Advanced Topics in Monetary Economics II 1

Transcription

1 Advanced Topics in Monetary Economics II 1 Carl E. Walsh UC Santa Cruz August 19-23, c Carl E. Walsh, Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

2 Course outline 1 New Keynesian monetary policy models and nominal rigidities; 2 Optimal policy and the zero lower bound; 3 Uncertainty and frictions in labor markets; 4 The open economy; 5 Monetary and fiscal interactions. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

3 The basic new Keynesian model 1 Households purchasing consumption goods and supplying labor services. 2 Monopolistic competition characterizing goods and labor market. 3 Sticky and stagged adjustment of prices and wages. 4 A nominal interest rate employed as the instrument of monetary policy. 5 Agents behave optimally given constraints they face. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

4 Households Continuum of households indexed by h whose utility is a function of a composite consumption good C t (h), real money balances M t (h)/p t, and leisure 1 H t (h), where H t (h) is the time (hours) devoted to market employment. Household preferences given by [ E t β i C t+i (h) 1 σ + γ ( ) ] 1 b Mt+i χ H t+i (h) 1+η. i=0 1 σ 1 b P t+i 1 + η The consumption aggregate that yields utility is defined as [ 1 C t (h) = 0 ] θ c jt (h) θ 1 θ 1 θ dj θ > 1. where c jt (h) is type j good. The parameter θ governs the degree of imperfect competition in goods market. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

5 Household demand for individual final goods Regardless of the level of C t (h), always optimal to purchase the least cost combination of the individual goods to achieve C t (h): 1 [ 1 min p jt c jt (h)dj + ψ t {C t (h) {c jt } 0 0 The demand for good j can be written as ] θ } c jt (h) θ 1 θ 1 θ dj c jt (h) = ( pjt ψ t ) θ C t (h). (1) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

6 Households From the definition of the composite level of consumption, this implies C t = = Solving for ψ t, [ 1 0 ] θ c θ 1 θ 1 θ jt dj ( ) 1 θ [ 1 ψ t 0 [ (pjt 1 ) ] θ 1 θ θ = C t 0 ψ t ] θ pjt 1 θ θ 1 dj Ct. [ 1 P t ψ t = 0 dj θ θ 1 ] 1 pjt 1 θ dj 1 θ. (2) The Lagrangian multiplier is the appropriately aggregated price index for consumption and hence ( ) θ pjt c jt (h) = C t (h). P t Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

7 Firms Continuum of firms of measure 1. Firms maximize profits, subject to three constraints: 1 The first is the production function summarizing the technology available for production. For simplicity, ignore capital, so output is a function solely of labor input L jt and an aggregate productivity disturbance Z t : c jt = Z t L jt, E(Z t ) = 1. 2 The second constraint on the firm is the demand curve each faces: c jt = ( pjt P t ) θ C t. 3 The third constraint is that each period some firms are not able to adjust their price. This nominal rigidity is what leads to the Keynesian features of the new Keynesian model. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

8 Price adjustment: micro facts Bils and Klenow (JPE, 2004) median duration between price changes is 4.3 months for items in the U.S. CPI with wide variation in frequency across different categories of goods and services. Klenow and Kryvtsov (QJE 2008) price changes large on average, but a significant fraction of price changes are small. variations in the size of price changes, rather than variation in the fraction of prices that change, can account for most of the variance of aggregate inflation. Nakamura and Steinsson (QJE 2008) five facts: sales have a significant effect on estimates of the median duration between price changes (US CPI) excluding sales increases median duration between changes from 4.5 months to 10 months. the frequency of price changes follows a seasonal pattern. the probability the price of an item changes (the hazard function) declines during the first few months after a change in price. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

9 Price adjustment Alternative approaches Time dependent pricing models Calvo fixed probability of adjusting each period. Taylor s model of fixed-length contracts State dependent price models Dotsey, King, and Wolman (1999, 2006) Dotsey and King (2005) Golosov and Lucas (2007) Gertler and Leahy (2008) Costain and Nakov (2011) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

10 Price adjustment: the basic TDP Calvo model Each period, the firms that adjust their price are randomly selected: a fraction 1 ω of all firms adjust while the remaining ω fraction do not adjust. The parameter ω is a measure of the degree of nominal rigidity; a larger ω implies fewer firms adjust each period and the expected time between price changes is longer. For those firms who do adjust their price at time t, they do so to maximize the expected discounted value of current and future profits. Profits at some future date t + s are affected by the choice of price at time t only if the firm has not received another opportunity to adjust between t and t + s. The probability of this is ω s. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

11 Price adjustment The firm s decision problem First consider the firm s cost minimization problem, which involves minimizing W t N jt subject to producing c jt = Z t N jt. This problem can be written as min N t W t N t + ϕ n t (c jt Z t N jt ). where ϕ n t is equal to the firm s nominal marginal cost. The first order condition implies W t = ϕ n t Z t, or ϕ n t = W t /Z t. Dividing by P t yields real marginal cost as ϕ t = W t / (P t Z t ). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

12 Price adjustment The firm s decision problem The firm s pricing decision problem then involves picking p jt to maximize E t ( ) ω i pjt i,t+i Π, ϕ i=0 P t+i, c t+i t+i ) 1 θ [ ( = E t ω i pjt i,t+i i=0 P t+i ( ) ] θ pjt ϕ t+i C t+i, P t+i where the discount factor i,t+i is given by β i (C t+i /C t ) σ and profits are [( ) ] pjt Π(p jt ) = c jt+i ϕ t+i c jt+i P t+i Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

