**BIS Working Papers **

**No 733**

**A risk-centric model of demand recessions and macroprudential policy**

by Ricardo J Caballero and Alp Simsek

**Monetary and Economic Department **

July 2018

JEL classification: E00, E12, E21, E22, E30, E40, G00, G01, G11

Keywords: Risk gap, output gap, risk-premium shocks, aggregate demand, liquidity trap,“rstar”, Sharpe ratio, monetary and macroprudential policy, heterogeneous beliefs, speculation, endogenous volatility

This publication is available on the BIS website (www.bis.org).

*© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.*

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)

**Foreword **

The 16th BIS Annual Conference took place in Lucerne, Switzerland, on 23 June 2017. The event brought together a distinguished group of central bank Governors, leading academics and former public officials to exchange views on the topic “Long for long or turning point?”. The papers presented at the conference and the discussants’ comments are released as BIS Working Papers. BIS Papers no 98 contains the opening address by Jaime Caruana (Former General Manager, BIS) and remarks by Alan Blinder (Princeton University) and Philip Lowe (Reserve Bank of Australia).

**A risk-centric model of demand recessions and macroprudential policy**

by Ricardo J Caballero and Alp Simsek

**Abstract **

When investors are unwilling to hold the economyís risk, a decline in the interest rate increases the Sharpe ratio of the market and equilibrates the risk markets. If the interest rate is constrained from below, risk markets are instead equilibrated via a decline in asset prices. However, the latter drags down aggregate demand, which further drags prices down, and so on. If investors are pessimistic about the recovery, the economy becomes highly susceptible to downward spirals due to dynamic feedbacks between asset prices, aggregate demand, and potential growth. In this context, belief disagreements generate highly destabilizing speculation that motivates macroprudential policy.

JEL Codes: E00, E12, E21, E22, E30, E40, G00, G01, G11

Keywords: Risk gap, output gap, risk-premium shocks, aggregate demand, liquidity trap, ìrstar,îSharpe ratio, monetary and macroprudential policy, heterogeneous beliefs, speculation, endogenous volatility.

- MIT and NBER. Contact information: caball@mit.edu and asimsek@mit.edu. We thank Jaroslav Borovicka, Gauti Eggertsson, Emmanuel Farhi, Kristin Forbes, Mark Gertler, Zhiguo He, Plamen Nenov, Zilu Pan, Matthew Rognlie, Martin Schneider, Larry Summers, Jaume Ventura, Olivier Wang, Nathan Zorzi, and participants at the CCBS conference hosted by the Bank of England and MacCalm, Paris School of Economics, the BIS, NBER EFG, NBER AP, 2018 AEA meetings, the Bank of Spain, the FED, Boston College, MIT, Princeton University, the Wharton conference on Liquidity and Financial Fragility, the Harvard/MIT Financial Economics Workshop for their comments. Simsek acknowledges support from the National Science Foundation (NSF) under Grant Number SES-1455319. First draft: May 11, 2017

**1. Introduction**

Risk-premium spikes put strong downward pressure on risky asset prices which, if not contained, have the potential to trigger severe recessions. By the same token, prolonged periods of low risk- premium can trigger speculation that, if not contained, can exacerbate the contraction associated to a reversal of the benign risk conditions. Central banks are acutely aware of the connection between risk markets and macroeconomic outcomes. For example, Cieslak and Vissing-Jorgensen (2017) conduct a textual analysis of 184 FOMC minutes during the 1994-2016 period and find extensive reference to stock market developments, which in turn had significant explanatory power for target rate changes. The rationale for these reactions points to the negative impact of severe stock markets declines on aggregate consumption and investment.

There is an extensive literature at the intersection of macroeconomics and corporate finance that highlights a variety of financial frictions and balance sheet mechanisms that capture the connection between asset prices and macroeconomics. The bulk of the mechanisms in this literature operate through the supply side of the economy, as the tightening of financial constraints reallocates resources from productive to less productive agents in the economy. In this paper we provide a model that complements this literature by removing all financial frictions and focusing on an aggregate demand channel and its interactions with speculation in causing recessions during severe risk-premium shocks. Correspondingly, we also study the implications of this mechanism for monetary and macroprudential policy.

For this purpose, we develop a continuous time macrofinance model with risk-premium shocks and speculative motives due to belief disagreements with respect to the transition probabilities between risk-on and risk-off environments. The supply side of the (model-)economy is a stochastic AK model with capital-adjustment costs and sticky prices. The demand side has risk-averse consumer-investors that demand the goods and risky assets. There are no financial frictions. Instead, we focus on “interest-rate frictions”: factors that might constrain or delay the adjustment of the risk-free interest rate to shocks. For concreteness, we work with a zero lower bound on the interest rate, but our mechanisms are also applicable if the interest rate is constrained for other reasons such as fixed exchange rates or currency unions.

To fix ideas, consider a shock that increases volatility. We interpret this as a stand-in for various factors that increase the risk premium such as a rise in actual or perceived risks, risk aversion, irrational pessimism, or financial frictions. The common denominator of these “risk premium shocks” is that they exert a downward pressure on risky asset prices without a change in current productivity (the supply-determined output level). A risk gap develops, in the sense that the economy generates too much risk relative to what investors are willing to absorb at the pre-shock level of prices and interest rates. The natural response of the economy is a decrease in the interest rate, which increases the Sharpe ratio of capital (the risk-adjusted expected return on capital in excess of the interest rate) and restores equilibrium in risk markets. This also keeps the asset prices high and ensures a supply-determined equilibrium in goods markets.

If there is a lower bound on the interest rate, the economy loses its natural line of defense.

Instead, the risk markets are equilibrated via a decline in asset prices, which increases the Sharpe ratio via an increase in expected capital gains. However, the decline in asset prices lowers consumption through a wealth effect and investment through a standard valuation (marginal-Q) channel. This reduces aggregate demand and output, that is, the economy experiences a demand recession.

Dynamics play a crucial role in our environment, as the recession is exacerbated by two feedback mechanisms. First, when the risk-premium shock is somewhat persistent, the decline in future demand lowers expected profits, which exerts further downward pressure on asset prices. Second, the decline in current investment lowers the growth of potential output, which reduces expected profits and asset prices (even if there is no demand recession in future periods). In turn, the decline in asset prices feeds back into current consumption and investment, generating scope for severe spirals in asset prices and output. Figure 1 provides a graphical illustration of these dynamic feedback mechanisms. As the figure suggests, These feedbacks are especially powerful when investors are pessimistic and interpret the risk-premium shock as a lasting one. In this case, it takes a large drop in current asset prices to increase investors’ Sharpe ratio and stabilize the risk markets. If instead investors are optimistic about the recovery, then they don’t anticipate strong feedbacks and a limited asset price drop is sufficient to restore equilibrium. Hence, the degree of optimism is a critical state variable in our economy, not only because optimism has a direct impact on asset valuations, but also because it weakens the dynamic feedbacks.

These dynamic effects are exacerbated by belief disagreements which motivate macroprudential policy. We focus on disagreements about the likelihood of transitions between recessions (the high risk-premium state) and booms (the low risk-premium state). We interpret these disagreements as capturing more broadly heterogeneous valuations for risky assets. With disagreements, the economy’s degree of optimism depends on the share of wealth in the hands of optimistic and pessimistic investors. The value of rich optimists for the economy as a whole is high during recessions since they raise asset valuations, which in turn increases aggregate demand. However there is nothing in the economy that ensures this allocation of wealth. Disagreements also lead to speculation which makes the economy effectively extrapolative. During the boom, optimists sell put options that pay in case there is a transition to recession. This enriches optimists if the boom persists but impoverishes them in the state of the economy that needs them the most. Conversely, during the recession, optimists buy call options. This increases optimists’ wealth in case there is a recovery but depletes their wealth if the recession lingers. That is, through relative wealth effects the economy becomes extrapolative: booms breed optimism and recessions breed pessimism.

Speculation during the boom causes damage, because the extrapolation that it induces has asymmetric effects on the economy. If the boom persists, then the interest rate rises to neutralize the effect of greater optimism on asset prices and output (to prevent overheating). However, if the economy transitions into recession, the interest rate is constrained and greater pessimism translates into lower prices and output. This motivates macroprudential policy that restricts speculation during the boom. Intuitively, optimists’ risk taking is associated with aggregate demand externalities. The depletion of optimists’ wealth during a demand recession depresses asset prices and aggregate demand. Optimists (or more broadly, high valuation investors) do not internalize the effect of their portfolio risks on asset valuations during future demand recessions, which leads to excessive risk taking from an aggregate point of view. We show that macroprudential policy that makes optimistic investors behave as-if they were more pessimistic can lead to a Pareto improvement (that is, we evaluate investors’ welfare according to their own beliefs).

Speculation during the recession also exacerbates the dynamic feedbacks. If the economy transitions into the boom, then the interest rate (optimally) rises to neutralize the effect of greater optimism on asset prices and output. However, if the recession persists, the interest rate is constrained and greater pessimism translates into strong feedbacks and (much) lower prices and output. Moreover, the anticipation of this feature lowers asset prices and output immediately. Investors “overweight” low probability paths dominated by pessimists, because these paths feature strong feedback effects. This suggests that restricting speculation via macroprudential policy can also be useful during the recession. However, macroprudential policy also depresses aggregate demand immediately, which can be easily offset by the interest rate policy during the boom but not during the recession. Hence, we find that macroprudential policy is naturally procyclical. The damage from speculation during the recession strengthens the case for macroprudential policy but it does not undo the procyclicality of the policy.

We also find that the drop in asset prices during the recession has implications for “rstar” during the boom. The fear of a switch into a recession driven by a rise in the risk premium raises the expected capital loss as well as the risk premium in the boom state—and considerably so when pessimism or speculation is high and the feedbacks are strong. The interest rate then has to decline also in the boom state so as to increase the Sharpe ratio and equilibrate the risk markets. Hence, our model can generate low interest rates together with low volatility—similar to the current macroeconomic environment—because investors fear downward price spirals triggered by a spike in the risk premium.

While we work with exogenous volatility shocks—to capture various factors that induce time- varying risk premium—the model also generates endogenous price volatility (jumps). Without interest rate rigidities, the interest rate policy optimally mitigates the impact of risk-premium shocks on asset prices. When the interest rate is constrained, these shocks translate into price volatility. With belief disagreements, speculation exacerbates endogenous price volatility further by creating fluctuations in investors’ wealth shares.

Literature review.

The interactions between risk shocks and interest rate lower bounds is also a central theme of the literature on structural safe asset shortages and safety traps (see, for instance, Caballero and Farhi (2017); Caballero et al. (2017b)). We extend this literature by focusing on dynamics and analyzing the risk-on and-off phases of recurrent business cycles driven by risk-premium shocks and speculation. We also develop an integrated analysis of interest-rate and macroprudential policies in this environment. More broadly, the literature on liquidity traps is extensive and has been rekindled by the Great Recession (see, for instance, Tobin (1975); Krugman (1998); Eggertsson and Woodford (2006); Eggertsson and Krugman (2012); Guerrieri and Lorenzoni (2017); Werning (2012); Hall (2011); Christiano et al. (2015); Eggertsson et al. (2017); Rognlie et al. (2017); Midrigan et al. (2016); Bacchetta et al. (2016)). Both the safe asset shortage literature and our paper extend the liquidity traps literature by focusing on the risk aspects (both shocks and mechanisms) behind the drop in the natural rate below its lower bound. In recent work, Del Negro et al. (2017) provide a comprehensive empirical evaluation of the different mechanisms that have put downward pressure on interest rate and argue convincingly that risk and liquidity considerations played a central role (see also Caballero et al. (2017a)). At the cyclical level, Pflueger et al. (2018) document that risk appetite (as measured by the performance of high vol/beta relative to low vol/beta assets) is an important driver of rates and real activity in the U.S.

Our paper is also related to a large New Keynesian literature that investigates the sources of demand shocks that might drive business cycles. A strand of the literature emphasizes “noise” about future expectations (see, for instance, Lorenzoni (2009); Blanchard et al. (2013)). Ilut and Schneider (2014) emphasize “confidence” about future expectations, which they model as changes in ambiguity (or Knightian uncertainty). Gourio (2012) develops a model in which time-varying disaster risk is observationally equivalent to “discount factor shocks,” which would affect aggregate demand (although his is a real business cycle model that does not feature the aggregate demand channel). These shocks can be viewed as modern formulations of Keynesian “animal spirits.” Aside from the modelling novelty (ours is a continuous time macrofinance model), we provide an integrated treatment of these and related forces and refer to them as “risk premium shocks” to emphasize their close connection with asset prices, and use the framework to explore the interactions of these shocks with interest rate frictions and speculative forces. Another strand of the New Keynesian literature emphasizes the role of financial frictions and nominal rigidities in driving business cycle fluctuations, and emphasizes this as a major contributing factor to the Great Recession (see, for instance, Bernanke et al. (1999); Curdia and Woodford (2010); Gertler and Karadi (2011); Gilchrist and Zakrajsek (2012); Christiano et al. (2014)). Like this literature, we focus on episodes with high risk premia but we emphasize that these episodes can be driven by many other factors than financial frictions (in fact, in our model there are no financial frictions).

Our paper is related to a large finance literature which documents that the risk premium on various asset classes varies over time, and investigates the reasons behind this fact (see Cochrane (2011); Campbell (2014) for recent reviews). We show that, when the interest rate is constrained, an increase in the (aggregate) risk premium generates a demand recession. Moreover, as we illustrate in Section 2, this result applies regardless of whether time-varying risk premium is driven by changes in risk attitudes, actual risks, irrational beliefs, or even financial frictions. Hence, our paper illustrates how a large number of empirically-relevant “finance” shocks can also affect macroeconomic outcomes. In this sense our paper is also related to a strand of the literature emphasizes the role of “risk shocks” in exacerbating financial frictions (see, for instance, Christiano et al. (2014); Di Tella (2012)). We share with this literature the emphasis on risk, but we focus on changes in aggregate risk or risk attitudes—as opposed to idiosyncratic uncertainty—which increases risk premia even in absence of frictions. More broadly, there is an extensive recent empirical literature documenting the importance of uncertainty shocks in causing and worsening recessions (see, for instance, Bloom (2009)).

