneous agents since Kaplan, Moll and Violante (2018) broughttogether the re-emergence of Two-Agent models. As summarized by Coenen et al. (2012),these class of models are extensively used by central banks to analyze fiscal and monetary

1Pesquisa de Orçamentos Familiares (2008-2009) - POF, in Portuguese.

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policy. Specifically on fiscal policy, Drautzburg and Uhlig (2015) is a great example of amodel where some agents are borrowed constraint and behave in a “hand-to-mouth” fashion,consuming their current labor income at all times, while the other agent have full access tofinancial markets and capital investments.

Croce, Nguyen and Schmid (2012) study the effect of fiscal policies on long-term growthwhen agents are uncertain about the distribution of future fiscal shocks. Agents have atendency to put more probability in the worse scenario of fiscal distortions. The authors usea stochastic version of an exogenous growth model, assuming that the government finances itsspending on debt and distortionary taxes on labor income. The greater the uncertainty, theworse the agents’ view, making future tax expectations higher, discouraging work. In turn,Bi, Leeper and Leith (2013) expands the Bertola and Drazen (1993) model with uncertaintyabout the timing and composition of fiscal consolidation, seeking to explain empirical studiesthat find an expansionary outcome, with an acceleration in production growth after a fiscalcontraction, as a cut of government spending.

While Bi, Leeper and Leith (2013) consider uncertainty about systematic parts of the fiscalrule to study the timing and composition of a fiscal consolidation, Fernández-Villaverde et al.(2015) and Born and Pfeifer (2014) consider the uncertainty in fiscal policy as a variant shockin the time. Similar in many aspects, these two works differ in the format of the shocks andsome specifications of the model. The results are qualitatively close, nevertheless Born andPfeifer (2014) note that the significance of uncertainty shocks is relatively small. On theother hand, the results of Fernández-Villaverde et al. (2015) demonstrate that the results ofa volatility shock on fiscal policy are not only significant but also greater in a Zero LowerBoundary scenario (ZLB). The fall in interest rates allows to reduce the counter-cyclical effectof shocks of fiscal volatility in the level of economic activity. For the Brazilian case, wherethere is still a lot of room for lowering the interest rate level, this perspective suggests thatthe responses tend to be less corrosive at the production level, for example. It is also worthhighlighting the work of Basu and Bundick (2017), which, instead of considering uncertaintyin the fiscal policy rule, considers uncertainty on the demand side.

As highlighted by Moura (2015), few papers in Brazil analyze the issue of fiscal policy,and even fewer debate about volatility shocks on fiscal instruments. Among the few papersusing a dynamic equilibrium model to discuss fiscal stimulus in Brazil, there are de Carvalhoand Valli (2011), Moura (2015) and Cavalcanti and Vereda (2015), which still proves to be asubject devoid of investigation.

The rest of this paper is organized as follows. In section 2, we present the estimationof tax and spending process with stochastic volatility. In Section 3, we present a baselineNew Keynesian model with fiscal volatility and we detailed computation and calibration tothe US economy on Section 4. Section 5 illustrates the effects of fiscal volatility shocks onthe US economy and Section 6 compare results with Brazil. Section 7 presents the study ofa pseudo-optimal fiscal policy and section 8 concludes. A technical appendix offers furtherdetails.

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2 Measuring Fiscal Policy UncertaintyOur analysis focus on three fiscal policy instruments, government spending as a share of out-put (gt), tax rate on labor income (τwt ) and tax rate on capital income (τ kt ). To understandthe impact uncertainty on the instruments, we adopt the idea introduced by Bloom (2009)that uncertainty follows a stochastic volatility process. However, fiscal policy rule can behighly sensitive to the data. Most of the literature about fiscal policy is centered on gov-ernment spending, but Leeper, Plante and Traum (2010) formulate fiscal rules for capital,labor and consumption that would be empirically estimated and also providing better datafitting. Since then, some other alternatives emerged on the literature, such as those used byFernández-Villaverde et al. (2015) or Born and Pfeifer (2014).

The fiscal rule suggested by Leeper, Plante and Traum (2010) for capital and labor taxallows a response to the current cyclical position of the economy and to changes in the levelof government debt. Furthermore, they also allow the shocks to be serially correlated, inorder to capture the persistent nature of exogenous changes in instruments. Born and Pfeifer(2014), in turn, assume that tax rates follow an AR(2) process and allow for a responseto the lagged debt-to-output ratio and lagged output position. As in Fernández-Villaverdeet al. (2015) and Born and Pfeifer (2014), Leeper, Plante and Traum (2010) assume that theresponse of theses fiscal instruments to output cycles and debt will be positive, while for thegovernment spending these responses should be negative.

For x ∈ {g, τ k, τ l}, the law of motion for each instrument used by Fernández-Villaverdeet al. (2015) is given by

xt − x =ρx(xt−1 − x) + ρx,yỹt−1

+ ρx,b

(bt−1yt−1

− bsys

)+ exp{σx,t}εx,t, εx,t ∼ N (0, 1),

(1)

where the log standard deviation of each policy instrument (σx,t) has a time-varying stochasticvolatility process,2

σt = (1− ρσ)σs + ρσσt−1 + (1− ρ2σ)1/2ησut, ut ∼ N (0, 1). (2)

Equation (3) assumes that demean fiscal policy reacts to its previous value xt−1, laggedoutput gap ỹt−1 and the lagged debt-to-output ratio (bt−1/yt−1) deviation from its meanvalue. Idiosyncratic shocks to uncertainty are independent of the first movement fiscal policyshocks εx,t, which follows a normal distribution with zero mean and one standard deviation.The stochastic volatility process σx,t on Equation (2) allows a shock to the second moment ut,which will be interpreted as an increase on uncertainty about the future time path of fiscalpolicy, and ησ will be the unconditional standard deviation of the fiscal volatility shock.The higher is the term (1 − ρ2σ)1/2ησ, stronger is the evidence of a time-varying volatility.Furthermore, σt follows and AR(1) process with steady-state value of σs.

We explore several specification for our data. As initially explored by Gavin and Per-otti (1997), it is well known that fiscal policy in emerging markets, and specifically Latin

2We omitted the subscribe x on the stochastic volatility process for the sake of clearness, but parameterand σ were estimated according to the fiscal instrument.

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American countries, not necessarily have the same fiscal policy reaction of advanced coun-tries. Specifically for Brazil, the literature is centered on the understanding of governmentspending reaction to output and debt,3 mainly due to the lack of data about other fiscalinstruments, as we highlight on Section 6. The same happens for other developing countries,which cannot take advantage of the effective tax rates methodology developed by Mendoza,Razin and Tesar (1994) for industrial countries.

For x ∈ {g, τ k, τ l}, the law of motion for each instrument used for Brazil is given by

xt − x =ρx(xt−1 − x) + ρx,y(ỹt−1)+ ρx,bb̃t−1 + exp{σx,t}εx,t, εx,t ∼ N (0, 1),

(3)

where b̃t−1 is the lagged deviation of debt from its trend and σx,t is the log standard deviationof each policy instrument follows Eq. 2. This rule can be seen as a mixture of those in Leeper,Plante and Traum (2010) and Fernández-Villaverde et al. (2015). However, these authorsassume that capital and labor income tax will be procyclical reaction to debt and output, andcountercyclical for government spending. Considering results in Gavin and Perotti (1997),Vegh and Vuletin (2015) and Campos and Cysne (2019) we relaxed the hypothesis of acountercyclical reaction of the government spending to the output. At the same time, werelaxed the the procyclicality assumption on the reaction of labor and capital tax to laggeddebt, since direction for this relation are absent on the literature of developing countries.Further discussion on fiscal policy rules is presented on Appendix A.