13 Price adjustment All firms adjusting in period t face the same problem, so all adjusting firms will set the same price. Let pt be the optimal price chosen by all firms adjusting at time t. The first order condition for the optimal choice of pt is ( ) ( 1 p E t ω i ) 1 θ ( ) ( i,t+i [(1 θ) t 1 p ) ] θ + θϕ t i=0 p jt P t+i t+i pt P t+i Using the definition of i,t+i, ( p ) t = P t ( ) ( ) θ θ Et i=0 ω i β i (C t+i /C t ) σ ϕ Pt+i t+i P Ct+i t ( ) θ 1 θ 1. E t i=0 ω i β i (C t+i /C t ) σ Pt+i P Ct+i t Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

14 Price adjustment The first order condition can be expressed as ( p ) t = H t. F t where H t = and P t ( ) θ ( ) θ ( ) E t θ 1 ω i β i (C t+i /C t ) σ Pt+i θ ϕ t+i C t+i = i=0 P t θ 1 F t = E t ( ) θ 1 ω i β i (C t+i /C t ) σ Pt+i C t+i = C t + ωβe t F t+1 i=0 P t Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

15 The case of flexible prices With flexible prices, all firms adjust each period, so ω = 0. This implies So H t = ( p ) t P t ( ) θ ϕ θ 1 t C t ; F t = C t = H ( ) t θ = ϕ F t θ 1 t = µϕ t. But with all firms setting the same price, pt = P t and ϕ t = W ( ) t /P t θ 1 = < 1. Z t θ Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

16 The case of sticky prices When prices are sticky (ω > 0), the firm must take into account expected future marginal cost as well as current marginal cost when setting p t. The aggregate price index is an average of the price charged by the fraction 1 ω of firms setting their price in period t and the average of the remaining fraction ω of all firms who set prices in earlier periods. Because the adjusting firms were selected randomly from among all firms, the average price of the non-adjusters is just the average price of all firms that was prevailing in period t 1. Thus, the average price in period t satisfies P 1 θ t = (1 ω)(p t ) 1 θ + ωp 1 θ t 1. (3) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

17 Alternatives: SDP models SDP models allow price behavior to be influenced by an intensive and an extensive margin after a large shock, those firms that adjust will make, on average, bigger adjustments (this is the intensive margin) and more firms will adjust (this is the extensive margin). Size of price changes among firms adjusting can vary and fraction of firms adjusting can vary. Basic intuition highway road repairs. Caplin and Leahy (1987) Golosov and Lucas (2007) emphasize that firms most likely to adjust are those furthest from their desired price this is called the selection effect. The selection effect acts to make the aggregate price level more flexible than might be suggested by simply looking at the fraction of firms that change price. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

18 Firm specific shocks Most models of price adjustment developed for use in macroeconomics have assumed that firms only face aggregate shocks. This generally implies that all firms that do adjust their price choose the same new price as they all face the same (aggregate) shock. The Dotsey, King, and Wolman model features firm-specific shocks to the menu cost, but these shocks only influence whether a firm adjusts, not how much it changes prices. In contrast, Golosov and Lucas (2007) and Gertler and Leahy (2008) have emphasized the role of idiosyncratic shocks in influencing which firms adjust prices and in generating a distribution of prices across firms. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

19 Assessing models of nominal price rigidities Klenow and Kryvtsov (QJE 2008) No model fits all the micro facts Calvo does surprisingly well except for the declining hazard rate Variations in the size of price changes, rather than variation in the fraction of prices that change, can account for most of the variance of aggregate inflation. Carlsson and Nordström Skans (AEJ Macro 2012) Uses Swedish firm-level data to compare models of staggered price adjustment with models of flexible prices but imperfect information (sticky information, rational inattention to macro factors) Concludes Calvo explains data best. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

20 Trend inflation Standard NKPCs in DSGE models add ad hoc assumptions about indexation. Example: Justiniano, Primiceri, and Tambalotti (2011) assume non-optimizing firms and households set P t (j) = P t 1 (j)π ι p t 1 π1 ι p ; W t (j) = W t 1 (j) (π t 1 e z t ) ι w π 1 ι w where π is steady-state inflation and z t is the productivity shock. Estimated values of ι p, ι w 0 but this still imposes indexation to steady-state inflation. Costly relative price and wage dispersion depends on and [π t ι p π t 1 (1 ι p ) π] 2 (π t π) 2 [π w,t ι w (π t 1 + z t ) (1 ι w ) π] 2 (π w,t π) 2. Trend inflation doesn t matter. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

21 Trend and cyclical inflation 12.5 INFLATION_PCE INFLATION_TREND DETRENDED_INFL Figure: U.S. inflation (pce), trend inflation (hp), and cyclical inflation Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

22 Calvo with trend inflation Coibion and Gorodnichenko (AER 2011) ( 1 ωπ θ 1 π t = ωπ θ 1 ) p t where pt is the relative price set by adjusting firms, Π is the steady-state inflation rate, ω is the Calvo paramter (fraction of firms not resetting prices), and θ is the elasticity of substitution across goods. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

23 Calvo with trend inflation The optimal reset price is ( 1 + θη 1 ) p t = ( 1 + η 1) (1 γ 2 ) j=0 γ j 2 E tx t+j ( ) +E t γ j 2 γj 1 (g t+j i t+j 1 ) j=0 +E t j=0 { γ j [ ( θ 1 + η 1 )] } γ j 1 θ π t+j where γ 2 = ωr 1 gπ θ, γ 2 = γ 1 Π 1+θ η, g t is the growth rate of output, x is the output gap, and i is the nominal interest rate. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

24 Trend inflation If Calvo model (and micro evidence) taken seriously, one can t rely on indexation. Ascari and Ropele (2007), Cogley and Sbordone (2008), Coibion, Gorodnichenko, and Wieland (2011), Alves (2010). Indexation to trend inflation: P 1 θ t = (1 ω) (P t ) 1 θ + ωp 1 θ t 1 Π δ(1 θ) where δ is the degree of indexation. Steady-state output gap is not equal to zero if δ < 1. Coibion, Gorodnchenko and Wieland show that ( ) 1+η Ȳ Ȳ f = 1 ωβ 1 Π (1 δ)θ(1+η) ( 1 ωβ 1 Π (1 δ)(θ 1) 1 ω 1 ω Π (1 δ)(θ 1) ) 1+ηθ θ 1. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