At a methodological level, our paper belongs in the new continuous time macrofinance literature started by the seminal work of Brunnermeier and Sannikov (2014, 2016a) and summarized in Brun- nermeier and Sannikov (2016b) (see also Basak and Cuoco (1998); Adrian and Boyarchenko (2012); He and Krishnamurthy (2012, 2013); Di Tella (2012); Moreira and Savov (2017); Silva (2016)). This literature seeks to highlight the full macroeconomic dynamics induced by financial frictions, which force the reallocation of resources from high-productivity borrowers to low-productivity lenders after a sequence of negative shocks. While the structure of our economy shares many similarities with theirs, in our model there are no financial frictions, and the macroeconomic dynamics stem not from the supply side (relative productivity) but from the aggregate demand side. Related, in recent work, Brunnermeier and Sannikov (2014) also obtain endogenous price volatility but our model makes the additional prediction that volatility will be higher when the interest rate policy is constrained. This prediction lends support to the many unconventional tools aimed at reducing downward volatility, which the major central banks put in place once interest-rate policy was no longer available during the Great Recession.

Our results on macroprudential policy are related to a recent literature that analyzes the implications of aggregate demand externalities for the optimal regulation of financial markets. For instance, Korinek and Simsek (2016) show that, in the run-up to deleveraging episodes that coincide with a zero-lower-bound on the interest rate, welfare can be improved by policies targeted toward reducing household leverage. In Farhi and Werning (2017), the key constraint is instead a fixed exchange rate, and the aggregate demand externality calls for ex-ante regulation but also ex-post redistribution, in the form of a fiscal union. In these papers, heterogeneity in agents’ marginal propensities to consume (MPC) is the key determinant of optimal macroprudential policy. The policy works by reallocating wealth across agents and states in a way that high-MPC agents hold relatively more wealth when the economy is more depressed due to deficient demand. The mechanism in our paper is different and works through heterogeneous asset valuations. In fact, we work with a log-utility setting in which all investors have the same marginal propensity to consume. The policy operates by transferring wealth to optimists during recessions, not because optimists spend more than other investors, but because they raise the asset valuations and induce all investors to spend more (while also increasing aggregate investment).

Beyond aggregate demand externalities, the macroprudential literature is also extensive, and mostly motivated by the presence of pecuniary externalities that make the competitive equilibrium constrained inefficient (e.g., Caballero and Krishnamurthy (2003); Lorenzoni (2008); Bianchi and Mendoza (2013); Jeanne and Korinek (2010)). The friction in this case is not “nominal” and interest rate rigidities, but market incompleteness or collateral constraints that depend on asset prices (see Davila and Korinek (2016) for a detailed exposition). Macroprudential policy typically improves outcomes by mitigating fire sales that exacerbate financial frictions. The policy in our model also operates through asset prices but through a different channel. We show that a decline in asset prices is damaging not only because of the fire-sale reasons emphasized in this literature, but also because it lowers aggregate demand through standard wealth and investment channels. Moreover, our analysis does not feature the incomplete markets or collateral constraints that are central in this literature.

Our results are also related to a large literature that analyzes the effect of belief disagreements and speculation on financial markets (e.g., Lintner (1969); Miller (1977); Harrison and Kreps (1978); Varian (1989); Harris and Raviv (1993); Chen et al. (2002); Scheinkman and Xiong (2003); Fostel and Geanakoplos (2008); Geanakoplos (2010); Simsek (2013a,b); Iachan et al. (2015)). One strand of this literature emphasizes that disagreements can exacerbate asset price fluctuations by creating endogenous fluctuations in agents’ wealth distribution (see, for instance, Basak (2000, 2005); Cao (2017); Xiong and Yan (2010); Kubler and Schmedders (2012); Korinek and Nowak (2016)). Our paper features similar forces but explores them in an environment in which output is not necessarily at its supply-determined level due to interest rate rigidities. In fact, our framework is similar to the models analyzed by Detemple and Murthy (1994); Zapatero (1998), who show that speculation between optimists and pessimists (with log utility) can increase the volatility of the interest rate. In our model, these results apply when the interest rate is unconstrained but they are modified if the interest rate is downward rigid. In the latter case, speculation translates into (ineff cient) fluctuations in asset prices as well as aggregate demand. We show that these fluctuations depress the current level of aggregate demand, which translates into low output and asset prices during recessions. We also show that macroprudential policy that restricts speculation can generate a (Pareto) improvement in social welfare even if the planner respects investors’ individual beliefs.

The rest of the paper is organized as follows. In Section 2 we present an example that illustrates the main mechanism and motivates the rest of our analysis. Section 3 presents the general environment and defines the equilibrium. Section 4 characterizes the equilibrium in a benchmark setting with homogeneous beliefs. This section illustrates how risk premium shocks can lower asset prices and induce a demand recession, and how optimism helps to mitigate the recession. It also illustrates how the drop in asset prices during the recession lowers the interest rate during booms. Section 5 characterizes the equilibrium with belief disagreements, and illustrates how speculation exacerbates the recession. Section 6 establishes our normative results in two steps. Section 6.1 characterizes the value functions and illustrates the aggregate demand externalities. Section 6.2 analyzes the effect of introducing risk limits on optimists, and presents our results on (procyclical) macroprudential policy. Section 7 concludes. The (online) appendix contains the omitted derivations and proofs.

**2. A stepping-stone example**

Here we present a simple (largely static) example that illustrates the workings of the basic aggregate demand mechanism and speculation, and that serves as a stepping stone into our main (dynamic) model.

A two-period risk-centric aggregate demand model. Consider an economy with two dates, t 2 {0,1g, a single consumption good, and a single factor of production—capital. For simplicity, capital is fixed (i.e., there is no depreciation or investment) and it is normalized to one. Potential output is equal to capital’s productivity, zt, but the actual output can be below this level due to a shortage of aggregate demand, yt < zt. For simplicity, we assume output is equal to its potential at the last date, yi = zi, and focus on the endogenous determination of output at the previous date, yo < zq. We assume the productivity at date 1 is uncertain and log-normally distributed so that,

We also normalize the initial productivity to one, zo = 1, so that g denotes the (objective) expected growth rate of productivity, and a denotes its volatility. output.

There are two types of assets. There is a “market portfolio” that represents claims to the output at date 1 (the return to capital as well as profits), and a risk-free asset in zero net supply. We let Q and rk = log ZQ denote, respectively, the price and the log return of the market portfolio, and rf denote the log risk-free interest rate.

There are two types of investors, optimists and pessimists, denoted by the superscript i 2 {o,pg. They are identical except possibly their assessment of productivity growth. Specifically, type i investors believe productivity at date 1 is distributed according to log zi. We assume go > gp so that optimists perceive greater growth. Investors have dogmatic beliefs in the sense that they wouldn’t change their beliefs if they learn about the other type’s beliefs through prices or other means (formally, they “agree to disagree”). We normalize the mass of each belief type to one so that i = o and i = p denotes, respectively, the representative optimist and pessimist. However, we allow investors to differ in their initial wealth shares: specifically, we let a* 2 [0, 1] (with ao + ap = 1) denote the wealth share of type i investors, which will be the key state variable in this economy.

Type i investors are initially endowed with the fraction, a*, of the initial output as well as the same fraction, a* of the claims on future output. They choose how much to consume, c0, and how many assets to hold, a0. They allocate a fraction of their assets, !k’*, to the market portfolio, and the residual fraction, 1 — !k;*, to the risk-free asset. We assume investors have Epstein-Zin preferences with the discount factor given by e~p, the elasticity of intertemporal substitution (EIS) equal to 1, and the relative risk aversion coefficient (RRA) given by 7. We relegate the formal description of the investors’ problem to the appendix (see problem (A.2)). Asset market clearing condition requires,

that is, total amount of wealth invested in the market portfolio equals the value of the market portfolio.

The supply side of the economy is described by New-Keynesian firms that have preset fixed prices. These firms meet the available demand at these prices as long as it does not exceed their marginal costs (see Appendix A.2.2 for details). These features imply that output is determined by the aggregate demand for goods (consumption) up to the capacity constraint,

Since prices are fully sticky, the real interest rate is equal to the nominal interest rate, which is controlled by the monetary authority. We assume that the interest rate policy attempts to replicate the supply-determined output level. However, there is a lower bound constraint on the interest rate, rf > 0. Thus, the monetary policy is described by, rf = max (rf *, 0), where rf* is the natural interest rate that ensures output is at its potential, yo = Zo.

To characterize the equilibrium, first note that there is a tight relationship between output and asset prices. Specifically, the assumption on the EIS implies that each investor consumes a fraction of her lifetime income,

Aggregating this equation across investors, and using the aggregate resource constraint (3), we obtain the following output-price relation,

Intuitively, asset prices increase aggregate wealth and consumption, which in turn leads to greater output.

Next note that asset prices must be also consistent with equilibrium in risk markets. Combining Eqs. (2) , (4) and (5), the asset market clearing condition can be rewritten as,

that is, investors’ wealth-weighted average portfolio weight on the market portfolio is equal to one. In Appendix A.1, we show that, up to a local approximation, each investor’s optimal weight on the market portfolio is determined by,

In words, the optimal portfolio risk (left side) is proportional to “the perceived Sharpe ratio” on the market portfolio (right side). The Sharpe ratio captures the reward per risk, where the reward is determined by the risk premium: the (log) expected return in excess of the (log) risk free rate. This is the standard risk-taking condition for mean-variance portfolio optimization, which applies exactly in continuous time and approximately in the two-period model.

Combining Eqs. (6) and (7), with the observation that rk = log Q, we obtain the risk balance condition

In words, the equilibrium in asset markets requires the perceived Sharpe ratio on the market portfolio (right side) to be sufficiently large to convince the investors to hold the risk generated by the productive capacity (left side). Note that the perceived Sharpe ratio depends on the wealth share of optimists and pessimists.

Next consider the supply-determined equilibrium in which output is equal to its potential, yo = zo = 1. Eq. (5) reveals that this requires the asset price to be at a particular level, Q* = e~p. Combining this with Eq. (8), the interest rate needs to be at a particular level,

Intuitively, the monetary policy needs to lower the interest rate to a sufficiently low level to induce sufficiently high asset prices and aggregate demand to clear the goods market.

Now suppose the initial parameters are such that rf* > 0, so that the equilibrium features Q* , rf* and supply-determined output, yo = zo = 1. Consider a “risk-premium shock” that raises the volatility, a, or risk aversion, 7. The immediate impact of this shock is to create an imbalance in the risk-market equilibrium condition (8). The economy produces too much risk (left side) relative to what investors are willing to absorb (right side). In response, the monetary policy lowers the risk-free interest rate (as captured by the decline in rf* ), which increases the risk premium and equilibrates the risk market condition (8). Intuitively, the monetary authority lowers the opportunity cost of risky investment and induces investors to absorb risk.

Next suppose the shock is sufficiently large so that the natural interest rate becomes negative, rf* < 0, and the actual interest rate becomes constrained, rf = 0. In this case, the risk market condition is reestablished with a decline in the price of the market portfolio, Q. This increases the expected return on risky investment, which in turn induces investors to absorb risk. However, the decline in Q reduces aggregate wealth and induces a demand-driven recession. Formally,

Note also that, in the constrained region, asset prices and output become sensitive to beliefs about

future prospects. For instance, a parallel upward shift in investors’ expected growth rates, g° and gp (optimism) increases asset prices and mitigates the recession. In fact, while we analyzed “risk premium shocks” that raise a or 7, Eqs. (8) and (9) reveal that “pessimism shocks” that lower investors’ g° and gp would qualitatively lead to the same effects. Moreover, risk-premium shocks also lead to expected future demand recessions in the dynamic model, which depresses expected profits, which reinforces the decline in current asset prices, and so on.

Eq. (9) reveals further that asset prices and output become sensitive to investors’ wealth shares, a° and ap. For instance, an increase in optimists’ wealth share, a°, mitigates the recession. Unlike beliefs (which are exogenous), investors’ wealth shares in a dynamic setting are endogenously determined and can be influenced by policy. As we will see, macroprudential policy in our setting will improve outcomes precisely because it will increase optimists’ wealth share, a°, during demand recessions.

Time-varying risk premium and demand recessions. The finance literature has documented that the risk premium on most asset classes moves over time. For example, Campbell and Shiller (1988) show that a decrease in the price to dividend ratio of stocks predicts high expected stock market returns as well as high (realized) equity risk premium. Bollerslev et al. (2015) show that changes in the variance risk premium further helps to predict the expected stock market returns. There are also “return predictability” results for treasury yields, corporate bonds, foreign exchange (the carry trade), and so on, which illustrate that time-varying risk premium is a pervasive phenomenon (see Cochrane (2011); Campbell (2014) for recent reviews). There is disagreement in the literature about what drives the time-varying risk premium. The “behavioral” strand emphasizes psychological factors (e.g., Shiller et al. (2014); Greenwood and Shleifer (2014)), which in our model could be mapped into changes in the perceived g, a in excess of their objective values. The “rational” strand emphasizes risk attitudes (e.g., Campbell and Cochrane (1999)), long-run risks (e.g., Bansal and Yaron (2004)), or disaster risks (e.g., Bansal and Yaron (2004); Barro (2006); Gabaix (2012)), which could be mapped into changes in 7 or a. Our analysis illustrates that time-varying

risk premium can generate a demand recession regardless of its source.

To see why the result applies generally, note that the risk premium shock exerts downward pressure on asset prices without a change in the current level of potential output. In view of the relationship between asset prices and aggregate demand, this type of shock exerts recessionary pressures regardless of its source. When the interest rate is constrained, these shocks lead to a demand recession. In the dynamic model, we will generate time-varying risk premium from shocks to a, as this leads to a tractable analysis, but we view these shocks as as capturing the more general forces behind the time-varying risk premium that we observe in the data.

**3. General environment and equilibrium**

In this section we introduce our general environment and define the equilibrium. In subsequent sections we will characterize this equilibrium in various special cases of interest. We start by describing the production and investment technology, as well as the risk-premium shocks that play the central role in our analysis. We then describe the firms’ investment decisions, followed by the investors’ consumption and portfolio choice decisions. Then, we introduce the nominal and the interest rate rigidities that ensure output is determined by aggregate demand. We finally introduce the goods and asset market clearing conditions and define the equilibrium.

