The New Keynesian Model: Main Functions
Transcript of The New Keynesian Model: Main Functions
The New Keynesian Model: Main Functions
Vivaldo M. Mendes
ISCTE � Lisbon University Institute
November 2018
(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 1 / 34
I �What�s is the New Keynesian Model?
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What is the New Keynesian Model?
The Old and the New Keynesian Models
1 The new model is the old Keynesian model2 With the usual nominal and/or real rigidities in price setting
Prices are sticky (the baseline version)Nominal wages are rigid downwardsStaggered contractsCapacity utilization constraints
3 Without the problems that pushed the model to serious trouble inthe early 70s
1 "forward looking/rational expectations" instead of "adaptiveexpectations"
2 built upon sound microeconomic foundations3 no permanent trade-o¤ between in�ation and unemployment4 no room for stag�ation
4 A new message: rules instead of discretion by Central Banks in themanagement of monetary policy
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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 BusinessCycle literature
3 Contrary to RBC, there is a key role to monetary policy and a minorrole for �scal policy
4 Four basic predictions (the old model up-side-down):1 the instrument of monetary policy ought to be the short term interestrate,
2 policy should be focused on the control of in�ation,3 in�ation can be reduced by aggressively increasing short term interestrates (see next �gure).
4 the central bank should conduct monetary policy adopting a strategy ofcommitment in a forward-looking environment, instead of discretion.
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What is the New Keynesian Model?
Active interest rate policy by the Fed
The FED now reacts much more aggressively to in�ation than in the "oldtimes"
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What is the New Keynesian Model?
A picture of the New Keynesian model
1 On the demand side:1 the IS function is forward looking
xt = �ϕ(rt � Etπt+1) + Etxt+1 + µt
2 the LM function, where the central bank now controls the interestrate, not the money supply:
rt = L(?, ?)
2 In the supply side: the Aggregate Supply function is also forwardlooking:
πt = β � Etπt+1 + λxt + υt
3 (xt) output gap, (πt) in�ation rate, (Etπt+1) expected value at t ofin�ation at t+ 1, (µt) an exogenous demand shock, (υt) anexogenous supply shock.(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 6 / 34
The New IS function
The New IS function
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The New IS function
The IS function in three steps
1 You may �nd a very ellaborate way of deriving the IS function2 There is a simple and intuitive way to derive it
1 Firstly, get the Euler equation2 Second, log-linearize the Euler equation3 Third, express the Euler equation in terms of percentage deviationsfrom the steady state
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The New IS function
The Marginal Rate of Substitution (MRS)
1 Assume a two period economy, with the household�s objective ofmaximizing intertemporal utility from consumption U(C0, C1)
maxC0,C1
u(C0) + βu(C1)
2 The marginal rate of substitution (MRS) of intertemporalconsumption is given by
dU(C0, C1) = 0u0(C0) � dC0 + βu0(C1) � dC1 = 0
3 From where we can get
MRS =dC1
dC0= � u0(C0)
β � u0(C1)
4 See next Figure(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 9 / 34
The New IS function
The utility trade-o¤ between current vs future consumption
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The New IS function
The Relative Price of intertemporal consumption
1 The households budget constraints in each period is given by
P0C0 +A0 = W0
P1C1 = W1 +A0(1+ r0)
P(�) is the price of consumption goods, W(�) is wage income andsavings (A0) are invested in period 0.
2 The two constraints can be consolidated by cancelling out A0
P1C1 = W1 + (W0 � P0C0) (1+ r0)
3 We can obtain the relative price (RP) of future consumption in termscurrent consumption (see next �gure).
RP =dC1
dC0= �P0
P1(1+ r0) = �
(1+ r0)
(1+ π1)
4 π1 is the rate of in�ation between t0 and t1; P1 = (1+ π1)P0.
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The New IS function
The costs of current vs future consumption
W1
W0
slope
C1
C0
� (1+r0 )/ (1+π1 )
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The New IS function
The maximization of intertemporal utility: graphically
� u�(C0 )/ β u�(C1 )
Equal slopes
C1
C0
U0 U1 U2
=� (1+r0 )/ (1+π1 )C*1
C*0
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The New IS function
The Euler equation
1 Therefore, the maximization of utility is given by the condition
MRS = RP
� u0(C0)
β � u(C1)= � (1+ r0)
(1+ π1)
u0(C0) = β ��
u0(C1)(1+ r0)
(1+ π1)
�(1)
