The Effects of Monetary Policy Announcements
Summary and Keywords
The effects of news (i.e., information innovations) are studied in dynamic general equilibrium models where liquidity matters. As a leading example, news can be announcements about monetary policy directions. In three standard theoretical environments—an overlapping generations model of fiat currency, a new monetarist model accommodating multiple payment methods, and a model of unsecured credit—transition paths are constructed between an announcement and the date at which events are realized. Although the economics is different, in each case, news about monetary policy can induce volatility in financial and other markets, with transitions displaying booms, crashes, and cycles in prices, quantities, and welfare. This is not the same as volatility based on selffulfilling prophecies (e.g., cyclic or sunspot equilibria) studied elsewhere. Instead, the focus is on the unique equilibrium that is stationary when parameters are constant but still delivers complicated dynamics in simple environments due to information and liquidity effects. This is true even for classicallyneutral policy changes. The induced volatility can be bad or good for welfare, but using policy to exploit this in practice seems difficult because outcomes are very sensitive to timing and parameters. The approach can be extended to include news of real factors, as seen in examples.
Keywords: news, monetary policy, forward guidance, inflation, interest rates
Introduction
Many people have argued in recent years that news—that is, innovations in information that can affect expectations—can be important in macroeconomics. Cochrane (1994) is an early advocate for the relevance of news about productivity, policy, energy prices, regulation, international factors, and sectoral shifts. News about productivity is the main focus of Beaudry and Portier (2004, 2006, 2007), who try to identify information shocks in the data and show how they can be used as impulses in macro models. Other notable work includes Jaimovich and Rebelo (2009), Barsky and Sims (2011, 2012) and SchmidttGrohe and Uribe (2012), but as there are too many papers to discuss each individually, we refer readers to the introductory piece by Krusell and McKay (2010) and the extensive survey by Beaudry and Portier (2014).
Our focus in this article concerns announcements by central banks, including the Federal Reserve System in the United States, as well as the European Central Bank, the Bank of England, and others. In particular, we take a theoretical perspective by analyzing the impact of monetary policy (and other) news in dynamic general equilibrium models where liquidity plays a central role. This draws on our recent research, as reported in Gu, Han, and Wright (2018b), but we also add new material. To describe the general approach, first, note that a standard way to capture changes in policy or other exogenous variables in dynamic economic theory is to have these variables follow stochastic processes and trace out responses to different realizations. Instead, given the topic at hand, here we consider changes in variables at date ${t}_{1}$ that are known to be coming as of ${t}_{0}\le {t}_{1}$ but are unanticipated at $t<{t}_{0}$. If there seems to be a tension between this and the usual concept of rational expectations, note that even a supporter of that concept as staunch as Sargent (1993) argues that we should be willing to entertain the possibility of surprises. In any case, the approach captures totally unexpected events (${t}_{0}={t}_{1}$) and perfect foresight (${t}_{0}\to \infty $) as special cases.
To show the generality of the methods, we pursue the issues in three standard environments from the literature. The first is an overlapping generations (OG) model that allows fiat currency to be valued. In particular, it is very close to the simple specification in Azariadis (1981).^{1} We then turn to the New Monetarist framework, as presented in the recent survey by Lagos, Rocheteau, and Wright (2017) and the book by Rocheteau and Nosal (2017).^{2} This framework borrows from search theory by modeling exchange as a process where agents trade explicitly with each other, not merely along budget lines, which accommodates multiple payment instruments (standard OG models are somewhat too Walrasian for that). It also allows us to construct more realistic numerical examples. Last, we consider models of unsecured credit with endogenous debt limits following the basic approach in Kehoe and Levine (1993).^{3} In fact, we also present cases that have endogenous debt limits plus currency.
For each specification, we show how to construct the response to news, with announcements about monetary policy being a leading example. The results may be surprising: even in very simple settings, news at ${t}_{0}$ can induce complicated dynamics, including booms, busts, and cycles with amplitudes that increase or decrease as we get close to ${t}_{1}$. This does not rely on sticky prices, disparate beliefs, or multiple equilibria where agents use news to coordinate beliefs, as in other papers on related topics (see Gu et al., 2018b, for references). Moreover, it is critical to emphasize that the results are not the same as the volatile equilibria based on selffulfilling prophecies—for example, endogenous cyclic, chaotic, or stochastic (sunspot) equilibria—that have been extensively studied elsewhere (again see Gu et al., 2018b, for references). While it is well known that these models have multiple equilibria, some of which display complex dynamics, the focus here is on the unique equilibrium that is stationary for fixed parameters, with the transition constructed from ${t}_{0}$ to ${t}_{1}$ by backward induction. Although this path is pinned down uniquely, it can look very different depending on parameters and timing, with runups, crashes, or oscillations in prices, quantities, and welfare.
As may also be surprising, complicated dynamic responses can occur even for news about policies that are neutral in the long run, or even neutral in the short run, absent the announcement effects emphasized here. This includes a classically neutral onetime increase in the money supply, providing a stark counterexample to the classical macro position, commonly associated with Friedman, Phelps, Lucas, or Sargent and Wallace, that money injections matter if and only if they are unanticipated (e.g., see Mishkin, 1982, for a discussion and a review of the evidence at the time). To be clear, it is of course true in some contexts that unanticipated money injections have real effects while anticipated ones do not; what we show is that the exact opposite can be true in other contexts. One conclusion from this is that market reactions to Federal Reserve System news does not logically imply that money is nonneutral.^{4}
More generally, in terms of monetary policy, there are implications for the notion of forward guidance, defined as communication about the likely future course of policy (see the website of the Board of the Governors of the Federal Reserve System for discussion). The relevance of this practice is typically taken for granted, as evidenced by Blinder, Ehrmann, Fratzscher, de Haan, and Jansen (2009), who aver that “ the view that monetary policy is, at least in part, about managing expectations is by now standard fare both in academia and in central banking circles ” (p. 911). While the usual motive for this practice is to avoid large reactions when they occur, we show that announcing changes in advance can induce rather than reduce volatility. To be clear, the claim is not that advance warning always leads to volatile reactions but that it can lead to volatile reactions, which is different from conventional wisdom.^{5}
Another result is that newsinduced volatility may improve welfare, even for policies that are neutral, or even for those that are unambiguously bad, in the long run. However, while this form of guidance may enhance welfare in theory, exploiting it is too sensitive to timing and parameter values to be manageable in practice. Note also that the news does not have to be about a change happening in the future: the actual change could happen now, but the effects might only kick in after a while. A policy of printing currency can be implemented immediately, for example, but there could be long and variable lags as the new cash works its way into the system via government expenditures, transfers, tax cuts, or open market operations. That would make the timing especially delicate. Additionally, no news, or the lack of an announcement, can be just as important as news, depending on what was anticipated. Indeed, what matters for the dynamics is expectations, but the maintained hypothesis is that expectations are influenced by announcements.
