The Review of Economic Studies
, Volume 85 (4) – Oct 1, 2018

/lp/ou_press/wealth-and-volatility-1NPIzhDu86

- Publisher
- Oxford University Press
- Copyright
- Â© The Author(s) 2018. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
- ISSN
- 0034-6527
- eISSN
- 1467-937X
- DOI
- 10.1093/restud/rdx074
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- See Article on Publisher Site

Abstract Between 2007 and 2013, U.S. households experienced a large and persistent decline in net worth. The objective of this article is to study the business cycle implications of such a decline. We first develop a tractable monetary model in which households face idiosyncratic unemployment risk that they can partially self-insure using savings. A low level of liquid household wealth opens the door to self-fulfilling fluctuations: if wealth-poor households expect high unemployment, they have a strong precautionary incentive to cut spending, which can make the expectation of high unemployment a reality. Monetary policy, because of the zero lower bound, cannot rule out such expectations-driven recessions. In contrast, when wealth is sufficiently high, an aggressive monetary policy can keep the economy at full employment. Finally, we document that during the U.S. Great Recession wealth-poor households increased saving more sharply than richer households, pointing towards the importance of the precautionary channel over this period. 1. Introduction Between 2007 and 2013, a large fraction of U.S. households experienced a large and persistent decline in net worth. Figure 1 plots median real net worth from the Survey of Consumer Finances (SCF), for the period 1989–2013, for households with heads between ages 22 and 60. Between 2007 and 2010, median net worth for this group roughly halved and no recovery is evident in 2013. In relation to income, the decline is equally dramatic: the median value for the net worth to income ratio fell from 1.58 in 2007 to 0.92 in 2013. Figure 1 View largeDownload slide Median household net worth in the U.S. Note: Sample includes households with heads between ages 22 and 60. Figure 1 View largeDownload slide Median household net worth in the U.S. Note: Sample includes households with heads between ages 22 and 60. The objective of this article is to study the business cycle implications of such a large and widespread fall in wealth. We argue that falls in household wealth (driven by falls in asset prices) leave the economy more susceptible to confidence shocks that can increase macroeconomic volatility. Thus, policymakers should view low levels of household wealth as presenting a threat to macroeconomic stability. Figures 2 and 3 provide some suggestive evidence for this message. Figure 2 View largeDownload slide Household net worth since 1920. Figure 2 View largeDownload slide Household net worth since 1920. Figure 3 View largeDownload slide Wealth and volatility. Note: Standard deviations of GDP growth are computed over 40-quarter rolling windows. Observations for net worth are averages over the same windows Figure 3 View largeDownload slide Wealth and volatility. Note: Standard deviations of GDP growth are computed over 40-quarter rolling windows. Observations for net worth are averages over the same windows Figure 2 shows a series for the log of total real household net worth in the U.S. from 1920 to 2013, together with its linear trend. The figure shows that over this period there have been three large and persistent declines in household net worth: one in the early 1930s, one in the early 1970s, and the one that started in 2007. All three declines marked the start of periods characterized by deep recessions and elevated macroeconomic volatility.1 Figure 3 focuses on the post-war period, for which we can obtain a consistent measure of macroeconomic volatility. We measure volatility as the standard deviation of quarterly real GDP growth over a 10-year window. The figure plots this measure of volatility for overlapping windows starting in 1947.1 (the values on the $$x$$-axis correspond to the end of the window), together with wealth, measured as the deviation from trend (the difference between the solid and dashed lines in Figure 2) averaged over the same 10-year window. The figure reveals that periods when wealth is high relative to trend, reflecting high prices for housing and/or stocks, tend to display low volatility in aggregate output (and hence employment and consumption). Conversely, periods in which net worth is below trend tend to be periods of high macroeconomic volatility. For example, during windows ending in the late 1950s and early 1980s, wealth is well below trend and volatility peaks; conversely, in windows ending in the early 2000s and late 1960s, wealth is well above trend and volatility is low. There are many possible explanations for a correlation between wealth and volatility, some of which we discuss in Section 1.1. The novel idea of this article is that the value of wealth in an economy determines whether or not the economy is vulnerable to economic fluctuations driven by changes in household optimism or pessimism (animal spirits). When wealth is low, consumers are poorly equipped to self-insure against unemployment risk, and hence have a precautionary saving motive which is highly sensitive to unemployment expectations. Suppose households come to expect high unemployment. With low wealth, the precautionary motive to save will increase sharply, and households’ desired expenditure will fall. In an environment in which demand affects output (because, say, of nominal rigidities), this decline in spending rationalizes high expected unemployment. Suppose, instead, that households in the same low wealth environment expect low unemployment. In this case, because perceived unemployment risk is low, the precautionary motive will be weak, consumption demand will be relatively strong, and hence equilibrium unemployment will be low. Thus, when asset values are low, economic fluctuations can arise due to self-fulfilling changes in expected unemployment risk. In contrast, when the fundamentals are such that asset values are high, consumers can use wealth to smooth consumption through unemployment spells, and thus the precautionary motive to save is weak irrespective of the expected unemployment rate. Thus, high wealth rules out a confidence-driven collapse in demand and output. One additional important issue is the role of monetary policy, and in particular whether the monetary authority can, by cutting the nominal interest rate, sufficiently stimulate household spending to prevent self-fulfilling confidence crises. We will show that aggressive monetary policy can indeed stabilize the economy, but only when household liquid wealth is sufficiently high. This article is broadly divided into two parts. In the first part we develop our theoretical analysis, while in the second we provide micro empirical evidence supporting the importance of the precautionary motive for aggregate spending. The theory part develops a simple model of a monetary economy in which precaution-driven changes in consumer demand can generate self-fulfilling aggregate fluctuations. The model has three key ingredients. First, unemployment risk is imperfectly insurable, so that changes to anticipated unemployment change the strength of the precautionary motive to save. Second, there is a nominal rigidity, so that precaution-driven changes to consumer demand can translate into changes in equilibrium unemployment. Specifically, we assume sticky nominal wages (as in Rendahl (2016), or Midrigan and Philippon (2016)) so that changes to consumer demand, by affecting the price level, can influence real wages and labour demand. Third, there is a monetary authority that controls the nominal interest rate, and can thereby affect aggregate demand. Importantly, however, the monetary authority’s ability to stabilize the economy is constrained by the zero lower bound (ZLB) on interest rates. Labour in the model is indivisible, so if real wages are too high to support full employment, a fraction of potential workers end up unemployed. We rule out explicit unemployment insurance (UI), but assume that households own an asset (housing) that can be used to smooth consumption in the event of an unemployment spell. We avoid the numerical complexity associated with standard incomplete markets models ($$e.g.$$Huggett (1993) or Aiyagari (1994)) by assuming that individuals belong to large representative households. However, the household cannot reshuffle resources from working to unemployed household members within the period. This preserves the precautionary motive, which is the hallmark of incomplete markets models.2 We will heavily exploit one model property: higher liquid wealth ($$i.e.$$ higher house prices, or a greater ability to borrow against housing) makes desired precautionary saving (and thus consumption demand) less sensitive to the level of unemployment risk. We first show that if fundamentals are such that household liquid wealth is relatively high, then the monetary authority can stabilize the economy at full employment by promising to cut rates aggressively should unemployment ever materialize. The intuition is that high liquid wealth implies a weak precautionary motive, so that the monetary authority can always promise enough stimulus to undo any precaution-driven slump in demand. When fundamentals are such that liquid wealth is low, in contrast, high unemployment can arise in equilibrium, even if the central bank is very aggressive. To see this imagine that households come to expect high unemployment. In this case, because of low liquid wealth, the precautionary motive to save would strengthen, and aggregate demand would fall. The monetary authority will try to increase aggregate demand by lowering the nominal rate, but if the precautionary motive is strong enough, then even a zero nominal rate will be insufficient to restore aggregate demand, and the initial expectation of high unemployment will be validated. After characterizing equilibria in the model, we show that the theory can be applied to help us better understand some features of the Great Recession of 2007–9 in the U.S. A parameterized version of the model displays an equilibrium in which the economy experiences a persistent recession featuring an extended period at the ZLB. This ZLB recession is triggered by a non-fundamental negative shock to unemployment expectations, rather than by exogenous shocks to credit or patience, as in most of the existing literature. A change in expectations also drives the emergence of the ZLB in Schmitt-Grohé and Uribe (2017). However, in that paper the path to the lower bound involves deflationary expectations, while in our model—as in the data—the ZLB binds because the equilibrium real interest rate is low, and not because the economy is experiencing deflation. In the second part of the paper, we use micro data from the Consumer Expenditure Survey (CES) and the Panel Study of Income Dynamics (PSID) to document that, around the onset of the Great Recession, low net worth households increased their saving rates by significantly more than high net worth households. This pattern is especially remarkable when considered alongside a second finding, which is that low wealth households suffered much smaller wealth losses during the recession. This new evidence indicates that the precautionary motive, in the context of sharply eroded home equity wealth and rising unemployment risk, was a key driver of consumption dynamics during the recession. 1.1. Related literature On the theory side, there is a long tradition of models in which self-fulfilling changes in expectations generate fluctuations in aggregate economic activity (see Cooper and John (1988), for an overview). A classic early contribution is Diamond (1982), which generates multiplicity using a thick market externality. Chamley (2014) constructs a model in which different equilibria are supported by differences in the strength of the precautionary motive to save, as in our model. In the low output equilibrium, individuals are reluctant to buy goods because they are pessimistic about their future opportunities to sell goods and because credit is restricted. In Kaplan and Menzio (2016), multiplicity is driven by a shopping externality: when more people are employed, the average shopper is less price sensitive, thereby increasing firms’ profits and spurring vacancy creation. Benigno and Fornaro (2016) argue that expectations of low demand can be self-fulfilling as weak expectations lead to low profits, low innovation investment, low growth, and a stagnation trap at the ZLB. Bacchetta and Van Wincoop (2016) note that with strong international trade linkages, expectations-driven fluctuations will necessarily tend to be global in nature. In Farmer (2013, 2016) households form expectations—tied to asset prices—about the level of output, and wages in a frictional labour market adjust to support the associated level of hiring. A model recession is driven by a self-fulfilling fall in expected asset prices, which depresses spending via a wealth effect channel. In our theory, in contrast, what drives reduced spending, causing a recession, is a self-fulfilling increase in expected unemployment, which reduces spending via a precautionary channel. Two recent papers related to ours are Auclert and Rognlie (2017) and Ravn and Sterk (2017b). Both study environments in which agents face uninsurable idiosyncratic risk and where changes in the precautionary motive can impact the equilibrium interest rate and output. Both papers consider the possibility of multiple equilibria, where an increase in idiosyncratic risk depresses aggregate demand, which increases unemployment, which validates the increase in idiosyncratic risk. One important difference between these papers and ours is that we focus on the role of household liquid wealth in determining when self-fulfilling fluctuations can arise. Guerrieri and Lorenzoni (2009, 2017), Challe and Ragot (2016), and Midrigan and Philippon (2016) all emphasize the role of precautionary savings as a mechanism that amplifies fundamental shocks. Ravn and Sterk (2017a), den Haan et al. (2016), and Challe et al. (2017) have in common with our paper that weak demand can amplify unemployment risk, which in turn can feed back into weak demand. In Beaudry et al. (2017), the precautionary savings channel amplifies a negative demand shock—via higher unemployment risk—but in their model, the impetus to low demand is excessively high past wealth accumulation, whereas we emphasize vulnerability when wealth is low. However, none of these papers considers the possibility of self-fulfilling precaution-driven fluctuations. Our article also adds to the literature exploring the causes and consequences of hitting the ZLB on nominal interest rates. In Eggertsson and Woodford (2003), Christiano et al. (2011), Werning (2012), and Rendahl (2016), what drives the economy into the ZLB is increased saving due to a temporary exogenous shock to households’ patience. In Eggertsson and Krugman (2012), Guerrieri and Lorenzoni (2017), Eggertsson and Mehrotra (2014), and Midrigan and Philippon (2016), additional saving arises due to a tightening of leverage constraints. In Caballero and Farhi (2017), it is the interaction of aggregate risk with a shortage of safe assets. In contrast to all these papers, in our model no exogenous fundamental shocks are required to hit the ZLB. Instead, in a low liquid wealth environment, a negative shock to expectations can endogenously generate an increase in unemployment risk and a sharp decline in equilibrium interest rates. Benhabib et al. (2001, 2002) describe a related expectations-driven equilibrium path to the ZLB, but in their model this path is associated with falling inflation, rather than rising unemployment. Our emphasis on the role of asset values in shaping the set of possible equilibrium outcomes is shared by the literature on bubbles in production economies. Martin and Ventura (2016) consider an environment in which credit is limited by the value of collateral. Alternative market expectations can give rise to credit bubbles, which increase the credit available for entrepreneurs and therefore generate a boom (see also Kocherlakota (2009)). Hintermaier and Koeniger (2013) link the level of wealth to the scope for equilibrium multiplicity in an environment in which sunspot-driven fluctuations correspond to changes in the equilibrium price of collateral against consumer borrowing. There are also papers that emphasize a link between asset values and volatility, with causation running from volatility to asset prices. For example, Lettau et al. (2008) point out that higher aggregate risk should drive up the risk premium, and hence lower prices, on risky assets such as housing and equity. In our model, asset prices are the primitive, and the level of asset prices determines the possible range of equilibrium output fluctuations ($$i.e.