25 Trend inflation Steady-state output gap is not equal to zero. Coibion, Gorodnchenko and Wieland show that ( ) 1+η X 1+η Ȳ t = Ȳ f = 1 ωβ 1 Π (1 δ)θ(1+η) ( 1 ωβ 1 Π (1 δ)(θ 1) 1 ω 1 ω Π (1 δ)(θ 1) X is increasing for very low but positive levels of trend inflation but then decreases. Optimal rate of inflation is positive but small. Interestingly, downward wage rigidity lowers the optimal inflation rate and makes ZLB less likely (will return to this point). ) 1+ηθ θ 1. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

26 Equilibrium conditions for basic sticky price NK model Euler condition: C σ t MRS = real wage: Marginal cost: W t Optimal price setting: ( ) Pt = βr t E t Ct+1 σ P t+1 χn η t C σ t P t = W t P t = ϕ t ( p ) t = P t Z t ( θ θ 1 ) Ht where H t = Ct 1 σ ϕ t + ωβe t H t+1 ; F t = Ct 1 σ + ωβe t F t+1 Aggregate price index: P 1 θ t F t = (1 ω)(p t ) 1 θ + ωp 1 θ t 1 Goods market clearing: t 1 Y t = C t ( ) pt (i) θ Price dispersion: t = di 1 Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283 P t

27 The role of price dispersion Output is Y t = c jt dj = Z t N jt dj = Z t N t But Y t = ( ) θ pjt c jt dj = C t dj = C t t C t = t 1 Y t. P t Since t 1, price dispersion means more has to be produced to achieve a given level of C t. More work effort required to produce a given C t. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

28 Benchmarks: effi cient and flexible-price output With flexible prices, = 1 Y t = C t, Y t = Z t H t and χ t H η t Ct σ = χ ( ) t Y f η ( ) ( ) ( ) ( t /Z t 1 Wt 1 1 ( ) Y f σ = t µ w = t P t µ w ϕ t Z t = t µ w t µ t Y f t = ( Z 1+η t χ t µ w t µ t ) 1 σ+η ( ) 1 1 : Y f σ+η = Y = χµ w µ Effi cient level of output requires = 1 and µ t = µ w t = 1: Y e t = ( ) Z 1+η 1 σ+η t : Y e = χ t ( ) 1 1 σ+η χ > Y f ) Z t Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

29 Benchmarks: effi cient and flexible-price output Log linearizing effi cient output around the effi cient steady state level yields ( ) ( ) ( ) Y ŷt e e log t 1 + η 1 Y e = ẑ t ˆχ η + σ σ + η t. Log linearizing flexible-price output around the effi cient steady state level yields ( Y ŷt f f log t = Y e ( 1 + η η + σ where ( ) Y x f log Y e ) ( Y f = log t ) ẑ t Y f ( 1 σ + η ) ( Y f + log Y e ) ) ( ˆµ t + ˆχ t + ˆµ w t ) + x ( ) ( ) Y 1 = log Y e = (µ + µ w ) 0. σ + η Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

30 Obtaining a linearized version of the model The demand side with sticky prices (and flexible wages) Linearize the Euler condition around steady-state: ( ) 1 ĉ t = E t ĉ t+1 (î t E t π t+1 ). σ Goods market equilibrium (no capital) is ŷ t = ĉ t ( ˆ t = 0 to first order) so the Euler condition becomes ( ) 1 ŷ t = E t ŷ t+1 (î t E t π t+1 ). σ This is the expectational IS curve. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

31 Demand and the output gap Output gap: gap between output and effi cient output: ( ) xt e = log (Y t /Yt e ) = log Y t /Y f log(yt e /Y e ) log (Y ) e /Y f = ŷ t ŷ e t x. So from Euler condition, ŷ t ŷ e t = E t (ŷ t+1 ŷ e t+1) ( ) 1 (î t E t π t+1 ) + (E t ŷt e ŷt e ), σ or ( ) 1 xt e = E t xt+1 e (r t rt e ), σ where r t = î t E t π t+1 and rt e σ ( E t ŷt+1 e ŷ t e ). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

32 Inflation adjustment Using the first order condition for price setting by firms and approximating around a zero average inflation, flexible-price equilibrium, π t = βe t π t+1 + κ ( ˆϕ t + ˆµ t ) where (1 ω) [1 βω] κ = ω Equation (??) is often referred to as the New Keynesian Phillips curve. Obtain similar reduced form under Taylor (1979) price adjustment or quadratic costs ala Rotemberg (1982 ). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

33 Real marginal cost and the output gap The firm s real marginal cost is equal to the real wage it faces divided by the marginal product of labor: ϕ t = (W t /P t ) Z 1 t. Because nominal wages have been assumed to be completely flexible, the real wage must equal the marginal rate of substitution between leisure and consumption: ˆω t = ŵ t ˆp t = ηĥ t + σŷ t + χ t + ˆµ w t. Recalling that ĉ t = ŷ t, ŷ t = ĥ t + ẑ t, ˆϕ t + ˆµ t = [( ηĥ t + σŷ t + ˆχ t + ˆµ w ) ] t ẑt + ˆµt [ ( ) 1 + η = (η + σ) ŷ t ẑ t + η + σ ( ) = (σ + η) ŷ t ŷt f = (σ + η) x f t = (σ + η) = (σ + η) x e t + ˆµ t + ˆµ w t. ( ) ] 1 ( ˆµ σ + η t + ˆχ t + ˆµ wt ) ( ) ŷ t ŷt e + ŷt e ŷt f Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