Potential output and risk-premium shocks. The economy is set in infinite continuous time, t 2 [0,1), with a single consumption good and a single factor of production: capital. Let kt;S denote the capital stock at time t and the aggregate state s 2 S. Suppose that, when fully utilized, kt;S units of capital produces Akt;S units of the consumption good. Hence, Akt;S denotes the potential output in this economy. Capital follows the process,

Here, q,S = denotes the investment rate, ' (h,s) denotes the production function for capital (that will be specified below), and d denotes the depreciation rate. The second equation defines the expected growth rate of capital (and potential output). The term, dZt, denotes the standard Brownian motion, which captures “aggregate productivity shocks.

The states, s 2 S, differ only in terms of the volatility of aggregate productivity, aS. For simplicity, suppose there are only two states, s 2 f1, 2g, with a\ < (see the extended working paper version Caballero and Simsek (2017b) for the general formulation with an arbitrary number of states). State s = 1 corresponds to a low-volatility state, whereas state s = 2 corresponds to a high-volatility state. At every instant, the economy in state s transitions into the other state s0 = s according to a Poisson process.

Remark 1 (Interpreting the Volatility Shocks). As we explain in Section 2, we use the volatility shocks to capture the time variation in the risk premium due to various unmodeled subjective or objective factors (such as irrational beliefs, risk aversion, long-run risks, disaster risks, Knightian uncertainty, or financial frictions). The variance parameters, {ct2} , could be viewed as the exogenous shifters of the risk premium due to these unmodeled factors.

Transition probabilities and belief disagreements. We let A), denote the Poisson transition probability in state s (into the other state) according to investor i 2 I. These probabilities will play a central role in the analysis, as they capture investors’ optimism or pessimism. For instance, an investor with low A2> is pessimistic in the sense that she expects the high risk conditions to persist. Likewise, an investor with high A1 is pessimistic in the sense that she believes that, even though the economy currently features low risk, the high risk conditions are around the corner. We will set up the model for investors with heterogeneous beliefs (and in fact, this will be the only possible source of heterogeneity). We will first analyze the special case with common beliefs (Section 4) and then investigate the effect of belief disagreements and speculation (Section 5). When investors disagree, they have dogmatic beliefs: that is, they know each others’ beliefs and they agree to disagree. We use these types of belief disagreements to capture a broad array of reasons that generate heterogeneous valuations and trade in financial markets, ranging from a literal interpretation to institutional factors (see Remark 3 in Section 5).

Investment and the growth-price relationship. There is a continuum of identical firms that manage capital. These firms rent capital to production firms (that will be described below) to earn the instantaneous rental rate, Rt;S. They also make investment decisions to maximize the value of capital. Letting QtyS denote the price of capital, the firm’s investment problem can be written as,

Under standard regularity conditions for ' (i), investment is determined by the optimality condition, ' (it,.s) = 1/Qt,S We will work with the special and convenient case proposed by Brunnermeier and Sannikov (2016b): ' (i) = /log ^^ + 1^. In this case, we obtain the closed form solution,

The parameter, , captures the sensitivity of investment to asset prices.

The log price level, qt,s, will simplify some of the expressions below. Combining Eq. (12) with Eq. (10), we also obtain an expression for growth,

In particular, unlike in the two period model, the expected growth rate of capital (and potential output) is now endogenous and depends on asset prices. Lower asset prices reduce investment, which in turn translates into lower growth and lower potential output in future periods. This mechanism will be a source of amplification.

Capital price and return. As before, we assume there is a “market portfolio” that represents a claim on aggregate capital (more specifically, a claim on the firms that manage capital). The return on this portfolio depends on (among other things) the evolution of the value of aggregate capital, Qt,skt,s. We next describe how Qt,skt,s evolves and how this translates into the return.

Absent transitions, the price of capital follows an endogenous but deterministic process

When investors have common beliefs (Section 4), the endogenous price drift will be zero, ^Qs = 0: that is, the price of capital will be fixed within low and high risk states, {Qi,Q2}. With belief disagreements (Section 5), there will be room for price dynamics due to changes in investors’ wealth shares. Combining Eqs. (10) and (14), the aggregate wealth (conditional on no transition) evolves according to

It follows that, absent state transitions, the volatility of the market portfolio is given by, Us. Likewise, the expected return on this portfolio conditional on no transition is given by,

Here, the first term can be thought of as the “dividend yield,” which captures the instantaneous rental rate of capital, Rt,s, as well as the investment costs. The second component is the (expected) capital gain conditional on no transition, which reflects the expected growth in aggregate wealth due to the growth of capital or price drift.

Eqs. (14 — 16) describe the prices and returns conditional on there not being a state transition. If there is a transition at time t from state s into state s' = s, then the price of capital jumps from Qt;S to a potentially different level, Qt S'. Therefore, the aggregate wealth also jumps from Qt,Skt,S to a potentially different level, Qt;S'kt;S, and the investors that hold the market portfolio experience instantaneous capital gains or losses that will be reflected in their portfolio problem.

Consumption and portfolio choice. There is a continuum of investors denoted by i 2 I, who are identical in all respects except possibly their beliefs about state transitions, AS, and who continuously make consumption and portfolio allocation decisions. Each investor has access to three types of assets. First, she can invest in the market portfolio that we described above. Second, she can invest in a risk-free asset with return, rf S. The risk-free asset is in zero net supply. Third, the investor can also invest in a contingent Arrow-Debreu security that trades at the (endogenous) instantaneous price pS S, and that pays 1 dollar if the economy transitions to the other state s' = s. These securities are also in zero net supply, and they ensure that the financial markets are complete.

Specifically, at any time t and s, investor i has some financial wealth denoted by at S. She chooses her consumption rate, denoted by c* S; what fraction of her wealth to allocate to capital, denoted by S; and what fraction of her wealth to allocate to the contingent security, ! S*. The residual fraction, 1 — S* — !S*, is invested in the risk-free asset. For analytical tractability, we assume the investor has log utility. The investor then solves a relatively standard portfolio problem. Appendix A.2.1 states the problem formally and derives the optimality conditions using recursive techniques. In view of log utility, the investor’s consumption is a constant fraction of her wealth,

Less obviously, the investor’s optimal portfolio allocation to capital is determined by,

Intuitively, she invests in capital up to the point at which the risk of her portfolio (left side) is equal to “the

Sharpe ratio” of capital (right side). This is similar to the optimality condition in the two period model (cf. Eq. (7)) with the difference that the dynamic model also features state transitions. Our notion of the Sharpe ratio accounts for potential revaluation gains or losses from state transitions (the term, ^t;Q ^M) as well as the adjustment of marginal

The portfolio weight, !St S*, is implicitly determined as the level that ensures that this equality holds. The investor buys contingent securities up to the point at which the price-to-probability ratio of a state (or the state price) is equated to the investor’s relative marginal utility in that state. Note that replacing (19) into (18) shows that investors allocate identical portfolio weights to capital, s (which will be equal to one in equilibrium), and express their differences in beliefs through their holdings of contingent securities.

Equilibrium in asset markets. Asset markets clearing requires that the total wealth held by investors is equal to the value of aggregate capital before and after the portfolio allocation decisions,

Contingent securities are in zero net supply, which implies,

The market clearing condition for the risk-free asset (which is also in zero net supply) holds when conditions (2°) and (21) are satisfied.

Nominal rigidities and aggregate demand. The supply side of our model features nominal rigidities similar to the standard New Keynesian model. We relegate the details to Appendix A.2.2 and describe the main implications relevant for our analysis. There is a continuum of monopolistically competitive production firms that rent capital from investment firms and produce intermediate goods (which are then converted into the final good). For simplicity, these production firms have preset nominal prices that they never change. The firms meet the available demand (as long as they find it optimal to do so). In equilibrium, these features imply that output is determined by aggregate demand,

Here, % s denotes the instantaneous factor utilization rate for capital. We assume firms can increase factor utilization for free until 'qts — 1 and they cannot increase it beyond this level (we relax the latter assumption in the extended working paper version). Aggregate demand corresponds to the sum of aggregate consumption and aggregate investment.

There are also lump sum taxes on the production firms’ profits combined with linear subsidies

This also implies, yt,s — Rt,skt,s, that is all output accrues to the investors in the form of return to capital, which simplifies our analysis. Combining this expression with Eqs. (16), and using Eqs. (22) and (17), we also obtain the instantaneous (expected) return to capital conditional on no transition as,

Hence, in equilibrium, the dividend yield from capital is the same as the consumption rate p.

Output-price relationship. Our analysis so far implies that there is a one-to-one relationship between output and the price of capital as in the two period model (cf. Eq. (5)). Specifically, combining Eqs. (17) and (20) implies that aggregate consumption is a constant fraction of aggregate wealth, Jj ctj.sdi — pQt, skt,s. Plugging this into Eq. (22), and using the investment equation (11), we obtain,

Intuitively, output per capital (or factor utilization) depends on asset prices, because consumption depends on asset prices through a wealth effect and investment depends on asset prices through a standard marginal-Q channel. Rewriting this expression, we obtain,

This illustrates that full factor utilization, % s — 1, obtains only if the price of capital is at a particular level q* = q (1). This is the efficient price level that ensures that the implied consumption and investment clears the goods market. Likewise, the economy features a demand recession, ^t s < 1, if and only if the price of capital is strictly below q* .

Interest rate rigidity and monetary policy. Our assumption that production firms do not change their prices implies that the aggregate nominal price level is fixed. The real risk-free interest rate is then equal to the nominal risk-free interest rate, which is determined by the interest rate policy of the monetary authority. We assume there is a lower bound on the nominal interest rate,

In practice, this type of constraint emerges naturally from a variety of factors. The zero lower bound in particular can be motivated by the presence of cash in circulation (which we leave unmodeled for simplicity). Since cash offers zero interest rate, the monetary authority cannot lower the interest rate (much) below zero—a constraint that appeared to be binding for major central banks in the aftermath of the Great Recession.

We assume that the interest rate policy focuses on replicating the level of output that would obtain absent nominal rigidities subject to the constraint in (26). Appendix A.2.2 illustrates that, without nominal rigidities, capital is fully utilized, % s = 1. Thus, we assume the interest rate policy follows the rule,

Here, rf’s* is recursively defined as the (instantaneous) natural interest rate that obtains when the (instantaneous) utilization is given by r/t s = 1, and the monetary policy follows the rule in (27) at all future times and states.

Remark 2 (Interpretation of Price Stickiness). Our assumption that the aggregate nominal price (or inflation) level is fixed is extreme. However, we should note that making the prices more flexible does not necessarily circumvent the bound in (26). In fact, if monetary policy follows an inflation targeting policy regime, then limited price flexibility leads to price deflation during a demand recession, which strengthens the bound in (26) and exacerbates the recession (see Werning (2012); Korinek and Simsek (2016); Caballero and Farhi (2017) for further discussion). We could capture this mechanism by allowing for some price flexibility, which would introduce a standard New-Keynesian Phillips curve into the model as in Werning (2012). We have chosen not to emphasize the deflationary spiral mechanism since the analysis is already involved with several other- amplification mechanisms related to the endogeneity of (real) asset prices.

Equilibrium in the goods market. Combining Eq. (27) with the output-price relationship (25), the goods market side of the economy can be summarized with,

In particular, the equilibrium at any time and state takes one of two forms. If the natural interest rate is nonnegative, then the interest rate policy ensures that the price of capital is at the efficient

level, qt;S = q*, capital is fully utilized, iqts = 1, and output is equal to its potential, yt,s = Akt,s.

Otherwise, the interest rate policy is constrained, rfs = 0, the price of capital is at a lower level, qt,s < q*, and output is determined by aggregate demand according to Eq. (25). We can now define the equilibrium as follows.

Definition 1. The equilibrium is a collection of processes for allocations, prices, and returns such that capital and its price evolve according to Eqs. (10) and (14), investment firms maximize (cf. Eqs. (17), the growth rate is given by Eq. (13), investors maximize (cf. Eqs. (17 — 19)), asset markets clear (cf. Eqs. (20) and (21)), output is determined by aggregate demand (cf. Eqs. (22) and (25)), the return to capital (conditional on no transition) is given by Eq. (24), the interest rate policy follows the rule in (27), and the goods market clears (cf. Eq. (28)).

For future reference, we also note that the first-best equilibrium without interest rate rigidities implies that the price of capital is at its efficient level at all times and states, qt , s = q*. This also implies that the growth rate of output and the expected return to capital are constant and given by, respectively, g = fiq* — 5 and rk = p + fiq* — 5 (see Eq. (24)). We next turn to the characterization of equilibrium with interest rate rigidities.

**4. Common beliefs benchmark and amplification mechanisms**

In this section, we analyze the equilibrium in a benchmark case in which all investors share the same belief, that is, AS = As for each i. We also normalize the total mass of investors to one so that individual and aggregate allocations are the same. We use this benchmark to establish two amplification mechanisms that have no counterparts in the two period model. We also establish the comparative statics of the equilibrium with respect to investors’ (common) belief, and illustrate that amplification mechanisms are especially powerful when investors are pessimistic.

In view of the linear structure of the model, we conjecture that the price and the interest rate will remain constant within states, Qt,s = Qs and rf s = rf (in particular, there is no price drift, Ets = 0). Since the investors are identical, we also have s = 1 and ! s = 0. In particular, the representative investor’s wealth is equal to aggregate wealth, at,s = Qt, skt , s. Combining this with Eq. (18) and substituting for rk s from Eq. (24), we obtain the following risk balance conditions,

These equations are the dynamic counterpart to Eq. (8) in the two period model. They say that, in each risk state, the total risk in the economy (the left side) is equal to the Sharpe ratio perceived by the representative investor (the right side). Note that the Sharpe ratio accounts for the fact that the aggregate wealth (as well as the marginal utility) will change in case there is a state transition.

The equilibrium is then characterized by finding four unknowns, (Qi,rf, Q2,rfj, that solve the two equations (29) together with the two goods market equilibrium conditions (28). We solve these equations under the following parametric restriction.

Assumption 1. o^{2} > P + V’q* _ 5 > o.

