程序代写代做代考 flex Lecture 6:

Lecture 6:
New Keynesian Models
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020

Introduction
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Starting Point: the RBC Model
• In the RBC model, prices (goods prices and wages) adjust to ensure equilibrium in all markets.
• If we add a monetary sector to the RBC model, nominal disturbances only affect nominal prices and wages, leaving the real side of the economy unchanged.
• This is the classical dichotomy.
• Introducing some market imperfections may not be enough to break the classical dichotomy.
• e.g., employment models from last week are real models • adding imperfect competition alone will not be enough
• To break the classical dichotomy, we need some kind of nominal imperfection.
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Incomplete Nominal Adjustment
• With incomplete (or, sluggish) nominal adjustment, changes in aggregate demand for goods at a given level of prices affect the amount that firms produce.
⇒ Nominal (e.g., monetary) disturbances cause changes in employment and output, at least in the short run.
• In addition, real shocks affect aggregate demand at a given price level.
⇒ Incomplete price adjustment creates a new channel through which real shocks affect output.
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Outline
1. Imperfect competition and price setting (i) The model
(ii) The effects of nominal price rigidity
2. Dynamic New Keynesian models (i) Building blocks
(ii) The Calvo model and the New Keynesian Phillips curve (iii) The canonical New Keynesian model
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Imperfect Competition
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Assumptions
• Continuum of differentiated goods indexed by i ∈ [0, 1].
• Each good is produced by single firm who is a monopolist. • Firm i’s production function is
Yi =Li.
Labor Li hired in perfectly competitive market. Output Yi sold in imperfectly competitive market.
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Assumptions: Household Utility
• Households’ utility depends on consumption and labor: U=C−1Lγ, γ>1,
γ
• C is a CES aggregate of the individual goods: 􏰒􏰗 1 (η−1)/η 􏰓η/(η−1)
C=Ci di ,η>1. 0
• η is the elasticity of substitution between varieties.1 • This formulation for C is due to Dixit and Stiglitz (1977)
1Define the marginal rate of substitution between varieties i and j, MRSi,j = (∂C/∂Ci)/(∂C/∂Cj). Then, it can be shown that the elasticity of substitution, defined as d ln(Cj /Ci )/d ln MRSi ,j , is equal to η.
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Further Assumptions
• No investment, government, or international trade. Hence: Y=C
• To analyze the effects of monetary changes, we assume aggregate demand is
Y = M. (1) P
Interpretations:
• (1) implies inverse relationship between prices and output. • M can be thought of as a generic variable affecting
aggregate demand rather than money.
• It’s possible to derive (1) from more complete models: e.g.,
real money balances in the utility function, cash-in-advance constraint, or central bank targets nominal GDP.
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Household’s Problem
• Household’s problem can be solved in two stages.
(1) Choose C and L given a price index P for aggregate
consumption and wage W
(2) Choose varieties Ci given prices Pi of the individual varieties
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Household’s Problem: Stage 1
• Choose C and L: 􏰆1􏰇
max C− Lγ s.t. PC=WL+R C,L γ
where P is a price index for C, W is the nominal wage, and
R is nominal profit income from firms.2
• Substitute for C using budget constraint:
􏰆WL+R 1 􏰇 max − Lγ
• FOC:
P
LPγ
W−Lγ−1=0 ⇒ (2) Labor supply elasticity is 1/(γ − 1). Since Y = L, this also
establishes a relationship between Y and W /P.
2From the stage 2 problem, we will see what the price index must be.
􏰂W 􏰃1/(γ−1) L=P
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Household’s Problem: Stage 2
• Given C, choose varieties Ci:
L= • FOC for Ci:
􏰗 1 00
η η −1η−1 η−1 􏰒􏰗1 η−1 􏰓η−1 η−1
􏰓
max
􏰒􏰗 1 (η−1)/η 􏰓η/(η−1) 􏰗 1 Ci di s.t.