This rest of the article is organized as follows. The OG model is presented and shows how to construct transition paths after information innovations in a relatively simple context. The next section derives similar results in the New Monetarist framework, with multiple means of payments, allowing us to make rigorous the effects of monetary policy on, for example, the stock market or the housing market. The following section takes up models of credit and the last section concludes.
An Overlapping Generations Model
We start with a standard OG model, similar to Azariadis (1981). Although it can be generalized in several ways, we keep it simple, to show how the most rudimentary formulation can generate complicated dynamic responses to news.
Setup and Equilibrium
A large number of twoperiodlived agents are born at each date $t=0,1,2\dots $ There is also a large number of agents alive only at $t=0$, called the initial old. For simplicity, assume the same number of agents are born each period, so there is no population growth. Also for simplicity, assume each agent can convert labor to a numeraire consumption good at a oneforone rate when young, but not when old, and values consumption only when old. Also assume that goods cannot be stored across periods. The initial old are endowed with an intrinsically useless object that can be stored called fiat money. Injections or extractions of money in subsequent periods are conducted through lumpsum transfers or taxes. Agents trade in perfectly competitive (Walrasian) markets.
Lifetime utility of an agent born at $t$ is ${\mathcal{l}}_{t}+\beta u\left({x}_{t+1}\right)$, where ${\mathcal{l}}_{t}$ is labor when young and ${x}_{t+1}$ is consumption when old. Assume $u$ is twice continuously differentiable with ${u}^{\prime}>0$, ${u}^{\u2033}<0$, and $u(0)=0$, and notice that $\beta \in \left(\mathrm{0,1}\right)$ is included to facilitate comparison with other models presented later, even though it could obviously be subsumed in the notation. Then the problem of an agent at $t$ can be written
where ${m}_{t}$ is money acquired from production when young, ${\varphi}_{t}$ is the price of money in terms of goods, or the inverse of the nominal price level, and ${\tau}_{t+1}$ is a transfer (if $\tau >0$) or tax (if $\tau <0$) of money to generation $t$ at $t+1$. The firstorder condition for ${m}_{t}>0$ is
equating the marginal cost of a dollar to its marginal benefit next period.
The aggregate money supply per old agent at $t$ is ${M}_{t}$, where ${\pi}_{t}$ is the rate of monetary expansion, so that ${M}_{t}=\left(1+{\pi}_{t}\right){M}_{t1}$, and the consolidated monetaryfiscal authority’s budget constraint is ${\tau}_{t+1}={\pi}_{t+1}{M}_{t}$. Denote real balances by ${z}_{t}\equiv {\varphi}_{t}{m}_{t}$, and notice that money market clearing entails ${m}_{t}={M}_{t}$. Then, multiplying the LHS of (1) by ${M}_{t}$ and the RHS by ${M}_{t+1}/\left(1+{\pi}_{t+1}\right)$, we get a difference equation
This says the real value of money—which in this simple economy is the same as total liquidity—at $t$ depends on its value at $t+1$, adjusted by discounted marginal utility and the money growth rate. It is this forwardlooking condition that can lead to interesting dynamics, and versions of similar equations will play a significant role in all of the models analyzed later.
Any bounded solution to (2) constitutes an equilibrium path for ${z}_{t}$.^{6} A nonmonetary equilibrium always exists where ${z}_{t}=0$ for all $t$ (intuitively, if future generations do not value money, the current generation will not accept it). We are interested in monetary equilibria, where ${z}_{t}>0$ for some $t$, which then implies ${z}_{t}>0$ for all $t$. If ${\pi}_{t}=\pi $ is time invariant then it makes sense to define a stationary monetary equilibrium (SME) as a nonzero steady state of the system given in (2), that is, a solution ${z}_{s}>0$ to ${u}^{\prime}\left({z}_{s}\right)=\beta /(1+\pi )$. Under standard conditions, SME exists and is unique.
The left column of Figure 1 plots three examples of $f\left(z\right)$ constructed using the utility function^{7}
The examples all use $A=0.1$, $b=0.1$, and three values for $\gamma $ corresponding to low, medium, and high curvature, ${\gamma}_{L}=0.5$, ${\gamma}_{M}=2.5$, and ${\gamma}_{H}=3.5$ (Appendix A/Table 1 lists the parameter values used for all experiments here while Appendix B contains all the figures). There are always two intersections of $f\left(z\right)$ and the ${45}^{o}$ line, autarky at $z=0$, and the unique SME at $z={z}_{s}>0$. As is well known from the literature, the mapping $f$ does not have to be monotone. As show in Figure 1, ${\gamma}_{L}$ implies ${f}^{\prime}({z}_{s})>0$, ${\gamma}_{M}$ implies $0>{f}^{\prime}\phantom{\rule{0.2em}{0ex}}({z}_{s})>1$, and ${\gamma}_{H}$ implies $1>{f}^{\prime}\phantom{\rule{0.2em}{0ex}}({z}_{s})$. As shown, in all cases ${f}^{\prime}\left(0\right)>0$, as can be proved under standard conditions.
Hence, in addition to SME, even with all parameters (including $\pi $) constant, there are dynamic monetary equilibria: in each case, there exist ${z}_{0}>0$ such that, starting at ${z}_{0}>0$, the path ${z}_{t+1}={f}^{1}\left({z}_{t}\right)$ stays bounded. The simplest such equilibria have ${z}_{t}\to 0$ monotonically, which can be understood in terms of hyperinflation as a selffulfilling prophecy. It is also possible to have ${z}_{t}\to 0$ nonmonotonically, at least for ${\gamma}_{M}$ and ${\gamma}_{H}$. Additionally, for ${\gamma}_{H}$ we have ${f}^{\prime}\left({z}_{s}\right)<1$, and that implies there exist twoperiod cycles, where ${z}_{1}=f\left({z}_{2}\right)>{z}_{s}$ and ${z}_{2}=f\left({z}_{1}\right)<{z}_{s}$. As $\gamma $ increases further there emerge cycles of higher order, and it can be shown that for $\gamma $ above some threshold there exist cycles of every order as well as chaotic dynamics. Moreover, it can be shown that ${f}^{\prime}\left({z}_{s}\right)<1$ implies there exist sunspot equilibria, where $z$ fluctuates stochastically around ${z}_{s}$.^{8}
This is all well established in the literature. Intuitively, the economics behind these results can be described in terms of the nonmonotonicity of behavior today with respect to the value of money tomorrow: if ${\varphi}_{t+1}$ is higher then, on the one hand, young agents work more at $t$ due to the usual substitution effect, but, on the other hand, they might work less due to the income effect, and the net impact is ambiguous. Hence the supply of goods at $t$, and therefore the value of money at $t$, can be nonmonotone in its value at $t+1$. In these examples, as $\gamma $ increases the slope of ${z}_{t}=f\left({z}_{t+1}\right)$ becomes smaller around ${z}_{s}$. When this slope becomes less than $1$, even with parameters constant there exist equilibria that display cyclic, chaotic, or stochastic behavior, although there always coexists the SME.
