Speculative Runs on Interest Rate Pegs

We analyze a new class of equilibria that emerges when a central bank conducts monetary policy by setting an interest rate (as an arbitrary function of its available information) and letting the private sector set the quantity traded. These equilibria involve a run on the central bank's interest target, whereby money grows fast, private agents borrow as much as possible against the central bank, and the shadow interest rate is different from the policy target. We argue that these equilibria represent a particular danger when banks hold large excess reserves, such as is the case following periods of quantitative easing. Our analysis suggests that successfully managing the exit strategy requires additional tools beyond setting interest-rate targets and paying interest on reserves; in particular, freezing excess reserves or fiscal-policy intervention may be needed to fend off adverse expectations.


Introduction
Until the last few years, most central banks (CBs) around the world conducted monetary policy by setting targets for short-term interest rates. Maneuvering interest rates as a way to achieve low and stable inflation is now regarded as a success story, and it is widely expected that it will return to be the dominant tool of monetary policy as soon as the economy and inflation recover enough to warrant moving away from the zero lower bound on nominal interest rates.
The aftermath of quantitative easing implies subtle differences for interest-rate management that have however potentially dramatic implications for the control of the price level. Taking the Federal Reserve System as an example, before 2008, day-to-day implementation of a given interest-rate target was entrusted to open-market operations undertaken by the trading desk of the Federal Reserve Bank of New York; the trading desk retained full control of the quantity of monetary base available for transactions. In the aftermath of quantitative easing, the monetary base is much larger than what is demanded purely for transaction reasons, and, during the period of exit, control of interest rates is expected to be achieved by setting a price, the interest paid on bank reserves. By setting an appropriate interest on reserves, it is expected that banks will not attempt to lend out their funds on deposit at the CB, which could unleash inflation if all private agents tried to convert their (large) money holdings into consumption at once. This strategy acknowledges that the Federal Reserve System does not have direct control over the quantity of excess reserves that will be converted into cash. In this paper, we argue that purely setting interest on reserves is an insufficient tool to achieve price stability; we show that this policy is subject to "runs." A CB that persisted using simply interest on reserves as its policy tool in the face of a run would face hyperinflation. Other, more likely exit scenarios in such adverse circumstances involve freezing excess reserves or fiscal-policy intervention; these scenarios deserve thus further attention.
We conduct our analysis in a simple environment that features flexible prices and a standard cash-in-advance constraint, where the intuition for our results is simple and transparent; however, our results would extend to models with frictions. In this setup, we introduce a CB that sets the one-period interest rate; this interest rate need not be fixed, but rather may depend in arbitrary 1 ways on all the information that the CB has at the moment it makes its decision. The private sector is free to choose quantities traded with the CB, up to a limit. In the case of interest on reserves, this limit is zero: banks cannot hold negative reserves. More generally, the CB could (and does) allow borrowing, but this is limited, typically by collateral requirements. We show that setting a policy rate in this way leads to multiple equilibria. Some of the equilibria are familiar and common to the environments where limits to money growth are not considered. 1 However, new equilibria emerge, where money growth and inflation are higher. These equilibria involve a run on the CB's interest target: the private sector borrows as much as possible from the central bank, money in circulation grows fast, and the shadow interest rate in the private market is different from the policy rate.
In our environment, the severity of a run is affected by the size of the trades that the private sector can undertake against the CB. In the case of quantitative easing and interest on reserves, this is determined by the size of the CB's balance sheet. More generally, if government bonds are an important source of collateral to borrow from the CB, fiscal policy plays a prominent role in defining the characteristics of equilibria that feature runs. 2 This is a new channel by which excessive deficits affect price stability, and it is independent of the familiar unpleasant monetarist arithmetic of Sargent and Wallace [16] and of the fiscal theory of the price level (Leeper [12], Sims [17], Woodford [19]). In fact, we deliberately rule out these alternative channels of monetary-fiscal interaction by postulating fiscal rules that ensure long-term budget balance independently of the path of inflation.
In an extension of our model, we consider what happens if the central bank sets interest rates in a (possibly narrow) sliver of the market, rather than standing ready to buy and sell a large swathe of securities at a set price. When no run occurs, we show that the equilibrium remains the same independently of the size of the market in which the central bank operates. But, if a run 1 Examples of these equilibria are those identified by Benhabib, Schmitt-Grohé and Uribe [5,6] and those discussed in Cochrane [8]. In those equilibria, the Fisher equation linking interest rates and expected inflation remains valid, while the speculative runs that we identify involve high inflation and severe monetary distortions coexisting with low (official) nominal interest rates. 2 Arguably, the size of a CB's balance sheet is a measure of "in-house" fiscal policy run by the monetary authorities, since it involves managing the magnitude of the CB's interest-bearing liabilities.
2 occurs, the consequences are more limited the more circumscribed this market is. This suggests a rationale for why central banks may find it attractive to set targets only for very short-term interest rates, but refrain from doing the same for a broad spectrum of the yield curve.
Our model sheds light on two historical episodes. In the more extreme case, the policy of the Reichsbank during the German hyperinflation fits well within our model. As mentioned by Sargent [14], the German Reichsbank discounted Treasury and commercial bills at fixed nominal interest rates in 1923; these rates were far too low to equilibrate loan markets given expected inflation, and a run of precisely the type that we describe occurred: the policy added fuel to the hyperinflation by causing the Reichsbank to greatly increase the money supply and transferring this money to the government and to those private entities lucky enough to borrow from the Reichsbank at the official discount rate.
In a more relevant example for the current situation, the Federal Reserve System successfully managed an "exit strategy" from quantitative easing once before. During the Great Depression, commercial banks accumulated sizeable excess reserves deposited with the Federal Reserve, that lasted through the early 1950s. In the 1940s, up to the Treasury accord of 1951, the Fed managed monetary policy by pegging interest rates. However, at various points this led to inflationary tensions. As discussed by Eichengreen and Garber [10], the Fed did not rely purely on interest rates to subdue them, but it rather adjusted required reserves, an instrument that offered direct control over the quantity of funds that would be available to start a full-blown speculative attack.
In this light, our analysis suggests a new role for the "twin-pillar" doctrine of paying attention to The description above assumes that the CB sets its interest rate as the discount factor on government bonds, as was the case for the German Reichsbank. The same equations apply in the case of modern central banks in the aftermath of quantitative easing, simply by relabeling variables appropriately. Specifically, in this case −T t represent seigniorage payments from the central bank to Treasury (that are in turn rebated to households), and R t is the interest on excess reserves; in this case, "money" represents cash or required reserves that banks must hold if they expand their deposits, and "bonds" are excess reserves that do not provide liquidity services and are held only if they offer the same return as competing private assets. 4 In what follows, we will continue to use the "Reichsbank labels," except when deriving remarks that specifically apply to interest on reserves.
After the asset markets, a goods market opens. In the goods market, households produce the consumption good using their own labor for the use of other households (but, as usual, not their own household) and the government. Each household has one unit of time and a constantreturns-to-scale technology that converts units of time into units of the consumption good one for one. Households use money to purchase units of the consumption good produced by other households. The government uses either money or bonds (it is immaterial which) to purchase G t = G ≥ 0 units of the consumption good. Figure 1 summarizes the events within each period. 5 4 When we follow this interpretation, we still separately assume that Treasury taxes and debt are set so that the fiscal theory of the price level does not apply. 5  gov't purchases G t Figure 1: Timeline of events within period t.
Let M t denote the amount of money in circulation at the end of the asset market in period t, after taxes are paid. Let B t−1 be the nominal amount of government bonds payable at date t. (If B t−1 < 0 then it represents a debt that households owe the government at date t.) The households start with initial nominal claims W −1 against the government. 6 Consider a price sequence {P t , R t ,R t } ∞ t=0 , where P t is the nominal price of a unit of the consumption good at date t, R t is the nominal risk-free rate between period t and t + 1 at which the government trades with private agents, andR t is the rate at which households trade with with the initial condition In what follows, we use lower-case letters to indicate individual household choices and uppercase variables to indicate aggregates: as an example, m t are individual money holdings, and M t are aggregate money holdings. In equilibrium, lower and upper-case variables will coincide, since we consider a representative household.
Households are subject to a cash-in-advance constraint: their consumption must be purchased with money. A household's path is given by {c t , y t ,b t , b t , m t } ∞ t=0 , whereb t are holdings of privately- 6 These claims represent money and maturing bonds, before paying period 0 taxes.