34 Real marginal cost and the output gap Using these results, the inflation adjustment equation becomes π t = βe t π t+1 + κx f t = βe t π t+1 + κx e t + κ µ t (4) where κ = (η + σ) κ = (η + σ) (1 ω) (1 βω) /ω and µ t ˆµ t + ˆµ w t. This inflation adjustment or forward-looking Phillips curve relates output, in the form of the deviation around the level of output that would occur in the absence of nominal price rigidity, to inflation x f t. Or, to the effi ciency gap x e t and price and wage markups ˆµ t + ˆµ w t. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

35 Real marginal cost and the output gap Generalizing production and assuming firm specific labor markets Assume Y t (i) = Z t N 1 a t, 0 < a < 1. Assume firms producing aboce (below) average amount must pay higher (lower) wages. Then the inflation adjustment equation becomes π t = βe t π t+1 + ˆκx f t = βe t π t+1 + ˆκx e t + κ µ t where ( ) ( ) ψ + σ (1 ω) (1 βω) ψ + σ κ = κ = 1 + ψθ ω 1 + ψθ where ψ = (a + η) /(1 a). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

36 The simple sticky price, flex wage no wage markup shock model Two equation system Consistent with x e t π t = βe t π t+1 + κxt e + κ ˆµ t ( ) 1 = E t xt+1 e (i t E t π t+1 rt e ) σ optimizing behavior by households and firms budget constraints market equilibrium Two equations but three unknowns: x e t, π t, and i t need to specify monetary policy Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

37 The transmission channel Narrow view monetary policy affects aggregate demand because of nominal rigidities. Other channels: Open economy exchange rate channel. Cost channel effects on the supply side. Micro evidence: Barth and Ramey NBER Annual 2002, Gaiotti and Secchi JMCB 2006, Macro: Ravenna and Walsh JME 2006, Rabanal IMF Effects on labor costs when employment relationships last for multiple periods. Nonprice credit channels. Effects of monetary policy income distribution: Gorodnichenko, et. al. 2012) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

38 Solving the model for the rational expectations equilibrium: role of policy Suppose i t is a function of the exogenous shocks r e t and ˆµ t. Write system as [ ] [ ] β 0 Et π t+1 E t xt+1 e = 1 σ 1 or [ Et π t+1 E t x e t+1 ] [ = 1 β 1 σβ [ 1 κ 0 1 κ β 1 + κ σβ ] [ πt ] [ πt x e t x e t ] [ κ σ ] [ κ σ ] [ ] [ ˆµ t i t r e t ˆµ t i t r e t ] ] or E t Z t+1 = MZ t + Ne t, e t = [ ˆµ t i t r e t ] Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

39 Solving the model for the rational expectations equilibrium There exists a locally unique, stationary rational expectations equilibrium if and only if the number of eigenvalues of M outside the unit circle is equal to the number of forward-looking variables (two). Condition is not satisfied! So a policy that just sets i t as a function of exogenous shocks rt e ˆµ t does not ensure a unique rational expectations equilibrium. Condition of indeterminacy. and Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

40 Solving the model for the rational expectations equilibrium Suppose i t = r e t + δ π π t. Write system as [ ] [ ] β 0 Et π t+1 E t xt+1 e = or 1 σ 1 [ Et π t+1 E t x e t+1 ] [ = [ 1 κ β βδ π 1 σβ κ β 1 + κ σβ ] [ πt x e t ] [ πt ] [ κ σ x e t ] [ κ + 0 ] [ ˆµt ] ˆµ t δπ t ] Two eigenvalues outside the unit circle if and only if δ > 1: the Taylor Principle. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

41 The Taylor Principle Policy based on responding solely to exogenous disturbances does not ensure a unique equilibrium. Policy must respond to endogenous variables and policy must respond suffi ciently strongly to inflation. Reasonable intuition: if inflation rises, increase i suffi ciently to raise the real interest rate and engineer a contraction to bring inflation back down. If policy also responds to the output gap, then Bullard and Mitra (JME 2002) show condition becomes κ(δ π 1) + (1 β)δ x > 0. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

42 The Taylor principle Stabilizing the economy or promising to blow up the world (Cochrane 2010)? Suppose we have a model displaying superneutrality. The monetary side of the model is summarized by i t = r t + E t π t+1 : Fisher equation i t = r t + π + δ (π t π ) + z t : Policy rule where z t = ρz t 1 + v t is an exogenous stationary process and r t is exogenous with respect to inflation and the nominal interest rate. Eliminating the nominal interest rate yields a difference equation for inflation: E t π t+1 = (1 δ)π + δπ t + z t When does it imply a unique stationary (bound) equilibrium? Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

43 The Taylor principle Equilibrium can be written as E t π t+1 = (1 δ)π + δπ t + z t. π t+1 = (1 δ)π + δπ t + z t + ε t+1, E t ε t+1 = 0 For arbitrary π 0 and sunspots, ε t+1 there is an equilibrium path for inflation. Solving forward, ( 1 δ π t = δ ( 1 δ = δ ( 1 δ ) π + ) π ( ) 1 E t π t+1 δ ( ) 1 j ( ) 1 z t δ δ ) J E t π t+j j=0 ) ε t + lim J ( 1 δ ( ) 1 z t δ j=0 ( ρ δ ) j ( ) 1 ε t δ Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

44 The Taylor principle ( 1 δ π t = δ ( 1 δ = δ ( 1 δ ) π + ) π ( ) 1 E t π t+1 δ ( ) 1 j ( ) 1 z t δ δ ) J E t π t+j j=0 ) ε t + lim J ( 1 δ ( ) 1 z t δ j=0 ( ρ δ ) j ( ) 1 ε t δ If δ > 1 so 1/δ < 1 (the Taylor Principle), there is a unique locally bounded equilibrium inflation rate given be all paths explode except one: π t = π ( ) 1 z t + lim δ ρ J ( ) 1 J ( ) 1 E t π t+j = π z t. δ δ ρ Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