Equation (3) and (2) are estimated for each instrument independently and the posteriordistribution of the parameters are characterized using the Random Walk Metropolis-Hastingsalgorithm, with a sequential Monte Carlo method to obtain a numerical estimation of the like-lihood. We draw 400,000 times from the posterior using a random walk Metropolis-Hastings.The draw was run after a search for appropriate initial conditions and an additional 100,000burn-in draws.4 We tune the covariance scaling factor of the proposal density according tothe rule used in Herbst and Schorfheide (2019), to induce the appropriate acceptance ratioof proposals. The tuning function allows us to calibrate the covariance scaling factor duringthe burn-in period and, after achieving the target acceptance ratio, we keep making fineadjustments every 1,000 draws until the last burn-in draw. Each evaluation of the likelihoodis performed with 40,000 particles.5

We present on Table 1 parameters estimated by Fernández-Villaverde et al. (2015) for theUS economy together with estimations for Brazil. Details about data and further analysisfor these results are presented on Section 6.

3See Campos and Cysne (2019).4All programs needed for the Metropolis-Hastings and the sequential Monte Carlo estimation were coded

in C++-17 and compiled in GCC-9.1 with fully parallelized algorithm by using Intel TBB library to run onOSX-based machines. On a 2.6 GHz Intel Core i7 with six physical cores, each draw from the posterior usingthe sequential Monte Carlo with 40,000 particles takes around 0.107 seconds. That implies a total of about16h for each simulation of 500,000 draws.

5See Appendix B for details about the estimation procedure.

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Table 1: Fiscal Policy Instruments Parameters

United States BrazilLabor Capital Government

SpendingLabor Capital Government

Spending

ρx 0.99 0.98 0.99 0.88 0.63 0.91[0.98, 0.99] [0.92, 0.99] [0.65, 0.97] [0.80, 0.95] [0.47, 0.77] [0.87, 0.96]

ρx,y 0.040 0.043 -0.004 0.047 0.11 0.021[0.008, 0.045] [0.003, 0.099] [-0.018, 0.00] [0.004, 0.159] [0.02, 0.23] [-0.025, 0.066]

ρx,b 0.003 0.003 -0.008 0.01 -0.002 -0.012[0.000, 0.008] [0.003, 0.099] [-0.018, 0.00] [-0.01, 0.04] [-0.03, 0.03] [-0.021, -0.006]

σs -6.01 -4.89 -6.20 -4.10 -4.56 -5.59[-6.20, -5.81] [-7.35, -6.87] [-6.53, -5.71] [-4.74, -3.79] [-5.15, -4.33] [-5.99, -5.38]

ρσ 0.46 0.65 0.92 0.28 0.20 0.21[0.33, 0.58] [0.39, 0.86] [0.78, 0.99] [0.03, 0.69] [0.02, 0.69] [0.02, 0.58]

ησ 0.820 0.400 0.180 0.27 0.16 0.20[0.70, 0.97] [0.22, 0.58] [0.10, 0.29] [0.02, 0.93] [0.02, 0.88] [0.02, 0.65]

Note: Parameters estimated following a Bayesian approach by combining the likelihood function with flat priors and samplingfrom the posterior with a Markov Chain Monte Carlo. For each parameter, we report the posterior mean and, in brackets, a90 percent probability interval.

3 ModelThe quantitative analysis will be based on a Two-Agent model with Epstein-Zin (EZ) prefer-ences and fiscal policy volatility shocks. Hand-to-mouth (HtM) agents do not have access tocapital or bonds market, while bondholders (or savers) own capital in the economy, receivefirms profits and has access to financial market, buying governmental bonds to smooth con-sumption. This distinction not only changes the way each agent behaves, mainly because ofthe inability of HtM agents to insure against unexpected shocks, but also allows that somefiscal instruments have direct or indirect impact according to agents type. For instance, thegovernment levy capital duty, which is only part of Bondholders budget, and labor, whichwill have a direct impact on both members type of the household. The model also containsseveral frictions that introduce nominal and real rigidities in prices, wages, capital and invest-ments. Great part of the model is build on the framework developed in Fernández-Villaverdeet al. (2015, hereafter FGKR) and Schmitt-Grohé and Uribe (2005).

3.1 Households

Preferences. All households are assume to to have preferences given by an EZ aggregatorbetween the period utility Ut and the continuation Vt+1:

V 1−ψt = (1− β)Ut(ct, ht)1−ψ + βEt(V1−γt+1 )

1−ψ1−γ , (4)

where Ut = eξtcηt (1 − ht)1−η. The discount factor β ∈ (0, 1) and the elasticity of leisureη have the same value for both agents.6 At the same time, the risk aversion γ and the

6Domeij and Flodén (2006) present some empirical micro evidence that the elasticity of leisure will bedifferent for constrained and unconstrained agents (See Table 4 in their paper), however, the functional formused here already give us flexibility to define risk aversion and the intertemporal elasticity of substitutiondeferentially. The addition of a third difference on agents preference could make their relationship poorly

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inverse intertemporal elasticity of substitution ψ are calibrated according to the type of eachagent. The consumption and hours worked will be agent-specific, as will become clear on thepresentation of budget constraints. We omitted a superindex denoting cht , hht or cbt , hbt , forHtM or bondholders, respectively, in favor of a clearer notation. The intertemporal preferenceshock ξt follows ξt = ρξξt−1 + σξεξ,t, where εξ,t ∼ N (0, 1).

Labor Supply. We assume that households supplies differentiated labor service to theproduction sector through a representative labor aggregator.7. Each type of agent belongs toa different labor packer that are proportionally aggregate and supplied to the firms. That is,each type of household supplies labor hj,t to a continuum of unions j ∈ [0, 1], and firms regardeach household’s labor service as an imperfect substitute for the labor of other households.Denoting the aggregate demand for composite labor services as hd,t, The bundling technologyis

hd,t =

(∫ 10

hεw−1εw

j,t dj

) εwεw−1

,

where εw > 1 measures the elasticity of substitution. The labor aggregator minimizes thecost of producing a given amount of the aggregate labor index,8 taking each households wagewj,t as given, where wt is the aggregate nominal wage. Thus, the demand for labor servicesis

hj,t =

(wj,twt

)−εwhd,t. (5)

Considering this result, it is straightforward to define the aggregate wage index as wt =(∫ 1

0w1−εwj,t dj)

11−εw , and the aggregate labor supply as the sum of labor by variety, ht =

∫ 10hj,tdj.

Budget Constraint. A fraction µ of the agents in the economy are Hand-to-Mouth,without access to any kind of asset of the economy. The budget constraint of the HtM agentis given by:

cht + Ωht = (1− τwt )

∫ 10

wj,thhj,tdj − ACwj,t, (6)

where Ωt are lump-sum taxes. Wages are subject to a quadratic adjustment cost ACwj,t =φw2

(wj,twj,t−1

− zs)2yt, where zs is the steady-state of the productivity shock zt. A perfect

competitive labor packer aggregate the different types of labor hj,t into homogeneous labor,

so hj,t = hd,t∫ 1

0

(wj,twt

)−εwdj.