5
issued bonds maturing in period t + 1. 7 In addition, households are potentially constrained in their holdings of government securities to a set B t . We will first explore the case in which B t is the entire real line, and we will then explore the implications of setting a limit to private indebtedness against the government.
A household path is feasible if for all t > 0 b t together with the initial condition b 0 and the no-Ponzi condition Equation (6) imposes that households cannot borrow more than the present value of working 1 unit of time while consuming nothing, holding no money in every period after t, and maximally exploiting any price discrepancy between government-issued and private securities. This present value is evaluated at the sequence of intertemporal prices {R s } ∞ t=0 . When B t = R, a no-arbitrage condition will ensureR t+j = R t+j , making the corresponding term disappear from (6). When limits to household indebtedness against the government are present, we will study equilibria where government securities have a different price than equivalent privately-issued securities, in which case household can profit from the mispricing (at the expense of the government), and the corresponding profits are part of their budget resources. 8 Facing 7 In equilibrium,b t ≡ 0. 8 Of course, in equilibrium the aggregate profits of the households from this activity are matched by lump-sum taxes that the government has to impose, so that in the aggregate this limited arbitrage opportunity is a zero-sum game.
, and given initial nominal wealth, a household's problem is to choose subject to (3), (4), (5), (6), and b t ∈ B t . We assume that u is continuously differentiable, that both consumption and leisure are normal goods, and that the following conditions hold: and Equation (8) is a standard Inada condition; it will ensure an interior solution to our problem.
Equation (9) imposes that the marginal disutility of labor is bounded away from zero in equilibria in which production is also bounded away from zero.