When this restriction holds (and additional assumptions are satisfied), there is an equilibrium in which the low-risk state 1 features positive interest rates, efficient asset prices, and full factor utilization, rf > 0, qi = q* and ri = 1, whereas the high-risk state 2 features zero interest rates, lower asset prices, and imperfect factor utilization, rf = 0, q2 < q* and r2 < 1. In particular, the analysis with common beliefs reduces to finding two unknowns, (q2,rf), that solve the two risk balance equations (29) (after substituting qi = q* and rf =0).

Equilibrium in the high-risk state. Using our conjecture, the risk balance equation (29) for the high-risk state s = 2 can be written as,

In view of Assumption 1, if the price were at its efficient level, Q2 = Q*, the risk (the left side) would exceed the Sharpe ratio (the right side). As in the two period model, the economy generates too much risk relative to what the investors are willing to absorb at the constrained level of the interest rate. As before, the price of capital, Q2, needs to decline to equilibrate the risk markets. Unlike in the two period model, however, the decline in the price of capital does not necessarily increase the Sharpe ratio, due to two destabilizing amplification mechanisms.

Amplification mechanisms. The first amplification mechanism comes from the output-price relation (cf. Eq. (25)). If the dividends from capital were kept constant, a decline in the current asset price would increase the dividend yield as well as the return—a stabilizing force. However, in our model the dividends are not constant and they are increasing in the current price of capital. A lower asset price level reduces output and economic activity, which reduces the rental rate of capital (see Eq. (23)), which in turn lowers dividends. In fact, the dividend yield term in Eq. (30) can be better understood by writing it as, ^QQ = p (see also Eq. (16)). It does not depend on the price because the cash flows in the numerator also decline proportionally with the price level. Hence, the output-price relation overturns an important stabilizing force from price declines, and opens the door for amplification of these declines.

The second amplification mechanism comes from the growth-price relation (cf. Eq. (13)). In particular, a decline in the current asset price also lowers investment, which reduces the expected growth of potential output and dividends, which in turn lowers the return to capital. The strength denotes the capital gains and Qt,s denotes the marginal utility adjustment when there is a representative investor Qt.s' (see (18)).

of this effect depends on the sensitivity of investment to asset prices, captured by the term 092. Figure 1 in the introduction presents a graphical illustration of the two amplification mechanisms.

In view of these amplification mechanisms, one might wonder how the risk market ever reaches equilibrium once the price, Q2, starts to fall below its efficient level, Q*. The stabilizing force is captured by the last term in Eq. (30), A2 ^1. A decline in the price of capital increases the

expected capital gain from transition into the recovery state s = 1, which tends to increase the expected return to capital as well as the Sharpe ratio. Note that the stabilizing force is stronger when investors are more optimistic and perceive a higher transition probability into the recovery state, A2. In fact, to ensure that there exists an equilibrium with positive prices, we need a minimum degree of optimism, which is captured by the following assumption.

Assumption 2. A2 > All1™, where All1™ is the unique solution to the following equation over the range A2 > 0:

Assumption 2 ensures that there is a unique positive solution to Eq. (30) (see Appendix A.3). When the assumption holds as strict inequality, the decline in prices increases the Sharpe ratio. In this case, the stabilizing capital gains force dominates the destabilizing endogenous output and growth mechanisms. When the condition is violated, a lower price level would lower the return further, which would trigger a downward spiral that would lead to an equilibrium with zero asset prices and output. When the condition holds as equality, the stabilizing force barely balances the destabilizing mechanisms. As we will see below, the price and output in this case is very low and also very sensitive to further changes in beliefs.

Equilibrium in the low-risk state. Using our conjecture, the risk balance equation (29) for the low-risk state s = 1 can be written as,

Given 92, this equation determines the interest rate, rf. Intuitively, given the expected return on

capital (that depends on 92, among other things), the interest rate adjusts to ensure that the risk-

balance condition is satisfied with the eff cient price level, 9i = 9* . For our conjectured equilibrium,

we also require that the implied interest rate to be nonnegative, rif > 0. The following parametric condition ensures that this is the case.

Assumption 3. Ai < A0ax (92), where A0ax (92) > 0 denotes the unique solution to the following

equation with q2 < q* that solves Eq. (30):

That is, we need pessimism in the low-risk state (captured by the transition probability) to be sufficiently low so that the fear of a transition into the high-risk state does not push the economy into the interest rate lower bound. As expected, greater equilibrium price level in the high-risk state, q2, increases the upper bound for pessimism, Amax (q2).

Proposition 1. Consider the model with two states, s 2 {1, 2g, with common beliefs and Assumptions 1-3. The low-risk state 1 features a nonnegative interest rate, efficient asset prices and full factor utilization, rf > 0, qi = q* and ^1 = 1, whereas the high-risk state 2 features zero interest rate, lower asset prices, and a demand-driven recession, rf = 0, q2 < q*, and ^2 < 1. The price level in state 2 is characterized as the unique solution to Eq. (30), and the risk-free rate in state 1 is characterized by Eq. (31).

Comparative statics for the high-risk state. We next establish comparative statics of the equilibrium, starting with the high-risk state. First consider how a change in optimism, A2, affects the price of capital, q2. Implicitly differentiating Eq. (30), we obtain,

Here, the inequality follows since the denominator is nonnegative in view of Assumption 2 (see Appendix A.3). Hence, the effect of optimism on the price is determined by its direct effect on the expected return to capital captured in the numerator, which is positive. Intuitively, greater optimism increases the expected capital gains, which increases the asset price.

Next consider this expression for the special case in which optimism is at its lowest allowed level, A2 = Amin, so that Assumption 2 holds as equality. In this case, the denominator in Eq. (32) is zero, and we have = 1. Hence, in the neighborhood of A2 = A!)1™, the recession is deep, and asset prices and output are extremely sensitive to further changes in beliefs due to the destabilizing endogenous output and growth mechanisms.

More generally, as Eq. (32) illustrates, the destabilizing mechanisms are weaker when investors are optimistic about recovery. Hence, optimism in this model raises asset prices not only because of its direct impact on asset valuations, but also because it weakens the destabilizing feedback effects. Figure 2 illustrates these results for a particular parameterization.

Comparative statics for the low-risk state. Note also that, as illustrated by Eq. (31), these changes that reduce the price in the high-risk state, q2, also reduce the interest rate in the low-risk f state, rf. Lower prices in state 2 also lower asset prices and aggregate demand in state 1, which is countered by a lower interest rate. Moreover, the interest rate in the low-risk state is also influenced

by the beliefs in this state. Specifically, we have

Figure 3 illustrates this result for a particular parameterization. For this exercise, we set A2 = Amm so that the recession is severe and q2 is low. We also set the exogenous shifter of the risk premium in the boom state to be much lower than in the recession state, a2 = 0.01 < ^2 = 0.1 (so as to capture the current low volatility environment). This choice ensures that the first-best level of the interest rate in the boom state is quite high, rf * ' 7%. This is also the interest rate that obtains in equilibrium when pessimism is extremely low so there is no recession risk. The figure illustrates that, starting from this benchmark, small doses of pessimism can considerably lower the risk-free interest rate, rf. In particular, the equilibrium interest rate becomes zero for A^^ ' 0.09, i.e., when the representative investor assigns about 9% probability to a risk-driven recession in a given year.

How can a relatively small chance of a recession lower the interest rate by several percentage points? The intuition is that, as we discussed above, the price during the recession, q2, is lowered considerably due to the destabilizing forces triggered by a combination of a risk shock and pessimism. The fear of a downward price spiral lowers the interest rate during the boom. Figure 3 decomposes this effect further into a component that reflects the expected capital loss from the jump into the recession, and an additional component that reflects the jump risk premium. Note

that both components are sizeable. In particular, risk premium can be elevated even when the exogenous shifters of the risk premium such as volatility are low.

Endogenous Jump Volatility An important aspect of equilibrium is that it features endogenous volatility in asset prices. To establish this formally, we fix some At > 0 and consider the proportional change in the value of capital over this time interval, defined as,

Corollary 1. For any s 2 {1, 2g, the instantaneous (unconditional) variance of capital is,

This is strictly greater than the instantaneous variance that would obtain in the first-best equilibrium without interest-rate frictions, fi2:..

Intuitively, when there is a shock to the risk premium, the interest rate policy changes the rate to mitigate the impact of the shock on asset prices. Interest rate rigidities reduce the ability of the policy to lean against risk premium shocks, which leads to endogenous volatility. As we will see in the next section, speculation exacerbates endogenous volatility further, because it generates endogenous fluctuations in the effective belief that determines asset prices.

**5. Belief disagreements and speculation**

We next consider the equilibrium with belief disagreements. We show that speculation induced by belief disagreements creates further amplification and worsens the recession. While investors’

beliefs are exogenously fixed, the extent of their speculation can be influenced by policy, which motivates our analysis of welfare and macroprudential policy in the next section.

We restrict attention to two types of investors, “optimists” and “pessimists”, with beliefs denoted by, { (Ap,Ap) }je{0pg. We normalize the mass of each belief type to one so that i = o and i = p denotes, respectively, the representative optimist and pessimist. We assume the beliefs satisfy the following.

Assumption 4. Ap > Ap and Ap < Ap.

This assumption ensures that optimists are more optimistic than pessimists in either state. Specifically, when the economy is in the high-risk state, optimists find the transition into the low-risk state relatively likely (Ap > Ap); when the economy is in the low-risk state, optimists find the transition into the high-risk state relatively unlikely (Ap < Ap).

Remark 3 (Interpreting Persistent Belief Disagreements). The essence of this assumption is that there are some investors that value risky assets more than others, and that they do so across most environments. This could be interpreted literally as differences in beliefs, in which case it is supported by an extensive psychology literature that documents the prevalence of optimism, as well as its heterogeneity and persistence—since it is largely a personal trait (see Carver et al. (2010) for a review). The assumption could also be interpreted as capturing in reduced form other fundamental reasons for heterogeneous valuations, such as differences in risk tolerance or (perceived) Knightian uncertainty, which are likely to be persistent. Finally, the assumption could capture institutional reasons for heterogeneous valuations, such as capacity or mandates for handling risk. Investment banks, for example, have far larger capacity to handle and lever risky positions than pensioners and money market funds. Our qualitative results are robust to the exact source of heterogeneous valuations, as long as this heterogeneity is persistent across booms and recessions.

To characterize the equilibrium, we define the wealth-weighted average transition probability,

Here, at,s denotes optimists’ wealth share, and it is the payoff-relevant state variable in this economy. The

notation, As (at,s), describes the wealth-weighted average belief in state s as a function of optimists’ wealth share, and At,s denotes the belief at time t and state s. This belief is central to the analysis because the following analogue of the risk balance condition (29) holds in this setting (see Appendix A.4),

In particular, the equilibrium in risk markets is determined according to the wealth-weighted average belief. When at,s is greater, optimists exert a greater influence on asset prices.

It remains to characterize the evolution of optimists’ wealth share, at,s (and thus, the evolution of At,s). In Appendix A.4, we solve for investors’ positions and find that = ^t’S = 1. That is, investors continue to have the same exposure to the market portfolio, which is equal to one in equilibrium. Intuitively, since investors disagree about the jump probabilities, they settle these disagreements by adjusting their holdings of contingent securities as opposed to their exposure to the diffusion risk. In fact, we have the following closed form solution for optimists’ equilibrium contingent positions [cf. Eq. (A.22)],

Optimists take a positive position on a contingent security whenever their belief for the transition probability

exceed the weighted average belief. In view of Assumption 4, we further have, !2° < 0 and u^’2° > 0. In the boom (low-risk) state, optimists sell put options since they think transition into the recession (high-risk) state is unlikely. In the recession state, they buy call options since they believe the transition into the boom state is likely.

Consistent with this interpretation, we also find that optimists’ wealth share evolves according to [cf. Eqs. (A.23) and (A.24)],

Here, ext,s — d<0t’s denotes the derivative with respect to time. In the boom state, optimists’ wealth share

drifts upwards due to the profits they make from selling put options, but it makes a downward jump if there is a transition into the recession state. In the recession state, optimists’ wealth share drifts downwards due to the cost of the call options they purchase, but it makes an upward jump if there is a transition into the boom state. Figure 4 illustrates the dynamics of optimists’ wealth share for a particular parameterization and a particular realization of uncertainty.

These observations also imply that the weighted-average belief in (33) (that determines asset prices) is effectively extrapolative. As the boom state persists, and optimists’ wealth share increases, the aggregate belief becomes increasingly more optimistic. After a transition to the recession state, the aggregate belief becomes more pessimistic. Conversely, the aggregate belief becomes more pessimistic as the recession persists, and it becomes more optimistic after a transition into the boom. As we will see, these endogenous extrapolation dynamics and their anticipation are behind the amplification mechanism in this setting.

The equilibrium is then characterized as follows. Regardless of the level of asset prices and output, Eq. (36) determines the evolution of investors’ wealth shares. This in turn determines the weighted average belief, as well as its evolution [cf. Eq. (33)]. Given the characterization for the weighted-average belief, the equilibrium is determined by jointly solving the risk balance equation (34) and the goods market equilibrium condition (28). Solving these equations is slightly more involved than before since the weighted-average belief is generally not stationary, which implies the

price of capital might also have a nonzero drift, p,Qs (although a

To make progress, we suppose Assumptions 1-3 from the previous section hold according to both belief types. This ensures that, regardless of the wealth shares, the low-risk state 1 features a positive interest rate, efficient price level, and full factor utilization, rf 1 > 0 , qt, 1 = q* ,rt 1 = 1, and the high-risk state 2 features a zero interest rate, a lower price level, and imperfect factor f utilization, rf 2 = 0,qt,2 < q* , r 1 < 1. We next characterize this equilibrium starting with the high-risk state.

Equilibrium in the high-risk state. Consider the risk balance equation (34) for state s = 2. After substituting the return to capital from (24), and using pQ2 = dq;'1J=t = qt,2, we obtain,

This expression is similar to its common-beliefs counterpart, Eq. (30), except for the term, qt,2, which captures the price drift conditional on no transition. This term enters the risk balance condition since it affects the expected return on capital. A negative price drift lowers the expected return and exerts a downward pressure on the equilibrium price. Conversely, a positive price drift increases the return and exerts an upward pressure.

