Global Dynamics at the Zero Lower Bound

Transcription

1 Global Dynamics at the Zero Lower Bound William T. Gavin Alexander W. Richter Benjamin D. Keen Nathaniel A. Throckmorton June 9, ABSTRACT This article presents global solutions to standard New Keynesian models to show how economic dynamics change when the nominal interest rate is constrained at its zero lower bound (ZLB). We focus on the canonical New Keynesian model without capital, but we also study the model with capital, with and without investment adjustment costs. Our solution method emphasizes accuracy to capture important expectational effects of hitting the ZLB and returning to a positive interest rate. Although we do not model the large scale asset purchases used by the Fed since 9, our results suggest the economy may have trouble recovering if the interest rate remains at zero. At the ZLB, higher levels of technology lower employment and weaken aggregate demand, regardless of whether technology or discount factor shocks drive the interest rate to zero. Given the unconventional dynamics at the ZLB, we evaluate how monetary policy affects the likelihood of encountering the ZLB. We find that the probability of hitting the ZLB depends importantly on the monetary policy rule. A policy rule based on a dual mandate, such as the one proposed by Taylor (99), is more likely to cause ZLB events when the central bank places greater emphasis on output stabilization. Keywords: Monetary Policy; Zero Lower Bound; Global Solution Method JEL Classifications: E; E; E8; E6 Gavin, Research Division, Federal Reserve Bank of St. Louis, P.O. Box, St. Louis, MO (gavin@stls.frb.org); Keen, Department of Economics, University of Oklahoma, 8 Cate Center Drive, 7 Cate Center One, Norman, OK 79 (ben.keen@ou.edu); Richter, Department of Economics, Auburn University, Haley Center, Auburn, AL 689 (arichter@auburn.edu); Throckmorton, Department of Economics, Indiana University, S. Woodlawn, Wylie Hall, Bloomington, IN 7 (nathrock@indiana.edu). The authors thank Javier Birchenall and other seminar participants at the University of California, Santa Barbara. We also thank seminar participants at the Federal Reserve Bank of St. Louis and Renmin University in Beijing, China, and participants at the Midwest Macroeconomic meetings for helpful comments. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of St. Louis or the Federal Reserve System.

2 US Federal Funds Rate (%) Japan Overnight Rate (%) Employment to Population (%) US Japan Figure : U.S. and Japanese interbank lending rates (left panel) and employment-to-population percentages (right panel). Sources: Board of Governors of the Federal Reserve System, the Bank of Japan, the U.S. Bureau of Labor Statistics, and the Organisation for Economic Co-operation and Development. INTRODUCTION During the 8 financial crisis, the Fed flooded the market with money, adding about $6 billion in excess reserves to an economy that normally operates with about $ billion. This drove money market interest rates to their zero lower bound (ZLB). Four years after the crisis, the market remains saturated with money, money market interest rates remain near zero, and the economy is stagnant. The Fed has used two unconventional monetary policy tools to keep money market interest rates near zero and lower longer-term interest rates. The first tool is large scale asset purchases. As of December these purchases have added over $ trillion to the Fed s balance sheet, increasing excess reserves to $. trillion. The second tool is forward guidance, which signals to the public the Fed s expected policy rate path. The target has remained between and basis points since December 8, but the Fed has repeatedly changed its directive, generally to extend the length of time it expects to keep its policy rate near zero. Policymakers have been increasingly disappointed about the economy s failure to fully recover from the 8-9 recession. They continue to add to the stock of excess reserves, ensuring that the policy rate will remain at zero far into the future. Figure shows U.S. data from 99- for the federal funds rate and the employment-to-population percentage. The federal funds rate (left panel) has varied between 6. percent and zero since 99 and has been held below basis points since the fourth quarter of 8. During this period, the inflation rate has been at or below the Fed s long-run inflation objective, which led policymakers to shift their focus from the inflation target to the real economy. In policy statements, the Fed has given the weak state of the labor market (right panel) and the unusually slow recovery as a justification for forward guidance. Figure also plots Japanese data. The Bank of Japan sharply lowered its policy rate in 99, reaching basis points in 99. Since then it has remained between and basis points, while the employment-to-population percentage steadily fell from 6 percent to about 7. percent. The economy slightly rebounded in the mid-s, but following the financial crisis interest rates were reduced and the employment-to-population percentage fell further.

