Monetary Theory: The Basic New Keynesian Model Behzad Diba University of Bern April 2011 (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 1 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations 1 The New Keynesian Phillips curve relating in‡ation to the output gap (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations 1 2 The New Keynesian Phillips curve relating in‡ation to the output gap A "Dynamic IS Equation" linking the evolution of aggregate demand (and the output gap) to the nominal interest rate (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations 1 2 3 The New Keynesian Phillips curve relating in‡ation to the output gap A "Dynamic IS Equation" linking the evolution of aggregate demand (and the output gap) to the nominal interest rate A monetary-policy rule for setting the nominal interest rate (or, a money-supply rule plus a money-demand speci…cation) (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations 1 2 3 The New Keynesian Phillips curve relating in‡ation to the output gap A "Dynamic IS Equation" linking the evolution of aggregate demand (and the output gap) to the nominal interest rate A monetary-policy rule for setting the nominal interest rate (or, a money-supply rule plus a money-demand speci…cation) These equations determine the real interest rate, in‡ation, and the output gap (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 Three Equations The basic New Keynesian model with Calvo price setting simpli…es to a system of three equations 1 2 3 The New Keynesian Phillips curve relating in‡ation to the output gap A "Dynamic IS Equation" linking the evolution of aggregate demand (and the output gap) to the nominal interest rate A monetary-policy rule for setting the nominal interest rate (or, a money-supply rule plus a money-demand speci…cation) These equations determine the real interest rate, in‡ation, and the output gap We will derive the …rst two equations and put them together with a policy rule below (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 2 / 12 The Output Gap De…ne the natural level of output as the level in the ‡exible-price equilibrium, and let ytn denote the logarithm of natural output (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 3 / 12 The Output Gap De…ne the natural level of output as the level in the ‡exible-price equilibrium, and let ytn denote the logarithm of natural output For example, in our model with iso-elastic utility and the production function yt = at + (1 α)nt , (1) 0 < α < 1, we got ytn = (1 α)[log(1 α) µ] + ϕ + α + σ ασ ϕ+1 ϕ + α + σ ασ at (2) relating ytn to productivity shocks (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 3 / 12 The Output Gap De…ne the natural level of output as the level in the ‡exible-price equilibrium, and let ytn denote the logarithm of natural output For example, in our model with iso-elastic utility and the production function yt = at + (1 α)nt , (1) 0 < α < 1, we got ytn = (1 α)[log(1 α) µ] + ϕ + α + σ ασ ϕ+1 ϕ + α + σ ασ at (2) relating ytn to productivity shocks We de…ne the output gap as s yt = yt ytn , and de…ne recessions as periods with a negative output gap (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 3 / 12 The Output Gap De…ne the natural level of output as the level in the ‡exible-price equilibrium, and let ytn denote the logarithm of natural output For example, in our model with iso-elastic utility and the production function yt = at + (1 α)nt , (1) 0 < α < 1, we got ytn = (1 α)[log(1 α) µ] + ϕ + α + σ ασ ϕ+1 ϕ + α + σ ασ at (2) relating ytn to productivity shocks We de…ne the output gap as s yt = yt ytn , and de…ne recessions as periods with a negative output gap Comparison to real-world measures, and the role of productivity shocks (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 3 / 12 Real Marginal Cost Recall that with ‡exible prices real marginal cost is constant at the steady-state value e 1 mc = log , e which is the inverse of the monopoly markup (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 4 / 12 Real Marginal Cost Recall that with ‡exible prices real marginal cost is constant at the steady-state value e 1 mc = log , e which