Income Inequality, Financial Crises, and Monetary Policy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Income Inequality, Financial Crises, and Monetary Policy Isabel Cair´o and Jae Sim 2018-048 Please cite this paper as: Cair´o, Isabel, andJaeSim(2018). “IncomeInequality, FinancialCrises, andMonetaryPolicy,” Finance and Economics Discussion Series 2018-048. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2018.048. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Income Inequality, Financial Crises, and Monetary Policy∗ Isabel Cair´o† Jae Sim‡ May 2018 Abstract Weconstructageneralequilibriummodelinwhichincomeinequalityresultsininsufficientaggregate demand,deflationpressure,andexcessivecreditgrowthbyallocatingincometoagentsfeaturinglow marginal propensity to consume, and if excessive, can lead to an endogenous financial crisis. This economygeneratesdistributionsforequilibriumpricesandquantitiesthatarehighlyskewedtothe downside due to financial crises and the liquidity trap. Consequently, symmetric monetary policy rules designed to minimize fluctuations around fixed means become inefficient. A simultaneous reductionininflationvolatilityandmeanunemploymentrateisfeasiblewhenanasymmetricpolicy rule is adopted. JEL Classification: E32, E44, E52, G01 Keywords: income inequality, credit, financial crises, monetary policy ∗The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. We thank Adrien Auclert, Robert Barsky, Larry Christiano, James Costain,JordiGal´ı,WouterdenHaan,MichaelKumhof,AlbertoMartin,PascalPaul,andFedericoRavennaforvaluable comments and suggestions. We are also grateful to colleagues and to seminar and conference participants of USC Dornsife, Annual Inflation Targeting Conference at the Banco Central Do Brasil, Empirical Macro Workshop at the University of Ghent, Danmarks Nationalbank, the CEPR-ESSIM Conference at the Banco de Espan˜a, the European CentralBank/BankofCanadaJointAnnualConference,theFederalReserveBankofBoston/CEBRAJointConference, theSocietyforEconomicDynamics,theSocietyforComputationalEconomics,theEuropeanEconomicAssociation,the EuropeanWinterMeetingoftheEconometricSociety,theSpanishEconomicAssociation,theBoardofGovernorsofthe Federal Reserve System, the Federal Reserve Bank of Kansas City, and the Federal Reserve Bank of Chicago. †Board of Governors of the Federal Reserve System. Email: isabel.cairo@frb.gov ‡Board of Governors of the Federal Reserve System. Email: jae.w.sim@frb.gov

“It’s revealing that in public debates, advocates for workers...have tended to favor the continuationofeasymoney,whilethetypicalop-edabouttheadverseeffectsofeasymoneyonthedistribution of income and wealth is written by a hedge fund manager, banker, or right-wing politician...That political alignment—workers’ groups in support of easy money, financiers in favor of higher interest rates—is of course the historical pattern in the United States, going back to William Jennings Bryan and beyond.” – Bernanke (2017) 1 Introduction ThedistributionofincomeintheUnitedStateshasbecomemoreunequaloverthelastseveraldecades, reaching unprecedented levels as documented by Atkinson et al. (2011). While macroeconomic implications of inequality have recently become the center of discussions by policymakers (Yellen, 2014; Bernanke, 2017), the academic literature on the link among inequality, the macroeconomy, and monetarypolicyisstillinitsinfancy. Thispapercontributestothisliteraturebyexploringapossibilitythat income inequality may be at the root of many macroeconomic problems that the U.S. economy could be facing such as secular stagnation, deflation pressure, excess credit growth, and financial crises. Using a novel theoretical model, we show that an economy suffering from such problems may be subject to disproportionately large downside risks, making traditional monetary policy framework aiming to minimize the volatilities of endogenous quantities and prices around fixed means inefficient. One of the main contributions of this paper is to uncover robust monetary policy rules that can correct the biases of the means due to the left-skewed distributions and improve macroeconomic outcomes. Income inequality may play three essential roles in the macroeconomy, each of which will be an important building block in the construction of our theoretical model. First, in an economy that runs below its production capacity, income inequality may directly affect aggregate demand if there exists substantial heterogeneity of marginal propensity to consume (MPC) and MPCs are negatively correlated with income levels.1 According to Jappelli and Pistaferri (2014), the average MPC of the most affluent income group is substantially lower than the MPCs of lower-income groups. In fact, half of the most affluent income group identified by Jappelli and Pistaferri (2014) has the MPC out of transitoryincomeequaltozero. Iftheeconomyallocatesagrowingshareofnationalincometoagroup with the lowest MPC, it can produce insufficient aggregate demand.2 In this regard, Summers (2015), in his effort to revive the underconsumption theory of Hansen (1939), points to income inequality as one of the most important factors driving secular stagnation. The concern for the link between income inequality and underconsumption dates back to almost a century ago: “...society was so framed as to throw a great part of the increased income into the control of 1JohnMaynardKeynesconsideredincomedistributionasacentralelementofhistheoryofeffectivedemand: “Moreover, each level of effective demand will correspond to a given distribution of income.”(Keynes,1936)because“People’s propensity to spend (as I call it) is influenced by many factors such as the distribution of income, their normal attitude to the future and - tho probably in a minor degree - by the rate of interest.” (Keynes, 1937) 2ThenegativecorrelationbetweenMPCsandincomelevels(orliquidwealthholdings)anditsimplicationforaggregate demandandstabilizationpoliciesarebroadlysupportedbyagrowingbodyofempiricalliterature: Blundelletal.(2008); Parker et al. (2013); Broda and Parker (2014); Kaplan and Violante (2014); Carroll et al. (2017); and Auclert (2017). 1

Figure 1: Income Inequality and Credit-to-GDP Ratio 100 90 80 70 60 50 40 30 1971 1984 1998 2012 )tnecrep ni( PDG-ot-tiderC 24 22 20 18 16 14 12 10 8 )tnecrep ni( %1 pot erahs emocnI Credit-to-GDP (left axis) Income share top 1% (right axis) Sources: Household credit-to-GDP ratio is from the Financial Accounts of the United States, Federal Reserve System. The income share of top 1 percent is from the 2016 update of Piketty and Saez (2003). the class least likely to consume it. The new rich...preferred the power which investment gave them to the pleasures of immediate consumption...And so the cake increased; but to what end was not clearly contemplated... the virtue of the cake was that it was never to be consumed, either by you nor by your children after you.” – Keynes (1919) Second, another important macroeconomic consequence of income inequality is the deflation pressure. In particular, the labor income share, which is empirically related to income inequality, has been decliningformorethanthreedecadesforseveralreasonsstudiedbyElsbyetal.(2013), Karabarbounis and Neiman (2013), Solow (2015), and Koh et al. (2016). Why does that matter for inflation? The labor income share represents a good measure of real marginal costs and accurately captures inflation dynamics, according to the New Keynesian literature such as Gal´ı and Gertler (1999), Gal´ı et al. (2001), Woodford (2001), and Sbordone (2002). Thus, if the decline of the labor share continues as real wage growth fails to catch up with labor productivity growth, it will be much more difficult for a central bank to achieve a certain inflation target. Third, andperhapsmostimportantly, incomeinequalityplaysacrucialrolefortheoverallstability ofthefinancialsystem. Whenagrowingshareofnationalincomeisallocatedtoagroupwiththelowest MPC, unused income has to be stored in financial claims. Those claims may represent investment in real assets such as physical capital or borrowing of lower-income groups. In this latter case, a link between income inequality and excessive credit expansion that endangers the soundness of financial systemmightarise. Indeed, Figure1showsapositivecorrelationbetweenincomeinequality, measured by the income share of the top 1 percent earners based on data collected by Piketty and Saez (2003), and the household sector credit-to-GDP ratio. During the 1960s and 1970s, both time series moved more or less sideways. Since the early 1980s, however, both series continued to rise over more than 2

three decades. In 2007, on the eve of the Global Financial Crisis, the credit-to-GDP ratio reached an unprecedented level and the top 1 percent income share reached 24 percent, a level unseen since 1928, which was also on the eve of the Great Depression. Many researchers view the credit growth in the late 1990s and early 2000s as “excessive” and responsible for the financial vulnerability that ended up in the Global Financial Crisis. For example, Drehmann et al. (2010) and Drehmann et al. (2011) point out that excessive credit growth is a highqualityindicatorforthelikelihoodoffinancialinstability. Jord`aetal.(2011)andSchularickandTaylor (2012)establishedastatisticallinkbetweencreditgrowthandtheprobabilityoffinancialcrises. Thus, if excessive credit growth is a good predictor of financial crises, and if income inequality is the main driver of excessive credit growth, income inequality would then predict financial crises. Indeed, Paul (2017), in an empirical study using a macro-financial database of 17 advanced economies over the 1870–2013 period, finds that income inequality outperforms credit in the predictive power of crises. In this paper, we build a macroeconomic model in which the three aforementioned channels of inequality-macroeconomy nexus play important roles in determining aggregate demand and financial vulnerability. To create a meaningful degree of income inequality, we follow the Kaleckian tradition in which there are two agents, shareholders and workers, with their income shares determined by bargaining power, not by their marginal contribution to production. Weassignso-calledWeberian preferencestotheshareholderssuchthattheyearndirectutilityfrom accumulating financial assets. The Weberian preferences, which we adopt from Kumhof et al. (2015) (KRW henceforth), play two crucial roles in creating links among income inequality, deflation pressure and financial instability. First, the Weberian preferences lower the MPC of the shareholders such that any innovations that make income distribution more favorable to them lead to underconsumption and deflation pressure. Second, the Weberian preferences contribute to financial instability by inducing shareholders to over-accumulate credit to a point where borrowers have strong incentives to partially renege on their debt obligation. As in KRW, a financial crisis erupts endogenously when the leverage of the borrowers reaches unsustainable levels. While the endogenous financial crisis mechanism is borrowed from KRW, our model economy is fundamentally different from theirs. First, KRW is an endowment economy in which income distribution is determined by exogenous forces. In contrast, our model is a production economy whose income distribution is determined by bargaining powers of the two agents in sharing the production rents. Second, more importantly, KRW is a real business cycle economy, and hence Say’s Law holds: supply always creates its demand regardless of the degree of income inequality. In contrast, our economy features nominal rigidities, which allow for a link between income inequality and aggregate demand. Finally, the presence of nominal rigidities permits monetary policy intervention and the size of crises becomes endogenous. This feature enables us to study the dilemma of the monetary authority whose stabilization function is severely distorted during crises by the zero lower bound (ZLB) constraint. The distributions of equilibrium prices and quantities in such an economy are highly skewed to the downside owing to the two important sources of nonlinearity: occasional eruptions of financial crises and the liquidity trap. These two sources of nonlinearity work only to the downside, making 3

the unconditional means of inflation and output deviate to the downside from the levels that would prevail in the absence of crises and the ZLB constraint. This generates first order welfare losses, the magnitude of which we show is an increasing function of the strength of the Weberian preferences. In line with the consensus widely held among monetary economists we find that appointing Rogoff (1985)’s conservative central banker that “places a large, but finite, weight on inflation” for such an economy may maximize the welfare of a society if the central banker’s policy choice is not subject to the ZLB constraint. One benefit of a conservative central banker is that she provides a decisive monetary stimulus during crises. However, in the presence of the ZLB constraint, we find that such a conservative central banker may minimize the inflation volatility only at the costs of increasing the probability of financial crises and the deflation bias, and shifting the distributions of inflation and output more to the downside. ThetwosidesofRogoff(1985)’sconservativecentralbankerhighlighttheinefficiencyofasymmetric monetary policy rule in the sense that a simultaneous reduction in inflation volatility and mean unemployment rate is feasible when an asymmetric policy rule is adopted. In this paper, we find that an asymmetric policy rule that prescribes a lenient response against inflation during normal times, but provides decisive accommodation during financial crises by temporarily yet persistently raising the inflation target can correct the deflation bias and improve macroeconomic outcomes. We also find that such an asymmetric policy rule can bring large welfare gains when implemented in the form of price-level targeting instead of inflation targeting. We emphasize that monetary policy cannot eliminate income inequality. However, we find that optimal monetary policy breaks the link between income inequality and aggregate demand, which exists only in the presence of nominal rigidities under a suboptimal monetary policy rule. To the extent that optimal monetary policy achieves this goal, the economy should behave as if Say’s Law applied. While income inequality still creates a tendency of underconsumption, optimal monetary policy offsets the fall in aggregate demand by sufficiently lowering the real interest rate to recover the aggregatedemandat thepotentiallevel ofoutput. Whether ornotactualmonetary policy canachieve this depends on the frequency of the binding ZLB constraint. Finally, we find that the optimized monetary policy rules for a loss function that assigns a large weight to the reduction in unemployment improves the welfare of workers only at the cost of bringing welfare losses to shareholders. Overall, these findings suggest that monetary policy can have distributional consequences, in line with the empirical results of Coibion et al. (2017). The rest of the paper is organized as follows. Section 2 provides a brief summary of related literature. Section 3 develops the model, which is then calibrated in Section 4. Section 5 shows the model dynamics using impulse response functions, highlighting the role of financial crises and the ZLB constraint. Section 6 discusses our main findings regarding the linkage between income inequality, aggregate demand, and financial stability. Section 7 analyzes the relationship between monetary policy and the macroeconomy using alternative monetary policy rules. Section 8 concludes. A complete description of the non-stochastic steady state, the solution method, the complete system of equations and additional results are described in Appendix A, B, C, and D, respectively. 4

2 Related Literature Our paper contributes to three strands of literature. First, it contributes to the theoretical literature thatformalizesthelinkbetweenincomeinequalityandaggregatedemand. AuclertandRognlie(2016), Kaplan et al. (2018) and McKay and Reis (2016) construct models that feature rich heterogeneity a la Bewley-Huggett-Aiyagari and nominal rigidities. Auclert and Rognlie (2016) show that a permanent increase in income inequality can lead to a permanent Keynesian recession. Kaplan et al. (2018) show that monetary policy transmission channel can be fundamentally different for the rich and the poor. In turn, McKay and Reis (2016) and Mitman et al. (2017) analyze the large stabilization role of fiscal and social insurance policies in an economy suffering from insufficient aggregate demand due to income inequality. While our modeling of income inequality is much more stylized compared with this literature, our stylized model allow us to study the link among income inequality and financial vulnerability in the sense of endogenous financial crises and the policy dilemma facing a monetary authority trying to secure price stability and financial stability under the ZLB constraint, which are absent in this literature. Finally, none of this literature exploits the bargaining power between firms and workers in a search and matching framework as a driver of income inequality.3 Second, our paper contributes to the theoretical literature that analyzes the causes and consequencesofasymmetricdistributionsofequilibriumpricesandquantities. Inparticular,ourtwosources of nonlinearity are able to generate what Adrian et al. (2016) and Adrian and Duarte (2016) call “vulnerablegrowth”, i.e., highlyleft-skeweddistributionofGDPgrowthcorrelatedwithfinancialleverage. Our contribution is to generate such vulnerable growth in a completely structural way. The nature of asymmetric distributions in our framework is comparable to Dupraz et al. (2017), who generate an asymmetric distribution of the unemployment rate by combining downward nominal wage rigidity and search frictions but abstract from income inequality and financial crises. Relatedly, Kocherlakota (2000) and Jensen et al. (2017) show that occasionally binding financial constraints in a highly leveraged economy can be an important source of business cycle asymmetries. While our paper also highlights the role of credit in generating asymmetric distributions, Kocherlakota (2000) and Jensen et al. (2017) abstract from nominal rigidities, which prevent the study of monetary policy. Importantly, to the best of our knowledge, the interaction between financial crises and the binding ZLB constraint in our paper is only found in Guerrieri and Lorenzoni (2017). In particular, they model an heterogeneous-agent incomplete-market economy in which deleveraging after a shock to borrowing capacity forces the real interest rate to fall to zero even with flexible prices. However, credit crunch is exogenous in Guerrieri and Lorenzoni (2017), whereas financial crises arise endogenously in our paper. Finally, our paper contributes to the theoretical literature that analyzes the distributional consequences of monetary policy. In particular, our results are in line with what Gornemann et al. (2016) find in assessing the welfare implications for “Main Street” and “Wall Street”of increasing the mone- 3Krueger et al. (2016) is unique in this literature. They analyze the role of endogenous changes in income/wealth inequalityasanamplification/propagationchannelinanequilibriumbusinesscycleframeworkwithoutnominalrigidities. What is remarkable in their work is that the model and the calibration strategy allow Krueger et al. (2016) to match the stylized fact in the data that the bottom 40% of households hold no net worth. 5

tary policy reaction to the unemployment gap in a New Keynesian model with heterogeneous agents in incomplete market. Even though our modeling of income distribution is much more stylized than Gornemann et al. (2016), our framework allows us to study optimal monetary policy when the economy features occasional financial crises and is subject to the ZLB constraint. Our paper is also related to Auclert (2017) that analyzes the different redistribution channels through which monetary policy affects macroeconomic aggregates due to heterogeneity in MPCs. In particular, our model economy features both the earnings heterogeneity channel and the Fisher channel of monetary policy described in Auclert (2017), but in a model economy that features financial crises and ZLB constraint. 3 The Model The model features two groups of agents: shareholders, denoted by superscript K, and workers, denoted by superscript W. Each group contains a continuum of agents and forms a large family that shares the budget and consumption. The population shares of shareholders and workers are denoted by χ and 1−χ, respectively. We assume a segmented asset market structure such that only shareholders own production firms and accumulate physical capital. Shareholders also accumulate private bonds and government bonds. Workers do not participate in capital markets and the only instrument available to them to smooth consumption is borrowing from the private bond market. In equilibrium, shareholders lend money to workers. Monetary policy determines the interest rate on government bonds. While only shareholders accumulate government bonds, monetary policy also affects workers’ consumption profiles due to general equilibrium effects. 3.1 Workers Preferences of workers are specified as: UW = E (cid:88) ∞ (βW)t (cid:40) (cW t −scW t−1 )1−1/σc (cid:41) , (1) t t 1−1/σ c t=0 where βW is the time discount factor, cW denotes per capita consumption level of workers, s is the t degree of external habit formation, and σ is the elasticity of intertemporal substitution. Per capita c composite consumption is given by cW = (cid:104) (cid:82)1 cW(i)1−1/γdi (cid:105)1/(1−1/γ) , where γ is the elasticity of subt 0 t stitution between different good varieties denoted by i. Workers earn wage incomes by providing labor when employed and search for new jobs and earn unemployment benefits when unemployed. We assume that the only financial instrument that workers can use to smooth consumption is issuance of defaultable discount bonds. We denote per capita private bond issuance of workers by b and the price of the discount bond by qB. If borrowers do t t not default, the bond delivers a real return of E [1/π ] to lenders in the next period, where π is t t+1 t the gross inflation rate. If borrowers default, lenders get back only (1−h)E [1/π ], where h is the t t+1 6