3 Why do ZLB events matter? Friedman (969), Kocherlakota (), and others argue that a nominal interest rate equal to zero is ideal because it promotes an optimal level of real cash balances. In the long run, that policy generates an inflation rate equal to the inverse of the risk-free real interest rate, which is inconsistent with the Federal Reserve s percent inflation rate target. Williams (9) claims central banks should embrace the ZLB during periods of economic weakness, since it demonstrates that monetary policymakers will do everything possible to stimulate an underemployed economy. According to Williams (9), the failure to hit the ZLB in past recessions was a sign of a suboptimal policy response to economic conditions. In recent years, Japan has come the closest of any major industrialized country to embracing the ZLB, during which the Japanese economy has endured anemic economic growth and slight deflation. Their experience generated a significant amount of research on the effects of the Bank of Japan s policy [see, for example, Krugman (998), Posen (998), Hoshi and Kashyap (), Eggertsson and Woodford (), and Braun and Waki (6)]. Many arguments against the ZLB are motivated, in part, by the recent Japanese experience. One set of arguments is based on the possibility of multiple equilibria at the ZLB. Schmitt-Grohé and Uribe () show that when a central bank follows a Taylor-type policy rule, the consequences of hitting the ZLB may include moving to an undesirable low output/low inflation equilibrium. Summers (99) argues that the inflation target should not be set to zero, but rather to some higher number precisely to avoid hitting the ZLB when conducting countercyclical policy. He argues that the central bank s ability to achieve its employment and output targets are constrained when the interest rate is pegged at zero. Reifschneider and Williams (), Chung et al. (), and Coibion et al. () discuss the optimal policy when ZLB events are possible and provide analysis of the welfare losses during ZLB events. Chung et al. () argue that the literature understates the probability of hitting the ZLB because past analyses have not taken proper account of model uncertainty, including uncertainty about the shock processes hitting the economy. A practical criticism is that a low nominal interest rate target may be misinterpreted by households. Bullard () notes that attempting to stimulate the economy by promising to keep the interest rate at zero may backfire as inflation expectations may fall rather than rise. Indeed, Del Negro et al. () provide evidence that recent promises to maintain the ZLB for an extended period have been interpreted as a signal that the central bank believes the economic outlook has worsened. A major contribution of this study is that it examines economic dynamics at the ZLB using global solutions to nonlinear New Keynesian models. Most of the existing literature uses loglinearized New Keynesian models. However, using log-linearized models creates the potential for large approximation errors. For example, Braun et al. () argue that log-linearized models often lead to incorrect inferences about existence of equilibrium, uniqueness, and local dynamics. Braun et al. () and Fernández-Villaverde et al. () also provide explicit examples of the mistakes See Bullard () for a summary of this argument and further references. The paper closest to ours is Fernández-Villaverde et al. (), which uses different methods to generate global solutions to the New Keynesian model with capital and a larger set of shocks. It studies the properties of the equilibrium and fiscal multipliers at the ZLB. Basu and Bundick () use global solutions to a nonlinear New Keynesian model to show that the effect of uncertainty on aggregate demand is magnified at the ZLB. Within the policy arena, three influential papers that use linearized New Keynesian models to study the consequences of ZLB are Eggertsson and Woodford (), Gertler and Karadi (), and Werning (). Papers that address the likelihood of hitting the ZLB in linearized models include Reifschneider and Williams (), Hatcher (), Chung et al. (), and Gavin and Keen ().

4 resulting from log-linearized models evaluated at the ZLB. Our paper avoids these problems by using the global nonlinear solution procedure described in Richter et al. (). Our results suggest being at the ZLB may actually delay the recovery. We present global solutions to standard New Keynesian models to show how economic dynamics change when the nominal interest rate is pegged at its ZLB. We find that when the ZLB binds, positive technology shocks, which would normally aid the recovery, have unconventional effects. At the ZLB, higher levels of technology lower employment and weaken aggregate demand, regardless of whether technology or discount factor shocks drive the interest rate to zero. While no one believes interest rates fell to zero in 8 due to a series of positive technology shocks, our main interest is to learn how the economy reacts to technology shocks when the ZLB binds. Our findings provide a compelling reason to avoid the ZLB that is directly relevant for forward guidance. We show how the dual mandate established in the Taylor rule affects the probability and frequency of ZLB events. Our results indicate that the probability of hitting the ZLB rises when the central bank places more emphasis on output stabilization and falls when there is more emphasis on price stability. In the New Keynesian model, the policies that reduce the likelihood of hitting the zero lower bound also tend to deliver higher welfare. Most of the work on the ZLB is done in models without capital. accumulation is a key feature because it gives households another margin to smooth consumption. This saving channel reduces the volatility of consumption and the nominal interest rate, decreasing the frequency of ZLB events. Finally, we show that investment adjustment costs inhibit this mechanism and increase the frequency of ZLB events. Section briefly describes the alternative models. Section describes the calibration and solution procedure, and sections through 6 present the results. These sections report on the model solutions, the economic dynamics at the ZLB, the likelihood of hitting the ZLB, and the welfare consequences of ZLB events. Section 7 concludes. ECONOMIC MODELS This section presents three alternative models. The baseline specification is a New Keynesian model with Rotemberg (98) price adjustment costs. Model assumes stochastic processes for the discount factor and technology but does not include capital. Models and incorporate capital accumulation into Model, and Model also includes investment adjustment costs.. MODEL : BASELINE A representative household chooses sequences{c t,n t,b t } t= to maximize expected lifetime utility, given by, { } E β t logc t χ n+η t, () +η t= Although we set the lower bound on the policy rate equal to zero, these same unconventional dynamics would occur if the bound was set to a small but positive value. The key is the existence of a lower bound, which prevents the Fed from responding to inflation. This is important because the Fed has not targeted a policy rate equal to zero. There are some caveats to this pessimistic conclusion. First, we have not explicitly modeled the Fed s unconventional policies. The large scale asset purchase program seems to have kept deflation at bay. Second, Wieland () uses structural VAR evidence to argue that these unconventional dynamics did not occur following shocks to the earthquake/tsunami in Japan or recent oil supply shocks. He also shows that a New Keynesian model with financial frictions does not display unconventional dynamics, but it does imply that expansionary monetary and fiscal policies will be less effective at the ZLB than others have found. His results do not preclude the possibility that the ZLB may prevent positive technology shocks from leading to a recovery as they would in normal times.