The Bondholder agent can trade one-period riskless bonds, invest on capital and receive

identifiable.7Following the assumption of Erceg, Henderson and Levin (2000), the labor aggregator can be interpreted

as an union, labor packer or employment agency.8It solves the following problem:

maxhj,t

wt

(∫ 10

hεw−1εw

j,t dj

) εwεw−1

−∫ 10

wj,thj,tdj.

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the profits of the firms, facing the following budget constraint:

cbt + xt + bt + Ωbt = (1− τwt )

∫ 10

wj,thbj,tdj + (1− τ kt )rkt utkt−1 +

Rt−1bt−1Πt

− ACwj,t + Ft, (7)

where xt is the investment level, ut is th rate of utilization of capital, Ωt are lump-sum taxesand Ft are the profits of the firms in the economy. Besides the labor taxes τwt levied on bothagents, Bondholder also face capital tax τ kt , since she is the owner of capital ranted by a raterkt to the intermediate firms.

Investment decisions are subject to an adjustment cost. The law of motion of physicalcapital is described by

kt = (1− δ(ut))kt−1 +(

1− S[xtxt−1

])xt, (8)

where the adjustment cost of investments assumes a quadratic form S[

xtxt−1

]= κ

2

(xtxt−1− zs

)2,

and δ(ut) = δ+ δ1(ut−1)+ 12δ2(ut−1)2 is the depreciation rate that depends on the capacity

utilization rate.Agents Aggregation. Firms cannot distinguish the type of each agent on the labor

market neither their decision on consumption level, thus, assuming that a fraction µ of theagents in the economy are HtM, the aggregation of these variables is given by

ht = µhht + (1− µ)hbt , (9)

ct = µcht + (1− µ)cbt . (10)

The bondholder chooses processes for ct, bt, ut, kt, xt, and wj,t so as to maximize its utilityfunction (4) subject to the budget constraints, wage adjustment costs, the law of motion ofcapital (8) and demand for labor services (5). On the other hand, HtM agent only choosect and wj,t, subject to related constraints. Appendix C presents the equilibrium conditionsassociated with households problems.

3.2 Firms

The model firms consists of perfectly competitive final-goods-producing firms and a con-tinuum of monopolistically competitive intermediate goods producers. At every time t, acompetitive firme produces the final consumption good ydt using the intermediate goodsyi,t, i ∈ [0, 1] and the technology ydt = (

∫ 10y

(εp−1)/εpi,t di)

εp/(εp−1), where εp is the substitutionelasticity. Expenditure minimization leads to the final input demand function given by

yi,t =

(pi,tpt

)−εpyd,t,

where pi,t is the price of the intermediate goods and pt is the price index for final goods,pt = (

∫ 10p

1−εpi,t di)

1/(1−εp).

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The monopolistic firm produces intermediate good using a production function yi,t =kαi,t(zthi,t)

1−α. Each i intermediate good producer rent labor hi,t from both agents and capitalki,t−1 from the saver agent to produce intermediate good yi,t. Producers maximize profits bysetting prices subject to a quadratic adjustment cost. Since firms are owned by bondholders,firms use their stochastic discount factor:

maxpi,t+s

Et∞∑s=0

[∂Vt/∂c

bt+s

∂Vt/∂cbt

] [pi,t+spt+s

yi,t+s −mci,t+syi,t+s − ACpi,t+s]

(11)

subject to the final input demand function yi,t and a quadratic adjustment cost for prices,

ACpi,t =φp2

(pi,tpi,t−1

− Πs)2yi,t, where Πs is the steady state level of inflation.

3.3 Government and monetary policy

The government issue bonds bt every period to satisfy its flow budget constraint, given by

bt =Rt−1bt−1

Πt+ gt − τwt wtht − τ kt rkt kt−1

− zt[φd,b

(bt−1zt−1ys

− bsys

)+ µΩh + (1− µ)Ωb

],

(12)

where we consider that lump-sum tax stabilize de debt-to-output ratio, imposing a passivefiscal/ active monetary regime. The government expenditure gt, labor and capital taxes (τwtand τ kt ) evolve endogenously according to thre rules defined in Equations (3) and (2).

The monetary authority controls the short term interest rates following a modified Taylorrule that responds to deviations of inflation with respect to the steady state, gap of observedoutput and shocks of uncertainty. We consider results in Fernández-Villaverde et al. (2015),where the authors highlight that a standard Taylor Rule does not generates the same effectsfrom empirical estimations for the inflation and interest rates. They propose the rule givenby

Rt = Rs

(Rt−1Rs

)φR [(ΠtΠs

)φΠ (ytys

)φy (eσe,teσe,s

)φσ]1−φRexp{σmΦt}, (13)

where the monetary policy shock Φt follows a N (0, 1) process. The parameters φR is asmoothing parameters to capture gradual movements of in interest rates, φΠ and φy cap-ture the responsiveness of the interest rate to deviations of inflation from its steady stateand output gap from its trend, and φσ captures the reaction of the monetary authority touncertainty shocks.

4 Quantitative Analysis

4.1 Solution and Calibration

The characterization of equilibrium conditions of the model is relatively standard and isrelegated to the Appendix C. The model presented in this paper needs a higher-order ap-

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proximation. As explained by Fernández-Villaverde et al. (2011), second-moment shocks thatcan generate precautionary behavior does not show up in a first-order approximation (which isequivalent to the traditional log-linearizion), and second-order approximation would capturethe volatility effect only indirectly, via cross-product terms. Hence, to explore the direct roleof the volatility shocks in the model, we need to consider cubic-terms. Another issue is that athird-order approximation of this model gives explosive sample paths, so we use the pruningmethod for third-order approximation developed by Andreasen, Fernández-Villaverde andRubio-Ramírez (2018).

Table 2: Summary Calibration and Fixed Parameters

Description Value Source

Preferencesβ Time-discount factor 0.9959 Calibrationγh HtM Risk Aversion 370.34 Calibrationγb Bondholder Risk Aversion 661.83 Calibrationψh HtM Inverse IES 0.40 Vissing-Jørgensen (2002)ψb Bondholder Inverse IES 1.25 Vissing-Jørgensen (2002)µ Fraction of HtM agents 0.40 Kaplan, Violante and Weidner (2014)η Consumption preferences 0.4034 EndogenousΩh HtM lump-sum tax steady state -0.2263 EndogenousΩb Lump-sum tax steady state 0.3019 EndogenousTechnologyα Capital Share 0.36 FGKRεp Demand elasticity goods 21.0 FGKRεw Demand elasticity labor 21.0 FGKRδ Steady state depreciation 0.0451 Calibrationδ1 Rate of capital depreciation 0.01 Jaimovich and Rebelo (2009)δ2 Rate of capital depreciation 0.0139 Calibrationκ Investment cost 0.9419 CalibrationTaylor RuleφR Smoothing of past interest rate 0.7584 CalibrationφΠ Response to inflation deviations 1.6966 Calibrationφy Response to output gap deviations 0.0863 Calibrationφσ Response to capital tax volatility 0.005 FGKRΠs Steady state inflation 1.0090 Calibrationσm Standard dev. of the monetary shock 2.0e-4 Calibration

Note: parameters fixed prior to the estimation are referred with their corresponding source. Parameters estimated with SMMprocedure are tagged with "Calibration". The consumption preference parameter η is tagged with “Endogenous” because wenormalize the steady state of hours worked to 1/3 using η. We also set Ωh and Ωb endogenously to achieve a balanced HtMand governmental budget constraint, respectively.

