Unanticipated negative demand/supply shocks can result in lower levels of inflation and even deflation. In such a situation, central banks usually try to lower nominal and thereby real interest rates in order to stimulate the economy out of a deflationary spiral. At very low levels of inflation or deflation, a negative interest rate may even be required. However, since there is a zero lower bound (hereafter referred to as ZLB) on interest rates, the economy might then become caught in a liquidity trap of prolonged deflation and recession. Though the practical possibility of these circumstances was debated for decades after it was first hypothesised by Keynes (1936), the recent experience of Japan (shown in Fig. 1) has offered proof that it can indeed occur. The question of what central banks can do to escape such circumstances has therefore been transformed from a theoretical curiosity to one of practical importance to policy-makers.
There appears to be considerable agreement among researchers and policymakers regarding ways to avoid the zero lower bound and minimize the risk of hitting it. Many recommendations have been put forward including positive symmetric inflation targets which act as buffers, and forward-looking policies which enable pre-emptive actions based on forecasts. Svensson (2003) even suggests that a set of emergency measures to be prepared and declared in advance, which can be used at pre-announced indications of an imminent liquidity trap. However on the subject of how to escape from a liquidity trap there is no such consensus. There are a multitude of practical proposals to climb out of a liquidity trap: taxes on money to lower the ZLB on interest rates, money transfers to the private sectors, open market purchases, and writing options to establish a ceiling on long-term interest rates. In this paper we explore some of the main issues with formulating and implementing monetary policy in a liquidity trap, and for the sake of brevity, we discuss only proposals which appear to be the especially promising among those which exclusively use monetary instruments.
When interest rates are immobilised at zero, what can the central bank do to battle deflationary pressures? The 'reflex' theoretical answer is that it can expand the monetary base via open-market operations in Treasury bills and more unconventional assets (Benhabib, Schmitt-Grohé and Uribe, 2002; Bernanke, 2000; Clouse et al., 2003; Meltzer (2001); Orphanides and Wieland, 2000). However this is ineffective because short-term bills and base money become approximate substitutes at the ZLB and open-market purchases have no direct effect on the public's wealth. The fall in the value of the public's holdings of Treasury debt will compensate for their increased holding of the monetary base, and since the nominal rate cannot be reduced any further there is no potential for future capital gains through Treasury bill holdings. The effects of open market operations on liquidity are limited because they do not change the actual sum of Treasury bills and reserves (now approximate substitutes) being held, only its composition. Moreover, in cash-in-advance, money-in-the-production-function, money-in- the-utility-function, shopping-time models of money demand, the marginal value of liquidity is given by the nominal interest rate (Clouse et al 2003) so when the latter is frozen at zero, open market operations do not produce any marginal benefit from additional liquidity and therefore have no effect on aggregate demand.
'Unconventiona;' monetary policy such as open-market purchases of unorthodox assets such as long-maturity treasury bonds that do not promise the same (state-contingent) returns as money could still stimulate aggregate demand by reducing the relative supply of assets with a preferred maturity and decreasing the risk premium in the interest rate on those assets. However the empirical evidence on these 'portfolio effects' has been mixed: although previous studies of Operation Twist and similar alterations to the maturity composition of the government debt suggested negligible effects on risk premiums (Holland 19690. However, more recently Bernanke, Reinhaart and Sack (2004) offered evidence from event study analysis that large changes in the relative supplies of securities may have significant effects on their yields.
Even if the nominal interest rate is frozen at ZLB the central bank can influence the real rate if it can affect inflation expectations. If the private sector believes that the central bank will keep interest rates low (and prices high) even after deflationary pressures subside, the real rate will fall and this stimulates aggregate demand. Therefore the key to effective monetary policy when at the ZLB is the management of higher inflation expectations. Committing to maintaining lower nominal rates for any given price level in the future should generate higher inflation expectations, lower real rates and a milder recession during the liquidity trap (see Krugman, 1998; Reifschneider and Williams, 2000; Eggertsson and Woodford, 2003; Jung, Teranishi and Watanabe, 2005; Adam and Billi, 2006).
However, beliefs about inflation in the private-sector are not easy to manipulate. The central bank's promises of high future price levels may not be seen as credible by the private sector agents who may doubt the ability or the intention of the central bank to generate inflation in the future. This is a classical dynamic inconsistency problem: the central bank may promise high future inflation to get out of the liquidity trap, but once out it will be tempted to renege on the promise and keep prices low by raising interest rates - investors know this and will discount any promises the central bank makes in the first place. This is aggravated by the fact that the deflationary shocks that give rise to this problem are so rare that it is difficult for a central bank to develop a reputation for dealing with them competently.
