Deflation, Depression, and the Zero Lower Bound

Transcription

1 Deflation, Depression, and the Zero Lower Bound Sebastien Buttet and Udayan Roy December 6, 2011 Abstract We analyze the dynamic paths of output, interest rates, and inflation in a simple New Keynesian model of the business cycle that explicitly incorporates the zero lower bound on nominal interest rates. We show that two long-run equilibria exist, one stable and the other unstable, and we characterize the conditions under which the economy plunges into a deflation-induced depression following a contractionary demand shock. We study the policy-maker s ability to (i) keep an economy out of a deflationary spiral and (ii) end a deflationary spiral that has already begun. Qualitatively, expansionary fiscal policy is an adequate response to both (i) and (ii), while expansionary monetary policy specifically, an increase in the target inflation rate is a partial answer to (i) only. Quantitatively, our calibration exercise suggests that a modest rise in the Fed s inflation target from two to three percent is enough to prevent the start of a deflationary spiral. On the other hand, only large fiscal stimulus packages have a chance to pull the economy out a deflationary spiral, when it is already in one. JEL Classification: E12, E52, E62 Key Words: Deflation, Depression, Zero Lower Bound, Monetary Policy, Fiscal Policy Economics Department, C.W. Post Campus, Long Island University, Brookville, NY udayan.roy@liu.edu. Phone: Thanks to Veronika Dolar for helpful comments. All errors are ours. 1

2 1 Introduction Just a few years before the financial crisis hit, economists and central bankers generally agreed that business cycles were a phenomenon of the past. Robert Lucas, in his presidential address to the American Economic Association, declared that the problem of depression prevention which gave birth to the field of macroeconomics during the Great Depression was solved (Lucas, 2003). In another famous speech, the current chairman of the Federal Reserve System attributed a large part of the decline in macroeconomic volatility over the past thirty years the so-called Great Moderation period to improved monetary policy and better use of the Taylor rule (Bernanke, 2004). The Great Recession of 2008 challenged the neoclassical synthesis and forced policymakers to take the twin threats of deflation and depression seriously. There was a tight link between deflation and depression during the Great Depression years and many believe that deflation contributed to the severe output losses in the U.S. and other countries between 1929 and 1932 (Bernanke and Carey, 1996; Eichengreen and Sachs, 1985). In addition, memories of the Great Depression and to a lesser extent the Japanese Lost Decade of the 1990s, continue to influence the decisions of policymakers who, to this day, remain reluctant to adopt any policy that may cause deflation (Bernanke, 2002). For example, the Friedman rule, which calls for a modicum of anticipated deflation (Friedman, 1969), is rarely implemented by central banks, even though it is well-known that zero nominal interest rates lead to an optimal allocation of resources under fairly broad conditions (Chari et al., 1996; Cole and Kocherlakota, 1998; Lagos, 2010). Outside of the Great Depression years, however, the empirical link between deflation and depression is much weaker. Analyzing more than one hundred years of data on inflation and output growth for 17 countries, Atkeson and Kehoe (2004) show that ninety percent of episodes with deflation did not lead to depression, while thirty percent of depression episodes had no deflation. According to Rolnick and Weber (1997), there is only weak evidence supporting a long-run positive relationship between inflation and output growth rates for countries using fiat money systems. Finally, higher inflation rates are associated with lower output growth for the years after WWII (Atkeson and Kehoe, 2004; Benhabib and Spiegel, 2009). 2

3 Why is the correlation between inflation and output growth tight during the Great Depression but weak in other time periods? Why do central banks rarely implement the Friedman rule? Are policymakers over-concerned about episodes of mild deflation? To answer these questions and contribute to the deflation-depression debate more generally, we extend a simple 3-equation New Keynesian model of the business cycle, by adding of a non-negativity constraint on nominal interest rates. To be sure, a large and rapidly growing body of empirical and theoretical research has already studied the impact that the zero lower bound has on the economy, including the restrictions it places on the use of the conventional tools of monetary and fiscal policy (e.g., Eggertsson and Woodford (2003); Bernanke et al. (2004); Woodford (2010); Correia et al. (2011)). What sets our research apart, however, is that we concentrate our analysis on the deflation-depression link and fully characterize the conditions under which a deflation-induced depression occurs. The properties of our model without the zero lower bound are well known (Mankiw, 2010; Carlin and Soskice, 2006). Output and real interest rates are pro-cyclical and, in the absence of shocks, all variables converge toward a unique stable equilibrium, where the economy is operating at full capacity and inflation is equal to the central bank s inflation target. The introduction of the zero lower bound changes the model dynamics along several dimensions. First, the correlation between output and real interest rates becomes negative, which implies that changes in real interest rates reinforce (rather than dampen) changes in output when the zero lower bound is binding. Second, we find that an additional unstable equilibrium exists when the zero lower bound is binding. Third, there is a one-to-one relationship between the inflation level and whether the zero lower bound is binding. We show that a critical threshold for inflation exists such that the zero lower bound becomes binding when inflation falls below the threshold. Finally, the inflation reaction function, which links inflation across two consecutive time periods, is no longer linear but is kinked at the inflation threshold. The negative feedback loop between output and inflation is the mechanism that leads to a deflation-induced depression, as previously explained by Fisher (1933) or Krugman (1998). In normal times, when nominal interest rates are positive, the central bank 3