An interest rate policy
In this section, we construct equilibria for an economy in which the government/monetary authority sets an interest rate rule, without imposing limits to household trades with the central bank (i.e., B t = R) . In particular, suppose the central bank offers to buy or sell any amount of promises to pay $1 at date t + 1 for 1/(1 + R t ) < 1 dollars at date t. We assume that nominal interest rates remain strictly positive (R t > 0); this is purely to save notation. In the equilibria featuring runs that are the object of study in this paper, the cash-in-advance constraint will always be binding, even if the policy target is R t = 0.
Using Svensson and Woodford's [18] language, the interest-rate rule is here used as a reaction function: the central bank adopts the interest rate as its instrument, and sets it as a function of everything that is observable up to that point in time. We allow for arbitrary history dependence, so in particular this assumption encompasses Taylor rules that depend on past inflation. In section 4.2 we discuss the role of this assumption in the broader context of alternatives, and we also explain why it may be particularly appropriate in the wake of the policy of quantitative easing pursued by many central banks across the developed world in recent years.
We suppose that the government sets a "Ricardian" fiscal rule, i.e., a rule that ensures that the present-value budget constraint of the government (and hence the transversality condition of the agents) holds whenever all other competitive equilibrium conditions are met, independent of the price level. We choose such a fiscal policy because we are interested in the set of equilibria that can arise when money is not directly backed by tax revenues, as it happens instead when the fiscal theory of the price level holds. We will specify below a class of fiscal rules that satisfies sufficient conditions for this requirement.
solves the household's problem taking {P t ,R t , R t , T t } ∞ t=0 as given, and such that markets clear for all t ≥ 0: andB In order for the household problem to have a finite solution, it is necessary that the prices of government and private assets be the same:R When (12) fails, households can exploit the difference in price to make infinite profits. In addition to (6) and (12), necessary and sufficient conditions from the household optimization problem yield the following conditions for all t ≥ 0: and the transversality condition Substituting (10) and (12) into (13), we obtain We now turn to constructing equilibria. The initial price level, P 0 , is not determined. For each initial price P 0 , one can use the interest rate rule R t and equations (1), (2), (10), (14), (15), and (17) to sequentially solve for a unique candidate equilibrium allocation and price system. 9 That is, first the fiscal policy rule determines T 0 . Given T 0 , the interest-rate rule determines R 0 , equation (17) solves for C 0 , equation (10) then implies Y 0 , and equation (15) implies M 0 .
Finally, equation (2) determines B 0 . With all time-0 variables now determined, the fiscal policy rule determines T 1 , the monetary policy rule determines R 1 , which by no arbitrage is equal tô As in period 0, equation (17) solves then for C 1 and equation (10) for Y 1 .
Knowing C 1 and Y 1 , equation (14) can be solved for P 1 , and equation (15) for M 1 . Equation (1) then yields B 1 , and from there the process continues to period 2 and on.
To verify whether the candidate equilibrium allocation and price system we derived above is an equilibrium, we need only to check that the household transversality and no-Ponzi conditions (6) and (16) hold. To this end, we will restrict fiscal policy to a (broad) class which ensures the policy is Ricardian (Assumption A2); but a necessary step to do so is to ensure that the present-value of seigniorage remains finite. For now, we achieve this by imposing an upper bound on nominal interest rates: The appendix studies alternative ways of ensuring that the present value of seigniorage remains finite even when Assumption 1 is violated. 10 9 The Inada condition and the assumptions of normal goods ensure that an interior solution can be found and that (17) is strictly monotone in C t . In our analysis, we do not rule out explosive paths, for the reasons highlighted in Cochrane [8]. 10 In the cases considered in the appendix, it may not be possible to find equilibria with a perfectly anticipated 9 As a specific class of Ricardian fiscal policies, we assume T t satisfies Assumption 2 There exist finite B > 0 and T such that Essentially, we require that if real debt is neither too high nor too low, taxes may be any function of past information subject only to a uniform bound in real terms. But when real debt exceeds a threshold (in absolute value), taxes cover at least a fraction α of debt, putting the brakes to a debt spiral. As an example, one simple rule that belongs to this general class is We relegate the proof that (6) and (16) hold (and thus the candidate equilibrium is an equilibrium) to the appendix.
The construction above establishes results that are well known from Sargent and Wallace [15], Woodford [20], and reemphasized by Cochrane [8]. Under an interest rule, the initial price level P 0 is indeterminate, but, once a value of P 0 is specified, there exists a unique deterministic equilibrium allocation and price system. Moreover, in the deterministic equilibrium, in any period in which the nominal rate set by the central bank is low, so is inflation. As an example, if R t = 1 β − 1 for all t ≥ 0, then inflation is exactly zero in all periods. When uncertainty is present, sunspot equilibria arise; we discuss such equilibria in the appendix. But, even in that case, a low official interest rate translates into a limit on expected inflation. To see this, note that the intratemporal optimization condition (13) and the market clearing condition (10) still hold in a world with sunspots, so equation (17) still holds. Thus consumption and labor in each period are pinned down by the interest rate policy. If R t is constant then consumption and labor are constant. If R t = 1 β − 1, the stochastic version of the consumption Euler equation becomes run on the central bank's interest rate peg, such as the one we will study in section 4, but there will instead be equilibria where runs occur with positive probability.
The expected real value of a dollar remains constant into the future. Furthermore, if we assume a bound on how fast the price level can drop (i.e., we impose P t /P t+1 < 1/ almost surely ∀t), then the law of large numbers will apply, and average inverse inflation over long horizons will be 0: In the next section, we show that a very different type of equilibrium emerges when households are not allowed to borrow unlimited funds from the central bank. In this equilibrium, a low interest rate set by the central bank is accompanied by high expected inflation from the outset, and private-market interest rates diverge from the policy rate.