$$ macroeconomic volatility). Lustig and van Nieuwerburgh (2005) and Favilukis et al. (2017) build models that share a key transmission channel with ours: house prices affect households’ borrowing ability and change household exposure to idiosyncratic risk. However, their focus is on understanding asset price dynamics given aggregate risk, rather than explaining aggregate risk itself. Our emphasis on the role of confidence is also a feature of Angeletos and La’O (2013) and Angeletos et al. (2016) in which sentiment shocks ($$i.e.$$ shocks to expectations about other agents’ behavior) can lead to aggregate fluctuations. On the empirical side, our model is related to a large literature that relates individual expenditures to labour income risk and to wealth, to assess the importance of the precautionary motive for consumption dynamics. Using British micro data, Benito (2006) finds that more job insecurity (using both model-based and self-reported measures of risk) translates into lower consumption. Importantly for the mechanism in our model, he finds that this effect is stronger for groups that have little household net worth. Engen and Gruber (2001) exploit state variation in UI benefit schedules and estimate that reducing the UI benefit replacement rate by 50% for the average worker increases gross financial asset holdings by 14%. Carroll (1992) argues that cyclical variation in the precautionary savings motive explains a large fraction of cyclical variation in the savings rate. Carroll et al. (2012) find that increased unemployment risk and direct wealth effects played the dominant roles in accounting for the rise in the U.S. savings rate during the Great Recession. Mody et al. (2012) similarly conclude that the global decline in consumption was largely due to an increase in precautionary saving. Alan et al. (2012) exploit age variation in savings responses in U.K. data to discriminate between increased precautionary saving driven by larger idiosyncratic shocks versus the direct effects of tighter credit. They conclude that a time-varying precautionary motive was the key factor: tighter credit, in their model, mostly affects the young, whereas all age groups increased saving. Mian and Sufi (2010, 2016) and Mian et al. (2013) use county-level data to show that consumption declines during the Great Recession were larger in counties with lower initial net worth, evidence again consistent with a heightened precautionary motive. Baker (2017) uses household-level U.S. data to show that consumption responses to income shocks are muted for households with high levels of liquid wealth. Jappelli and Pistaferri (2014) reach a similar conclusion using Italian survey data. Finally, Kaplan et al. (2014) argue that the number of households for whom the precautionary motive is strong might be much larger than would be suggested by conventional measures of net worth, since there is a large group of households with highly illiquid wealth. The rest of the article is organized as follows. Section 2 describes the model, and Section 3 characterizes how confidence crises can arise, depending on the parameters that determine household liquid wealth and the stance of monetary policy. Section 4 contains an application to the U.S. Great Recession, and Section 5 discusses some policy implications. Section 6 presents our microeconomic evidence, and Section 7 concludes (see Supplementary Material). 2. Theory There are two goods in the economy: a perishable consumption good, $$c$$, produced by a continuum of identical competitive firms using labour, and housing, $$h$$, which is durable and in fixed supply. There is a continuum of identical households, each of which contains a continuum of measure one of potential workers. Households and firms share the same information set and have identical expectations. 2.1. Firms Firms are perfectly competitive, and the representative firm produces using indivisible labour according to the following technology: \begin{equation} y_{t}=n_{t}^{\alpha }, \label{eq:prodn} \end{equation} (2.1) where $$y_{t}$$ is output and $$n_{t}$$ is the number of workers hired. The curvature parameter $$\alpha \in \left( 0,1\right) $$ determines the rate at which the marginal product of labour declines as additional workers are hired. Firms take as given the price of output $$p_{t}$$ and must pay workers a nominal wage $$w_{t}$$ that rises over time at a constant exogenous net rate $$\gamma _{w}.$$ The price of output and the wage are both expressed relative to a nominal numeraire (money). Thus, firms solve a static profit maximization problem: \begin{equation} \max_{n_{t}\geq 0}\left \{ p_{t}y_{t}-w_{t}n_{t} \right\} \label{eq:pi_max} \end{equation} (2.2) subject to equation (2.1). The first-order condition to this problem is \begin{equation} \alpha n_{t}^{\alpha -1}=\frac{w_{t}}{p_{t}}. \label{eq:firm_foc} \end{equation} (2.3) Thus, fixing the nominal wage $$w_{t}$$, a higher price level $$p_{t}$$ implies a lower real wage $$w_{t}/p_{t}$$ and higher employment $$n_{t}$$. The representative firm’s profits, which we denote $$\varphi _{t}$$, can be interpreted as the returns to a fixed non-labour factor.3 We will not model an explicit market for stocks, but simply assume that households collect profits at the end of the period. 2.2. Households Households are infinitely lived. They can save in the form of housing and government bonds. At the start of each period, the head of the representative household sends out its members to look for jobs in the labour market and to purchase consumption. If the representative firm’s labour demand $$n_{t}$$ is less than the unit mass of workers looking for jobs in the representative household, then jobs are randomly rationed, and the probability that any given potential worker finds a job is $$n_{t}.$$ Let \begin{equation} u_{t}=1-n_{t} \label{eq:u} \end{equation} (2.4) denote the unemployment rate. Because each household has a continuum of members, this is both the fraction of unemployed workers in any given household, and the aggregate unemployment rate. Within the period, it is not possible to transfer wage income from household members who find a job to those who do not. Thus, unemployed members must rely on savings to finance consumption. If wealth is low or illiquid, it will not be possible to equate consumption between employed and unemployed household members. At the end of the period, all the household members regroup, pool resources, and decide on savings to carry into the next period. More precisely, the representative household seeks to maximize \begin{equation} E_{0}\sum \limits_{t=0}^{\infty }\left( \frac{1}{1+\rho }\right) ^{t}\left \{ \left( 1-u_{t}\right) \frac{\left( c_{t}^{w}\right) ^{1-\gamma }}{1-\gamma }+u_{t}\frac{\left( c_{t}^{u}\right) ^{1-\gamma }}{1-\gamma }+\phi \frac{\left( h_{t-1}\right) ^{1-\gamma }}{1-\gamma }\right \} , \label{eq:hh_obj} \end{equation} (2.5) where $$\rho $$ is the household’s rate of time preference. The values $$c_{t}^{w}$$ and $$c_{t}^{u}$$ denote household consumption choices that are potentially contingent on whether an individual household member is working (superscript $$w$$) or unemployed (superscript $$u$$) at date $$t.$$ The parameter $$\phi $$ defines the weight on utility from housing consumption, which is common across all household members. The parameter $$\gamma $$ controls households’ willingness to substitute consumption inter-temporally, and their aversion to risk. For most of the analysis, we will focus on the especially tractable case in which $$\gamma =1$$, so that the utility function is additive in logs. In that case, the utility function is effectively Cobb–Douglas between housing and non-housing consumption, a specification consistent with Davis and Ortalo-Magné (2011). Within the period, when intra-period transfers are ruled out, household members face budget constraints specific to their employment status: \begin{eqnarray} p_{t}c_{t}^{u} &\leq &\psi p_{t}^{h}h_{t-1}+b_{t-1}, \label{eq:cu} \\ \end{eqnarray} (2.6) \begin{eqnarray} p_{t}c_{t}^{w} &\leq &\psi p_{t}^{h}h_{t-1}+b_{t-1}+w_{t}, \label{eq:cw} \end{eqnarray} (2.7) where $$h_{t-1}$$ and $$b_{t-1}$$ denote the household’s holdings of housing and nominal one-period government bonds, and where $$p_{t}^{h}$$ is the nominal price of housing. Bonds are assumed to be perfectly liquid, so they can be used dollar-for-dollar to finance consumption. Housing is imperfectly liquid within the period, so any household member can only use a fraction $$ \psi \in (0,1)$$ of home value to finance current consumption. The simplest interpretation of $$\psi $$ is that it captures the maximum loan-to-value ratio for home equity loans. For simplicity we have assumed that stocks are perfectly illiquid, so stock wealth cannot be tapped to finance consumption while unemployed.4 The only difference between the within-period constraints for unemployed versus employed household members is that the employed can also access wage income $$w_{t}.$$ Assets are (optimally) identically distributed between working and unemployed household members because unemployment is randomly allocated within the period. The household budget constraint at the end of the period takes the form \begin{equation} \left( 1-u_{t}\right) p_{t}c_{t}^{w}+u_{t}p_{t}c_{t}^{u}+p_{t}^{h}h_{t}+ \frac{b_{t}}{1+i_{t}}\leq \left( 1-u_{t}\right) w_{t}+\varphi _{t}+p_{t}^{h}h_{t-1}+b_{t-1}. \label{eq:hh_bc} \end{equation} (2.8) The left-hand side of equation (2.8) captures total household consumption and the cost of housing and bond purchases. The price of bonds is $$(1+i_{t})^{-1}$$, where $$i_{t}$$ is the nominal interest rate. The first term on the right-hand side is earnings for workers, the second is nominal firm profits, and the last two reflect the nominal values of housing and bonds purchased in the previous period. Note that each household solves an identical problem, and therefore chooses the same asset portfolio. The equilibrium cross-household wealth distribution is therefore degenerate. Thus, this model of the household is a simple way to introduce idiosyncratic risk and a precautionary motive, without having to keep track of the cross-sectional distribution of wealth as in standard incomplete-markets models. 2.3. Monetary authority The monetary authority sets the nominal interest rate $$i_{t}$$ paid on government bonds, which are in zero net supply. In our simple model, inflation per se has no direct impact on real allocations. The only reason inflation matters is via its impact on the real wage, which in turn impacts unemployment. Given this, we adopt a simple monetary rule in which the central bank responds only to deviations in unemployment from its optimal value (zero).5 It follows a simple rule of the form \begin{equation} i_{t}=i^{CB}(u_{t})=\max \left \{ \left( 1+\gamma _{w}\right) \left( 1+\rho -\kappa u_{t}\right) -1,0\right \}. \label{eq:tr} \end{equation} (2.9) Note that if $$u_{t}=0$$, then the net real interest rate is equal to the rate of time preference $$\rho $$. The parameter $$\kappa $$ defines how aggressively the monetary authority cuts nominal rates in response to unemployment. The ZLB constraint rules out negative nominal rates. One way to micro-found the assumption that the monetary authority can impose a rule of the form in equation (2.9) is to explicitly model money, and derive a mapping from changes in the money supply to changes in the nominal rate. In Appendix B we develop this extension formally, introducing money in the utility function and as an additional source of liquidity in households’ budget constraints. We then describe the conditions under which the baseline model described above can be interpreted as the “cashless limit” of an underlying monetary economy. 2.4. Household problem Households take as given the unemployment rate $$u_{t}$$, and prices $$\left \{ p_{t},p_{t}^{h},w_{t},i_{t}\right \} $$. They form expectations over the joint distribution of future prices and future unemployment, and given these expectations choose $$\left \{ c_{t}^{w},c_{t}^{u},b_{t},h_{t}\right \} $$ to maximize equation (2.5) subject to equations (2.6), (2.7), (2.8) and $$\left \{ c_{t}^{w},c_{t}^{u},b_{t},h_{t}\right \} \geq 0.$$ Most of our analysis will consider perfect foresight equilibria, in which households take as given a known sequence for $$\left \{ u_{t},p_{t},p_{t}^{h},w_{t},i_{t}\right \} .$$ The first-order conditions (FOCs) that define the solution to this problem can be combined to give two inter-temporal conditions: one for bonds and one for stocks. The condition for bonds is \begin{equation} \left( c_{t}^{w}\right) ^{-\gamma }\frac{1}{1+i_{t}}=\frac{1}{1+\rho }E_{t} \frac{p_{t}}{p_{t+1}}\left[ \left( 1-u_{t+1}\right) \left( c_{t+1}^{w}\right) ^{-\gamma }+u_{t+1}\left( c_{t+1}^{u}\right) ^{-\gamma } \right] , \label{eq:bond_foc} \end{equation} (2.10) where \begin{equation} c_{t+1}^{u}=\left \{ \begin{array}{ccc} c_{t+1}^{w} & \text{if} & c_{t+1}^{w}\leq \psi \frac{p_{t+1}^{h}}{p_{t+1}} h_{t}+\frac{b_{t}}{p_{t+1}} \\ \psi \frac{p_{t+1}^{h}}{p_{t+1}}h_{t}+\frac{b_{t}}{p_{t+1}} & \text{if} & c_{t+1}^{w}>\psi \frac{p_{t+1}^{h}}{p_{t+1}}h_{t}+\frac{b_{t}}{p_{t+1}}. \end{array} \right. \text{ } \label{eq:cu_rule} \end{equation} (2.11) This condition is easy to interpret. The real return on the bond (gross real interest rate) is the gross nominal rate divided by the inflation rate between $$t$$ and $$t+1$$. The marginal value of an extra real unit of wealth at $$t+1$$ is the average marginal utility of consumption within the household, that is, the unemployment-rate-weighted average of workers’ and unemployed members’ marginal utilities. Equation (2.11) indicates that these two marginal utilities will be equal if household liquidity is sufficient to equate consumption within the household (so that constraint 2.6 is not binding). Otherwise, unemployed workers will consume as much as possible, but within-household insurance will be imperfect. Note that if $$c_{t+1}^{u}=c_{t+1}^{w}$$ with probability one, then the FOC looks just as it would in a representative agent model. In contrast, if there is a positive probability that both $$c_{t+1}^{u}<c_{t+1}^{w}$$ and $$u_{t+1}>0$$, then households have a stronger incentive to save. In particular, there is then an active precautionary motive: higher next-period wealth loosens the liquidity constraint for the unemployed, and improves insurance within the household. The FOC for housing is \begin{equation} P_{t}^{h}\left( c_{t}^{w}\right) ^{-\gamma }=\frac{1}{1+\rho } E_{t}P_{t+1}^{h}\left[ \left( 1-u_{t+1}\psi \right) \left( c_{t+1}^{w}\right) ^{-\gamma }+u_{t+1}\psi \left( c_{t+1}^{u}\right) ^{-\gamma }\right] +\frac{1}{1+\rho }\phi \left( h_{t}\right) ^{-\gamma }, \label{eq:house_foc} \end{equation} (2.12) where $$P_{t}^{h}\equiv p_{t}^{h}/p_{t}$$ is the price of housing relative to consumption. The real financial return on housing is the change in this real price. In addition, an extra unit of housing delivers marginal utility $$\phi \left( h_{t}\right) ^{-\gamma }$$ to all household members. Similarly to the bond, an additional unit of housing is differentially valued by employed versus unemployed household members. However, because housing is imperfectly liquid, an extra real unit of housing wealth can only be used to finance an additional $$\psi $$ units of consumption by unemployed workers. 