As fascinating as these results may be, the interest here is in something completely different. Instead of looking at the many monetary equilibria where $z$ fluctuates even while parameters are constant, consider focusing on the unique one with the property that $z$ is constant if parameters are constant and asking what happens when parameters, including monetary policy parameters, change over time. In particular, supposed it is announced, or for any other reason it is anticipated, at ${t}_{0}$ that there will be a change at ${t}_{1}\ge {t}_{0}$. In the class of equilibria under consideration there is a unique transition path for $z$ between the news at ${t}_{0}$ and the event at ${t}_{1}$, after which it reverts $z$ to a constant given there are no additional parameter changes. However, while the transition is unique, it can be quite complex, and its qualitative properties can depend dramatically on parameter values.
Transition Path
An elementary example involves a onetime level increase in $M$ engineered by lump sum transfers. Initially, suppose all agents take $\pi $ as fixed, and the economy is in the unique SME, $z={z}_{s}$. Then at ${t}_{0}$ it is announced (and believed) that $\pi $ will change at ${t}_{1}\ge {t}_{0}$ to ${\pi}_{{t}_{1}}={\pi}^{\prime}>\pi $, after which it reverts to ${\pi}_{t}=\pi $ for all $t>{t}_{1}$. After the reversion, the economy returns to the same SME, and that provides a terminal condition to uniquely pin down the transition by backward induction. As a special case, if ${t}_{0}={t}_{1}$, the policy is a complete surprise, and the jump in $M$ causes $\varphi $ to drop so that $z=\varphi M$ and all other real variables remain the same. This is classical neutrality.
Now suppose ${t}_{0}<{t}_{1}$, so the injection is anticipated. At the ultimate date of the monetary injection we revert to the original SME, ${z}_{{t}_{1}}={z}_{s}$. Using (2) and ${\pi}_{{t}_{1}}={\pi}^{\prime}$, at the penultimate date ${t}_{1}1$ we have
The economics is straightforward: at ${t}_{1}1$ the value of currency ${\varphi}_{{t}_{1}1}$ is low because agents know it will be debased at ${t}_{1}$. At ${t}_{1}$ this is not a problem, because while ${\varphi}_{{t}_{1}}$ is low ${M}_{{t}_{1}}$ is high, and the offsetting effects cancel so that ${z}_{{t}_{1}}={z}_{s}$ (neutrality). However, at ${t}_{1}1$ agents are using today’s money but paying tomorrow’s prices, making liquidity scarce.
Given ${z}_{{t}_{1}1}$, at the antepenultimate date ${z}_{{t}_{1}2}$ is again given by (2), and continuing in this way back to $t={t}_{0}$ we generate the entire path. This is shown in the left panels of Figure 1 for an example with the policy is announced five periods in advance (for news arriving further in advance, just keep iterating). The arrows show time moving backward from ${t}_{1}$ to ${t}_{0}$; the paths in real time are shown in the right panels. In the first example, with ${f}^{\prime}\left({z}_{s}\right)>0$, ${z}_{t}$ falls monotonically until ${t}_{1}$, then jumps back to ${z}_{s}$. In the second, with ${f}^{\prime}\left({z}_{s}\right)\in (1,0)$, ${z}_{t}$ displays increasing oscillations before finishing up at ${z}_{s}$. In the third, with ${f}^{\prime}\left({z}_{s}\right)<1$, ${z}_{t}$ displays decreasing oscillations before finishing at ${z}_{s}$. Thus, as claimed earlier, while the transition is unique, it can be complex and can look quite different depending on parameters.
While the responses are different in the three examples, there is always a sizable jump in ${z}_{t}$ from ${t}_{1}1$ to ${t}_{1}$, for reasons explained already. This is important: given a jump in real balances coincident with the jump in the nominal money supply, a naive observer might jump to the conclusion that injecting cash causes an increase in liquidity and that implies nominal prices must be sticky. That would be a mistake. It is true that liquidity goes up with the cash injection, but not because the price level fails to respond—to the contrary, in fact, the price level has already responded in anticipation of the injection.
In summary, in the simplest of monetary models these examples show how announcement effects can be complicated and highly dependent on parameters, despite being anchored by a fixed terminal condition, ${z}_{{t}_{1}}={z}_{s}$. This is true even for the simplest of monetary policies, a lump sum injection, which is neutral in the standard sense—that is, in terms of comparative statics or, equivalently, in real time if the injection comes as a complete surprise.^{9} While a permanent increase in the level of $M$ is the same as a one timetime increase in the growth rate $\pi $, one can of course also consider a permanent change in $\pi $, which is not neutral (it results in a permanent decrease in $z$). One can also consider multiple and staggered announcements. Additionally, one can model policy as following a stochastic process for $\pi $ and make announcements in terms of changes in that process. One can also study the effect of news about real factors. The same methods apply, but instead of going into the details we at this point switch to a different class of models that allow us to make additional points.
A New Monetarist Model
While OG models allow for valued fiat money, the frictionless nature of Walrasian competitive markets puts many restrictions on what one can do—for example, it is not easy to introduce an equally safe asset with a different rate of return and have it compete with money as a payment instrument. Another concern is that it is hard to evaluate the welfare impact of policy interventions in OG environments, compared to those with a representative agent, because policies generally affect different generations differently. Moreover, while one can always construct examples, one might worry with OG models about their realism, at least for shortterm policy analysis. For all these reasons we move to a framework based on Lagos and Wright (2005) and Rocheteau and Wright (2005).
Setup and Equilibrium
There is a large number of agents that live forever. In each period $t=0,1,2...$ they interact in two distinct markets: first there is a decentralized market (DM), with frictions detailed later; then there is a frictionless centralized market (CM). There are two permanently different types of agents called buyers and sellers, based on what they do in the DM, although they are the same in terms of the CM. These agents may be households, firms, or financial institutions, and what they exchange can be goods, inputs, or assets; all that matters is that in the DM buyers and sellers meet bilaterally and at random, and the latter can provide something the former wants. The meeting process is summarized by a probability $\alpha $ that a buyer meets a seller, since $n\alpha $ is then the probability a seller meets a buyer, given the buyerseller ratio $n$. In the CM, all agents work, consume a numeraire good, and adjust asset positions in competitive markets.
Period payoffs for buyers and sellers are $u(q)+U\left(x\right)\mathcal{l}$ and $c(q)+U(x)\mathcal{l},$ where $q$ is the object being traded in the DM, $x$ is the CM numeraire good, and $\mathcal{l}$ is CM labor, and for simplicity $x$ is produced oneforone with $\mathcal{l}$. Here $u\left(q\right)$ can be the utility from consuming $q$ and $c\left(q\right)$ the disutility of producing it, or $u\left(q\right)$ can be output of $x$ from using $q$ as an asset/input and $c\left(q\right)$ the opportunity cost of giving it up. Goods are nonstorable, and $U$, $u$ and $c$ are twice continuously differentiable with ${U}^{\prime},{u}^{\prime},{c}^{\prime}>0$ and ${U}^{\u2033},{u}^{\u2033}<0\le {c}^{\u2033}$. Also, $u(0)=c(0)=0$. Agents discount between the CM and DM at $\beta $, but not between the DM and next CM, without loss of generality.