One of the key parameters in our model is the proportion of borrowing constraint agents inthe economy. Following Kaplan, Violante and Weidner (2014), we assume that 40 percent of

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the households are HtM (µ = 0.4)9. Some parameters are fixed before the estimation becausedata moments can be uninformative and the literature have proper estimation of them. Wenormalize the steady state of hours worked to 1/3 using η. The intertemporal elasticity ofsubstitution (IES) is set to 2.5 for bondholders and 0.8 for HtM agents (ψ = 0.40 and 1.25,respectively)10, considering results in Vissing-Jørgensen (2002). The depreciation parameterδ1 is used to set the steady state capital utilization to 1 from its first order condition. Thequadratic cost of changing nominal prices is set to φp = 200.0, which implies all intermediate-products prices are fixed for about six quarters in a linearized Calvo setting, and the measureof wages rigidity is set to φw = 2000.0, both parameters following FGKR. The parameterscalibrated with data are the discount factor β, risk-aversion of each type of agent, i.e., γh andγb, two parameters related to the depreciation rule, δ and δ2, the parameter κ that controlsde curvature to the adjustment cost of investment function, and five parameters of the Taylorrule, φR, φΠ, φy, σm and Πs. We present other fixed and calibrated parameters on Table 2.

Since our model has a large number of variables, the computation required to find theo-retical moments is computationally inefficient. Thus, the only difference on our calibrationprocedure in comparison to Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) isthat instead of calculate theoretical moments we compute them via simulation. We appliedthe Simulated Method of Moments (SMM) to calibrate selected parameters of the model. Asdescribed by Ruge-Murcia (2012), SMM is an appropriate method to calibrate parameters ofa non-linear model, since it is computationally efficient and delivers accurate estimates evenfor short sample series. We present model fit for selected moments on Table 3. We matchthe average of interest rates and inflation, standard deviation, auto-correlation with one lagand correlation of all variables with output and the correlation between interest rates andinflation, summing up 26 moments to calibrate 11 parameters. With a few exceptions, themodel does a reasonable job on matching the mean, standard deviation and auto-correlation,while results for the correlation are mixed. As detailed on Section 2, the parameters for thefiscal rule are exogenously calibrated with a Stochastic Volatility model. Data for the USeconomy are taken from St. Louis Fed’s FRED database, and time span 1970:Q1–2014:Q2 isthe same used in Fernández-Villaverde et al. (2015) for the sake of results comparison. Dataseries used to match with moments of variables yt, ct, xt, wt, ht, ut, Rt and Πt are the RealGross Domestic Product (GDPC1), Real Personal Consumption Expenditures (PCECC96),Real Gross Private Domestic Investment (GPDIC1), Compensation Per Hour (HCOMPBS),Hours of All Persons (HOABS), Capacity Utilization of Total Industry (TCU), EffectiveFederal Funds Rate (FEDFUNDS) and Implicit Price Deflator (GDPDEF).

9Since there is no consensus about the fraction of Hand-to-Mouth agents on the US economy, we alsopresent results for a sensitivity test on this parameter (Figure 2).

10Due to the utility function definition, the IES is equal to 1/ψ. See Swanson (2012) and Gourio (2012).

12

Table 3: Model Fit - selected moments

Moments Data Model Moments Data Model

MeanRt 0.8885 0.8912 Πt 1.4028 1.2852Standard Deviation Auto-Correlationyt 1.5278 1.8796 corr(yt, yt−1) 0.8779 0.9178ct 1.2481 0.7818 corr(ct, ct−1) 0.8909 0.8343xt 7.0489 6.9979 corr(xt, xt−1) 0.8532 0.9511wt 0.9329 0.2090 corr(wt, wt−1) 0.7074 0.9694ht 1.9474 1.6701 corr(ht, ht−1) 0.9276 0.8315ut 3.2484 2.4141 corr(ut, ut−1) 0.8993 0.8970Rt 0.9365 0.7355 corr(Rt, Rt−1) 0.9701 0.9842Πt 0.6079 0.7192 corr(Πt,Πt−1) 0.9038 0.9349Correlationcorr(yt, ct) 0.8796 0.7932 corr(yt, ut) 0.8777 0.6230corr(yt, xt) 0.9214 0.8232 corr(yt, Rt) 0.2021 -0.4609corr(yt, wt) 0.1196 0.5958 corr(yt,Πt) 0.1160 -0.2736corr(yt, ht) 0.8702 0.8959 corr(Πt, Rt) 0.6651 0.9113

Note: All data, except nominal interest rates and inflation, are in logs, HP-filtered, and multiplied by 100 to express them inpercentage deviation from trend. Nominal interest rates and inflation are directly expressed in percentage points. We omittedmodel fit for the mean and auto-correlation for a lag of five in the interest of space.

5 The Effect of Fiscal Volatility Shocks

5.1 Estimation Results

Parameters calibrated with SMM in Table 2 are the same chosen by FGKR, with exceptionof the risk-aversion γ of each type of agent. A few differences can be notice when comparingresults for the parameters related to the Taylor Rule, specifically in what concerns the re-sponsiveness of the monetary authority to deviations of the steady state inflation φΠ, whichtakes the value of 1.35 in their results in comparison with 1.69 in ours. This difference is canbe explained by a slightly higher estimation of the steady state inflation in our model (anannualized inflation rate 0.55 percent higher) and lower standard deviation of the innovationto the monetary policy shock σm.

Our estimation for the risk-aversion parameters are in tune with other estimations in themacroeconomics representative agent literature. These parameters are in contrast with theempirical micro literature, where confronting individuals with specific risk prospects, Barskyet al. (1997) obtain a measure of the relative risk-aversion parameter ranging from 3.8 to 15.7,while other macro studies with a representative agent with Epstain-Zin preferences estimatevalues ranging from 80.0 to 600.0, depending on the model structure.11 The Bondholderagent presents a parameter value of 661.83, in line with Andreasen, Fernández-Villaverde

11See van Binsbergen et al. (2012), Rudebusch and Swanson (2012), Andreasen, Fernández-Villaverde andRubio-Ramírez (2018), for example.

13

Figure 1: IRF to a Fiscal Policy Uncertainty Shock

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bt), aggregate consumption and product (ct and yt). Second line, HtM, Saver and aggregate hours worked

(hht , hbt and ht) and wages (wt). Finally, investments (xt), marginal cost (mct), inflation (Πt) and interest rates (Rt).

and Rubio-Ramírez (2018). At the same time, the HtM agent presents a low parameter of370.34, which is reasonable with the fact that she does not have assets, such as capital orbonds. This result is in line with Guiso and Paiella (2008) self-selection argument, where theauthors highlight that more risk-averse agents select themselves into occupations with low-income risk. However, the reasoning of this results it is not completely straight forward. Wewill expand the discussion when we present risk aversion results for Brazil, where Bondholdersare less risk-averse then HtM agents.

5.2 Equilibrium Dynamics

The estimated model allows us to investigates the outcome of fiscal policy volatility shockon business cycles fluctuations and observe heterogeneous effects between constraint andunconstrained agents. As pointed out by Debortoli and Galí (2017), a TANK model can ap-proximate well the aggregate results of a New-Keynesian model with heterogeneous agents.This is relevant for the analysis of policy transmissions, highlighting indirect transmissionchannels absent on a standard representative agent model. Thus, we focus on the differenteffect of fiscal uncertainty on constraint and unconstrained agents as well as aggregate con-sequences of these shocks. Since volatility shocks on instruments others than capital tax ratepresent smaller responses, we overlook their results on this section.