Limits to Central Bank Lending
Suppose now we impose the additional constraint on the households that B t ≥ 0, t ≥ 0: households are not allowed to borrow from the government/central bank (or, equivalently, they are allowed to borrow from the central bank only by posting government bonds as collateral). That the borrowing limit is precisely zero is not central to our analysis, but simplifies exposition somewhat. In this section, we construct additional deterministic equilibria which do not exist when With the no-borrowing limit we just imposed, the official rate R t only becomes a lower bound for the private-sector rateR t . When households are at the borrowing limit with the central bank, private nominal interest rates may exceed the official rate. The no-arbitrage condition All other equilibrium conditions remain the same, except that the private rateR t replaces the government rate R t in equation (17): The allocation of section 3 remains part of an equilibrium even when the central bank limits its lending, provided that households have nonnegative bond holdings in all periods. For a given sequence of prices, interest rates, consumption and work levels, household holdings of government debt in this equilibrium depend on the sequence of taxes. Government debt will be strictly positive in each period t > 0 if and only if the following condition is satisfied: whereĉ(R) is the consumption implied by equation (21) It is straightforward to see that there are fiscal rules that satisfy (22) and Assumption 2. 11 We assume that fiscal policy is run by one such rule.
In period 0, government debt will be nonnegative if An interior equilibrium will only exist if which we will assume. While P 0 can take any positive value in section 3, now equation (23) imposes a ceiling.

Additional Equilibria: A Single Run
The simplest equilibrium that may arise when a limit to private indebtedness is introduced is a deterministic run, where B s = 0 for a single date s > 0. In the case of the German Reichsbank, this is an equilibrium in which all of the government debt is monetized in period s. Under the QE interpretation, this is an equilibrium in which households demand enough cash (and, in a richer model, banks expand their deposits so much) that all of the excess reserves are converted into cash (and required reserves).
The conditions under which such a simple equilibrium exists are stringent. This is not surprising: it is true in all models of runs. To use a fixed exchange-rate regime as an example, 11 As an example, and α ∈ (0, 1). equilibria in which a fixed exchange rate collapses at a perfectly anticipated time exist only in very specific circumstances. In the appendix, we accordingly extend the analysis to probabilistic runs, where the date at which inflationary expectations take off and a run occurs is not perfectly known ahead of time. Such equilibria exist under much more general conditions.