2.5. Equilibrium An equilibrium in this economy is a probability distribution over quantities $$\left \{ u_{t},n_{t},y_{t},\varphi _{t},\right.$$$$\left.c_{t}^{w},c_{t}^{u},h_{t},b_{t}\right \} $$ and prices $$\left \{ i_{t},p_{t},p_{t}^{h},w_{t}\right \} $$ that satisfies, at each date $$t$$, the restrictions implied by equations (2.1), (2.2), (2.3), (2.4), (2.10), (2.11), (2.12), the policy rule equation (2.9), the law of motion $$w_{t+1}=(1+\gamma _{w})w_{t}$$, and the following three market-clearing conditions: \begin{equation} \left( 1-u_{t}\right) c_{t}^{w}+u_{t}c_{t}^{u}=y_{t}, \label{eq:GMC} \end{equation} (2.13) \begin{equation} h_{t}=1, \label{eq:HMC} \end{equation} (2.14) \begin{equation} b_{t}=0. \label{eq:BMC} \end{equation} (2.15) The second of these reflects an assumption that the aggregate supply of housing is equal to one, while the third reflects the fact that government bonds are in zero net supply. 3. Characterizing Equilibria In this section, we show that the number of model steady states and their stability properties depend on the level of liquid household wealth, defined by the parameters $$\phi $$ and $$\psi $$, and on the aggressiveness of monetary policy, defined by the parameter $$\kappa $$. To preview the key results, when liquid wealth is high, and the precautionary motive to save is therefore relatively weak, an aggressive monetary policy ensures that full employment is the unique model steady state. Furthermore, this steady state is locally unstable, in the sense that it is not possible to construct sunspot shocks that feature temporary deviations from full employment. When liquidity is low, in contrast, richer equilibrium dynamics arise, and no value for the monetary policy $$\kappa $$ guarantees full employment. When policy is sufficiently aggressive, the model features multiple steady states, including one in which the interest rate is zero and unemployment is strictly positive. When policy is sufficiently passive, full employment is the unique steady state, but this steady state is locally stable, so that non-fundamental shocks to confidence can induce temporary recessions. 3.1. Steady states: general properties We start by describing some general properties of model steady states. Steady states are equilibria in which all real model variables are constant. In this section we therefore drop time subscripts. In any steady state, the price inflation rate will equal the rate of wage growth $$\gamma _{w}$$. The model is especially tractable in the case of logarithmic utility $$(\gamma =1).$$ Thus, we consider that special case in the remainder of this section. See Appendix A for detailed derivations of the following results. Result 1: Full employment steady state. Irrespective of parameter values, the model always features a full employment steady state in which \begin{eqnarray*} u &=&0,\text{ }y=1, \\ \frac{1+i}{1+\gamma _{w}}-1 &=&\rho , \\ P^{h} &=&\frac{\phi }{\rho }. \end{eqnarray*} This is the only efficient allocation, given that utility is strictly increasing in consumption, and there is no utility cost from working. Note that the net real interest rate is simply the household’s rate of time preference, and the real price of housing is the present value of full employment implicit rents. At this point it is useful to define a new parameter $$\lambda \equiv \psi \frac{\phi }{\rho }.$$ This is simply the value of liquid wealth in the full employment steady state. This parameter determines the degree of within-household risk sharing. Risk sharing within the household is perfect in steady state if $$c^{u}=c^{w}=y$$ (consumption of unemployed and employed workers is equalized), and imperfect if $$c^{u}=\psi P^{h}<y<c^{w}.$$ Result 2: Risk sharing in steady state. Risk sharing is perfect in any steady state if and only if $$\lambda \geq 1.$$ In addition, if $$\lambda \geq 1$$ and $$\kappa >0$$, then full employment is the unique steady state.6 The intuition for the parametric condition $$\lambda \geq 1$$ is as follows. With perfect risk sharing, the model collapses to a representative agent environment, and the real house price in a steady state with output $$y$$ is proportional to the representative agent’s consumption, $$P^{h}=\left( \phi /\rho \right) y.$$ The maximum an unemployed worker can consume $$\psi P^{h}$$ is then equal to $$\psi (\phi /\rho )y$$, which is larger than per capita output $$y$$ if and only if $$\lambda \geq 1.$$ Note that $$\lambda $$ can be larger than one either because the fundamental value of housing is high ($$i.e.$$$$\phi /\rho $$ is high) or because it is easy to borrow against housing ($$i.e.$$$$\psi $$ is high). It is also easy to see why $$\kappa >0$$ guarantees that full employment is the unique steady state. With perfect risk sharing, the only real interest rate $$r=\frac{1+i}{1+\gamma _{w}}-1$$ consistent with households optimally choosing constant consumption is $$r=\rho .$$ If $$\kappa >0$$, the central bank sets $$r<\rho $$ whenever $$u>0$$, thus ruling out steady states with $$u>0.$$ For the rest of the article, we will focus on the region of the parameter space in which $$\lambda <1$$, so that risk sharing is imperfect. We start our analysis by exploring how imperfect risk sharing affects asset pricing, taking as given a constant unemployment rate $$u.$$ We will then move to ask which values for $$u$$ are consistent with the central bank’s policy rule. Result 3: Steady-state house prices with imperfect risk sharing. Given $$\lambda <1$$, the household FOC for housing implies the following steady-state relationship between the unemployment rate $$u$$ and the real house price $$P^{h}:$$ \begin{equation} P^{h}=\underset{\text{fundamental component}}{\underbrace{\frac{\phi }{\rho } (1-u)^{\alpha }}}\times \underset{\text{liquidity component}}{\underbrace{ \frac{u+\phi }{\lambda u+\left( 1-\left( 1-\lambda \right) u\right) \phi }}}. \label{eq:real_hp} \end{equation} (3.16) The first term in this expression is the “fundamental” component of house value, defined as the market-clearing price $$\left( \phi /\rho \right) y$$ in a representative agent version of the model. This fundamental value declines monotonically with steady-state unemployment. The second term, which is larger than one given $$\lambda <1$$, reflects the “liquidity” premium embedded in equilibrium house prices. House prices exceed their fundamental value because housing serves a role in providing insurance within the household. The liquidity term is always increasing in the unemployment rate given $$\lambda <1$$. At $$u=0$$, the steady-state house price is increasing in $$u$$ if $$\lambda <\frac{1+(1-\alpha )\phi }{1+\phi }$$. Thus, if liquidity is sufficiently low, a marginal increase in unemployment risk at $$u=0$$ increases households’ willingness to pay for housing because the marginal additional liquidity value of housing wealth outweighs the marginal loss in fundamental value. For higher values for unemployment, the fundamental component of home value comes to dominate, and house prices decline in the unemployment rate. As $$u\rightarrow 1$$, the steady-state real house price converges to zero. An illustrative example of the steady-state house price implied by equation (3.16) is plotted in Figure 4. Figure 4 View largeDownload slide Real house prices as a function of unemployment. Figure 4 View largeDownload slide Real house prices as a function of unemployment. Result 4: Steady-state interest rates. Given $$\lambda <1$$, the household FOCs for bonds and housing, along with the market-clearing conditions for those two markets, imply the following steady-state relationship between the unemployment rate $$u$$ and the interest rate $$i$$: \begin{equation} i=i(u)=\left( 1+\rho \right) \left( 1+\gamma _{w}\right) \left( \frac{u+\phi }{u\left( 1+\frac{\rho }{\psi }-\phi \right) +\phi }\right) -1. \label{eq:iu_ss} \end{equation} (3.17) The gross nominal interest rate $$1+i(u)$$ is equal to $$(1+\rho )(1+\gamma _{w})$$ at $$u=0$$ and is declining and convex in $$u$$ for all $$u\in \lbrack 0,1] $$. Equation (3.17) can be derived starting from the steady-state version of the household FOC for bonds, recognizing that a binding liquidity constraint implies $$c^{u}=\psi P^{h}$$, and then substituting in the steady-state expression for $$P^{h}$$ in equation (3.16). The function $$i(u)$$ describes the interest rate at which households will optimally choose zero bond holdings (and hence the market for bonds will clear) given an unemployment rate $$u.$$ Implicit in this expression is that for each value for $$u$$, the corresponding constant real house price clears the market for housing. The market-clearing interest rate $$i(u)$$ varies with unemployment because the unemployment rate determines the strength of the household’s precautionary motive. In fact, it does so through two channels. First, the unemployment rate mechanically determines the fraction of household members who will be liquidity constrained. Second, the unemployment rate also affects the steady-state house price, and thus the consumption differential between employed and unemployed household members. Result 4 indicates that when there is no unemployment risk, the steady-state real interest rate is simply the household’s rate of time preference, while increasing the steady-state unemployment rate always implies a lower interest rate. Why does increasing unemployment always reduce the market-clearing interest rate? As unemployment rises, and average income thus declines, a reader might expect that low income levels would eventually depress desired saving, thereby translating into a higher interest rate. Indeed, this is the standard equilibrating mechanism in many models of the ZLB in which a depressed level of output dampens desired saving when the real interest rate cannot decline. This effect is present in our model, but there are two additional channels via which the unemployment rate affects the equilibrium interest rate $$i$$. First, as we have already noted, higher $$u$$ increases the precautionary demand for assets, pushing down $$i$$. Second, on the supply side, increasing $$u$$ changes the equilibrium real price of housing, and thus the aggregate supply of wealth. The relative importance of these different factors varies with the unemployment rate. At low values for $$u$$, the precautionary effect dominates: increasing unemployment risk generates strong additional demand for savings, which outweighs the traditional channel whereby depressed income reduces desired saving. Thus, $$i$$ declines with $$u$$. At high values for $$u$$, the traditional channel is still present, but now it is dominated by the endogenous house price effect. Increasing $$u$$ sharply reduces house prices (Figure 4), and thus the effective supply of assets. As a result, $$i$$ continues to decline with $$u$$.7 Holding fixed the unemployment rate $$u$$, it is immediate that higher nominal wage inflation $$\gamma _{w}$$ translates one-for-one into a higher market-clearing nominal interest rate. A higher value for the credit parameter $$\psi $$ also raises the interest rate by improving insurance and weakening the precautionary motive. A stronger taste for housing $$\phi $$ has the same effect: a higher $$\phi $$ translates into a higher house price, thereby improving insurance and weakening precautionary demand for bonds. Interestingly, the impact of increasing the rate of time preference $$\rho $$ is ambiguous. At $$u=0$$, increasing $$\rho $$ (impatience) increases the market-clearing interest rate, the conventional effect. At higher unemployment rates, however, making households more impatient actually lowers the interest rate. The logic is that greater impatience depresses house prices, thereby worsening insurance and strengthening the precautionary motive. A model steady state is a pair $$(i,u)$$ that satisfies both the market-clearing condition (3.17) and the policy rule (2.9). Figure 5 plots the two equations for a set of parameters that satisfy the condition $$\lambda <1$$ (imperfect risk sharing). The dotted line is the policy rule, which kinks where the nominal rate hits the ZLB. The solid line plots equation (3.17), that is, the set of ($$i,u$$) pairs at which both the housing market and the bond market clear. Figure 5 View largeDownload slide Steady states. Figure 5 View largeDownload slide Steady states. A steady state is a point at which these two lines intersect, so that markets clear at exactly the interest rate dictated by the central bank’s policy rule. In the graphical example, there are three steady states: one at $$u=0$$, and two with positive unemployment rates. 3.2. Equilibrium dynamics The tractability of the model delivers a complete characterization of how many model steady states exist in different regions of the parameter space, and closed-form expressions for equilibrium prices and quantities in each of those steady states. We now explore the nature of equilibrium dynamics around each steady state. We first establish a result on how the stability of the full employment steady state depends on the choice for the parameter $$ \kappa $$, which defines the responsiveness of monetary policy to unemployment. We define a steady state to be locally stable (unstable) if there exist (do not exist) perfect foresight equilibrium paths in which unemployment starts away from but converges to its steady-state value. Definition: Monetary policy is aggressive if $$\kappa > (1+\rho )\left( \frac{1-\lambda }{\lambda }\right) $$ and is passive otherwise. The aggressive monetary policy definition is the algebraic counterpart to the property that the policy rate function $$i^{CB}(u)$$ declines more rapidly with unemployment than does the market-clearing rate expression $$i(u)$$ at $$u=0$$ (Figure 5). Result 5: Monetary policy and stability of steady states. If monetary policy is aggressive (passive), then the full employment steady state is locally unstable (stable). Thus, if policy is passive, then there exist equilibria in which (1) the economy starts at full employment, (2) a non-fundamental sunspot “confidence” shock drives an unexpected jump in unemployment, and (3) the economy then converges back to full employment. We develop intuition for Result 5 in the next section. We now move to analyse how the set of model equilibria changes with the parameters that determine the level of liquid household wealth. In particular, we will partition the parameter space into two regions. Definition: Liquid wealth is high if $$\psi >\frac{\rho }{\left( 1+\rho \right) \left( 1+\gamma _{w}\right) (1+\phi )-1}$$ and is low otherwise. 