Figure 1 presents the GIRFs to a positive two-standard-deviations innovation to a fiscalvolatility shock to the capital income tax.12 Bondholder household invest less because of theincreased probability of a high tax rate on capital income, and the lower marginal cost (highermarkups) imply that firms will produce less output and require less capital. The high decline

12We used the closed-form solution in Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) tocompute GIRFs for a third-order approximation.

14

of marginal costs is related with the reaction of the monetary authority to the uncertaintyshock, increasing the interest rate more than it would on a scenario where the Taylor Ruledoes not account for reaction to second moment shocks. This prompt reaction of marginalcosts decreases labor demand by firms and hours worked by both agents. However, the non-HtM agent can decrease their labor supply without losing as much consumption as the HtMagent. At the same time, the HtM household would have to decrease consumption moredrastically if she reduce labor supply by the same rate of Bondholders.13 The proportionalitybetween effects on consumption and hours worked is due to the utility kernel used, sincenon-separability induces that individuals working less hours will dedicate more time to homeproduction, decreasing its the aggregate level.

The contraction in output and investments presents a slightly higher drop than results inFGKR. Moreover, together with investment rates, the uncertainty shock generates a decreasefollowed by an overshooting reaction, in line with Bloom (2009). Firms diminish investmentrates after face uncertainty about future capital taxes, which fosters an wait-and-see atti-tude. The rapid activity slowdown bounced back after five quarters, a different reactionin comparison to the actual shock, which did not return to its steady state value after 21quarters (see Figure E.1). It is also worth noticing that the decrease on consumption facedby HtM has the same size generated by an increase on labor taxes but with different shape,since it generates a long lasting reaction after an actual increase of labor taxes (Figure E.2).The output reaction due to an uncertainty about capital tax is also grater than the effectsof an actual increase on labor taxes or government spending (Figure E.3), which shows thatvolatility shocks should not be neglected by policy makers.

The effect of a change on the fraction of hand-to-mouth agents on the economy canbe seen on Figure 2. Kaplan, Violante and Weidner (2014) highlight that estimation ofthis parameter in the literature are highly susceptible to the definition of a hand-to-mouthindividual. This analysis allows us to understand to impact of this parameter on the modeland which variables are more sensitive to the fraction of borrowed constraint household on theeconomy. Furthermore, ranging µ from 0.1 to 0.8 by increments of 0.1, Figure 2 encompassall possible values for µ referred in Kaplan, Violante and Weidner (2014) as well as somefar-fetched cases.

The fraction of hand-to-mouth agents on the economy does not alter aggregate effectson investments and marginal costs, while other key aggregate variables, such as output andconsumption, are sensitive the this ratio. The effect on output can vary from -0.4 to -0.6percent, but its duration is almost independent of tha ratio of HtM to non-HtM agents. Asimilar analysis can be made for the aggregate consumption, which is mainly leaded by HtMagents, that consume less as long as work less hours. The smaller the portion of Bondholderson the economy, the bigger is pressure of HtM agents on wages, which makes the Bondholderdecreases labor supply. At the same time, since HtM does not have any asset to substituteincome from labor, she cannot decrease hours worked without losing consumption, which isnot the case for Bondholders, who face an almost unchangeable consumption reaction. Thisis possible because they will tune capital utilization to not lose so much income after the

13Domeij and Flodén (2006) estimate that individuals with no assets will increase their labor supply ifwages goes down. We did not get this effect in our model so far, since we cannot distinguish supply anddemand labor responses.

15

Figure 2: Sensitivity to µ

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uncertainty shock. Furthermore, a stochastic volatility shock to the capital tax rate resemblethe results for an income-risk shock of the heterogeneous agent model in Bayer et al. (2019).Even that we cannot account for effects related to the distribution of wealth and income, ouraggregate results are more inline with data, which shows a slower and slightly hump-shapedrecovery. This results reinforce the statement of Debortoli and Galí (2017) that a TANKmodel can be viewed as a framework able to account aggregate results of HANK models.

6 Fiscal Volatility Shocks on a Developing CountryIn this section we aim to explore the effects of fiscal volatility shocks on a developing country.In this way it would be possible to understand if the sensitivity of the model to the shareof hand-to-mouth agents on the economy is suitable to changes according to country specificcharacteristics, as well as identify which aggregate effect are more sensitive to a differentmacroeconomic scenario. The reaction of fiscal instruments to the debt-to-output ratio andthe output gap are some of the key differences that can emerge on the comparison betweencountries. Also, the time-varying process for the standard deviation is correlated with na-tional political events, such as fiscal reforms proposals on the congress, national public deficitadministration and/or financial crises. Furthermore, parameters internally calibrated withthe model, such as those concerning the Taylor Rule, are considerably sensitive to the inputdata. As demonstrated in the previous section, the monetary policy is one of the main chan-nels of fiscal policy volatility transmission, and response to this shock can have alternativeeffects due to its calibration.

The current juncture of Brazil’s national accounts, discussions on a proposal by the gov-

16

ernment to reform social security and increased levels of public indebtedness have raiseduncertainty about the evolution of the fiscal policy. The Brazilian economy is facing fiscaldifficulty which should be seen within the context of a hike in deficit since 2011. Beforethat, there are two moments that it is possible to identify the need for fiscal reform, as in1999 and 2003. However, both episodes are related to exchange rate shocks, while since 2011the debt increase has occurred due to increased government spending. The primary resultof the public sector began to deteriorate more specifically in August 2011, with a fall inextraordinary revenues observed in previous years. In addition, it is possible to recognize anacceleration of this process, given that the gross debt of the general government has grownrapidly, from around 50 percent of GDP in 2014 to more than 70 percent in 2017.14 Through-out 2017, i.e., several proposals for reforms were sent to the national congress, generatingdoubts about fiscal adjustment directions. In addition to the limitation of public spending,there is a long debate about social security system reform, the need to raise taxes to financethe government deficit and changes in the loans rate of the National Bank for Economic andSocial Development (BNDES, in Portuguese). All these events are potential sources of fiscalvolatility.

6.1 Estimation Results

The reaction of fiscal policy instruments and the stochastic volatility process follow descrip-tion on Section 2 and estimated parameters on Table 1. To calibrate others parameters ofthe model, Born and Pfeifer (2014) and Fernández-Villaverde et al. (2015) construct theiraggregate effective tax rates using national account information following the measurementof relevant aggregate tax rates in Mendoza, Razin and Tesar (1994) and Jones (2002). How-ever, the aggregation of these data may be more challenging if we consider Brazilian data,as discussed by Azevedo and Fasolo (2015). Since the construction of aggregate tax rates isalready worth a whole study, we took advantage of Azevedo and Fasolo (2015) using theirdatabase for estimation. We adjusted the labor income tax by the unemployment rate, sincethe period of analysis exhibit a considerable decrease of unemployment levels in Brazil from18.9 to 11.0 percent, which generates an upward trend on labor tax income.