Assumption 3
Define We assume that fiscal policy satisfies the following stronger version of (22): where u y (Ḡ) is defined in (9).
Equation (22) guaranteed that in each period there are positive bonds/excess reserves that can be converted into money and initiate a speculative run. The stronger condition (25) ensures that, after a period in which a run occurred and thus previous government debt was monetized, there are enough new bonds (or the monetary base is sufficiently large) that the economy can return to a path where households hold positive amounts of government debt and equation (14) holds. This assumption is required for the run to last a single period.
be determined as in the equilibrium of section 3, with P 0 satisfying (23), and let fiscal policy satisfy Assumption 2. A necessary and sufficient condition for the existence of a different (deterministic) equilibrium in which B s = 0 is that the following equation admits a solution forR s > R s : A sufficient condition (based on preferences alone) for (26) to have a solution withR s > R s is Proof: The proof works by construction. Starting from an arbitrary price level P 0 that satisfies (23), the equilibrium allocation, price system, and government policy are solved as in section 3 up to period s − 1. Specifically, we use the interest rate rule R t and the fiscal policy rule with equations (10), (14), (15), and (17) to sequentially solve for the unique candidate equilibrium allocation and price system.
In period s, in order forR s > R s to be an equilibrium, the constraint B s ≥ 0 must be binding, Furthermore, equations (14) and (21) require Substituting (28) into (29), we obtain (26) From period s+1 onwards, the allocation and price system is once again uniquely determined (sequentially) by the interest rate rule R t , the fiscal policy rule, and equations (10), (14), (15), and (17). Equation (25) ensures that the resulting sequence for government debt is strictly positive. Once again, the proof that (6) and (16) hold is relegated to the general proof in the appendix.
Given these assumptions, one equilibrium of this economy is a steady state: In each period When all other households are expected not to roll over their debt, each household expects thus a high money supply and high resulting inflation; in response to this expectation, the optimal strategy is not to roll over nominal debt, validating the run.
To go beyond this basic intuition and understand why a perfectly anticipated run requires specific assumptions about preferences, we inspect the evolution of the run in greater detail.
First, notice that, if a run occurs, the private-sector interest rateR t must be greater than the interest rate set by the central bank, which is constant at 1/β − 1; in our example, this occurs in period 2, as shown in the left panel of Figure 2. The intratemporal optimality condition (21) implies that consumption decreases in period 2, when the run occurs (right panel of Figure 2).  With consumption down and the money supply up, the price level must jump up so that the (binding) cash-in-advance constraint holds, as shown by the left panel of Figure 3. Whether such a candidate allocation can be supported as an equilibrium depends on whether these changes can be made consistent with the household Euler equations for leisure and consumption, which are respectively (14) and Specifically, in order to have a perfectly anticipated run in period 2, and not before, it must be the case that households are willing to lend to the government in period 1 (or, to keep excess reserves deposited at the central bank, depending on the interpretation) even though the nominal interest rate by the central bank is constant and expected inflation between period 1 and period 2 is high. Since households expect a consumption drop between periods 1 and 2, this can be the case, but only if either the drop in consumption (and, by market clearing, in the labor supply) is very steep or the intertemporal elasticity of substitution of consumption is sufficiently low.
Equation (21) implies that the consumption drop is steeper, the less curvature there is in the marginal disutility of labor and in the marginal utility of consumption. So, less curvature in u y (c, c + G) unambiguously helps in satisfying equation (32). Less curvature in u c (c, c + G) has an ambiguous effect, since (for givenR s ) it creates a bigger drop in consumption, but it also implies a greater intertemporal elasticity of substitution. The second effect turns out to be the relevant one, so that a perfectly anticipated run can happen when the curvature is low and hence the functionĉ is not very responsive to R. From these observations, we can thus understand the role of assumption A2. We can also understand why a run can happen under much weaker assumptions if it occurs with probability smaller than one, as described in the appendix: in this case, the potentially negative effect of a run on the households' willingness to save between periods 1 and 2 is tempered by the lower probability of the occurrence. In the limit, as the probability of a run goes to 0, households are content to save at the rate 1/β − 1 between periods 1 and 2 when the no-run allocation remains at the steady state throughout.
Next, we consider the other intertemporal choice that households face in their decision to save between periods 1 and 2, i.e., their labor supply. Because of the cash-in-advance timing, this decision is related to the household labor supply in periods 0 and 1, as shown by equation (14).
Since the allocation and inflation are at the no-run steady state values in these two periods, the relevant Euler equation for leisure is automatically satisfied. For this reason, the intertemporal elasticity of substitution of leisure does not play the same role as the one of consumption in determining whether a perfectly anticipated run can occur.
Having discussed the economic forces that lead households to save between periods 0 and 1, we next consider the elements that pertain to the private-market interest rate between periods 1 and 2, in the period of the run. This time, it is simpler to start from the Euler equation for labor, equation (14). The relevant margin of choice for households is their labor supply in period 1 (paid in period 2) vs. period 2. Here, it is straightforward to see why households optimally choose not to invest in government bonds in period 2 at the nominal rate 1/β − 1. First, the nominal wage (which is equal to the price level) increases from period 1 to period 2, which yields an incentive to postpone labor when the nominal interest rate does not adjust correspondingly.
Second, the equilibrium features actually a lower labor supply (which tracks consumption) in period 2 than in period 1, providing a further incentive not to save in period 1 and to postpone work. Both of these channels imply that the interest rate offered by the government within the equilibrium allocation is too low for households to be willing to lend to the government, and that the private-market interest rate that justifies the labor decision is instead higher. Similarly, on the consumption side (where the relevant margin is once again shifted one period forward), households look forward to an increase in consumption between periods 2 and 3, and hence they require a higher real interest rate to be willing to save than the one offered by the government.
This is particularly true because further inflation occurs between periods 2 and 3, as we establish next, in our discussion of how the run ends. After the run ends, households resume lending to the government at the rate R 3 = 1/β − 1 in period 3. With a fixed nominal interest rate, inflation between period 2 and 3 must adjust so that households find it optimal to increase their labor supply between the crisis period 2 and the return to normalcy in period 3. By equation (14), this requires further inflation between periods 2 and 3. The increase in both prices and production (and consumption) between periods 2 and 3 implies that money supply must also grow. Since the crisis wiped out government debt, households cannot acquire this additional money by selling government debt. If the run is to last a single period, fiscal policy must generate enough new nominal liabilities at the beginning of period 3, as implies by Assumption 3; this is achieved through a tax cut. From that point onward, output and consumption return to their pre-run steady state, while government debt (in real terms) converges back to the steady state gradually.