45 The Taylor principle and liquidity traps Figure: Dynamic Instability under the Taylor Principle π(t+1) π π(t) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

46 The Taylor Principle Stabilizing the economy or promising to blow up the world (Cochrane 2010)? Any equilibrium other than π t = π blows up when δ > 1. Arbitrary number of equilibria when δ < 1. Separate issue Is δ identifiable? Empirical work has focused on estimating δ to determine whether it exceeds 1. But what matters for determinacy is what the central bank would do in situations not observed. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

47 The Taylor principle and liquidity traps Benhabib, Schmit-Grohé, and Uribe (2001, 2002) Figure: Liquidity Trap Case π(t+1) π π π(t) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

48 Optimal policy: Topics to cover 1 Policy objectives 2 Optimal policy under discretion and commitment 3 The zero lower bound 4 Uncertainty Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

49 Policy in forward-looking models: welfare Quadratic approximation Woodford demonstrates that deviations of the expected discounted utility of the representative agent around the level of steady-state utility can be approximated by E t β i V t+i 1 i=0 2 ΩE t [ β i π 2 t+i + λ (x t+i x ) 2]. (5) i=0 x t is the gap between output and the output level that would arise under flexible prices, and x is the gap between the steady-state effi cient level of output (in the absence of the monopolistic distortions) and the steady-state level of output. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

50 Policy weights Theory says something about the weights in the loss function: where E t β i V t+i 1 i=0 2 ΩE t [ β i π 2 t+i + λ (x t+i x ) 2], i=0 More general case, Ω = Ȳ U c κθ, and λ = κ (σ + η) θ = κ θ. (σ + ψ) Ω = Ȳ U c κ (1 + ψθ) θ, and λ = κ (1 + ψθ) θ, ψ = a + η 1 a. Greater nominal rigidity (larger ω) reduces λ (recall κ = (1 ω) (1 ωβ)/ω. Loss function endogenous. Calvo specification implies λ is small Taylor specification leads to larger weight on output gap. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

51 Policy implications of price stickiness When the price level fluctuates, and not all firms are able to adjust, price dispersion results. This causes the relative prices of the different goods to vary. If the price level rises, for example, two things happen. 1 The relative price of firms who have not set their prices for a while falls. They experience in increase in demand and raise output, while firms who have just reset their prices reduce output. This production dispersion is ineffi cient. 2 Consumers increase their consumption of the goods whose relative price has fallen and reduce consumption of those goods whose relative price has risen. This dispersion in consumption reduces welfare. The solution is to prevent price dispersion by stabilizing the price level by keeping inflation equal to zero. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

52 Woodford versus Friedman The basic new Keynesian model suggests price stability (i.e., zero inflation) is optimal. Zero inflation eliminates ineffi cient price dispersion. Milton Friedman argued that a zero nominal rate of interest is optimal. Zero nominal rate eliminates ineffi ciency in money holdings. Optimal inflation is negative (deflation) at rate equal to real rate of interest. Khan, King, and Wolman (2000) analysis model with both distortions and conclude optimal inflation is closer to zero than to the Friedman rule. Schmidt-Grohe and Uribe (2009), Coibion, Gorodnichenko, and Wieland (2012, REStudies): optimal π still small even when ZLB taking into account. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

53 Basic model eliminating the steady-state distortion Assume objective is to minimize 1 2 ΩE t β i ( π 2 t+i + λx 2 ) t+i. i=0 Note that x has been set equal to zero in loss function Fiscal subsidy to offset distortion from monopolistic competition. If x = 0, can t use first order approximations to structural equations to obtain a correct second order approximation to the representative agent s welfare. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

54 Policy Implication of forward-looking models The basic new Keynesian inflation adjustment equation took the form π t = βe t π t+1 + κx t, where x is output relative to flexible-price output and real marginal cost is (σ + η)x t. That is, there is no additional disturbance term. π t = βe t π t+1 + κx t π t = κ i=0 β i E t x t+i The absence of a stochastic disturbance implies there is no conflict between a policy designed to maintain inflation at zero and a policy designed to keep the output gap equal to zero. Just set x t+i = 0 for all i; keeps inflation equal to zero: Blanchard and Galí s divine coincidence. Productivity shocks don t appear with only prices sticky, real wage is flexible. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

55 Cost shocks Assume π t = βe t π t+1 + κx t + e t where e represents an inflation or cost shock. Then π t = κ i=0 β i E t x t+i + i=0 β i E t e t+i Cannot keep both x and π equal to zero: trade-offs must be made. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

56 Sources of cost shocks Stochastic wedge between marginal rate of substitution and real wage. Wedge between flexible-price output and effi cient level of output. Stochastic markups Endogenous sources Sticky nominal wages. Cost channel. Exchange rate movements, imperfect pass through. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

57 Decompositing output Consider the following decomposition of output: ( ) ( ) ( Yt Y f Y t = t Y pot ) t Yt f Yt pot Yt e Yt e where Y e t is the effi cient level of output and Y f t under flexible prices and wages, Y pot t constant markups and flexible prices). In terms of log deviations around the steady state, y t y e t In baseline NK model, xt fpot steady-state markups). So = x t + x fpot t = 0 and x pot t is the level of output is potential output (output with + x pot t. is a constant (related to σ 2 y y e = σ2 x and minimizing output around the effi cient level is the same as minimizing x t (Blanchard-Galí divine coincidence ). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