To solve for the equilibrium, we combine Eqs. (36) and (37) to obtain a differential equation,

This system describes the joint evolution of the price and optimists’ wealth share, (qt,2, at,2), conditional on there not being a transition. In Appendix A.4, we show that this system is saddle path stable. In particular, for any initial wealth share, at2 2 (0,1), there exists a unique equilibrium price level.

When at;2 = 1, the solution satisfies qt;2 = q°.

Note also that the equilibrium system in (38) is stationary, which implies that the equilibrium price can be written as a function of optimists’ wealth share, that is, qt;2 = q2 (a) for some function q2 : [0,1] ! [qp,q°]. In particular, we can eliminate time from the system in (38) (using the observation, qt2 = q2 (a) at2), to obtain,

This provides an equivalent characterization of the price function as a solution to a differential equation in a-domain, together with the boundary conditions, q2 (0) = qp and q2 (1) = q°. In Appendix A.4, we further show that the price function, q2 (a), is strictly increasing in a. As in the previous section, greater optimism increases the asset price.

Amplification from speculation. We next present the main result in this section, which illustrates that speculation creates further amplification. To this end, we define q^ (a) as the solution to the risk balance equation in the common-beliefs benchmark [cf. Eq. (30)] when all investors share the wealth-weighted average belief, A2 (a). Comparing the equilibrium price with this benchmark

isolates the effect of speculation. In the appendix, we show that

That is, the equilibrium with speculation always features a lower equilibrium price (and a more severe recession).

Intuitively, speculation reshuffles optimists’ wealth across states so that they become wealthier in case there is a transition into the boom state but they become poorer if the recession persists longer [cf. Eq. (36)]. The increase in optimists’ wealth in the boom state does not increase asset prices since it is neutralized by monetary policy, which increases the interest rate and keeps the price of capital at its efficient level. However, the decline in optimists’ wealth in the recession state causes damage. Specifically, conditional on no transition, optimists’ wealth share and the asset price drift downwards, at;2 < 0 and qt;2 < 0. Moreover, as illustrated by Eq. (37), the damage is anticipated by investors and lowers their expected return to capital. Thus, the current price falls further to equilibrate the risk balance condition, which leads to a more severe recession.

Figure 5 illustrates the price function, 92 (a), for a particular parameterization. We chose the parameters so that pessimists’ transition probability in state 2 is at the lowest allowed level, Af = (see Assumption 2). This implies that, when optimists’ wealth share is low, asset prices and output are very low due to the destabilizing feedbacks that we discussed in the previous section. The figure also illustrates that the price with belief disagreements differs sharply from the (appropriate) common beliefs benchmark. When investors share the same belief, there is no speculation and optimism improves the price considerably. With belief disagreements, optimism has a smaller impact since it comes bundled with speculation. This suggests that it is enough to have one group of highly pessimistic investors to unleash destabilizing dynamics.

Equilibrium in the low-risk state. Following similar steps for the risk balance condition for the low-risk state s = 1, we obtain,

Here, rf (a) denotes the interest rate when optimists’ wealth share is equal to a. The interest rate depends on (among other things) the weighted average transition probability into the high- risk state, Ai (a), as well as the price level that would obtain after transition, 92 (a0). The latter depends on the wealth-share of optimists after transition, a0, which is smaller than a since optimists are selling put options. For our conjecture to be valid, we also require that rf (a) > 0 for each a. This condition holds because Assumptions 1-3 hold for pessimists (as well as optimists).

It is easy to check that the interest rate function, rf (a), is increasing in optimists’ wealth share, a, for two reasons. First, smaller a makes the wealth-weighted average belief assign a higher probability to a transition into the recession state, which decreases the interest rate (even if qt;2 were kept constant). This effect is reminiscent of the analysis in Hall (2016), who argues that the decline in the wealth share of relatively optimistic (and risk tolerant) investors can explain some of the decline in the interest rate in recent years. In our model, there is a second effect that operates in the same direction because the severity of the recession is endogenous. In particular, smaller a also reduces the price after a transition into the recession state, q2 (a0), which further lowers the interest rate. The following result summarizes the equilibrium characterization.

Proposition 2. Consider the model with two beliefs types. Suppose Assumptions 1-3 hold for each belief, and that beliefs are ranked according to Assumption 4. Then, optimists’ wealth share evolves according to Eq. (36). The equilibrium prices and interest rates can be written as a function of optimists’ wealth share, qi (a) , rf (a), q2 (a), rf (a). At the high-risk state, rf (a) = 0 and q2 (a) solves the differential equation (37) with q2 (0) = qp and q2 (1) = qf. At the low-risk state, qi (a) = q* and rf (a) is given by Eq. (41). The equilibrium price and interest-rate functions are increasing in optimists’ wealth share. Moreover, speculation reduces the price and exacerbates the recession in the high-risk state, that is, the price function satisfies the inequality in (40).

Dynamics of equilibrium. We next fix investors’ beliefs and simulate the equilibrium for a particular realization of uncertainty over a 50-year horizon. We choose the (objective) simulation belief to be in the “middle” of optimists’ and pessimists’ beliefs in terms of the relative entropy distance, which ensures that there is a non-degenerate long-run wealth distribution in which neither optimists nor pessimists permanently dominate. Figure 6 illustrates the evolution of equilibrium variables (except for optimists’ wealth share, which we plot in Figure 4). For comparison, the dashed line plots the equilibrium that would obtain in the common-beliefs benchmark if all investors shared the “middle” simulation belief. For another comparison, the dotted line plots the first-best equilibrium that would obtain without interest rate rigidities.

The figure illustrates two points. First, consistent with our benchmark analysis in the previous section, the interest rate is more compressed and the price of capital is more volatile than in the first-best equilibrium. In the high-risk state, the interest rate cannot decline sufficiently to close the risk gap, which leads to a drop in asset prices. This also lowers output as well as investment

Figure 6: The evolution of the equilibrium variables with interest rate rigidities and belief disagreements (solid line), with rigidities and common beliefs (dashed line), and without rigidities (dotted line) over the medium run (50 years).

and expected growth. In the low-risk state, the fear of transition into the recessionary high-risk state keeps the interest rates lower than in the first-best benchmark.

Second, consistent with our analysis in this section, these effects are more powerful when investors have belief disagreements. In fact, the common beliefs benchmark is not too far from the first-best equilibrium since we have calibrated the “middle” belief to be relatively optimistic (in particular, it comfortably satisfies Assumptions 2 and 3 in the previous section). The figure shows that belief dispersion around this relatively optimistic level can by itself create considerable damage. This illustrates the amplification caused by speculation and motivates the analysis of macroprudential policy that restricts speculation, which we turn to next.

**6. Welfare analysis and macroprudential policy**

In this section we establish our normative results on macroprudential policy. To this end, we first characterize investors’ value functions in equilibrium. This establishes the determinants of welfare in this setting and illustrates the aggregate demand externalities. We then show that, when investors have belief disagreements, the equilibrium can be Pareto improved by macroprudential policy that restricts optimists’ risk taking. Throughout, we work with the model with two belief types, {o,p}, that we analyzed in the previous section.

In Appendix A.2.1, we show that the value function can be written as,

Here, v\ s denotes the normalized value function per unit of capital stock. An investor that has twice the capital chooses the same portfolio weights and consumes twice the consumption state-by-state, which leads to the functional form in (42).

In Appendix A.5, we further characterize v| s as the solution to the following differential equation

system

This expression illustrates the determinants of welfare. When there is a demand-driven recession (e.g., in the high-risk state s = 2), a lower equilibrium price, qt , s, reduces investors’ welfare since it is associated with lower factor utilization, % s. Note that welfare declines due to a decline in current consumption (captured by the term, log p + qt , s) as well as a decline in investment and consumption growth (captured by the term, ipqt,s — S = gt,s). The variance, of,, also affects welfare through its influence on the risk-adjusted consumption growth. Finally, speculation among investors with belief disagreements also affects (perceived) welfare. This is captured by the term, — (As — At, s) + As log ^, which is zero with common beliefs, and strictly positive with disagreements.

To facilitate our analysis of macroprudential policy, we also break down the value function into two components,

Here, v*s denotes the first-best value function that would obtain if there were no interest rate rigidities. It is characterized by solving Eq. (43) with the efficient price level, qt , s = q*, for each t, s. The residual, wt, s = vt,s — v*s, denotes the gap value function, which captures the loss of value due to interest rate rigidities and demand recessions. Using Eq. (43), the gap value function is characterized as the solution to the following differential equation

This illustrates that the gap value captures the loss of welfare due to the price deviations from the efficient level. As we will see, the gap value functions are useful to understand the marginal effect of macroprudential policy on social welfare.

When investors share the same belief, the value function and its components are stationary, e.g., vt , s = vs. In Appendix A.5, we calculate these values in closed form (see Eq. (A.28)) and find that they depend on a weighted average of the price of capital, (qs)se{i 2}, as well as the variance terms,

Figure 7: The equilibrium value functions for each state and belief type. The solid lines are the actual value functions, vls (a), the dotted lines are the first-best value functions, vl’* (a), and the dashed lines (in the bottom panels) are the gap value functions, wl (a). (CTs)l2{l2}, in the two states. The weights reflect time discounting and transition probabilities: They can be thought of as the “discounted expected time” the investor spends in one state relative to another. We show that the value in the recession state is lower than in the boom state, V2 < vi, precisely because the investor expects to spend more discounted time in state 2 that features both lower price of capital and higher risk relative to the other state. For the same reason, we find that the gap value is negative in both states but more so in the recession state, W2 < wi < 0.

With belief disagreements, the value function is not necessarily stationary since the price might have a drift. Recall that the equilibrium price in the high-risk state is a function of optimists’ wealth share, q2 (a). In Appendix A.5, we show that the equilibrium values and its components can also be written as a function of optimists’ wealth share, |vl (a) , vl’* (a) , ws (a). We also characterize these value functions as solutions to differential equations in a-domain. Figure 7 illustrates the numerical solution for the equilibrium plotted in the earlier Figure 5.

The bottom panels of Figure 7 show that the gap value functions are increasing in the wealth share of optimists, a, which illustrates the aggregate demand externalities. Greater a increases the effective optimism, which in turn leads to a greater equilibrium asset price in the high-risk state (see Figure 5). This improves the gap value function in this state by raising the aggregate demand and bringing the economy closer to the first-best equilibrium (see Eq. (45)). It also improves the gap value function in the low-risk state, because the economy can always transition into the high-risk state, and these transitions are less costly when a is greater. Hence, increasing optimists’ wealth share is always associated with positive aggregate demand externalities. Individual optimists that take risks (or pessimists that take the other side of these trades) do not internalize their effects on asset prices, which leads to inefficiencies and generates scope for macroprudential policy.

The top panels of Figure 7 illustrate that the first-best value functions are increasing in a for pessimists but they are decreasing in a for optimists. These effects can be understood via pecuniary externalities in contingent security markets. Increasing the wealth of optimists increases the price of contingent securities that optimists purchase, while decreasing the price of contingent securities that pessimists purchase. This creates negative pecuniary externalities (or crowd-out effects) on optimists, and positive pecuniary externalities on pessimists.

Finally, note that the actual value function is the sum of the first-best and the gap value functions. For pessimists, the actual value is always increasing in a, since the two components move in the same direction. For optimists, this is not necessarily the case since the gap value is increasing in a whereas the first-best value is decreasing.

6.2. Macroprudential policy

We capture macroprudential policy as risk limits on optimists. Suppose, the planner can induce optimists to choose (instantaneous) allocations as if they have less optimistic beliefs. Specifically, optimists are constrained to choose allocations as-if they have the beliefs, A°’pl = ^A°’pl, , that satisfy, Al’pl > A° and A^’p1 < A!).16 Pessimists continue to choose allocations according to their own beliefs. Throughout, we use AS’pl to denote investors’ as-if beliefs and AS to denote their actual beliefs (for pessimists, the two beliefs coincide). We also use the notations, Apls = at;SA°’pl + (1 — at;S) Ap’pl and As (a) to represent the weighted average as-if belief.

In Appendix A.6, we show that the planner can implement this policy by imposing inequality restrictions on optimists’ portfolio weights, while allowing them to make unconstrained consumption- savings decisions. Specifically, the policy constrains optimists from taking too low a position on the contingent security that pays in the high-risk state,(restrictions on selling “put options”). It also constrains optimists from taking too high a position on the contingent security that pays in the low-risk state, !^’2° < (restrictions on buying “call options”). Finally, the policy also constrains optimists’ position on capital not to exceed the market average, < 1 (since otherwise optimists start to speculate by holding more capital).

Remark 4 (Banks and Macroprudential Policy). In practice, most macroprudential policies are implemented through banks, especially large ones. If the banks are interpreted as the high valuation investors in the economy, perhaps because of their greater risk tolerance or capacity (see Remark 3), then our policy applies directly to their balance sheets. If instead the borrowers of the banks are interpreted as the high valuation investors, then strictly speaking the policy applies to borrowers’ balance sheets. However, under the realistic assumption that the borrowers have little choice but to obtain risk exposure via banks, the policy can still be implemented through banks by limiting their lending to their optimistic borrowers (e.g., real estate investors in the run-up to the housing bubble) or other high-valuation borrowers (e.g., hedge funds). The key aspect of macroprudential policy in our environment is that it restricts high valuation investors’ exposure to recession risks.

The characterization of equilibrium with policy is the same as in Section 5. In particular, Eqs. (36) and (37) continue to hold with the only difference that investors’ beliefs are replaced with their as-if beliefs, A(,’p1.

To characterize the optimal policy, we assume the planner respects investors’ individual beliefs, that is, investors’ expected values in equilibrium are calculated according to their own beliefs, A),. To trace the Pareto frontier, we also allow the planner to do a one-time wealth transfer among the investors at time zero. In Appendix A.6, we show that the planner’s Pareto problem can then be reduced to,

Hence, the planner maximizes a wealth-weighted average of investors’ normalized values. The relative wealth shares reflect the planner’s relative Pareto weights

We further characterize v(, s as the solution to a differential equation [cf. Eq. (A.35)]. This is the analogue of Eq. (43) with the only difference that the portfolio weights on contingent securities (and the payoffs from these positions) are calculated according to investors’ as-if beliefs, A)’p1, whereas the transition probabilities are calculated according to their actual beliefs, A). As before, we also decompose the value function into first-best and gap value components, v| s = v^’* + w\ s.