Other Equilibria, and Alternative Government Strategies
By repeating the steps outlined in section 4.1, it is easy to construct equilibria in which runs occur repeatedly, and it is also possible to construct equilibria in which runs last for more than one period. The conditions under which such equilibria exist are similar to those for a single run (in particular, Assumptions 1, 2, and 3 are sufficient conditions). In more general cases, the appendix considers stochastic equilibria, where runs can emerge with probability less than 1. In these stochastic run equilibria, even when the official interest R t is constant, the levels of consumption and labor are not constant because the effective interest rate in the household optimization conditions,R t , is not constant. Further, when R t = 1 β − 1, it is no longer the case that on average, P t /P t+1 = 1. Setting a low nominal rate no longer guarantees low average real depreciation of the currency.
In the simple example above, we assumed that the central bank sets an unconditional interest rate peg. However, our results hold even when interest rates exhibit arbitrary dependence on the past; in particular, Taylor rules that depend on past inflation fit our framework well. 15 Our results would of course change if the interest rate is not simply set by the central bank, by standing ready to trade at the given rate, but is instead simply a target to be attained through different, potentially more complicated (and unspecified) rules. Following Svensson and Woodford [18], central banks in the past may have adopted interest-rate rules as "targeting rules," but may have then implemented those rules in different ways. 16 Indeed, until 2008, most central 15 In the case of "active" Taylor rules, Assumption 1 (that nominal interest rates are bounded) is violated, which implies that a deterministic run would not exist. However, stochastic runs would continue to exist if preferences are such that the present value of seigniorage revenues remains finite, as discussed in the appendix. 16 As an example, Atkeson et al. [3], following the methods in Bassetto [4], devise more sophisticated strategies to achieve unique implementation by reverting to money supply rules when the inflation rate deviates from its target. But, as is well known (see e.g. Woodford [19]), money supply rules may also be subject to multiple equilibria. Alternatively, uniqueness can be attained by strategies where currency is explicitly backed by fiscal revenues, as in the fiscal theory of the price level.
banks did not set a fixed rate at which they were willing to trade with the private sector, but rather they relied on controlling the monetary base day to day through open-market operations to achieve their target; this might be the reason we have not observed any of the runs described here in the recent past.
If the interest-rate rules adopted by central banks in the past were mere targeting rules, and not true reaction functions, our analysis still provides several new insights: • A more complete specification of the low-level reaction function is essential to understand how central banks successfully kept inflation in check and prevented runs. Most likely, this did not involve passively supplying the money required to achieve the interest rate target, which would be equivalent to the strategies described below, but would instead be closer to the "twin-pillar doctrine," as discussed in Lucas [13].
• The advent of quantitative easing may have created a danger of runs that was not previously present. Since 2008, the large amount of excess reserves held by commercial banks has implied that the chief instrument to attain the interest-rate target is the rate paid by the central bank on excess reserves. Paying interest on reserves is also an essential element of the planned exit strategy, while central banks gradually reduce the size of their balance sheet (see e.g. Bernanke [7]). The strategy of paying interest on reserves is well captured by our section 4.1 if we simply reinterpret B t as the central bank's own interest-bearing debt (excess reserves), rather than the entire stock of government debt. By relying on intereston-reserves as its primary tool to achieve the interest-rate target, a central bank stands ready to exchange cash for reserves at the given interest rate (which can be a function of anything that the central bank has observed in the past): this is precisely the strategy that creates the possibility of a run of the type that we discussed.
• The Federal Reserve System faced exit from a situation of large excess reserves held by the banking sector once before, in the 1940s. Eichengreen and Garber [10] argue that the Fed controlled liquidity during those years by changing reserve requirements, which then allowed it to stabilize inflation expectations and therefore support stable interest rates.
To the best of our knowledge, changes in reserve requirements have not been cited as one of the tools that will be adopted during the exit from quantitative easing. Our analysis suggests instead that they would be an essential tool in preventing runs on the interest rate set for excess reserves and the associated inflationary consequences.

Debt and the Severity of Runs
In Rather than relying on the fiscal authorities to restrict the pool of available bonds, an alternative strategy for the central bank to mitigate the consequences of a run is to peg rates on only a subset of the bonds. In practice, this is a relevant scenario for at least two reasons: 1. In the real world, there is long-term debt, whose price is not directly targeted by the central bank; 2. It is unlikely that the central bank would be willing to monetize the entire amount of government debt; rather, the bound after which a CB would stop accommodating a run is likely to be tighter.
Here, we consider the case in which there are two types of bonds, "red" bonds and "blue" bonds, both with one-period maturity, whose only difference stems from their treatment by the central bank. 17 We assume that, when asset markets open, the central bank sets the interest rate on red bonds, being willing to purchase or sell them at a rate R t (which may depend on past history, as before). In contrast, blue bonds are auctioned. From the fiscal perspective, red bonds and blue bonds are identical: both constitute a promise to deliver a dollar to the holder at the beginning of the subsequent period. We assume that taxes are set according to a fiscal policy rule that satisfies Assumptions 2 and 3, where B t refers to the total amount of bonds (red and blue). In addition, we need to specify a rule that describes the supply of blue bonds at auction, as a function of past history. Letting B B t be the amount of blue bonds being auctioned in period t and maturing in period t + 1, we assume that this rule satisfies the following assumption: 18 It is straightforward to prove that Assumption 4 is sufficient for the existence of an interior equilibrium, in which private agents hold a strictly positive amount of red bonds. The allocation and price system in this equilibrium coincides with the one computed in Section 3. In this equilibrium, blue bonds, red bonds, and privately-issued bonds are perfect substitutes from the household perspective, and trade at the same interest rate. That the central bank targets a narrower segment of the bond market is thus immaterial for its ability to control inflation and real activity.
In the event of a run, the presence of blue bonds makes a difference. Households again perceive blue bonds, red bonds, and privately-issued bonds as perfect substitutes. But if a run occurs in period t, the interest rate R t sanctioned by the central bank for red bonds is lower than the private-sector rateR t , and consequently households do not buy any red bonds. At the same time, if a positive amount of blue bonds is offered at auction, households will bid for them, at the interest rateR t . The evolution of money supply in period t will thus be governed by the following equation: 19 Ceteris paribus, the sale of blue bonds reduces the monetization of maturing government debt, alleviating the consequences of the run. We can illustrate this point using our numerical example once again. Let all the parameter values, the initial conditions, and the rules for T t and R t be those of Section 4, but assume that, in each period, blue bonds are supplied according to the following rule: B B rates, rather than trying to impose an entire yield curve on the market. Even in recent times, when several central banks have tried to affect the yield curve by policies of "quantitative easing," it is noteworthy that they chose to do so by setting an interest rate target for the short end, and a quantity target for their purchases of longer-term securities. 21 (It is also noteworthy that the Fed's attempt to peg the entire yield curve in the 1940's ultimately led the Fed to be the sole purchaser of short-term Treasury debt.)