3.2.1. High liquid wealth equilibria The high liquid wealth definition ensures that the steady-state market-clearing nominal interest rate is positive at any unemployment rate. In particular, high liquidity ensures that $$i(u)>0$$ at $$u=1$$ and thus, given Result 4, at all $$u\in \left[ 0,1\right] $$. Recall that higher wage inflation $$\gamma _{w}$$ or a stronger taste for housing $$\phi $$ both push up market-clearing interest rates, and thus expand the set of values for $$\psi $$ that satisfies the high liquidity definition. Result 6: Uniqueness with high liquid wealth. If liquid wealth is high and monetary policy is aggressive, then full employment is the only steady state. If liquid wealth is high and monetary policy is passive, then there may be a second steady state with positive unemployment. Figure 6 illustrates Result 6 graphically. The solid curve labelled $$i(u)$$ plots the market-clearing condition, while the steeper dotted line represents the policy rule $$i^{CB}(u)$$ when the central bank is aggressive. In this case there is a unique steady state at full employment. The intuition for steady-state uniqueness is that with high liquid wealth, the precautionary motive to save is relatively weak, and the interest rate that clears markets is always positive. With an aggressive monetary authority, the policy rate therefore always falls below the market-clearing rate, except at full employment. Figure 6 View largeDownload slide High liquid wealth equilibria. Figure 6 View largeDownload slide High liquid wealth equilibria. Moreover, around the full employment steady state, $$i^{CB}(u)$$ is steeper than $$i(u)$$, which ensures that this steady state is locally unstable (Result 5). Thus, confidence-driven fluctuations in unemployment cannot arise in a high liquid wealth environment with aggressive policy. Intuitively, the slope of the $$i(u)$$ function is a measure of how rapidly equilibrium demand declines with unemployment risk. A steeper slope indicates demand is more sensitive to unemployment, in the sense that a larger decline in the interest rate is required to maintain the market-clearing level of demand as the unemployment rate rises. If $$ i^{CB}(u) $$ is steeper than $$i(u)$$, the central bank is promising to cut rates by more than is required to support demand in the event of a marginal increase in unemployment. It follows if the unemployment rate were to increase marginally, demand would exceed supply if the unemployment rate were expected to remain constant or to decline. In fact, the only way a small increase in unemployment could be supported would be if households were to subsequently expect unemployment to keep increasing. It follows that with aggressive policy, the only non-explosive equilibrium path is permanent full employment.8 The flatter dashed line in Figure 6, labelled “$$i^{CB}(u)$$ Passive”, represents instead a passive policy rule. The key problem with a passive monetary policy is that at full employment the policy rule $$i^{CB}(u)$$ is flatter than the market-clearing condition $$i(u)$$. This means a fall in consumer demand, induced by pessimistic expectations about the path of unemployment, is not sufficiently counteracted by lower policy rates. Thus, there is an equilibrium path in which unemployment jumps and then steadily declines (Result 5). At each point along this path, the precautionary motive to save is offset by a combination of (1) slightly lower nominal rates and (2) positive expected growth looking forward. One such path is represented in the figure by the dotted path with arrows.9$$^{,}$$10 The main lesson we draw from this case is that with high liquid wealth, confidence crises can be avoided with appropriately aggressive monetary policy. 3.2.2. Low liquid wealth equilibria If liquid wealth is low, the precautionary motive to save is relatively strong, and the $$i(u)$$ function is negative for sufficiently high unemployment rates. This implies that the ZLB will constrain the central bank’s ability to counteract a confidence crisis. Our next result summarizes the set of equilibria that can arise in this case. Result 7: Multiplicity with low liquid wealth. If liquid wealth is low and monetary policy is aggressive, there are always at least two steady states: full employment, and a second ZLB steady state in which \begin{eqnarray*} u &=&u^{+}=\frac{\phi \left( 1+\gamma _{w}\frac{(1+\rho )}{\rho }\right) }{\frac{1-\psi }{\psi }-\frac{\phi }{\rho }-\gamma _{w}\frac{(1+\rho )}{\rho }}\\ y &=&y^{+}=(1-u^{+})^{\alpha } \\ i &=&0 \\ P^{h} &=&\frac{\phi }{\rho }y^{+}\times \frac{1}{1-\psi \left( 1+\phi\right) \left( 1+\gamma _{w}\frac{(1+\rho )}{\rho }\right) }. \end{eqnarray*} The ZLB steady state is locally stable. If liquid wealth is low and monetary policy is instead passive, and sufficiently so, then full employment is the unique steady state. Figure 7 describes the set of possible equilibria when liquid wealth is low. The key difference, relative to the high liquid wealth case, is that aggressive monetary policy no longer guarantees steady-state uniqueness. The logic is that if households expect permanent unemployment equal to $$u^{+}$$, their precautionary motive will be so strong that even at a zero nominal interest rate, they will choose consumption equal to $$y^{+}$$, thereby validating the expected unemployment rate. Figure 7 View largeDownload slide Low liquid wealth equilibria. Figure 7 View largeDownload slide Low liquid wealth equilibria. Moreover, the positive unemployment ZLB steady state is locally stable, in the sense that there exist equilibrium paths with higher or lower unemployment that converge to it, like the dotted ones marked with arrows. As in the earlier discussion of dynamics around the full employment steady state, local stability arises because at $$u=u^{+}$$, the $$i(u)$$ curve is steeper than the $$i^{CB}(u)$$ curve, and thus monetary policy is locally too passive in how it responds to unemployment. Because the ZLB steady state is locally stable, lots of sunspot equilibria exist. For example, there is an equilibrium in which the economy starts out with full employment, but at some date an unexpected shock to expectations causes households to coordinate on a path for unemployment in which the unemployment rate jumps and then increases over time towards $$u^{+}$$. An aggressive central bank will set the policy rate to zero along this entire transition, but this stimulus is exactly offset by a combination of a strong precautionary motive and the expectation that unemployment will continue to rise until the economy ends up at the $$u^{+}$$ steady state. Thus, when liquid wealth is low, an aggressive monetary policy cannot rule out confidence-driven crises that lead to a “stagnation” steady state. The case of passive monetary policy is similar to the high liquid wealth case. When policy is passive, sunspot equilibria around the full employment equilibrium (the dotted path marked with arrows) are possible. Whether or not stagnation equilibria are also possible under passive policy depends on just how passive policy is: a sufficiently passive policy ensures that the line “$$i^{CB}(u)$$ Passive” never crosses the curve $$i(u)$$ and thus that full employment is the unique (albeit sunspot-vulnerable) steady state. The finding that aggressive monetary policy guarantees local uniqueness around full employment, but introduces a stable ZLB steady state is reminiscent of the analysis in Benhabib et al. (2001 and 2002). They show that an aggressive response to inflation around its target creates induces an additional, stable, low inflation, and zero interest rate equilibrium. One key difference is that our focus is on the response of the central bank to unemployment, while Benhabib et al. focus on the response to inflation. Moreover, our ZLB steady state features high unemployment, while their’s features low inflation. Another difference is that we emphasize the role of household liquid wealth, showing that with high wealth, aggressiveness can indeed guarantee global uniqueness, while the zero interest rate equilibrium only appears in economies with low wealth. 3.2.3. Global dynamics To conclude the analysis, in Figure 8 we numerically explore the global equilibrium dynamics, including real house price dynamics, that arise in the low liquid wealth case.11 The Figure 8A displays possible equilibria in the case of aggressive monetary policy. There are no paths converging to full employment in this case, but there are equilibrium paths converging to the stagnation steady state. The path converging from the left displays some interesting features. In the initial phase of the equilibrium path (the vertical part), unemployment is low and nearly constant but house prices are gradually increasing. To an outside observer, the economy would appear to be experiencing a house price bubble: a steadily increasing price with no change in fundamentals. What sustains this “bubble-like” path as an equilibrium outcome is the fact that in the long run, agents expect the economy will converge to the stagnation steady state with permanent unemployment, and here house prices will remain relatively high because they will have a high value as a source of liquidity. Figure 8B instead depicts the equilibrium paths that arise under passive monetary policy. Recall that in this parameterization (corresponding to Figure 7), policy is sufficiently passive that the model has a unique steady state at full employment. The path plotted represents equilibria in which unemployment starts out positive, but gradually declines to zero. Note that if an unexpected sunspot shock were to cause the economy to jump from full employment to a point on this path, one would observe a jump in unemployment, but little change in house prices. This is possible because when unemployment is temporarily high, the fundamental value of housing is low and the liquidity value is high, while this decomposition is gradually reversed as unemployment declines. We explore such a scenario more fully below when we use the model to construct a narrative for the Great Recession in the U.S. Figure 8 View largeDownload slide Global dynamics with low liquid wealth. (a) Aggressive policy; (b) Passive policy. Figure 8 View largeDownload slide Global dynamics with low liquid wealth. (a) Aggressive policy; (b) Passive policy. 3.2.4. Alternative models of wage stickiness At this point it seems important to discuss the importance of our assumption of exogenous nominal wage growth ($$i.e.$$ wage stickiness) in generating “stagnation” equilibria under aggressive policy. With a fully flexible Walrasian labour market, inelastic desired labour supply would guarantee permanent full employment. Our assumption of sticky nominal wages is therefore important for introducing the possibility that weak demand can depress the price level and push the equilibrium real wage above the value at which firms are willing to hire all workers.12 But what about intermediate cases, where wages are not fully flexible, but where wage growth does respond (negatively) to the unemployment rate? Recent work by Schmitt-Grohé and Uribe (2017) and Benigno and Fornaro (2016) has highlighted that stagnation equilibria can still arise in such cases. In particular, the economy can converge to a steady state in which the unemployment rate is positive and nominal wages fall at a constant rate, but where prices fall at the same rate, and the real wage remains too high to clear the labour market. Thus, the economy can get stuck in a “permanent” slump with a zero nominal interest rate. In fact, the more inversely sensitive wage inflation is to the unemployment rate, the harder it becomes for an aggressive central bank to avoid the deflationary steady state, because low nominal policy rates are undercut by wage and price deflation. Thus, with sufficient nominal wage flexibility, stagnation ZLB equilibria can exist even in the absence of a precautionary motive. 3.3. Policy takeaway The key conclusions from this section are two. First, if household liquid wealth is high, then the monetary authority can easily avoid confidence-driven recessions. By promising to respond aggressively to unemployment, the central bank can keep the economy at full employment, the efficient allocation. Second, if liquidity is low, then setting a monetary policy rule that avoids indeterminacy is much more difficult. As in a high liquidity environment, aggressive monetary policy rules make the economy immune to sunspot shocks that generate temporary fluctuations around the full employment steady state. However, the fact that in the low liquid wealth case the model has multiple steady states (Result 7) indicates that an aggressive monetary policy cannot avoid sunspot shocks that set the economy on paths leading to a permanent slump. In contrast, a passive policy rule, while it renders the economy vulnerable to sunspot shocks around full employment, has the advantage that it can rule out permanent slumps. The idea is that in a permanent slump, households have a strong precautionary motive, and thus a low interest rate is required to support demand equal to constant expected output. If monetary policy is sufficiently passive, interest rates will never fall low enough, and thus markets can clear with positive unemployment only if the economy is expected to gradually recover.13 So what should the monetary authority do in a low liquid wealth environment? No simple rule in the class we have considered guarantees that the economy will remain glued to full employment. Recent work in related models has shown that multiplicity can be eliminated if the monetary authority commits to a policy that switches from an interest rate rule to a money rule when the economy heads towards the ZLB steady state (see Benhabib et al. (2002) and Atkeson et al. (2010)). Our setup is different in that multiplicity is due to expectations about high unemployment rather than expectations of a deflationary spiral. Nonetheless, the fact that high liquid wealth ensures that the ZLB equilibrium does not exist (see Result 6) suggests that an aggressive interest rate policy coupled with a commitment to liquidity creation in the event that the economy should ever head towards the ZLB steady state might restore uniqueness. Exploring this conjecture formally is an interesting future research direction. In Section 5 we will explore the role for non-monetary macroeconomic stabilization tools. 4. Application: The Great Recession The dashed lines in Figure 9 show time paths for (1) the unemployment rate, (2) house prices, (3) inflation, and (4) the federal funds rate in the U.S. between $$2006$$ and $$2016$$. The house price series plotted is the Case-Shiller U.S. National Home Price Index, deflated by the GDP deflator, and relative to a 2% trend growth rate for the real price.14 Between the start of 2007 and the end of 2008, house prices fell by 30% relative to trend, largely accounting for the sharp fall in median net worth documented in Figure 1. The rise in the unemployment rate was concentrated in the second half of 2008 and the first half of 2009. Thus, the fall in house prices began well before the most severe portion of the recession. Figure 9 View largeDownload slide The Great Recession: Model versus data. (a) Unemployment (U $$+$$ 4.5% in model); (b) Real house price; (c) Inflation; (d) Nominal interest rate. Figure 9 View largeDownload slide The Great Recession: Model versus data. (a) Unemployment (U $$+$$ 4.5% in model); (b) Real house price; (c) Inflation; (d) Nominal interest rate. Although our model is highly stylized, we would like to know whether it can replicate these features of the U.S. Great Recession. In particular, we are interested in constructing and comparing to data an equilibrium path in which a sunspot shock, in the context of a low liquid wealth and passive monetary policy environment, triggers a jump in unemployment followed by a gradual perfect foresight recovery. This exercise requires picking values for the model parameters. We set the curvature parameter on consumption and housing to $$\gamma =3$$. This implies an inter-temporal elasticity of substitution of consumption of one-third, which is in the middle of existing empirical estimates based on micro data (Havranek, 2015). This choice will imply a relatively strong precautionary motive. The parameter $$\rho $$ defines the steady state full employment annual real interest rate, and we set it to $$2.5\%.$$15 The parameter $$\gamma _{w}$$ defines the steady state inflation rate, and we set it to $$2.0\%.