haircut associated with the default. Thus, the actual payment can be denoted as: b l = (1−hδB) t−1 , (2) t t π t where δB ∈ {0,1} is the default indicator that takes 1 upon default and 0 otherwise. t Default involves pecuniary and non-pecuniary costs, the latter in terms of direct loss of borrowers’ utility. The size of the pecuniary default cost is given by a fraction ν of aggregate output y that t t follows the following process: ν = ρ ν +γ δB, (3) t ν t−1 ν t where the impact effect of a default is given by γ and ρ governs the decay rate in the absence of ν ν further defaults. Default occurs when the borrowers’ utility gain from defaulting is greater than the utilitycostofdefaultχ , whichfollowsaniidprocesswithitscumulativedistributionfunctiondenoted t by Ξ(·). Formally, default occurs when χ < UD −UN, where UD and UN denote the workers’ values t t t t t of default and non-default, which will be given formal definitions below. The probability of default is then simply given by: pδ ≡ prob(δB = 1) = Ξ(UD −UN). (4) t t t t The distribution of the utility cost of default χ is assumed to follow a modified logistic distribution t as in KRW:  (cid:37)   if χ < ∞  Ξ(χ t ) = 1+exp(−ςχ t ) t , (5)  1 if χ = ∞  t where 0 < (cid:37) < 1. The parameters (cid:37), ς, γ and ρ are calibrated to match the empirical evidence on ν ν financial crises. Per capita budget constraint for workers can then be expressed as: 1 (cid:20)(cid:90) 1 (cid:21) cW = qBb −l + w (i)n (i)di+(1−χ−n )bU −ν y , (6) t t t t 1−χ t t t t t 0 (cid:82)1 wherew (i)n (i)iswageincomeoftheworkersemployedbyfirmi,n = n (i)diistotalemployment, t t t 0 t and bU are unemployment benefits. Denoting the shadow value of workers’ budget constraint by ΛW t and using the probability of default, the first-order conditions (FOCs) for workers can be expressed as: ΛW = (cW −scW )−1/σc (7) t t t−1 and (cid:20)ΛW 1 (cid:21) qB = βWE t+1(1−hpδ ) . (8) t t ΛW t+1 π t t+1 For later use, we define the stochastic discounting factor of workers as mW ≡ (βW)s−tΛW/ΛW. t,s s t Regardless of the default decision, the consumption level of the workers is determined by equation (6) at any point in time. However, we need to construct a few auxiliary devices to analyze the probability of default, which relies on hypothetical values of default and non-default. To construct 7

the borrowers’ hypothetical values of default and non-default, we define the consumption levels under default and non-default, cD and cN as: t t b 1 cD = qBb −(1−h) t−1 + [w n +(1−χ−n )bU −(ρ ν +γ )y ] (9) t t t π 1−χ t t t ν t−1 ν t t and b 1 cN = qBb − t−1 + [w n +(1−χ−n )bU −ρ ν y ]. (10) t t t π 1−χ t t t ν t−1 t t Denoting the continuation values conditioned upon default and non-default by VD and VN, ret t spectively, we define the value of default UD and the value of non-default UN as the sum of one period t t utility and the continuation value in each case: (cD −scW )1−1/σc UD = t t−1 +VD (11) t 1−1/σ t c and (cN −scW )1−1/σc UN = t t−1 +VN. (12) t 1−1/σ t c To construct the conditional continuation values, we define cNN, cND, cDN and cDD as the cont+1 t+1 t+1 t+1 sumptionvaluetomorrowinthefollowingfourcases. cNN, consumptionlevelinthecaseofnon-default t+1 tomorrow after non-default today is given by: b 1 cNN = qB b − t + [w n +(1−χ−n )bU −ρ ρ ν y ]. (13) t+1 t+1 t+1 π 1−χ t+1 t+1 t+1 ν ν t−1 t+1 t+1 cND, consumption level in the case of default tomorrow after non-default today is: t+1 b 1 cND = qB b −(1−h) t + [w n +(1−χ−n )bU −(ρ ρ ν +γ )y ]. (14) t+1 t+1 t+1 π 1−χ t+1 t+1 t+1 ν ν t−1 ν t+1 t+1 cDN, consumption level in the case of non-default tomorrow after today’s default is: t+1 b 1 cDN = qB b − t + [w n +(1−χ−n )bU −ρ (ρ ν +γ )y ]. (15) t+1 t+1 t+1 π 1−χ t+1 t+1 t+1 ν ν t−1 ν t+1 t+1 Finally, cDD, consumption level in the case of default tomorrow after today’s default is given by: t+1 b 1 cDD = qB b −(1−h) t + {w n +(1−χ−n )bU−[ρ (ρ ν +γ )+γ ]y }. (16) t+1 t+1 t+1 π 1−χ t+1 t+1 t+1 ν ν t−1 ν ν t+1 t+1 The continuation value under default today is then given by a weighted average of the value under default and the value under non-default tomorrow such that: (cid:34) (cid:32) (cid:33) (cid:32) (cid:33)(cid:35) (cDD −scD)1−1/σc (cDN −cD)1−1/σc VD = βWE pδ t+1 t +VD +(1−pδ ) t+1 t +VN . (17) t t t+1 1−1/σ t+1 t+1 1−1/σ t+1 c c 8

Thecontinuationvalueundernon-defaulttodaycanbeconstructedinasymmetricwaysuchthat:4 (cid:34) (cid:32) (cid:33) (cid:32) (cid:33)(cid:35) (cND −scN)1−1/σc (cNN −scN)1−1/σc VN = βWE pδ t+1 t +VD +(1−pδ ) t+1 t +VN . (18) t t t+1 1−1/σ t+1 t+1 1−1/σ t+1 c c 3.2 Shareholders We denote per capita consumption level of shareholders by cK, per capita government bond holdt ings of shareholders by bG, and per capita holdings of private bonds of shareholders by b (1−χ)/χ. t t Shareholders maximize the following intertemporal utility function: UK = E (cid:88) ∞ (βK)t (cid:40) (cK t −scK t−1 )1−1/σc +ψB [1+b t (1−χ)/χ]1−1/σ b +ψG (1+bG t )1−1/σg (cid:41) , (19) t t 1−1/σ 1−1/σ 1−1/σ c b g t=0 where βK is the discount factor of shareholders and the composite consumption good is given by cK = (cid:104) (cid:82)1 cK(i)1−1/γdi (cid:105)1/(1−1/γ) . The parameters ψB and ψG are strictly positive weights given to the t 0 t utility from asset holdings, while σ and σ determine how fast the marginal utility of financial asset b g holdings decline. These preferences exhibit “the spirit of capitalism” of Weber (2013) in that financial assets are not simply means to utility maximization but become direct goal of utility maximization. ThesepreferencesgeneralizeKRW’spreferencesbyintroducingexternalconsumptionhabitsandutility from holding government bonds in addition to private bonds. Note that government bonds are needed to generate a transmission channel for monetary policy. Per capita budget constraint for shareholders can then be expressed as: bG bG 1−χ 1 cK = t−1 − t +(l −qBb ) − {qK[k −(1−δ)k ]+r k +ΠY +ΠK −T +ν y }, t π 1+i t t t χ χ t t t−1 t t−1 t t t t t t t (20) where i is the nominal interest rate controlled by the monetary authority, r is the rental rate of t t capital, k /χ is per capita capital stock, δ is the depreciation rate of capital, qK is the relative price t t of capital, ΠY/χ is the per capita profit of intermediate-goods firms, ΠK/χ is the per capita profits of t t investment-goods firms, and T /χ is the per capita lump sum tax. t As indicated by the last term in the shareholders’ budget constraint, we assume that the pecuniary default cost of workers is transferred to shareholders in lump-sum fashion. In this sense, the haircut is notcompletelylost. Thedefaultprovidesalargeone-timereleaseintheworkers’budget,butgenerates a persistent cost. Hence the transfer of default costs can be viewed as debt restructuring.5 Denoting the shadow value of shareholders’ budget constraint by ΛK, the FOCs of shareholders t can be expressed as: ΛK = (cK −scK )−1/σc, (21) t t t−1 4The forward iteration of the recursive formulation given by equations (11)∼(18) can cover the entire event tree in the future. 5The expected present value of the lump-sum transfer of the pecuniary default costs amounts to 2/3 of the haircut, implying 66 percent recovery rate in our calibration. 9

(cid:20)ΛK 1 (cid:21) ψB (cid:20) (cid:18) 1−χ (cid:19)(cid:21)−1/σ b qB = βKE t+1(1−hpδ ) + 1+b , (22) t t ΛK t+1 π ΛK t χ t t+1 t (cid:20)ΛK (cid:18)r +(1−δ)qK (cid:19)(cid:21) 1 = βKE t+1 t+1 t+1 (23) t ΛK qKφ t t t and 1 (cid:20)ΛK φ (cid:21) ψG = βKE t+1 t + (1+bG)−1/σg, (24) 1+i t ΛK π ΛK t t t t+1 t where we assume that the economy is subject to Smets and Wouters (2007)’s risk premium shock to create aggregate demand disturbances, which is denoted by φ and follows an AR(1) process:6 t logφ = ρ logφ +(cid:15) , (cid:15) ∼ N(−0.5σ2,σ2). (25) t φ t−1 φ,t φ,t φ φ Note that ψB > 0 and ψG > 0 create liquidity premium for financial assets as opposed to real asset. Shareholders earn non-pecuniary benefits from holding financial claims in addition to pecuniary returns, which makes them to accept lower interest rate and provide more credit to financial markets. For later use, we define the stochastic discounting factor of shareholders as mK ≡ (βK)s−tΛK/ΛK. t,s s t 3.3 Symmetric Equilibrium Note that cD and cN defined by equations (9) and (10) differ from each other only in aspects related t t to default, i.e., l = (1−h)b /π and ν = ρ ν +γ in one case, and l = b /π and ν = ρ ν t t−1 t t ν t−1 ν t t−1 t t ν t−1 in the other case. Otherwise all elements of the budget constraint are symmetric across the two cases. This does not imply that macroeconomic variables, such as the market price of bond qB, total wage t income w n , and aggregate output y , are not affected by the crisis. As will be shown, the dynamics t t t of these macroeconomic variables are influenced by the crisis in an important way. However, individuals take the macroeconomic variables as given while making their individual default decision. The bond market is characterized as a competitive equilibrium with a continuum of agentsand“theactionsofasingleindividualarenegligible”(Aumann,1975). Inoursymmetricdefault or non-default equilibrium, every individual makes an identical choice, believing that their actions will notaffectmacroeconomicoutcomes. However,witheveryonemakingthesamechoice,defaultdecisions impact the economy in equilibrium. It is for the same reason that neither the borrower’s efficiency condition nor the lender’s efficiency condition (equations (8) and (22), respectively) incorporate the effect of increasing debt on the probability of default or the price of bond. In other words, both agents behave as if ∂pδ /∂b = ∂qB/∂b = 0 because they view their individual actions inconsequential for t+1 t t t the competitive equilibrium in debt market: they are price takers. In our model economy, δB ∈ {0,1}, has both endogenous and exogenous natures. δB is endogenous t t 6The risk premium shock appears in the denominator of Lucas tree equation (23) such that the expected real return falls. However,thesameriskpremiumshockappearsinthenumeratoroftheconsumptionEulerequation(24)suchthat theshockisequivalentwithmonetarypolicyshockimplementedby“nature”andnotbythemonetaryauthority. Fisher (2015) provides a structural interpretation of such a shock as “money demand shock.” 10

inthesensethattheprobabilityofδB = 1isendogenouslydetermined. However, giventheprobability t of default, whether or not the default occurs is stochastic. In this latter sense, δB can be viewed as t an exogenous variable. In simulating the model economy, we employ a ‘guess and verify’ strategy. We first simulate the economy under the guess that default does not occur today, i.e., we set δB = 0 and t simulate the model economy. The probability of default is computed as a result of this simulation. Equations (9)∼(12) are the devices that allow us to compute the probability of default under the assumption δB = 0. We then verify if our guess is valid. We take a random draw of the utility cost of t default χ . If the utility gain UD −UN is less than the random draw, we conclude that the guess was t t t correct. If not, i.e., if UD −UN > χ , we conclude that the default indeed happens today. We then t t t set the exogenous variable δB = 1 and simulate the economy again (see Appendix B for more details). t 3.4 Labor Market Thereexistsacontinuumoffirmsindexedbyi ∈ [0,1]andownedbyshareholders. Firmspostvacancies to hire new workers at the beginning of each period. The matching process between vacancies and unemployed workers is assumed to be governed by a constant returns to scale matching function: m(v ,u ) = ζv(cid:15)u1−(cid:15), (26) t t t t (cid:82) where v ≡ v (i)di denotes the total measure of vacancies and u denotes the measure of unemployed t t t workers searching for a job at the beginning of period t defined as: u = 1−χ−(1−ρ)n , (27) t t−1 with ρ ∈ (0,1) being the exogenous separation rate. The parameter ζ stands for matching efficiency and 1−(cid:15) is the matching function elasticity with respect to unemployment. The probability for an unemployed worker to meet a vacancy and the probability for a vacancy to meet with an unemployed worker are given by, respectively: m(v ,u ) p(θ ) = t t = ζθ(cid:15), (28) t u t t m(v ,u ) q(θ ) = t t = ζθ(cid:15)−1, (29) t v t t where θ ≡ v /u is labor market tightness. Note that firms consider these flow probabilities as given t t t when deciding optimal employment. 3.5 Firm Problem Each firm i produces a differentiated good, using an identical Cobb-Douglas production function with capital k (i) and labor n (i). The production technology is represented by: t t y (i) = z k (i)αn (i)1−α, (30) t t t−1 t 11

where z is an aggregate productivity shock following an AR(1) process: t logz = ρ logz +(cid:15) , (cid:15) ∼ N(−0.5σ2,σ2). (31) t z t−1 z,t z,t z z Firms make three decisions: a pricing decision subject to infrequent price adjustments, a hiring decision in a frictional labor market, and a capital rental decision. The timing of events for the firm’s problem is summarized as follows. At the end of time t − 1, a fraction ρ of workforce is exogenously separated from the firm. Then, aggregate shocks realize, firms make pricing decisions, which determines production scale. Firms then post vacancies v (i) at a flow vacancy posting cost ξ t per period, which will be filled with probability q(θ ). Then, firms make the capital rental decision, t which is assumed to be frictionless. It is assumed that vacancies posted at the beginning of the period can be filled in the same period before production takes place, and that recently separated workers can search and find a job in the same period. Thus, the law of motion for the firm’s workforce is given by: n (i) = (1−ρ)n (i)+q(θ )v (i). (32) t t−1 t t Afterthematchingprocessiscomplete,thewageisdeterminedthroughNashwagebargaining. Finally, production takes place, and wages, capital rents and dividends are paid. 3.5.1 Cost Minimization The firm’s cost minimization problem can be separated from the optimal pricing problem. A firm i minimizes its production costs subject to equations (30) and (32), formalized by the Lagrangian: L = E (cid:88) ∞ mK (cid:34) w (i)n (i)+ ϑ (cid:18) w s (i) −1 (cid:19)2 w (i)n (i)+r k (i)+ξv (i) (cid:35) (33) t t,s s s 2 w (i) s−1 s s s−1 s s−1 s=t ∞ (cid:88) +E mK J (i){n (i)−[(1−ρ)n (i)+q(θ )v (i)]} t t,s s s s−1 s s s=t ∞ (cid:88) +E mK µ (i){y (i)−z k (i)αn (i)1−α}, t t,s s s s s−1 s s=t where we assume that firms face quadratic adjustment cost of changing the real wage. µ (i) and t J (i) are the shadow values of constraints (30) and (32), respectively. Thus, µ (i) represents the real t t marginal cost of production and J (i) the real marginal value of employment. The associated FOCs t with respect to v (i), n (i), and k (i) are, respectively: t t t ξ = q(θ )J (i), (34) t t y (i) ϑ (cid:18) w (i) (cid:19)2 J (i) = µ (i)(1−α) t −w (i)− t −1 w (i)+(1−ρ)E [mK J (i)] (35) t t n (i) t 2 w (i) t−1 t t,t+1 t+1 t t−1 12

and y (i) t r = µ (i)α . (36) t t k (i) t−1 Equation (34) equalizes the costs of posting a vacancy with the expected benefit of employing a new worker, which is given by equation (35) and equals the gap between the cost reduction of an additional worker and the wage, plus the continuation value of the employment relationship. Given that the capital allocation decision is frictionless, equation (36) equals the real rental rate to the cost reduction due to renting an additional unit of capital. 3.5.2 Wage Bargaining The worker’s surplus at a firm i, denoted by W (i), is given by the sum of the surplus of the contract t wage w (i) over the worker’s outside option value w and the continuation value at firm i: t t W (i) = w (i)−w +(1−ρ)E [mW W (i)]. (37) t t t t t,t+1 t+1 The worker’s outside option w is given by the sum of unemployment benefits and the expected value t of finding a new job next period: (cid:20) (cid:90) 1 v (j) (cid:21) w = bU +(1−ρ)E mW p(θ ) t+1 W (j)dj . (38) t t t,t+1 t+1 v t+1 0 t+1 The matched firm and worker bargain over the wage to maximize the Nash product: w t (i) = argmaxW t (i)η tJ t (i)1−η t, (39) wt(i) where η is the bargaining power of the worker, which is assumed to follow an AR(1) process: t logη = (1−ρ )logη+ρ logη +(cid:15) , (cid:15) ∼ N(−0.5σ2,σ2). (40) t η η t−1 η,t η,t η η The solution to the Nash bargaining problem is given by the surplus sharing rule: ∂W (i) J (i) ∂J (i) t t t 0 = η +(1−η ) , (41) t∂w (i) W (i) t ∂w (i) t t t where ∂W (i)/∂w (i) = 1 ≡ −ΓW and t t t (cid:40) (cid:34) (cid:35)(cid:41) ∂J (i) (cid:18) w (i) (cid:19) ϑ (cid:18) w (i) (cid:19)2 t = −1−ϑ t −1 +(1−ρ)E mK t+1 −1 ≡ ΓJ. ∂w (i) w (i) t t,t+12 w (i) t t t−1 t We now define the generalized workers’ bargaining power Ω as: t η Ω ≡ t . t η +(1−η )ΓJ/ΓW t t t t 13