If there were perfect credit, the outcome would be the efficient ${q}^{\ast}$, where ${u}^{\prime}\left({q}^{\ast}\right)={c}^{\prime}\left({q}^{\ast}\right)$, with buyers in the DM promising sellers payment in the next CM. However, agents cannot commit to make payments and are anonymous in the DM, so such promises are not credible (but see later discussion). This implies an essential role for assets as payment instruments. We first consider fiat money as the only asset available to serve in this capacity; later we include other payment instruments.
Letting ${W}_{t}\left({m}_{t}\right)$ be the value function for a buyer with ${m}_{t}$ dollars in the CM, we have
where ${V}_{t+1}({\widehat{m}}_{t+1})$ is the continuation value in the next DM. Notice we distinguish between ${m}_{t}$, money taken into the CM at $t$, and ${\widehat{m}}_{t+1}$, money taken into the DM at $t+1$. Assuming interior solutions, the firstorder condition for ${x}_{t}$ is ${U}^{\prime}\left({x}_{t}\right)=1$, which pins down consumption of numeraire, and the firstorder condition for ${\widehat{m}}_{t+1}$ is
Note that (6) implies ${\widehat{m}}_{t+1}$ is independent of ${m}_{t}$, while the envelope condition ${{W}^{\prime}}_{t}\left({m}_{t}\right)={\varphi}_{t}$ implies the CM payoff is linear, results that greatly enhance tractability of the framework.^{10} The sellers’ problem is similar, and hence omitted, but their CM payoff is also linear. However, one difference is that since they have no use for money in the DM they chose to take ${\widehat{m}}_{t+1}=0$ out of the CM.
When buyers and sellers meet in the DM at $t$, they need to decide a quantity ${q}_{t}$, and a real payment ${p}_{t}$, subject to ${p}_{t}\le {z}_{t}$, where as usual ${z}_{t}\phantom{\rule{0.2em}{0ex}}$ denotes real balances, and the constraint is simply a feasibility condition saying that a buyer cannot give more than he has to a seller.^{11} While one can use a variety of mechanisms to determine these terms of trade—for example, different bargaining solutions, price posting, or price taking—for simplicity let us assume buyers make takeitorleaveit offers to sellers, since once again the goal is to show how even simple setups can generate complicated dynamic outcomes. Also, let $c\left(q\right)=q$, so that by virtue of the result that ${W}_{t}\left({m}_{t}\right)$ is linear, the terms of trade satisfy ${p}_{t}={q}_{t}$, which means the payment exactly covers sellers’ cost. Hence, a buyer’s DM value function satisfies
where the first term is the payoff from carrying cash into the next CM, and the second is the expected surplus from DM trade.
Differentiating (7) and using (6), we get the Euler equation. The result is given by
if ${z}_{t}<{q}^{\ast}$, that is, if the buyer does not have enough liquidity to get ${q}^{\ast}$, and ${\varphi}_{t1}=\beta {\varphi}_{t}$ otherwise. It is convenient to let
where $L\left({z}_{t}\right)$ is the liquidity premium, or the Lagrange multiplier on ${q}_{t}\le {z}_{t}$, and write the Euler equation as
If $\alpha =1$ then (10) is similar to (2), except now $f\left(z\right)$ is linear for ${z}_{t}\ge {q}^{\ast}$.^{12}
Transition Path
While our OG and searchbased theories are different in terms of economics, mathematically they are similar, and once again the key observation is that $f\left(z\right)$ can be nonmonotone. Here this results because the resale value of $z$ in the next CM is increasing while the liquidity premium is decreasing in $z$. One can plot (10) and trace out paths from a policy announcement to its implementation, as in the previous section, and the results are qualitatively similar, but here it is arguably easier to construct more realistic examples. While this is not meant to be a serious calibration, in Gu et al. (2018b) we argue similar examples are not unrealistic (for more on calibrating or estimating these kinds of models, see Aruoba [2011] or Aruoba & Schorfheide [2011]).
To this end, let the period length be a month and set $\beta =0.9959$ and $\pi =0.0041$, so the annualized real interest rate and inflation rate are both $5\%$ in SME. For the matching parameter, for the sake of illustration we somewhat arbitrarily set $\alpha =0.5$. For the cost function, let us use $c\left(q\right)=q$. For utility we again use the specification in (3), now with $A=1$ and $b=0.1$, and consider three values of $\gamma $, now with ${\gamma}_{L}=0.5$, ${\gamma}_{M}=4$, and ${\gamma}_{H}=8$, which deliver ${f}^{\prime}\left({z}_{s}\right)>0$, $0>{f}^{\prime}\left({z}_{s}\right)>1$, and ${f}^{\prime}\left({z}_{s}\right)<1$, respectively. Note that the CM function $U\left(x\right)$ actually does not matter for the results, and we can evaluate welfare by the representative buyer’s lifetime DM surplus. In fact, this takes the same units as consumption of numeraire, since in equilibrium ${U}^{\prime}\left(x\right)=1$.
The results are qualitatively similar to results from the OG model. If $f\left(z\right)$ is monotone increasing, news of a money injection sends the economy on an inflationary trajectory, lowering output and welfare along the way, before recovery when the money is actually injected. If ${f}^{\prime}\left({z}_{s}\right)<0$, there can be oscillations in output during the transition, and the increase in the DM surplus in some periods can outweigh the decrease in others, so in principle welfare can improve with the announcement of a future money injection. Indeed, as long as monetary policy is away from the Friedman rule, which here means as long as $\pi >\beta 1$, SME entails $q<{q}^{\ast}$ because the liquidity constraint binds; it may be slack, however, and we may get $q={q}^{\ast}$ in some periods during transitions. From a welfare perspective, this tells us something about when it is to best make announcements: if $f\left(z\right)$ is monotone the welfare loss is minimized by revealing a plan to increase $M$ as late as possible, and, symmetrically, the welfare gain is maximized by revealing a plan to decrease $M$ as soon as possible; if $f\left(z\right)$ is nonmonotone, however, this rule of thumb need not be valid.
The left column of Figure 2 plots transitions given a temporary change from $\pi $ to ${\pi}^{\prime}$ that generates a $1\%$ jump in $M$ at ${t}_{1}\phantom{\rule{0.2em}{0ex}}$ over its value along the path with $\pi $ fixed, announced at ${t}_{0}$, with the three rows corresponding to different values of $\gamma $. The news in these examples comes 12 periods (months) in advance of the injection, but the effects of decreasing or increasing the lead time can be seen simply by picking up the transition closer to ${t}_{1}$ or extending the iterations back further. Shown are liquidity ${z}_{t}$ and welfare ${S}_{t}$, with both normalized to $1$ in SME. Notice ${z}_{t}$ and ${S}_{t}$ both change roughly $1\%$ at their peak over the transitions. Hence, even for a policy that is neutral in the longrun, and neutral in the short run if it is a surprise, news about an upcoming change has real effects and they are quantitatively relevant.