$$ These two choices immediately imply a steady state full employment nominal interest rate of $$4.5\%$$, which is close to the effective federal funds rate in 2006 and 2007. We set labour’s share of production $$\alpha$$ to 0.7. We set the initial house price parameter $$\phi $$ to $$0.075$$, implying a full employment house price $$\phi /\rho $$ equal to three times full employment output. This was the ratio of home value to aggregate consumption at the peak of the housing boom in the first half of 2016. We then imagine a one-time permanent shock to $$\phi $$, which reduces $$\phi $$ from $$0.075$$ to $$0.05$$ in 2008. This implies a reduction in the full employment house price from $$3$$ to $$2$$, and is a simple way to force the model to replicate the one-third decline in house prices observed between 2006 and 2010. The key implication, in terms of the theory, is that this decline in $$\phi $$ will move the economy from the region of the parameter space in which liquidity is high—and where an aggressive central bank can ensure full employment—to the region in which liquidity is low—where confidence-induced recessions are always possible. We set $$\kappa =1.5$$, so that each 1 percentage point in the unemployment rate reduces the desired policy rate by 1.5 percentage points. Taylor rules are generally specified as responding to output gaps, rather than employment, and the two standard coefficients on the output gap in the literature are 0.5 (Taylor, 1993) and 1.0 (Taylor, 1999). Using a standard Okun’s law coefficient whereby a 1% output gap corresponds to 2 percentage points of unemployment, the corresponding unemployment coefficients are 1.0 and 2.0, respectively. Our choice of $$1.5$$ is intermediate. Given our other parameter choices, $$\psi $$ will determine the consumption level for unemployed households, and thus the strength of the precautionary saving motive. There are two possible ways to measure the strength of the precautionary motive and to thereby discipline the choice for $$\psi .$$ One approach is to note that the strength of the precautionary motive depends on the ratio of the consumption of the unemployed to the consumption of workers.16 Thus, one could set $$\psi $$ to try to replicate estimates of the consumption decline experienced by agents who suffer an unemployment shock. Chodorow-Reich and Karabarbounis (2016) estimate a decline in non-durables and services expenditure upon unemployment of around $$21$$%. An alternative way to calibrate $$\psi $$ is to note that, in the context of model equilibrium, the strength of the precautionary motive will be closely tied to the speed of economic recovery. To understand why this is the case, note first that we will calibrate the sunspot shock to generate a realistic increase in the unemployment rate at the start of the recession. In equilibrium, a given increase in unemployment must induce a commensurate decline in desired consumer spending. This reduction in demand can be obtained in different ways. If consumers have a strong precautionary motive (small $$\psi$$), even a short-lived increase in unemployment ($$i.e.$$ a fast expected recovery) is sufficient to support the required fall in demand. Alternatively, if consumers have a weaker precautionary motive (larger $$\psi$$) then they will only want to reduce spending by the same amount if they expect a slower recovery and thus lower permanent income. Thus, for a fixed increase in unemployment (given by the data) a larger $$\psi$$ (a weaker precautionary motive) will correspond to a slower equilibrium expected recovery. The fact that estimated empirical consumption declines upon unemployment are quite small points towards a relatively large value for $$\psi $$, as does the fact that the observed post-recession recovery in the U.S. was very slow. At the same time, however, to construct an equilibrium in which a recovery to full employment is even possible, monetary policy must be passive (Result 5). Given our other parameter values —and in particular the choice for $$\kappa$$— this imposes an upper bound on $$\psi$$ equal to $$0.37$$.17 Intuitively, $$\psi > 0.37$$ would imply such a weak precautionary motive that even if agents were to expect permanent stagnation they would still not reduce demand sufficiently to validate a marginal increase in unemployment. Given all these considerations, for our baseline calibration we set $$\psi =1/3$$, which is close to the maximum consistent with the existence of a recession followed by recovery.18 This choice implies that in the period when the recession hits, consumption of the unemployed is $$76$$% of that of workers, close to the Chodorow-Reich and Karabarbounis estimate. Figure 9 describes our simulation, model against data. The fundamental $$\phi $$ shock hits in $$2008$$, but we assume that agents initially remain optimistic, and the economy remains at full employment. Then, in 2009, we envision a one-off unanticipated sunspot shock, in which households suffer a collective loss of confidence, triggering a $$6.0$$ percentage point jump in the unemployment rate. The event that precipitated this loss of confidence was perhaps the collapse of Lehman Brothers in the fall of $$2008.$$ After the confidence shock, agents have perfect foresight about the future paths for prices and unemployment. When the shock hits, households become fearful of unemployment, which they expect to persist. Their desire to build a precautionary buffer increases demand for housing and bonds, and reduces demand for consumption goods. The central bank would like to counteract this fall in demand by cutting the nominal rate, but it can only cut rates $$4.5$$ percentage points before it is constrained by the ZLB. At a zero nominal rate, goods demand is still weak given the expected path for unemployment, and this weak demand translates into a fall in the price level. Falling prices in 2009 show up as weak inflation, as in the data. The fall in the price level in turn drives up the real wage, which reduces labour demand, which in turn validates the initial expectation of high unemployment. Over time, the economy gradually recovers, at a speed similar to the recovery observed in the data. During this recovery, below-normal interest rates and positive expected economic growth are factors encouraging spending, and these factors exactly offset the unemployment-related precautionary motive. Inflation during recovery is slightly above trend, supporting a gradual decline in the real wage. The nominal interest rate is at zero for seven years, as in the data, before gradually rising back to its steady-state level.19 Note that from 2009 onwards, house prices are relatively stable, as in the data, while unemployment steadily declines. This is a result of two contrasting effects of unemployment on equilibrium house prices: declining unemployment increases the fundamental component of house value, but reduces the liquidity component of house value, and thus on net, house prices do not move much. Overall, we take this exercise as providing suggestive evidence that a sudden loss of confidence in a low liquid wealth environment played a quantitatively important role in generating the Great Recession and subsequent slow recovery in the U.S. 5. Policy We now show how a simple UI scheme can be used to increase within-household insurance, and thereby help to rule out the possibility of confidence-driven fluctuations. We assume that the government can implement a UI scheme in which it taxes workers lump-sum a nominal amount $$T_{t}$$ and immediately transfers the resources raised to the unemployed in the form of a constant amount of consumption $$b$$. The budget constraint for this policy is \begin{equation*} (1-u_{t})T_{t}=u_{t}p_{t}b. \end{equation*} Such a policy redistributes from low marginal utility workers to high marginal utility unemployed workers — precisely the reallocation that households cannot achieve when liquid wealth is low — and thereby dampens households’ precautionary motive to save. Thus, the UI scheme will shift up the $$i(u)$$ function in Figure 5 and — because it reduces the liquidity value of housing wealth — shift down the house price function in Figure 4 (in this sense, public insurance crowds out private insurance). With a sufficiently generous UI policy, the government will be able to ensure that the $$i(u)$$ function lies above zero for all $$u$$, effectively shifting the economy from a low liquid wealth regime to a high liquid wealth regime. In the calibration described in our Great Recession application, a value of $$b=0.29$$ ($$i.e.$$ unemployment benefits are $$29$$% of full employment output) is sufficient to achieve this. Thus, UI and monetary policy are complementary policy tools when liquid wealth is low: UI effectively provides a substitute for low liquid private wealth, and an aggressive monetary policy can then be used to eliminate confidence-driven fluctuations. There is a literature exploring how redistributive policies can be used to counter a negative aggregate shock to the economy, for example by redistributing income from households with a low marginal propensity to consume to those with a high MPC (see, $$e.g.$$, Kekre (2017), or McKay and Reis (2016)). The element that is different here is that redistributive policies can work in our model by preventing a negative aggregate shock from emerging in the first place, by dampening the feedback loop from pessimistic expectations to weak demand, and thereby ruling out self-fulfilling expectations-driven fluctuations. Thus, redistribution is valuable as an ex ante policy that narrows the set of shocks that can arise in equilibrium, rather than as an ex post policy response to mitigate the response to exogenously given shocks. Note, however, that our simple model stacks the deck in favour of introducing public insurance because it exogenously assumes zero private UI, while allowing for any amount of non-distortionary public insurance. A more satisfactory theory of optimal public insurance would model both the frictions that create a role for public insurance in the first place, as well as the disincentives to search and work effort associated with public taxes and transfers. However, one finding that is likely to generalize is that public policies that increase insurance against idiosyncratic risk are especially valuable at times when (1) a strong precautionary motive is restraining aggregate demand and (2) monetary policy is constrained by the ZLB. 6. Microeconomic Evidence A key element of our model is that when households have low liquid wealth, their desired saving rate becomes more sensitive to perceived unemployment risk. When unemployment risk rises, wealth-poor households’ desire to save becomes much stronger, which translates into lower equilibrium employment, and rationalizes ex post the fear of high unemployment. Is this mechanism empirically relevant for understanding the decline in aggregate consumption during the Great Recession? We cannot easily test this mechanism in aggregate time series. But at the household level there is significant heterogeneity in wealth. In this section, we will exploit this heterogeneity to explore whether low-wealth households increased their saving by more than high-wealth households in response to rising unemployment risk during the Great Recession. We also investigate alternative potential drivers of differential saving rates between low- and high-wealth groups, such as differential wealth losses, future income prospects, or unemployment risk. We start by documenting novel evidence, based on data from the Consumer Expenditure Survey (CES) and the Panel Study of Income Dynamics (PSID), that at the onset of the recession lower wealth households exhibit systematically larger increases in their saving rates. This evidence is broadly consistent with a number of empirical studies showing that wealth is an important determinant of consumption responses to income shocks (Section 1.1). Collectively, this evidence lends support to the importance of wealth and the precautionary motive in understanding demand dynamics. 6.1. Rich and poor Before contrasting savings across wealth groups, we first verified that the dynamics of aggregate consumption, income, and wealth in our cross-sectional data capture the broad contours of national income and product accounts (NIPA) aggregates over the course of the Great Recession (see Appendix C for details). The sample we select for our analysis includes all households in the PSID and the CES with a head or spouse between ages 22 and 60, and which report income and consumption for at least two consecutive interviews over the period 2004–13. We focus on households of working age, since unemployment risk is most relevant for this group. For each data set and for each year $$t$$, we split households by wealth into two equal-sized groups: “rich” and “poor”. In constructing our wealth ranking, we measure wealth as net worth in period $$t$$ relative to a measure of consumption expenditure around period $$t$$. We focus on net worth relative to consumption (as opposed to just net worth) as this ratio better captures the extent to which a household can shield consumption from income declines, and is thus more closely connected to the strength of households’ precautionary motive.20 6.1.1. Summary statistics Table 1 reports, for the year $$2006$$, characteristics of the rich and poor groups in both the PSID and the CES. Table 1. Characteristics of rich and poor, 2006 PSID CES Poor Rich All Poor Rich All Sample size (unweighted) 3,443 2,508 5,951 1,915 1,960 3,875 Mean age of head 38.0 47.1 42.5 40.2 46.4 43.3 (0.32) (0.26) (0.19) (0.25) (0.24) (0.16) Head with college (%) 20.5 36.6 28.6 24.8 39.4 32.1 (1.0) (1.0) (0.8) (1.1) (1.2) (0.7) Mean household size 2.47 2.73 2.60 2.79 2.85 2.82 (0.04) (0.04) (0.02) (0.04) (0.04) (0.02) Mean household net worth ( $\$$) 12,017 619,831 315,997 11,967 338,535 175,259 (964) (49,297) (30,634) (1,155) (12,644) (7,450) Median household net worth ( $\$$) 5,000 265,000 66,000 1,800 187,102 58,765 (506) (6,789) (3,006) (294) (4,893) (2,222) Per capita disp. income ( $\$$) 14,951 28,476 22,052 18,739 30,184 24,401 (265) (728) (433) (334) (593) (385) Head unemployment rate (%) 4.2 1.0 2.5 1.0 0.3 0.6 (0.36) (0.19) (0.2) (0.3) (0.2) (0.1) Per capita cons. expenditure ( $\$$) 9,832 13,101 11,548 9,185 10,858 10,013 (186) (244) (170) (232) (188) (166) Saving rate (%) 34.2 54.0 47.6 51.0 64.0 59.0 (0.90) (0.95) (0.75) (1.18) (0.66) (0.59) PSID CES Poor Rich All Poor Rich All Sample size (unweighted) 3,443 2,508 5,951 1,915 1,960 3,875 Mean age of head 38.0 47.1 42.5 40.2 46.4 43.3 (0.32) (0.26) (0.19) (0.25) (0.24) (0.16) Head with college (%) 20.5 36.6 28.6 24.8 39.4 32.1 (1.0) (1.0) (0.8) (1.1) (1.2) (0.7) Mean household size 2.47 2.73 2.60 2.79 2.85 2.82 (0.04) (0.04) (0.02) (0.04) (0.04) (0.02) Mean household net worth ( $\$$) 12,017 619,831 315,997 11,967 338,535 175,259 (964) (49,297) (30,634) (1,155) (12,644) (7,450) Median household net worth ( $\$$) 5,000 265,000 66,000 1,800 187,102 58,765 (506) (6,789) (3,006) (294) (4,893) (2,222) Per capita disp. income ( $\$$) 14,951 28,476 22,052 18,739 30,184 24,401 (265) (728) (433) (334) (593) (385) Head unemployment rate (%) 4.2 1.0 2.5 1.0 0.3 0.6 (0.36) (0.19) (0.2) (0.3) (0.2) (0.1) Per capita cons. expenditure ( $\$$) 9,832 13,101 11,548 9,185 10,858 10,013 (186) (244) (170) (232) (188) (166) Saving rate (%) 34.2 54.0 47.6 51.0 64.0 59.0 (0.90) (0.95) (0.75) (1.18) (0.66) (0.59) Notes: Bootstrapped standard errors in parentheses. All statistics (except sample size) weighted using sample weights. Table 1. Characteristics of rich and poor, 2006 PSID CES Poor Rich All Poor Rich All Sample size (unweighted) 3,443 2,508 5,951 1,915 1,960 3,875 Mean age of head 38.0 47.1 42.5 40.2 46.4 43.3 (0.32) (0.26) (0.19) (0.25) (0.24) (0.16) Head with college (%) 20.5 36.6 28.6 24.8 39.4 32.1 (1.0) (1.0) (0.8) (1.1) (1.2) (0.7) Mean household size 2.47 2.73 2.60 2.79 2.85 2.82 (0.04) (0.04) (0.02) (0.04) (0.04) (0.02) Mean household net worth ( $\$$) 12,017 619,831 315,997 11,967 338,535 175,259 (964) (49,297) (30,634) (1,155) (12,644) (7,450) Median household net worth ( $\$$) 5,000 265,000 66,000 1,800 187,102 58,765 (506) (6,789) (3,006) (294) (4,893) (2,222) Per capita disp. income ( $\$$) 14,951 28,476 22,052 18,739 30,184 24,401 (265) (728) (433) (334) (593) (385) Head unemployment rate (%) 4.2 1.0 2.5 1.0 0.3 0.6 (0.36) (0.19) (0.2) (0.3) (0.2) (0.1) Per capita cons. expenditure ( $\$$) 9,832 13,101 11,548 9,185 10,858 10,013 (186) (244) (170) (232) (188) (166) Saving rate (%) 34.2 54.0 47.6 51.0 64.0 59.0 (0.90) (0.95) (0.75) (1.18) (0.66) (0.59) PSID CES Poor Rich All Poor