Note that if there is no real wage rigidity, ΓJ/ΓW = 1 and Ω = η . Using the generalized bargaining t t t t power, the surplus sharing rule (41) can be rewritten as a generalization of the standard Nash-wage bargaining condition: Ω J (i) = (1−Ω )W (i). (42) t t t t Without loss of generality, we focus on a symmetric equilibrium in which the real wage is equalized across all firms, i.e., w (i) = w . The surplus sharing condition (42), the cost minimization conditions t t (34) and (35), the worker’s surplus value function (37), and the worker’s outside option (38) jointly imply the following equilibrium wage:7 y ϑ (cid:18) w (cid:19)2 w = Ω µ (1−α) t +(1−Ω )bU −Ω t −1 w (43) t t t n t t 2 w t−1 t t−1 (cid:20)(cid:18) (cid:19) (cid:21) Ω ξ +(1−ρ)E Ω mK −(1−Ω )mW [1−p(θ )] t+1 . t t t,t+1 t t,t+1 t+1 1−Ω q(θ ) t+1 t+1 3.5.3 Optimal Pricing Each firm i produces a differentiated good and faces an identical isoelastic demand curve given by: (cid:18) P (i) (cid:19)−γ t y (i) = y , (44) t t P t where P (i) is the firm’s price, P is the aggregate price level, and y is the aggregate demand. t t t We assume that firms set prices according to a variant of the formalism proposed by Calvo (1983). In particular, each firm has a constant probability 1−ϕ to reset its price in any given period, which is independent across firms and time. Firms that cannot reset their price in a given period partially index their price to lagged inflation. Thus, the firm’s price in period t is (cid:40) P∗ with probability 1−ϕ P (i) = t t P (i)πε with probability ϕ t−1 t−1 where P∗ is the reset price, π = P /P is the inflation rate, and ε is the degree of indexation. t t t t−1 AfirmthatreoptimizesitspriceinperiodtchoosestheresetpriceP∗ thatmaximizestheexpected t present value of the profits generated while the price remains effective. Formally, P∗ satisfies: t P∗ = argmaxE (cid:88) ∞ ϕs−tmK (cid:32) P t ∗ s (cid:81) −1 πε−µ (cid:33)(cid:32) P t ∗ s (cid:81) −1 πε (cid:33)−γ y , (45) t P t ∗ t s=t t,s P s j=t j s P s j=t j s with 0 < ε < 1 and (cid:81)s−1πε = 1 for s ≤ t. j=t j Thefirstorderconditionto(45)isexpressedastheoptimalresetpriceinflationratep ≡ P∗/P : 0,t t t−1 p = PN/PD, (46) 0,t t t 7If we assume a single representative agent such that mK = mW = m and further assume a constant t,t+1 t,t+1 t,t+1 bargaining power and no real wage rigidity, the equilibrium wage is given by the more conventional wage equation: w =ηµ (1−α)y /n +(1−η)bU +η(1−ρ)E [m ξθ ]. t t t t t t,t+1 t+1 14

where PN and PD satisfy the following recursions: t t (cid:110) (cid:111) PN = π (1−ε)γ πεγγµ y +ϕE (cid:2) mK PN (cid:3) , (47) t t t t t t t,t+1 t+1 and (cid:110) (cid:111) PD = π (1−ε)(γ−1) π ε(γ−1) (γ −1)y +ϕE (cid:2) mK PD (cid:3) . (48) t t t t t t,t+1 t+1 CombiningtheaggregatepriceindeximpliedbytheDixit-Stiglitzaggregatorandthepartialindexation assumption, one can summarize the aggregate inflation dynamics as: (cid:104) (cid:105)1/(1−γ) π = (1−ϕ)p1−γ +ϕπ ε(1−γ) . (49) t 0,t t−1 3.6 Closing the Model 3.6.1 Investment-good Industry Weassumethatthereexistsarepresentativefirminacompetitiveinvestment-goodproducingindustry, which maximizes the following profit given a CRS technology: maxE (cid:88) ∞ mK (cid:40) qKx − (cid:34) x + κ (cid:18) x s −1 (cid:19)2 x (cid:35)(cid:41) . xs t t,s s s s 2 x s−1 s−1 s=t x is the investment level at time t and κ is the investment adjustment cost parameter. The first order t condition to the above problem is given by an investment Euler equation: (cid:40) (cid:34) (cid:35)(cid:41) (cid:18) x (cid:19) κ (cid:18) x (cid:19)2 qK = 1+κ t −1 −E mK t+1 −1 . (50) t x t t,t+12 x t−1 t 3.6.2 Government Budget Constraint We assume a balanced budget each period. Government spending is financed by lump sum taxes on shareholders and is composed of two elements: unemployment benefits and interest expenses on government debt. The balanced budget constraint then implies: (cid:18)bG bG (cid:19) T = (1−χ−n )bU +χ t−1 − t . (51) t t π 1+i t t In all computations considered in this paper, it is assumed that the amount of government bond issuance remains constant. This implies that when the monetary authority changes the interest rate, it is the demand for government bonds that should adjust to clear the bond market. 15

3.6.3 Market Clearing We define aggregate consumption as c = (1−χ)cW +χcK. Imposing a balanced budget condition t t t yields the following aggregate resource constraint: ϑ (cid:18) w (cid:19)2 κ (cid:18) x (cid:19)2 t t y = c +x +ξv + −1 w n + −1 x . (52) t t t t t−1 t t−1 2 w 2 x t−1 t−1 Equating equations (30) and (44) and integrating the equality yields: y = ∆−1z kα n1−α, (53) t t t t−1 t where (cid:18) P∗(cid:19)−γ (cid:18)P πε (cid:19)−γ ∆ = (1−ϕ) t +ϕ t−1 t−1 = πγ[(1−ϕ)p−γ +ϕπ−εγ] (54) t P P t 0,t t−1 t t is due to the price dispersion under staggered pricing.8 3.6.4 Monetary Policy In the absence of shocks that affect the output gap and inflation gap in opposite directions, such as price markup shocks, the model economy exhibits the divine coincidence.9 For this reason, the monetary authority is assumed to follow an inflation targeting regime in our baseline economy: (cid:26) (cid:18) πY −π∗(cid:19)(cid:27) i = max 0,i∗+ρ t , (55) t π 4 where i∗ is the steady state interest rate, πY ≡ π π π π is the annual inflation rate, and π∗ is t t t−1 t−2 t−3 the annual inflation target. As shown by equation (55), the ZLB constraint is imposed on the policy interest rate. Thus we can analyze, first, the effects of ZLB on the ability of the central bank to control the probability of financial crises and to stabilize inflation during financial crises; and second, to analyze the effects of the strength of inflation targeting given by ρ on the frequency of financial crises and binding ZLB π constraint. To satisfy the ZLB constraint, we use a combination of current monetary policy shocks and anticipated news shocks. For this purpose, we express the monetary policy rule (55) as: (cid:18) πY −π∗(cid:19) (cid:88) n i = i∗+ρ t + (cid:15) , (56) t π 4 j,t−j j=0 8Since we solve our model around zero trend inflation rate, the price dispersion term does not play any active role. 9We define the divine coincidence in terms of inflation gap and natural output gap – the distance of actual output from the output of flexible economy. This is different from Blanchard and Gal´ı (2007), where they define the welfare relevantoutputgapasthedistanceofactualoutputfromefficientlevelofoutputthatwillprevailwithoutthedistortions due to nominal/real rigidities and market power. For simplicity, we assume that the central bank is concerned about minimizing the output from their natural levels. However, as Blanchard and Gal´ı (2007) showed, the presence of real wage rigidity in our model may break down the divine coincidence when the central bank is concerned about the gap from the efficient level of output. 16

where (cid:15) is the current monetary policy shock and (cid:15) is the j−periods ahead news shock to the 0,t j,t−j policy rate. While both (cid:15) and (cid:15) are called “shocks”, we use them only to satisfy the ZLB 0,t j,t−j constraint, which can be expressed as a combination of i ≥ 0, E [i ] ≥ 0,..., and E [i ] ≥ 0, a t t t+1 t t+n totalofn+1constraints. WeensurethattheZLBconstraintissatisfiednotonlyinthecurrentperiod, but also in agent’s expectations such that economic agents understand the nature of the constraint. For a sufficiently large value of n, E [i ] ≥ 0 can be safely assumed to be non-binding. In each t t+n+1 period, n+1 shocks should be endogenously determined to satisfy the n+1 constraints.10 As an alternative monetary policy rule, we consider the following form of price-level targeting:11 n i = i∗+ρ Π t + (cid:88) (cid:15) , (57) t Π 4 j,t−j j=0 where Π is defined as the cumulative miss in inflation target, i.e.,12: t ∞ (cid:88) Π ≡ (πY −π∗) = Π +πY −π∗. (58) t t−s t−1 t s=0 4 Calibration The model is calibrated at a quarterly frequency. Table 1 summarizes our choices for the structural parameters determining preferences and default, which closely follow the calibration strategy in KRW. Table 2 summarizes the parameters pertaining to the structure of production, labor markets, nominal rigidities, monetary policy and exogenous shock processes. Preferences and Default: We use the same haircut due to default (h = 0.1) and the persistence of the default cost (ρ = 0.650.25) as in KRW. The output loss upon default is set to γ = 0.045, which ν ν is slightly above the KRW value of 0.04. For this choice, we target a cumulative output loss of 26 percent during the nine years following a financial crisis. This corresponds to the observed output loss suffered by the U.S. economy from 2008 to 2016 relative to its potential as estimated by the CBO in 2017.13 Regarding the parameters of the modified logistic distribution of the utility cost of default, we calibrate ς = 18 as in KRW and set (cid:37) = 0.055 to match a quarterly default probability of 1.3 percent, which is consistent with the annual default probability in the data computed by Schularick and Taylor (2012) on the eve of the Great Recession. 10See Appendix B for the solution method. Lind´e et al. (2016) recently used the same methodology. 11We do not consider nominal GDP targeting as an alternative monetary policy rule because there is no shock in the model that moves the output gap and the inflation gap in opposite directions. 12Notethatweusetheannualizedinflationratetodefinethecumulativemissinhittingtheinflationtarget. Ifweuse the quarterly inflation rate, and respecify the monetary policy rate as i = i∗+ρ Π + (cid:80)n (cid:15) this rule becomes t Π t j=0 j,t−j literally a price-level targeting rule. To see this, one could rewrite ∞ (cid:20) (cid:21) Π ≡ (cid:88) (π −π∗)=log P t −π∗+log P t−1 −π∗+...+log P 1 −π∗ =log P t . t t−s P P P P (π∗)t t−1 t−2 0 0 s=0 Since we use the 4-quarter annual rate, the formula becomes slightly different, which creates inconsequential difference. 13Note that this output loss can be considered a conservative estimate, given the significant downward revisions to potential output undertaken by the CBO during the years after the Great Recession. 17

Table 1: Parameters for Preferences and Default Parameter Target/Source Value Haircut KRW h=0.1 Persistence of default cost KRW ρ =0.650.25 ν Impact default cost Output loss during the γ =0.045 ν Great Recession Default cost parameter KRW ς =18 Default cost parameter Empirical default probability (cid:37)=0.055 Population share of shareholders 5 percent χ=0.05 Steady state interest rate Literature i∗=0.005 Utility weight on private bond Debt-to-income ratio of bottom 95 ψB=0.45 percent income earners = 1.5 Utility weight on government bond Same as private bond ψG=0.45 Discount factor of workers Literature βW=0.99 Discount factor of shareholders Income share of top 5 percent βK=0.89 income earners = 0.36 Wealth elasticity of shareholders KRW σ =1.09 b Wealth elasticity of shareholders Annual nominal interest σ =0.328 g rate of 2 percent Elasticity of intertemporal subs. Literature σ =1 c Elasticity of subs. between goods Literature γ =5 The steady state risk-free interest rate is set to 0.5 percent quarterly. In contrast to KRW, in which the risky private bond is the only financial investment, in our model economy the risk-free bond is an important financial investment vehicle for monetary policy transmission. Once we calibrate σ , g equation (24) and the target interest rate pin down the level of per capita government bond holdings. We calibrate the steady state debt-to-income ratio of workers as 150 percent, which is close to the debt-to-income ratio of the bottom 95 percent income earners on the eve of the Great Recession. This choice leads us to set a higher utility weight on private bond holdings of shareholders (ψB = 0.45). For the government bond holdings, we use the same utility weight ψG as with private bonds. We choose a slightly lower value of the discount factor for shareholders than KRW such that there is a greater incentive to invest in financial assets. This allows us to match 150 percent debt-to-income ratio. However, we use the same wealth elasticity as in their analysis and set σ = 1.09. With these b choices, the income share of shareholders is 0.36 in the steady state, a similar value observed on the eve of the Great Recession for the top 5 percent income earners. Also, the marginal propensity to save (MPS) of shareholders in our baseline economy equals 0.332, close to the value of 0.397 used in KRW for the top 5 percent income earners that matches empirical MPS.14 We specify a log utility, σ = 1, and an elasticity of substitution between differentiated goods of c γ = 5 to generate a 20 percent markup in steady state. We set the degree of habit formation equal to s = 0.45, which is close to the mean value of 0.43 from 597 estimates of habit formation in the literature reported by Havranek et al. (2017). Production: The capital share of production equals α = 0.2 to match a labor income share of 0.60 in steady state. This value is somewhat lower than the convention of 0.3 ∼ 0.4. However, a 14TocomputetheMPSoftopearnersinourmodeleconomy,wecomparetheincreaseinsavingsofshareholdersversus the increase in their income that occur in steady state when aggregate productivity rises 1 percent permanently. 18

Table 2: Parameters for Technology, Labor Markets, Nominal Rigidities and Shock Processes Parameter Target/Source Value Capital share of production Labor income share = 0.60 α = 0.2 Investment adjustment cost Relative volatility of investment κ = 12 Habit in consumption Literature s = 0.45 Depreciation rate Literature δ = 0.025 Separation rate CPS ρ = 0.37 Matching efficiency CPS, job finding rate ζ = 0.91 Matching function elasticity Literature (cid:15) = 0.5 Worker’s bargaining power Literature η = 0.5 Unemployment benefit bU/w = 0.83 bU= 0.5 Vacancy posting cost Literature ξ = 0.11 Real wage stickiness S.D. of real comp. 4-quarter growth of ϑ = 50 nonfarm business sector = 1.5 percent Inflation indexation Literature ε = 0.5 Price stickiness S.D. of inflation = 0.64 percent ϕ = 0.935 Taylor rule: Inflation gap Literature ρ = 1.5 π Persistence of technology shock Literature ρ = 0.85 z Persistence of bargaining power shock Near Random Walk ρ = 0.90 η Persistence of risk premium shock Literature ρ = 0.85 φ Std. Dev. of technology shock 1/3 of output variance share and ZLB frequency σ = 0.005 z Std. Dev. of bargaining power shock 1/3 of output variance share and ZLB frequency σ = 0.0135 η Std. Dev. of risk premium shock 1/3 of output variance share and ZLB frequency σ = 0.00075 φ conventional calibration results in a too-low labor share in our environment given the existence of rents due to market power, which are divided into different agents according to bargaining power. We set the coefficient of investment adjustment cost equal to κ = 12 to make investment three times more volatile than output as in the data. The depreciation rate of capital stock is set to δ = 0.025. Labor markets: The efficiency of the matching function is set to ζ = 0.91 to hit a quarterly job finding rate of 85 percent as in the Current Population Survey (CPS). The exogenous gross separation rate is calibrated to ρ = 0.37, so that the quarterly net separation rate equals 5.6 percent as in the CPS. For the elasticity of the Cobb-Douglas matching function with respect to vacancies, we follow the evidence reported in Pissarides and Petrongolo (2001) and set (cid:15) = 0.5. We follow much of the literature and set the steady state workers’ bargaining power to η = 0.5. Unemployment benefits equal bU = 0.5, which represent 83 percent of the equilibrium wage in steady state. Finally, we set the vacancy posting cost equal to ξ = 0.11, about 11 percent of labor productivity, essentially the same as in Hagedorn and Manovskii (2008) and very similar to other values used in the literature. Price and wage rigidities: Wesettherealwageadjustmentcostparameterϑ = 50tomatchthe standard deviation of the 4-quarter growth rate of real compensation in the non-farm business sector. Regarding the degree of nominal rigidities, we choose a low value for the probability of resetting the price (1−ϕ = 0.065) to be consistent with so-called flat Phillips curve. For the same reason, we select a substantial degree of price indexation (ε = 0.5). With these choices, together with the calibration of realwagerigidity, inflationinthebaselineeconomyisasvolatileasPCEpriceinflationinthepost-war 19

period. Monetary policy rule: As mentioned earlier, we let the monetary policy rule react only to the inflation gap given that the nature of the shocks considered are consistent with the so-called divine coincidence. In the baseline calibration, the inflation coefficient is set to ρ = 1.5 following Taylor π (1999). Notethatinaneconomywiththedivinecoincidence,thisparametrizationwithoutthereaction term for the output gap implies a more lenient monetary policy rule than in Taylor (1999). However, we study a range of values for ρ in our analysis. π Exogenous shock processes: The persistence parameters of the technology shock and the risk premium shock are chosen to be the same and equal to ρ = ρ = 0.85. We set the persistence of the z φ bargaining power shock to a slightly higher value of ρ = 0.9. The reason is that we view changes η in bargaining power as a slow-moving process representing changes in social norms and labor market institutions. We choose the volatility of the three exogenous shocks such that they have equal variance decompositionsharesforoutputinthelongrun,withoutconsideringfinancialcrisesorZLBconstraint. This gives us two restrictions. The third ones comes from matching the frequency of being at the ZLB of around 5 percent, as in Coibion et al. (2012). These three restrictions determine the values for σ , z σ , and σ . Note that we use the monetary policy shock only to satisfy the ZLB constraint. Overall φ η the volatility of the cyclical component of output in the baseline economy with financial crises and the ZLB constraint is 2.2 percent, slightly above the 1.6 percent observed in the United States during the post-war period. 5 Model Dynamics This section characterizes the dynamics of the model using impulse response functions. In doing so, we first assume that neither default nor a binding ZLB constraint occur. We then show how these two sources of nonlinearity modify model dynamics. 5.1 Without Crises and ZLB Figure2depictstheeffectsofapositivetechnologyshock(bluesolidline), anegativeshocktoworkers’ bargaining power (red dashed line), and a positive shock to the risk premium (black dash-dotted line) when neither default nor a binding ZLB constraint occur. Technology shock: Panel (a) shows that, despite the technology shock being positive, the response of aggregate output is negative. This is because our baseline monetary policy fails to provide enoughmonetaryaccommodation(seeGal´ı(1999)). Thepositivetechnologyshocklowersthemarginal costandgeneratesdeflationarypressureasshowninpanel(i). Inresponse,thepolicyrateisloweredin panel (k). However, monetary accommodation is not strong enough and, as a result, the real interest rate initially rises as shown in panel (l), which then leads to a decline of real investment (panel (d)) and aggregate output. The decline of aggregate demand and higher technology then lead to a fall in job creation, rising unemployment rate in panel (j), and a fall in the real wage in panel (h). Accordingly, workers’ 20