The right column of Figure 2 repeats the exercise for a $0.05\%$ temporary increase in $\pi $. As the change is smaller, the transition stays closer to the steady state and it is easier to generate oscillations during the transition—intuitively, as ${z}_{t}$ strays further from ${z}_{s}$ we are more likely to hit the linear branch of the dynamical system. The bottom right panel in particular shows a case where welfare is enhanced by announcing a money injection two periods in advance. Since this is true for policy that is neutral in the long run, by continuity it can also be true for policies that are in fact bad in the long run. This may seem paradoxical, but the economic idea is clear: news of a future liquidity shortage raises demand for money now, which raises $\varphi $ and hence liquidity in the short run. While it is probably impractical in reality to know the parameters and get the timing precisely enough to exploit this in practice, this shows how it could work in theory. See Gu et al. (2018b) for more on this simple model, including the effects of news about nonneutral policies, changing the period length, allowing news to diffuse slowly over time, and other extensions.
Alternative Means of Payment
To show how monetary news affects other markets, consider an economy where a real asset $a$ and fiat money $m$ can both be used in the DM. The real asset is in fixed supply, normalized to $1$ unit per buyer, its CM price is ${\psi}_{t}$, and it bears a dividend ${\rho}_{t}>0$ in terms of CM numeraire.^{13} The CM budget equation then becomes
Since $a$ pays a dividend and $m$ bears the inflation tax, $m$ and $a$ cannot coexist if they are perfect substitutes in transactions. Hence, assume $a$ and $m$ are imperfect substitutes because they are not equally acceptable in DM trade: ${\alpha}_{m}$ is the probability of meeting a seller who accepts only $m$; ${\alpha}_{a}$ is the probability of meeting one who accepts only $a$; ${\alpha}_{b}$ is the probability of meeting one who accepts both; and buyers do not know who they will meet until the CM has closed.^{14}
Also, as in the large literature following Kiyotaki and Moore (1997), we introduce a pledgeability parameter $\chi \le 1$, meaning that buyers can only use a fraction $\chi $ of their $a$ holdings in DM transactions. While Li, Rocheteau, and Weill (2012) endogenize $\chi $ using private information, in our experiments we take it as a parameter. Also, at this point it is worth emphasizing that there are two interpretations of the model. First, one can say that buyers simply hand over assets to sellers in exchange for $q$, treating $a$ analogous to $m$. Second, one can alternatively say that buyers do not hand over assets but promise sellers a payment in numeraire in the next DM, and if they renege they will be punished by having their assets confiscated. Hence, we can say $a$ serves as collateral.
Clearly, in this secured credit arrangement, a buyer will honor an obligation if and only if it is less than the value of the collateral, and therefore the seller will not accept promises of payments above the value of the collateral. Given this, rather than using secured credit, the buyer may as well hand over the assets in the DM, as in the first interpretation. In other words, in this and many other environments, if not in all environments, it is equivalent to think of assets as supporting deferred settlement—that is, as collateral—or as payment instruments facilitating immediate settlement—that is, as money. This is especially relevant for our present purposes because it implies that the effects of monetary policy news on assets other than currency applies to situations where these assets serve as collateral and not just situations where they assets serve as media of exchange.
One detail is that $\chi <1$ is sometimes interpreted in the literature as giving borrowers the ability to abscond with a fraction $1\chi $ of their assets in the outofequilibrium event of default, which might sound like it fits the deferred settlement story better. However, in the approach of Li et al. (2012), where $\chi $ is endogenized using private information, an equally compelling story is that the seller is worried about the quality of the buyer’s assets, and therefore a desire to have $\chi <1$ in the model does not wed one to the deferred settlement interpretation, although that interpretation is perfectly appropriate for our purposes.^{15}
Having said all that, let us denote a buyer’s DM liquidity by ${z}^{a}$ in an assetonly meeting, ${z}^{m}$ in a moneyonly meeting, and ${z}^{b}$ in an assetandmoney meeting, where ${z}_{t}^{a}=\chi {\widehat{a}}_{t}({\psi}_{t}+{\rho}_{t})$, ${z}_{t}^{m}={\varphi}_{t}{m}_{t}$, and ${z}_{t}^{b}={z}_{t}^{m}+{z}_{t}^{a}$. Consider for the sake of illustration the specification ${\alpha}_{a}=0$, so no seller accepts assets but rejects cash. Generalizing (10) the Euler equations for $\widehat{m}$ and $\widehat{a}$ yield a twodimensional system:
Let ${q}_{t}^{j}$ be the quantity traded in type $j$ DM meetings. In SME, in type $m$ meetings the constraint ${q}^{m}\le {z}_{s}^{m}$ binds and ${q}^{m}<{q}^{\ast}$ as long as we are away from the Friedman rule, that is, as long as $\pi >\beta 1$. Also, in SME, in type $b$ meetings ${q}^{b}\le {z}_{s}^{b}$ binds if $\rho $ is small and is slack if $\rho $ is big, and in the former case we have $q<{q}^{\ast}$ and $\psi >{\psi}_{F}\equiv \rho /r$ while in the latter we have ${q}^{b}={q}^{\ast}$ and $\psi ={\psi}_{F}$.
Let us focus on the case where $\rho $ is small, so the liquidity constraint binds in type $b$ meetings, and consider news at ${t}_{0}$ of a onetime increase in $\pi $ at ${t}_{1}$. Given $\rho $ is small, ${z}_{t}^{m}$ and ${z}_{t}^{a}$ are both affected during the transition. Figure 3 shows a $1\%$ increase in $M$ at ${t}_{1}$, with ${z}_{t}^{m}$ and ${z}_{t}^{a}$ in the left column and ${z}_{t}^{b}$ and ${S}_{t}$ in the right, normalized so ${z}_{s}^{m}={z}_{s}^{a}=1$. For the parameters in this example ${q}^{b}<{q}^{\ast}$ and $\psi >{\psi}_{F}$ in SME. The transitions for ${z}_{t}^{m}$ and ${S}_{t}$ are similar to the baseline model, except now monetary policy news affects the stock market. In the top row there is an initial jump in ${z}_{t}^{a}$ as agents compete for other assets to compensate for the drop in ${z}_{t}^{m}$, then both decline until ${z}_{t}^{a}$ gets to ${z}_{s}^{a}$ and ${z}_{t}^{m}$ jumps back to ${z}_{s}^{m}$ after bottoming out. The other rows are similar but display oscillations, with the bottom row especially volatile. This corresponds to equity markets reacting to news about monetary policy.