Pi Ci di = PC
Ci0 0 • Set up Lagrangian:
=
Cj η dj 􏰒􏰗 1 η−1
η
􏰒􏰗 1 (η−1)/η 􏰓η/(η−1) 􏰒
Ci di +λ PC−
∂L
∂Ci η−10
−λPi =0
PiCidi
Ci
⇒ Cηdj Cη=λP
0
−1 jii
1 􏰓η−1
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Household’s Problem: Stage 2
• From previous slide 􏰒􏰗 1
0
􏰒 η−1
• Note C = 􏰖 1 C η di
1
η−1 􏰓η−1 −1
Cηdj Cη=λP jii
η
􏰓η−1 η−1 η−1
⇒ C η = 􏰖 1 C η 0i 0i
di
􏰒􏰓11
Cη−1 η−1C−η=λP ⇒
(3)
ηii
• To solve for λ, plug (3) into budget constraint:
−η 􏰒􏰗1 1−η 􏰓−1
λ =P Pi di 0
PC =
􏰗1 −η−η
Pi Cλ Pi di ⇒ (4)
0
􏰏 􏰎􏰍 􏰐
Ci
C =Cλ−ηP−η ii
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Household’s Problem: Stage 2
• Use (4) to substitute for λ in (3):
P −η
Ci =C􏰖1P1−ηdjPi (5) 0j
• Substitute (5) back into definition of C:
η􏰄􏰅η 􏰒􏰗 1 η−1 􏰓η−1 PC 􏰗 1 􏰀 􏰁η−1 η−1
C= Cηdi =􏰖 0 i
P−ηηdi 1P1−ηdj 0 i
0j η
􏰗1 1−η 􏰒􏰗1 1−η 􏰓η−1 ⇒ Pi di=P Pi di
00
⇒
1 􏰒􏰗 1 1−η 􏰓1−η
P= Pi di 0
This is the price index corresponding to C. • Substitute for P in (5)
􏰂Pi 􏰃−η Ci=C P
(6)
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Firm’s Problem
• Real profits of producer i:
Ri = Pi Yi − W Li
As a monopolist, producer sets the price Pi , taking as given
the demand for its good.
• Substitute for the demand function (6) and use that Yi = Li
(by the production function), Y = C and Yi = Ci :
Ri 􏰂Pi 􏰃1−η W 􏰂Pi 􏰃−η
P=PY−PPY • FOC for Pi/P:
􏰂Pi 􏰃−η W 􏰂Pi 􏰃−η−1 (1−η)PY+ηPP Y=0
PPP
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Firm’s Problem
• FOC (from previous slide):
􏰂Pi 􏰃−η W 􏰂Pi 􏰃−η−1
(1−η)PY+ηPP Y=0 • Solving for Pi/P:
Price set as a markup over marginal cost.
Pi = η W P η−1P
(7)
Markup is higher when elasticity of demand, η, is smaller (i.e., closer to 1, since η > 1)
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Equilibrium
• Use Y = L and household’s FOC in (2), W /P = Lγ−1, then the optimal price becomes:
P i∗ η
P =η−1Y
γ − 1
• By symmetry Pi = P, ∀i. Hence,
􏰂η − 1􏰃1/(γ−1)
Y= η (8) • Then use aggregate demand to determine P
P=M= M (9) Y [(η − 1)/η]1/(γ−1)
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Welfare
• Level of labor L ̄ (using C = L = Y ) which maximizes welfare: max L ̄ − 1 L ̄γ ⇒ L ̄ = 1
γ
• Hence from (8), Y = 􏰀 η−1 􏰁1/(γ−1) < 1. η When producers have market power, the equilibrium output is less than the socially optimal level. When producers have market power, then higher labor supply elasticity (lower γ) ⇒ smaller output 18/46 Shift in Aggregate Demand • Suppose the economy is initially in its flexible price equilibrium. From (8) and (9): 􏰂η − 1􏰃1/(γ−1) M Y = η and P = [(η − 1)/η]1/(γ−1) • Suppose there is an increase in M to M′ • With flexible prices, P increases, Y unaffected. • Imperfect competition alone does not imply monetary non-neutrality. Change in M ⇒ proportional change in nominal wage and all nominal prices, so that output and the real wage are unaffected. 19/46 Shift in Aggregate Demand: Nominal Rigidity • Suppose instead, no firms adjust their prices M′ 􏰂Pi 􏰃−η Y ′ = and Y ′ = Y ′ PiP Since Pi and P are unchanged, Y and Yi increase • Conditional on no other firms changing their prices, if one individual firm could change its price, it would choose Pi′ η ′γ−1 = (Y ) usingY =L,(2)and(7) P η−1􏰏 􏰎􏰍 􏰐 W/P Firm wants to increase Pi /P as it faces higher marginal costs • If increase in profits from new price less than a menu cost equal to Z, no firms adjust prices. 