Figure 2: Impulse Response Functions: Without Crises and ZLB (a) Output, pct (b) Cons. W, pct (c) Cons. K, pct (d) Investment, pct 0.4 0.2 0 0 0.2 0 -0.1 -0.1 -0.2 0 -0.2 -0.2 -0.4 -0.3 -0.3 -0.2 -0.6 0 20 40 0 20 40 0 20 40 0 20 40 (e) Income inequality, ppt (f) DTI ratio (W), ppt (g) Prob. of crisis, bps (h) Real wage, pct 1 1 0.2 0.4 0.5 0.5 0 0.2 0 0 -0.2 0 -0.5 -0.5 -0.4 0 20 40 0 20 40 0 20 40 0 20 40 (i) Inflation, ann. ppt (j) Unemployment, ppt (k) Nominal rate, ann. ppt (l) Real rate, ann. ppt 0.2 0 0 0.4 0.2 -0.2 0 -0.2 0 -0.4 -0.4 -0.2 -0.2 0 20 40 0 20 40 0 20 40 0 20 40 Technology shock Bargaining power shock Risk premium shock Notes: Thisfigureplotsimpulseresponsefunctionstoapositivetechnologyshock,anegativeshocktoworker’sbargaining power, and a positive risk-premium shock. It is assumed that neither default nor a binding ZLB constraint occur. consumption declines persistently in panel (b). To smooth out the decline in consumption, workers increase borrowing and their debt-to-income ratio increases in panel (f), which then raises the probability of crisis in panel (g). In contrast, the consumption level of shareholders increases persistently owing to the increased profits. The decline of real investment also contributes to the increase in the consumption of shareholders because the reduced investment level creates a slack in the budget constraint of this group. Finally, the combination of lower wage income and higher profits results in a substantial increase in income inequality defined as the income share of shareholders in panel (e), which is correlated with increases in the debt-to-income ratio of workers. Bargaining power shock: In the literature, a shock to workers’ bargaining power affects aggregate output through the labor market channel. For instance, a lower bargaining power leads to an improvementinthejobcreationcondition,expandingbothemploymentandinvestment(seeGertleret al. (2008) and Drautzburg et al. (2017)). Such a labor market channel is active in our model economy. However, the negative bargaining power shock also redistributes income from workers to shareholders. This redistribution may lead to a decline in aggregate demand if the MPS of shareholders is strong enough such that the reduced consumption of workers is not completely offset by the increased consumptionofshareholders. Panels(a)to(c)ofFigure2showthatthisisindeedthecaseinourbaseline economy. 21

Despitethefundamentalimprovementinthejobcreationcondition,employmentactuallydecreases because of the fall in aggregate demand. For the same reason, investment decreases. Since lower bargaining power reduces the real wage persistently, marginal cost declines and the economy faces a mild deflation pressure.15 Monetary policy reacts by lowering the nominal interest rate, but the real interest rate is slightly increased. Workers try to smooth out the decline in consumption by issuing morebonds. Sinceshareholdershavetoincreaselending,theydonotincreaseconsumptionsufficiently. The rise in income inequality generated by the negative bargaining power shock correlates with the increase in the debt-to-income ratio of workers. As the debt-to-income ratio goes up persistently, the probability of default rises. Risk premium shock: Panel (a) of Figure 2 shows that the risk premium shock leads to a persistentdeclineinaggregateoutput. Thisisdrivenbythedeclineintheconsumptionofshareholders through the consumption Euler equation and the decline in investment through the investment Euler equation. While this shock does not directly affect workers’ consumption through their consumption Euler equation, the shock also leads to a decline in their consumption because the decline of aggregate demand leads to higher unemployment and lower wages. The debt issuance of workers declines as the borrowing cost is elevated by this shock. The decline of income and consumption in the context of higher borrowing cost increases the incentive to default and the default probability goes up, but not substantially since the borrowers undergo deleveraging in response to the risk-premium shock. 5.2 With Crises and ZLB Figure 3 illustrates how occasional eruptions of financial crises make a difference in the response of endogenous variables to a given series of random shocks. It also makes clear how the ZLB constraint works in conjunction with the endogenous crisis mechanism. To create an environment in which a financialcrisisandabindingZLBconstraintmayoccur,weassumeasequenceofadverseriskpremium shocks that bring the economy into a recession.16 First,wefocusonthebluesolidlinewhichshowstheresponseoftheeconomyundertheassumption that a financial crisis does not occur during this episode. Whether or not a financial crisis occurs depends on the specific random draw that the economy is given for the utility cost of default and the difference between the values of default and non-default for workers. Panels (a)∼(c) of Figure 3 show that aggregate output and consumption for both types of agents decline about 1∼2 percent from their steady state values. In panel (d), investment in physical assets declines 5 percent. Panel (j) shows that unemployment rate rises up to 1 percentage point above its steady state level, bringing downward 15The change in aggregate demand due to redistribution of aggregate income among different income groups with heterogeneous MPCs is very much in the Kaleckian tradition: “Ifanincreaseinbargainingcapacityisdemonstratedbyspectacularachievements,thereisadownward shiftinfunctionf(p¯/p)andthemark-upsdecline. Aredistributionofnationalincomefromprofitstowages will take place then...But, as there is a redistribution of income from profits to wages...there is a rise in employment and output there”–(Kalecki (1971), p. 162). 16In particular, we assume (cid:15) =σ ×[ 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.75 0.6 0.45 0.3 0.15 ](cid:48). φ,1:12 φ 22

Figure 3: Nonlinear Effects of ZLB and Financial Crises (a) Output, pct (b) Cons. W, pct (c) Cons. K, pct (d) Investment, pct 2 0 0 10 0 0 -5 -5 -10 -2 -20 -10 -10 0 20 40 0 20 40 0 20 40 0 20 40 (e) Income inequality, ppt (f) DTI ratio (W), ppt (g) Asset prices, pct (h) Real wage, pct 4 5 0 2 10 0 0 -2 -5 -2 0 -10 -4 -4 -6 -10 -15 -6 0 20 40 0 20 40 0 20 40 0 20 40 (i) Inflation, ann. ppt (j) Unemployment, ppt (k) Nominal rate, ann. pct (l) Real rate, ann. ppt 0 6 2 4 -2 4 0 2 2 -4 0 -6 -2 0 0 20 40 0 20 40 0 20 40 0 20 40 No crises, no ZLB Crises, no ZLB Crises, ZLB Notes: We use a sequence of adverse risk-premium shocks such that the economy undergoes a recession. The blue solid line shows the case where a financial crisis does not occur and the ZLB is not binding. The red dashed line and the black dash-dotted line are the cases in which a financial crisis occurs in period 7 without and with the ZLB constraint, respectively. pressure on the real wage. Note that without a financial crisis, the ZLB constraint does not bind in this example, as shown by panel (k). Next, the red dashed line shows the case where the economy is given a particularly low realization of the random draw for the utility cost of default in period 7 and a financial crisis occurs. This case assumes that the ZLB constraint does not exist as indicated by the fact that the level of the nominal interest rate falls nearly to -2 percent. Overall, the responses of all endogenous variables are much stronger except the consumption level of shareholders, which increases in response to the crisis. The reason is that shareholders reduce investment in physical assets and financial assets up to a point where there exists a large surplus cash flow for increased consumption expenditures. Finally,theblackdash-dottedlineshowsthecasewhentheZLBconstraintexistsinadditiontothe crisis happening in period 7, and thus the ZLB constraint binds endogenously in response to the crisis. It is somewhat difficult to see in the figure, but the ZLB constraint does not bind immediately after the crisis erupts. This is because the response of inflation rate is hump-shaped. However, the ZLB constraintbindsinfutureperiodsinexpectationsimmediatelyandtheeconomyfacesanticipatednews shocks to monetary policy. In this environment, the economy undergoes a much deeper recession.17 17The sizes of these anticipated news shocks are large on the order of 3∼5 percentage points. 23

Aggregate output, for instance, can deviate as much as 6 percentage points from the case without the crisis and the binding ZLB constraint. In the presence of binding ZLB constraint, the size of the financial crisis roughly doubles in terms of output loss, unemployment increase and deflation.18 6 Simulation Results: Income Inequality, Aggregate Demand, and Financial Crises In this section, we first perform a thought experiment to highlight the link between income inequality and aggregate demand in the model. We then use stochastic simulations to analyze the key properties of the model economy. 6.1 Illustration of the Link between Income Inequality and Aggregate Demand Before describing the main results of the paper, we illustrate the key mechanism that provides links between income inequality, insufficient aggregate demand and deflation pressure. The fundamental reason why increases in income inequality lead to insufficient aggregate demand in the model is the composition of aggregate MPC: Shareholders have a strictly lower MPC than workers due to the Weberian preferences of “the spirit of capitalism” of shareholders. In this situation, the redistribution of national income towards workers leads to increases in aggregate demand. To build intuition, as Keynes (1936) did, let us assume that consumption is a linear function of income for each agent, where each agent’s income is given by fractions α and 1 − α of aggregate income y : cW = kWαy and cK = kK(1−α)y with 0 < kK < kW ≤ 1. Consider a simplified version of aggregateincomeidentitywithnoadjustmentcost: y = cW+cK+x. Fornowwetreatx“autonomous” componentofaggregatedemand. Combiningtheconsumptionfunctionsandaggregateincomeidentity yields x x y(α) = ≡ 1−[αkW +(1−α)kK] 1−k(α) where 1/(1−k(α)) is our version of what Keynes called “investment multiplier”. Now consider a new income redistribution α(cid:48) > α that increases the share of the income that accrues to the agent with the highest MPC. It is then clear that x x y(α(cid:48)) = > = y(α), 1−k(α(cid:48)) 1−k(α) 18NotethatthebindingZLBconstraintcanbethoughtofascontractionarymonetarypolicyshocksexecutedbynature rather than by the monetary authority. The difference between the black dash-dotted lines and the red dashed lines in panels (b) and (c) of Figure 3 show that the monetary policy transmission channel is starkly different for workers and shareholders for the same reason mentioned as for the effects of the risk premium shock. In the case of shareholders, the difference is mainly driven by the intertemporal substitution effect owing to the higher real interest rate during the binding ZLB episode. In contrast, the reduction in workers’ consumption is primarily driven by the general equilibrium effects of reduced job creation. While our model is a very stylized, two-agent general equilibrium model, this difference inmonetarytransmissionchannelsbetweenshareholdersandworkersisessentiallyidenticaltothatobtainedinthemuch richermodelofKaplanetal.(2018),withheterogeneousagentsandincompletemarketswithmultipleassetsofdifferent liquidities. 24

since k(α(cid:48)) = α(cid:48)kW + (1 − α(cid:48))kK > αkW + (1 − α)kK = k(α). For a given level of autonomous investment spending x, the ratio y(α(cid:48))/y(α) > 1 because [1−k(α)]/[1−k(α(cid:48))] > 1, which is the ratio of the marginal propensities to save in the two economies. This is what we call inequality multiplier. Notethatthesizeoftheinequalitymultiplierwillbegreaterthany(α(cid:48))/y(α)ifinvestmentresponds positively to the increase in aggregate demand, i.e., x = x +βy(α), β > 0 such that a x a y(α) = , 1−k(α)−β where x is the autonomous component of investment and 1/[1−k(α)−β] is our version of what Hicks a (1972) called “super-multiplier”. It can be shown that the inequality multiplier can be much greater if constructed with the super-multiplier. In order to illustrate that our model features this property, Figure 4 plots the effects of a fictitious government policy that redistributes part of shareholders’ income to workers. This redistribution is assumed to be a one-off event with no persistence. The per capita income transfer for workers is denotedbyω,andiscalibratedas10percentofwageincomeinthesteadystate. Percapitatransferfor shareholders is then given by −ω(1−χ)/χ. Note that the aggregate resource constraint is not affected because the population weighted sum of the transfer is zero, i.e. (1−χ)ω+χ[−ω(1−χ)/χ] = 0.19 Panel (a) of Figure 4 shows the marginal effects of income redistribution on consumptions of the two agents, i.e., dcW/ω for workers and dcK/[−ω(1−χ)/χ] for shareholders.20 Note that dcK/[−ω(1− χ)/χ] is positive because both the numerator and the denominator are negative. In other words, the transitory income shock for shareholders leads to a decline in their consumption. Panel (a) shows that the marginal effects of the transitory income shock are about six times larger for workers than for shareholders. Panel (b) displays the effects on aggregate output (blue solid line), investment (red dashedline)andinflationrate(blackdash-dottedline). Thechangeintheincomedistributioninfavor of those agents with greater MPC brings a persistent boom by changing the composition of aggregate MPC, even when the shock is completely transitory. Recall that the investment decision for capital accumulation is made by shareholders. Thus, the fact that the negative income shock for shareholders leads to an increase in investment is a remarkable result: while shareholders observe that the government confiscates part of their income in this experiment, they also expect a persistent boost in aggregate demand and increase investment spending.21 Note that the inequality multiplier is operative only to the extent that prices fail to clear the market immediately due to the mix of nominal rigidities and suboptimal monetary policy. Consider an income redistribution that leads a fall in aggregate demand. In an economy with no nominal rigidities, the real “rate of interest falls at a sufficiently rapid rate”(Keynes (1936), p. 31) to induce 19This experiment is for illustration purposes and such government redistribution does not occur in our simulations. 20Note that these marginal effects are not the MPC discussed in the microeconomics literature. This is because the income redistribution in this experiment brings large general equilibrium effects as shown by the aggregate output response in panel (b) of Figure 4. 21While our focus is on monetary policy issues, the transfer experiment shown in Figure 4 suggests much larger stabilization roles for tax-and-transfer and social insurance (see McKay and Reis (2016) and Mitman et al. (2017)). 25

Figure 4: Effects of a Transitory Income Transfer Shock (a) MPC in general equilibrium (b) Aggregate effects of income transfer 2.5 0.3 Workers 2 Output, pct Shareholders Investment, pct 0.25 1.5 Inflation, ann.ppt 1 0.2 0.5 0.15 0 0.1 -0.5 0.05 -1 0 -1.5 -0.05 -2 0 20 40 60 80 0 20 40 60 80 Notes: This figure plots the effects of a fictitious government policy that redistributes part of shareholders’ income to workers. This redistribution is assumed to be a one-time event with no persistence and is calibrated to 10 percent of wage income in the steady state. greater investment immediately (as well as consumptions of both agents, not just because of lower real interestratebutbecausetheincreaseininvestmentimpliesagreaterincome),completelyoffsettingthe decline of aggregate demand due to the increased income inequality. Say’s Law holds: supply creates its own demand and the channel between income inequality and aggregate demand disappears.22 In Section 7, we show that this prediction is satisfied in our model economy. 6.2 Main Results Table 3 summarizes the unconditional moments of the economy under the baseline monetary policy rule, which takes the form of inflation targeting with ρ = 1.5. The moments are computed using π 250,000 periods and 40 periods of news shocks to monetary policy such that the agents of the model understand the nature of the ZLB constraint.23 This exercise is meant to uncover important links between income inequality, low aggregate demand, deflation pressure and the likelihood of financial crises. The links have been conjectured but not yet quantified in the existing literature. In order to better understand the separate effects of financial crisis and the ZLB constraint, Table 3 reports four set of results, using an identical set of random draws for shocks in each column. The four columns show the cases (i) with no crises and no ZLB, (ii) with no crises and the ZLB, (iii) with crises and no ZLB, and (iv) with crises and the ZLB, which is our baseline case. In our explanation of the results, we focus on the baseline case for the sake of space. However, we will draw readers’ attention to alternative cases to emphasize the role of a particular mechanism. Lines 1∼3 of Table 3 show that income inequality, defined as the share of shareholders’ income relative to total income, is negatively correlated with aggregate output, consumption, and investment, 22How much fall in aggregate demand is offset by increases in consumption and investment due to the decline of real interestrateinthisscenariodependsonthedegreeofinvestmentadjustmentfriction,whichisarealadjustmentfriction. 23Increasing the number of simulation periods to 500,000 does not change the moments in any meaningful way. Also, using 60 periods of news shocks to monetary policy leads to roughly identical moments. 26