5 where /η is the Frisch elasticity of labor supply, c t is consumption of the final good, n t is labor hours, β, and β t = t i= β i for t >. β i is a time-varying subjective discount factor that evolves according to β i = β(β i /β) ρ β exp(ε β,i ), whereβ is the stationary discount factor, ρ β <, andε β,i N(,σ β ). We normalizeβ = β. The representative household s choices are constrained by c t +b t = w t n t +r t b t /π t +τ t +d t, whereπ t = p t /p t is the gross inflation rate,w t is the real wage,τ t is a lump-sum tax,b t is a oneperiod real bond, r t is the gross nominal interest rate, and d t are profits from intermediate firms. Solving the household s utility maximization problem yields the following optimality conditions w t = χn η tc σ t, () = r t E t {β t+ (c t /c t+ ) σ /π t+ }. () The production sector consists of monopolistically competitive intermediate goods firms who produce a continuum of differentiated inputs and a representative final goods firm. Each firm i [,] in the intermediate goods sector produces a differentiated good, y t (i), with identical technologies given by y t (i) = z t n t (i), where n t (i) is the level of employment used by firm i. z t represents the level of technology, which is common across firms and follows z t = z(z t / z) ρz exp(ε z,t ), where z is steady-state technology, ρ z <, and ε z,t N(,σz ). Each intermediate firm chooses its labor supply to minimize its operating costs,w t n t (i), subject to its production function. Using a Dixit and Stiglitz (977) aggregator, the representative final goods firm purchasesy t (i) units from each intermediate goods firm to produce the final good, y t [ y t(i) (θ )/θ di] θ/(θ ), where θ > measures the elasticity of substitution between the intermediate goods. Maximizing profits for a given level of output yields the demand function for intermediate inputs given by y t (i) = (p t (i)/p t ) θ y t, where p t = [ p t(i) θ di] /( θ) is the price of the final good. Following Rotemberg (98), each firm faces a cost to adjusting its price, which emphasizes the potentially negative effect that price changes can have on customer-firm relationships. Using the functional form in Ireland (997), real profits of firm i are [ (pt ) θ ( (i) pt (i) d t (i) = Ψ t p t p t ) θ ϕ ( ) ] pt (i) πp t (i) y t, where ϕ determines the magnitude of the adjustment cost, Ψ t is real marginal costs, and π is the steady-state gross inflation rate. Each intermediate goods firm chooses its price level, p t (i), to maximize the expected discounted present value of real profits E t k=t λ t,kd k (i), where λ t,t, λ t,t+ = β t+ (c t /c t+ ) σ is the stochastic pricing kernal between periods t and t+, and λ t,k k j=t+ λ j,j. In a symmetric equilibrium, all intermediate goods firms make the same decisions and the optimality condition becomes ( πt ) ϕ π πt π = ( θ)+θψ t +ϕe t [ ( πt+ ) λ t,t+ π πt+ π ] y t+. () y t

6 In the absence of price adjustment costs (i.e. ϕ = ), the real marginal cost of producing a unit of output equals(θ )/θ, which is the inverse of the firm s markup of price over marginal cost. Each period, the fiscal authority finances a constant level of discretionary spending, ḡ, by levying lump-sum taxes. The monetary authority sets policy according to r t = max{, r(π t /π ) φπ (y t /ȳ) φy }, whereπ is the inflation rate target and φ π and φ y are the policy responses to inflation and output. In this paper, the output gap is defined as the deviation of output from its steady state. We use this measure because we believe policymakers, in the short-to-medium term, assume potential output grows at a relatively constant rate. Potential output measures are revised in the long run following incoming information about shocks, but the revisions occur well after the temporary economic effects from sticky prices have dissipated. In our model, a positive technology shock causes output to rise relative to its steady state and inflation to fall. For our baseline calibration, the lower inflation dominates the higher output leading to a lower nominal interest rate. Alternatively, the output gap can be defined as the difference between actual output and the level of output in the absence of nominal frictions. Under this definition of the output gap, a positive technology shock would result in a negative output gap because price frictions would prevent actual output from rising as much as it would in the flexible price economy. Thus, the downward pressure on the nominal interest rate coming from low inflation would be reinforced by the additional downward pressure coming from a negative output gap. The aggregate resource constraint is given by c t + ḡ = [ ϕ(π t / π ) /]y t = ỹ t, where ỹ t includes the value added by intermediate firms, which is their output minus quadratic price adjustment costs. Equilibrium is characterized by the household s and firm s optimality conditions, the government s budget constraint, the bond market clearing condition (b t = ), and the aggregate resource constraint.. MODEL : BASELINE WITH CAPITAL Models adds capital accumulation to Model, but assumes a constant discount factor. Assuming, β t = β for all t, the household chooses sequences {c t,k t,i t,n t,b t } t= to maximize () subject to c t +i t +b t = w t n t +r k tk t +r t b t /π t +τ t, () k t = ( δ)k t +i t, (6) wherei t is investment,k t is the capital stock, andrt k is the real capital rental rate. The representative household s optimality conditions include (), (), and the consumption Euler equation, given by, = βe t {(c t /c t+ ) σ (rt+ k + δ)}. (7) Each firmi [,] in the intermediate goods sector produces a differentiated good,y t (i), with identical technologies given by y t (i) = z t k t (i) α n t (i) ( α), where k t (i) and n t (i) are the levels of capital and employment used by firm i. Every intermediate firm then chooses its capital and labor inputs to minimize its operating costs,r k t k t (i)+w t n t (i), subject to its production function. The firm pricing equation, (), remains unchanged, except that the definition of the marginal cost changes. The aggregate resource constraint is now given by c t +i t +ḡ = ỹ t.