We calibrate internally the same set of parameters of the model for the US economy usinga SMM procedure. Results are reported on Table 4 and a summary of fit for selected momentson Table D.1, which presents consistent results mainly for the standard deviation and auto-correlation, with exception of the moments for wages, as well as results for correlation that aremixed. Risk-aversion parameter for HtM agents are considerably higher then those presentedby others on the macroeconomic literature with Epstain-Zin preferences, while risk-aversionof Bondholders are line with literature. Comparing with our prior results for the US, theseparameters are switched, since HtM are less risk averse then Bondholders in the modelpreviously calibrated. This result for Brazil is contrary to the argument on the literature,

14Brazil is a developing country with a recent history of high levels of inflation and the historical level ofindebtedness tolerated by Brazil can still be considered low, when compared with other countries. Thus, adebt level of 70 percent of GDP can be seen as excessive, as Reinhart, Rogoff and Savastano (2003) and Bi,Shen and Yang (2016) illustrate.

17

Table 4: Summary Calibration for Brazil

Description Value Source

Preferencesβ Time-discount factor 0.9785 Calibrationγh HtM Risk Aversion 1329.10 Calibrationγb Bondholder Risk Aversion 373.46 Calibrationη Consumption preferences 0.4034 EndogenousΩh HtM lump-sum tax steady state -0.2263 EndogenousΩb Lump-sum tax steady state 0.3019 EndogenousTechnologyδ Steady state depreciation 0.0442 Calibrationδ2 Rate of capital depreciation 0.003 Calibrationκ Investment cost 0.9458 CalibrationTaylor RuleφR Smoothing of past interest rate 0.7611 CalibrationφΠ Response to inflation deviations 1.5701 Calibrationφy Response to output gap deviations 0.0097 CalibrationΠs Steady state inflation 1.0124 Calibrationσm Standard dev. of the monetary shock 0.0034 Calibration

Note: Parameters estimated with SMM procedure are tagged with "Calibration". The consumption preference parameter ηis tagged with “Endogenous” because we normalize the steady state of hours worked to 1/3 using η. We also set Ωh and Ωbendogenously to achieve a balance HtM and governmental budget constraint, respectively. The others parameters fixed priorthe estimation are reported on Section 4, and those estimated exogenously on Table 1.

which claims that risk-averse agents self-select themselves in a low income risk position.15There is a lack of studies on the literature to carry out this investigation for developingcountries, but one hypothesis is that in this countries agents are not on a HtM situationbecause they want to, but because they can get out of it. This would means that mosthand-to-mouth agents in emerging markets are poor, in contrast with the US, where Kaplan,Violante and Weidner (2014) highlight that many of these households are wealthy.

The curvature of the depreciation rule δ2 is almost flat for Brazil, since this parametergoes to zero. Parameters related to the Taylor Rule reinforce the Brazilian hyperinflationhistoric, once the steady state inflation is higher then the presented in the US. At the sametime, reaction to output gap deviations are lower, with close parameters for the smoothnessof interest rates. The reaction to deviations from the steady state inflation are slightly lowerin Brazil, which can be due to the higher steady state inflation and monetary shock standarddeviation. Theses differences, together with those results for the fiscal instruments shapedifferent reaction in response to a capital tax rate volatility shock.

15See Guiso and Paiella (2008) among others.

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Figure 3: IRF to a Fiscal Policy Uncertainty Shock for Brazil

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bt), aggregate consumption and product (ct and yt). Second line, HtM, Saver and aggregate

hours worked (hht , hbt and ht) and wages (wt). Finally, investments (xt), marginal cost (cmt), inflation (Πt) and interest rates

(Rt).

6.2 Response to Volatility Shocks

The GIRFs to a positive two-standard-deviations innovation to a fiscal volatility shock to thecapital income tax in Brazil are presented on Figure 3. The first remarkable difference withUS results (Figure 1) is related to the size of responses. We keep two-standard-deviationinnovations as in FGKR, where the authors assess that there is a probability of 10 percentof theses shocks hit the economy, but they argue that it cannot be considered an extremeevent, since they show evidence that it is likely that at least three or four of those eventshit the US economy from 1970 to 2014. Nevertheless, a positive two-standard-deviationsinnovation in Brazil generates effects slightly smaller is size comparable with those presentedpreviously for the US economy. The main reason for that is the size of unconditional standarddeviation ησ and the parameter ρσ, which controls the shock persistence. Since the databasefor Brazil is smaller then the one for the US, Table 1 shows that the confidence interval forthese parameters is wider, and there values are smaller.

Broadly speaking, the same mechanism takes place, working through the lowering ofmarginal cost and inflation. Yet, the monetary authority in Brazil reacts with lower intensitythan the US Fed, pushing up interest rates even with inflation going down, due to thepositive reaction to the uncertainty shock. We present on Figures D.1 and D.2 the role ofthis innovation on the Taylor Rule comparing the results of a standard Taylor Rule, withoutany direct response of the monetary authority to the shock, and the Taylor Rule on Eq. (13).While the effects on the US are relatively subtle, allowing the model to replicate empiricalevidences of a drop on inflation, as documented by FGKR, in Brazil the difference is muchstronger. Without a direct response of the monetary authority to uncertainty about capitalincome tax all observed effects would be negligible. Furthermore, this also shows that themodel can be extremely sensible to the parameter φσ that controls the responsiveness of

19

Figure 4: Sensitivity to µ in Brazil

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interest rates to volatility shocks.16Further effects that can be observed are the output decline, which takes more than 21

quarters to get back to its steady state level, a more persistent result compared with the USspawned by the high persistence of capital tax rate volatility. The decline of hours worked byHtM agents still bigger than Bondholders response, but it returns faster to the steady statelevel. The opposite occurs on the reaction consumption levels, where Bondholders present atypical hump-shaped response to uncertainty shock, retrieving equilibrium in five quarters,while HtM face a slow recovery.

Finally, we are able to observe similar results on the sensitivity to the fraction of bor-rowing constraint households on the economy (Figure 4). Since HtM agent does not havealternative sources of income, they push wages down, which makes Bondholders work less.The higher the fraction of borrowing constraint agents on this economy, higher will be thiseffect. Bondholders mange their budget to work less with lower capital utilization, as longas wages are too low. Also, for the set of parameters estimated for Brazil, investment levelis more sensitive to µ than in the US. The same happens with the consumption response ofBondholders, which is completely unresponsive to fiscal volatility shocks on the US.

16An essential next step on this analysis would be the implementation of an empirical model, which weaim to include on a upcoming version of this paper.

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7 Pseudo-Optimal Fiscal PolicyConsidering that countries can diverge on their fiscal policy rule and how they react to itsoutput growth or indebtedness, we perused a “Pseudo-Optimal” Fiscal Policy. Since we arenot solving a welfare Ramsey problem, we cannot call this exercise as an optimal policy pre-scription. In fact, we vary the parameter which captures the reaction of government spendingto previous debt and output gap, considering a range in line with the literature.17 We usedan equally spaced grid of 6,400 points for ρg,y ∈ {−0.2, . . . , 0.2} and ρg,b ∈ {0, . . . , 0.2},where for each combination we solved the model and computed the cumulative GIRFs forthree years. Thus, this procedure allows us to identify optimal combinations for the fiscalpolicy rule on government spending in a specific parametric space reported by the empiricalliterature.