Table 3: Key Moments under the Baseline Monetary Policy Rule No crises No crises Crises Crises No ZLB ZLB No ZLB ZLB 1. Corr(income inequality, y) -0.78 -0.88 -0.57 -0.82 2. Corr(income inequality, c) -0.61 -0.82 -0.48 -0.80 3. Corr(income inequality, x) -0.68 -0.75 -0.55 -0.77 4. Corr(income inequality, π) -0.97 -0.97 -0.80 -0.88 5. Corr(income inequality, b/y) 0.82 0.86 0.72 0.84 6. Corr(income inequality, pδ) 0.75 0.85 0.62 0.81 7. Corr(consumption inequality, y) -0.79 -0.82 -0.92 -0.90 8. Corr(consumption inequality, c) -0.72 -0.80 -0.89 -0.90 9. Corr(consumption inequality, x) -0.73 -0.72 -0.77 -0.74 10. Corr(consumption inequality, π) -0.94 -0.92 -0.92 -0.88 11. Corr(consumption inequality, b/y) 0.94 0.95 0.91 0.94 12. Corr(consumption inequality, pδ) 0.87 0.90 0.81 0.92 13. Corr(b/y, pδ) 0.98 0.99 0.96 0.98 14. E(pδ) 1.29 1.29 1.32 1.33 15. E(i), quarterly, percent 0.50 0.55 0.33 0.49 16. E(π), quarterly, percent 0.00 -0.03 -0.11 -0.23 17. E(y) 0.91 0.90 0.90 0.89 18. E(u), percent 5.88 6.32 6.73 8.15 19. E(debt-to-income ratio of workers) 1.52 1.53 1.55 1.56 20. S.D.(y), percent 0.91 1.82 1.61 4.36 21. S.D.(π), quarterly, ppts 0.30 0.38 0.35 0.64 22. corr(y,π) 0.74 0.87 0.79 0.93 23. Frequency of ZLB, percent - 3.24 - 5.41 24. Mean MP shock, annual bps - 1.01 - 3.22 25. Skewness(i) 0.01 0.52 -0.33 0.70 26. Skewness(π) 0.01 -1.13 -0.33 -2.04 27. Skewness(y) 0.02 -2.39 -0.96 -2.59 28. Skewness(u) 0.00 1.80 0.47 2.52 Notes: The moments are computed using 250,000 simulations with identical set of random draws for the shocks in each column. The baseline monetary policy is assumed in all cases. Mean MP shock is the annualized mean of monetary policy shocks (both contemporaneous and anticipated news shocks) required to satisfy the ZLB constraint. and the degrees of correlations are strong in the baseline case.24 Since correlation is not causation, one cannot state that the income inequality causes insufficiency of aggregate demand. Rather, causation runs in both directions. Owing to the lower MPC of shareholders, aggregate demand falls when a growing share of national income is distributed to those agents. Conversely when aggregate demand increases, labor income rises as more jobs are created and wage is boosted. Since labor income is the major income source for workers, a boost in aggregate demand leads to a decline in income inequality. As shown in lines 7∼9, the link between inequality and aggregate demand is also confirmed when inequality is measured in terms of consumption. These results indicate that secular stagnation due to 24Note that it is the case with no ZLB but with financial crises, shown in the third column of Table 3, where these correlations are the weakest. This is because without the ZLB constraint, financial crises tend to lower the overall comovementpropertyoftheeconomy. Asillustratedbytheredlineinpanel(c)ofFigure3,financialcrisisworkslikea financial shock that tends to lower the comovement between consumption and investment when the ZLB is not present. Thatis,shareholdersincreaseconsumptioninresponsetoafinancialcrisisasthecrisisreducesoverallaggregatedemand and they respond by reducing real investment substantially. The increase in consumption of shareholders in this case lowers the negative correlation between income inequality and aggregate demand. 27

income inequality and underconsumption, originally raised by Hansen (1939) and recently re-taken by Summers (2015), can be a real possibility. Line 4 of Table 3 shows a strong, negative correlation between income inequality and the inflation rate. Again, causation runs in both direction. Since income inequality is negatively correlated with aggregate demand, it is also negatively correlated with inflation. Conversely, deflation elevates the real debt burden of workers and redistributes income towards shareholders, increasing the degree of income inequality. We want to emphasize the role played by nominal rigidities. Without them, a negative correlation between income inequality and aggregate demand cannot exist in KRW’s original framework as Say’s Law holds regardless of the degree of income inequality. Lines5and11ofTable3showthatthestrongcorrelationbetweenincome/consumptioninequality andexcesscredit,definedasthedeviationsofcredit-to-GDPratiofromitssteadystate,suchasseenin thedata,canbereplicatedbyourmodel. Moreimportantly,lines6and12indicateastrongcorrelation between inequality and the probability of financial crises. This arises due to the near perfect linear relationship between excess credit and the probability of financial crises regardless of the presence of the ZLB constraint, as shown in line 13. Such a linear relationship replicates the empirical findings by Jord`a et al. (2011) and Schularick and Taylor (2012).25 Interestingly, the tie between income inequality and aggregate demand is much stronger in the presenceoftheZLBconstraint.26 Thisindicatesthatafailuretoprovideasufficientdegreeofmonetary accommodation during economic downturns strengthens the nexus between income inequality and aggregate demand. The presence of the ZLB constraint also slightly increases the probability of financial crises, due to slightly higher debt-to-income ratio of workers. Importantly, both financial crises and the presence of the ZLB constraint modify the distributions ofequilibriumquantitiesandprices. Inparticular, withnofinancialcrisesandnoZLBconstraint(first column of Table 3), the skewness of endogenous variables (see lines 26∼28) is basically zero, which should not be surprising given the assumed Gaussian shocks. This also implies that the mean of their distributions coincides with the non-stochastic steady state of each variable (see lines 16∼18). When financial crises are included but not the ZLB constraint (third column), the distributions of inflation, aggregate output and unemployment rate become skewed to the downside and their means deviate fromthenon-stochasticsteadystatevalues. Thus, financialcrisescreateskewnesstothedistributions. The combination of crises and the ZLB constraint (fourth column) creates extraordinary degrees of skewness for the endogenous variables. Since negative skewness assigns more probability mass to the downside of the economy, the mean levels of the inflation rate and aggregate output are substantially lower. Indeed, the presence of the ZLB constraint doubles the deflation bias created by financial crises and lowers the mean level of output by 1 percent.27 Figure 5 shows how the exact shapes of the 25The linear relationship also justifies the reduced form approach that assumes a (log) linear relationship between the excess credit and the probability of disaster without structural modeling, an approach recently taken by Gourio et al. (2018). In Appendix D, using our simulated data, we replicate the results of Paul (2017) that income inequality dominates credit in the predictive power of financial crises. 26Figure 11 in Appendix D shows the unconditional and conditional means of monetary policy shocks (both contemporaneous and anticipated news shocks) needed to satisfy the ZLB constraint in the baseline economy. 27Note that the skewness of the unemployment rate without the ZLB constraint is 0.47 in our economy, which is the 28

Figure 5: Skewed Distributions of Inflation and Output (a) Inflation rate (ppt) (b) Output (pct) 1.4 0.4 1.2 0.35 0.3 1 0.25 0.8 0.2 0.6 0.15 0.4 0.1 0.2 0.05 0 0 -3 -2 -1 0 1 2 -20 -15 -10 -5 0 5 No crises, no ZLB Crises, no ZLB Crises, ZLB Notes: The figure plots the distributions of inflation and output in deviations from their non-stochastic steady state depending on the presence of financial crises and the ZLB constraint using kernel density estimates of simulated data. distributions change depending on the presence of financial crises and the ZLB constraint using kernel density estimates of simulated data for inflation rate in panel (a) and aggregate output in panel (b). Additionally, the ZLB constraint makes the distribution of the policy rate more skewed to the upside (see lines 15 and 25), creating additional deflation bias. Both sources of nonlinearity also increase the volatility of inflation and output (see lines 20 and 21). Interestingly, the presence of the ZLB constraint strengthens the comovement property of the economy as measured by the correlation between aggregate output and inflation (see line 22). Note that, in our model economy, rational expectations imply that current inflation is the present value of future expected real marginal costs and current consumption is the negative present value of future expected real interest rates. When the ZLB constraint is expected to bind in the future, agents expect real marginal costs to be lower and the real interest rate to be higher. This situation leads to an immediate decline of both inflation and consumption in line with the results in McKay et al. (2016), strengthening the comovement property of the economy. The presence of the ZLB constraint therefore introduces nominal shocks, which are considered to be the ones with stronger comovement property in the DSGE literature.28 sameintheU.S.datafrom1948to1999. Theskewnessgoesupto0.62ifthesampleisexpandedto2017toincludethe binding ZLB episode. 28McKay et al. (2016) show that the precautionary saving motive in incomplete markets can make the response of aggregate consumption less sensitive to the forward guidance. Our model economy can be viewed as a two-agent incomplete market economy. The ZLB constraint is implemented by anticipated news shocks to monetary policy, which can be viewed as reverse forward guidance. Despite the stylized incomplete market setting, the forward guidance in our model economy is as powerful as in a representative agent economy for two reasons. First, shareholders in our economy are well-insured, making the forward guidance powerful. Second, once the forward guidance becomes powerful through reduction in consumption and investment of shareholders, this brings a large general equilibrium effects on labor market, which then puts direct downward pressure on workers’ consumption. Furthermore, correlation between credit and workers’ consumption is very close to zero in the baseline economy. This means that on average, the credit instrument is not very useful for workers’ consumption smoothing, making them more like the agents under extreme 29

The results in this section show that income inequality generates insufficient aggregate demand and financial fragility, creating deflation bias and elevating the mean unemployment rate. In our model economy, the negative link between income inequality and aggregate demand is created by the desire of shareholders to overaccumulate financial wealth, motivated by their Weberian preferences. In order to illustrate this mechanism, Figure 6 reports the effects of changing the weight given to the direct utility from holding financial assets, ψB. As the weight is lowered from the baseline value ψB = 0.45, which was chosen to match the observed debt-to-income ratio of workers, toward zero, both the standard deviation of inflation and mean unemployment rate fall linearly in panel (a), and the mean inflation rate is raised from -0.23 percent in the baseline towards zero in panel (d). Panel (c) show the effects on the debt-to-income ratio of workers and the probability of financial crises: they both converge to essentially zero as the “love of wealth” preferences disappear. In panel (b), as the deflation risk falls, the frequency of the ZLB constraint declines linearly and the size of the monetary policy shocks needed to satisfy the ZLB constraint also falls. One can also see that the mean level of aggregate output would be more than 2 percent higher with much weaker “love of wealth” preferences (panel d), and the distributions of inflation and the unemployment rate would be much less skewed to the downside (panel e). Finally, the mean levels of both income and consumption inequality would be much lower if the “love of wealth” preferences disappear (panel f). Overall, the results in Figure 6 indicate that the Weberian preferences are the key mechanism that creates a link between income inequality, deflation pressure, excess credit and financial fragility in our model economy. 7 Monetary Policy Strategies Thissectiondiscussesmonetarypolicystrategiesforaneconomythatsuffersfrominsufficientaggregate demand due to income inequality and faces occasional financial crises due to excess credit. We first describe the monetary policy dilemma facing a central bank that follows either an inflation targeting rule or a price-level targeting rule. We then examine the role of the ZLB constraint in generating the monetary policy dilemma. Finally, we discuss asymmetric monetary policy rules that can mitigate the degree of skewed distributions of endogenous quantities and prices due to occasional financial crises and binding ZLB constraint. 7.1 The Costs and Benefits of Appointing a Conservative Central Banker Clarida et al. (1999) showed that Rogoff (1985)’s conservative central banker is optimal in the context of an optimal monetary policy rule without commitment. As the weight on the output gap in the centralbank’slossfunctionconvergestozero,whichisequivalentwithassumingthedivinecoincidence duetothelackofmarkupshocks,theinflationreactioncoefficientshouldapproachinfinity. Aspointed out earlier, our model economy features the divine coincidence. However, when the economy faces disproportionately large downside risks due to financial crises and the stabilizing capacity of monetary asset illiquidity in Werning (2016) rather than the agents with precautionary savings in McKay et al. (2016). 30

Figure 6: The Role of the Weberian Preferences 0.65 0.6 0.55 0.5 0.45 0.4 0.35 6 7 8 9 Mean unemployment rate etar noitalfni .veD .dtS (a) 5.5 B=0.45 5 4.5 4 B=0.05 3.5 0 0.01 0.02 0.03 0.04 Mean monetary policy shocks BLZ ycneuqerF (b) 200 B=0.45 150 100 50 B=0.05 0 0 0.5 1 1.5 Mean default probability srekrow fo oitar ITD naeM (c) B=0.45 B=0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0.885 0.89 0.895 0.9 0.905 Mean output etar noitalfni naeM (d) -0.8 B=0.05 -1 -1.2 -1.4 -1.6 -1.8 -2 =0.45 -2.2 1.5 2 2.5 3 Skewness unemployment rate etar noitalfni ssenwekS (e) 0.35 B=0.05 0.345 0.34 0.335 B=0.45 0.33 0.35 0.355 0.36 0.365 0.37 Mean income inequality ytilauqeni noitpmusnoc naeM (f) B=0.45 B=0.05 Notes: Thefigureshowsthelociofthemomentsofselectedendogenousvariablesfordifferentvaluesoftheutilityweight on private bond ψB. The baseline economy uses ψB = 0.45. The horizontal axis in panel (b) shows the annualized unconditionalmeaninpercentofmonetarypolicyshocks(bothcontemporaneousandanticipatednewsshocks)required to satisfy the ZLB constraint. policy is severely distorted by the presence of the ZLB constraint, can one reach the same conclusion? We re-investigate the benefits and costs of a conservative central banker. Figure7showshowkeymomentsofthemodeleconomychangeasthecentralbankincreaseseither the inflation reaction coefficient (blue circles) or the reaction coefficient for price-level deviation (red triangles). Panel (a) uncovers an important trade-off: A conservative central banker achieves lower inflationvolatilityonlyatacostofelevatingthemeanunemploymentrate. Thisisduetothetendency of a conservative central banker to aggravate the skewed distributions of output (employment) and inflation as shown by panel (b). The welfare loss due to the increase in mean unemployment rate and the decrease in mean output may dominate any welfare gain from reducing the volatility of inflation. Panel (c) of Figure 7 presents another challenge facing a conservative central banker. The panel showshowdifferentdegreesofpolicyconservatismaffectsthefrequencyoftheZLBconstraint(vertical axis) and the unconditional mean of monetary policy shocks (both contemporaneous and anticipated) required to satisfy the ZLB constraint (horizontal axis). The former measures the time spent at the ZLB constraint while the latter measures how much damage is inflicted to the economy in terms of monetary policy shocks to satisfy the ZLB constraint. As it can be seen, the frequency of binding 31

Figure 7: The Consequences of Conservatism in Monetary Policy 0.7 0.6 0.5 0.4 0.3 6 7 8 9 Mean unemployment rate etar noitalfni .veD .dtS (a) 0 =1 =20 -1 -2 -3 =0.1 =4 -4 -4 -3 -2 -1 0 Skewness output etar noitalfni ssenwekS (b) 12 =0.1 10 =1 8 =4 6 4 2 =20 0 0 0.5 1 1.5 2 Mean monetary policy shocks BLZ ycneuqerF (c) =20 =4 =1 =0.1 10 0 -10 -20 -30 1.33 1.335 1.34 1.345 Mean default probability sisirc nopu ssol tuptuo evitalumuC (d) =4 0 =20 =0.1 -0.1 -0.2 =1 -0.3 0.88 0.89 0.9 0.91 Mean output Inflation targeting Price-level targeting etar noitalfni naeM (e) -0.6 =4 -0.65 =0.1 -0.7 -0.75 -0.8 =1 -0.85 =20 -0.9 -1 -0.8 -0.6 Corr(inc. inequality,output) )noitalfni,ytilauqeni .cni(rroC (f) =4 =0.1 =20 =1 Notes: The figure shows the loci of the moments of selected endogenous variables as the central bank increases either theinflationreactioncoefficientρ (inflationtargeting)orthereactioncoefficientforprice-leveldeviationρ (price-level π Π targeting). The horizontal axis in panel (c) shows the annualized unconditional mean in percent of monetary policy shocks (both contemporaneous and anticipated news shocks) required to satisfy the ZLB constraint. ZLB constraint increases monotonically for both inflation targeting and price-level targeting as policy conservatism is strengthened. Furthermore, the mean level of monetary policy shocks used to satisfy theZLBconstraint, whichcanbeinterpretedastheshadowvalueoftheZLBconstraint, alsoincreases monotonically with the degree of policy conservatism. Panel(d)ofFigure7showstwoaspectsofpolicyconservatismregardingtheprobabilityoffinancial crises (horizontal axis) and the cumulative output loss upon a financial crisis (vertical axis). For both rules, aconservativecentralbankertendstoincreasetheprobabilityoffinancialcrises. Thisisbecause strongreactionsinmonetarypolicyratestoinflationtendstoelevatedebt-servicingcostsofborrowers. Furthermore, the increase in unemployment rate due to hawkish monetary policy as shown in panel (a) also elevates the debt-to-income ratio and thus strengthens the incentive of borrowers to default. However,thesameconservativecentralbankerhasavirtueofprovidingamassiveamountofmonetary accommodation during a financial crisis, limiting the size of the cumulative output drop.29 It is notable that while the price-level targeting regime faces the same type of trade-off between 29Figure 12 in Appendix D replicates the experiment shown in Figure 3 when both financial crises and the ZLB constraint are present for a range of calibrated inflation targeting and price-level targeting rules. 32

stabilization of inflation volatility and reduction of the mean unemployment rate, it substantially improves the nature of the trade-off. This can be seen in the fact that the frontier associated with the price-level targeting is located much lower and more to the left in panel (a) of Figure 7. Furthermore, as shown in panel (b), the absolute degrees of skewness of the distributions are also much smaller with the price-level targeting as compared with the inflation targeting. Price-level targeting can also be much more effective in controlling the deflation bias. Panel (e) of Figure 7 shows the mean inflation rate on the vertical axis and the mean aggregate output on the horizontal axis. The results suggest that, in contrast to the inflation targeting, price-level targeting eliminatesthedeflationbiasalmostcompletelyregardlessofthedegreeoftheconservatismofmonetary policymakers. It also suggests that simply switching from the inflation targeting with ρ = 20 to the π price-level targeting with ρ = 0.1 can result in an almost 3 percent increase in the mean level of Π aggregate output, which has significant implications for the welfare of both types of agents.30 Panel (e) also suggests that, if the price-level targeting cannot be implemented, adopting the most lenient inflation targeting regime can minimize the deflation bias at the cost of increased inflation volatility. Finally, panel (f) of Figure 7 shows how monetary policy conservatism affects the correlation between income inequality and the inflation rate (vertical axis) and the correlation between income inequality and output (horizontal axis). The results point out that price-level targeting can be much more effective in breaking the link between income inequality and aggregate demand in the model. In fact, no level of inflation targeting is nearly as successful as the price-level targeting in breaking the tie between income inequality and aggregate demand. Summing up, the investigation of various aspects of the economy under alternative monetary policy rules leads to two conclusions regarding the desirability of monetary policy conservatism. First, regardless of optimality of a conservative central banker, price-level targeting outperforms inflation targeting in many dimensions such as containing the deflation bias created by occasional financial crises and binding ZLB constraint. Second, conservatism in monetary policy may lead to greater deflation bias and more frequent financial crises in an economy suffering from insufficient aggregate demand due to income inequality, though the same conservatism may have a benefit of providing a protection against downside risk conditioned upon an actual financial crisis. The second conclusion seems to suggest that in an economy in which an active link exists among income inequality, deflation pressureandfinancialcrises, amixofadovishcentralbankerandextraordinarystabilizationmeasures that are implemented only during financial crises may improve upon symmetric monetary policy rules. 7.2 The Role of the ZLB Constraint Before we move onto the analysis of optimal monetary policy, we want to emphasize that the trade-off facing a conservative central banker is due to the presence of the ZLB constraint. In other words, the presence of occasional financial crises alone does not create the trade-off in monetary policy. To 30Outputlossassociatedwithappointingaconservativecentralbankhastwosources: skeweddistributionsofproductioninputsandpricedispersionduetothedeflationbias. Thelatterdoesnotexistforprice-leveltargetingregardlessof policy conservatism as the rule eliminates the bias almost completely. 33