4 can afford to cut interest rates following a negative demand shock to provide short-term stimulus to the economy. When the zero-lower bound is binding, however, cutting rates is not feasible and real interest rates spike up as a result of lower inflation. Higher real rates in turn depress the economy further, which put further pressure on real rates, which depress the economy further, and so on and so forth. For a given set of parameter values and initial conditions, some interesting questions arise. First, to which of the two equilibria does inflation and output converge? Second, can the model generate a deflation-induced depression and if yes, are there equilibrium paths where deflation does not lead to depression? Third, how large should a negative demand shock be to trigger a depression? Finally, how effective are monetary and fiscal policy are at preventing or stopping the deflationary spiral? We find that answers to the first two questions depend on the sign of a simple expression, the sum of the inflation rate and the natural rate of interest. We show that, as long as the sum of the inflation rate and the natural real interest rate stays positive, the economy converges back to the long-run stable equilibrium, even when the zero lower bound is initially binding. On the other hand, a deflationary spiral starts when the sum of the inflation rate and the natural rate of interest becomes negative. One scenario that makes the sum of the inflation rate and the natural rate negative is a large contractionary demand shock that induces a sudden fall in the inflation rate. However, since this scenario has a small probability of occurrence, few equilibrium paths lead to a depression. Our theoretical results that (i) mild deflation does not lead to a depression and (ii) deflation and depression are closely linked when the economy is hit by a large demand shock, shed light on the aforementioned empirical findings of Atkeson and Kehoe (2004) and Benhabib and Spiegel (2009). Since the likelihood of a large negative demand shock is small, the model dynamics for output and inflation are consistent with the fact that the deflation-depression link is strong for the Great Depression years and weak for other years. The results (i) and (ii) also help explain why central banks rarely implement the Friedman rule as the optimal monetary policy. The Friedman rule, which calls for a small dose of anticipated deflation, takes the sum of the inflation rate and the natural 4

5 real interest rate uncomfortably close to zero, and thereby raises the probability of the economy plunging into a depression. We also study the effectiveness of monetary and fiscal policy in dealing with the deflationary spiral. We distinguish between prevention and remedy, as proposed by Ben Bernanke in his 2002 speech about deflation (Bernanke, 2002), and ask two simple questions: (i) What can be done to keep an economy away from the deflationary spiral? (ii) Can policy get an economy out of a deflationary spiral, if it is already in one? None of the channels of unconventional monetary policy at the zero lower bound are at work in our model. By monetary policy, we mean a change in the parameters of the Taylor rule, including the central bank s inflation target, and by fiscal policy, we mean changes in government spending and taxes. We show that expansionary fiscal policy is an adequate answer to both questions, while expansionary monetary policy specifically, an increase in the target inflation rate is a partial answer to (i) only. Finally, we simulate our model to illustrate how output and inflation respond to a fourperiod contractionary demand shock when the zero lower bound is binding. We do not see the simulation as a calibration exercise (we are not aiming to match any moment of the data), but our quantitative endeavor allows us to answer two important questions. First, how large should a demand shock be to trigger a depression? Second, how aggressive should the policy response be to prevent or cure a depression? We find that it takes an output loss of at least 24 percent over four consecutive periods to trigger a deflationinduced depression and that the likelihood that such large shocks occur is infinitesimal. To put things into perspective, output in the U.S. during the Great Depression contracted by 46 percent between 1929 and The answer to the second question is also encouraging. Everything else equal, a modest rise in the central bank s inflation target from 2 to 3 percent is enough to prevent the start of a deflationary spiral. On the other hand, only large fiscal stimulus packages (expressed as a fraction of the natural level of output) have a chance to get the economy out the deflationary spiral, when it is already in one. The remainder of the paper is organized as follows. In Sections 2 and 3, we introduce our model and characterize its long-run equilibria. We discuss the stability of these equilibria in Section 4. We reinterpret empirical findings in light of our theoretical results 5

6 in Section 5. We study the policy responses to a deflationary environment in Section 6 and present some numerical simulation results in Section 7. Finally, we offer concluding remarks in Section 8. 2 The Model We extend a simple New Keynesian model of the business cycle by explicitly incorporating a non-negativity constraint on nominal interest rates. The version of our model without the zero lower bound has replaced the IS-LM model as the mainstay of short-run analysis in macroeconomics textbooks and is variously referred to as the dynamic AD-AS (or DAD- DAS) model in Mankiw (2010), the AS/AD model in Jones (2011), and the 3-equation (IS-PC-MR) model in Carlin and Soskice (2006), where PC refers to the Phillips Curve and MR refers to the central bank s Monetary Policy Rule. Equilibrium in the market for goods and services is given by Y t = Ȳt α(r t ρ) + ɛ t (1) where Ȳt denotes the natural or full-employment level of output, r t is the real interest rate, ρ is the natural real interest rate of interest, α is a positive parameter representing the responsiveness of aggregate expenditure to the real interest rate, and ɛ t represents both demand shocks (say, from changes in private-sector optimism or animal spirits ) and the government s fiscal policy stance. This equation is essentially the well-known IS Curve. The key feature of (1) is the negative relationship between the real interest rate r t and output Y t. When the real interest rate increases, borrowing becomes more expensive and savings yields a higher reward. As a result, firms engage in fewer investment projects and consumers save more and spend less. These effects reduce the demand for goods and services. The shock ɛ t represents exogenous shifts in demand that arise from changes in consumer and/or business sentiment the so-called animal spirits and changes in fiscal policy. When the government implements an economic stimulus (an increase in government expenditure or a tax cut), ɛ t is positive, while fiscal austerity means a negative ɛ t. Note that the economy operates at full capacity (Y t = Ȳt) when there are no demand 6