is the inverse of the monopoly markup Recall also that, with sticky prices, real marginal cost is positively related to output because an increase in output increases the real wage, and may also lead to diminishing marginal product of labor (mpn in the textbook) if we have a …xed factor in the production function (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 4 / 12 Real Marginal Cost Recall that with ‡exible prices real marginal cost is constant at the steady-state value e 1 mc = log , e which is the inverse of the monopoly markup Recall also that, with sticky prices, real marginal cost is positively related to output because an increase in output increases the real wage, and may also lead to diminishing marginal product of labor (mpn in the textbook) if we have a …xed factor in the production function For example, with our iso-elastic speci…cation, with (1), we get mc ct = (Institute) σ+ α+ϕ 1 α s yt Monetary Theory: The Basic New Keynesian Model April 2011 4 / 12 Marginal Cost and Output mct = (wt pt) mpnt = ( yt + ' nt) (yt '+ yt = + 1 Under ‡exible prices '+ mc = + ytn 1 =) ytn = where y where yt ( ytn log(1 ))(1 ) +'+ (1 ) > 0 and =) mc ct = yet is the output gap nt) log(1 ) 1+' at log(1 1 1+' at 1 y + ya '+ + 1 ya at 1+' +'+ (1 (yt log(1 ) (6) ) (7) ). ytn) (8) Real Marginal Cost and the Output Gap Note that this equation, mc ct = σ+ α+ϕ 1 α s yt , relates the proportional deviation of real marginal cost from its steady-state value to the output gap (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 5 / 12 Real Marginal Cost and the Output Gap Note that this equation, mc ct = σ+ α+ϕ 1 α s yt , relates the proportional deviation of real marginal cost from its steady-state value to the output gap 1 with α = 0, the connection involves the preference parameters σ and ϕ, re‡ecting the e¤ect of output on real wages (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 5 / 12 Real Marginal Cost and the Output Gap Note that this equation, mc ct = σ+ α+ϕ 1 α s yt , relates the proportional deviation of real marginal cost from its steady-state value to the output gap 1 2 with α = 0, the connection involves the preference parameters σ and ϕ, re‡ecting the e¤ect of output on real wages with 0 < α < 1, we have the additional e¤ect of the diminishing marginal product of labor (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 5 / 12 Real Marginal Cost and the Output Gap Note that this equation, mc ct = σ+ α+ϕ 1 α s yt , relates the proportional deviation of real marginal cost from its steady-state value to the output gap 1 2 with α = 0, the connection involves the preference parameters σ and ϕ, re‡ecting the e¤ect of output on real wages with 0 < α < 1, we have the additional e¤ect of the diminishing marginal product of labor Either way, an expansion of output raises real marginal cost, and this erodes the monopoly markup (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 5 / 12 Real Marginal Cost and the Output Gap Note that this equation, mc ct = σ+ α+ϕ 1 α s yt , relates the proportional deviation of real marginal cost from its steady-state value to the output gap 1 2 with α = 0, the connection involves the preference parameters σ and ϕ, re‡ecting the e¤ect of output on real wages with 0 < α < 1, we have the additional e¤ect of the diminishing marginal product of labor Either way, an expansion of output raises real marginal cost, and this erodes the monopoly markup New prices, in Calvo’s model, rise with real marginal cost (in response to the erosion of the markup), and this leads to in‡ation (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 5 / 12 The New Keynesian Phillips Curve (NKPC) More precisely, we saw that (with Calvo price setting) in‡ation dynamics are governed by π t = βEt π t +1 + λmc ct with a coe¢ cient λ > 0 that is inversely related to the degree of price rigidity θ (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 6 / 12 The New Keynesian Phillips Curve (NKPC) More precisely, we saw that (with Calvo price setting) in‡ation dynamics are governed by π t = βEt π t +1 + λmc ct with a coe¢ cient λ > 0 that is inversely related to the degree of price rigidity θ Combining