Figure 8: Liquidity Trap and the Trade-off Facing a Conservative Central Banker 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 6 7 8 9 Mean unemployment rate etar noitalfni .veD .dtS (a) Inflation targeting 0.4 =1 =20 0.3 =1 =1 0.2 =20 0.1 =20 0 5.5 6 6.5 7 Mean unemployment rate etar noitalfni .veD .dtS (b) Price-level targeting =0.1 =4 =0.1 =0.1 =4 =4 Baseline Crises, no ZLB No crises, ZLB Note: Thefigureshowsthelociofthemomentsofselectedendogenousvariablesasthecentralbankincreaseseitherthe inflationreactioncoefficientρ inpanel(a)(inflationtargeting)orthereactioncoefficientforprice-leveldeviationρ in π Π panel (b) (price-level targeting). illustrate that, Figure 8 shows what happens to the trade-off between inflation stabilization and bias reduction in the unemployment rate if there is no ZLB constraint (red triangles) and if there is no financial crises (black squares) with panel (a) showing the case of inflation targeting and panel (b) the case of price-level targeting. For comparison, the two panels also show the cases with both ZLB and financial crises (blue circles). Figure 8 clearly indicates that if there is no ZLB constraint (red triangles), the trade-off does not exist. In contrast, as indicated by black squares in the two panels, evenwithoutfinancialcrises, aconservativecentralbankerstillfacesthesametypeofmonetarypolicy dilemmainthepresenceoftheZLBconstraint.31 Inreality, financialcrisescontributetothemonetary policy dilemma only to the extent that financial crises make the ZLB constraint more likely to bind for a given monetary policy rule. From a monetary policymaker’s perspective, financial crises are no different from large ‘negative’ demand disturbances. As such, they can be completely accommodated by sufficiently lowering the interestrate. Hence, thedesirabilityofconservativecentralbankersismaintainedinthisenvironment. In fact, complete elimination of inflation volatility and bias in the unemployment rate can be achieved byaveryconservativecentralbankerbothintheinflationtargetingregimeandtheprice-leveltargeting regime. This experiment also makes it clear that the rationale for price-level targeting hinges on the presence of the ZLB constraint, not on the presence of endogenous financial crises. The inefficiency of inflation targeting in the presence of the ZLB constraint is clearly due to its principle of bygones are bygones. Notethatwehaveemphasizedthelinkbetweenincomeinequalityandaggregatedemand. However, such link cannot exist in the model without nominal rigidities. In other words, such link exists only 31This shows that the presence of ZLB constraint creates trade-off in implementing the dual mandate, which is consistent with what Nakata and Schmidt (2015) find in a canonical New Keynesian model. 34

Table 4: Selected Moments under Optimal Monetary Policy with Financial Crises but without the ZLB Constraint Inflation targeting Price-level targeting ρ =1e+10 ρ =1e+10 π Π 1. Corr(income inequality, y) -0.02 -0.02 2. Corr(income inequality, π) 0.00 0.00 3. E(π), quarterly, percent 0.00 0.00 4. S.D.(π) 0.00 0.00 5. E(income share of shareholders) 0.36 0.36 6. E(debt-to-income ratio of workers) 1.57 1.57 7. E(pδ) 1.32 1.32 Notes: The moments are computed using 250,000 simulations with identical set of random draws for the shocks in each column, with financial crises but without the ZLB constraint. when nominal rigidities exist and monetary policy is suboptimal in eliminating their effects. To verify this claim, in Table 4, we ignore the ZLB constraint and set the monetary policy coefficients at an extremely large level, which is optimal given the lack of cost-push shock. Lines3and4ofTable4showthatintheabsenceoftheZLBconstraint, suchmonetarypolicyrules eliminate completely the effects of nominal rigidities, making both mean and standard deviation of inflation equal to zero. Lines 1 and 2 confirm that in an economy with no effects coming from nominal rigidities, there exists no comovement between income inequality and output on the one hand and between income inequality and inflation on the other hand. This is not because the optimal monetary policy eliminates income inequality. As shown by lines 5∼7, the main financial characteristics of the economy are no different from the baseline economy. This shows that optimal monetary policy cannot eliminate financial crises due to income inequality, but can well insulate the business cycle from the effects of inequality. 7.3 Dual Mandate and Optimal Monetary Policy This section studies optimal monetary policy when the central bank has a dual mandate to stabilize the inflation rate and minimize the bias in the unemployment rate. This dual mandate is justified in ourmodeleconomybecauseoccasionalfinancialcrisesandbindingZLBconstraintgenerateatrade-off between the two objectives (see panel (a) of Figure 7), even when the economy is not subject to shocks that move inflation and output gap in opposite directions. These considerations support the following form of loss function: L = λE[π−E(π)]2+(1−λ)[E(u)−u∗]2, (59) where λ ∈ [0,1] is a preference parameter for the central bank and u∗ is the level of the unemployment rate in the absence of nominal rigidities, aggregate shocks and financial crises. We restrict our study within the boundary of optimized simple rules that minimize the loss function (59). By varying λ from 0 to 1 and optimizing the monetary policy rules to minimize the loss function for a given λ, we can derive an “efficient policy frontier” (EPF) in the spirit of Taylor (1979), but between inflation volatility and unemployment bias, instead of between inflation volatility and 35

unemployment volatility. We start by optimizing ρ for the symmetric inflation targeting rule (56) and ρ for the symmetric π Π price-leveltargetingrule(57). Inaddition,wealsostudythebenefitsofadoptingasymmetricmonetary policy rules with the potential of mitigating the degree of skewness in the distributions of endogenous quantitiesandpricesthatoccasionalfinancialcrisesandbindingZLBconstraintgenerate. Specifically, we optimize the inflation targeting rule (56) and price-level targeting rule (57) under the assumption that the inflation target of these policy rules follows a time-varying law of motion: π∗ = (1−ρ )π∗+ρ π∗ +σ δB, (60) t π∗ π∗ t−1 π∗ t where 0 ≤ ρ < 1 and σ > 0.32 We refer to the alternative monetary policy rules as asymmetric π∗ π∗ rules because the inflation target deviates from its long run value π∗ only to the upside when the crisis indicator δB is activated. Under the two asymmetric policy rules, the central bank announces t an increase in the inflation target by σ in response to an eruption of a financial crisis and brings π∗ it back to the long-run target π∗ only gradually. As shown previously, a conservative central banker has the benefit of mitigating the effects of financial crises only at the cost of increasing the degrees of skewed distributions and thereby increasing the mean level of the unemployment rate. The asymmetric policy rules allow the economy to have such policy conservatism only during financial crises. In our optimization, we fix σ = 0.01 so that the annual inflation target is raised by 1 percentage point π∗ immediately after a financial crisis hits the economy. We then optimize (ρ ,ρ ) for asymmetric infla- π π∗ tion targeting and (ρ ,ρ ) for asymmetric price-level targeting. If such an asymmetric adjustment Π π∗ of inflation targeting brings more harm than good, our optimization procedure will choose a value for ρ close to zero. π∗ Figure 9 compares four EPFs: inflation targeting (blue circles); asymmetric inflation targeting (black squares); price-level targeting (red triangles); and asymmetric price-level targeting (green pluses).33 The pink diamond bullet locates the moments of the baseline economy. The cyan circled bullet marks the location of socially optimal allocation where inflation volatility is zero and the level of mean unemployment rate is given by u∗. Since these are all efficient frontiers, all frontiers are located inward from the baseline economy. It is not difficult to see how the asymmetric inflation targeting rule improves upon the symmetric counterpart in terms of the reduced mean unemployment rate. Even when the central bank assigns zero weight on the unemployment bias term, which can be found at the right-most point of the EPF, it achieves lower unemployment than the level achieved when the symmetric inflation targeting rule is optimized under the full weight on the bias term. What is remarkable in Figure 9 is that the strict price-level targeting rule can improve upon both inflation targeting rules in terms of reducing the variance of inflation and the mean unemployment 32Whilewecallequation(60)time-varyinginflationtarget,thesametermcanbeinterpretedastime-varyingintercept adjustment with the opposite sign. That is, a monetary policy rule that adjusts the constant term i∗ downward in response to financial crises. 33In this optimization, we impose 0.5 ≤ ρ ≤ 20 and 0.01 ≤ ρ ≤ 4. The upper bounds never bind, while the lower π Π bounds bind in some instances (see Tables 5 and 6). 36

Figure 9: Efficient Policy Frontiers: Standard Deviation of Inflation Rate and Mean Unemployment Rate 0.8 0.7 0.6 0.5 0.4 0.3 0.0 5.5 6 6.5 7 7.5 8 8.5 Mean unemployment rate etar noitalfni .veD .dtS λ=0 λ=0.5 λ=0 λ=0 λ=0.5 λ=1 λ=1 λ=0 Baseline IT λ=0.5 ITA λ=0.5 λ=1 λ=1 PT PTA LR optimal Note: IT, ITA, PT, and PTA stand for inflation targeting, asymmetric inflation targeting, price-level targeting and asymmetric price-level targeting, respectively. The pink diamond bullet locates the moments of the baseline economy. Thecyancircledbulletmarksthelocationofsociallyoptimalallocationwhereinflationvolatilityiszeroandthelevelof the unemployment rate is given by its long-run natural rate. rate regardless of the weights given to the loss function. In other words, when the weight given to the variance of inflation rate is zero, it still achieves much lower inflation volatility than any points in the two EPFs of the inflation targeting rules. Furthermore, when the weight given to the bias term is zero, it still achieves lower unemployment rate than most of the points on the two EPFs of inflation targeting rules. Finally, a large reduction in the mean unemployment rate is possible if the central bank adopts the asymmetric price-level targeting rule and assigns a nontrivial weight on the bias correction term. In particular, when a strictly positive weight is given to the bias correction term, the asymmetric price-level targeting can achieve a large reduction in the mean level of the unemployment rate on the order of 2.2 percentage points versus the baseline economy. In fact, when the bias reduction is given a weight close to λ = 0.5, it attains the long-run natural rate of unemployment. Assigning a weight greater than this level to the bias correction results in a large increase in inflation volatility without further reduction in the unemployment rate beyond its long-run natural rate.34 Table 5 shows the optimized coefficients of symmetric inflation and price-level targeting rules for selected values of λ together with key moments of the model economy under each policy rule. Not 34In Appendix D, we consider an alternative loss function that penalizes the level of unemployment rather than the biasterm. Inthiscase,theEPFshowsthatalargerreductioninthemeanlevelofunemploymentrateontheorderof4 percentage points from the baseline economy is possible at some costs of inflation bias and increased inflation volatility. 37

Table 5: Optimized Rules: Inflation Targeting vs. Price-Level Targeting Inflation targeting Price-level targeting 1. λ 0.00 0.50 1.00 0.00 0.50 1.00 2. ρ 0.50 0.77 12.51 - - - π 3. ρ - - - 0.04 0.40 3.87 Π 4. ρ - - - - - - π∗ 5. E(u) 7.63 7.72 8.41 6.23 6.40 6.57 6. E(π) -0.18 -0.19 -0.25 0.00 0.00 0.00 7. S.D.(u) 5.91 5.44 5.09 4.28 2.95 2.81 8. S.D.(π) 0.71 0.66 0.59 0.44 0.31 0.30 9. Skewness(u) 1.02 1.54 3.44 0.76 1.95 2.12 10. Skewness(π) -0.76 -1.14 -3.40 -0.61 -1.50 -1.54 11. E(pδ) 1.32 1.33 1.34 1.33 1.33 1.34 12. Frequency of ZLB., pct. 2.08 3.42 9.74 0.77 4.02 9.35 13. Mean MP. shock, bps. 0.37 1.00 30.85 0.75 16.48 176.39 14. Loss function value 3.10 3.88 3.46 0.13 0.61 0.88 15. Welfare of workers, CE 0.01 0.01 -0.03 0.38 0.37 0.38 16. Welfare of shareholders, CE -0.29 -0.24 0.47 -2.23 -2.24 -2.23 17. Aggregate welfare, CE -0.01 -0.01 -0.01 0.25 0.24 0.25 Note: Weimpose0.5≤ρ π ≤20and0.01≤ρ Π ≤4intheoptimization,whileρ π∗ andσ π∗ aresetequaltozero. MeanMP shock is the annualized mean of monetary policy shocks (both contemporaneous and anticipated news shocks) required to satisfy the ZLB constraint. CE stands for consumption equivalent that needs to be given to the baseline economy to achieve the same level of welfare. surprisingly, as the loss function assigns a greater weight on the stabilization of the inflation gap, the reactioncoefficienttotheinflationgapandprice-levelgapareincreased(lines2and3).35 Consequently, the frequency of a binding ZLB constraint and the sizes of anticipated monetary policy shocks are substantially higher (lines 12 and 13), and the flip side is a larger bias in the mean unemployment rate (line 5). The price-level targeting rule dominates inflation targeting in terms of bias reduction in the unemployment rate regardless of the value of λ. Furthermore, the optimized price-level targeting rules eliminate deflation bias in contrast to inflation targeting rules (line 6). Finally, higher order moments shown in lines 7∼10 also confirm the improved stabilization properties of the price-level targeting rule in terms of variance reduction of the unemployment rate and the inflation rate and in terms of correcting skewed distributions of these key macroeconomic variables. Overall, smaller loss function values (line 14) associated with price-level targeting summarize the better performance of this rule against the inflation targeting rule in our model economy 36 Lines 15 and 16 in Table 5 report the consumption equivalents that need to be given to the agents in the baseline economy to be indifferent between the economy under the optimized policy rules and thebaselineeconomy. Assuch, positivenumbersrepresentwelfareimprovementandnegativenumbers welfaredeteriorationwithrespecttothebaselineeconomy.37 Whatisnotableaboutthewelfareresults 35ItisnotablethatsocalledTaylorprincipleisnotappliedinthecaseofinflationtargeting. Infact,inflationtargeting rule with λ=0 hits the lower bound ρ =0.50. π 36Lossfunctionvaluescannotbecomparedacrossdifferentvaluesofλastheyrepresentdifferentpreferences. However for a given λ, loss function values can be compared for different policy rules. 37All welfare values are based on a first order approximation. The presence of multiple anticipated news shocks to satisfy the ZLB constraint in expectations creates a large number of auxiliary variables to limit the lag structure of the 38