58 Decomposing gaps In general, σ 2 y y e = σ2 x + 2σ xx fpot + t.i.p. and minimizing σ 2 x does not necessarily minimize σ 2 y y e. Covariances can matter. Optimal policy may not involve minimizing y t yt e since resulting fluctuations in price and wage inflation may be too costly. Theory of second best in the face of multiple distortions. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

59 Decomposing gaps Which fluctuations in y t are due to fluctuations in x t, which due to xt fpot, which to xt pot, which to yt e? Which fluctuations in y t are effi cient? Which are ineffi cient? Chari, Kehoe, and McGrattan (2009) emphasized need to distinguish between effi cient and ineffi cient sources of fluctuations. May be hard to identify sources of fluctuations. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

60 Gaps and wedges Effi cient requires that mrs t = mpl t Shocks that cause wedge between these two can be Exogenous Effi cient: preference shifts (χ t ) Ineffi cient: shifts in desired markups to wages and prices ( ˆµ w t, ˆµp t ) Endogenous Ineffi cient: nominal wage and price rigidities (ϕ w t, ϕp t ). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

61 The case of flexible prices and wages Flex-price/wage output With flexible prices and wages, ω t = mpl t ˆµ p t ; ω t = mrs t + ˆµ w t Suppose marginal rate of substitution is and marginal productivity is mrs t = η ˆn t + σĉ t + ˆχ t mpl t = ŷ t ˆn t = ẑ t. Then, the steady-state labor equilibrium condition yields η ˆn t + σĉ t + ˆχ t + ˆµ w t = ω t = ẑ t ˆµ p t. Now using the fact that ŷ t = ẑ t + ˆn t and ŷ t = ĉ t, the flexible-price equilibrium output ŷt f can be expressed as ( ) ( ) 1 + η 1 ŷt f = ẑ t ( ˆµ p t η + σ η + σ + ˆµw t + ˆχ t ). (6) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

62 Measuring wedges: sticky prices and wages The wedges ϕ w t and ϕ p t are defined by ω t = mpl t ˆµ p t ϕp t ω t = mrs t + ˆµ w t + ϕ w t The effi ciency wedge is mrs t mpl t = ( ˆω t ˆµ w t ϕ w t ) ( ˆω t + ˆµ t + ϕ p t ) = ( ˆµ w t + ˆµ t ) (ϕ w t + ϕ p t ) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

63 Measuring wedges The observed wedge: (η ˆn t + σĉ t ) (ŷ t ˆn t ) = ( ˆµ t + ˆµ w t + ˆχ t ) + (ϕ w t + ϕ p t ) LHS is observable: RHS isn t. With log utility and ŷ = ĉ, ( ) 1 ˆn t = ( ˆµ 1 + η t + ˆµ w t + ˆχ t ϕ w t ϕ p t ) so employment gives a direct measure of the labor wedge but not its sources. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

64 Effi ciency gaps Galí, Gertler, and López-Salido (2002) define the ineffi ciency gap as the gap between the household s marginal rate of substitution between leisure and consumption (mrs t ) and the marginal product of labor (mpl t ). They divide this gap into the wedge between the real wage and the marginal rate of substitution, which they label the wage markup, and the wedge between the real wage and the marginal product of labor (the price markup). mrs t mpl t = (mrs t ω t ) + (ω t mpl t ) Based on United States data, they conclude the wage markup accounts for most of the time series variation in the ineffi ciency gap. Consistent with the importance of nominal wage rigidity. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

65 Empirical estimates of the wedges: Galí, Gertler, and Lopez-Salido (REStat 2007) Assume σ = η = 1, so lw t = (ˆn t + ĉ t ) (ŷ t ˆn t ) GGLS_PRICE GGLS_WAGE Figure: Galí, Gertler, and Lopez-Salido s price gap and wage gap. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

66 Empirical estimates of the wedges: Galí, Gertler, and Lopez-Salido (REStat 2007) GGLS_GAP GGLS_WAGE GGLS_PRICE UNRATE Figure: Galí, Gertler, and Lopez-Salido s effi ciency, wage, and price gaps and US unemployment rate (right axis). Karabarbounis (2013) looks at OECD countries: it s the wage gap, Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

67 Decomposing gaps using DSGE models Is it a good shock or a bad shock? (Chari, Kehoe, and McGrattan 2009) Suppose there are shocks to the disutility of labor and to the wage markup: mrs t = χ t Nη t C σ t = (W t /P t ) µ w t µw t χ t Nη t C σ t = W t P t What matters is µ w t χ t one is a bad shock (µw t ) and one is a good shock (χ t ). Linearized: η ˆn t + σĉ t (ŵ t ˆp t ) = ( ˆχ t + ˆµ w t ) Can we identify the two shocks? Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

68 Explaining the labor wedge Galí, Smets and Wouter (2011): using unemployment as an observable (see exercise). Sala, Söderström, and Trigari (2010): alternatives with markups and preferences as source of persistence. Justiniano, Primiceri, and Tambalotti (2012): measurement error and persistence: If wage series (ŵt m ˆp t ) measures (ŵ t ˆp t ) with error, then η ˆn t + σĉ t = (ŵt m ˆp t ) (et m + ˆχ t + ˆµ w t ) Use multiple (2) series ŵ1,t m and ŵ 2,t m to reduce measurement error. Assume χ t is AR(1) and µ w t is i.i.d. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

69 Sala, Söderström, and Trigari (2010) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

70 Summary Gaps/wedges can arise from effi cient shocks and ineffi cient shocks. Implications for policy are different, so identifing nature of shocks it important. Serious identification issues: Is the labor wedge due to preference shocks, wage markup shocks, or measurement error? Karabarbounis (NBER 19015, 2013) does Galí, Gertler, Lopez-Salido exercise for several OECD countries. Concludes ω mpl 0, but mrs ω = 0. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