Rich All Sample size (unweighted) 3,443 2,508 5,951 1,915 1,960 3,875 Mean age of head 38.0 47.1 42.5 40.2 46.4 43.3 (0.32) (0.26) (0.19) (0.25) (0.24) (0.16) Head with college (%) 20.5 36.6 28.6 24.8 39.4 32.1 (1.0) (1.0) (0.8) (1.1) (1.2) (0.7) Mean household size 2.47 2.73 2.60 2.79 2.85 2.82 (0.04) (0.04) (0.02) (0.04) (0.04) (0.02) Mean household net worth ( $\$$) 12,017 619,831 315,997 11,967 338,535 175,259 (964) (49,297) (30,634) (1,155) (12,644) (7,450) Median household net worth ( $\$$) 5,000 265,000 66,000 1,800 187,102 58,765 (506) (6,789) (3,006) (294) (4,893) (2,222) Per capita disp. income ( $\$$) 14,951 28,476 22,052 18,739 30,184 24,401 (265) (728) (433) (334) (593) (385) Head unemployment rate (%) 4.2 1.0 2.5 1.0 0.3 0.6 (0.36) (0.19) (0.2) (0.3) (0.2) (0.1) Per capita cons. expenditure ( $\$$) 9,832 13,101 11,548 9,185 10,858 10,013 (186) (244) (170) (232) (188) (166) Saving rate (%) 34.2 54.0 47.6 51.0 64.0 59.0 (0.90) (0.95) (0.75) (1.18) (0.66) (0.59) Notes: Bootstrapped standard errors in parentheses. All statistics (except sample size) weighted using sample weights. Although the levels of many of these statistics differ across data sets (due in part to different coverage, sample design, and exact question asked), differences between rich and poor are comparable across the two data sets. With respect to demographics, the wealth-poor group is younger and less educated. The most striking difference between the two groups, not surprisingly, is in terms of wealth. Median net worth for the poor group is near zero in both surveys, while for the rich group it is around $\$$ 265,000 in the PSID and around $\$$187,000 in the CES.21 This dramatic difference suggests that the precautionary saving motive for the poor group should be much stronger than for the rich group, while the rich group likely experiences much larger capital losses when housing and stock prices fall. The wealth-poor group also has a higher (pre-recession) unemployment rate and has a little more than half the average income of the rich group. At the same time, the poor group has a much lower saving rate, so that differences in consumption expenditures between the two groups are relatively small (around 20% in the CES and 30% in the PSID).22 Overall the table suggests that the two groups are quite different in nature for a variety of reasons. This is why our interest is not in the levels of their respective saving rates, but rather in how their saving rates change differentially at the onset of the Great Recession, when the labour income risk faced by both groups increases dramatically. 6.1.2. Changes in saving rates We now compute the object of interest: changes over time in the saving rates of households in different groups of the net worth distribution. It is important that the set of households in each wealth group is fixed when we measure the change in the saving rate between two successive periods, so that the change in the measured saving rate reflects a true change in household savings behavior, and is not a result of a change in the composition of the groups. Fortunately, both the PSID and CES data sets have a panel dimension: in the PSID, households are re-interviewed every two years, while in the CES they are interviewed for four consecutive quarters, and in their first and last interviews, they report both income and consumption. Let $$i$$ denote a group of households (say, households with wealth below the median in 2006) and $$t\_t+1$$ denote a time interval (say, 2006–8). The change in group $$i$$’s saving rate over the interval $$t\_t+1$$, $$\Delta s_{i,t\_t+1}$$, is computed as \begin{equation} \Delta s_{i,t\_t+1}=\frac{E(Y_{i,t+1})-E(C_{i,t+1})}{E(Y_{i,t+1})}-\frac{ E(Y_{i,t})-E(C_{i,t})}{E(Y_{i,t})}, \label{srate} \end{equation} (6.18) where $$Y$$ denotes household disposable income, $$C$$ denotes household consumption expenditures, and $$E$$ denotes the group average (see Appendix C for more details). Figure 10 plots changes in saving rates computed from the PSID (top panels) and from the CES (bottom panels). The two panels on the left (A and C) show the evolution over time of the changes in saving rates of the rich and poor groups. Both panels show that in “normal” times, there are no significant differences in saving rate changes between the two wealth groups, while at the onset of the Great Recession wealth-poor households increased their saving rate significantly more than wealth-rich households.23 Figure 10 View largeDownload slide Changes in saving rates for rich and poor. (a) PSID over time; (b) PSID by net worth quintile; (c) CES over time; (d) CES by net worth quintile. Note: the thin dashed lines delimit bootstrapped 2 standard error bands. Figure 10 View largeDownload slide Changes in saving rates for rich and poor. (a) PSID over time; (b) PSID by net worth quintile; (c) CES over time; (d) CES by net worth quintile. Note: the thin dashed lines delimit bootstrapped 2 standard error bands. The two panels on the right (B and D) disaggregate more finely by net worth, and report the changes in saving rates for the five quintiles of the net worth distribution. Here we focus on just two time periods: a pre-recession period, and the period when the economy enters recession. The plots show that as the economy moves into recession, saving rates increase for all wealth quintiles, but the increase in the saving rate is a declining function of initial net worth, with the bottom quintiles increasing their saving rates significantly, while the top quintiles experience much smaller increases. The fact that during the Great Recession poor households increased their saving rates much more than the rich points towards an important precautionary motive in the face of rising unemployment risk. However, there are other potential explanations for this especially sharp increase in the saving rate of the poor. In particular, it is possible that the poor were more adversely affected by deteriorating future income prospects, by rising unemployment, or by falling asset values. To control for these alternative explanations for the observed increase in saving, we make use of the long panel nature of the PSID. Having a long panel is important as it allows us to assess whether the change in saving during the recession is related to these post-recession outcomes. For this exercise, we select all the households that are in the sample continuously from 2004 through 2012 and that satisfy the age restrictions specified above. This leaves us with a sample of 3,773 households, which we then divide into rich and poor groups using as a cutoff (weighted) net worth at the end of 2006 (relative to average consumption expenditures over the period 2006–8). In Figure 11 we track the evolution of several variables for these two groups over time. Figure 11A reports saving rates, and confirms the findings of Figure 10: as the recession hits, both groups increased their saving rates, but the poor increased saving much more than the rich. Figure 11B and C report the paths for the two components of the saving rate: disposable income and consumption expenditures. Over the period 2008–12, disposable income growth of both rich and poor slows down, but it is the income of the rich that slows more during and after the recession. This suggests that the disproportionate increase in saving of the poor cannot be attributed to their expecting relatively weak future income growth. Figure 11D shows that during and after the recession, both groups faced similar increases in unemployment risk, suggesting that differential unemployment risk cannot explain why the poor increased their saving rate more.24Figure 11C shows that consumption growth of the rich declines more than consumption of the poor. This is consistent with the evidence presented in Petev et al. (2011) that high-wealth households reduced consumption expenditures the most during the recession, and it is also consistent with Parker and Vissing-Jorgensen (2009), who show that in recent recessions, consumption expenditures of high-income households have been more cyclical than those of low-income households. Note, however, that while the rich group chooses a somewhat larger reduction in consumption, they also face a much larger slowdown in income, so that their saving rate does not increase as much as the saving rate of the poor. Finally, Figure 11E shows the path of wealth as a fraction of disposable income. Note that, relative to 2006, net worth of the rich declines by over 100% of their disposable income, while the poor actually experience an increase in net worth. Thus, the larger increase in saving by the poor cannot be attributed to the poor having experienced larger wealth losses during the recession. Note also that households in the poor group increase their net worth by around $$50$$% of their income during the recession. This finding provides independent evidence that this group really is increasing saving. In addition, the net worth data confirm the popular narrative of deleveraging during the Great Recession: low-wealth households cut expenditure to reduce debts and rebuild eroded balance sheets. Figure 11 View largeDownload slide Rich and poor: a long panel analysis. (a) Saving rates; (b) Disposable income; (c) Consumption expenditures; (d) Unemployment rate; (e) Net worth to income ratio. Figure 11 View largeDownload slide Rich and poor: a long panel analysis. (a) Saving rates; (b) Disposable income; (c) Consumption expenditures; (d) Unemployment rate; (e) Net worth to income ratio. Overall, the evidence from the panel analysis in Figure 11 supports our hypothesis that the differential changes in saving rates shown in Figure 10 reflect a strong precautionary motive to save on the part of wealth-poor households in the face of rising unemployment risk. The magnitudes of the observed changes in saving rates are economically relevant. For example, in the PSID over the period 2006–8, poor households increased their saving rate by about $$4$$ percentage points more than rich households. If we attribute this difference to a stronger precautionary motive, then given that the poor account for about one-third of total disposable income (Table 1), we can conclude that increased precautionary saving by the poor reduced aggregate consumption by $$1/3\times 4\%$$$$\simeq $$$$1.3\%$$ of aggregate disposable income. 7. Conclusions The message of this article is that when household wealth is low, a decline in consumer confidence can be self-fulfilling because with low wealth, higher unemployment risk implies a large increase in the precautionary motive to save, rationalizing low equilibrium consumption and output. We argued that the decline in U.S. house prices in 2007 and 2008 reduced U.S. household net worth and left the U.S. economy vulnerable to just such a self-fulfilling wave of pessimism. We developed a simple model to explore the conditions under which confidence-driven fluctuations are possible and to illustrate the connections between the level of liquid wealth, the monetary policy rule, and the nature of confidence-driven fluctuations that can arise. Precautionary motives that vary with wealth and unemployment risk play the key role in these links. These precautionary motives are central in standard inter-temporal consumption theory and are consistent with consumption patterns observed in micro data during the course of the Great Recession. An obvious project for future work is to develop a richer, more quantitative version of the model. One interesting extension would be to introduce household heterogeneity in net worth, to better understand the implications for aggregate precautionary demand of the extremely uneven distribution of net worth in the U.S. A richer model of labour markets, in which desired labour supply plays a role in the long-run adjustment process, is another direction for future research. Finally, it would be interesting to introduce capital in the environment, since an increase in perceived unemployment risk would both increase the supply of saving and reduce the demand for investment, thereby changing the contours of the macroeconomic response. APPENDIX A. Proofs We first describe the steady state versions of the model equilibrium conditions, which are useful for proving several of the following results. Consider a steady state with a constant unemployment rate $$u.$$ The market-clearing conditions are \begin{equation} h=1 \label{eq:ss_h} \end{equation} (A.1) \begin{equation} b=0 \label{eq:ss_b} \end{equation} (A.2) and \begin{equation} c^{w}(u)=\frac{y(u)-uc^{u}(u)}{1-u}, \label{eq:ss_cw} \end{equation} (A.3) where \begin{equation*} y(u)=(1-u)^{\alpha }. \end{equation*} Given the market-clearing conditions equations (A.1) and (A.2), the steady state versions of the household’s FOCs for houses and bonds are \begin{equation} P^{h}(u)c^{w}(u)^{-\gamma }=\frac{P^{h}(u)}{\left( 1+\rho \right) }\left[ \left( 1-\psi u\right) c^{w}(u)^{-\gamma }+\psi uc^{u}(u)^{-\gamma }\right] + \frac{\phi }{\left( 1+\rho \right) } \label{eq:ss_hp} \end{equation} (A.4) \begin{equation} c^{w}(u)^{-\gamma }\frac{1}{1+i(u)}=\frac{1}{\left( 1+\rho \right) }\frac{1}{ \left( 1+\gamma _{w}\right) }\left[ \left( 1-u\right) c^{w}(u)^{-\gamma }+uc^{u}(u)^{-\gamma }\right], \label{eq:ss_ip} \end{equation} (A.5) where $$P^{h}(u)=p^{h}_{t}(u)/p_{t}(u)$$ is the steady state real house price, and \begin{equation*} c^{u}(u)=\max \left \{ c^{w}(u),\psi P^{h}(u) \right \} . \end{equation*} The steady-state version of the firm’s FOC is \begin{equation*} W(u)=\alpha (1-u)^{\alpha -1}, \end{equation*} where $$W(u)=w_{t}/p_{t}(u)$$ is the real wage. The central bank policy rule is \begin{equation*} i(u)=\max \left \{ \left( 1+\gamma _{w}\right) \left( 1+\rho -\kappa u\right) -1,0\right \} . \end{equation*} Result 1: Full employment steady state At full employment, the policy rule implies $$i=(1+\gamma _{w})(1+\rho )-1.$$ It is immediate that at this interest rate and $$u=0$$, the bond FOC is satisfied. Then, from the housing FOC, the real house price is given by $$ P^{h}=\phi /\rho .$$ Result 2: Risk sharing in steady state To prove that if risk sharing is perfect then $$\lambda \geq 1$$, note that with perfect risk sharing the real house price equals the present value of implicit rents, $$P^{h}=\phi y/\rho .$$ A necessary condition for risk sharing to be perfect is that the value of liquid wealth has to be at least as high as per capita output, $$i.e.$$ \begin{equation*} \psi P^{h}\geq y. \end{equation*} Substituting in the expression for $$P^{h}$$ yields \begin{equation*} \lambda \equiv \psi \phi /\rho \geq 1. \end{equation*} We now prove, by contradiction, that if $$\lambda \geq 1$$, then risk sharing is perfect. Assume that $$\lambda \geq 1$$ and that risk sharing is imperfect. If risk sharing is imperfect, then the real price of housing (using Result 3 below) is \begin{equation*} P^{h}=\frac{\phi }{\rho }y\frac{u+\phi }{\lambda u+\left( 1+\left( \lambda -1\right) u\right) \phi }. \end{equation*} In addition, a necessary condition for risk sharing to be imperfect is $$\psi P^{h}<y.