2 Eq. (1) is the Euler equation, already known from previousmaterials.
3 It relates current to future consumption in an optimal manner4 And as we will show, it is also the basis for the new IS function
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The New IS function
The Euler equation with uncertainty
1 Until now, we have assumed no uncertainty about the future level ofconsumption (C1) and in�ation (π1)
2 With uncertainty about C1 and π1, we have just to add anexpectations operator (E0),
3 Now, the Euler equation looks like
u0(C0) = β � E0
�u0(C1)
(1+ r0)
(1+ π1)
�4 Intuition 1: " r0 )" u0(C0))# C0
5 Intuition2: " E0π1 )# u0(C0))" C0
6 Why the term on the left hand side has no expectations operatorattached?
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The New IS function
Log-linearize the Euler equation
1 To further simplify the presentation, assume that the utility functionis CRRA for t = 0, 1, ...:
U(ct) =C1�σ
t1� σ
) u0(Ct) = C�σt
2 With this, the Euler equation can be written as
C�σt = β � Et
�C�σ
t+1(1+ rt)
(1+ πt+1)
�3 Apply logs to the previous equation
�σ ln Ct = ln β+ ln�
Et
�C�σ
t+1(1+ rt)
(1+ πt+1)
��(2)
= ln β+ ln Et
�(1+ rt)
(1+ πt+1)
�� σ ln EtCt+1 (3)
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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 Therefore, the second term on the right hand-side of equation (3) canbe written
ln Et
�(1+ rt)
(1+ πt+1)
�� Et [ln(1+ rt)� ln(1+ πt+1)] (4)
� rt � Etπt+1 (5)
4 and equation (3) will come as
�σ ln Ct = ln β|{z}+�0
(rt � Etπt+1)� σ ln EtCt+1 (6)
5 Notice: as (1+ rt) is known at t, no expectations operator here.(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 17 / 34
The New IS function
The Euler equation as % deviations from the steady state
1 To simplify exposition, let�s use small letters to express variables inlog values, that is
ct = ln Ct
2 Therefore, eq. (6) can be written as
ct = Etct+1 �1σ(rt � Etπt+1) (7)
3 Assume for simplicity: no investment in the economy (capitalremains constant over the short run), no government expenditures
4 Then the log of consumption is equal to the log of output
ct = yt
5 The linearized Euler equation (7) can be written as
yt = Etyt+1 �1σ(rt � Etπt+1) (8)
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The New IS function
The Euler eq. as % deviations from the steady state (II)
1 Let�s de�ne the output gap (xt) as the di¤erence between the loglevel of output (yt) and the log level of potential output, or thesteady state level, (y)
xt = yt � y2 After some rearrangement, eq. (8) can be rewritten as
xt = Etxt+1 �1σ(rt � Etπt+1) + µt (9)
3 where we added µt as exogenous demand shocks: µt = ρµµt�1 + εt,
with εt � (0, σ2ε) and 0 < ρµ < 1.
4 If families expect future output gap to increase (" Etxt+1), currentdemand increases and the current output gap will increase (" xt)
5 If expected future in�ation increases (" Etπt+1) more than theincrease in current interest rates (" rt), the increase in rt does notconstrain the level of current economic activity (" xt). See Fig.5.(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 19 / 34
The New IS function
The New IS function
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The New IS function
The New AS function (or theNew Keynesian Phillips Curve)
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The New IS function
1 As shown in the opening slides, the New AS function, or the NewKeynesian Phillips curve (NKPC),can be written as
πt = β � Etπt+1 + λxt + υt (10)
2 where υt is a supply shock, υt = ρυυt�1 + εt, with εt � (0, σ2υ), and
0 < ρυ < 1.3 To derive the New AS function can be a hard task4 But as we did for the IS function, there is also an intuitive way: let�sfollow the latter to derive the NKPC
5 Be aware that there are more formal and sophisticated ways to getthere, and these are required if you want to know all details, but inthe end you get to the same results, so for now this su¢ ces ...