71 Optimal policy in the basic model When forward-looking expectations play a role, discretion leads to a stabilization bias even though there is no average inflation bias. Minimize subject to 1 2 ΩE t β i ( π 2 t+i + λx 2 ) t+i i=0 π t = βe t π t+1 + κx t + e t. Notice the Euler condition imposes no constraint use it to solve for i t once optimal π t and x t have been determined. This would not be case if central bank cares about interest rate volatility. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

72 Discretion The policy problem Policy maker takes expectations as given leads to period by period maximization. Problem is to pick π t and x t to minimize taking E t π t+1 as given. 1 ( π 2 2 t + λxt 2 ) + ψt (π t βπ t+1 κx t e t ) The first order conditions can be written as π t + ψ t = 0 (7) λx t κψ t = 0. (8) Eliminating ψ t, λx t + κπ t = 0 this is a targeting rule. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

73 Discretion Behavior of the interest rate From the IS equation, Using solution, i t = i t = E t π t+1 + σ (E t x t+1 x t ) + r n t. [ ( κ ) ] Aρ σ (ρ 1) e t + rt n = Be t + rt n. λ Shifts in natural rate of interest r n are fully offset. So optimal policy involves i responding to shocks, but adopting a rule of the form i t = Be t + r n t does not ensure a unique rational expectations equilibrium. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

74 Precommitment When forward-looking expectations play a role, discretion leads to a stabilization bias even though there is no average inflation bias. Under optimal commitment, central bank at time t chooses both current and expected future values of inflation and the output gap. Minimize subject to 1 2 ΩE t β i ( π 2 t+i + λx 2 ) t+i i=0 π t = βe t π t+1 + κx t + e t. Notice the Euler condition imposes no constraint use it to solve for i t once optimal π t and x t have been determined. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

75 Optimal precommitment The central bank s problem is to pick π t+i and x t+i to minimize [ ] 1 E t β i ( π 2 i=0 2 t+i + λxt+i 2 ) + ψt+i (π t+i βπ t+i+1 κx t+i e t+i ). The first order conditions can be written as π t + ψ t = 0 (9) E t ( πt+i + ψ t+i ψ t+i 1 ) = 0 i 1 (10) E t ( λxt+i κψ t+i ) = 0 i 0. (11) Dynamic inconsistency at time t, the central bank sets π t = ψ t and promises to set π t+1 = ( E t ψ t+1 ψ t ). When t + 1 arrives, a central bank that reoptimizes will again obtains π t+1 = ψ t+1 the first order condition (9) updated to t + 1 will reappear. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

76 Timeless precommitment An alternative definition of an optimal precommitment policy requires the central bank to implement conditions (10) and (11) for all periods, including the current period so that π t+i + ψ t+i ψ t+i 1 = 0 i 0 λx t+i κψ t+i = 0 i 0. Woodford (1999) has labeled this the timeless perspective approach to precommitment. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

77 Timeless precommitment Under the timeless perspective optimal commitment policy, inflation and the output gap satisfy ( ) λ π t+i = (x t+i x t+i 1 ) (12) κ for all i 0. Woodford (1999) has stressed that, even if ρ = 0, so that there is no natural source of persistence in the model itself, a > 0 and the precommitment policy introduces inertia into the output gap and inflation processes. This commitment to inertia implies that the central bank s actions at date t allow it to influence expected future inflation. Doing so leads to a better trade-off between gap and inflation variability than would arise if policy did not react to the lagged gap. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

78 Illustrating commitment versus discretion in the simple NK model 0 Output Gap 0.8 Inflation Nominal interest rate Cost Shock Taylor Rule Optimal Discretion Full Commitment Optimal Policy Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

79 Improved trade-off under commitment The difference in the stabilization response under commitment and discretion is the stabilization bias due to discretion. Consider a positive inflation shock, e > 0. A given change in current inflation can be achieved with a smaller fall in x if expected future inflation can be reduced: π t = βe t π t+1 + κx t + e t Requires a commitment to future deflation. By keeping output below potential (a negative output gap) for several periods into the future after a positive cost shock, the central bank is able to lower expectations of future inflation. A fall in E t π t+1 at the time of the positive inflation shock improves the trade-off between inflation and output gap stabilization faced by the central bank. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

80 The case of a persistent cost shock 0 Output Gap 1.2 Inflation Nominal interest rate Cost Shock Taylor Rule Optimal Discretion Full Commitment Optimal Policy Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

81 Illustrating policy effects in S-W model: inflation shock 0 Output gap 0 Actual output Inflation Nominal interest rate Estimated Taylor Rule Optimal Commitment Policy Simple Taylor rule Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

82 Is the ZLB a constraint? Figure: Distribution of the federal funds rate, Jan July Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

83 The zero lower bound Causes Non-fundamentals-based liquidity traps Expectationally driven Fundamentals-based liquidity traps (will address later) Negative shock to the equilibrium (Wicksellian) real rate of interest Potential constraint on monetary policy Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

84 The Taylor principle and liquidity traps Non-fundamentals-based liquidity traps Benhabib, Schmit-Grohé, and Uribe (2001, 2002) have argued that deflationary paths cannot be ruled out if policy satisfies the Taylor Principle. Multiple steady-state equilibria in forward-looking expectational models. The argument is based on the observation that the nominal rate of interest cannot fall below zero. Explosive deflations would eventually force the nominal interest rate to zero, but the nominal rate is then prevented from falling further. They argue that simple and seemingly reasonable monetary policy rules that follow the Taylor Principle in changing the nominal interest rate more than one-for-one is response to changes in inflation can introduce the possibility the economy will be caught in a deflationary liquidity trap. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