$$ Substituting in this inequality, the expression for $$P_{h}$$ yields \begin{equation*} \lambda \frac{u+\phi }{\lambda u+\left( 1+\left( \lambda -1\right) u\right) \phi }<1. \end{equation*} If $$\lambda=1$$ there is an immediate contradiction. If $$\lambda>1$$ the inequality implies \begin{equation*} u>1, \end{equation*} which is impossible. Thus it cannot be that risk sharing is imperfect and $$\lambda \geq 1$$. Result 3: Steady-state house prices with imperfect risk sharing Assuming imperfect risk sharing, \begin{equation} c^{u}(u)=\psi P^{h}(u). \label{eq:ss_cu} \end{equation} (A.6) Imposing $$\gamma =1$$ (logarithmic utility), the steady state house price equation (A.4) can then be written as \begin{equation*} \frac{P^{h}(u)}{c^{w}(u)}=\frac{\phi +u}{\rho +\psi u}. \end{equation*} Spending by workers, using (A.3), is \begin{equation} c^{w}(u)=\frac{(1-u)^{\alpha }-u\psi P^{h}(u)}{1-u}. \label{eq:ss_pcw} \end{equation} (A.7) Substituting this into the previous equation, one can solve out for $$ P^{h}(u):$$ \begin{eqnarray*} P^{h}(u) &=&\frac{\phi }{\rho }(1-u)^{\alpha }\frac{u+\phi }{\frac{\psi \phi }{\rho }u+\left( 1+\left( \frac{\psi \phi }{\rho }-1\right) u\right) \phi } \\ &=&\frac{\phi }{\rho }(1-u)^{\alpha }\frac{u+\phi }{\lambda u+\left( 1+\left( \lambda -1\right) u\right) \phi }. \end{eqnarray*} Result 4: Steady-state interest rates The steady-state version of the bond FOC can be written, after substituting in market-clearing conditions, as \begin{eqnarray*} \frac{1}{c^{w}}\frac{1}{1+i(u)} &=&\frac{1}{1+\rho }\frac{1}{1+\gamma _{w}} \left( \frac{1-u}{c^{w}}+\frac{u}{c^{u}}\right) \\ \frac{1}{1+i(u)} &=&\frac{1}{1+\rho }\frac{1}{1+\gamma _{w}}\left( \left( 1-u\right) +u\frac{c^{w}}{c^{u}}\right). \end{eqnarray*} Now substitute in (A.7) and (A.6): \begin{equation} \frac{1}{1+i(u)}=\frac{1}{1+\rho }\frac{1}{1+\gamma _{w}}\left( (1-u)+\frac{u \frac{(1-u)^{\alpha }-u\psi P^{h}(u)}{1-u}}{\psi P^{h}(u)}\right). \label{eq:ss_i} \end{equation} (A.8) Substituting in equation (A.4) gives the expression in the text: \begin{equation} 1+i(u)=\left( 1+\rho \right) \left( 1+\gamma _{w}\right) \left( \frac{u+\phi }{u\left( 1+\frac{\rho }{\psi }-\phi \right) +\phi }\right). \label{eq:ss_itext} \end{equation} (A.9) It is immediate to show that if $$u=0$$, then $$1+i(u)=(1+\rho )(1+\gamma _{w}). $$ Differentiating equation (A.9) with respect to $$u$$, it is straightforward to show that the equilibrium interest rate $$i(u)$$ is declining in $$u$$ given $$\lambda <1.$$ Similarly, $$\lambda <1$$ ensures that the second derivative of $$i(u)$$ with respect to $$u$$ is positive, so $$i(u)$$ is a convex function of $$u$$. Result 5: Monetary policy and stability of steady states If monetary policy is aggressive, then the full employment steady state is locally unstable, in the sense that there are no perfect foresight equilibrium paths that converge to this steady state in which the initial unemployment rate is positive. If monetary policy is passive, then the full employment steady state is stable, and sunspot shocks are possible. Assuming perfect foresight, the FOC for bonds can be written as \begin{equation*} \frac{1}{c_{t}^{w}}\frac{1+\rho }{1+i_{t}}=\frac{1}{\frac{p_{t+1}}{p_{t}}} \left[ \frac{1-u_{t+1}}{c_{t+1}^{w}}+\frac{u_{t+1}}{c_{t+1}^{u}}\right]. \end{equation*} From the firm’s FOC, \begin{equation*} \frac{p_{t+1}}{p_{t}}=\frac{(1+\gamma _{w})(1-u_{t+1})^{(1-\alpha )}}{ (1-u_{t})^{(1-\alpha )}}. \end{equation*} Assuming the unemployed are borrowing constrained and imposing market clearing, we have \begin{eqnarray*} c_{t}^{u} &=&\psi P_{t}^{h} \\ c_{t}^{w} &=&(1-u_{t})^{\alpha -1}-\frac{u_{t}}{1-u_{t}}\psi P_{t}^{h}, \end{eqnarray*} where $$P_{t}^{h}$$ is the real house price. Finally, the policy rule (assuming the ZLB is not binding) is \begin{equation*} i_{t}=\left( 1+\gamma _{w}\right) \left( 1+\rho -\kappa u_{t}\right) -1. \end{equation*} Substituting all these expressions into the bond FOC gives \begin{eqnarray*} &&\frac{1}{(1-u_{t})^{1-\alpha }\left( (1-u_{t})^{\alpha -1}-\frac{u_{t}}{ 1-u_{t}}\psi P_{t}^{h}\right) }\frac{1}{\left( 1+\gamma _{w}\right) \left( 1+\rho -\kappa u_{t}\right) } \\ &=&\frac{1}{1+\rho }\frac{1}{(1+\gamma _{w})(1-u_{t+1})^{1-\alpha }}\left[ \frac{1-u_{t+1}}{(1-u_{t+1})^{\alpha -1}-\frac{u_{t+1}}{1-u_{t+1}}\psi P_{t+1}^{h}}+\frac{u_{t+1}}{\psi P_{t+1}^{h}}\right] \end{eqnarray*} or \begin{equation*} \frac{1}{\left( 1-u_{t}(1-u_{t})^{-\alpha }\psi P_{t}^{h}\right) }\frac{ 1+\rho }{\left( 1+\rho -\kappa u_{t}\right) }=\frac{1-u_{t+1}}{ 1-u_{t+1}(1-u_{t+1})^{-\alpha }\psi P_{t+1}^{h}}+\frac{u_{t+1}}{ (1-u_{t+1})^{1-\alpha }\psi P_{t+1}^{h}}. \end{equation*} Let $${\rm LHS}(u_{t},P_{t}^{h})$$ and $${\rm RHS}(u_{t+1},P_{t+1}^{h})$$ denote the left- and right-hand sides of this equation. Now take a first-order Taylor series approximation to this equation around $$u=0$$ and $$P^{h}=\phi /\rho $$. Thus, \begin{eqnarray*} &&\frac{\partial }{\partial u_{t}}{\rm LHS}(u_{t},P_{t}^{h})|_{\left( 0,\phi /\rho \right) }\times u_{t}+\frac{\partial }{\partial P_{t}^{h}} {\rm LHS}(u_{t},P_{t}^{h})|_{\left( 0,\phi /\rho \right) }\times \left( P_{t}^{h}- \frac{\phi }{\rho }\right) \\ &\simeq &\frac{\partial }{\partial u_{t+1}}{\rm RHS}(u_{t+1},P_{t+1}^{h})|_{\left( 0,\phi /\rho \right) }\times u_{t+1}+\frac{\partial }{\partial P_{t+1}^{h}} {\rm RHS}(u_{t+1},P_{t+1}^{h})|_{\left( 0,\phi /\rho \right) }\times \left( P_{t+1}^{h}-\frac{\phi }{\rho }\right). \end{eqnarray*} It is straightforward to verify that the partial derivatives with respect to $$P_{t}^{h}$$ and $$P_{t+1}^{h}$$ are zero, and that \begin{equation*} \frac{\partial }{\partial u_{t}}{\rm LHS}(u_{t},P_{t}^{h})|_{\left( 0,\phi /\rho \right) }=\psi \frac{\phi }{\rho }+\frac{\kappa }{1+\rho } \end{equation*} \begin{equation*} \frac{\partial }{\partial u_{t+1}}{\rm RHS}(u_{t+1},P_{t+1}^{h})|_{\left( 0,\phi /\rho \right) }=\frac{\phi }{\rho }\psi -1+\frac{1}{\frac{\phi }{\rho }\psi }. \end{equation*} Thus, at $$u=0$$ and $$P^{h}=\phi /\rho $$, a first-order approximation to the bond FOC gives \begin{equation*} \frac{u_{t+1}}{u_{t}}\simeq \frac{\psi \frac{\phi }{\rho }+\frac{\kappa }{1+\rho }}{\frac{\phi }{\rho }\psi -1+\frac{1}{\frac{\phi }{\rho }\psi }}. \end{equation*} Now suppose we contemplate the perfect foresight dynamics for $$ u_{0}=\varepsilon >0$$, where $$\varepsilon $$ is small so the approximation around $$u=0$$ is good. Do the dynamics of the system take us towards full employment, or does unemployment tend to grow over time? Unemployment will revert to full employment if $$\frac{u_{t+1}}{u_{t}}<1$$, which in turn is satisfied if \begin{eqnarray*} \psi \frac{\phi }{\rho }+\frac{\kappa }{1+\rho } &<&\frac{\phi }{\rho }\psi -1+\frac{1}{\frac{\phi }{\rho }\psi } \\ \kappa &<&(1+\rho )\left( \frac{1-\frac{\psi \phi }{\rho }}{\frac{\psi \phi }{\rho }}\right) =(1+\rho )\left( \frac{1-\lambda }{\lambda }\right). \end{eqnarray*} This is precisely the condition that defines passive monetary policy. Result 6: Uniqueness with high liquid wealth We want to prove that if liquid wealth is high and monetary policy is aggressive, then full employment is the only steady state. First, recall that if liquid wealth is high, then $$i(u)>0$$ for all $$u.$$ Second, recall that aggressive monetary policy ensures $$\kappa >-\frac{ \partial i(u)}{\partial u}$$ at $$u=0$$, so that the $$i^{CB}(u)$$ is steeper than $$i(u)$$ at $$u=0.$$ Because $$i(u)$$ is convex (Result 4) while $$i^{CB}(u)$$ is linear, it follows immediately that $$i(u)>i^{CB}(u)$$ for all $$u\in (0,\rho /\kappa ]$$ (the point where $$i^{CB}(u)$$ kinks). Now, recall again that high liquidity ensures that $$i(u)>0$$ for all $$u.$$ Thus, $$i(u)>i^{CB}(u)=0$$ for all $$u\in (\rho /\kappa ,1].$$ Combining these results, the only value for $$u$$ at which $$i(u)=i^{CB}(u)$$ is $$u=0.$$ Result 7: Multiplicity with low liquid wealth When liquidity is low and monetary policy is aggressive, there are two steady states: full employment, and a second steady state in which \begin{eqnarray*} u &=&u^{+}=\frac{\phi \left( 1+\gamma _{w}\frac{(1+\rho )}{\rho }\right) }{ \frac{1-\psi }{\psi }-\frac{\phi }{\rho }-\gamma _{w}\frac{(1+\rho )}{\rho }} \\ y &=&y^{+}=(1-u^{+})^{\alpha } \\ i &=&0 \\ P^{h} &=&\frac{\phi }{\rho }y^{+}\times \frac{1}{1-\psi -\phi \psi -\psi \gamma _{w}(1+\phi )\frac{(1+\rho )}{\rho }} \end{eqnarray*} This solution for $$u^{+}$$ comes from setting $$i=0$$ in equation (A.9) and solving for $$u.$$ The solution for $$P^{h}$$ then follows from substituting the solution for $$u$$ into equation (A.4). Great Recession Simulation Sensitivity How sensitive is the shape of the recovery to alternative parameter values? We have experimented with alternative values for $$\psi $$ and $$\kappa $$. With a lower value for $$\kappa $$ — a less aggressive monetary policy — the recovery is faster. A lower value for $$\kappa $$ implies higher interest rates during recovery, encouraging saving. In equilibrium, this stronger incentive to save is offset by the fact that a swifter recovery implies a stronger incentive to borrow. With a lower value for $$\psi $$ — that is, less liquidity — the recovery is again faster, consistent with the intuition described in the main text. Figure A1 plots our baseline against two alternatives: (1) $$\kappa =1.0$$ (instead of $$1.5)$$, and (2) $$\psi =0.25$$ (instead of $$1/3$$). Figure A1 View largeDownload slide The Great Recession: Alternative calibrations. (a) Unemployment (U$$+$$4.5% in model); (b) Real house price; (c) Inflation; (d) Nominal interest rate. Figure A1 View largeDownload slide The Great Recession: Alternative calibrations. (a) Unemployment (U$$+$$4.5% in model); (b) Real house price; (c) Inflation; (d) Nominal interest rate. B. An Explicit Monetary Model In our baseline model, we did not model money explicitly. We simply assumed that the central bank could dictate the nominal interest rate. To better understand the mechanics of interest rate control, it is useful to explicitly introduce money, and to trace out the details of how changes in the money supply allow the central bank to achieve its interest rate targets. In the monetary model we lay out here, households are willing to hold both interest-bearing bonds and non-interest-bearing money because real money balances deliver direct utility. In particular we add a new term to the period utility function, so that households now seek to maximize \begin{equation*} E_{0}\sum \limits_{t=0}^{\infty }\beta ^{t}\left \{ \left( 1-u_{t}\right) \frac{\left( c_{t}^{w}\right) ^{1-\gamma }}{1-\gamma }+u_{t}\frac{\left( c_{t}^{u}\right) ^{1-\gamma }}{1-\gamma }+\phi \frac{\left( h_{t-1}\right) ^{1-\gamma }}{1-\gamma }+v\left( \frac{m_{t}}{p_{t}}\right) \right \} , \end{equation*} where $$m_{t}$$ denotes nominal balances, and $$p_{t}$$ is the price of consumption in units of money. We assume that the function $$v\left( \frac{ m_{t}}{p_{t}}\right) $$ is increasing, differentiable, and concave, and constant above some threshold $$\bar{M}$$, so that $$v^{\prime }\left( \frac{ m_{t}}{p_{t}}\right) =0$$ for $$\frac{m_{t}}{p_{t}}\geq \bar{M}.$$ We will assume that money, like bonds, is perfectly liquid. The government can change the money supply by distributing new cash $$\Delta _{t}$$ in a lump-sum fashion. If $$\Delta _{t}<0$$, the government imposes a lump-sum nominal tax on households, payable in cash. The household budget constraints are now \begin{eqnarray*} p_{t}c_{t}^{u} &\leq &\psi p_{t}^{h}h_{t-1}+b_{t-1}+m_{t-1}+\Delta _{t}, \\ p_{t}c_{t}^{w} &\leq &w_{t}+\psi p_{t}^{h}h_{t-1}+b_{t-1}+m_{t-1}+\Delta _{t}, \end{eqnarray*} and \begin{equation*} (1-u_{t})p_{t}c_{t}^{w}+u_{t}p_{t}c_{t}^{u}+p_{t}^{h}\left( h_{t}-h_{t-1}\right) +\frac{1}{1+i_{t}}b_{t}+m_{t}\leq \left( 1-u_{t}\right) w_{t}+\varphi _{t}+b_{t-1}+m_{t-1}+\Delta _{t}. \end{equation*} We now get the following inter-temporal conditions for houses, $$ h_{t}$$, bonds, $$b_{t}$$, and money, $$m_{t}$$, respectively: \begin{eqnarray*} \frac{p_{t}^{h}}{p_{t}}(c_{t}^{w})^{-\gamma } &=&\frac{1}{1+\rho }E_{t}\frac{ p_{t+1}^{h}}{p_{t+1}}\left[ \left( 1-u_{t+1}\psi \right) (c_{t+1}^{w})^{-\gamma }+u_{t+1}\psi (c_{t+1}^{u})^{-\gamma }\right] +\frac{ \phi }{1+\rho }(h_{t})^{-\gamma}, \\ (c_{t}^{w})^{-\gamma }\frac{1}{p_{t}}\frac{1}{1+i_{t}} &=&\frac{1}{1+\rho } E_{t}\frac{1}{p_{t+1}}\left[ (1-u_{t+1})(c_{t+1}^{w})^{-\gamma }+u_{t+1}(c_{t+1}^{u})^{-\gamma }\right] , \\ \frac{1}{p_{t}}(c_{t}^{w})^{-\gamma }-\frac{1}{p_{t}}v^{\prime }\left( \frac{ m_{t}}{p_{t}}\right) &=&\frac{1}{1+\rho }E_{t}\frac{1}{p_{t+1}}\left[ (1-u_{t+1})(c_{t+1}^{w})^{-\gamma }+u_{t+1}(c_{t+1}^{u})^{-\gamma }\right] . \end{eqnarray*} Note that the right-hand sides of the first-order conditions for bonds and money are identical, so the left-hand sides must also be equal, which implies \begin{equation*} v^{\prime }\left( \frac{m_{t}}{p_{t}}\right) =(c_{t}^{w})^{-\gamma }\frac{ i_{t}}{1+i_{t}}. \end{equation*} This condition is intuitive. The higher the interest rate is, the higher the marginal utility of real balances must be ($$i.e.$$ the smaller must be real balances), so that households remain indifferent between earning interest on bonds versus enjoying utility from money. In equilibrium, the quantity of money $$m_{t}$$ that households choose to hold in period $$t$$ — money demand — must equal money supply, which is the quantity of money previously in circulation plus the quantity of new money injected: \begin{equation*} m_{t}=m_{t-1}+\Delta _{t}. \end{equation*} Suppose that at some date $$t$$, the central bank wants to cut the nominal rate. It can achieve this by injecting more money into the economy – raising $$\Delta _{t}$$. Given a sticky wage, the price level will increase by less than the money supply, so higher nominal balances mean higher real balances. Higher real balances reduce the marginal utility of saving additional money, and, by making money less attractive as a savings vehicle, allow the bond market to clear at a lower nominal rate $$i_{t}$$. The money injection will increase consumer demand via two channels. First, lower interest rates will encourage households to spend rather than save. Second, higher real balances will provide additional liquidity to unemployed household members, and reduce the precautionary motive to save. This second channel is absent in the baseline model, in which money is not modeled explicitly. Higher consumer demand will tend to push up the price level, and given a sticky nominal wage, this will expand labour demand and output.25 Note that to push the nominal interest rate all the way to zero requires increasing the money supply above the point at which the marginal utility of real balances hits zero. When money provides no direct utility on the margin, households will be indifferent between saving in the forms of bonds versus money only if bonds — like money — pay no interest. In general, this explicit monetary model will not generate allocations identical to those in the model without money laid out in the main text, even given the same interest rate rule. The reason is that when real balances are in positive net supply, the total supply of assets in the economy is increased, and this will improve within household insurance. However, if the utility function over real balances $$v\left( \frac{m}{p}\right) $$ has the property that satiation is reached at a very small value for $$\bar{M}$$, and if the monetary authority never expands the money supply beyond the point required to deliver its target rate $$i^{CB}(u)$$ — that is, if real balances never exceed $$\bar{M}$$, which delivers $$i=0$$ — then equilibrium real balances will always be very small, and thus the allocations in the models with and without money will be very similar. In particular, as $$\bar{M}\rightarrow 0$$, and the economy converges to the cashless limit, allocations in the economy with money will converge to those in the economy without money. Even if $$\bar{M}$$ is small, however, hitting the ZLB does not necessarily mean the central bank is out of ammunition. In particular, it can still increase the money supply, and at a zero nominal rate, households may be willing to hold large real balances even if the marginal utility from additional money is zero. Moreover, such policies could be effective in stimulating demand by improving insurance, and reducing precautionary saving. Thus, the model with money could be used to explore unconventional monetary policies that temporarily expand the money supply when the economy is at the ZLB. This is something we leave for future work. C. Empirical Analysis C.1. Total Household Net Worth Total net worth of U.S. households is computed as the sum of the following components of the Financial Accounts of the U.S. (Z1 release): (1) households and non-profit organizations; real estate at market value minus home and commercial mortgages, (2) households and non-profit organizations; corporate equities, (3) households and non-profit organizations; Treasury securities, including U.S. savings bonds. For the period 1920–44, these series are not available from