7 shocks (ɛ t = 0) and when the real interest rate is equal to the natural real interest rate (r t = ρ). The ex-ante real interest rate in period t is determined by the Fisher equation and is equal to the nominal interest rate i t minus the inflation expected in the next period: r t = i t E t π t+1 (2) Inflation in the current period, π t, is determined by a conventional Phillips curve augmented to include the role of expected inflation, E t 1 π t, and exogenous supply shocks, ν t : π t = E t 1 π t + φ(y t Ȳt) + ν t (3) with φ is a positive parameter. Inflation depends on expected inflation because some firms set prices in advance. When these firms anticipate high inflation, they expect their own costs to be rising quickly and that their competitors will be implementing price hikes. The expectation of high inflation induces these firms to announce price increases for their products, which in turn cause high actual inflation in the economy. Conversely, when firms expect low inflation, they forecast that their costs and competitors prices will rise only modestly. In this case, they keep price increases in check, which leads to low inflation. The business cycle also affects inflation that goes up when output rises above its natural level, since the parameter φ is positive. When the economy is booming and output rises above its natural level, firms experience increasing marginal costs and so they raise their prices. Conversely, when the economy is in recession and output is below its natural level, marginal costs fall and firms cut prices, which results in low inflation. The random variable ν t captures all other influences on inflation other than inflation expectations (which are captured in the term E t 1 π t ) and short-run economic conditions (which are captured in the term φ(y t Ȳt)). For example, an increase in oil prices would mean a positive value for ν t since higher input prices might force firms to raise the price of their products. Inflation expectations play a key role in both the Fisher equation (2) and the Phillips curve (3). To keep the model tractable analytically and because the role of expectations was analyzed in great detail in Eggertsson and Woodford (2003, 2004), we assume that 7

8 inflation in the current period is the best forecast for inflation in the next period 1. That is, agents have adaptive expectations with: E t π t+1 = π t (4) Finally, we complete the description of the model with a rule for monetary policy. Dynamic New Keynesian models assume that the central bank sets a target for the nominal interest rate, i t, based on the inflation gap and the output gap according to the Taylor rule (Taylor, 1993). Specifically, it is assumed that i t = π t +ρ+θ πt (π t πt )+θ Y t (Y t Ȳt), where the central bank s policy parameters θ πt and θ Y t are non-negative. We, however, wish to explicitly incorporate the fact that nominal interest rates need to be non-negative. Therefore, our monetary policy rule is: i t = max{0, π t + ρ + θ πt (π t πt ) + θ Y t (Y t Ȳt)}. (5) Although the central bank sets a target for i t, its true influence on the economy works through the real interest rate, r t. When inflation in period t is equal to the central bank s target (π t = πt ) and output is at its natural level (Y t = Ȳt), the last two terms in equation (5) are equal to zero, implying that the real rate is equal to the natural rate of interest (r t = ρ). As inflation rises above the central bank s target (π t > πt ) or output rises above its natural level (Y t > Ȳt), the Taylor rule ensures that the real rate rises to bring down inflation and/or output. Conversely, if inflation falls below the central bank s target (π t < πt ) or output falls below its natural level (Y t < Ȳt), the real interest falls to provide a stimulus to the economy. For given values of the model s period-t parameters (α, ρ, φ, θ πt, θ Y t, πt, and Ȳt), its period-t shocks (ɛ t and ν t ), and the pre-determined inflation rate (π t 1 ) for period 1 Following seminal papers by Lucas (1972, 1976), there is a long tradition in the field of macroeconomics to build models where agents have rational expectations. More recently, a line of research by Hansen and Sargent (2008) explores a class of mechanisms for expectations formation based on robust control, which captures the idea that the decision-maker has an imperfect understanding of how the economy works. Sims (2003, 2006) most recent work explores a new theory for expectations formation based on rational inattention, which captures agents limited capacity to process information. Finally, Chow (2011) provides strong empirical evidence for supporting the adaptive expectations hypothesis in economics and points out that the rational expectations hypothesis was embraced by the economics profession too quickly. 8

9 t 1, one can use the the model s five equations to solve for its five period-t endogenous variables (Y t, r t, i t, E t 1 π t, and π t ). Once π t is determined, the process can be repeated for period t + 1, and so on and on. To study the model s dynamic properties algebraically as opposed to numerically we will simplify the model by assuming a constant target inflation rate: that is, πt = π for all t. 3 Long-Run Equilibria An equilibrium sequence is any intertemporal sequence of values for the model s endogenous variables such that the model s equations (1) (5) are satisfied. A long-run equilibrium is any equilibrium sequence such that, in the absence of shocks (ɛ t = ν t = 0 for all t), the inflation rate is constant (π t 1 = π t = π t+1 =...). It is then straightforward to identify the two long-run equilibria of the model. First, one can check that if π t 1 = π and there are no shocks, then π t+s = π, Y t+s = Ȳt+s, r t+s = ρ, and i t+s = ρ + π for s = 0, 1, 2,.... We will refer to this long-run equilibrium as the orthodox equilibrium. To ensure that the orthodox long-run equilibrium does not run afoul of the zero lower bound on the nominal interest rate we assume ρ + π > 0 or, equivalently π > ρ. Second, one can check that, if π t 1 = ρ and there are no shocks, then π t+s = ρ, Y t+s = Ȳt+s, r t+s = ρ, and i t+s = max{0, ρ + ρ + θ π,t+s ( ρ π ) + θ Y,t+s (Ȳt+s Ȳ t+s )} = 0 for s = 0, 1, 2,.... We will refer to this long-run equilibrium as the deflationary equilibrium. To summarize, we have the following: Proposition 1. Once the DAD-DAS model is generalized to allow the zero lower bound on the nominal interest rate to be binding, it has two long-run equilibria: an orthodox equilibrium that is obtained when π t 1 = π > ρ and a deflationary equilibrium that is obtained when π t 1 = ρ. 9