To show the approach is not limited to studying monetary policy announcements, consider news about real assets. Information about ${\rho}_{{t}_{1}}$ revealed at ${t}_{0}<{t}_{1}$ changes ${z}_{{t}_{1}}^{a}$ directly; this then affects ${z}_{{t}_{1}1}^{m}$, and the economy follows a transition path given by (11) and (12). Figure 4 plots the transition when it becomes known at ${t}_{0}$ there will be a onetime fall in $\rho $ at ${t}_{1}$, which is similar to news about an increase in $M$, except the patterns in ${z}_{t}^{m}$ and ${z}_{t}^{a}$ are reversed. In general, the lesson is that news about one market can influence all markets. One can therefore extend the analysis by adding other assets, such as foreign currencies or home equity, to see how monetary policy announcements affect foreign exchange and housing markets. See Gu et al. (2018b) for details, but, briefly, with two currencies, news about monetary or real factors in one country induces dynamics in each country’s price level and output, as well as their interest and exchange rates. Similarly, when home equity can be used as collateral, news can induce dynamics in house prices, including booms, crashes, and cycles, as well as goods prices and output.
Unsecured Credit
Announcement effects are not unique to monetary economies. We discussed secured credit to show how monetary policy news affects asset markets, but one can also consider models of unsecured credit following Kehoe and Levine (1993) and many others. Here, to facilitate comparison to the other environments discussed in this article, we employ the version in Gu, Mattesini, Monnet, and Wright (2013a, 2013b). To ease the presentation, we start with a pure credit model, without money, and then later reintroduce money.
Setup and Equilibrium
Assume as before that agents cannot commit to repay debts, but now defaulters can be punished by taking away future credit (as opposed to taking away some of their assets, as in secured credit theories). In particular, we suppose that at each date $t=0,1,2\dots $ there are two subperiods, and two goods $x$ and $q$. Agents called debtors produce $x$ and consume $q$ in the first subperiod while other agents called creditors produce $q$ in the first subperiod but want to consume $x$ only in the second. Debtors can store or otherwise invest $x$ across subperiods at a return $R$ per unit; creditors cannot store or invest it across subperiods; and, for simplicity, no goods can be stored across periods, only across subperiods.
Agents meet randomly each period, where $\alpha $ is the meeting probability of a debtor. Given these preferences and technologies, a desirable arrangement is for a creditor to produce $x$ for a debtor in the first subperiod in exchange for a promise that he will deliver $Rx$ to the creditor in the second. The payoff from this arrangement is $Rxc\left(q\right)$ for the creditor, and $u\left(q\right)x$ for the debtor, assuming he does not consume any of his own output. However, the debtor can consume $Rx$ in the second subperiod for extra utility $\lambda Rx$. We impose $\lambda R<1$, so that it is not efficient for a debtor in the first subperiod to produce $x$ for his own consumption in the second subperiod, but he might opportunistically consume it after the cost is sunk. Thus, $\lambda =0$ implies a debtor’s promise to deliver the goods in the second subperiod is credible, but if $\lambda >0$ he may be tempted to renege.^{16}
If a debtor reneges on an obligation, he gets caught with probability $\mu $, in which case he is punished by future autarky, with a payoff normalized to $0$ (while $\mu =1$ is fine, it turns out that allowing $\mu <1$ opens up some interesting possibilities, and so we keep things general). The incentive condition at $t$ for a debtor to honor his debt obligation, called the repayment constraint, is therefore
where ${V}_{t+1}$ is the continuation value as long as he has never been caught reneging. Let us rewrite (13) as ${R}_{t}{x}_{t}\le {D}_{t}$, where ${D}_{t}\equiv \beta {\mu}_{t}{V}_{t+1}/\lambda $ is the endogenous debt limit.
Here we adopt Walrasian price taking, which might make more sense if there are multilateral instead of bilateral meetings, so let us now assume large numbers of debtors and creditors meet each period.^{17} For a debtor, who has both a budget and a repayment constraint, we have
where ${p}_{t}$ is the price of ${q}_{t}$ in terms of ${x}_{t}$. Clearly, ${u}^{\prime}\left({q}_{t}\right)={p}_{t}/{R}_{t}$ if ${p}_{t}{x}_{t}<{D}_{t}$ and ${q}_{t}={D}_{t}/{p}_{t}$ otherwise. For a creditor, who faces no repayment constraint, we simply have ${c}^{\prime}\left({q}_{t}\right)={p}_{t}$.
Let ${q}_{t}^{\ast}$ solve ${u}^{\text{'}}\left({q}_{t}^{\ast}\right)={c}^{\text{'}}\left({q}_{t}^{\ast}\right)/{R}_{t}$. Then in equilibrium, ${q}_{t}={q}_{t}^{\ast}$ and ${x}_{t}={c}^{\prime}\left({q}_{t}^{\ast}\right){q}_{t}^{\ast}/{R}_{t}$ if ${D}_{t}\ge {c}^{\prime}\left({q}_{t}^{\ast}\right){q}_{t}^{\ast}$, while ${q}_{t}={D}_{t}/{c}^{\prime}\left({q}_{t}\right)$ and ${x}_{t}={D}_{t}/{R}_{t}$ if ${D}_{t}<{c}^{\prime}\left({q}_{t}^{\ast}\right){q}_{t}^{\ast}$. Write ${q}_{t}=g\left({D}_{t}\right)$ when ${D}_{t}\le {c}^{\prime}\left({q}_{t}^{\ast}\right){q}_{t}^{\ast}$ binds, then use ${D}_{t}=\beta {\mu}_{t}{V}_{t+1}^{b}/\lambda $ and (14) to write
where a debtor’s trade surplus is
with $u\circ g\left(D\right)$ denoting the composite, $u(g(D))$.
Hence, the outcome is given by a simple difference equation, similar to the monetary economies studied earlier, except the debt limit $D$ replaces real balances $z$ as the relevant measure of liquidity. In particular, $D=0$ is always a stationary equilibrium, with no credit, analogous to the nonmonetary equilibria that always exists with fiat currency. There also exists a stationary equilibrium with $D>0$ under reasonable conditions. Moreover, unsecured credit economies can also have multiple cyclical, chaotic, and stochastic equilibria, where ${D}_{t}$ fluctuates over time as a selffulfilling prophecy, as in monetary economies although for different economic reasons.^{18} However, similar to the analysis of monetary economies in previous sections, we ignore these outcomes to focus on something different—reactions to news in the equilibrium where ${D}_{t}=D>0$ is constant as long as parameters are constant.