7 Constant of Relative Risk Aversion σ Inflation Target π. Frisch Elasticity of Labor Supply /η Inflation Coefficient: MP Rule φ π. Elasticity of Substitution between Goods θ 6 Output Coefficient: MP Rule φ y. Steady State Government Spending Share ḡ/ȳ. Stationary z Rotemberg Adjustment Cost Coefficient ϕ 8. Persistence ρ z.8 Leisure Preference Parameter χ.7 Shock Standard Deviation σ z. Depreciation Rate δ. Stationary Discount Factor β.99 Cost Share of α. Discount Factor Persistence ρ β.8 Investment Adjustment Cost ν. Discount Factor Standard Deviation σ β. Table : Baseline calibration. MODEL : BASELINE WITH CAPITAL AND INVESTMENT ADJUSTMENT COSTS Model adds investment adjustment costs to Model. now evolves according to [ k t = ( δ)k t +i t ν ( ) ] it, i t where ν measures the size of the adjustment costs. Optimality yields a new consumption Euler equation, which replaces (7) in Model, and an investment Euler equation, given by, q t = βe t {(c t /c t+ ) σ( )} rt+ k +q t+ ( δ) [ = q t ν ( ) it ν i ( ) ] { ( ) σ ( ) ( ) } t it ct it+ it+ +βνe t q t+, i t i t i t c t+ i t i t whereq t is Tobin sq. The aggregate resource constraint is the same as in Model, except that both investment and output now include resources lost to investment adjustment costs. CALIBRATION AND SOLUTION TECHNIQUE The models in section are calibrated at a quarterly frequency and the parameters are given in table. The risk-free real interest rate is set equal to percent annually, which implies a stationary quarterly discount factor,β, equal to.99. We set the persistence of the discount factor,ρ β, equal to.8 and the standard deviation of the shock, σ β, equal to.. We follow Fernández-Villaverde et al. () who chose these parameters so that a discount factor shock has a half life of about quarters and an unconditional standard deviation of. percent. The Frisch elasticity of labor supply, /η, is set to, which is consistent with Peterman () who estimates that the Frisch elasticity lies between.9 and.. The leisure preference parameter, χ, is calibrated so that steadystate labor equals / of the available time. s share of output, α, is set to. and the depreciation rate, δ, equals. percent per quarter. The investment adjustment cost (IAC) parameter, ν, is calibrated to., which follows Christiano et al. (). The elasticity of substitution between intermediate goods, θ, is set to 6, which corresponds to an average markup of price over marginal cost equal to percent. The costly price adjustment parameter, ϕ, is calibrated to 8., which is consistent with a Calvo (98) price-setting specification where prices change on average once every four quarters. In the policy sector, the steady-state gross inflation rate, π, is set to., which implies an annual inflation rate target of percent. The steady-state ratio of government spending to output is calibrated to percent. In our baseline case, the coefficients on inflation and output in the policy rule are set to. and., which is consistent with Taylor (99). 6