20/46 Building Blocks of Dynamic New Keynesian Models 21/46 Overview • Like the RBC model, the New Keynesian model is a general equilibrium model, built up from microeconomic foundations. • However, the New Keynesian model introduces nominal rigidities, so that monetary disturbances have real effects. • To generate nominal rigidity, two ingredients are needed: • Imperfect competition (price-setting firms) • Frictions in nominal price adjustment (e.g., Calvo) 22/46 Overview of Assumptions • Time is discrete. • Imperfectly competitive firms produce output using labor as only input. • No government purchases, no investment, no international trade • Households maximize utility, taking as given paths of the real wage and the real interest rate. • Firms, owned by households, maximize the present discounted value of profits, subject to constraints on price setting. • Central bank determines the path of the real interest through its conduct of monetary policy. 23/46 Household’s Problem • Representative household’s lifetime utility: ∞ 􏰈βt[U(Ct)−V(Lt)], 0<β<1 t=0 C is a CES aggregate of individual differentiated consumption goods, with the elasticity of substitution η > 1. • Assume functional forms:
C 1−θ
U(Ct) = t , θ > 0
1−θ
V(Lt)=BLγt, B>0,γ>1
γ
• W is the nominal wage and P is the price level.
Formally, P is the price index associated with the CES consumption good; However, we will approximate p = ln P as the average of firms’ log prices.
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Household’s Problem: Labor Supply
• FOC for labor supply:
V′(Lt) = U′(Ct)Wt
Pt • Substituting for V′(Lt) and U′(Ct):
BLγ−1 =C−θWt t tPt
• UsingthatYt =Lt andCt =Yt:
Real wage is positively related to output.
Wt =BYθ+γ−1 Pt t
(10)
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Household’s Problem: Euler Equation
• The Euler equation is
U′(Ct) = β(1 + rt)U′(Ct+1)
• Substitute for U′(·):
C−θ =β(1+r)C−θ.
t tt+1 • Taking logs and dividing by θ give
lnCt =lnCt+1−1ln[β(1+rt)] θ
• UsingagainthatCt =Yt,ln(1+rt)≈rt,wegetthe new Keynesian IS curve,
, a≡−ln(β)/θ (11) This implies an inverse relationship between rt and Yt.
lnYt =a+lnYt+1−1rt θ
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Firm’s Problem
• Firm i’s production function is Yit = Lit
• Firm i faces the demand function Y = Y 􏰀 Pit 􏰁−η
it tPt
• Real profits are
P W 􏰂P 􏰃1−η W 􏰂P 􏰃−η
R = itY − tY =Y it − t it itPitPittP PP
ttttt
• We study the firm’s price setting decision at time 0.
• Firms are owned by households – hence, firms discount
profits in period t using λt ≡ βtU′ (Ct)/U′ (C0)
• The length of time a given price is in effect is random. Let qt denote the probability that the price the firm sets in period 0 is still in effect in period t.
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Firm’s Problem: Price Setting
• To set price Pi , firm solves:
max E 􏰈 qtλtRit
Pi
where Rit is profits in period t if Pi is still in effect.
• Substituting for Rit:
􏰄∞􏰅 t=0
􏰈∞ 􏰂Pi􏰃1−η Wt􏰂Pi􏰃−η qtλtYt  − 
maxE
Pit=0 Pt PtPt
∞
Pi
max E 􏰈 q λ Y Pη−1􏰀P1−η − W P−η􏰁
tttt i ti t=0
∞􏰂η−1􏰃 maxE 􏰈q λ Y Pη−1 P1−η − P∗P−η
tttt i ti Pi t=0 η
P∗ =
t η−1 t
η W , price which maximizes profits in period t.