this with mc ct = σ+ α+ϕ 1 α s yt we get the New Keynesian Phillips Curve s π t = βEt π t +1 + κ yt with κ = λ σ+ (Institute) α+ϕ 1 α >0 Monetary Theory: The Basic New Keynesian Model April 2011 6 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 inverse relation, in the data, between in‡ation and unemployment (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 3 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation an in‡ation-unemployment (in‡ation-output) trade-o¤ for policymakers? (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 3 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation an in‡ation-unemployment (in‡ation-output) trade-o¤ for policymakers? The Phelps-Friedman "Expectations-augmented" Phillips Curve (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 3 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation an in‡ation-unemployment (in‡ation-output) trade-o¤ for policymakers? The Phelps-Friedman "Expectations-augmented" Phillips Curve the New Classical rendition (with Rational Expectations) π t = Et 1 π t + κ (yt (Institute) ytn ) , κ > 0 Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 3 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation an in‡ation-unemployment (in‡ation-output) trade-o¤ for policymakers? The Phelps-Friedman "Expectations-augmented" Phillips Curve the New Classical rendition (with Rational Expectations) π t = Et 1 π t + κ (yt ytn ) , κ > 0 may arise from the Lucas aggregate-supply curve yt = ytn + φ (pt Et 1 pt ) , φ > 0 (setting κ = 1/φ) (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Digression: Phillips Curves, Old and New The original Phillips Curve 1 2 3 inverse relation, in the data, between in‡ation and unemployment the traditional Keynesian interpretation an in‡ation-unemployment (in‡ation-output) trade-o¤ for policymakers? The Phelps-Friedman "Expectations-augmented" Phillips Curve the New Classical rendition (with Rational Expectations) π t = Et 1 π t + κ (yt ytn ) , κ > 0 may arise from the Lucas aggregate-supply curve yt = ytn + φ (pt Et 1 pt ) , φ > 0 (setting κ = 1/φ) Contrast with the forward-looking NKPC π t = βEt π t +1 + κ (yt (Institute) ytn ) , κ > 0 Monetary Theory: The Basic New Keynesian Model April 2011 7 / 12 Dynamic IS Equation The consumers’Euler equation governs the evolution of aggregate demand, given our market-clearing condition Ct = Yt (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 8 / 12 Equilibrium Goods markets clearing Yt(i) = Ct(i) for all i 2 [0; 1] and all t. R1 1 1 Letting Yt di Y (i) 0 t 1 , Yt = Ct for all t. Combined with the consumer’s Euler equation: yt = Etfyt+1g 1 (it Etf t+1 g ) (5) Dynamic IS Equation The consumers’Euler equation governs the evolution of aggregate demand, given our market-clearing condition Ct = Yt De…ne the natural real interest rate rtn as the real rate in the ‡exible-price equilibrium (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 8 / 12 Dynamic IS Equation The consumers’Euler equation governs the evolution of aggregate demand, given our market-clearing condition Ct = Yt De…ne the natural real interest rate rtn as the real rate in the ‡exible-price equilibrium Recall that rtn depends on productivity, but not on the prevailing nominal interest rate (nor anything else that monetary policy can a¤ect) (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 8 / 12 Dynamic IS Equation The consumers’Euler equation governs the evolution of aggregate demand, given our market-clearing condition Ct = Yt De…ne the natural real interest rate rtn as the real rate in the ‡exible-price equilibrium Recall that rtn depends on productivity, but not on the prevailing nominal interest rate (nor anything else that monetary policy can a¤ect) For example, in our iso-elastic speci…cation leading to (2), we have ytn = and rtn = ρ + σ (Institute) 1 n (r σ t ρ) + Et ytn+1 ϕ+1 ϕ + α + σ ασ Et (∆at +1 ) Monetary Theory: The Basic New Keynesian Model April 2011 8 / 12 IS Equation and the Output Gap Combining the Euler equations in the two equilibria (with ‡exible and sticky prices), we can relate the evolution of the output gap to the interest-rate gap rt rtn (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 9 / 12 New Keynesian Phillips Curve t where = + '+ . 