85 The Taylor principle and liquidity traps Suppose we have a model displaying superneutrality. The monetary side of the model is summarized by i t = r n t + E t π t+1 i t = g(π t ) where rt n is exogenous with respect to inflation and the nominal interest rate and g(π) is the policy rule. For simplicity, let Then combining these equations, g(π t ) = r n t + π + δ (π t π ) = r n t + (1 δ)π + δπ t E t π t+1 = (1 δ)π + δπ t There is a unique stationary equilibrium inflation rate equal to π. However, if the time t inflation rate is not equal to π, the inflation process is unstable. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

86 The Taylor principle and liquidity traps A simple example π(t+1) π π π(t) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

87 The zero lower bound Fundamentals-based liquidity traps Optimal policy an a basic new Keynesian model implies the policy interest rate moves one-for-one with the equilibrium real (natural, Wicksellian) interest rate: i t = r n t + π + δ (π t π ) 0. Negative shock to rt n this. Deflationary trap: could require i t to be negative ZLB prevents r t = i t E t π t+1 = E t π t+1 Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

88 Policy at the ZLB If i = 0, short-term government debt and money perfect substitutes. Altering B + M doesn t causes returns to adjust. Consider the nominal budget constraint of a household holding money plus short-term government debt, denoted B yielding nominal return i b : P t Y t + (1 + i b t )B t + M t P t C t + P t T t + B t+1 + M t+1. The budget constraint can be expressed in real terms as ( ) ( ) i Y t rt b b F t C t + T t + t m t + F t+1, 1 + π t where F t b t + m t equals real holds of financial assets. If the nominal interest is zero, 1 + r b = 1/(1 + π), ( ) 1 Y t + F t C t + T t + F t+1, 1 + π t and m drops out. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

89 Is the ZLB a constraint? Is the ZLB a constraint on monetary policy? Conventional model based on an expectational IS relationship: ( ) 1 x t = E t x t+1 (i t E t π t+1 rt n ) σ Interest rates both current and expected future matter: ( ) ( ) 1 1 x t = (i t E t π t+1 ) E t σ σ (i t+i π t+1+i ) i=1 ( ) 1 + E t σ rt+i n, i=0 Reflects a narrow view of the transmission mechanism no role for quantitative easing or credit easing policies. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

90 Conventional instruments at the ZLB Even at the ZLB, policy has the potential to influence real spending if it can affect expectations of future real interest rates. Eggertsson and Woodford (2003) If i t = 0 and is expected to remain at zero until t + T, then x t = ( 1 σ + ) T ( 1 σ i=0 ) E t E t π t+1+i rt+i n. i=0 ( ) 1 E t σ (i t+i π t+1+i ) i=t +1 Raising expected future inflation or committing to lower future nominal rates can stimulate current spending. Cost of ZLB low in linear models when central bank is credible (Nakov 2008) Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

91 Promising future inflation Optimal policy at the ZLB involves promising future inflation. This is done by keeping interest rates low even when the ZLB no longer binds (Eggertsson and Woodford 2003). Central banks have been reluctant to promise higher future inflation Contrasts with recommendations made to the Bank of Japan: Krugman (1998), McCallum (2000), Svensson (2001, 2003), and Auerbach and Obstfeld (2005) Bernanke (2000) versus Bernanke (2010). Communicating clearly the conditional nature of future interest rate paths cited as concern. Commitment requires promises be fulfilled have to deliver higher future inflation. Central banks may lack the credibility to steer future expectations (Bodenstein, Hebden, and Nunes 2010). Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

92 Promising future inflation: Bodenstein, Hebden, and Nunes (JME 2011) A lack of credibility makes it harder to stabilize in the face of the ZLB; With less credibility, future promises must be more extreme; Promising low interest rates for an extended period of time may be a sign of a lack of credibility; Promising low interest rates in the future and also promising no inflation is an inconsistent policy. When current credibility is low, the central bank has to promise lower future rates, but that means the cost of fulfill promises is higher, raising the incentive to deviate. Central banks with low credibility face the greatest temptation to break their promises. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

93 Reforming inflation targeting Raising the inflation target Raising the inflation target raises average nominal interest rates and makes hitting the ZLB less likely. Trade-off steady-state loss of higher inflation against ability to improve stabilization. Schmidt-Grohe and Uribe (2009), Billi (AEJ: Macro 2011), Coibion, Gorodnichenko, and Wieland (2012, REStudies): optimal π still small when ZLB taken into account. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

94 Replacing inflation targeting Price level targeting and the ZLB Optimal policy at the ZLB can be implemented via a time-varying price-level target (Eggertsson and Woodford 2003). Intuition under commitment, central bank has many tools even if current policy rate at zero. Can promise future low interest rates and a boom. This generates expectations of future inflation which raises current inflation and lowers current real interest rate. Promise to generate inflation to achieve price-level target. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

95 Price level targeting Vestin (2006) shows price level targeting can replicate the timeless precommitment solution if the central bank is assigned the loss function pt 2 + λ PL xt 2 in an environment of discretion. Under timeless precommitment, ( ) ( ) λ λ π t = (1 L)p t = (x t x t 1 ) p t = x t κ κ Price level targeting makes inflation expectations act as an automatic stabilizer. Walsh (2003) adds lagged inflation to the inflation adjustment equation and shows that the advantages of price level targeting over inflation targeting decline as the weight on lagged inflation increases.. Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283

96 Reforming IT: Using expectations as automatic stabilizers under PLT π t = βe t π t+1 + κx t + e t Price level targeting Svensson (1999), Vestin (2006), Walsh (2003), Cateau, et. al (2008), Dib, et. al. (2008), Kryvstov, et. al. (2008). Central banks may find it easier to commit to objectives than to future policy actions. Distorting objectives in a discretionary environment can improve outcomes (Walsh 1995) Outcomes to shocks under discretion PLT relative to IT Disturbance Price-level targeting Inflation shocks Potentially Better Demand shocks, no ZLB Same Demand shocks, ZLB Better Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center August 19-23, / 283