In relation to this, Eggertsson and Woodford (hereafter E&W) (2003) put forward the irrelevance proposition that for a given policy rule in operation at the ZLB, in the model's rational-expectations equilibrium quantitative easing has no effect on the deflationary price-level path. This is because if the public perceives the target as price stability, the use of a Taylor rule implies a reversal of expansionary policy when deflationary pressures recede. Since many central banks openly follow the Taylor rule, the public will expect an interest rate hike as soon as inflation forecasts exceed the implicit target. Krugman (1998) explores monetary expansion in the form of an increased future money supply. Theoretically unlimited purchases of domestic and foreign government debt and assets could increase the monetary base infinitely. However once again we face the problem of time inconsistency: when the economy is in a liquidity trap and the interest rate is fixed at zero, the demand for monetary base is perfectly elastic and any excess liquidity can be easily absorbed by the private sector. However, once we escape and face a positive nominal rate the demand for money will shrink drastically, and this will motivate the central bank to pursue radical reduction of the monetary base. Increasing the money supply is thereof impotent if the public expects it to revert to some constant value when the deflationary pressures abate. Eggertsson (2006) shows more generally that a discretionary central bank which minimises a standard loss function depending on inflation and output gap, always has an incentive ex-post to renege on its inflation promise in order to achieve low inflation. 0
The verdict from these irrelevance results is that monetary policy is most effective if it can influence inflation expectations. Auerbach and Obstfeld (2005) show that the adoption of a new policy rule at ZLB - either in the form of a one-time jump in the base path or an increase in its slope (the rate of growth) - which enables the monetary expansion to be perceived as permanent can raise inflation expectations and reduce the real rate. However the question remains of how the central bank can develop and implement a credibly permanent policy for monetary expansion, and effectively signal this commitment to the private sector. For the rest of this paper we therefore evaluate the effectiveness of monetary policy proposals in raising inflation expectations on the basis of the credibility and verifiability of their commitment being permanent during and after the deflationary period.
Svensson (2001), Eggertsson (2003), Bernanke (2003), E&W (2003) and Wolman (2005) propose an announcement of an upward-sloping target path which, in accordance with the optimality of future overshooting, should be constructed with a price gap such that the path begins higher than the current point and then rises at 1-2% every year. If a central bank undershoots inflation in one year, it must compensate for it though higher inflation in future years in order to return to the intended price-level trajectory, leaving expectations of long-term average inflation unaffected. Long-term expectations are often more important because, as E&W (2003) emphasise, it is the expected future path of short-term real rates which affects aggregate demand, i.e. spending and investment decisions are based upon long-term real rates. Moreover, if deflation pushes the price level further below the target, inflation expectations will rise in order to get back on the target path, automatically lowering the real rate even if the nominal rate is frozen at ZLB.
Critics question the point of announcing a price-level target when the primary means by which this can be achieved (the interest rate) is immobilised. E&W(2004) argue that the target is not simply a promise about future price levels but rather a commitment regarding the conditions under which the policy to maintain low-interest-rates will be abandoned. Therefore the bank's credibility remains in tact even if it fails to hit the target for several periods as long as it maintains low interest rates. Additionally, awareness of what the target is and how far the central bank is from achieving it allows agents to form informed expectations of how long policy must remain loose. It has been shown through simulations that depending on the sophistication of the targeting rule the optimal solution can be closely replicated.
McCallum (2000, 2002, 2003) and Svensson (2001, 2002, 2003) recommend foreign-exchange interventions in the form of devaluing the domestic currency in order to stimulate the economy by boosting demand in export and import-competing industries when the nominal interest rate is frozen at zero. Of even greater importance, as noted by Svensson (2001), devaluing and pegging the domestic currency at a depreciated rate are concrete actions which can act as a conspicuous and credible commitment to a higher future price level. If the peg is credible, since the initial depreciation has shifted the expected exchange-rate upwards, private sector agents must believe in a higher future exchange-rate and so internal consistency necessitates that they must then also expect a higher price level in the future.
The depreciation and peg also credibly commit the central bank to their promises of future inflation. In the case of a combined government and central bank, wherein the monetary authority is subordinate to the fiscal authority, Eggertsson (2003) argues that nominal liabilities confirm the government's incentive to generate inflation in the future to reduce the real value of public debt and, thereby, distortionary taxation. Even in the case of independent central banks which are more common in advanced, industrialised economies, Jeanne and Svensson (2007) argue that there exists a balance-sheet incentive to keep inflation promises. An independent central bank would never allow its capital to fall below a certain level in order to avoid being at the mercy of the government's capital injections. Since undoing the currency depreciation would imply a capital loss in the future on the bank's foreign-exchange reserves, this minimum capital level provides a lower floor to the future exchange rate (a ceiling to future intentional appreciation). Therefore a central bank can credibly commit to a higher future price level by managing its capital in a way that ensures it reaches the minimum capital level for the exchange-rate level that is consistent with the desired higher future price level.