10 4 Stability of Equilibrium The issue of stability inevitably arises: How does the economy behave for arbitrary values of π t 1? Will it converge to one of the two long-run equilibria? If so, which one? To analyze the stability properties of the DAD-DAS model, we need to solve for the model s endogenous variables for the case in which the zero lower bound on the nominal interest is not binding and again for the case in which the zero lower bound is binding. 4.1 Zero Lower Bound is Non-Binding When the zero lower bound is not binding, the monetary policy rule (5) becomes i t = π t + ρ + θ πt (π t π t ) + θ Y t (Y t Ȳt). (6) Therefore, the economy is described by equations (1) (4) and (6). Using these equations, it is straightforward, if tedious, to derive expressions for the model s endogenous variables in terms of the parameters of the model and the pre-determined inflation rate of the previous period. Specifically, it can be shown that equilibrium output is equilibrium inflation is Y t = Ȳt + αθ πt(πt π t 1 ν t ) + ɛ t, (7) 1 + α(θ πt φ + θ Y t ) π t = (1 + αθ Y t) (π t 1 + ν t ) + αθ πt φπt + φɛ t, (8) 1 + α(θ πt φ + θ Y t ) and the equilibrium nominal interest rate is 1 i t = ρ+ 1 + α (θ πt φ + θ ) {(1 + θ πt + αθ Y t )(π t 1 + ν t ) (1 αφ)θ πt πt + (θ Y t + (1 + θ πt )φ)ɛ t }. Y t As the Fisher equation (2) and adaptive expectations (4) imply r t equilibrium real interest rate can be derived from equations (8) and (9) above. (9) = i t π t, the It follows from (7) that, in the usual case where the zero lower bound on the nominal interest rate is not binding, output (Y t ) is directly related to full-employment output (Ȳt), the demand shock (ɛ t ), and the central bank s target inflation (π t ), and inversely related to the previous period s inflation (π t 1 ), the inflation shock (ν t ), the responsiveness 10

11 of inflation to output in the Phillips Curve (φ), and the responsiveness of the nominal interest rate to output in the monetary policy rule (θ Y t ). It follows from (8) that inflation (π t ) is directly related to π t 1, ν t, π t, and ɛ t. We will see in section 6 that these comparative static results for inflation play an important policy role in the dynamic stabilization of the economy. Figure 1 graphs the dynamic mapping from π t 1 to π t for given values of all the parameters in equation (8) and the shocks set at zero (ɛ t = ν t = 0). It can be checked that this graph which we will call the inflation mapping curve must intersect the 45- degree line at π t 1 = π t, because π t 1 = π t orthodox long-run equilibrium of Proposition 1. implies π t = π t 1 = π t. This represents the Note also that the inflation mapping curve must be flatter than the 45-degree line as its slope is a positive fraction: that is, π t / π t 1 = (1 + αθ Y t )/(1 + αθ Y t + αθ πt φ) (0, 1). As is shown in Figure 1, this feature of the inflation mapping curve when the zero lower bound on the nominal interest rate is not binding and there are no shocks implies that the orthodox long-run equilibrium of Proposition 1 is stable. (The stability of the deflationary long-run equilibrium will be discussed below.) To guarantee the intuitive result that a positive demand shock increases output in equation (14), we assume that 1 αφ > 0 in this paper. It then follows from (9) that the nominal interest rate, assuming it is not at the zero lower bound, is directly related to ρ and to all exogenous variables and shocks that are directly related to inflation. Note that these results prevail only when i t, as given by (9), is non-negative. It can be checked that (9) implies > where i t = 0 if and only if π > t 1 πc t 1, (10) π c t 1 (1 αφ)θ πtπ t (1 + αθ πt φ + αθ Y t )ρ (θ πt φ + θ Y t + φ)ɛ t 1 + θ πt + αθ Y t ν t (11) is the critical value for the previous period s inflation rate such that the equilibrium nominal interest rate is given by equation (9) when π t 1 π c t 1, and is zero when π t 1 π c t 1. 11

12 Figure 1: Dynamics without the zero lower bound on the nominal interest rate The dynamic mapping from π t 1 to π t when there are no shocks (ɛ t = ν t = 0) and the zero lower bound on the nominal interest rate is non-binding is shown here. For any given value of π t 1, the inflation mapping curve and the 45-degree line can be used to trace inflation in subsequent periods. The orthodox long-run equilibrium (π t 1 = π ) can be seen to be stable. 12

