The New Keynesian Model: Introduction

The New Keynesian Model: Introduction Vivaldo M. Mendes ISCTE Lisbon University Institute 13 November 2017 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 1 / 39

Summary 1 What is the New Keynesian Model? 2 The New IS function 3 The New AS function or the NKPC 4 The Central Bank s behavior: introduction (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 2 / 39

What is the New Keynesian Model? I What s is the New Keynesian Model? (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 3 / 39

What is the New Keynesian Model? The Old and the New Keynesian Models 1 The new model is the old Keynesian model 2 With the usual nominal and/or real rigidities in price setting Prices are sticky (the baseline version) Nominal wages are rigid (downwards) Staggered contracts Utilization capacity constraints 3 Without the problems that pushed the model to serious trouble in the early 70s 1 "forward looking/rational expectations" instead of "adaptive expectations" 2 built upon sound microeconomic foundations 3 no permanent trade-off between inflation and unemployment 4 no room for stagflation 4 A new message: rules instead of discretion by Central Banks in the management of monetary policy (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 4 / 39

What is the New Keynesian Model? Four basic predictions 1 The same usual functions (IS, LM, Aggregate Supply)... 2 Quantitative simulations: crucial element like in the Real Business Cycle literature 3 Contrary to RBC, there is a key role to monetary policy and a minor role for fiscal policy 4 Four basic predictions (the old model up-side-down): 1 the instrument of monetary policy ought to be the short term interest rate, 2 policy should be focused on the control of inflation, 3 inflation can be reduced by aggressively increasing short term interest rates (see Figure 1). 4 the central bank should conduct monetary policy adopting a strategy of commitment in a forward-looking environment, instead of discretion. (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 5 / 39

What is the New Keynesian Model? Figure 1 Active interest rate policy by central banks The FED now reacts much more aggressively to inflation than in the "old times".12 IPIPIB Indice Preços Implicito no PIB (trimestral).1.08 "FED Funds Rate": 4.8% "FED Funds Rate": 9.7%.06.04.02 1960 1965 1970 1975 1980 1985 1990 1995 2000 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 6 / 39

A picture: Old vs New Keynesian model A picture of the Old Keynesian model 1 On the demand side: 1 the IS function Y D = C(Y T + ) + I( r, Y + ) + G 2 the LM function, where the central bank controls the nominal money supply (M): M/P = L( r, Y + ) 2 On the supply side: the Aggregate Supply function with sticky 3 wages: Y S = f (A +, N +, P/P e + ) ( Y D ) aggregate demand, (Y) real income, (T) income taxes, (r) nominal interest rate, (G) government expenditures, (P) price level, (A) labor productivity, (N) Employment (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 7 / 39

3 (x t ) output gap, (π t ) inflation rate, (E t π t+1 ) expected value at t of inflation at t + 1, (µ t ) an exogenous demand shock, (υ t ) an exogenous supply shock. (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 8 / 39 A picture: Old vs New Keynesian model A picture of the New Keynesian model 1 On the demand side: 1 the IS function is forward looking x t = ϕ(r t E t π t+1 ) + E t x t+1 + µ t 2 the LM function, where the central bank now controls the interest rate, not the money supply: r t = L(?,?) 2 In the supply side: the Aggregate Supply function is also forward looking: π t = β E t π t+1 + λx t + υ t

The New IS function II The New IS function (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 9 / 39

The New IS function The IS function in three steps 1 The New Keynesian Model has three main equations: 1 the IS function (the demand side of the economy) 2 the AS function, 3 and a rule for monetary policy 2 Let s start with the IS function. 3 Three steps there is a simple and intuitive way to derive the IS: 1 Firstly, get the Euler equation 2 Second, log-linearize the Euler equation 3 Third, express the Euler equation in terms of percentage deviations from the steady state (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 10 / 39

The New IS function The Marginal Rate of Substitution (MRS) 1 Assume a two period economy, with households maximizing intertemporal utility from consumption U(C 0, C 1 ) maxu(c 0 ) + βu(c 1 ) C 0,C 1 2 The marginal rate of substitution (MRS) of intertemporal consumption is given by 3 From where we can get 4 See Figure 2. du(c 0, C 1 ) = 0 u (C 0 ) dc 0 + βu (C 1 ) dc 1 = 0 MRS = dc 1 dc 0 = u (C 0 ) β u (C 1 ) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 11 / 39