The literature on optimal fiscal policy with recursive preferences is summarized by Karan-tounias (2018), who demonstrates that optimal fiscal policy prescriptions change dramaticallydepending on the Household preferences. According to Karantounias (2018), the governmentshould manipulates the returns of the government portfolio to minimize the welfare costsof taxes. In this way, the government would absorb the fiscal risk in order to attenuatesutility volatility by taxing lass in bad times. However, he also stresses that the connectionbetween risk-aversion and intertemporal elasticity of substitution is a key factor for this re-sult. What we aim to understand in this section is how should the government behave inorder to minimize utility loses generated by fiscal uncertainty. extending the understandingin Karantounias (2018) to the relationship between state-contingent debt and the existenceof the different types of agents on the economy.

The first thing we need to understand is what are the effects of capital income tax volatilityshocks on output when we considering the government reacting in many different ways, anda remarkable result on Figure (E.4) is that regardless of the government policy decision,uncertainty will have a negative impact on output. Figure (E.4) presents the response ofgovernment spending and output gap to a capital income tax volatility shock, where in theleft column is the three dimensional representation of all combinations of ρg,y and ρg,b, i.e, thereaction of government spending to previous debt and output gap, and on the left, its contourrepresentation. The level of the plot is cumulative GIRFs three years after the realizationof the uncertainty shock. It is possible to identify two quasi-optimal results for the output.First, when the parameter which indicates government behavior owing to past debt is closeto -0.05 and the parameter for response to the output gap is negative, i.e., a counter-cyclicalbehavior. In this case, the output would have decreased around -1.1 percent given the steadystate level. On the other hand, a pro-cyclical government could achieve a similar result if ithad a looser concern with previous debt level. The main distinction would be that in thefirst case the government would end up with a net spending close to zero, while in the secondcase the government is decreasing its spending level.

At the same time, when we make the same analysis to households consumption level onFigure (E.5), there is a striking distinction on the outcome of concerning the governmentalbehavior. The first thing to notice is that HtM household will always face a negative response

17See Vegh and Vuletin (2015).

21

on its consumption level, while the Bondholder is more responsive to fiscal policy. Concerningboth spots where the government decision is less harmful to output, while the first one resultsin a contraction of consumption, the second one can even delivery a positive 0.1 percentdeviation from its steady state level for the Bondholder, and a slightly less negative reactionof the HtM consumption level. It happens because when the government decides to shrinkits spending, income taxes are also reduced, which decreases the the effects of uncertainty onHouseholds.

This result is in line with Karantounias (2018), since the optimal policy in this exercisewould be if the government decided to absorb the risk, spending less and decreasing itsindebtedness level. At the same time, we show how different kind of agents are affected bythe government fiscal policy reaction paramenters. With a lower debt level, the tax level oncapital would slightly decrease, while capital income tax would be lower proportionally tothe alternative outcome, when the government decides to spend more in reaction to the loweroutput gap. With lower tax on both capital and labor, the Bondholder agent ends up with ahigher consumption level after three years, while the HtM agent will only be affected by thelower level of labor tax.

8 ConclusionsIn this paper, we examine the potential effects of an increased uncertainty about fiscal policyon economic activity and distinct impacts on hand-to-mouth agents and a household withaccess to assents. Through the lens of a medium scale Two-Agent New Keynesian model, wefind that labor market movements are essential to differentiate the responses on both typesof agents in the economy. The model is also able to reproduce empirical evidences that HtMagents reduce labor supply less than individuals with savings, but face a higher decrease onconsumption level. Theses results are ballooned in a scenario where a majority of householdsare borrowing constraint.

We reinforce the discussion started by Debortoli and Galí (2017), who argue that a TANKmodel can capture the differences in average consumption between constrained and uncon-strained households, even thought it is silent about consumption heterogeneity within thesubset of unconstrained households. In particular, results of fiscal policy volatility shock aresimilar to those presented by Bayer et al. (2019) as income risk on a fully heterogeneousmodel with incomplete markets.

Finally, on the last section of this paper we compare results for the US economy witha developing country, Brazil. This comparison allows us to understand to what extend adifferent set of parameters estimated for fiscal rules can affects our results. While the broadanalysis keeps the same, we are able recognize some country specific reactions, such as astrong drop of interest rates due to a lower level of inflation. Also, we highlight that thelabor market channel is key to understand fiscal volatility shocks, which would be mutedon a otherwise representative agent model. There is a mechanism in the model that makesagents on the borrowing constraint willing to work for a lower wage than those with someliquid asset or higher borrowing limits, whose will lower their hours worked. However, theseiterations are unobserved on the aggregate level.

22

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26

A Fiscal Policy RulesUnlike monetary policy, which follows a Taylor-rule that can be estimated for a wide rangeof countries, fiscal policy lacks a regular rule. Even for the United States, there is still fewworks on rules-based fiscal policy. Taylor (2000) discusses the relevance of a rule in thiscontext distinguishing discretionary changes in taxes and spending and change due to theautomatic stabilizers. The first is related to legislative and executive actions, posed by thegovernment, while the second are automatic changes in the fiscal instruments according tononcyclical factors, increasing spending on programs as unemployment compensations anddecrease on tax revenue as employment and income falls in the recession.

Taylor (2000) focus on discretionary policy, such as short-term impacts, i.e. deviationsfrom potential GDP, that are relevant for countercyclical fiscal policy. In his view, the fis-cal policy should work as monetary policy, keeping real GDP close to the potential GDP.However, Gavin and Perotti (1997) shows that this is not the case for Latin American coun-tries, where government spending is usually procyclical, with some evidence of asymmetricbehavior, making public spending even more procyclical during bad times. Recession arethus associated with collapses in public spending. The authors raise a few hypothesis for thisbehavior, but this is a feature that persists over the years for many emerging economies, asdemonstrated by Alesina, Campante and Tabellini (2008) and Vegh and Vuletin (2015).

Among the hypothesis raised by the literature, Alesina, Campante and Tabellini (2008)focus on political distortions to explain this theses effects, in a way that voters face corruptgovernments that can appropriate part of tax revenues for political rents. Another possibilitymentioned in Gavin and Perotti (1997) is the voracity effect, which arises because interestgroups compete for a share of tax revenue. In this case, any fiscal surplus will increasethe pressure on the government to increase investments, but ends up resulting on wastefulspending. An underling fact on this arguments is that some agents will have more politicalpower to appropriate public assets and keep them on the strong group, driving the procyclicalfiscal policy.

While this fact can be incorporated in a model as a matter of sing and parameter es-timation of the policy instrument rule, some other characteristics of a fiscal rule are moresensible to the data. Leeper, Plante and Traum (2010) have shown that different feedbackrules matter for policy analysis, Kliem and Kriwoluzky (2014) demonstrate the importanceof variables which the fiscal instrument is reacting. Even among the literature of macroe-conomic models for the United States, there is a lack of consensus about the fiscal rules.Comparing Leeper, Plante and Traum (2010), Born and Pfeifer (2014), Drautzburg and Uh-lig (2015), Fernández-Villaverde et al. (2015) and Leeper, Traum and Walker (2017), theyall share common features, such as a reaction to deviations from potential GDP and/or debtlevel. However, its not straightforward to define if the government is reacting to both at thesame time, if it is reacting to the debt-to-GDP or debt cycles, among some other character-istics observed by the authors. Thus, we incorporate in our model both the procyclicalityand the composition discussion of the fiscal policy rule on a developing country.