Discussion
In We should also emphasize that recent large excess reserves by banks have the potential of making mitigating such runs more difficult. Put simply, it is one thing for the central bank to stop buying the debt of the fiscal authority to stop a run. It is altogether a different thing for a central bank to refuse to honor its own debts. Suddenly increasing the reserve requirements of banks to fight a run may be seen as the central bank refusing to honor its commitment to deliver currency on demand to the holder of the reserve.
Can such runs actually happen? In the simple setup that we described, in the event of a run, households force the central bank to its bound in a single period. In practice, the unfolding of a run would be slowed by a number of frictions that may prevent all households from running at once with all of their nominal wealth; these frictions may take the form of limited participation in bond markets (see e.g. Grossman and Weiss [11], Alvarez and Atkeson [1], and Alvarez, 21 In our simple model, of course, quantitative easing would have no effect on the equilibrium allocation and prices. But our results would apply equally well to richer environments where a preferred habitat is present.

24
Atkeson, and Edmond [2]), noisy information about other households' behavior, or the presence of long-term bonds whose price is not pegged by the central bank. We leave the modeling of more slowly evolving runs to future research.

APPENDIX
A Analysis of the General Stochastic Case

A.1 The Environment with Sunspots
We modify the environment described in section 2 by introducing a sunspot variable s t in each period. Without loss of generality, s t is i.i.d. with a uniform distribution on [0, 1]. Its realization at time t is observed before any action takes place. All variables with a time-t subscript are allowed to be conditional on the history of sunspot realizations {s j } t j=1 . We assume that the government only trades in one-period risk-free debt, but we allow the households to trade state-contingent assets, and we denote by a t+1 the amount of nominal claims that a household purchases in period t maturing in period t + 1 (conditional on the sunspot realization s t+1 ). Without uncertainty, a t+1 ≡b t . Equation (3) is thus replaced by where Q t+1 is the stochastic discount factor of the economy. For the later analysis, it is convenient to defineR t := 1/E t Q t+1 − 1. This definition is consistent with the notation that we used in the main text for the deterministic case:R t is the one-period nominal risk-free rate in the market for private credit.
In period 0, the household budget constraint becomes The no-Ponzi condition (6) generalizes to With these changes, an equilibrium is defined as in section 3; the market-clearing condition (11) becomes 26 The conditions characterizing an equilibrium are given by (10), (15), (21), (38), the stochastic Euler equation the transversality condition, which in the stochastic case becomes 22 and finally the no-arbitrage condition for interest rates. This last condition statesR t = R t when (20) when B t ≥ 0 is imposed.
In the main text, we adopted Assumption 1 to ensure that seigniorage revenues remain bounded and hence that the present-value budget constraint of the households is well defined.
When Assumption 1 is violated, such as in the case of Taylor rules that have no upper bound on the interest rate, an alternative (sufficient) condition that we can adopt is given by When Assumption 5 is adopted, often perfectly anticipated runs will fail to exist (but probabilistic runs will continue to occur). We prove this proposition in 3 steps. First, we prove that A t+1 , as defined in (37), is well defined. Second, we prove that (40) holds, and finally that (37) holds.

A.2 Verification of the Transversality and no-Ponzi conditions
22 See Coşar and Green [9].

A.2.1 A t+1 is well defined.
We work backwards on the individual components of the sum defining A t+1 in equation (37).
From (20) we obtain 23 Next, use (39) to get which proves that the second piece of the infinite sum defining A t+1 is well defined. From Assumption 2, we have |T t+j+1 | ≤ T P t+j + |B t+j |, and so We analyze equation (45) in pieces. Using (44), we have 23 If the borrowing limit is not 0, the expression in (42) would not be 0, but it can be proven that A t+1 is nonetheless well defined. 24 We can interchange the order of the sum and the expectations since all elements of the sum have the same sign.
To work on the sum of debt, notice first that equation (1) continues to hold even if we replace R t byR t . This is because B t = 0 in the periods and states of nature in whichR t > R t . If We can then use (1), (15), (39), and (47) to get Using (43) and (48), we obtain (for j > 1) 29 Collecting all terms, equations (44), (46), and (50) imply (51)

A.2.3 Equation (37) holds.
The same steps used to prove (52) can also be used to prove As previously noted, equation (1) continues to hold even if we replace R t withR t , since the two values only differ when B t = 0. We can then iterate (1) forward, taking expectations conditional on time-t + 1 information, and use (54) to obtain which completes the proof. Equation (55) relies on G < 1 (government spending must be less than the maximum producible output) and on This completes the proof of proposition 2.

B Other Equilibria of the Stochastic Economy
The perfectly anticipated run described in section 4.1 relies on strong assumptions about preferences. As an example, if we assume that preferences are given by u(c t , y t ) = c 1−σ 1−σ − y ψ , such an equilibrium will always fail to exist for σ ≤ 1, since a solution to (26) cannot be found (witĥ R > R). Nonetheless, even for these preferences other equilibria that feature runs exist, provided that the occurrence of a run is sufficiently small. Moreover, these equilibria exist even when the central bank sets no upper bound to its interest rate (provided, of course, that preferences are such that the present value of seigniorage remains finite).
As is known since Sargent and Wallace [15], even without considering runs, setting monetary policy as an interest rate rule leaves open the possibility of sunspot equilibria. But equilibria with runs are qualitatively very different from these sunspot equilibria. In a standard environment where B = R and no runs can occur, the nominal interest rate is closely related to expected (inverse) inflation, so that setting the nominal interest rate still allows the central bank a considerable degree of control, at least over long periods of time. This relationship between nominal interest rates and expected inflation is lost in equilibria that feature runs, and the dangers from relying purely on the nominal interest rate as a policy instrument are correspondingly more acute.