We also show that the value function as well as its components can be written as a function of optimists’ wealth shares. As in the case without policy, we denote the equilibrium price functions with fqs (o)gs, individuals’ value functions with i vls (o) ,vl’* (o) ,wls (o). The planner’s value function is then a wealth-weighted average of individual value functions, vpl (o) = ov°s (o) + (1 — o) vp. We also break this into first-best value and gap value, vpl (o) = vpl ’* (o) + wpl (o).

A key observation is that the marginal impact of the policy on the planner’s first-best value function is zero,

This is because our model features complete markets and no frictions other than interest rate rigidities. Hence, the First Welfare Theorem applies to the first-best allocations that also correct for these rigidities (and features efficient output). This in turn implies that the marginal impact on the first-best value must be zero, since otherwise the first-best allocations could be Pareto improved by appropriately changing optimists’ as-if beliefs. It follows that the marginal impact of the policy is determined by its marginal impact on the planner’s gap value function, wpl (o) = ow° (o) + (1 — o) wp (o).

It remains to characterize how the policy affects investors’ gap value functions. In Appendix

where a' = a=o°Si—. This follows from the earlier equation (45) after replacing optimists’ wealth dynamics from Eq. (36) when they act according to their as-if beliefs, A!’?1. Note how the transition probability is calculated according to actual beliefs, Af,. The policy influences the perceived gap values not because it changes investors’ beliefs, but since it changes optimists’ wealth dynamics, which in turn affects asset prices and the output gaps relative to the first-best. We next describe the effect of macroprudential policy in the boom state s = 1, assuming that there is no intervention in the other state. We then analyze the polar opposite case of macroprudential policy in the recession state s = 2, assuming no intervention in the boom state.

6.2.1. Macroprudential policy during the boom

Suppose the economy is currently in the boom state s =1. The planner can use macroprudential policy in the current state, A^’p1 > Ai (she can induce optimists to act as if transition into the recession is more likely), but not in the other state A^’?1 = A2 (she cannot influence optimists’ actions in the recession state). Finally, suppose we are in the special case in which the beliefs satisfy, Ai = Ai = Ai (so investors disagree only in the recession state). We obtain a sharp result for this case, and we show in numerical simulations that the result also applies when AO < Ai.

Proposition 3. Consider the model with two beliefs types that satisfy Ai = A?. Consider the macroprudential policy in the boom state, AO’?1 > Ai (and suppose A^’?1 = A2). The policy increases the gap value according to each belief, that is,

The result shows that macroprudential policy improves the gap value function according to optimists as well as pessimists. Therefore, it also increases the wealth-weighted average gap value. In view of Eq. (47), it also increases the social welfare and leads to a Pareto improvement.

To obtain a sketch proof for the result, consider the differential equation (48) for the boom state s = 1 and an arbitrary belief type i 2 {o,p}. Differentiating this expression with respect to policy,

Here, A i denotes investors’ common belief in state 1 (by assumption). The second line uses a' =a=°-p—. The two terms inside the brackets capture the direct effects of macroprudential policy on social welfare. Macroprudential policy effectively induces optimists to purchase more insurance (or sell fewer puts). This reduces optimists’ relative wealth share in the boom state s = 1 but improves their relative wealth share in the recession state s = 2. Moreover, using the equilibrium prices, one unit of decline in wealth share in the boom state is associated with one unit of increase in expected wealth share in the recession state.

Next note that the gap value function in either state is increasing in optimists’ wealth share @W9c(a) j @W@c(a) > 0 (see Figure 7). Hence, macroprudential policy always involves a trade-off. Intuitively, optimism is a scarce resource that could also be utilized immediately or in the future. Moving optimism across states via macroprudential policy is always associated with costs as well as benefits. However, the typical situation is such that optimism increases the social welfare more in the recession state s = 2, where it provides immediate benefits, as opposed to the boom state s = 1, where its benefits are realized in case there is a future transition into the recession. For the special case with Al = Ap, we in fact have @W@1c(a) = . Combining this with Eq. (49) provides a sketch-proof of Proposition 3. The actual proof in Appendix A.6 relies on the same idea but uses recursive techniques to establish the result formally.

The left panel of Figure 8 illustrates the result by plotting the change in the planners’ value functions in the boom state resulting from a small macroprudential policy change (specifically, we start with the equilibrium with AO = 0.03 and set Al’pl = 0.0305). Note that the policy reduces the planner’s first-best value function, since it distorts investors’ allocations according to their own beliefs. However, the magnitude of this decline is small, illustrating the First Welfare Theorem (cf. Eq. (47)). Note also that the policy generates a relatively sizeable increase in the planner’s gap value function. This increase is sufficiently large that the policy also increases the actual value function and generates a Pareto improvement, illustrating Proposition 3.

Macroprudential policy improves welfare by internalizing the aggregate demand externalities. In the recession state s = 2, optimists improve asset prices, which in turn increases aggregate demand and brings output closer to the first-best level. Individual optimists do not internalize these general equilibrium effects, and therefore, they take too much risk from a social point of view. Macroprudential policy increases optimists’ insurance purchases (or reduces their insurance sales), which increases their wealth in the recession state and improves aggregate outcomes. The result is reminiscent of the analysis in Korinek and Simsek (2016), in which macroprudential policy improves outcomes by inducing households that have a high marginal propensity to consume (MPC) to bring

Figure 8: The left panel illustrates the effect of a small change in macroprudential policy in the boom (low-risk) state on the planner’s value functions. The right panel illustrates the effect of larger policy changes.

more wealth into states in which there is a demand-driven recession. However, the mechanism here is different and operates via asset prices. In fact, in our setting, all investors have the same MPC equal to p. Optimists improve aggregate demand not because they spend more than pessimists, but because they increase asset prices and induce all investors to spend more, while also increasing aggregate investment and hence growth.

As this discussion suggests, the parametric restriction, A^ = Ap, is useful to obtain an analytical result but it does not play a central role. We suspect that Proposition 3 also holds absent this assumption, even though we are unable to provide a proof. In our numerical simulations, we have not yet encountered a counterexample. The results displayed in Figure 8 actually correspond to our earlier parameterization that features A1 < Ap.

Proposition 3 concerns a small policy change. The right panel of Figure 8 illustrates the effect of larger policies by plotting the changes in the planner’s value as a function of the size of the policy (starting from no policy, Al’p1 = A°). For this exercise, we fix the optimists’ wealth share at a particular level, a = 1/2. Note that, as the policy becomes larger, the gap value continues to increase whereas the first-best value decreases. Moreover, the decline in the first-best value is negligible for small policy changes but it becomes sizeable for large policy changes. The (constrained) optimal macroprudential policy obtains at an intermediate level, Al’p1’* > A^.

The figure also illustrates that the constrained optimal policy intervention is not too large (specifically, we have Al’p1’* = 0.04 where A1 = 0.03). This is typically the case in our numerical simulations. The reason is that speculation generates high perceived utility for investors. Since macroprudential policy restricts speculation, the perceived costs quickly rise with the degree of the

Figure 9: The left panel illustrates the effect of macroprudential policy in the boom state on social welfare, when all investors’ value is calculated according to respectively optimists’ or pessimists’ belief. The panels on the right illustrate the effects on respectively the first-best and the gap value functions.policy intervention, which implies that the optimal intervention is not too large.

Macroprudential policy according to a belief-neutral criterion. When we interpret belief disagreements literally (see Remark 3), it is questionable whether the utility from speculation should be counted toward social welfare. A recent literature argues that the Pareto criterion is not the appropriate notion of welfare for environments with belief disagreements. If investors’ beliefs are different due to mistakes (say, in Bayesian updating), then it is arguably more appropriate to evaluate their utility according to the objective belief—which is common across the investors. Doing so would remove the speculative utility from welfare calculations, and it could lead to a constrained optimal policy that is much larger in magnitude. While reasonable, this approach faces a major challenge in implementation: whose belief should the policymaker use?

In recent work, Brunnermeier et al. (2014) offer a belief-neutral welfare criterion that circumvents this problem. The basic idea is to require the planner to evaluate social welfare according to a single belief, but also to make the welfare comparisons robust to the choice of the single belief. Specifically, their baseline criterion says that an allocation is belief-neutral superior to another allocation if it increases social welfare under every belief in the convex hull of investors’ beliefs. Proposition 3 suggests their criterion can also be useful in this context since macroprudential policy increases the gap value according to each belief—that is, the gap-reducing welfare gains are belief neutral.

For a formal analysis, fix some h 2 [0,1] and let vls ^a; AXfpl, Xft^ denote the value function for an individual when the planner implements policy, X1fpl, and evaluates utility under the beliefs,

Ap = Ap + h (Ap — Ap).19 As before, define the planner’s value function, vp1 ^a; Ap,p1, Ah^, as the wealth-weighted average of individual’s value functions. Then, given the wealth share a (that corresponds to a particular Pareto weight), the policy, A°’p1, is a belief-neutral improvement over some other policy, A/^ , as long as it increases the planner’s value according to each h 2 [0,1].

Figure 9 illustrates the belief-neutral optimal policy in the earlier example. The left panel plots the effect of the policy on the social welfare (given a = 1/2) when the planner evaluates all investor’s values under respectively pessimists’ belief and optimists’ belief. The social welfare evaluated under intermediate beliefs lie in between these two curves. As the figure suggests, tightening the policy towards Ap’p1,neutra1 = 0.1 constitutes a belief-neutral improvement. In particular, the belief-neutral criterion supports a much larger policy intervention than the Pareto criterion (cf. Figure 8).

The right panel provides further intuition by breaking the social welfare into its two components, vp1 = vp1’* + wp1. The top right panel shows that tightening macroprudential policy towards the belief, A0°pl’flrst = 0.1, generates a belief-neutral improvement in the “first best” social welfare, vp1’*. Speculation induces investors to deviate from the optimal risk sharing benchmark in pursuit of perceived speculative gains. However, these speculative gains are transfers from other investors, and they do not count towards social welfare when investors’ values are evaluated under a common belief (regardless of whose belief is used). Hence, if there were no interest rate rigidities, a belief- neutral planner would eliminate almost all speculation.20

The bottom right panel shows the effects of policy on the gap value, wp1, which captures the reduction in social welfare due to interest rate rigidities. Tightening the macroprudential policy towards the belief, Ap,p1,gap = 0.07 increases the gap value according to both optimists and pessimists (illustrating Proposition 3). Beyond this level, tightening the policy improves the gap value according to pessimists but not according to optimists—who perceive smaller benefits from macroprudential policy since they find the transition into state 2 unlikely.

It follows that, up to the level, Ap,p1’gap = 0.07—which constitutes a sizeable policy intervention—there is no conflict in belief-neutral policy objectives. Tightening the policy helps to rein in speculation while also improving the gap value, according to any belief. This might be a natural choice for a planner who focuses exclusively on closing the output gaps relative to the first best while remaining agnostic about whether speculation improves or reduces social welfare. Beyond this level, tightening the policy continues to generate belief-neutral welfare gains by reducing speculation and improving risk sharing, but it also reduces the gap value according to optimists.

Dynamics of equilibrium with policy. We next consider how macroprudential policy affects the dynamics of equilibrium variables. Figure 10 illustrates the evolution of equilibrium over a 50-year horizon when the planner implements the (belief-neutral) gap-value maximizing policy,

Figure 10: The evolution of the equilibrium variables without macroprudential policy (solid line) and with macroprudential policy in the boom state (dotted line) over the medium run (50 years).

A'1’pl,gap = 0.07. For comparison, the figure also replicates the evolution of the equilibrium variables without policy from Figures 4 and 6. Note that macroprudential policy ensures optimists’ wealth share drops relatively less when there is a transition into the high-risk state. This in turn leads to greater asset prices and higher growth rate in the high-risk state. However, macroprudential policy is not without its drawbacks. As the period between years 5-15 illustrates, the policy slows down the growth of optimists’ wealth share when the economy remains in the low-risk state.

The effect of macroprudential policy on the interest rate in the low-risk state is rather subtle. On the one hand, for a fixed level of optimists’ wealth share, the policy lowers the interest rate as it lowers aggregate demand. On the other hand, the policy also preserves optimists’ wealth over time, which increases the interest rate. In our simulation in Figure 10, the latter effect dominates and macroprudential policy leads to a higher interest rate over time.

6.2.2. Macroprudential policy during the recession

The analysis so far concerns macroprudential policy in the boom state and maintains the assumption that A°’pl = A^. We next consider the polar opposite case in which the economy is currently in the recession state s = 2, and the planner can apply macroprudential policy in this state, A'2’pl < A^ (she can induce optimists to act as if the recovery is less likely), but not in the other state, A',pl = A'. We obtain a sharp result for the special case in which optimists’ wealth share is sufficiently large.

Proposition 4. Consider the model with two belief types. Consider the macroprudential policy in

the recession state, A°pl < AO (and suppose Al’pl = A1). There exists a threshold, a < 1, such that if a 2 (a, 1], then the policy reduces the gap value according to each belief, that is,

Thus, in contrast to Proposition 3, macroprudential policy in the recession state can actually reduce the social welfare. The intuition can be understood by considering two counteracting forces. First, as before, macroprudential policy in the recession state is potentially valuable by reallocating optimists’ wealth from the boom state s = 1 to the recession state s = 2. Intuitively, optimists purchase too many call options that pay

if there is a transition to the boom state but that impoverish them in case the recession persists. They do not internalize that, if they keep their wealth, they will improve asset prices if the recession lasts longer.

However, there is a second force that does not have a counterpart in the boom state: Macroprudential policy in the recession state also affects the current asset price level, with potential implications for social welfare. It can be seen that making optimists less optimistic in the recession state shifts the price function downward, fg2(a)A < 0 (as in Figure 2 for common beliefs). Hence,the price impact of macroprudential policy is welfare reducing. Moreover, as optimists dominate the economy, a ! 1, the price impact of the policy is still first order, whereas the beneficial effect from reshuffling optimists’ wealth is second order. Thus, when optimists’ wealth share is sufficiently large, the net effect of macroprudential policy is negative, illustrating Proposition 4.