The New IS function Figure 2 The utility trade-off between current vs future consumption C 1 U0 U1 U2 slope u(c 0 )/ β u(c 1 ) C 0 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 12 / 39

The New IS function The Relative Price of intertemporal consumption 1 We have been concerned about "preferences" of consumers about C 1 and C 0 2 Now let s move on to the "costs" associated with the consumption in each period 3 The households budget constraints in each period is given by P 0 C 0 + A 0 = W 0 P 1 C 1 = W 1 + A 0 (1 + r 0 ) P is the price of consumption goods, W is wage income and savings (A 0 ) are invested in period 0. 4 The two constraints can be consolidated by cancelling out A 0 P 1 C 1 = W 1 + (W 0 P 0 C 0 ) (1 + r 0 ) 5 This is called the intertemporal budget constraint of our representative consumer (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 13 / 39

The New IS function The Relative Price of intertemporal consumption (cont.) 1 We obtained this intertemporal constaint P 1 C 1 = W 1 + (W 0 P 0 C 0 ) (1 + r 0 ) 2 I can ask this question: what is the price I have to pay if I consume 1 more unit now, having (obviously) to consume less in the future? 3 Notice that this is a relative price (RP) of current consumption in terms of future consumption (figure 3 RP = dc 1 = dc 0 Pr ice of C 0 {}}{ P 0 (1 + r 0 ) P 1 }{{} Pr ice of C 1 4 π 1 is the rate of inflation between t 0 and t 1 : P 1 = (1 + π 1 )P 0 = (1 + r 0) (1 + π 1 ) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 14 / 39

The New IS function Figure 3 The costs of current vs future consumption C 1 slope W 1 (1+r 0 )/ (1+π 1 ) W 0 C 0 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 15 / 39

The New IS function Figure 4 The maximization of intertemporal utility C 1 U0 U1 U2 Equal slopes C* 1 u(c 0 )/ β u(c 1 ) = (1+r 0 )/ (1+π 1 ) C* 0 C 0 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 16 / 39

The Euler Equation The New IS function 1 Therefore, the maximization of utility is given by the condition MRS = RP u (C 0 ) = (1 + r 0) β u(c 1 ) (1 + π 1 ) [ u (C 0 ) = β u (C 1 ) (1 + r ] 0) (1 + π 1 ) (1) 2 Eq. (1) is the Euler equation, already known from previous materials. 3 It relates current to future consumption in an optimal manner 4 And as we will show, it is also the basis for the new IS function (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 17 / 39

The New IS function The Euler Equation with uncertainty 1 Until now, we have assumed no uncertainty about the future level of consumption (C 1 ) and inflation (π 1 ) 2 With uncertainty about C 1 and π 1, we have just to add an expectations operator (E 0 ), 3 Now, the Euler equation looks like [ u (C 0 ) = β E 0 u (C 1 ) (1 + r ] 0) (1 + π 1 ) 4 Intuition 1: r 0 u (C 0 ) C 0 5 Intuition2: E 0 π 1 u (C 0 ) C 0 6 Why the term on the left hand side has no expectations operator attached? (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 18 / 39

The New IS function Log-linearize the Euler Equation 1 To simplify things, assume the utility function is CRRA for t = 0, 1,...: U(c t ) = C1 σ t 1 σ u (C t ) = Ct σ 2 With this, the Euler equation can be written as [ ] Ct σ = β E t Ct+1 σ (1 + r t ) (1 + π t+1 ) 3 Apply logs to the previous equation [ ]) σ ln C t = ln β + ln (E t Ct+1 σ (1 + r t ) (2) (1 + π t+1 ) [ ] (1 + rt ) = ln β + ln E t σ ln E t C t+1 (3) (1 + π t+1 ) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 19 / 39

The New IS function Log-linearize the Euler equation (continued) 1 We know that for a small constant ξ, ( 1 < ξ < 1), we get ln(1 + ξ) ξ 2 Applying the expectations operator, implies that E ln(1 + ξ) E(ξ) ln E(1 + ξ) 3 Then, the 2nd term on the right hand-side of eq.(3) can be written [ ] (1 + rt ) ln E t E t [ln(1 + r t ) ln(1 + π t+1 )] (4) (1 + π t+1 ) r t E t π t+1 (5) 4 and equation (3) will come as σ ln C t = ln β }{{} + (r t E t π t+1 ) σ ln E t C t+1 (6) 0 5 Notice: as (1 + r t ) is known at t, no expectations operator here. (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 20 / 39