13 The line segment BC in Figure 2 graphs the dynamic mapping from π t 1 to π t in equation (8) for π t 1 π c t 1 when there are no shocks and the central bank has a constant target inflation (π t = π ). A quick comparison of Figures 1 and 2 emphasizes the fact that the introduction of the zero lower bound on the nominal interest rate sharply restricts the domain over which equation (8) is applicable. Note, as before, that the coefficient of π t 1 on the right hand side of equation (8) is a positive fraction, which explains why BC is drawn flatter than the 45-degree line. At π t 1 = π, BC intersects the 45-degree line, indicating the orthodox long-run equilibrium of section 3. It can be checked from (11) that, in the absence of shocks, π c t 1 is a convex combination of π and ρ. Our assumption that π + ρ > 0 then implies π > π c t 1 > ρ, as shown in Figure 2. In the absence of shocks (ɛ t = ν t = 0), equation (8) reduces to π t π t 1 = αθ πt φ 1 + αθ πt φ + αθ Y t (π t π t 1 ). (12) Note that whenever the central bank s target inflation rate, π t, exceeds (respectively, is less than) the previous period s inflation, π t 1, the current period s inflation rises (respectively, falls), while still remaining less (respectively, greater) than the target inflation. In other words, if there are no shocks and the target inflation is constant and π t 1 π c t 1, the inflation rate converges over time to the central bank s target inflation rate. Under adaptive expectations and in the absence of shocks, the Phillips Curve yields φ (Y t Ȳt) = π t π t 1 = αθ πt φ 1 + αθ πt φ + αθ Y t (π t π t 1 ). The convergence of inflation to target inflation therefore implies the convergence of equilibrium output to the full-employment output. Therefore, according to the Taylor Rule (5 or 6), the nominal interest rate set by the central bank will converge to i = π +ρ > 0. And, by the Fisher equation (2) the real interest rate converges to ρ. We have, therefore, established the following: Proposition 2. When there are no shocks and π t 1 π c t 1, the economy converges to the orthodox long-run equilibrium. 13

14 4.2 Zero Lower Bound is Binding It follows from (10) that, when π t 1 < πt 1, c the zero lower bound on the nominal interest rate becomes binding: that is, i t = 0. (13) In this case, recall that the economy is described by equations (1) (4) and (13). Using these equations, it is straightforward, as before, to derive expressions for the endogenous variables at time t in terms of the parameters of the model and the pre-determined inflation rate of the previous period. Specifically, equilibrium output is Y t = Ȳt + α(π t 1 + ν t + ρ) + ɛ t 1 αφ (14) and equilibrium inflation is π t = 1 1 αφ (π t 1 + ν t ) + φ 1 αφ (αρ + ɛ t). (15) Note that the coefficient of π t 1 in (15) exceeds unity. This is why the segment AB in Figure 2, which shows the dynamic link between π t 1 and π t for π t 1 π c t 1 when there are no shocks, is steeper than the 45-degree line. Also, it can be checked that equating the right hand sides of equations (8) and (15) yields π t 1 = π c t 1, thereby confirming the continuity of the mapping of π t 1 to π t at π t 1 = π c t 1 in Figure 2. The contrast between equation (7), when the zero lower bound is not binding, and equation (14), when the zero lower bound is binding, is striking. It follows from (14) that now π t 1, ν t, and φ are directly related to output. Moreover, output is now directly related to the responsiveness of aggregate demand to the real interest rate (α) and to the long-run real interest rate (ρ), and unrelated to all monetary policy tools (π t, θ Y t, and θ πt ). The ineffectiveness of monetary policy follows from the fact that the nominal interest rate is now at the zero lower bound and monetary policy is, therefore, unable to influence the nominal interest rate. It follows from equation (15) that inflation is directly related to π t 1, ν t, φ, and ɛ t, as it is in equation (8), which describes inflation when the zero lower is not binding. Now, however, α and ρ have a direct effect on inflation, and the monetary policy tools have no effect at all (for the reason discussed in the previous paragraph). 14

15 Subtracting π t 1 from both sides of equation (15) yields π t π t 1 = αφ 1 αφ (π t 1 + ρ). (16) This confirms our earlier result that when there are no shocks and π t 1 = ρ, the economy is in the deflationary long-run equilibrium of section 3. However, equation (16) also reveals the unstable nature of the deflationary long-run equilibrium. If there are no shocks and if π t 1 < ρ, equation (16) implies that inflation keeps falling and, by equation (14) drags output down with it, indefinitely. This is the dreaded deflationary spiral. On the other hand, if there are no shocks and if π c t 1 > π t 1 > ρ, inflation and output both keep rising. In short, we have established the following: Proposition 3. Assume there are no shocks. The deflationary long-run equilibrium is unstable. As inflation keeps rising when there are no shocks and π c t 1 > π t 1 > ρ, equation (11) implies that inflation will, in finite time, exceed π c t 1 and that, therefore, the zero lower bound on the nominal interest rate will switch from binding to non-binding. Therefore, propositions 2 and 3 can be combined to yield the following: Proposition 4. Assume there are no shocks. If π t 1 > ρ, the economy converges to the orthodox long-run equilibrium. If π t 1 = ρ, the economy stays in the deflationary long-run equilibrium. If π t 1 < ρ, the economy stays in a deflationary spiral. Proposition 4, which is graphically illustrated in Figure 1, helps explain why central banks rarely implement the Friedman rule as the optimal monetary policy. The Friedman rule, which calls for a small dose of anticipated deflation, makes the sum of inflation and natural rate of interest very close to zero, which makes the economy vulnerable to further shocks and raises the probability that the economy plunges into a depression. 5 Reinterpretation of Empirical Findings In the introduction, we cited the empirical findings of Atkeson and Kehoe (2004), who show that, while the deflation-depression link is strong for the Great Depression years, 15

16 Figure 2: Stability ABC represents the dynamic mapping from π t 1 to π t when there are no shocks (ɛ t = ν t = 0). AB represents an economy in which the zero lower bound on the nominal interest rate is binding, and BC represents the same economy when the zero lower bound on the nominal interest rate is non-binding. For any given value of π t 1, ABC and the 45-degree line can be used to trace inflation in subsequent periods. The orthodox long-run equilibrium (π t 1 = π ) can be seen to be stable, whereas the deflationary long-run equilibrium (π t 1 = ρ) can be seen to be unstable. 16