This analysis also suggests that, even when the policy in the recession state exerts a net positive effect, it would typically increase the welfare by a smaller amount than a comparable policy in the boom state. Figure 11 illustrates this by plotting side-by-side the effects of a small policy change in either state. The left panel replicates the value functions from the earlier Figure 8, whereas the right panel illustrates the results from changing optimists’ belief in the recession state by an amount that would generate a similar distortion in the first-best equilibrium as in our earlier analysis. Note that a small macroprudential policy in the recession state has a smaller positive impact when optimists’ wealth share is small, and it has a negative impact when optimists’ wealth share is sufficiently large.

It is useful to emphasize that macroprudential policy does not have an adverse price impact in the boom state due to the interest rate response. Intuitively, as macroprudential policy reduces the demand for risky assets, the interest rate policy lowers the rate to dampen its effect on asset prices and aggregate demand. In the recession state, the interest rate is already at zero, so the interest

Figure 11: The left (resp. the right) panel illustrates the effect of a small change in macroprudential policy in the boom (resp. the recession) state.rate policy cannot neutralize the adverse effects of macroprudential policy.

Taken together, our analysis in this section provides support for procyclical macroprudential policy. In states in which output is not demand constrained (in our model, the boom state s = 1), macroprudential policy that restricts high valuation investors’ (in our model optimists’) risk taking is desirable. This policy improves welfare by ensuring that high valuation investors bring more wealth to the demand-constrained states, which in turn increases asset prices and output. Its adverse price effects are countered by a reduction in the interest rate. In contrast, in states in which output is demand constrained (in our model, the recession state s = 2), macroprudential policy has counteracting effects on social welfare. While the policy has the same beneficial effects as before, it also lowers asset prices and aggregate demand, which cannot be countered by the interest rate. The latter effect reduces the overall usefulness of macroprudential policy, and it could even reduce social welfare.

**7. Final Remarks**

We provide a macroeconomic framework where risk- and output-gaps are joint phenomena that feed into each other. The key tension in this framework is that asset prices have the dual role of equilibrating risk markets and supporting aggregate demand. When the dual role is inconsistent, the risk market equilibrium prevails. Interest rate policy works by taking over the role of equilibrating risk markets, which then leaves asset prices free to balance the goods markets. However, once interest rates reach a lower bound, the dual role problem reemerges and asset prices are driven primarily by risk market equilibrium considerations. This reduces aggregate demand and triggers

a recession, which then feeds back negatively into asset prices. The drop in asset prices during recessions also reduces interest rates during booms. In this environment, the role of macroprudential regulation is to preserve the wealth of high-valuation investors during recessions, so as to reduce the gap between the asset prices that equilibrate the risk and goods markets when the interest rate policy is no longer available.

Interest rate cuts work in our model by improving the market’s Sharpe ratio. From this perspective, any policy that reduces perceived market volatility and sudden asset price drops should have similar effects, which renders support to the many such policies implemented during the aftermath of the subprime and European crises.

In the model we take the interest rate friction to be a stark zero lower bound constraint, which can be motivated with standard cash-substitutability arguments. In practice, this constraint is neither as tight nor as narrowly motivated: Central banks do have some space to bring rates into negative territory, especially when macroeconomic uncertainty is rampant, but there are also many other frictions besides cash substitutability that can motivate downward rigidity in rates once these are already low (see, e.g., Brunnermeier and Koby (2016) for a discussion of the “reversal rate”, understood as a level of rates below which the financial system becomes impaired). The broader points of the dual role of asset prices and their interactions with aggregate demand constraints during recessions would survive many generalizations of the interest rate friction. Similarly, one could also imagine situations that motivate ceilings on interest rates, in which case asset prices would overshoot and the productive capacity would become stretched.

In the main text, we also did not take a stand on whether optimists or pessimists are right about the transition probabilities. The reason is that core of our analysis does not depend on this. For example, we could think of optimists as rational and pessimists as Knightians (see, e.g., Caballero and Krishnamurthy (2008); Caballero and Simsek (2013)). Absent any direct mechanism to alleviate Knightian behavior during severe recessions, the key macroprudential point that optimists may need to be regulated during the boom survives this alternative motivation.

As we noted earlier, our modeling approach belongs to the literature spurred by Brunnermeier and Sannikov (2014), although unlike that literature our analysis does not feature financial frictions. However, if we were to introduce these realistic frictions in our setting, many of the themes in that literature would reemerge and become exacerbated by aggregate demand feedbacks. For instance, in an incomplete markets setting, optimists take leveraged positions on capital, and by doing so they induce endogenous volatility in asset prices and the possibility of tail events following a sequence of negative diffusion shocks that make the economy deeply pessimistic (we analyze the incomplete markets case in a companion paper, Caballero and Simsek (2017a)).

The model omits many realistic healing mechanisms that were arguably relevant for the Great Recession (as well as other deep recessions). For example, a financial crisis driven by a reduction in banks’ net worth is typically mitigated over time as banks earn high returns and accumulate net worth (see Gertler et al. (2010); Brunnermeier and Sannikov (2014)). Likewise, household or firm deleveraging eventually loses its potency as debt is paid back (see Eggertsson and Krugman (2012);

Guerrieri and Lorenzoni (2017)). Investment hangovers gradually dissipate as the excess capital is depleted (see Rognlie et al. (2017)). While these healing mechanisms are useful to understand the aftermath of the Great Recession, they raise the natural question of why the interest rates seem unusually low and the recovery (especially in investment) appears incomplete almost ten years after the start of the recession. Our paper illustrates how high risk premium (due to objective and subjective risk factors) can drag the economy’s recovery.

Conversely, the model also omits many sources of inertia that stem from financial markets. Throughout we have assumed that risk markets clear instantly while goods markets are sluggish. In practice, risk markets have their own sources of inertia as portfolios are adjusted infrequently, financial institutions avoid or delay mark-to-market losses, and so on.

Finally, one feature of the aftermath of the subprime crisis is the present high valuation of risky assets, which could appear to contradict the higher required equity risk premium observed in the data. The model offers a natural interpretation for such a combination: While we focused exclusively on changes in the required risk-premium, there is also evidence that during this period both p and p have declined due to a variety of factors such as a worsening of the wealth inequality and an increase in monopoly rents (see, e.g., Gutierrez and Philippon (2016)). Equation (25) shows that such declines require a higher valuation to obtain full factor utilization, which is achieved via a drop in “rstar.” Moreover, the latter generates a feedback as it increases the fragility of the economy by reducing the distance to the ZLB. Thus, in our framework high valuations raise the risk of the economy not so much because of “irrational exuberance” (as the high valuations are needed to support full employment) but because of the low level of the interest rate needed to support them, and hence the reduced monetary policy ammunition to deal with further recessionary shocks.

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**Comments by Mohamed A El-Erian1**

It is such a pleasure and honour to be here. Huge thanks to Claudio, Jaime, Hyun Song and other BIS colleagues for inviting me to this wonderful event. I greatly appreciate the opportunity to participate in such a stimulating conference. It is also a privilege to be asked to provide some comments in this session on Ricardo Caballero's three interesting papers.

My remarks will be organised around the what, why, and so what of relating to - and here I am quoting Ricardo - "the world economy experienced a prolonged period of risk intolerance". I will approach this from the perspective of a capital market observer and participant who has followed closely the impact of unconventional monetary policies, including large-scale asset purchases by systemically important central banks.

My particular interest today is in the issue, and again I quote, that "the key tension that asset prices have the dual role of equilibrating financial markets and supporting aggregate demand". Moreover, given the forward-looking spirit of this gathering, I will also pose the question of what may happen if we run the current policy regime forward - with particular emphasis on cumulative effects and feedback loops which, I believe, are taking some economies closer to tipping points. Indeed, rather than signal a generalised increase in risk intolerance, as Ricardo suggests, we could well be in the midst of a new period of excessive risk-taking - one in which high financial risk-taking has become dangerously decoupled from economic risk-taking, opening up two potential outcomes depending on the policy response beyond the world of central banking. They carry similar probabilities but very different outcomes when it comes to economic, financial, institutional, political and social factors.

The What

When it comes to the what, Ricardo's work documents:

- The secular decline in real interest rates to persistently negative real levels and, in some cases, nominal too;
- Shows that it has been generalised across maturities and advanced economy jurisdictions - the so-called Triffin dilemma for fixed income securities that serve, or are perceived to serve as a true "store of value";
- That it is part of what he suggests "may well be a recurrent global safety traps environment", reflecting both supply and demand influences that have reduced the availability of safe assets around the world, and in advanced economies in particular; and

- That the phenomenon has both a public and private component.

Given my own perspective, I find it interesting that the analysis places a lot less weight on the information content of spreading contradictions on the ground especially if we consider the dramatic changes in traditional historical correlations.

There are, in my opinion, signals rather than noise and they include:

- The behaviour of both the term premium on government bonds and the flattening of the yield curve versus the valuation of stocks;
- The decoupling of financial risk-taking (high) from economic risk-taking (low);
- The promise of ample liquidity inherent in the proliferation of ETFs in traditionally less liquid asset classes (such as high-yield corporates and emerging markets) versus experience, particularly at times of change in the paradigms governing the general market behaviour;
- The discrepancy between soft and hard data;
- The unusual uncertainty in national politics and cross-border economic relations versus notably low actual and implied market volatility;
- The persistence of negative interest rates in systems that are now struggling to provide longer-term financial protection products to households (such as life insurance and retirement); and, more generally and linking to the earlier sessions; and
- The step decline in r* in a system built on the presumption of a notably higher level.

The Why

On the why, Ricardo's emphasis is primarily on structural and secular issues such as demographics and regulation that, in his analytical framework, have resulted in the supply of safe assets not keeping up with global demand. There is little discussion of the role of policy issues and choices, including the earlier pursuit by emerging economies of high self-insurance and, since the global financial crisis, central banks being de facto forced into a prolonged period of "being the only game in town" policy-wise. The latter has led to unprecedented policy experimentation, with both rates and balance sheets, turning quite a few previous improbables and unthinkables into reality.

With that, I wonder whether the analysis may underplay the role of policy choices that have accentuated the structural and secular factors that are correctly identified. These policy issues speak not only to trade-offs within the direct scope of monetary policy as mentioned by Jamie in his introductory remarks but also, and more importantly, the excessive reliance on central banks when compared to other policymaking entities that can pursue structural reform, more responsive fiscal policies, targeted debt reduction measures and greater global policy coordination.

The So What

All of which takes us to the so what.

Ricardo shows that the "volatility stabilisation" of financial assets has coincided with a period of increasing return on capital and, more importantly, that the actual and potential consequences are a mix of benign, concerning and damaging. It is a mix that gets more worrisome the closer we get to, and stay at, the effective lower bound. There, "any further intensification in the shortage of safe assets has destabilising macroeconomic consequences" - including pushing the global economy further away from its potential.

In terms of solution, Ricardo's work identifies four channels:

- through the exchange rate mechanism;
- through greater issuance of public debt;
- through the production of private safe assets; and
- through changes in the regulatory framework.

Ironically, I get to some of Ricardo's policy prescriptions, and add a few, but using a different route that involves some reverse causality compared with his analytical framework. Importantly, rather than reflect a "natural" process driven just by secular and structural factors, what Ricardo identifies may well also involve the impact of a prolonged phase of non-commercial activities in financial markets: directly through interest rate-setting and balance sheet management, and indirectly through forward guidance.

A notable part of what Ricardo picks up is the persistent use of a partial instrument for desirable growth and economic well-being objectives, and doing so for a long time and in such a way that Ben Bernanke's characterisation of the "benefits, costs and risks" have gradually evolved in a more worrisome fashion. In sum, rather than serve as the bridge to more comprehensive policies, as originally intended, unconventional monetary policy has become too much of a destination.

Considered through this prism, the shortage of safe assets is not just an outcome but also part of the transmission mechanism of central banks' policies. And it has unintended consequences, together with collateral damage.

All this speaks to the urgent importance of a policy hand-off: from prolonged excessive reliance on central banks to a broader policy response that deploys pro-growth structural reforms, more active fiscal policy where there is room, targeted debt reduction, and better regional and global economic architecture and cooperation.

If the hand-off occurs in a timely fashion, the "Triffin Dilemma" that Ricardo identifies would be solved in an orderly fashion as low growth and insufficiently inclusive growth gives way to high and more inclusive growth, as artificial financial stability becomes genuine, and as the scope for a "beautiful normalisation" (to adapt Ray Dalio's term) becomes more of a reality.

However, if the hand-off remains elusive, low growth would risk turning into periodic recessions, artificial financial stability would give way to unsettling volatility, the effective lower bound would become more of a binding constraint, and central bank effectiveness would erode further.

**Conclusion**

The data on government yields may be seen by some to reflect a period of risk intolerance which is supported by a decline in the supply of safe assets and an increase in demand. But the underlying influences go beyond that, reflecting an unbalanced policy mix that has put way too much of the burden on unconventional monetary policy. The result is another period of excessive financial risk-taking that is continuously fuelled by the liquidity trade.

It is a configuration that speaks to the contradictions that I mentioned at the start of my presentation. And the longer it persists, the greater the likelihood that, for central bank policy, what Ricardo labels the "safe assets shortage conundrum" risks going from being an outcome to becoming a notable problem in itself.

Thank you very much.

Comments by David Laidler

To a retired professor of monetary economics and its history, the topic of this session is irresistible. As a retiree, he is a member of a rentier class currently undergoing euthanasia by persistently low interest rates; as a monetary economist he is intellectually challenged by the efforts of economists like Ricardo Caballero to understand this phenomenon; and as a historian he is fascinated by certain similarities between the fruits of those efforts, and ideas that were current in earlier years.

