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