B.1 Sunspot Equilibria with no Runs
We can construct sunspot equilibria recursively as follows. For any arbitrary initial price P 0 , the variables R 0 , T 0 , C 0 , Y 0 , M 0 , and B 0 are determined as in section 3. The time-0 variables and the policy rules determine R 1 and T 1 , also as in section 3, which then pin down C 1 and Y 1 ; this implies that C 1 and Y 1 are known as of period 0. 25 But now the deterministic Euler equation (14) is replaced by its stochastic counterpart, (39). In an equilibrium with no runs, we know thatR 1 = R 1 . Substituting this into (39), rearranging and taking expected values we obtain We can then pick P 1 as an arbitrary function of the sunspot s 1 , subject to the single restriction (56) on its expected value. Given the realization of s 1 and thus P 1 , equation (15) determines (1) yields B 1 , and the process can be repeated for period 2.
Provided that either Assumption 1 or 5 hold, Proposition 2 ensures that the transversality and no-Ponzi conditions are satisfied for the sequences that we constructed: as discussed in Cochrane [8], in this model only fiscal policy can provide a boundary condition to rule out some of these arbitrary paths. 26 While sunspot equilibria imply that inflation is indeterminate, equilibria that feature no runs still display remarkable similarities across each other. As an example, suppose that monetary policy sets R t ≡ 1/β −1 unconditionally. It is straightforward to verify that equation (17) implies a constant allocation, and that (39) implies (18): the expected real value of a dollar remains constant. Equation (18) and the assumption of a uniform bound on P t /P t+1 in turn imply (19).
B.2 A Probabilistic Run in Period s > 0.
We now construct an equilibrium where a run occurs in period s with probability φ ∈ (0, 1).
As was the case in section 4.1, fiscal policy plays an important role in ensuring that households have enough nominal wealth to acquire their desired money balanced, and we assume that (25) holds. Starting from an arbitrary initial price level P 0 , we construct recursively a deterministic allocation and price system up to period s − 1 as we did in section 4.1. For period s, we consider an equilibrium with just two realizations of the allocation and price level: with probability φ, the price level is P H s and a run occurs (R H s > R s ), and with probability 1 − φ the price level is P L s and the private nominal interest rate coincides with the public one:R L s = R s . In order for R H s > R s to be an equilibrium, the constraint B s ≥ 0 must be binding, which implies Given any arbitrary valueR H s > R s , and given the predetermined time-s − 1 variables and the fiscal policy rule for T s , equation (57) can be solved for P H s /P s−1 , the level of inflation that will occur if a run on the interest rate peg materializes in period s. As was the case in section 4.1, sincê c is a decreasing function and taxes satisfy (22), inflation in the event of a run will necessarily be strictly greater than inflation in the equilibrium in which no run can take place.
To determine P L s /P s−1 , we rely on the household Euler equation ( Generically, this equation can be solved for P L s /P s−1 . However, we need to ensure that the solution is nonnegative, and that it entails nonnegative bond holdings, i.e., that A sufficient condition for both is that φ be sufficiently small. 27 Ifû y does not decline too fast with R, then equation (58) will imply that P L s /P s−1 is lower than in the deterministic equilibrium with no runs. Because of this, the possibility of a run may cause the central bank to under shoot inflation while the run is not occurring, further undermining inflation stability.
From period s onwards, the characterization of the equilibrium proceeds again deterministically and recursively, separately for the branch that follows P H s and P L s ; this follows the same steps as in section 4.1. The construction of the equilibrium is completed by Proposition 2 that ensures that the transversality and no-Ponzi conditions are satisfied for the sequences that we constructed.
The nature of the equilibrium that we constructed is quite different from those of section B.1.
To see this more in detail, consider again the case in which the central bank sets R t ≡ 1/β − 1 in every period. It is now no longer true that consumption is then fixed. If a run occurs, the relevant shadow cost of consumption in equation (21) is R H s , and consumption drops. This also implies that consumption is not predetermined, but it depends on the realization of the sunspot.
With the constant interest rate above, and taking into account that the run occurs in period s only, consumption is the same in periods s − 1 and s + 1 and we thus find We know that β(1 +R s ) ≥ β(1 + R s ) = 1, and the inequality is strict with probability φ. This implies 1 > E s−1 P s−1 P s+1 .
When runs can occur, setting the nominal interest rate is not sufficient to even control the expected real value of a dollar.

B.3 Recurrent Runs
We can generalize the example of subsection B.2 to construct equilibria in which runs can occur in any number of periods. As an example, there are equilibria in which runs occur with i.i.d.
probability φ in each period. Once again, we construct the allocation and price system recursively, as we did in section B.2. In each period t, the history of runs up to period t − 1 is taken as given, and (57) and (58) are used to solve for P H t /P t−1 and P L t /P t−1 . To contrast these equilibria with the usual sunspot equilibria where no runs occur, consider again the interest rule R t ≡ 1/β − 1, and assume that preferences are linear in leisure, i.e., u(c, l) = v(c) − l. In this case, equation (39) becomes 1 = βE s (1 +R s+1 ) P s P s+1 =⇒ 1 > E s P s P s+1 .
We then get lim T →∞ 1 T T s=1 P s P s+1 < 1 almost surely: if runs are a recurrent event, average inverse inflation is necessarily less than 1 over long horizons.