1 Et f t+1 g + yet (9) Dynamic IS equation 1 yet = Etfe yt+1g (it Etf t+1 g rtn) (10) where rtn is the natural rate of interest, given by rtn = Missing block: + + n Etf yt+1 g ya Et f at+1 g description of monetary policy (determination of it). IS Equation and the Output Gap Combining the Euler equations in the two equilibria (with ‡exible and sticky prices), we can relate the evolution of the output gap to the interest-rate gap rt rtn Monetary policy a¤ects the output gap through its e¤ect on the interest-rate gap: yt ytn = Et (yt +1 ytn+1 ) i.e. s s yt = Et (yt +1 ) (Institute) 1 [it σ 1 [rt σ Et ( π t + 1 ) rtn ] , rtn ] Monetary Theory: The Basic New Keynesian Model April 2011 9 / 12 Interest-rate Rules We can close the model by specifying an interest-rate rule for monetary policy (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 10 / 12 New Keynesian Phillips Curve t where = + '+ . 1 Et f t+1 g + yet (9) Dynamic IS equation 1 yet = Etfe yt+1g (it Etf t+1 g rtn) (10) where rtn is the natural rate of interest, given by rtn = Missing block: + + n Etf yt+1 g ya Et f at+1 g description of monetary policy (determination of it). Interest-rate Rules We can close the model by specifying an interest-rate rule for monetary policy For example, near the steady state with zero in‡ation, a Taylor Rule s it = ρ + φπ π t + φy yt + υt delivers determinacy if it satis…es the Taylor Principle, φπ > 1, and φy 0 (the necessary condition for determinacy, presented in the textbook, is a bit weaker) (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 10 / 12 Equilibrium under a Simple Interest Rate Rule it = + t + y yet + vt (11) where vt is exogenous (possibly stochastic) with zero mean. Equilibrium Dynamics: yet t where AT and 1 + y+ combining (9), (10), and (11) = AT Etfe yt+1g + BT (b rtn Etf t+1g 1 + ( + y) ; BT vt ) (12) 1 Uniqueness () AT has both eigenvalues within the unit circle Given 0 and y 0, (Bullard and Mitra (2002)): ( is necessary and su¢ cient. 1) + (1 ) y >0 Interest-rate Rules We can close the model by specifying an interest-rate rule for monetary policy For example, near the steady state with zero in‡ation, a Taylor Rule s it = ρ + φπ π t + φy yt + υt delivers determinacy if it satis…es the Taylor Principle, φπ > 1, and φy 0 (the necessary condition for determinacy, presented in the textbook, is a bit weaker) Here, υt represents a "monetary-policy shock" and may, for example, follow an exogenous AR(1) process (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 10 / 12 Interest-rate Rules We can close the model by specifying an interest-rate rule for monetary policy For example, near the steady state with zero in‡ation, a Taylor Rule s it = ρ + φπ π t + φy yt + υt delivers determinacy if it satis…es the Taylor Principle, φπ > 1, and φy 0 (the necessary condition for determinacy, presented in the textbook, is a bit weaker) Here, υt represents a "monetary-policy shock" and may, for example, follow an exogenous AR(1) process Although we can solve this simple model analytically, we usually rely on a calibration and stochastic simulations to assess the quantitative implications of our models (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 10 / 12 E¤ects of a Monetary Policy Shock Set rbtn = 0 (no real shocks). Let vt follow an AR(1) process vt = v vt 1 + "vt Calibration: v = 0:5, = 1:5, y = 0:5=4, = 0:99, = ' = 1, = 2=3, = 4. Dynamic e¤ects of an exogenous increase in the nominal rate (Figure 1): Exercise: analytical solution Figure 3.1: Effects of a Monetary Policy Shock (Interest Rate Rule)) E¤ects of a Technology Shock Set vt = 0 (no monetary shocks). Technology process: at = a 1 + "at: ya (1 a) at Implied natural rate: rbtn = Dynamic e¤ects of a technology shock ( Exercise: AR(1) process for at a at = 0:9) (Figure 2) Figure 3.2: Effects of a Technology Shock (Interest Rate Rule) Money-supply Rule Alternatively, we may specify a policy rule that sets the money supply– e.g., an AR(1) process for money growth– and solve the (simple) model or simulate (richer) models (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 