McCallum (2000, 2002, 2003) proposes a moving exchange-rate target which is a function of current inflation and output gap, such that lower (higher) than target inflation or output gaps increase (decrease) the rate of depreciation. Simulations and impulse reaction functions show that such a policy rule, if perceived to be credible by the private sector, can result in macroeconomic stabilization. While this is similar to the 'devaluation and peg' proposal of Svensson (2001) in that both require the central bank to make how many ever unsterilized exchange market purchases it takes to make the exchange rate take on a desired value, the latter's solution is more general and considered by many to be superior mainly because it generates a simple arbitrage equilibrium and does not depend on any portfolio-balance effects for which empirical support has thus far been weak.
It is interesting to consider whether using the depreciation and peg option alone or price-level path targeting alone are equally successful policies. Coenen and Wieland (2003) find that switching to a price-level target path can be equally effective in generating inflationary expectations. In fact, compared to an exchange rate peg, the nominal rate remains at zero for a longer time resulting in lower long-term real rates and greater real depreciation. However the issue of credibility is more worrying under the latter approach because while it can be verified daily whether the exchange-rate peg is being maintained by the central bank, the price-level target path and resulting price gap cannot be directly observed. Once Coenen and Wieland (2003) account for this imperfect credibility by allowing for a share of market participants to believe that the government will renege and pursue an inflation target instead in the future, the benefits of switching to price-level targeting are reduced substantially. Therefore the exchange mechanism remains the most effective mechanism by which to raise inflation expectations, and as Svensson (2004) demonstrated is likely to work in both small and large economies. Fortunately, foreign exchange interventions and price-level targeting are not necessarily mutually exclusive. Svensson (2001) received much acclaim upon combining them in proposing his 'Foolproof Way' to escape a liquidity trap.
However a more recent and equally attractive proposal comes in the form of McCallum's (2006) 'automatic' rule which uses as the instrument a weighted average of the interest rate and depreciation rate of the nominal exchange rate. The small weight attached to the depreciation term should be inconsequential in normal circumstances but at the ZLB it will call for strong adjustments. He conducts simulations to study this rule within McCallum and Nelson's (1999) small open economy model and finds it has strong stabilizing properties when the interest rate is immobilized at the ZLB while it does not impede monetary policy when the ZLB constraint is irrelevant. The best way to establish credibility would be for the central bank to publicly announce its commitment to this rule and establish a track record of following it before the economy enters the trap. It can further be inferred that regular use of the rule would keep the economy out of a ZLB situation automatically since private sector agents should not expect deflation in the long-term. Ankita: what does this new method add - is it an improvement, a complement or a substitute to Svensson's foolproof way?
It is important to remember that all policies which require foreign exchange interventions have non-negligible beggar-thy-neighbour aspects and are thus politically contentious in an international context. Moreover, if several important regions are simultaneously in a liquidity trap, as Krugman (2009) loosely speculated in the recent global recession, these policies would result in a race within these regions to simultaneously depreciate relative to one another.
 The intuition: in a world with money and government bonds, if the government offered negative interest rates nobody would be willing to swap their money for bonds when money carries with it a certain stream of zero nominal interest and no other costs. The lower bound on interest rates can be derived from the properties of the function of the model representing the transaction-facilitating properties of money (such as liquidity and anonymity) together with any costs (such as storage and security) associated with holding money. For the purpose of this essay we assume all of these to cancel out and equal zero.
 Expectations matter for several reasons. First, as discussed by Krugman (1998), inflationary expectations lower the real interest rate associated with the immobilised nominal rate. Second, a commitment to low rates for a long period of time could raise current spending by influencing the core determinants of long-term rates of interest and exchange. Third, the mere expectation of a boom could raise current spending due to accelerator and permanent-income mechanisms.
 This does not mean that open market operations are entirely ineffective as a tool of policy-making. Auerbach and Obstfeld (2005) show that they have microeconomic fiscal benefits which are compatible with standard macroeconomic stabilization objectives. Assuming positive long-term interest rates are expected at some point and that the expansion of the money supply is perceived as permanent, open market purchases reduces the real value of future public debt and hence the excess burden of the requisite taxes.
 Conventional wisdom stipulates that price-level targeting results in more variability in short-term inflation and/or output gap. However this has recently been challenged by Svensson (1999b), Vestin (2003), Cecchetti and Kim (2003), and Batini and Yates (2003), who show that a combination of inflation and price-level targeting may very well reduce short-term inflation and/or output-gap variability, in addition to reducing long-term price-level uncertainty.