8 Steady-state technology, z, is normalized to. The likelihood of hitting the ZLB depends critically on the parameters of the technology process (σz and ρ z). When we set these parameters to values typically used in quantitative New Keynesian and Neoclassical models, determinacy is not guaranteed on the entire state space of our models. A determinate solution requires that σz and ρ z are not too large for a given coefficient on the output gap in the policy rule. 6 Thus, we set ρ z =.8 andσ z between and. percent per quarter, depending on the model, which pushes the standard deviation ofz t toward values that are common in the literature. When analyzing the effects of ZLB spells in a linear model, researchers typically assume a large preference or natural rate shock hits the economy to drive the nominal interest rate to zero [Eggertsson and Woodford (), Jung et al. (), Erceg and Linde (), Christiano et al. ()]. The resulting ZLB event then lasts a predetermined duration. 7 Once the ZLB event ends, the nominal interest rate becomes positive and market participants do not expect it to return to zero. This type of experiment relies on unrealistically large shocks and arbitrarily fixes the frequency and duration of ZLB events. Additionally, Braun et al. () and Fernández-Villaverde et al. () show that linear solutions lead to significant approximation errors at the ZLB. Our global nonlinear solution captures expectational effects and allows us to perform simulations without giving households perfect foresight about the end of a ZLB event. Moreover, Richter et al. () show that the Euler equation errors from our global nonlinear solution are small near the ZLB. The models are simulated using draws from the distributions for the discount factor and technology shocks. The state space is discretized to minimize extrapolation of the policy functions during the simulation. As an example, we plot the simulated distributions of the state variables for Model in figure and show that they are contained within the bounds of the state space. 8 We simulate the model for, periods to obtain an accurate sample of ZLB events. Panel (a) shows the unconditional distributions of technology, the discount factor, and the nominal interest rate. The state space for technology lies within ±8.8 percent of the steady-state value, which is normalized to unity in our simulations. The state space of the discount factor lies between ±.9 percent of the steady state, which is equal to Over these states, the net nominal interest rate is distributed over a range of to percent, with a large mass (. percent of the simulated quarters) between and basis points. The steady-state quarterly rate is. percent. Panel (b) shows the distribution of the discount factor and technology conditional on the ZLB binding. Fernández-Villaverde et al. () also find that high levels of technology are associated with low interest rates. The reason is that for many monetary policy rules (including Taylor rules 6 The ZLB is equivalent to a fixed interest rate regime with a truncated distribution on the nominal interest rate. The solution is not determinate with an insufficient probability of returning to an interest rate rule that aggressively responds to inflation. Davig and Leeper (7) discuss determinacy in linear models where the monetary coefficient follows a -state Markov process. Richter and Throckmorton () discuss determinacy in a nonlinear model and show that determinacy imposes a clear tradeoff between the frequency and duration of ZLB events. 7 Erceg and Linde () use an alternative linear solution method that endogenizes the duration of the ZLB as a function of the government spending shock. However, the algorithm adds a sequence of current and future innovations so that the ZLB is satisfied conditional on the state. Performing a monte carlo simulation with such a method is infeasible, and therefore it too cannot be used to characterize the distribution of ZLB events. They also rely on a series of large preference shocks to hit the ZLB and there is no expectation to return after exiting. 8 For conciseness, we do not show the distributions for Models and, but they are available upon request. 9 These bounds are chosen so that they encompass percent of the probability mass of the distribution of technology and the discount factor in the state and to minimize extrapolation of the policy functions in simulations of the model. We also specify a very dense discretized state space, so the location of the kink in the policy function is accurate. This is particularly important since it affects the frequency and duration of ZLB events. 7

9 .. Discount Factor.. Interest Rate (a) Model : Unconditional Discount Factor (b) Model : Conditional on ZLB Figure : Model distributions as a percentage of a, period simulation. The variables are in percent deviations from steady state. The dashed lines are the bounds of the state space. The solid lines are the theoretical unconditional distributions of the state variables scaled for comparison with the distributions conditional on the ZLB. and fixed money growth rules), high levels of technology are associated with low inflation and low nominal interest rates. If technology is high enough and the central bank follows a Taylor rule, the nominal interest rate will hit its ZLB. Kiley () uses U.S. data to show that periods of high labor productivity growth have been associated with relatively low inflation and shows that this result could be caused by the Fed s policy rule. MODEL : STATES OF THE ECONOMY, ECONOMIC DYNAMICS, AND THE ZLB This section shows the nonlinear solution to Model as a function of the two state variables, the discount factor and technology. Our analysis assumes monetary policy is based on Taylor s (99) original specification withφ π =. and φ y =. when r t >. All of the variables are given in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are presented in levels. Price adjustment costs are measured as a percent of output. Figure shows three-dimensional contour plots of the non-predetermined variables over the entire state space. These policy functions give a complete picture of the model solution. The shaded areas represent the states of technology and the discount factor where the (net) nominal interest rate equals zero. This region illustrates that the nominal interest rate only hits the ZLB when either the technology state or the discount factor state are unusually high. Since these maps contain a lot of information and can be difficult to read, figure plots two-dimensional representations of two alternative cross sections of the contour plots. In figures and, the solid (black) line shows the cross section where the discount factor state is fixed at its stationary value (β =.99) and the 8