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The New IS function
Assumptions of the New AS function
1 The crucial part of the model: nominal rigidity known as "CalvoPricing", with four main assumptions:
2 A1. In each period there is a proportion of �rms that do not resettheir prices: θ.
3 A2. There is monopolistic competition in the goods market: �rms setprices (the frictionless price, p�t ) with a markup (`) over marginalcosts (mct). In logs we have
p�t = `+mct (11)
4 A3. Because �rms know that the price they set today remainconstant until some time in the future, they set prices today (zt) tominimize the loss in pro�ts for not resetting their prices until then
5 A4. Real marginal costs in logs (mct � pt) are a linear function ofthe output gap in logs (xt)
mct � pt = γxt
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The New IS function Minimizing the Loss function
Minimizing the Loss function
1 Given those assumptions, the Loss function L(zt) is given by
L(zt) =∞
∑n=0(θβ)n � Et (zt � p�t+n)
2| {z }expected loss in pro�ts
(12)
2 where 0 < β < 1 is a discount factor, and zt is the log price thatminimizes the loss of pro�ts due to no change in prices until t+ n
3 Notice: θn is the probability of having the price �xed until t+ n4 To minimize L(zt) with respect to zt, we set
∂L∂zt
= 0
(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 24 / 34
The New IS function Minimizing the Loss function
Minimizing the Loss function (continued)
1 To minimize L(zt) = ∑∞n=0(θβ)n � Et
�zt � p�t+n
�2 wrt zt, we get
∂L∂zt
= 0
2∞
∑n=0(θβ)n � Et (zt � p�t+n) = 0 (13)
∞
∑n=0(θβ)n � zt| {z }= 1
1�θβ �zt
=∞
∑n=0(θβ)n � Etp�t+n
zt = (1� θβ)∞
∑n=0(θβ)n � Etp�t+n (14)
2 The optimal price set by �rms (zt) is a weighted average of the pricesthat would be set in the future if there were no price rigidities
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The New IS function Minimizing the Loss function
Minimizing the Loss function (continued)
1 We can use eq (11) to substitute for p�tp�t = `+mct
zt = (1� θβ)∞
∑n=0(θβ)n � Etp�t+n
2 The result comes as
zt = (1� θβ)∞
∑n=0(θβ)n � Et (`+mct+n) (15)
3 Once we have zt, it is easy to obtain the level of the aggregate pricelevel (pt)
pt = θpt�1 + (1� θ)zt
or
zt =1
1� θ(pt � θpt�1) (16)
4 pt�1 is last period�s aggregate price level, zt is the new reset price(Vivaldo M. Mendes) The New Keynesian Model: Main Functions November 2018 26 / 34
The New IS function Minimizing the Loss function
Minimizing the Loss function (continued)
1 In order to continue we have to apply a trick. Remember from the"Solution to Rational Expectations Models" that a sequence like
yt = a∞
∑n=0
bn � Etxt+n
2 ... is a solution to the �rst order stochastic di¤erence equation ifjbj < 1
yt = axt + bEtyt+1
3 Applying this reasoning to (15), this eq. can be re-written (videappendix)
zt = θβ � Etzt+1 + (1� θβ) (`+mct) (17)4 So equating both eq. (16) and (17), we get
11� θ
(pt � θpt�1) = θβ � 11� θ
(Etpt+1 � θpt) + (1� θβ) (`+mct)
(18)5 See next slide for details
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The New IS function Minimizing the Loss function
Minimizing the Loss function (continued)
1 Eq. (16) was given by
zt =1
1� θ(pt � θpt�1)
2 Eq.(15) was given by
zt = θβ � Etzt+1 + (1� θβ) (`+mct)
3 Equating both we get
11� θ
(pt � θpt�1) = θβ � 11� θ
(Etpt+1 � θpt) + (1� θβ) (`+mct)
4 Leading to, after some rearrangement
πt|{z}=pt�pt�1
= βEtπt+1 +(1� θ) (1� θβ)
θ(`+ mct � pt)| {z }
real marg.cost
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The New IS function Minimizing the Loss function
Minimizing the Loss function (continued)
1 Therefore, the New Phillips Curve can be written
πt = βEtπt+1 +(1� θ) (1� θβ)
θ(`+ mct � pt)| {z }
real marg.cost
2 Now the �nal step. According to Assumption 4 real marginal costsare a log linear function of the output gap (xt)
mct � pt = γxt
3 Therefore, the AS function can �nally come out as
πt = β � Etπt+1 + λxt
4 with λ =γ(1� θ) (1� θβ)
θ. Add technological shocks
υt = ρυυt�1 + εt, and there you have: eq (10).
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The New IS function Minimizing the Loss function
The New Phillips Curve
π
x
CP1
x0
π 0
B
A
CP0
π 1
Subida nasexpectativasinflacionistas
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The New IS function Minimizing the Loss function
The Central Bank behavior
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The New IS function Minimizing the Loss function
The Central Bank behavior
1 Until now, we have 3 endogenous variables fxt+s, πt+s, rt+sgs=∞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 marketdetermines rt, (the old view)
2 The central bank controls the rt, and the market determines the levelof money in the market (the new view).
4 Next: what is better, the old view or the new view.
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The New IS function Minimizing the Loss function
Appendix
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The New IS function Minimizing the Loss function
Appendix: explaining how eq. (21) is obtained from (19)
if jbj < 1 and (θβ) < 1, then
yt = axt + bEtyt+1
#solution is :
yt = a∞
∑n=0
bn � Etxt+n
zt = (1� θβ)∞
∑n=0(θβ)n � Et (`+mct+n)
#solution to :
zt = θβ � Etzt+1 + (1� θβ) (`+mct)
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