11 / 12 Money-supply Rule Alternatively, we may specify a policy rule that sets the money supply– e.g., an AR(1) process for money growth– and solve the (simple) model or simulate (richer) models The practical motivation for this speci…cation is weaker because major central banks use the nominal interest rate as their policy instrument (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 11 / 12 Equilibrium under an Exogenous Money Growth Process mt = m mt 1 + "m t (13) Money market clearing where lt mt b bit lt = ybt = yet + ybtn (14) (15) bit pt denotes (log) real money balances. Substituting (14) into (10): (1 + ) yet = Furthermore, we have Etfe yt+1g + b lt + b lt 1 =b lt + t Etf mt t+1 g + rbtn ybtn (16) (17) Equilibrium dynamics 3 2 3 2 n 3 2 Etfe yt+1g yet rbt AM;0 4 t 5 = AM;1 4 Etf t+1g 5 + BM 4 ybtn 5 b b mt lt 1 lt 1 where 2 3 2 3 1+ 0 0 1 1 0 5 ; AM;1 4 0 0 5 ; BM AM;0 4 0 1 1 0 0 1 (18) 2 3 1 0 40 0 0 5 0 0 1 1 Uniqueness () AM AM;0 AM;1 has two eigenvalues inside and one outside the unit circle. E¤ects of a Monetary Policy Shock Set rbtn = ytn = 0 (no real shocks). Money growth process mt = where m 2 [0; 1) Figure 3 (based on m mt m 1 + "m t = 0:5) E¤ects of a Technology Shock Set mt = 0 (no monetary shocks). Technology process: at = Figure 4 (based on Empirical Evidence a = 0:9). a at 1 + "at: (19) Figure 3.3: Effects of a Monetary Policy Shock (Money Growth Rule) Figure 3.4: Effects of a Technology Shock (Money Growth Rule) Money-supply Rule Alternatively, we may specify a policy rule that sets the money supply– e.g., an AR(1) process for money growth– and solve the (simple) model or simulate (richer) models The practical motivation for this speci…cation is weaker because major central banks use the nominal interest rate as their policy instrument But the ability of various models to generate liquidity e¤ects, under money-supply rules, has motivated some academic researchers (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 11 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks The NK model’s household block (maximizing utility over time) makes it very di¤erent from traditional Keynesian models (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks The NK model’s household block (maximizing utility over time) makes it very di¤erent from traditional Keynesian models The NK concept of stabilization policy is fundamentally di¤erent from the traditional notion of stabilizing (de-trended) output (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks The NK model’s household block (maximizing utility over time) makes it very di¤erent from traditional Keynesian models The NK concept of stabilization policy is fundamentally di¤erent from the traditional notion of stabilizing (de-trended) output the output target ytn depends on productivity (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks The NK model’s household block (maximizing utility over time) makes it very di¤erent from traditional Keynesian models The NK concept of stabilization policy is fundamentally di¤erent from the traditional notion of stabilizing (de-trended) output the output target ytn depends on productivity as an illustration, consider the e¤ects of a productivity shock in the simple model with one-period price rigidity (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12 New Keynesian (NK) Insights The core RBC framework plays an important role in the NK model, and in its implications for the e¤ects of productivity (and …scal) shocks The NK model’s household block (maximizing utility over time) makes it very di¤erent from traditional Keynesian models The NK concept of stabilization policy is fundamentally di¤erent from the traditional notion of stabilizing (de-trended) output the output target ytn depends on productivity as an illustration, consider the e¤ects of a productivity shock in the simple model with one-period price rigidity The NK model o¤ers fundamentally new insights about the output-in‡ation "trade-o¤" facing the central bank, as we will see in Chapter 4 (Institute) Monetary Theory: The Basic New Keynesian Model April 2011 12 / 12
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