27

B Stochastic Volatility Estimation ProceduresThe stochastic volatility model estimated for the fiscal policy instruments can be representedin a generic for as

xt = Ψ(xt−1, yt−1; θ) + exp{σt}εx,t, εx,tt ∼ N (0, 1).σt = Φ(σt−1, ut; θ), ut ∼ N (0, 1),

where θ is a vector of parameters and yt represents the exogenous variables with explanatorypower over fiscal policy, such as output gap and debt-to-output ratio. In this section wedescribe briefly the procedures to estimate the parameters in θ and the particle filter proce-dure to approximate the likelihood function. Doucet and Johansen (2011), Creal (2012) andHerbst and Schorfheide (2015) are excellent references for a more formal exposition aboutparticle filters.18 Herbst and Schorfheide (2015) is also useful for a better understand ofthe estimation method, together with Robert and Casella (2004) and Gamerman and Lopes(2006).

As described in Robert and Casella (2004), a Metropolis-Hastings algorithm starts withthe objective density. A conditional density q(θi−1|θ) defined with respect to the domi-nating measure for the model is then chosen. The Metropolis-Hastings algorithm can beimplemented in practice when q(·|θ) is easy to simulate and is either explicitly available orsymmetric. The random walk chain is a common option and practical implementation ofMetropolis-Hastings algorithm when you cannot find a good approximation density for theposterior. The Random Walk Metropolis-Hastings algorithm (RWMH) takes the followingsteps:

1. Chose starting value, θ0;

2. Take a candidate draw ϑ from the candidate generating density, q(ϑ|θi−1);

3. Calculate the acceptance probability, α(θi−1, ϑ);

4. Set θi = ϑ with probability α(θi−1, ϑ) and θi = θi−1 with probability 1− α(θi−1, ϑ);

5. Repeat steps 2 to 4 NMH times.

The RWMH has a proposal distribution with a random walk form, so it generates candi-dates draws according to

ϑ = θi−1 + ϕ,

where ϕ is the increment random variable with mean zero and variance cV . We use a normalproposal distribution with V = 1 and set cn adaptively to achieve a desired acceptance rate.Tuning the proposal covariance scaling factor according to the rule proposed by Herbst andSchorfheide (2019),

cn = cn−1f(1−Rn−1), f(x) = 0.95 + 0.10e20(x−0.40)

1 + e20(x−0.40),

18Fernández-Villaverde et al. (2011) represent an example of particle filter application in economics andthe authors also deliver an applied explanation about SMC procedures on their appendix.

28

where Rn−1 is the rejection rate in the past draw and f(x) considers a target acceptance rateof 30 percent, updating the value of cn every 1,000 draws. Notice that once the acceptancerate is equal to the target, the scaling factor will be immutable. We keep tuning it untilthe last burn-in draw. This procedure allow us to achieve an acceptance rate close to thetarget without a very exhaustive search and inside the burn-in period, saving a significantcomputational time.

The acceptance probability α(·, ·) ensures that the chain moves in the correct direction.Giving the symmetric nature of the proposal distribution, the acceptance probability takesthe form

α(θi−1, ϑ) = min

{p(xT |ϑ)p(xT |θi−1)

, 1

},

in a way that a draw ϑ is accepted with probability one if the posterior at ϑ has higher valuethan the posterior at θi−1. Considering the ratio between accepted draws and total draws,we have the acceptance rate. As Gamerman and Lopes (2006) and Herbst and Schorfheide(2015) mention, there is no consensus on the literature for the optimal acceptance rate,suggesting, as a generic rule, an acceptance rate around 24 to 44 percent, which justifies thetarget set on the tuning function.

To obtain the estimate of the likelihood that will determine acceptance probability, weuse a particle filter with a swarm of 40,000 particles. There is no rule to guide our choicefor the number of particles, but the researcher faces a clear trade-off between accuracy andestimation time. In our case, moving from 20,000 to 40,000 particles, the filter takes 73milliseconds more to evaluate the likelihood of a simulated database with 100 observations(93 and 166 milliseconds, respectively), which means that for a 500,000 draws estimationthe procedure can takes roughly extra 7 hours. On the other hand, the higher the numberof particles, the lower will be the log-likelihood variance, which is an important accuracysignaling, since this is a simulation Monte Carlo procedure.

Taking the staking representation of all T observation of the observable variable as xTand given that the state-space representation has a Markov structure, the likelihood functioncan factorized as a function:

p(xT ; θ) =T∏t=1

p(xt|xt−1; θ),

which depends on the sequence {p(σt|xt−1; θ)}Tt=1 that we cannot characterize analytically.The particle filter substitutes the density p(σt|xt−1; θ) by an empirical draw from it, repre-senting the distribution of the hidden state vector σt conditional on time t information xtthrough a swarm ofM particles simulations {σit|t−1,W it }Mi=1, where σit|t−1 is a draw at momentt condition in t− 1 and W it is the weight of each particle based on their success in predictingthe time t observation, measured by p(xt|σit, θ). A well known problem in the literature is that{σit|t−1,W it }Mi=1 suffers from particle degeneracy, in the sense that, as de number of iterationsincreases all the probability mass will be allocated to one particle. To mitigate this problem,we resampled the particles at the end of each step. Thus, a new population of particles arereplicated from the existing population in proportion th their importance wights.19

19See Creal (2012) for a more detailed discussion.

29

C Equilibrium ConditionsThe stochastic discount factor for each agent is:

mt+1 =∂Vt/∂ct+1∂Vt/∂ct

=Uc,t+1Uc,t

Uψt

Uψt+1β

V ψ−γt+1(EtV 1−γt+1

)ψ−γ1−γ

,

which can be rewritten considering the FOC w.r.t. ct,

mt+1 =λt+1λt

βV ψ−γt+1(

EtV 1−γt+1)ψ−γ

1−γ. (C.1)

where λt is the Lagrangian multiplier associated with the budget constraint.Focusing on a symmetric equilibrium, the first-order conditions of the saver problem of

maximizing expected utility with respect to ct, bt, xt, kt, ut and wj,t are

λt = (1− β)(1− ψ)U−ψt Uc,t, (C.2)1

Rt= Et

{mt+1

(1− τ kt+1

Πt+1

)}(C.3)

1 =

Et

[mt+1qt+1κ

(xt+1xt− zs

)(xt+1xt

)2]

+ qt

[1− κ

2

(xtxt−1

− zs)2− κ

(xtxt−1

− zs)

xtxt−1

](C.4)qt = Et

{mt+1

[qt+1(1− δ(ut+1)) + (1− τ kt+1)ut+1rkt+1

]}(C.5)

qt =(1− τ kt )rktδ′(ut)

(C.6)

φwyt

(wtwt−1

− zs)

wtwt−1

=

φwEt{mt+1yt+1

(wt+1wt− zs

)wt+1wt

}+

1− ηη

ct1− ht

εwht + (1− εw)(1− τwt )wtht,(C.7)

where qt is the multiplier associated with the investment adjustment constraint. The FOC ofthe HtM agent will be equivalent to Equations C.2 and C.7, since she does not have accessto the bond market and cannot invest on capital.

Equilibrium conditions associated to firm’s problem will be marginal cost and FOC forfactor inputs,

mct =

(wt

zt(1− α)

)1−α(rktα

)α(C.8)

kt−1ht

=αwt

rkt (1− α), (C.9)

30

which are associated with the first stage of the problem, and the expended Phillips Curve isgiven by

φpΠt(Πt − Πs) =(1− εp) + εpmct +φpεp

2(Πt − Πs)2

+ φpEt{mt+1