The New IS function The Euler Equation: % deviations from the steady state (I) 1 To simplify exposition, let s use small letters to express variables in log values, that is c t = ln C t 2 Therefore, eq. (6) can be written as c t = E t c t+1 1 σ (r t E t π t+1 ) (7) 3 Assume for simplicity: no investment in the economy (capital remains constant over the short run), no government expenditures 4 Then the log of consumption is equal to the log of output c t = y t 5 The linearized Euler equation (7) can be written as y t = E t y t+1 1 σ (r t E t π t+1 ) (8) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 21 / 39

The New IS function The Euler Eq.: % deviations from the steady state (II) 1 Let s define the output gap (x t ) as the difference between the log level of output (y t ) and the log level of potential output (ȳ) x t = y t ȳ 2 Subtracting ȳ from both sides of eq. (8), this can be rewritten as x t = E t x t+1 1 σ (r t E t π t+1 ) (9) 3 Add an exogenous demand shock (µ t ) : µ t = ρ µ µ t 1 + ɛ t with ɛ t (0, σ 2 ɛ) and 0 < ρ µ < 1. 4 And finally we arrived at the standard IS eq. of the New Keynesian model x t = E t x t+1 1 σ (r t E t π t+1 ) + µ t (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 22 / 39

The New IS function The New IS Equation 1 The standard new IS equation x t = E t x t+1 1 σ (r t E t π t+1 ) + µ t 2 If households expect future output gap to increase ( E t x t+1 ), current demand increases and the current output gap will increase ( x t ) 3 If expected future inflation increases ( E t π t+1 ) more than the increase in current interest rates ( r t ), the increase in r t will not in fact constrain the level of current economic activity ( x t ). 4 See Fig.5. (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 23 / 39

The New IS function Figure 5 The New IS function r IS 0 IS 1 Efeitos de subida em E t x t+1 r 0 A B x 0 x 1 x (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 24 / 39

The New AS function or the NKPC III The New AS function (or the New Keynesian Phillips Curve) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 25 / 39

The New AS function or the NKPC Introducing the New AS function or the NKPC 1 In the opening slides, the New AS function, or the New Keynesian Phillips curve (NKPC) looked like π t = β E t π t+1 + λx t + υ t (10) where υ t is a supply shock, υ t = ρ υ υ t 1 + ε t, with ε t (0, σ 2 υ), and 0 < ρ υ < 1. 2 To derive the New AS function can be a hard task 3 But as we did for the IS function, there is also an intuitive way: let s follow the latter to derive the NKPC 4 Be aware that there are more formal and sophisticated ways to get there, and these are required if you want to know all details, but in the end you get to the same results, so for now this suffi ces... (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 26 / 39

The New AS function or the NKPC Assumptions of the New AS function 1 The crucial part of the model: nominal rigidity known as "Calvo Pricing", with four main assumptions: 2 A1. In each period there is a proportion of firms that do not reset their prices: θ. 3 A2. There is monopolistic competition in the goods market: firms set prices (the frictionless price, pt ) with a markup (l) over marginal costs (mc t ). In logs we have p t = l + mc t (11) 4 A3. Because firms know that the price they set today remain constant until some time in the future, they set prices today (z t ) to minimize the loss in profits for not resetting their prices until then 5 A4. Real marginal costs in logs (mc t p t ) are a linear function of the output gap in logs (x t ) mc t p t = γx t (12) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 27 / 39

The New AS function or the NKPC Minimizing the Loss function Minimizing the Loss function 1 Given those assumptions, the Loss function L(z t ) is given by L(z t ) = n=0 (θβ) n E t (z t pt+n) 2 }{{} expected loss in profits (13) 2 β is a time discount factor 3 θ n is the probability of having the price fixed until t + n 4 z t is the log price that minimizes the loss of profits due to no change in prices until t + n 5 To minimize L(z t ) with respect to z t, we set L z t = 0 (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 28 / 39

The New AS function or the NKPC Minimizing the Loss function Minimizing the Loss function (continued) 1 To minimize L(z t ) = n=0(θβ) n E t ( zt p t+n) 2 wrt zt, we get 2 n=0 L z t = 0 (θβ) n E t (z t p t+n) = 0 (14) n=0 (θβ) n z t } {{ } = 1 θβ 1 t = n=0 z t = (1 θβ) (θβ) n E t p t+n n=0 (θβ) n E t p t+n (15) 2 The optimal price set by firms (z t ) is a weighted average of the prices that would be set in the future if there were no price rigidities (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 29 / 39