**Caballero's message**

Professor Caballero's central message is that a growing concern with "risk" and a concomitant rise in the demand for "safe" stores of value has been the main driver of the world-wide fall in interest rates on high-quality bonds that began even in the 1990s, but gathered particular momentum after 2008. And this message has a corollary: that because growing antipathy to risk seems to be a structural phenomenon, and because rates have now reached rock bottom, other mechanisms - constricted aggregate demand for goods and services, sluggish output growth, or even outright contraction etc - are replacing further interest rate falls as means of equilibrating a potentially ever-growing demand for safe assets with an inadequate supply. Or to put it in traditional terms: those "dark forces of time and ignorance" as John Maynard Keynes called them, with which economic agents must always cope as they plan for their future, having seemed to be manageable since the Great Depression and the war which followed it, have recently begun once again to wreak havoc with orderly economic progress.

Caballero's message is intriguing and plausible, and historians of monetary thought will find it particularly attractive because within it are embedded new variations on certain ideas left over from the literature generated by that Great Depression, something of a rarity in today's economics.

Here I am referring first of all to the insight that output variations might take over as equilibrating mechanisms when other variables become stuck, an idea that underpinned the Richard Kahn (1931) - Jens Warming (1932) - Keynes (1936) multiplier (see David Laidler (1999, pp 172-177, 250-253)) and lay at the very heart of the so-called Keynesian Revolution; but second, and to my mind more importantly, to the significance which Caballero attaches to the interactions of the supply and demand for a particular subset of available stores of value that have the capacity to help agents to deal with the challenges posed by those above-mentioned

University of Western Ontario. These comments are based on Caballero's paper "On the Macroeconomics of Risk Intolerance", that was also included in Caballero's presentation at the 16th BIS Annual Conference. dark forces", a focus that recalls the theory of what in the 1930s was often called liquidity preference".

**Safe assets and liquidity preference**

To be sure, Caballero's "safe" asset - "a simple debt instrument that is expected to preserve its value during adverse systemic events" - at first sight differs quite a bit from the currency and bank deposits whose capacities to provide protection against risk were analysed in the inter-war years by, among others, Frederick Lavington (1921), John Hicks (1935) and, once more Keynes (1930, 1936) (see Laidler (1999, pp 139-42)). The instruments that fit Caballero's definition best, high-quality bonds, are less immediately negotiable and hence, in uncertain times are less useful to ordinary agents - firms households and the like - than currency and deposits, even if they are more likely to remain valuable in the face of shocks to the financial system overall. But Caballero's broader ideas about safety are easily extended to give some weight to immediate acceptability, while expectations about value preservation can never be held with certainty, as his own informal discussion makes clear. It is no surprise, then, that his story of how the demand for safe assets is prone to rise in uncertain times bears more than a passing resemblance to the above-mentioned older accounts of the role of "liquidity preference" (sometimes, in simplified discussions, but not always, synonymous with the "demand for money" (see Laidler (1999, pp 283-7)) in the mechanics of depression and stagnation.

Basically, agents hold stocks of safe assets to enable them to meet their "survival constraints" - to borrow Hyman Minsky's (1954) phrase - in the face of unforeseen adverse economic shocks. And the more agents fear such events, the larger will be the quantities of such assets that they wish to hold. As Caballero himself is careful to point out, these shocks take different forms at different times and for different agents, so fear of them gives rise to demands for assets with characteristics vis-a-vis price predictability, marketability, rate of return etc, that are also different. "Safety" is not just a matter of the objective characteristics possessed by an asset. It also has to do with how these characteristics match up to the specific risks that particular agents believe they face, and the availability to them of other means of dealing with them. Not all of these risks are systemic, but they all give rise to what are fundamentally precautionary demands for assets of various sorts. Which specific assets fill the bill best is a matter of particular situations.

So how do those safe securities issued by a select few governments on which Caballero focuses fit in to this broader picture? Large financial institutions certainly like them, particularly those heavily engaged in international transactions, because their market is deep and active, and for large participants they are easily disposed of at short notice. Crucially, non-reserve currency central banks, on whom domestic institutions rely to provide lender of last resort services, will also want to hold them, particularly in a world lacking a reliable international "central bankers' central bank" on which they can count for support when their own survival constraints start to bind. In short, because of the precautionary services they can provide to internationally significant private institutions and central banks, Caballero's safe securities contribute to the liquidity and stability of the international financial system, and hence, crucially, of the national financial systems that are linked to it.

It is thus possible to link traditional discussions of the monetary experiences of national economies to the issues that particularly concern Caballero, and he is right to note that, in the 1960s, debates about the role of the so-called Triffin dilemma in the workings of the Bretton Woods system, did precisely that; though I would argue that in this case he perhaps overemphasises the international element in the causes of the monetary problems of that time. In my view, the final demise of the gold exchange standard in the early 1970s had more to do with the destructive domestic fiscal and monetary policies pursued by the United States as it tried to finance two wars - one on poverty and the other in Vietnam - than with the stresses created within the Bretton Woods system by limits on the stock of monetary gold. In a similar vein I would also be inclined nowadays to put more stress than does Caballero on the effects of the domestic monetary policies pursued by, among others, the Fed, the ECB and the Bank of England on the economic performance of national economies, than on problems within the international financial system per se.

**Further issues**

Even so, the currently high prices of Caballero's "safe" securities are a conspicuous feature of the financial landscape and they do pose questions that are intellectually interesting and policy-relevant in their own right, so his analysis should certainly command careful attention. He argues that, without policy attention, rock-bottom interest rates are likely to persist because their underlying causes are structural. First, the demand for safe assets even in normal times grows roughly proportionally with the world economy - see in particular Caballero et al (2017b); second, and crucially, agents in that economy have become, and continue to become, more risk averse, significantly exacerbating the effects of this normal growth in demand - see in particular Caballero et al (2017a); and, finally, these forces are at play in an economy where, if anything, the capacity of the supply of safe assets to respond to growing demand is seriously limited by various institutional and political constraints. This case is well made and coherent, but it does raise a few further issues.

First, if I read Caballero et al (2017b) correctly, some of its argument is based - either for analytical simplicity or because the authors believe it to be true - on the proposition that the demand for "safe" assets grows roughly proportionally with the overall level of activity in the world economy. I wonder about this, because if, as I have argued above, this demand is fundamentally precautionary, then surely one would expect it to be subject to economies of scale - see Francis Edgeworth (1888), Knut Wicksell (1898) and a myriad others. There are many obvious reasons why the precautionary demand for safe assets might have grown over the last couple of decades not just from movements along, but also upward shifts of, its demand function, so perhaps its upward trend in future might become more subdued than in the past as the economic environment becomes more stable, and thus lessen the stresses to which Caballero points.

Second, the analysis seems to neglect another important feature of certain older treatments of the demand for precautionary assets. The risks that matter for this demand are those that agents perceive, and these are not independent of the time, effort and other resources that they devote to gathering and processing relevant information about the likely future course of events. Holding stocks of precautionary assets is thus, on the margin, a substitute for engaging in such activities, because doing so reduces the costs incurred when adverse shocks are encountered. Interactions between risk and the demand for "safe" assets thus run in two directions, not one. This consideration too would tend to make the effects of constraints on the supply of safe assets less acute.

Two questions also arise concerning the evidence presented in Caballero et al (2017a) about the intensity of risk aversion and its growth in recent years, findings that provides an important empirical basis for the rest of his analysis.

To begin with, the four series presented there on ex ante real yields on US Treasury securities (90-day, three-year, five-year and 10-year) do not measure truly "safe" real interest rates. These data have been constructed by subtracting estimates of the expected inflation rate from ex ante nominal yields to maturity, and the estimates in question come from the Michigan Survey of Consumers, by way of FRED (See Caballero et al (2017a), fig. 1 panel a, fn.). Now the Michigan Survey itself provides data based on questions about inflation expectations over two time horizons, the next year, and the next five to 10 years, but FRED publishes only the first of these. So: just how reliable, and hence safe, are estimates of ex ante real rates over three months, three, five and 10 years when they are constructed by subtracting inflation expectations over one year from the safe nominal yields to these various maturities? Furthermore, Michigan's published estimates of expected inflation are the median values of individual responses to their monthly questionnaire. These responses are subject to a great deal of dispersion around measures of central tendency in any month, and to volatility in that dispersion over time too. So, just how risk-free are real interest rates that are estimated using such a series? And has the element of risk that remains attached to them been constant over time?

Finally, we should be uncomfortable with the fact that an aggregate production function and estimates of the aggregate stock of capital play major inter-related roles in generating this paper's key empirical findings. To be sure, these are ubiquitous features of today's macroeconomics, not least in research on the determinants of that elusive variable r*, much discussed in other sessions of this conference, and much discussed in the older literature too - see Laidler (1999, pp 29-31, 53-57). But we all know (or ought to know) that the conditions under which these entities exist are vanishingly unlikely. Joan Robinson (1953-4) was right about this point during the so-called "Cambridge controversies" - see Avi Cohen and Geoffrey Harcourt (2003) for a retrospective survey - and her arguments had the backing of many others, including Wicksell (1893), Gustav Cassel (1918), and their Swedish successors, as well as a pair of Fishers - Irving (1907) and Franklin (1969).

But evidently economists cannot do without these constructs and have mostly been inclined to ignore her.

In some respects this is fair enough: we do have to get on with our economics, and if our starting points are not always quite right, perhaps it is still permissible to assume that the world behaves more or less "as if" they were, and then proceed. But the "as if" defence can be treacherous: it might be reasonable to analyse the behaviour of Milton Friedman's "expert billiard player" at the pool table "as if" this person was indeed a competent mathematician - see Friedman (1953) - but I'm not so sure that I would have much confidence in predicting that same agent's performance in a calculus test using this assumption.

By analogy, analysis deploying an aggregate production function probably is adequate when, for example, pinning down the supply side of output gap measures in empirically based monetary policy models; it is surely better than fitting simple time trends as people used to do in the 1970s. But the basic theoretical issues with this particular construct stem from the dependence of the relative price weights used to construct an index of "capital" on the behaviour of the very rate of interest that we wish it to help us explain. Might this fact not render an exercise, which deploys an aggregate production function in analysing the reasons why the discrepancy between estimates of the productivity of capital in the aggregate and the economy-wide safe real rate of interest has increased over time, as problematic as one which confronts an expert billiard player with a calculus test?

In short the critical stylised facts around which Caballero's analysis is constructed, may not be quite as securely anchored in either careful measurement of the variables concerned or sound microeconomic theory as they at first appear to be.

**Errors of optimism and pessimism**

Finally, even accepting Caballero's proposition that the world has become increasingly "risk-intolerant" in recent years, questions about just what this signifies remain. Does it mean that agents' tastes for taking well specified objective risks have been and perhaps still are on the move? This interpretation seems to underlie Caballero et al (2017a), at least on my reading. If so, then indeed there is no particular reason to expect such a fundamental change in tastes to be reversed and future policy choices should take account of this consideration. Or does it mean that, though tastes vis-a-vis risk remain the same, the objective risks to which agents are exposed in today's economy are growing? This interpretation seems to fit better with some of the discussion of Caballero et al (2017b), once again on my reading. But perhaps this does not matter much since this phenomenon would be potentially complementary in its effects and policy implications to an increase in risk aversion per se.

But a third possibility puts in an appearance, yet again on my own reading, in Cabellero and Simsek (2017) where agents are divided between "optimists" and "pessimists" and the implications of such heterogeneity are analysed: namely, that we are dealing with fundamentally subjective expectations about the likely

evolution of events. Expectations of this kind can and do systematically differ from their "objective" equivalents, if indeed the latter concept makes sense - a matter best not pursued further here! - and they can also differ across agents, as Caballero and Simsek stress. But they can change over time too, as individuals revise their views in the light of experience, their own and that of others. It was this possibility that Lavington (1922) and Arthur Pigou (1927) were prominent in stressing in the 1920s, and they accorded an important role in economic fluctuations to the influence of contagious and cumulative "errors of optimism and pessimism" - see Laidler (1999) pp 84-86.

This way of looking things not only opens up the possibility that the increases in "risk intolerance" whose consequences we are now seeing might have arisen in a cumulative manner from recent painful experience, but also, and more importantly, that more favourable future experience could halt and then reverse them. Clearly, to take this possibility seriously would considerably change the view of recent history and of the future policy menu that Caballero embraces.

Hindsight reveals no shortage of "errors of optimism" before 2008: believing that low and stable inflation was sufficient to guarantee asset market stability; believing that the elimination of currency risk inherent in the arrival of the euro also eliminated all those other risks that had long been inherent in the borrowing practices of certain member governments; believing that the risks inherent in American NINJA mortgages could somehow be made to disappear by securitization; forgetting that the mere creation of new dot-com companies did not guarantee a market for their products, and, later, that the construction of new suburban and retirement housing did not guarantee its salability etc.

And hindsight also suggests that, when history revealed these errors, the reaction was to act upon newly formed errors of pessimism: a hasty scramble out of real and now perceived to be risky assets into safe financial assets; a reluctance on the part of the authorities to provide the latter (including but not limited to those that figure in Caballero's story) in sufficient amounts to meet a jump in private demand for them; an accompanying reluctance on the part of too many economists who should have known better to question this hesitant response, on the grounds that what had already been grudgingly done along such lines threatened the imminent onset of hyper-inflation etc.

But, the passage of time has begun to reveal these errors too. So-called ultra-easy monetary policy did not create hyper-inflation. Indeed some of us would complain that over-emphasis on the level of nominal interest rates as indicators of the stance of monetary policy, and an almost wilful propensity to ignore the messages being imparted by the growth rates of those safe assets included in broader measures of domestic money supplies - perhaps another error of pessimism - led to monetary policy remaining too tight for far too long, in the US and the UK, but even more notably in the euro zone - see Congdon (ed) (2017) for a statement of this case.

Now that monetary policy has become easier, however, and economies are on the mend, perhaps expectations will shift towards optimism, and those ever present "dark forces of time and ignorance" will once again begin to seem less threatening than in recent years. If expectations do shift, then today's apparently chronic shortages in the supply of "safe" assets relative to demand will begin to slacken. And perhaps also those many and various, not to say popular, suggestions that the structure of the economy has changed permanently and for the worse, might themselves begin to look like errors of pessimism that helped perpetuate the very malaise that they sought to diagnose. But this is to indulge in foresight - always riskier than hindsight - so in the meanwhile readers are advised to pay careful attention to Professor Caballero's always stimulating research.

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