17 there is little evidence supporting a statistical relationship between deflation and depression outside of the Great Depression years. Here we briefly summarize their empirical strategy and revisit their findings in light of our results. The authors use the following simple specification to estimate the relationship between inflation and growth: y it = α + βπ it + ɛ it (17) where y it represents average annual growth for country i during five-year period t, π it represents average annual inflation for country i during five-year period t, and ɛ it represents an i.i.d. normal disturbance term. The data set consists of historical data on inflation and output growth rates for a panel of 17 different countries. Every data series runs for more than a hundred years and ends in the year The countries, with the year the data starts given in parenthesis, are: Argentina (1885), Australia (1862), Brazil (1861), Canada (1870), Chile (1908), Denmark (1871), France (1820), Germany (1830), Italy (1867), Japan (1885), the Netherlands (1900), Norway (1865), Portugal (1833), Spain (1849), Sweden (1861), the United Kingdom (1870), and the United States (1820). For all countries except Australia and Denmark, the data up to 1980 are taken from Rolnick and Weber (1997). The data for Australia and Denmark up to 1980 are taken from Backus and Kehoe (1992). The data from 1980 onwards are from the International Financial Statistics of the International Monetary Fund. Table 1 shows the estimation results for different time periods. For the Great Depression years, the point estimate for the slope β is equal to 0.4, implying that, on average, a one percentage point lower inflation rate is associated with a drop in growth of.4 of a percentage point, say, from 3.40 to 3.00 percent. When the entire time period is considered, however, the value for β drops to 0.08, implying a much weaker link between inflation and growth between 1820 and Finally, the point estimate of β is slightly negative for the years after WWII. Our theoretical results in Proposition 4 help explain the weak relationship between inflation and output growth outside of the Great Depression years and the tight deflationdepression link during the Great Depression years. When inflation falls below ρ and a 17

18 depression starts, our model clearly predicts a tight link between deflation and depression (see Figure 2). However, the probability of a large demand shock starting a depression is very small. Thus, many equilibrium paths involve a mild case of deflation but no depression, implying a small correlation between inflation and output growth even when the zero lower bound is binding. Table 1: Regression of Average Growth and Average Inflation (Std Dev in parenthesis) Episode Constant α Slope β Great Depression Episode ( ) 1.15 (1.24).40 (.28) Other Episodes (Excluding Great Depression) Including World Wars 3.02 (.19).04 (.03) Excluding World Wars 2.96 (.15).10 (.03) Before Great Depression (Pre-1939) 2.35 (.18).11 (.04) After WWII (Post-1949) 4.00 (.28) -.03 (.04) All Episodes ( ) 2.74 (.17).08 (.03) Source: Atkeson and Kehoe (2004) Using our model, we calculate the slope of the linear relationship between output growth and inflation, when the zero lower bound is binding and when it is not. That is, we seek to find the parameter β such that where y t = Y t+1 Y t Y t y t = γ + βπ t + u t (18) and u t is a disturbance term. Assuming that the natural output level is constant over time and normalizing it to one, we use equations (7) and (8) to calculate the slope β when the zero lower bound is not binding and equations (14) and (15) when it is binding. When the zero lower bound is not binding, we find that the value of β is approximately equal to: β P OS = α 2 θ 2 πφ (1 + αθ Y )(1 + α(θ π φ + θ Y )) 18 (19)

19 On the other hand, when the zero lower bound is binding, the slope β is approximately equal to: β ZLB = Some interesting remarks are in order. α2 φ 1 αφ (20) First, if one uses the values of α = 1 and φ = 0.25 as we will in our simulations, we find that β ZLB = 0.33 when the zero lower bound is binding, which is very close to the 0.4 estimate in Atkeson and Kehoe (2004) for the Great Depression years, certainly a period of time when the zero lower bound was binding. Second, we find that β P OS = if one sets θ Y = θ π = 0.5 for the Taylor rule as suggested by Taylor (1993). As a result, we have β ZLB > β P OS, which is also in line with the results in Table 1. For example, the estimated slope for the Great Depression years, when the zero lower bound is binding, is 0.4, while for the years excluding the Great Depression but including the wars, the slope is β = Policy Responses to Deflationary Spirals Given that a deflationary spiral is an undesirable outcome, (i) what can be done to keep an economy away from it, and (ii) what can be done to get an economy out of a deflationary spiral if it is already in one? None of the channels of unconventional monetary policy at the zero lower bound are at work in our model. 2 By monetary policy, we mean a change in the parameters of the Taylor s rule, including the Fed s inflation target, and by fiscal policy, we mean increased government spending. We will show that expansionary fiscal policy is an adequate answer to both questions, and expansionary monetary policy specifically, an increase in the target inflation rate is an answer to (i). 2 Bernanke et al. (2004) discuss the policy tools that central banks can use when the zero lower bound is binding, including balance sheet expansion through quantitative easing ; changing the composition of the Fed s balance sheet through, for example, the targeted purchases of long-term bonds as a means of reducing the long-term interest rate; or using communication to shape public expectations about the future course of interest rates. 19