Suppose agents initially believe ${\mu}_{t}=\mu $ and ${R}_{t}=R$ are constant for all $t$, and we are in steady state with $D={D}_{s}>0$. Then at ${t}_{0}$ they get news that at ${t}_{1}$ the monitoring probability ${\mu}_{t}$ will be lower for just one period, making it harder to identify and punish defaulters. This implies ${D}_{{t}_{1}}<{D}_{s}$, and the transition back to ${t}_{0}$ is determined by iterating on (15). Nonmonotone dynamics occur if ${f}^{\prime}\left({D}_{s}\right)<0$, similar to monetary models (although the economic channel is somewhat different, as discussed in Gu et al., 2013b). The left column of Figure 5 shows the results, where in the middle (bottom) row, for example, news that credit conditions will deteriorate in the future sets off oscillations in ${D}_{t}$ with increasing (decreasing) amplitude, before recovering to ${D}_{s}$. The right panel shows similar results for bad news about future productivity, in this case captured by a fall in ${R}_{t}$. In both experiments, along with ${D}_{t}$, the terms and amount of lending, as well as output and welfare, vary during the transition.
Money and Unsecured Credit
Next, to illustrate how monetary policy news and unsecured credit interact, we integrate KehoeLevine credit and the model of money presented earlier. To this end, assume buyers now produce the CM numeraire $x$ in DM meetings, but sellers have no use for it until the next CM and only buyers can store it, with gross return of $R=1$ to reduce notation. As in the purecredit economy, a buyer produces $x$ in the DM and promises to deliver it in the CM but can opportunistically divert it for a payoff $\lambda x$. If the debt limit is small, buyers may acquire cash to boost their liquidity in the DM.
On the equilibrium path, as buyers pay off debts at the beginning of the CM, the value function $W\left({m}_{t}\right)$ is still given by (5). Again imposing Walrasian pricing and $c\left(q\right)=q$, given any debt limit ${D}_{t}$, we have ${V}_{t}\left({\widehat{m}}_{t}\right)=\alpha \left[u\left({q}_{t}\right){q}_{t}\right]+{W}_{t}\left({\widehat{m}}_{t}\right),$
where ${q}_{t}={d}_{t}+{\varphi}_{t}{\widehat{m}}_{t}$ and ${d}_{t}\le {D}_{t}$. The Euler equation for ${\widehat{m}}_{t+1}$leads to
which is similar to (10) except the liquidity premium here is based on total liquidity, ${D}_{t}+{z}_{t}$. The repayment constraint in the CM is
where we again let $\mu $ be the probability of getting caught. This reduces to
Emulating the analysis of the purecredit model, we rewrite ${V}_{t+1}$ using ${D}_{t}$ as
There is always a nonmonetary equilibrium, which reduces to a purecredit model, and there can exist monetary equilibria if the endogenous debt limit is tight. Focusing on the latter, transitions after news are complicated by interactions between money and credit. Figure 6 shows the impact news about a onetime $1\%$ increase in the money supply. The transition for ${z}_{t}$ again displays intricate dynamics, depending on parameters, and now news about monetary policy also induces dynamics in ${D}_{t}$. Figure 7 shows the impact of news about a onetime decrease in ${\mu}_{t}$. Given the news that the monitoring probability will be lower in the future, the debt limit tightens now, which raises the demand for money, but on net total liquidity falls. The credit market can experience a gradual deterioration, or cycles, or even a period of positive momentum before crashing in the penultimate period.
The result is related to that in the previous section, where assets and cash are also substitutes in the payment process, like money and credit, but there is a difference: here news that leads to lower ${D}_{t}$ tends to increase ${z}_{t}$, as agents substitute across payment methods, but news that leads to lower ${z}_{t}$ tends to decrease ${D}_{t}$. The latter effect occurs because lower ${z}_{t}$ reduces equilibrium payoffs, which tightens the debt limit for some, if not all, parameters. In particular, real credit conditions—the amount and terms or lending—can depend in complicated ways on news about changes in monetary policy even if the changes are neutral in the usual sense. Of course, news about nonneutral policies can also be analyzed, as can any of the other extensions mentioned earlier.
Conclusion
This article has presented several related but distinct models of liquidity and used them to illustrate the effects of announced changes in monetary policy. This includes effects on goods, assets, and credit markets. It is well known that there are multiple equilibria in serious models of liquidity, but that is not what the presentation emphasized. It instead concentrated on the unique equilibrium with the property that the endogenous variables are constant when parameters are constant and asked what could happen in response to news about future events, including future monetary policy moves. Hence, the intricate dynamic paths displayed here were not based on multiplicities or selffulfilling prophecies and were in fact determined uniquely by backward induction. Our approach can also be extended to study the effects of news of real factors, and we briefly discussed a few examples along those lines. We can also consider alternative monetary policies such as nominal interest rate targeting. A more general and extensive discussion can be found in Gu et al. (2018b).
While we focused mainly on very simple announcements about oneperiod changes in the money growth rate, one can obviously consider more complicated announcements, including permanent changes in the money growth rate or oneperiod and permanent changes in nominal interest rates, which are not necessary neutral when they come as a complete surprise. As an example that may have some current relevance, suppose we engineer a big increase in the supply of currency at date ${t}_{0}$, but credibly promise to reverse it with a a big decrease at date ${t}_{1}$. There will not generally be a big increase in the price level at ${t}_{0}$, ostensibly in contradiction to the quantity theory. Indeed, liquidity rises with the cash injection at ${t}_{0}$, but it is critical to understand that any beneficial effects come from the promise of decreasing $M$ at ${t}_{1}$, not from the act of increasing $M$ at ${t}_{0}$. Moreover, a related finding is that when $M$ is injected at ${t}_{1}$ prices do not increase oneforone if the policy was known at ${t}_{0}<{t}_{1}$. The reason is not that prices are sticky—to the contrary prices already rose prior to the injection of $M$ due to the announcement effect.
Finally, while in general the results differed across the various environments, there are some general lessons. First, announcements of changes in policy might sometimes reduce volatility, as seems to be the conventional wisdom, but it is easy to generate cases where announcements instead induce volatility in financial and goods markets. Second, the transitions can look very different depending on parameter values and can involve runups, crashes, and cycles in liquidity, prices, output, and welfare. Third, induced volatility can in principle improve welfare, although in practice this is almost certainly too sensitive to timing and parameter values to be exploited as a policy instrument. Finally, we repeat that even in environments where by design the quantity theory and classical neutrality hold, in the sense that a surprise increase in $M$ has no real effects, news about a future increase in $M$ can have big real effects, which might be a useful word of caution about interpreting data.
Acknowledgments
We thank participants in many seminars for their input. Wright thanks the Ray Zemon Chair in Liquid Assets at the Wisconsin School of Business for support. The usual disclaimers apply.
Appendix A: Parameters for Experiments
In all cases ${t}_{0}=0$, ${t}_{1}=12$ and $\beta =0.9959$. The other parameters are set as follows:
Table 1. Parameters for Experiments
Figure 2 
$\alpha =0.5,A=1,b=0.1,\gamma =0.5,4,8$, $\pi =0.0041$, 
${\pi}^{\prime}=0.0141\text{(L),}\phantom{\rule{0.2em}{0ex}}{\pi}^{\prime}=0.0046$ (R). 