10 Discount Factor Discount Factor Discount Factor Discount Factor GAVIN ET AL.: GLOBAL DYNAMICS AT THE ZERO LOWER BOUND β =.99 β =. Discount Factor Discount Factor..7.7 Consumption Labor Real Wage Rate. 8 8 Discount Factor Inflation Rate Adjusted Output Real Marginal Cost Discount Factor Discount Factor Nominal Interest Rate Ex ante Real Interest Rate Rot. Adjustment Cost Figure : Model non-predetermined variables as a function of technology and the discount factor states. All variables are in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are in levels. Adjustment costs are measured as a percent of output. The shaded region indicates where the ZLB binds. The solid (black) and dashed (blue) horizontal lines correspond to the cross sections whereβ =.99 andβ =. dashed (blue) line shows the cross section where the discount factor is held constant at, which is the minimum value where the ZLB binds when technology is at its steady state. In figure, the darker (entire) shaded region indicates where the ZLB binds when β =.99 (β = ). We begin by examining the cross section where the discount factor is fixed at its stationary value. First consider the region of the state space where the ZLB does not bind; for Model this includes states where technology ranges between 8.8 percent below and. percent above its steady state. When technology is below its steady state, workers are less productive and firms per 9

11 β =.99 β =. Consumption Inflation Rate Nominal Interest Rate Labor Hours. Adjusted Output Ex ante Real Interest Rate Real Wage Rate Real Marginal Cost Rot. Adjustment Cost Figure : Model non-predetermined variables as a function of the technology states. In the solid line the discount factor state (β ) is fixed at its deterministic steady state value and in the dashed line the discount factor state is fixed at. All variables are in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are in levels. Adjustment costs are measured as a percent of output. The dark (entire) shaded region indicates where the ZLB binds whenβ =.99 (β = ).

12 unit marginal cost of production is higher. At low levels of technology, firms raise their prices and reduce their output and labor demand. With less output available for consumption, the household works more to moderate the decline in consumption. The higher labor supply dominates the drop in labor demand so that the equilibrium level of labor is higher and the real wage is lower. The household also believes technology will gradually return to its steady state and, as a result, expects its future consumption to increase. Higher expected future consumption is reflected in an elevated ex-ante real interest rate. At higher levels of technology and before the ZLB binds, workers are more productive and firms choose lower prices and higher output. The household consumes more but also desires more leisure. In this part of the state space, the decline in the labor supply dominates the increase labor demand so that labor hours are lower and real wages are higher. The natural tendency for technology to return to the steady state means that households expect lower consumption in the future and observe a lower real interest rate. Next consider the states where the ZLB binds (the darker shaded region in figure ), which includes technology states that are more than. percent above the steady state. In this case, higher technology continues to lower per unit production costs. The firms react by lowering their prices and raising their output, which further increases their price adjustment costs. The additional decline in expected inflation combined with a zero nominal interest rate forces the ex-ante real interest rate to rise. The household elects to sharply reduce its consumption and increase its labor supply to capitalize on those increased returns. Firms respond to the reduction in consumption demand by further lowering their prices and decreasing labor demand. The drop in labor demand dominates the increase in labor supply, so that both total labor hours and the real wage drop. Now turn to the cross section where the discount factor is held constant at(dashed line). The ZLB region now includes all positive technology states. The main reason for showing this cross section is to highlight that the unconventional response of the economy to a positive technology shock at the ZLB does not depend on a high level of technology to drive the economy to its ZLB. The policy functions in this cross section display the same qualitative properties as the cross section where β =.99. Looking at the highest discount factor shown in figure (β =.8), it is clear that the same dynamics continue to apply even when technology is below its steady state. Indeed, this is the area of the state space that is often considered in ZLB studies. If there was an even higher discount factor shock as modeled by Fernández-Villaverde et al. (), Christiano et al. (), and Schmitt-Grohé and Uribe (), then these same dynamics would appear. Figure compares the impulse responses to a one-time percent positive technology shock under two cases the baseline case (dashed line), which is initialized at the stochastic steady state withβ at its deterministic steady state, and the ZLB case (solid line), where a sequence of discount factor shocks keep β constant and equal to (i.e., the minimum value where the ZLB binds when technology is at its steady state). The horizontal dash-dotted lines are the stochastic steady-state values of inflation and the (net) interest rates, which are not equal to the determinastic steady state due to expectational effects of hitting the ZLB. In short, this exercise compares the conventional responses to a positive technology shock when the ZLB never binds to the responses based on a counterfactual where the ZLB always binds due to successive discount factor shocks. Intuitively, the series of discount factor shocks can be thought of as a persistent reduction in consumer confidence, an ongoing global savings glut, or a decision by the Fed to hold the policy rate at zero. The results in the baseline case are standard. A persistent technology shock increases output and causes inflation and the nominal interest rate to fall. According to the Taylor rule, the nominal interest rate falls more than the inflation rate, so there is also a decline in the ex-ante real rate,