20 6.1 Avoiding a Deflationary Spiral From Proposition 4, we see that, if π t 1 ρ and there are no shocks from period t onwards, the economy will not enter a deflationary spiral. However, the possibility of adverse shocks in period t cannot be wished away. We need to ask what can be done in period t to neutralize adverse period-t shocks that can cause a deflationary spiral from period t + 1 onwards. We will consider two cases: Case 1A: π t 1 πt 1 c > ρ, and Case 1B: πt 1 c π t 1 ρ Case 1A: π t 1 πt 1 c > ρ Equation (10) implies that in this case the zero lower bound on the nominal interest rate is not binding. Therefore, equation (8) gives us the period-t inflation rate, π t. By Proposition 4, if there are no further shocks from period t+1 onwards, we need π t > ρ to avoid a deflationary spiral (from period t+1 onwards). It is clear from equation (8) that a large enough and negative inflation shock and/or demand shock (from falling private-sector confidence, for example) could lead to π t < ρ (and, therefore, a deflationary spiral). However, equation (8) also makes clear that this danger can be neutralized by expansionary fiscal policy (ɛ t ) and/or an increase in the central bank s target inflation (π ). (As is clear from equation (8), the effects of θ πt and θ Y t, which represents the responsiveness of the central bank s interest-setting rule to inflation and output, respectively, on inflation are ambiguous.) These issues can be explored graphically. Let us refer to ABC in Figure 2 as the inflation mapping curve. Figure 3 shows the effect on the inflation mapping curve of contractionary monetary policy in the form of a reduction in the central bank s inflation target, and Figure 4 shows the effect of fiscal austerity. In Figures 5 and 6, a reduction in private-sector demand lowers the inflation mapping curve from ABC to A 1 B 1 C 1, thereby threatening to reduce inflation below ρ and initiation a deflationary spiral. However, Figures 5 and 6 show that with a timely increase in the inflation target (as in Figure 3) or in the fiscal stimulus (as in Figure 4) we can avoid a deflationary spiral. 20

21 Figure 3: Monetary Policy When there is a decrease in π t, the mapping from π t 1 to π t changes from ABC to AB 1 C Case 1B: πt 1 c π t 1 ρ Equation (10) implies that in this case the zero lower bound on the nominal interest rate is binding. Therefore, π t is now obtained from equation (15). As in Case 1A above, a large enough and negative inflation shock and/or demand shock can initiate a deflationary spiral from period t + 1 onwards. However, as equation (15) makes clear, expansionary fiscal policy (ɛ t ) can neutralize the threat of a deflationary spiral, as was the case in Case 1A. Note that, in contrast to Case 1A, changes in π are now of no help as a preventive strategy. This is because monetary policy cannot affect an economy that is at the zero lower bound. To summarize, raising the inflation target can keep help us avoid a deflationary spiral, but only when such a spiral is already unlikely (Case 1A: π t 1 πt 1 c > ρ). Expansionary fiscal policy, on the other hand, works as a preventive in all cases. 21

22 Figure 4: Fiscal Policy When there is a decrease in ɛ t either because of a decrease in private-sector confidence or because of fiscal austerity the mapping from π t 1 to π t changes from ABC to A 1 B 1 C 1. 22

23 Figure 5: Monetary Policy to Avoid a Deflationary Spiral Suppose, as in Figure 4, a decrease in private-sector confidence lowers the inflation mapping curve from ABC to A 1 B 1 C 1, and suppose π t 1 = π as in the orthodox long-run equilibrium and is denoted by point D. In this case, π t would fall below ρ and a deflationary spiral would begin, unless preventive measures are taken. An increase in the central bank s inflation target, π, that raises the inflation mapping curve to A 1 B 2 C 2 (or higher) averts the spiral. (See Figure 3 for the effect of π on the inflation mapping curve.) 23

24 Figure 6: Fiscal Policy to Avoid a Deflationary Spiral Suppose, as in Figure 4, a decrease in private-sector confidence lowers the inflation mapping curve from ABC to A 1 B 1 C 1, and suppose π t 1 = π as in the orthodox long-run equilibrium and is denoted by point D. In this case, π t would fall below ρ and a deflationary spiral would begin, unless preventive measures are taken. The implementation of fiscal stimulus (ɛ t ) that raises the inflation mapping curve to A 2 B 2 C 2 (or higher) averts the spiral. (See Figure 4 for the effect of ɛ t on the inflation mapping curve.) 24

25 6.2 Getting Out of a Deflationary Spiral Assume π t 1 < ρ. Then, by Proposition 4 and assuming no further shocks from period t onwards, a deflationary spiral will begin in period t. We can, however, rescue this economy by doing something in period t to ensure π t > ρ. As the discussion immediately following equation (11) makes clear, we have π t 1 < ρ < πt 1, c which implies that the zero lower bound on the nominal interest rate is binding (i t = 0). Therefore, we can seek guidance from equation (15). It is clear from equation (15) that the only way out is expansionary fiscal policy (ɛ t ). As we saw before, monetary policy is useless when the nominal interest rate is stuck at zero. Figure 7 demonstrates the role of fiscal stimulus in rescuing and economy from a deflationary spiral. To summarize our discussion of policy responses to a deflationary spiral, we state the following: Proposition 5. A deflationary spiral can be prevented by raising the target inflation rate, but only when the previous period s inflation is high enough. Other monetary policy tools have an ambiguous preventive role. Expansionary fiscal policy can be used for both prevention and cure. 7 Simulations We present some numerical simulations of the model given the parameter values in Table 2. Table 2: Parameter Values Ȳ t α ρ φ θ π θ Y π The first three parameters (Ȳt, α, ρ) appear in the aggregate demand equation (1). A value of 100 for the natural level of output is convenient as fluctuations in Y t Ȳt can be interpreted as percentage deviation of output from its natural level. The parameter 25

26 Figure 7: Fiscal Policy to Recover from a Deflationary Spiral Suppose the mapping from π t 1 to π t is represented by ABC and π t 1 is given by point D and, therefore, is less than ρ. Therefore, a deflationary spiral has already begun. However, a fiscal stimulus (ɛ t ) that raises the inflation mapping curve to A 1 B 1 C 1 (or higher) pushes π t above ρ, thereby stopping the spiral. (See Figure 4 for the effect of ɛ t on the inflation mapping curve.) 26