The New AS function or the NKPC Minimizing the Loss function Minimizing the Loss function (continued) 1 We can use eq (11) pt = l + mc t to substitute for pt in eq.(15) z t = (1 θβ) 2 The result comes as z t = (1 θβ) n=0 n=0 (θβ) n E t p t+n (θβ) n E t (l + mc t+n ) (16) 3 We have just obtained z t : the price that each individual firm sets at t 4 Notice that this is not the same as the aggregate price level (p t ) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 30 / 39

The New AS function or the NKPC Minimizing the Loss function Obtaining the aggregate price level: pt 1 The aggregate price level is determined as a weighted average of those firms that change their prices at t, (z t ), and those that do not change their prices at t (the prices of which are kept unchanged from t 1 : p t 1 ) 2 That is p t = θp t 1 + (1 θ)z t 3 Solving for z t z t = 1 1 θ (p t θp t 1 ) (17) 4 In order to proceed we have to apply a trick here (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 31 / 39

One trick please The New AS function or the NKPC Minimizing the Loss function 1 Remember from the "Solution to Rational Expectations Models" that a sum like this y t = a n=0 b n E t x t+n 2... is a solution to the first order stochastic difference equation if b < 1 3 So if you have equation like (16) z t = (1 θβ) y t = ax t + be t y t+1 n=0 (θβ) n E t (l + mc t+n ) 4 Then, eq. (16) must be a solution to an equation like this z t = θβ E t z t+1 + (1 θβ) (l + mc t ) (18) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 32 / 39

The New AS function or the NKPC Solving for p (after the trick) Minimizing the Loss function 1 So, now we have two equations that display the dynamics of z t : eq. (17) and eq (18). z t = 1 1 θ (p t θp t 1 ) z t = θβ E t z t+1 + (1 θβ) (l + mc t ) 2 Equating both sides 1 1 θ (p 1 t θp t 1 ) = θβ 1 θ (E tp t+1 θp t ) + (1 θβ) (l + mc t ) 3 Defining π t = p t p t 1 4 We will arrive at (after some rearrangement) (1 θ) (1 θβ) π t = βe t π t+1 + (l + mc t p t ) θ (19) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 33 / 39

The New AS function or the NKPC Minimizing the Loss function The New Keynesian Phillips Curve: final steps 1 We have just obtained π t = βe t π t+1 + (1 θ) (1 θβ) (l + mc t p t ) θ }{{} real marg.cost 2 Now the final step. According to Assumption 4 real marginal costs are a log linear function of the output gap (x t ) mc t p t = γx t 3 Assume also that, for simplicity, that l does not change in the business cycle, so it does not affect the dynamics of π t 4 Then we can finally arrive at the NKPC (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 34 / 39

The New AS function or the NKPC The New Keynesian Phillips Curve Minimizing the Loss function 1 The NKPC function finally comes out as π t = β E t π t+1 + λx t γ(1 θ) (1 θβ) with λ =. θ 2 Add technological shocks and there you have: eq (10). υ t = ρ υ υ t 1 + ε t π t = β E t π t+1 + λx t + υ t 3 Basic intuition: see next figure (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 35 / 39

The New AS function or the NKPC Figure 5 The New Phillips Curve Minimizing the Loss function π CP 1 π 1 B CP 0 π 0 A Subida nas expectativas inflacionistas x 0 x (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 36 / 39

The Central Bank s behavior: introduction IV The Central Bank s behavior: introduction (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 37 / 39

The Central Bank s behavior: introduction The Central Bank behavior 1 Until now, we have 3 endogenous variables {x t+s, π t+s, r t+s } s= s=0, but only two equations (IS, AS or the New Phillips curve). 2 So we need another equation in order to have the model determined. 3 There are two major ways to close the model: 1 The central bank controls the money supply, and the market determines r t, (the old view) 2 The central bank controls the r t, and the market determines the level of money in the market (the new view). 4 Next: what is better, the old view or the new view. (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 38 / 39

The Central Bank s behavior: introduction Appendix: explaining how eq. (21) is obtained from (19) if b < 1 and (θβ) < 1, then solution is : y t = ax t + be t y t+1 y t = a n=0 b n E t x t+n solution to : z t = (1 θβ) n=0 (θβ) n E t (l + mc t+n ) z t = θβ E t z t+1 + (1 θβ) (l + mc t ) (Vivaldo M. Mendes) The New Keynesian Model: Introduction 13 November 2013 39 / 39