13 Baseline ZLB.6 Consumption. Inflation Rate. Nominal Interest Rate Labor Hours Real Wage Rate Adjusted Output Real Marginal Cost Ex ante Real Interest Rate Rot. Adjustment Cost 8 6 Figure : Model impulse responses to a one-time percent positive technology shock in period one. The baseline case (dashed line), where the ZLB does not bind, is initialized at the stochastic steady state with a discount factor equal to.99. In the ZLB case (solid line), the discount factor is constant and equal to one, so the ZLB always binds. The horizontal dash-dotted lines are the stochastic steady values of inflation and the (net) interest rates. The remaining variables are given in percent deviations from their respective stochastic steady state values. which increases consumption. A positive technology shock acts as a positive labor productivity shock, which decreases labor and raises real wages. The technology shock induces small price adjustment costs, so the increase in consumption is slightly lower than the increase in output. In the ZLB case, a positive technology shock has unconventional effects, as the impact on most of the real variables is the exact opposite of the impacts described in the baseline case. At the stochastic steady state with β fixed at, the higher discount factor imposes slight deflation. A positive technology shock leads to further deflation. With the nominal interest rate constrained at zero, the ex-ante real interest rate sharply rises. In response, consumption and labor both fall and the effect on output is dampened relative to the baseline case. The greater deflation is associated with a sharp rise in price adjustment costs. In both cases, the level of technology returns to its steady state about quarters after the initial impact of the shock. The Fisher relation holds in the stochastic steady state with β fixed at one. However, this is not readily apparent in figure since the expected inflation rate is not shown. Expected inflation is different than current inflation since the household expects mean reversion in the discount factor process.

14 MODELS AND : STATES OF THE ECONOMY AND THE ZLB This section shows how the model solution changes when capital accumulation is added to Model. In Model, the only way for the household to smooth consumption is by varying their labor supply. In Model, capital gives the household another margin to smooth consumption. This model contains two state variables the lagged capital stock (k ), which is endogenous, and technology (z ). Figure 6 shows the three-dimensional contour plots of the non-predetermined variables over the entire state space. Capturing a complete picture of the model solution is particularly important in models with an endogenous state variable. In Model, the discount factor and technology states are independent and, therefore, any one realization of the discount factor is just as likely at high and low technology states. In Model, the capital and technology states are not independent. At low (high) technology states, the capital state is most likely below (above) its steady-state value. The contour plots capture these endogenous dynamics and provide a better picture of the model solution. In general, the patterns for consumption, inflation, and the nominal interest rate are qualitatively similar to Model, however the ability to invest means consumption is less volatile. We begin by examining the behavior of the economy when the ZLB does not bind. Regardless of the capital state, higher technology states are associated with a lower marginal cost and lower inflation. Firms increase their production and labor demand. With more output available to divide between consumption and investment, both variables increase. To smooth its consumption across time, the household reduces its labor supply and increases its investment in capital. Whether higher technology states increase or decrease the equilibrium level of labor when r > depends on how the capital state co-moves with the technology state. When capital is held fixed at its steady-state value, the increase in labor demand dominates the decrease in labor supply, causing the equilibrium level of labor and the real wage rate to rise. Alternatively, if the capital state rises with technology, the decrease in labor supply dominates the increase in labor demand, causing the real wage rate to rise and the equilibrium level of labor to fall. These two alternative cross sections of the contour plots are shown in figure 6 and in figure 7, which plots a two-dimensional representation of these cross sections. The solid (black) line shows the cross section where capital is held fixed at its steady-state value and the dashed (blue) line shows the cross section where capital increases along the diagonal of the state space. In figure 7, the darker (entire) shaded region indicates the area of the state space where the ZLB binds in the steady-state (diagonal) cross section. The differences between the steady-state and diagonal cross sections are shown in figure 7. In the diagonal cross section where the capital state increases with the technology state, the marginal product of capital is lower in higher technology states. From the household s perspective, this makes investment less attractive as a consumption smoothing channel. The household responds by increasing consumption and decreasing labor supply more than in the cross section where the capital state is held fixed at its steady-state value. This is clear from the slopes of the investment, consumption, and labor policy functions. The policy function for the rental rate of capital is also qualitatively different between these cross sections. In the steady-state cross section, higher levels of technology and labor raise the marginal product of capital and the rental rate of capital due to complementarity. In the diagonal cross section, higher technology states are associated with more capital and less labor. The negative effects of capital and labor dominate the positive effects from technology so that the marginal product of capital and the capital rental rate decline whenr >. Another difference between these two cross sections is the behavior of the ex-ante real interest rate. Unlike Model, which only has one asset, Model has two assets capital and bonds. Ar-

15 .9.8. GAVIN ET AL.: GLOBAL DYNAMICS AT THE ZERO LOWER BOUND k = k k =k diag Consumption 8 8. Labor. 8 8 Wage Rate Inflation Rate Adjusted Output 8 8 Marginal Cost 8 8 Investment Interest Rate Ex ante Real Interest Rate Adjustment Cost Rental Rate Figure 6: Model non-predetermined variables as a function of capital and the technology states. The solid line indicates the cross section of the state space with capital in steady state, and the dashed line indicates the diagonal cross section where capital positively co-moves with technology in the state space. All variables are in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are in levels. Adjustment costs are measured as a percent of output. The shaded region indicates where the ZLB binds.

16 k = k k =k diag Consumption Inflation Rate Nominal Interest Rate Labor Hours Real Wage Rate Adjusted Output Real Marginal Cost Investment Ex ante Real Interest Rate Rot. Adjustment Cost Real Rental Rate Figure 7: Model non-predetermined variables as a function of technology. The solid line indicates the cross section of the state space with capital in steady state, and the dashed line indicates the diagonal cross section where capital positively co-moves with technology in the state space. All of the variables are in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are presented in levels. Adjustment costs are measured as a percent of output. The shaded region indicates where the ZLB binds.