27 α = 1 implies that a one percentage point increase in the real interest rate reduces the output gap by one percentage point. The natural rate of interest ρ is equal to 2 percent, in line with historical data. The parameter φ appears in the Philips curve in equation (3). The value of 0.25 implies that inflation rises by 0.25 percentage point when output is one percent above its natural level. Finally, the last three parameters affect the Taylor rule in equation (5). The value of one half for the coefficients θ π and θ Y stems directly from Taylor (1993). For each percentage point that inflation rises above the Fed s inflation target, the federal funds rate rises by 0.5 percent. For each percentage point that real GDP rises above the natural level of output, the federal funds rate rises by 0.5 percent. Since θ π is greater than zero, our equation for monetary policy follow the Taylor principle: whenever inflation increases, the central bank raises the nominal interest rate by an even larger amount 3. Finally, the inflation target π is equal to 2 percent, reflecting the Fed s concerns of keeping inflation under control, a policy that started thirty years ago under chairman Paul Volcker. We present below the paths for output, inflation, nominal and real interest rates when the economy is hit by a four-period contractionary demand shock. In the first experiment, where the demand shock ɛ t is equal to -3 in periods 3 to 6 and zero elsewhere, the zero lower bound is not binding in any period. The dynamics of the economy are well known, with output and real interest rates being pro-cyclical (Figure 8(a) to 8(e)). In response to the demand shock in period 3, output falls below the natural level and inflation is lower than the Fed s target. Monetary policy becomes accommodative and with nominal interest rates falling faster than inflation, declining real interest rates boost output between period 4 and 6. Finally, when the demand shock subsides in period 7, output overshoots and then converges to the natural level and real interest rates remain below the natural rate 3 Clarida et al. (2000) examine data on inflation, interest rates, and output and estimate the coefficients of the Taylor rule at different points in time. During the Volcker-Greenspan area, they find that θ π = 0.72, implying that the Taylor rule under these two chairmen satisfied the Taylor principle. On the other hand, during the pre-volcker era from 1960 to 1978, they find that θ π = The negative value for θ π means that the Taylor rule did not follow the Taylor principle. The Fed raised interest rates in response to rising inflation but not enough, implying that real interest rates fell. The authors believe that the wrong specification of the Taylor rule led to the Great Inflation of the 1970s and propose some tentative explanations for why the Fed was so passive in that earlier era. 27

28 of interest. (a) Demand Shock (b) Output (c) Inflation (d) Real Interest Rate (e) Nominal Interest Rate Figure 8: Zero Lower Bound Not Binding In the second experiment, we increase the demand shock to -5 between periods 3 to 6 (Figure 9(a)) so that the zero lower bound becomes binding between period 5 to 7 but the economy does not fall into a deflationary spiral, which on its own, is quite remarkable since the proposed shock represents a 20 percent output contraction over a one-year period. To provide context, output in the U.S. fell by 46 percent during the Great Depression between 1929 and The introduction of the zero lower bound changes the model dynamics. In particular, it implies that monetary policy has no traction when nominal interest rates are zero and the correlation between output and real interest rates becomes negative. In contrast to the first experiment, output declines between periods 28

29 4 to 6 since real interest rates surge due to lower inflation and the nominal interest rate is stuck at zero. Deflation is only transitory however, and with no additional negative demand shock in period 7, output surges back above the natural level. As the zero lower bound becomes non-binding, output, inflation, and real interest rates are all put back on a convergent path to the stable equilibrium. (a) Demand Shock (b) Output (c) Inflation (d) Real Interest Rate (e) Nominal Interest Rate Figure 9: Zero Lower Bound Binding - No Deflationary Spiral In the third experiment, we further increase the demand shock to -6 for four consecutive periods to induce a deflationary spiral characterized by sustained decline in output, deflation, a zero lower bound binding in every period, and rising real interest rates (Figures 10(a) to 10(e)). From period 3 to 7, output declines due to increases in the real interest rate. However, once the demand shock subsides in period 7, output returns to 29

30 levels very close to the natural output level which can be very deceiving to policy-makers. If one only observes the changes in per capita gross domestic product, it looks like the economy is getting back on track, whereas nothing could be further from the truth. When the zero lower bound is binding and especially if it has been binding for several consecutive quarters, it is of utmost importance that policy-makers monitor both inflation and output to prevent the start of a deflationary spiral. (a) Demand Shock (b) Output (c) Inflation (d) Real Interest Rate (e) Nominal Interest Rate Figure 10: Deflationary Spiral In the previous section, we derived an analytical result showing that raising the Fed s inflation target or introducing a fiscal stimulus package can both help to prevent the occurrence of a deflation spiral; however, once the deflation spiral has started, only fiscal policy will get the economy out of trouble. We now quantify how large the policy response 30

31 has to be, either to prevent the start of a depression or to stop the depression after it started. We model fiscal stimulus by a positive demand shock in period 7 that follows four consecutive negative demand shocks at -6. We find that a one-time stimulus package of 2 percent of the natural level of output is not enough to break the deflation spiral. As informally advocated by Krugman (2008) and by Koo (2009), Congress needs to take bold measures and pass a stimulus of at least 4 percent of the natural level of output so that the zero lower bound ceases to be binding and the economy exits the deflation spiral (Figures 11(a) to 11(e)). (a) Demand Shock (b) Output (c) Inflation (d) Real Interest Rate (e) Nominal Interest Rate Figure 11: Large Fiscal Stimulus Can Break a Deflationary Spiral In our last thought experiment, we ask by how much should the Fed raise the infla- 31