the source above, so we extend them as follows. The value of real estate is backcast using the growth rate of the value of total residential non-farm wealth in Grebler et al. (1956). Home mortgages are backcast using the growth rate of non-farm residential mortgage debt from Grebler et al. (1956), and commercial mortgages are backcast using the growth rate of non-farm commercial mortgage debt from the same source. The value of Treasury securities is backcast using the growth rate of the amount of public debt outstanding from the Treasury Department, and finally the value of corporate equities is backcast using the historical growth of the S&P 500 price index. C.2. Micro Data For each data set, our key variables are net worth, disposable income, and consumption expenditures. Below we first briefly describe the data sets, and then discuss the construction of these variables. The micro data used for the analysis are available on the authors’ websites. The Panel Study of Income Dynamics (PSID) is a panel of U.S. households, selected to be representative of the U.S. population, collected (from 1997) at a bi-annual frequency. Starting in 2004 the PSID reports, for every household in the panel, comprehensive consumption expenditure information, alongside information on income and wealth. Our panel includes all households that have at least one member between ages 22 and 60 years, that report yearly consumption expenditure of at least $\$$ 1,000, and that are in the panel for at least two consecutive interviews. The Consumer Expenditure Survey (CES) is a rotating panel of U.S. households, selected to be representative of the U.S. population, collected at a quarterly frequency. Households in the CES report information for a maximum of four consecutive quarters. Households report consumption expenditures in all four interviews, income information in the first and last interview, and wealth information in the last interview only. We use CES data from the first quarter of 2004 to the last quarter of 2013, and include all households that have at least one member between ages 22 and 60 years, that report yearly consumption expenditure of at least $\$$ 1,000, and that report consumption and income in the first and last interviews. The Survey of Consumer Finances (SCF) is triennial survey of U.S. households. The survey collects information on household income but focuses primarily on detailed information about household financial and non-financial assets and debts. Net Worth. In all three data sets, we construct net worth by summing all categories of financial wealth ($$i.e.$$ bank accounts, bonds, stocks) plus real estate wealth minus the value of any household debt (including mortgages, home equity loans, and other debts). The PSID and SCF have a more accurate record of wealth, and also report the values of individual retirement accounts (IRAs), family businesses, and vehicles. Our measure of net worth in PSID and SCF includes these variables. Disposable Income. In both the PSID and the CES data sets, we construct disposable income by summing all money income received by all members of the household, including transfers, and then subtracting taxes. In the PSID we compute taxes using the NBER TAXSIM utility, while in the CES we use taxes paid as reported by the household. Consumption Expenditure. In both data sets, we construct expenditure by summing the value of the purchases of: new/used cars and other vehicles, household equipment (including major appliances), goods and services used for entertainment purposes, food and beverages (at home and out), clothing and apparel (including jewelry), transportation services (including gasoline and public transportation), household utilities (including communication services such as telephone and cable services), education, and child care services. The two major categories that are excluded from our analysis are health expenditures and housing services. We exclude these categories to enable better comparison with NIPA data. Our key result regarding the differential behaviour of consumption rates between rich and poor (shown in Figure 10) survives with consumption measures that include these two categories. We also experimented with a narrower consumption measure that excludes food, transportation services, and utilities. The reason to consider excluding these categories is that households in the PSID are asked how much they spent on these categories in a typical week and not explicitly for the whole year (as for the other consumption categories). For this narrower consumption definition, the discrepancy between aggregate consumption expenditures in the PSID and the other two data sources (Panel A of Figure 1) is much smaller. We have also reproduced Figure 10 with this narrower consumption measure, and found that the patterns of changes in saving rates are very similar. Aggregates. Panel A of Figure C1 shows the dynamics of average per capita expenditures in the PSID and the CES against the equivalent measure in the NIPA. Figure C1B shows average per capita disposable income in the PSID and the CES versus NIPA personal disposable income. Figure C1C shows median household net worth in the PSID and the CES versus median net worth in the SCF. Our consumption concept includes all categories except expenditure on housing and health. Net worth includes net financial wealth plus housing wealth net of all mortgages (including home equity loans).26 The key message from the figure is that the dynamics of consumption, income, and wealth are broadly comparable across data sets. In particular, both micro data sets exhibit a marked reduction in consumption expenditure during the recession.27 Figure C1 View largeDownload slide Comparing aggregates across micro data sets. (a) Per capita consumption expenditures; (b) Per capita disposable income; (c) Median household net worth. Figure C1 View largeDownload slide Comparing aggregates across micro data sets. (a) Per capita consumption expenditures; (b) Per capita disposable income; (c) Median household net worth. Measuring changes in saving rates in the PSID and the CES. We now specify the details of the procedure used to produce Figure 10. PSID (Panels A and B) 1 For any year $$t$$ from 2004 to 2010, select households that (i) contain a head or spouse between ages 22 and 60 years, and (ii) report disposable income and consumption expenditures and net worth in year $$t$$ and year $$t+2.$$ 2 Rank households by net worth in year $$t$$ relative to the average of consumption expenditures in years $$t$$ and $$t+2$$. We then divide the sample into two (for panel A) or five (for panel B) equal-size (weighted) groups, where the dividing lines between groups are the (weighted) quantiles of the empirical distribution of net worth relative to consumption expenditure. 3 For each group, compute the change in saving rates between years $$t$$ and $$t+2$$, where the change is defined in equation (6.18) in the text. Note that the identity of households in a group does not change when we compute changes in saving rates between year $$t$$ and $$t+2$$. 4 Move to year $$t+2$$ and repeat steps 1 through 3. Note that when we move to a new year and re-define rich and poor, the identity of the households in each group changes. Households that have, for example, low net worth in 2006 do not necessarily have low net worth in 2008. CES (Panels C and D) 1 For any year $$t$$ (e.g. 2006), select households that (i) contain a head or spouse between ages 22 and 60 years, (ii) are interviewed for the first time in year $$t$$ ($$e.g.$$ the first year in the sample), and (iii) report annual disposable income and quarterly consumption expenditures in their first and last interviews, and report net worth in their last interview. 2 Rank these households by net worth in their last interview (the only time wealth is reported) relative to the average of consumption reported in the first and last interview. Then divide the sample into two (for panel C) or five (for panel D) equal size (weighted) subgroups, where the dividing lines between groups are the (weighted) quantiles of the empirical distribution of net worth relative to consumption expenditure. 3 For each group, compute change in saving rates between the first and last interview, where the change is defined in equation (6.18) in the text. Note that because the first and last interviews are nine months apart, for some households the last interview is at the end of year $$t$$, while for the rest it is in year $$t+1$$. Also, since consumption expenditures are quarterly and income is yearly we annualize consumption expenditures. 4 Move to year $$t+1$$ and repeat steps 1 through 3. To produce Figure 10D, we follow the same procedure, except that to compute saving rate changes over a two-year window, for a given wealth quintile, we add together the quintile-specific changes in the saving rate over each of the two years in the window. In all figures, data are nominal and in levels. Alternative wealth definition In our baseline empirical analysis, we define our rich and poor groups based on the ratio of net worth to consumption, to get a sense of wealth relative to permanent income, which in turn ought to be a good proxy for the strength of a household’s precautionary motive. If we were to group based on wealth in year $$t$$ relative to consumption in year $$t$$, the low-wealth group would mechanically have low wealth and/or high consumption. However, both household consumption and household wealth have a mean-reverting component. Thus, looking forward, the low-wealth group based on this definition would tend to have rapid wealth and/or slow consumption growth between $$t$$ and $$t+1$$. If we were to instead group based on wealth in year $$t$$ relative to consumption in $$t+1$$, the low-wealth group would tend to have high consumption in $$t+1$$ and thus rapid consumption growth between $$t$$ and $$t+1$$. When we define wealth as net worth at $$t$$ relative to the average of consumption at $$t$$ and $$t+1$$, there is no obvious mechanical bias in implied consumption growth in either direction. In any case, to check that our results on saving rates are not driven by mean reversion, we have experimented with a wide range of alternative ways to split the sample. We have found that our results are not driven by the particular definition of wealth chosen. Changing the date at which we measure consumption in the net worth to consumption ratio does affect average differences between consumption growth and saving rates of rich and poor, but it does not have much of an effect on how the behaviour of the two groups diverges as the U.S. economy enters the Great Recession. The simplest and starkest way to make this point is to consider what happens if we do not normalize by consumption at all. Figure C2 shows that when we simply define wealth as net worth, the result that, during the recession, the wealth-poor increase their saving rate more than the wealth-rich is still present.28 Figure C2 View largeDownload slide Changes in saving rates for rich and poor under alternative definition of wealth. (a) PSID; (b) CES. Notes: Solid lines use net worth to expenditure ratio as cutoff. Dashed lines use just net worth Figure C2 View largeDownload slide Changes in saving rates for rich and poor under alternative definition of wealth. (a) PSID; (b) CES. Notes: Solid lines use net worth to expenditure ratio as cutoff. Dashed lines use just net worth The editor in charge of this paper was Gita Gopinath. Acknowledgements We thank our editor, Gita Gopinath, and four anonymous referees for excellent suggestions. We also thank seminar participants at several institutions and conferences, as well as Mark Aguiar, Christophe Chamley, Andrew Gimber, Franck Portier, Emiliano Santoro, and Immo Schott for insightful comments and discussions. We are also grateful to Joe Steinberg for outstanding research assistance. Perri thanks the European Research Council for financial support under Grant 313671 RESOCONBUCY. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Supplementary Data Supplementary data are available at Review of Economic Studies online. Footnotes 1. To construct a consistent series for net worth, we focus on three categories of net worth for which we can obtain consistent data throughout the sample: real estate wealth (net of mortgages), corporate securities, and government treasuries. See Appendix C for details on the construction of the series. 2. Challe and Ragot (2016) show that an alternative way to preserve a low-dimensional cross-sectional wealth distribution, while still admitting a precautionary motive, is to assume that utility is linear above a certain consumption threshold. 3. The technology can be re-interpreted as Cobb–Douglas, $$y_{t}=k^{1-\alpha }n_{t}^{\alpha }$$, where the fixed factor $$k$$ is equal to one. 4. Ownership of equities outside illiquid pension funds and retirement accounts is not widespread. In the 2010 SCF, 15.1% of households owned directly held stocks, and 8.7% owned pooled investment funds. 5. In our application to the Great Recession, we have constructed numerical examples where the nominal interest also responds to fluctuations in the inflation rate, $$p_{t}/p_{t-1}$$. In those examples, the equilibrium dynamics are essentially unchanged relative to our baseline policy rule. 6. If $$\lambda \geq 1$$ and $$\kappa =0$$, then there is a continuum of steady states, one for each $$u\in \left[ 0,1\right] .$$ In each such steady state, $$y=(1-u)^{\alpha }$$, $$\frac{1+i}{1+\gamma _{w}}-1=\rho $$, and $$P^{h}=\frac{\phi }{\rho }y.$$ 7. We have verified this intuition by considering an alternative model in which house prices are fixed exogenously, thereby shutting down the asset supply channel. In that alternative model, increasing unemployment initially pushes down the equilibrium interest rate, but at high unemployment rates the interest rate increases in unemployment, reflecting the fact that the traditional income effect on the demand for saving eventually outweighs the precautionary motive. 8. Note that there are no equilibrium paths in which unemployment steadily rises until the economy converges to 100% unemployment. The reason is that the high liquidity condition ensures $$i(u)>0$$ at $$u=1$$, while an aggressive policy rule ensures $$i^{CB}(u)=0$$ at $$u=1$$. 9. The result that expectations-driven fluctuations in unemployment are possible if the policy rate is insufficiently responsive to unemployment is closely related to the “Taylor principle” (1993) point that expectations-driven fluctuations in inflation are possible if the interest rate is insufficiently responsive to inflation (see, e.g., Clarida et al., 2000). 10. In the example plotted, under the passive interest rate policy, there is a second steady state with positive unemployment and a positive nominal rate. Note that if policy were sufficiently passive, then this second steady state would not be present and full employment would be the unique steady state. 11. We use the same parameter values used to plot Figure 7. The arrows indicate the direction of the equilibrium path, and the circles along the stable manifolds are one model period apart. 12. Hall (2005), Farmer (2013), Michaillat (2012), and Shimer (2012) all exploit the observation that in frictional search and matching models of the labour market, sticky wages can be consistent with an absence of unexploited gains from trade for individual worker-employer pairs. 13. The point that sticking to a high interest rate policy can rule out permanent slumps is sometimes labelled the “Neo-Fisherian” channel of monetary policy. It has been advocated, among others, by Schmitt-Grohé and Uribe (2017). In our model, a passive policy works because high interest rates are inconsistent with expectations of permanently depressed output. In Schmitt-Grohé and Uribe, the policy works because high interest rates are inconsistent with expectations of permanently low inflation. 14. This is the average growth rate for real GDP per capita between 1947 and 2007. It