Figure 3 
$A=0.25,b=0.45,\gamma =3,6,9$, ${\alpha}_{m}=0.01,0.001,1\times {10}^{4},{\alpha}_{b}=0.5,$ 
${\chi}_{a}=1$, $\pi =0.0041,\rho =2\times {10}^{4},{\pi}^{\text{'}}=0.0141.$ 

Figure 4 
${\rho}^{\prime}=0.8\rho .$ Other parameters follow Figure 3. 
Figure 5 
$\alpha =0.5,A=1,b=0.1,\sigma =0.2,0.5,0.8$, $\gamma =0,$ 
$\lambda =0.1,\mu =0.99,R=0.9,$ 

${\mu}^{\prime}=0.98,\phantom{\rule{0.2em}{0ex}}{R}^{\prime}=0.89.$ 

Figure 6 
$\alpha =0.5,A=0.1,b=0.15,\gamma =2,4$ or $6,\lambda =0.1,$ 
$\mu =2\times {10}^{4}$, $2\times {10}^{5}$ or $2\times {10}^{6}$, $\pi =0.0041$, ${\pi}^{\prime}=0.0141$ 

Figure 7 
$\lambda =0.9$, ${\mu}^{\prime}=0.8\mu .$ Other parameters follow Figure 6. 
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Notes:
(1.) It is not too difficult to apply the methods in more general OG models, as presented in, for example, Wallace (1980), Sargent (1987), or Azariadis (1993). However, a main point here is that news can generate complicated dynamics even in the simplest setups.
(2.) See Gu, Han, and Wright, et al. (2018a) for a relatively short summary. For recent work in the area that is concerned especially with central bank policy, see Williamson (2012, 2016), Andolfatto, Berentsen, and Waller (2016), Geromichalos and Herrenbrueck (2016), and Rocheteau, Wright, and Xiao (2018).
(3.) There are many wellknown extensions and applications of this approach including, for example, Alvarez and Jermann (2000). Our presentation follows Gu et al. (2013b). Another approach to credit follows Kiyotaki and Moore (1997), where different assets serve as collateral, but, as discussed later, one can understand that simply by reinterpreting the New Monetarist models that we cover.
(4.) Of course there are other papers (e.g., Fischer 1979) that argue money can be nonneutral due to a MudelTobin effect (and other effects). The point here is that one can generate nonneutral effects due to news even for a policy that is, by design, neutral absent announcement effects.
(5.) As evidence of the conventional wisdom, consider Blinder et al. (2009): “central bank talk increases the predictability of central bank actions, which should in turn reduce volatility in . . . financial markets” (p. 912). Or consider Matsumoto, Cova, Pisani, and Rebucci (2011): “one might conjecture that providing more information about future fundamentals in DSGE models (i.e., more information about the exogenous stochastic processes) would reduce asset price volatility” (p. 3). Again, these statements may be valid in some situations, but we show the opposite is true in other situations.
(6.) In case it is not obvious, (2) defines equilibrium paths because it embodies both optimization and market clearing, and boundedness is a standard condition in economic dynamics that can most easily be motivated in this setup by assuming that $x$ can be produced oneforone with $\mathcal{l}$ only up to a point $\overline{x}$, say, because total time available for $\mathcal{l}$ each period is limited.
(7.) This common example reduces to the usual case of constant relative risk aversion utility when $b=0$, but allowing $b>0$ means that $u\left(0\right)=0$ even if $\gamma >1$. It actually does not drive anything important here, although having $u\left(0\right)=0$, or at least having $u\left(0\right)>\infty $, is convenient in models with bargaining rather than Walrasian pricing.
(8.) See Azariadis (1993) for a textbook treatment of the mathematics underlying all the results described in this paragraph.
(9.) A different policy is a onetime level increase in $M$ engineered by proportional, instead of sump sum, transfers. These are neutral independent of announcements, intuitively, because sellers at ${t}_{1}1$ value money more when they know their receipts will be increased multiplicatively at ${t}_{1}$.
(10.) These results obtain in many models following Lagos and Wright (2005) and keep the analysis much more tractable than, say, Molico (2007), despite random matching in the DM. In fact, solving this model basically reduces to solving a sequence of twoperiod problems, similar to OG models, except rather than killing off agents after two periods, we let them continue, using the CM to start with a clean slate in the next DM.
(11.) This is a big difference between searchbased and Walrasian markets: since agents here are trading with each other, it is obvious that $p\le z$ is simply a feasibility constraint. While a similar constraint holds in OG models, one cannot interpret it this way because agents are trading with their budget lines, not each other.
(12.) This is because old agents always spend all their money in OG settings, while here buyers in the DM can save some for the next CM and never spend more than it takes to get ${q}^{\ast}$ in the DM.
(13.) One can interpret $a$ as a claim on “trees” and $\rho $ as “fruit” following standard assetpricing theory based on Lucas (1978). Such a real asset was introduced into in a New Monetarist model by Geromichalos, Licari, and SuárezLledó(2007); subsequently, others have added capital, housing, bonds, and others to the mix.
(14.) While Lester, Postlewaite, and Wright (2012) endogenize the $\alpha $s—that is, the acceptability of different assets—using private information, for the experiments here we take them as parameters. Still we emphasize that it would be difficult at best to talk about acceptability in models based on Walrasian markets, but it is natural in models where agents trade with each other.
(15.) It can be interesting to allow different ${\chi}_{a}$ and ${\chi}_{m}$ for the two assets, both less than $1$, even though the focus here is on ${\chi}_{m}=1$. In the informationbased approach of Li et al. (2012), for example, ${\chi}_{j}$ is smaller when it is easier to misrepresent the value of asset $j$, and hence ${\chi}_{m}<{\chi}_{a}$ emerges if it is easier to counterfeit $m$ that $a$, which includes the possibility of passing bad checks if one thinks about money broadly to include demand deposits.
(16.) This way of formalizing debtors’ incentive problem is similar in spirit to the “cash diversion” models of Biais, Mariotti, Plantin, and Rochet (2007) and DeMarzo and Fishman (2007), although that is somewhat of a misnomer, since those models do not really have “cash” (and neither do we, for now, but see later discussion).
(17.) Gu et al. (2013b) show the outcome depends on the mechanism determining the terms of trade in models with unsecured credit: interesting dynamics emerge with Walrasian pricing, and with generalized Nash bargaining if the debtors bargaining power is $\theta \in \left(\mathrm{0,1}\right)$, but not if $\theta =1$. Hence, we prefer to avoid the takeitorleaveit offers used earlier and therefore impose Walrasian pricing in this application, but if one wants bilateral trade, one can get similar results using generalized Nash bargaining with $\theta \in \left(0,1\right)$.
(18.) See Gu et al. (2013b), Carapella and Williamson (2015) and Bethune, Hu, and Rocheteau (2018a, 2018b) for details. Carapella and Williamson (2015) is especially interesting because, while the authors do not study dynamics, they construct asymmetric equilibria where debtors that are intrinsically identical are treated differently, and, moreover, they can get default in equilibrium when they add a little private information.