Table 6: Optimized Rules: Asymmetric Inflation Targeting vs. Asymmetric Price-Level Targeting Asymmetric Asymmetric Inflation targeting Price-level targeting 1. λ 0.00 0.50 1.00 0.00 0.50 1.00 2. ρ 0.68 1.22 2.11 - - - π 3. ρ - - - 0.01 0.96 0.98 Π 4. ρ 0.99 0.99 0.98 0.94 0.96 0.95 π∗ 5. E(u) 6.55 6.67 7.50 5.87 5.97 6.08 6. E(π) -0.03 -0.01 -0.13 0.06 0.08 0.06 7. S.D.(u) 5.29 4.91 4.77 5.64 2.73 2.74 8. S.D.(π) 0.64 0.60 0.57 0.60 0.30 0.30 9. Skewness(u) 1.21 1.82 2.56 0.42 1.67 1.76 10. Skewness(π) -0.83 -1.34 -2.10 -0.33 -0.81 -0.95 11. E(pδ) 1.33 1.34 1.34 1.33 1.34 1.34 12. Frequency of ZLB., pct. 3.99 5.70 6.23 0.16 6.05 5.92 13. Mean MP. shock, bps. 1.16 3.16 5.74 0.11 61.70 61.45 14. Loss function value 0.45 2.09 3.28 0.00 0.46 0.90 15. Welfare of workers, CE 0.48 0.61 0.22 0.65 0.58 0.54 16. Welfare of shareholders, CE -1.98 -2.11 -0.89 -2.83 -3.13 -2.95 17. Aggregate welfare, CE 0.36 0.48 0.17 0.47 0.39 0.36 Note: We impose 0.5≤ρ π ≤20, 0.01≤ρ Π ≤4 and 0.0≤ρ π∗ ≤0.99 in the optimization. σ π∗ is calibrated at 0.01. MeanMPshockistheannualizedmeanofmonetarypolicyshocks(bothcontemporaneousandanticipatednewsshocks) required to satisfy the ZLB constraint. CE stands for consumption equivalent that needs to be given to the baseline economy to achieve the same level of welfare. isthataparticularmonetarypolicyrulethatimprovesthewelfareofonetypeofagentmaynotimprove the welfare of the other type of agent. In particular, a rule that improves the welfare of workers in turn deteriorates the welfare of shareholders. Overall, these findings suggest that monetary policy can have distributional consequences, in line with the empirical results of Coibion et al. (2017). As for the population weighted welfare (line 17), the price-level targeting rule brings welfare gains as the rule is more effective in controlling deflation bias and unemployment bias. Finally, Table 6 summarizes the optimization results for the asymmetric inflation and price-level targeting rules. Qualitative features remain the same as in Table 5 except for three features. First, larger welfare gains can be achieved by using a time-varying inflation target during financial crises, and these gains are particularly larger for inflation targeting compared to price-level targeting. This resultwasexpectedgiventhattheEPFinFigure9movesinwardmoresubstantiallywhentheinflation targeting rule is combined with the time-varying inflation target. The symmetric price-level targeting rule already does a good job in correcting for asymmetric business cycles, thus the additional gains from adopting the asymmetric component are relatively modest. Second, the optimized coefficients ρ and ρ tend to be smaller with the time-varying inflation π Π target. Thistendencyconfirmsourconjecturethatamixofadovishcentralbankerandatime-varying inflation target is likely to be optimal in an economy that suffers from insufficient aggregate demand due to income inequality. The fact that the frequency of a binding ZLB constraint and the mean size model to one period. This large number of auxiliary variables are not suitable for second order approximation. 39

ofmonetarypolicyshocksrequiredtosatisfytheZLBconstrainttendtobelowerwiththeasymmetric policy rules also suggests that such opportunistic adjustment of the inflation target can be an effective tool in fighting the deflation bias imposed by the ZLB constraint. Third, the population weighted welfare gains indicate that a conservative central banker tends to reduce the size of overall welfare. In particular, the gains for workers generally decline as the weight given to inflation stabilization is increased. Given that wage income is the major income source for lowincomeearners, acrucialfactorfortheirwelfaregainsishoweffectiveaparticularmonetarypolicy rule is in reducing the bias in the unemployment rate. 8 Conclusion The increase in income inequality in the United States during the recent decades has become a reason for concern. This paper shows the important role that income inequality plays in the economy. By usingatwo-agentgeneralequilibriummodel, weshowthatincomeinequalitycangeneratelowaggregate demand, deflation pressure, excessive credit growth and financial instability. The main mechanism to deliver those results is the desire to accumulate financial wealth by shareholders motivated by their “love of wealth” preferences. The paper shows that both occasional eruptions of endogenous financial crises and the zero lower bound constraint on the nominal interest rate generate distributions of equilibrium prices and quantities that are highly skewed to the downside. In that environment, a conservative central banker faces a trade-off between stabilizing inflation volatility and increasing the mean level of the unemployment rate. This trade-off highlights the inefficiency of symmetric monetary policy rules in the sense that a simultaneous reduction in inflation volatility and the mean unemployment rate is feasible when an asymmetric policy rule is adopted. Our results highlight the close link between income inequality and financial instability. While the focus in this paper is on monetary policy issues, our findings suggest an important role for fiscal policy as a macroprudential tool. In particular, redistribution policies that ameliorate income inequality can potentially bring additional benefits in terms of preventing excessive credit growth, reducing the probability of financial crises and further improving macroeconomic outcomes. We leave the study of fiscal and macroprudential policies in our model economy for future research. References Adrian, Tobias and Fernando Duarte, “Financial Vulnerability and Monetary Policy,” Staff Reports 804, Federal Reserve Bank of New York 2016. , Nina Boyarchenko, and Domenico Giannone, “Vulnerable growth,” Staff Reports 794, Federal Reserve Bank of New York 2016. Atkinson, Anthony B., Thomas Piketty, and Emmanuel Saez, “Top Incomes in the Long Run of History,” Journal of Economic Literature, March 2011, 49 (1), 3–71. 40

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Online Appendix - Not Intended for Publication A Non-Stochastic Steady State A.1 Pinning Down the Endogenous Variables To determine the steady state of the model, we guess the values of the borrowing level b and the default probability pδ. We assume a zero inflation steady state, i.e., π =1. Equation (49) then satisfies: (cid:104) (cid:105)1/(1−γ) 1= (1−ϕ)p1−γ +ϕ , (A.1) 0 resulting in p =1. Equations (46)∼(48) imply: 0 γµyπγ/(1−βϕ) p = =1, (A.2) 0 (γ−1)yπ(γ−1)/(1−βϕ) which results in: γ−1 µ= . (A.3) γ In the steady state qK =1 from equation (50). Then, equation (23) determines: r =1/βK −(1−δ). (A.4) Using this in equation (36), we define the steady state output-capital ratio as: y 1/βK −(1−δ) ρ ≡ = . (A.5) y/k k µα From the production function, we have: y (cid:16)n(cid:17)1−α =z , k k or equivalently n (cid:16)ρ (cid:17)1/(1−α) y/k ρ ≡ = . (A.6) n/k k z Again from the production function, we derive output-labor ratio as: y (cid:16)n(cid:17)−α ρ ≡ =z =zρ−α. (A.7) y/n n k n/k These ratios can be used to determine the levels of output and capital once the level of employment n is determined. We determine the steady state job market tightness by calibrating the job finding rate from the data, i.e., p(θ)=p. Using equation (A.7), we simplify the expression for the wage as: ξ w =ηµ(1−α)ρ +(1−η)bU +(1−ρ)η[βK −βW(1−p)] . (A.8) y/n q Equations (34) and (35) imply in steady state: ξ ξ =µ(1−α)ρ −w+(1−ρ)βK . (A.9) q y/n q Equations (A.8) and (A.9) provide us with two equations for two unknowns, w and ξ/q. Substituting equation (A.8) in equation (A.9) and solving it for ξ/q yields: (cid:104) (cid:105) (1−η) µ(1−α)ρ −bU ξ y/n J = = . (A.10) q 1−(1−ρ)βK{1−η[1−(1−p)βW/βK]} 45

Using this in equation (43) yields the steady state wage. Since ξ/q =ξθ/p(θ)=ξθ/p, we have:  (cid:104) (cid:105)  (1−η) µ(1−α)ρ −bU p y/n θ =  . (A.11) ξ 1−(1−ρ)βK{1−η[1−(1−p)βW/βK]} This determines the matching function efficiency as: p ζ = . (A.12) θ(cid:15) Note that we pin down ζ endogenously by matching the job finding rate of the model with the data. This requires us to treat ζ as an endogenous variable, which always takes the same value as in the steady state. From equation (32), we have the steady state employment stock as: q q n= v = θu. (A.13) ρ ρ Substituting this expression in equation (27) yields: q u=1−χ−(1−ρ) θu. ρ Solving this for u yields 1−χ u= . 1+(1−ρ)qθ/ρ The vacancy posting is determined as: v =θu, and using this in equation (A.13) gives us the steady state employment stock. The levels of capital and output are then given by: y =ρ n, (A.14) y/n and k =ρ−1 n. (A.15) n/k In a steady state where a default event does not occur: 1 cW =cN =(qB−1)b+ [wn+(1−χ−n)bU], (A.16) 1−χ and 1−χ δk rk ΠY ΠK T cK =b(1−qB) −qK + + + + , (A.17) χ χ χ χ χ χ where ΠY =y−wn−rk−ξv, (A.18) and ΠK =qKx−x=0. (A.19) Substituting equations (A.18) and (A.19), the steady state version of the FOC for borrowing given by: qB =βB(1−pδh), (A.20) and the balanced budget constraint for government given by: (cid:18) (cid:19) 1 T =−(1−χ−n)bU +χbG 1− 1+i 46

into equation (A.17), we express the steady state consumption level of shareholders as: 1 cK(b)= {b[1−β (1−pδh)](1−χ)−qKδk+y−wn−ξv−(1−χ−n)bU}. (A.21) χ W Note that the right-hand side of the above has only one unknown, b. In order to pin down b, we equate the FOCs of the two agents regarding borrowing and lending, which results in ψ (cid:20) (cid:18) 1−χ (cid:19)(cid:21)−1/σb (βW −βK)(1−pδh)= 1+b . (A.22) [(1−s)cK(b)]−1/σc χ Our initial guess b should satisfy equation (A.22) given our guess pδ. We now discuss how the guess solution pδ can be validated. A.2 Probability of Default We envision a steady state in which there has been no default for a long time such that the lagged value of pecuniary default cost has converged to zero, i.e., ν = 0. In such an environment, if a default occurs today, −1 the value of default cost becomes ν = γ . If a default does not occur today, the value of default cost becomes ν ν = 0. If a default occurs again tomorrow after an episode of default today, ν(cid:48) = (1+ρ )γ . If a default does ν ν not occur tomorrow after an incidence of default, ν(cid:48) =ρ γ . If a default occurs tomorrow after no incidence of ν ν default today, ν(cid:48) =γ . If a default does not occur either today or tomorrow, ν(cid:48) =0. Note that all of these four ν cases are hypothetical in that a default does not occur in our steady state. We denote four cases of consumption level tomorrow under the four scenarios by cDD, cDN, cND and cNN. These are given by: 1 cDD =[qB−(1−h)]b+ [wn+(1−χ−n)bU −(1+ρ )γ y], (A.23) 1−χ ν ν 1 cDN =(qB−1)b+ [wn+(1−χ−n)bU −ρ γ y], (A.24) 1−χ ν ν 1 cND =[qB−(1−h)]b+ [wn+(1−χ−n)bU −γ y], (A.25) 1−χ ν and 1 cNN =(qB−1)b+ [wn+(1−χ−n)bU]. (A.26) 1−χ Note that all right-hand side variables are already determined above under the initial guess solution of pδ. However, this guess solution needs to be verified. In the steady state, the continuation values of default and non-default are given by: (cid:18) (cDD−scD)1−1/σc (cid:19) (cid:18) (cDN −scD)1−1/σc (cid:19) VD =βWpδ +VD +βW(1−pδ) +VN (A.27) 1−1/σ 1−1/σ c c and (cid:18) (cND−scN)1−1/σc (cid:19) (cid:18) (cNN −scN)1−1/σc (cid:19) VN =βWpδ +VD +βW(1−pδ) +VN . (A.28) 1−1/σ 1−1/σ c c By subtracting equation (A.27) from equation (A.28), one can derive ∆V(pδ) as: ∆V(pδ)=VD−VN (cid:18) (cDD−scD)1−1/σc (cND−scN)1−1/σc (cid:19) =βWpδ − 1−1/σ 1−1/σ c c (cid:18) (cDN −scD)1−1/σc (cNN −scN)1−1/σc (cid:19) +βW(1−pδ) − . 1−1/σ 1−1/σ c c 47

Figure 10: Monetary Policy and Government Bond Market Clearing 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 1.33 1.34 1.35 1.36 1.37 1.38 1.39 Per capita government bond holdings etar tseretnI Increase in MU B-C = Excess Demand C B Monetary Policy Risk Premium A Monetary Policy Rate Demand for Government Bond Supply of Government Bond Hence ∆U(pδ) is given by: (cD−scW)1−1/σc (cN −scW)1−1/σc ∆U(pδ)= − +∆V(pδ). 1−1/σ 1−1/σ c c Note that we express ∆U and ∆V functions of our initial guess pδ. The default probability is then given by: (cid:37) pδ = (A.29) 1+exp(−ς∆U(pδ)) Equation(A.29)canthenbeviewedasafixedpointproblem. Hence, thesteadystateneedstobepinneddown by the simultaneous equations (A.22) and (A.29). Once this is done, the consumption level of shareholders can be determined by equation (A.21). Using this consumption level, we can then evaluate the marginal utility of shareholders and then finally determine the level of government debt consistent with the calibrated risk-free bond rate i: (cid:26) ΛK (cid:20) 1 (cid:21)(cid:27)−σg bG = −βK −1. (A.30) ψG (1+i∗)φ A.3 Monetary Policy Transmission Equation (A.30) describes the shape of demand for government bonds. Figure 10 shows that the demand for government bonds is an increasing function of the risk-free rate i∗ and the risk premium φ. Since the supply of government bonds is fixed, the equilibrium interest rate is determined as the intersection of the upward sloping demand curve and the vertical supply curve as marked by point A in Figure 10. When either the policy rate is raisedortheriskpremiumiselevated,thehorizontallineshiftsfromtheblacksolidlinetotheblackdottedline. Giventhatthesupplyofgovernmentbondsisfixed,thisgeneratesexcessdemandforgovernmentbonds,marked byB-C.Toclearthemarket, themarginalutilityofshareholdersΛK hastoincrease, i.e., theconsumptionlevel of shareholders has todecline suchthat the demand curve moves to the left as indicated by the red dashedline. Thus the new equilibrium is found at point C. 48

B Solution Method Themodelisinherentlynonlinearfortworeasons: theZLBconstraintandthebinarynatureofdefault. However, given the large number of state variables, we simulate the model piecewise linearly. This section illustrates how such simulations can be executed. Suppose that we solve the model linearly, ignoring the ZLB constraint and default. In our model economy, δB ∈{0,1}, has both endogenous and exogenous natures. δB is endogenous in t t the sense that the probability of δB =1 is endogenously determined. However, given the probability of default, t whether or not the default occurs is stochastic. In this latter sense, δB can be viewed as an exogenous variable. t Step 1. We linearize the model around the non-stochastic steady state in which we assume that the economy has not experienced a crisis for a long period of time and the default cost ν has converged to zero. We can express the state space representation for the system of equations as: (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) s As +B(cid:15) (cid:15)EX t = t−1 t , (cid:15) = t (B.1) x Cs t (cid:15)EN t t t and (cid:15)EX =[ (cid:15) (cid:15) (cid:15) ](cid:48) t z,t η,t φ,t (B.2) (cid:15)EN =[ δB (cid:15) (cid:15) ··· (cid:15) ](cid:48) t t 0,t 1,t n,t wheres andx arestateandpolicyvectorswithdimensionNotethat(cid:15)EX followsGaussiandistributions. t t t Although(cid:15)ENcanbeviewedas‘shocks’fromamechanicalpointofview,wetreat(cid:15)EN asendogenousvarit t ablessincethesizesoftheshocksshouldbedeterminedendogenously. Inparticular,thecontemporaneous and anticipated news shocks to monetary policy should satisfy (cid:26) (cid:20) (cid:18) πY −π∗(cid:19)(cid:21) (cid:88)n (cid:27) (cid:15) =max 0,− i∗+ρ t − (cid:15) , (B.3) k,t π 4 j=k+1 j,t−j for k = 0,...,n. In words, we use the current monetary policy shock (k = 0) to satisfy the ZLB today and the news shocks (k > 0) to satisfy the ZLB in expectations. Note that E [i ] and E [π ] for t t+j−1 t t+j j = 1,...,n should be consistent with (cid:15) and (cid:15) in the sense that when (cid:15) and (cid:15) are given to the 0,t j,t 0,t j,t economy, the agents’ expectations about E [i ] and E [π ] for j = 1,...,n should be identical with t t+j−1 t t+j those showing up on the right hand sides of equation (B.3). Step 2. In each period, we simulate the economy with random draws for (cid:15)EX under the assumption that t (cid:15)EN =0. The system of equations will return the default probability t (cid:37) pδ(0) =Ξ(∆UW)= , t t 1+exp(−ς∆U ) t and the endogenous component of monetary policy (cid:18) πY −π∗(cid:19) i(0) =i∗+ρ t . t π 4 Step 3. We then need to check if a default occurs and/or the ZLB is violated. To see if the ZLB is violated or not, we simply need to check if i is negative. We also need to check whether or not E [i ] ≥ 0 is t t t+j satisfied. Adefaultcanbedetectedinthefollowingway. Takearandomdrawfromauniformdistribution over a support [0,1]. If this draw is strictly greater than pδ(0), we conclude that a default did not occur. t If the draw is strictly less than pδ(0), we conclude that a default did occur. There can be four cases: (i) t no default, no ZLB violation; (ii) default, no ZLB violation; (iii) no default, ZLB violation; (iv) default, ZLB violation. Case (i). Since there is no default and the ZLB is not violated, we move to the next period. Case (ii). Since a default occurs, we simulate the economy again with δB = 1 and check if no ZLB t violation is still valid both in current period and in expectations. If the ZLB constraint binds or is expected to bind in future, then move to Case (iv). Otherwise we move to the next period. 49

Case (iii). SincetheZLBconstraintisviolated(i(0) <0),wherethesuperscript(0)indicatesthenumber t of iterations in the ZLB loop, we set (cid:15)(1) = −i(0) (cid:15)(1) = max{0,−E [i(0) ]} to satisfy E [i(1) ] ≥ 0. 0,t t j,t t t+j t t+j We then simulate the economy again and check if the ZLB in current period or in expectations is satisfied. This process cannot be done in one step because once the monetary policy shock or news shock is given to the system, the economy reacts to the shocks, and the initial amounts of monetarypolicyshockstosatisfytheZLBmaynotbeenough. Iftheconstraintisstillnotsatisfied, we set (cid:15)(2) =−i(0)−i(1) and (cid:15)(2) =−E [i(0) ]−E [i(1) ], and simulate the economy again and check 0,t t t j,t t t+j t t+j if the ZLB is satisfied in current or in future expectations. If the ZLB is finally satisfied after S simulations, the final level of the monetary policy shock is given by: S S (cid:15)(S) =− (cid:88) i(s) and (cid:15)(S) =− (cid:88) E [i(s) ] for j =1,...N. (B.4) 0,t t j,t t t+j s=0 s=0 An important thing in this iteration is to check whether or not a default occurs owing to the endogenous reaction of the economy to the monetary policy shock to satisfy the ZLB for a given simulationnumberj. Ifadefaultoccursforthisreason, thenmovetoCase(iv). Ifnot, movetothe next period. Case (iv). In this case a crisis occurs and the ZLB binds. Hence we set δB = 1 and determine the size t of the monetary policy shock according to equation (B.4). For a given simulation number j, we need to check if a crisis occurs or not. In principle, it is logically possible, although not likely, that the economy gets itself out of the default mode due to the monetary policy shock. If this happens, move to Case (iii). C System of Equations There are variables 50 endogenous variables within the system:   cK b l c π qB i qK k r t t t t t t t t t t  T t ΛK t ΛW t w t n t ν t y t cW t mK t,t+1 ∆U t    X t ≡  pδ t v t u t θ t p(θ t ) q(θ t ) z t mW t,t+1 J t µ t    w t W t η t p 0,t P t N P t D x t φ t ΓW t ∆ t  cD cN ΠY ΠK cDD cDN cND cNN Ω ΓJ t t t t t t t t t t The following provides the complete list of 50 equations for 50 endogenous variables: b l =(1−hδB) t−1, (C.1) t t π t c =χcK +(1−χ)cW, (C.2) t t t (cid:18)bG bG (cid:19) T =(1−χ−n )bU +χ t−1 − t , (C.3) t t π 1+i t t ΛK =(cK −scK )−1/σc, (C.4) t t t−1 ΛK mK =βK t+1, (C.5) t,t+1 ΛK t (cid:20) (cid:18)r +(1−δ)qK (cid:19)(cid:21) 1=E mK t+1 t+1 , (C.6) t t,t+1 qKφ t t 1 (cid:20) φ (cid:21) ψG =E mK t + (1+bG)−1/σg, (C.7) 1+i t t,t+1π ΛK t t t+1 t 50