17 Consumption Labor Hours Real Wage Rate Inflation Rate Adjusted Output Marginal Cost Investment Nominal Interest Rate Ex ante Real Interest Rate Rot. Adjustment Cost Rental Rate Figure 8: Model non-predetermined variables as a function of technology. All of the variables are in percent deviations from their deterministic steady state, except inflation and the (net) interest rates, which are in levels. Adjustment costs are measured as a percent of output. The capital stock and investment states (k, i ) are fixed at their deterministic steady state values. The shaded region indicates where the ZLB binds. 6

18 bitrage implies the expected rates of return on investment and bonds are equal. Thus, the expected rental rate of future capital moves positively with the current ex-ante real interest rate. In the steady state (diagonal) cross section, the real interest rate rises (falls) in higher technology states. It is interesting that in both cross sections, the ex-ante real interest rate falls in technology states that are high, but not high enough for the ZLB to bind. In these states, the unconventional dynamics that occur at the ZLB cause the household to expect the rental rate of capital to fall and consumption growth to slow. Both of these effects cause the real interest rate to fall before the ZLB is hit. In the steady-state (diagonal) cross section, the ZLB binds when technology is more than. (.) percent above its steady state. The qualitative properties of the policy functions when the ZLB binds are nearly identical across all possible cross sections. The mechanism that distorts the economy is essentially the same as Model. As the real marginal cost continues to decline in higher technology states, inflation falls. With the nominal interest rate pegged at zero, the ex-ante real interest rate rises. When the household s demand falls, both consumption and investment decrease. Firms respond to the lower demand by further reducing their prices and sharply cutting their labor demand, which causes the equilibrium level of labor and the real wage rate to fall. Lower consumption and investment pushes down output, despite the high technology state. As output falls, the household sharply reduces its investment to smooth consumption. Our results make clear that Model faces the same unconventional dynamics as Model ; equilibrium labor, investment, and consumption all fall sharply at the ZLB. Such a rapid adjustment in investment is at odds with the data and motivates us to include investment adjustment costs (IAC) in Model. This model contains three state variables lagged capital stock (k ) and lagged investment (i ), which are both endogenous, and technology (z ). With three state variables, there is no easy way to present the entire solution. Although the capital and investment states are both endogenous and likely to vary with the technology state, they are significantly less volatile. This reduction means the qualitative properties of the policy functions are less variable across alternative cross sections. Figure 8 plots a specific cross section of the model solution, where the capital and investment states are fixed at their respective steady-state values. The policy functions show that properties of Model are much closer to Model than Model, as both the equilibrium labor supply and the real interest rate are lower in higher technology states. The ZLB binds (the shaded region) in technology states that are more than. percent above steady state. At the ZLB, agents would again like to save more as the real interest rate rises. In this model, they can save by investing, but the costs of adjusting investment reduce the volatility of investment so much that consumption is almost as volatile as total output. Also, investment appears to adjust less than consumption as the economy moves to higher levels of technology. 6 THE LIKELIHOOD OF HITTING THE ZLB AND WELFARE This section examines the likelihood of hitting the ZLB in Models and using, quarter simulations of the models. The results are not strictly comparable across models because they are based on different assumptions about the shocks, but they provide a qualitative indication for how the frequency and duration of ZLB events differ. Our main result is that policymakers can reduce the likelihood of hitting the ZLB by de-emphasizing the dual mandate. This can be accomplished Although the rental rate of capital initially falls in higher technology states, the household expects capital to decline and, therefore, expects the rental rate of capital to increase in the future, which is consistent with a rising real interest rate. 7

19 ZLB Binds ZLB Spells Std. Dev. (% of mean) φ y % of quarters Average Longest Output Inflation Nom. Int. Rate (a) Model : No capital, technology and discount factor shocks. φ π =.,ρ z =.8, σ z =.9, ρ β =.8, and σ β = (b) Model :, only technology shocks. φ π =., ρ z =.8, and σ z =.9. Table : Volatility implications of a dual mandate. ZLB Binds ZLB Spells Std. Dev. (% of mean) φ π % of quarters Average Longest Output Inflation Nom. Int. Rate (a) Model : No capital, technology and discount factor shocks. φ y =., ρ z =.8, σ z =.9, ρ β =.8, and σ β = (b) Model :, only technology shocks. φ y =., ρ z =.8, and σ z =.9. Table : Volatility implications of alternative weights on the inflation gap. by either lowering the weight on the output gap (φ y ) or raising the weight on inflation (φ π ). Table shows the effect of reducing the weight on output while holding the weight on inflation constant at φ π =.. We begin with the original Taylor (99) specification, φ y =., and reduce this coefficient by increments of.. In Model, the ZLB binds in 8. percent of the quarters in our simulation whenφ y =.. This value monotonically falls withφ y and equals.6 8