(cid:20) 1 (cid:21) ψB (cid:20) (cid:18) 1−χ (cid:19)(cid:21)−1/σb qB =E mK (1−hpδ ) + 1+b , (C.8) t t t,t+1 t+1 π ΛK t χ t+1 t (cid:20) (cid:21) 1 qB =E mW (1−hpδ ) , (C.9) t t t,t+1 t+1 π t+1 1 cW =qBb −l + [w n +(1−χ−n )bU −ν y ], (C.10) t t t t 1−χ t t t t t ν =ρ ν +γ δB, (C.11) t ν t−1 ν t ΛW =(cW −scW )−1/σc, (C.12) t t t−1 ΛW mW =βW t+1, (C.13) t,t+1 ΛW t (cD−scW )1−1/σc (cN −scW )1−1/σc ∆U = t t−1 − t t−1 (C.14) t 1−1/σ 1−1/σ c c (cid:34) (cid:32) (cid:33)(cid:35) (cDD −scD)1−1/σc (cND −scN)1−1/σc +βWE pδ t+1 t − t+1 t t t+1 1−1/σ 1−1/σ c c (cid:34) (cid:32) (cid:33)(cid:35) (cDN −scD)1−1/σc (cNN −scN)1−1/σc +βWE (1−pδ ) t+1 t − t+1 t , t t+1 1−1/σ 1−1/σ c c b 1 cD =qBb −(1−h) t−1 + [w n +(1−χ−n )bU −(ρ ν +γ )y ], (C.15) t t t π 1−χ t t t ν t−1 ν t t b 1 cN =qBb − t−1 + [w n +(1−χ−n )bU −ρ ν y ], (C.16) t t t π 1−χ t t t ν t−1 t t b 1 cDD =qBb −(1−h) t−1 + [w n +(1−χ−n )bU −[ρ (ρ ν +γ )+γ ]y ], (C.17) t t t π 1−χ t t t ν ν t−2 ν ν t t b 1 cDN =qBb − t−1 + [w n +(1−χ−n )bU −ρ (ρ ν +γ )y ], (C.18) t t t π 1−χ t t t ν ν t−2 ν t t b 1 cND =qBb −(1−h) t−1 + [w n +(1−χ−n )bU −(ρ ρ ν +γ )y ], (C.19) t t t π 1−χ t t t ν ν t−2 ν t t b 1 cNN =qBb − t−1 + [w n +(1−χ−n )bU −ρ ρ ν y ], (C.20) t t t π 1−χ t t t ν ν t−2 t t (cid:37) pδ = , (C.21) t 1+exp(−ς∆U ) t u =1−χ−(1−ρ)n , (C.22) t t−1 p(θ )=ζθ(cid:15), (C.23) t t q(θ )=ζθ(cid:15)−1, (C.24) t t ξ =J , (C.25) q(θ ) t t 51

y ϑ (cid:18) w (cid:19)2 J =µ (1−α) t −w − t −1 w +(1−ρ)E [mK J ], (C.26) t t n t 2 w t−1 t t,t+1 t+1 t t−1 ΓW =−1, (C.27) t (cid:18) w (cid:19) (cid:40) ϑ (cid:34)(cid:18) w (cid:19)2 (cid:35)(cid:41) ΓJ =−1−ϑ t −1 +(1−ρ)E mK t+1 −1 , (C.28) t w t t,t+12 w t−1 t η Ω = t , (C.29) t η +(1−η )ΓJ/ΓW t t t t Ω W = t J , (C.30) t 1−Ω t t (cid:20) (cid:21) Ω w =bU +(1−ρ)E mW p(θ ) t+1 J , (C.31) t t t,t+1 t+1 1−Ω t+1 t+1 y ϑ (cid:18) w (cid:19)2 w =Ω µ (1−α) t +(1−Ω )bU −Ω t −1 w (C.32) t t t n t t2 w t−1 t t−1 (cid:20)(cid:18) (cid:19) (cid:21) Ω ξ +(1−ρ)E Ω mK −(1−Ω )mW (1−p(θ ) t+1 , t t t,t+1 t t,t+1 t+1 1−Ω q(θ ) t+1 t+1 PN =π(1−ε)γ(cid:8) πεγγµ y +ϕE (cid:2) mK PN (cid:3)(cid:9) , (C.33) t t t t t t t,t+1 t+1 PD =π(1−ε)(γ−1) (cid:110) πε(γ−1)(γ−1)y +ϕE (cid:2) mK PD (cid:3)(cid:111) , (C.34) t t t t t t,t+1 t+1 PN p = t , (C.35) 0,t PD t (cid:104) (cid:105)1/(1−γ) π = (1−ϕ)p1−γ +ϕπε(1−γ) , (C.36) t 0,t t−1 (cid:18) x (cid:19) (cid:40) κ (cid:34)(cid:18) x (cid:19)2 (cid:35)(cid:41) qK =1+κ t −1 −E mK t+1 −1 , (C.37) t x t t,t+12 x t−1 t i =i∗+ρ (cid:18) π t π t−1 π t−2 π t−3 −π∗(cid:19) + (cid:88) n (cid:15) , (C.38) t π 4 j,t−j j=0 logz =ρ logz +(cid:15) , (C.39) t z t−1 z,t logη =(1−ρ )logη+ρ logη +(cid:15) , (C.40) t η η t−1 η,t logφ =ρ logφ +(cid:15) , (C.41) t φ t−1 φ,t y =∆−1z kα n1−α, (C.42) t t t t−1 t ∆ =πγ[(1−ϕ)p−γ +ϕπ−vγ], (C.43) t t 0,t t−1 n =(1−ρ)n +q(θ )v , (C.44) t t−1 t t 52

k =(1−δ)k +x , (C.45) t t−1 t y r =µ α t, (C.46) t t k t v θ = t, (C.47) t u t ϑ (cid:18) w (cid:19)2 ΠY =y −w n − t −1 w n −r k −ξv , (C.48) t t t t 2 w t−1 t t t−1 t t−1 (cid:34) κ (cid:18) x (cid:19)2 (cid:35) ΠK =qKx − x + t −1 x , (C.49) t t t t 2 x t−1 t−1 bG bG 1−χ 1 cK = t−1 − t +(l −qBb ) − {qK[k −(1−δ)k ]+r k +ΠY +ΠK −T +ν y }. (C.50) t π 1+i t t t χ χ t t t−1 t t−1 t t t t t t t 53

D Additional Results D.1 Replication of Paul (2017) In what sense can the moments shown in Table 3 can be considered representative of reality? To answer this question, wereplicatetheempiricalexerciseofPaul(2017): wetestthepredictivepowerofcreditinforecasting the probability of financial crises, with and without controlling for income inequality using our simulated data. Specifically,wereplicatePaul(2017)’searlywarningsignalmodelusingthebinarydataofcrisisindicatorduring our stochastic simulation. The logit model is specified as: (cid:18) P[δB =1|·] (cid:19) log t =α +β Z +β X +u P[δB =0|·] k 1 t−4 2 t−4 t t whereZ istheincomeinequalityandX isothercontrolvariables. Thefollowingtableshowstheestimation t−4 t−4 results using our simulation data. Table 7: Replicating the Logit Estimation Using Simulated Data (1) (2) (3) Credit 1.342∗∗ - 1.007∗ t−4 (0.5927) - (0.597) [0.017] [0.013] Shareholders Income Share - 2.349∗∗∗ 2.111∗∗ t−4 - (0.774) (0.780) [0.030] [0.027] Note: Parentheses are for standard errors and square brackets are for marginal effects. AsinPaul(2017),thecreditvariablepredictsfinancialcrisesinoursimulatedeconomyasreportedincolumn (1). However, column (2) shows that the share of shareholders predicts financial crises as well. Finally, column (3) reports that once we control for income inequality, the credit variable loses its predictive power. These are the stylized facts emphasized by Paul (2017) and our model can replicates them quite well. In particular, the marginal effects of credit and income inequality shown in squared brackets are quite comparable to those reported in his study. D.2 Monetary Policy Shocks to Satisfy the ZLB Constraint Figure 11 shows the annualized unconditional and conditional means of monetary policy shocks (both contemporaneous and anticipated news shocks) that are used to satisfy the ZLB constraint in our baseline economy. The former measures unconditional tightening effects due to the ZLB constraint. As such, the sum of unconditional means represents a constant deflationary force facing the economy due to the ZLB constraint for any given period. The latter measures tightening effects conditioned upon any of 41 shocks being positive due to either a presently binding ZLB constraint or the past expectations that the constraint would be binding in the near term. As such, the sum of the conditional means represents a typical magnitude of tightening during a binding ZLB episode. D.3 Monetary Accommodation During Financial Crises Figure 12 replicates the experiment shown in Figure 3 when both financial crises and the ZLB constraint are present for ranges of inflation targeting rules between 1 and 20 (upper three panels) and price-level targeting rules between 0.1 and 4 (lower three panels) and shows the reactions of aggregate output, the inflation rate and the path of the policy rate. The results shown in Figure 12 appear to suggest that for both inflation targeting and price-level targeting rules, the more hawkish the central banker is, the more monetary accommodation she provides to the economy during a financial crisis. 54

Figure 11: Means of Monetary Policy Shocks to Satisfy the ZLB Constraint (a) Unconditional mean 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 Periods launna ,tnecreP (b) Conditional mean 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 Periods launna ,tnecreP Notes: The figure shows the annualized unconditional and conditional means of monetary policy shocks used to satisfy theZLBconstraintinourbaselineeconomywithfinancialcrisesandtheZLBconstraint. Theperiodmarkedas0onthe horizontal axis corresponds to contemporaneous shocks. The rest of the periods show the sizes of 40 anticipated news shocks at each horizon. Figure 12: Monetary Policy Accommodation During a Financial Crisis 2 0 -2 -4 -6 -8 -10 0 20 40 gnitegrat noitalfnI (a) Output, pct (b) Inflation, ann. ppt (c) Nominal rate, ppt 2 3 2.5 0 2 1.5 -2 1 0.5 -4 ρ=1 π 0 ρ=20 π -6 -0.5 0 20 40 0 20 40 2 0 -2 -4 -6 -8 -10 0 20 40 gnitegrat level-ecirP (d) Output, pct (e) Inflation, ann. ppt (f) Nominal rate, ppt 2 3 2.5 0 2 1.5 -2 1 0.5 -4 ρΠ=0.1 0 ρΠ=4 -6 -0.5 0 20 40 0 20 40 Notes: We use the same sequence of adverse risk-premium shocks as in Figure 3 when both financial crises and the ZLB constraint are present for different values of the inflation reaction coefficient ρ in the top panels and for different π reaction coefficient for price-level deviation ρ in the bottom panels. Π 55

D.4 Distributions of Key Endogenous Variables under Optimized Rules Figures 13 and 14 show the exact shapes of the distributions for inflation and output in deviations from their non-stochastic steady state under the optimized symmetric and asymmetric rules, respectively. Figure 13: Distributions of Inflation and Output under Optimized Inflation and Price-Level Targeting Rules with λ = 0.5 (a) Inflation rate (ppt) (b) Output (pct) 2.5 0.25 2 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 -3 -2 -1 0 1 2 -20 -10 0 10 Baseline Inflation targeting Price-level targeting Notes: The figure plots the distributions of inflation and output in deviations from their non-stochastic steady state depending on the presence of financial crises and the ZLB constraint using kernel density estimates of simulated data. Figure 14: Distributions of of Inflation and Output under Optimized Asymmetric Inflation and Price-Level Targeting Rules with λ = 0.5 (a) Inflation rate (ppt) (b) Output (pct) 2.5 0.25 2 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 -3 -2 -1 0 1 2 -20 -15 -10 -5 0 5 10 Baseline Asymmetric inflation targeting Asymmetric price-level targeting Notes: The figure plots the distributions of inflation and output in deviations from their non-stochastic steady state depending on the presence of financial crises and the ZLB constraint using kernel density estimates of simulated data. 56

D.5 Alternative EPF The EPFs shown in Figure 15 assume the following alternative loss function: L=λ{E[π−E(π)]2}1/2+(1−λ)E(u), λ∈[0,1], (D.1) in which the central bank penalizes the level of the unemployment rate in contrast to the bias minimization shown in the main text. This can be considered James Tobin’s loss function.38 Roughly speaking, the loss function (59) in the main text works like a constrained optimization of equation (D.1), i.e., minλ{E[π−E(π)]2}1/2+(1−λ)E(u) subject to E(u)≥u∗. This is the reason why the asymmetric price-level targeting in Figure 9 appears as a vertical line at E(u)=u∗ as the constraint starts binding at this level of the mean unemployment rate. Finally, Tables 8 and 9 report the optimized coefficients and key moments under both symmetric and asymmetric inflation and price-level targeting for selected values of λ. Figure 15: Efficient Policy Frontiers under Alternative Loss Function 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 Mean unemployment rate etar noitalfni .veD .dtS Baseline λ=0 IT ITA PT λ=0 PTA λ=0.5 λ=0 λ=0.5 λ=1 λ=1 λ=0 λ=0.5 λ=0.5 λ=1 λ=1 Note: IT, ITA, PT, and PTA stand for inflation targeting, asymmetric inflation targeting, price-level targeting and asymmetric price-level targeting, respectively. The diamond bullet locates the moments of the baseline. 38JosephE.Stiglitzisknowntohavesaid,“IdidapaperwhereIanalyzedtheoptimalunemploymentrate...Tobinwent lividovertheidea. Tohimtheoptimalunemploymentratewaszero”(http://www.bbc.com/news/magazine-15276765). 57

Table 8: Optimized Rules under Alternative Loss Function: Inflation Targeting vs. Price-Level Targeting Inflation targeting Price-level targeting 1. λ 0.00 0.50 1.00 0.00 0.50 1.00 2. ρ 0.50 1.34 12.47 - - - π 3. ρ - - - 0.04 0.62 3.92 Π 4. ρ - - - - - - π∗ 5. E(u) 7.63 7.96 8.41 6.23 6.43 6.57 6. E(π) -0.18 -0.21 -0.25 0.00 0.00 0.00 7. S.D.(u) 5.91 5.10 5.09 4.30 2.89 2.81 8. S.D.(π) 0.71 0.62 0.59 0.45 0.30 0.30 9. Skewness(u) 1.02 2.31 3.44 0.75 2.06 2.12 10. Skewness(π) -0.76 -1.82 -3.40 -0.61 -1.56 -1.54 11. E(pδ) 1.32 1.33 1.34 1.33 1.33 1.34 12. Frequency of ZLB., pct. 2.08 4.93 9.74 0.77 4.79 9.39 13. Mean MP. shock, bps. 0.37 2.58 30.74 0.72 26.44 178.78 14. Loss function value 7.63 7.08 5.88 6.23 4.73 2.97 15. Welfare of workers, CE 0.01 0.00 -0.03 0.38 0.37 0.38 16. Welfare of shareholders, CE -0.29 -0.05 0.47 -2.23 -2.24 -2.23 17. Aggregate welfare, CE -0.01 0.00 -0.01 0.25 0.24 0.25 Note: Weimpose0.5≤ρ π ≤20and0.01≤ρ Π ≤4intheoptimization,whileρ π∗ andσ π∗ aresetequaltozero. MeanMP shock is the annualized mean of monetary policy shocks (both contemporaneous and anticipated news shocks) required to satisfy the ZLB constraint. CE stands for consumption equivalent that needs to be given to the baseline economy to achieve the same level of welfare. Table 9: Optimized Rules under Alternative Loss Function: Asymmetric Inflation Targeting vs. Asymmetric Price-Level Targeting Asymmetric Asymmetric Inflation targeting Price-level targeting 1. λ 0.00 0.50 1.00 0.00 0.50 1.00 2. ρ 0.70 1.36 2.18 - - - π 3. ρ 0.01 0.43 1.05 Π 4. ρ 0.99 0.99 0.98 0.99 0.99 0.95 π∗ 5. E(u) 6.55 6.70 7.51 4.11 4.45 6.08 6. E(π) -0.02 -0.01 -0.13 0.32 0.32 0.06 7. S.D.(u) 5.26 4.87 4.77 5.45 3.07 2.74 8. S.D.(π) 0.64 0.59 0.57 0.59 0.37 0.30 9. Skewness(u) 1.24 1.93 2.59 0.34 0.92 1.76 10. Skewness(π) -0.86 -1.44 -2.14 -0.18 -0.03 -0.95 11. E(pδ) 1.33 1.34 1.34 1.34 1.35 1.34 12. Frequency of ZLB., pct. 4.07 5.96 6.32 0.69 5.98 6.10 13. Mean MP. shock, bps. 1.24 3.67 6.00 0.77 39.92 66.24 14. Loss function value 6.55 6.31 5.73 4.11 4.08 2.99 15. Welfare of workers, CE 0.49 0.63 0.23 1.70 1.58 0.54 16. Welfare of shareholders, CE -1.99 -2.11 -0.89 -5.61 -5.71 -2.95 17. Aggregate welfare, CE 0.37 0.50 0.17 1.33 1.22 0.36 Note: We impose 0.5≤ρ π ≤20, 0.01≤ρ Π ≤4 and 0.0≤ρ π∗ ≤0.99 in the optimization. σ π∗ is calibrated at 0.01. MeanMPshockistheannualizedmeanofmonetarypolicyshocks(bothcontemporaneousandanticipatednewsshocks) required to satisfy the ZLB constraint. CE stands for consumption equivalent that needs to be given to the baseline economy to achieve the same level of welfare. 58