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Firm’s Problem: Price Setting
• Re-write the firm’s problem as ∞
maxE 􏰈q λ Y Pη−1F(p ,p∗) pi ttttit
t=0
where pi = lnPi, pt∗ = lnPt∗ and
F(pi,pt∗)=e(1−η)pi −η−1ept∗−ηpi η
• Approximate F(·,pt∗) with a second-order Taylor approximation around pt∗:3
F(pi,pt∗)≃F(pt∗,pt∗)−K(pi −pt∗)2 3Second-order Taylor approximation of f (x) around a is
f(x)≈f(a)+f′(a)(x−a)+f′′(a)(x−a)2. Inthiscase,atpi =pt∗, 2
∂F(pi,pt∗)/∂pi = 0 and ∂2F(pi,pt∗)/∂pi2 < 0. Furthermore, technically K should depend on pt∗, but we assume it’s constant as well. 29/46 Firm’s Problem: Price Setting • Assume that variation in U′(Ct ) Y Pη−1 is negligible relative to ∗U′(C0) t t the variation in qt and pt . • Then the firm’s problem simplifies to4 ∞ min􏰈qtβtE[(pi −pt∗)2] pi t=0 ∞ min􏰈qtβt 􏰋(pi −E[pt∗])2 +Var(pt∗)􏰌 pi • FOC for pi implies ∞ βtqt pi=􏰈ω ̃tE[pt∗], ω ̃t=􏰔∞ βsq t=0 t=0 s=0 Optimal price is a weighted average of the profit-maximizing (12) s prices (or marginal cost) during the time the price is in effect. 4This uses that E[x2] = E[x]2 + Var(x). 30/46 Firm’s Problem: Price Setting • Using the household’s FOC (W /P = BY θ+γ−1, see (10)): ttt P t∗ η W t η θ + γ − 1 P =η−1P =η−1BYt tt 􏰂η􏰃 pt∗ =pt +ln 􏰏 􏰎􏰍 􏰐 η−1 ≡c +lnB+(θ+γ−1)yt 􏰏 􏰎􏰍 􏰐 ≡φ • Set c = 0: pt∗ − pt is proportional to yt . (13) p t∗ = p t + φ y t 31/46 The Central Bank • To close the model, we assume the central bank sets the interest rate, rt , following some rule conditional on the state of the economy. • For example, rt will be a linear function expectations of future inflation and output. • More specifics on this will be given later. 32/46 Calvo Model 33/46 Calvo Model for Price Setting • In each period, a fraction α ∈ (0, 1] of firms, drawn at random, are allowed to reset their price optimally. • The others are constrained to keep their prevailing price, and do not re-optimize. • The probability a firm is able to change its price is the same each period, regardless of when it last changed its price. • This assumption is due to Calvo (1983). • Advantages: • The model can easily accommodate any degree of price stickiness via different values of α. • It leads to a simple expression for the dynamics of inflation. 34/46 Calvo Model for Price Setting • Let xt be the price set by firms that set new prices in t. All firms will choose the same price. • This implies: Average price of firms who do not change their price is just pt =αxt +(1−α)pt−1 the average price from the last period. We are using the approximation that pt is the average of firm log prices. • Subtracting pt−1 from both sides gives πt =α(xt −pt−1) where πt = pt − pt−1 is inflation. 35/46 Calvo Model for Price Setting • From (12), the price chosen optimally by a firm at time t is 􏰈∞ βjqj xt = 􏰔∞ βsq Etpt+j. j=0 s=0 s • Calvo’s assumption implies that qj = (1 − α)j . Hence: ∞ x =[1−β(1−α)]􏰈βj(1−α)jEp∗ j=0 t t t+j • Re-write: x =[1−β(1−α)] p∗+􏰈βj(1−α)jEp∗ ∗ ∞  t t tt+j j=1  36/46 Calvo Model • Make substitution, k = j − 1, in the summation: 􏰆∞􏰇 k=0 ∞ β(1−α)[1−β(1−α)]􏰈βk(1−α)kE p∗ t t+k+1 k=0 􏰏 􏰎􏰍 􏰐 Etxt+1 • Subtract pt from both sides: xt−pt =[1−β(1−α)](pt∗−pt)+β(1−α)(Etxt+1−pt) x =[1−β(1−α)] p∗ +􏰈βk+1(1−α)k+1E p∗ t t xt =[1−β(1−α)]pt∗+ t t+k+1 37/46 Calvo Model • Using πt = α(xt −pt−1) and pt∗ = pt +φyt (see (13)): (xt −pt−1)−(pt −pt−1)=[1−β(1−α)](pt∗ −pt) 􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐 πt/α πt 􏰏 􏰎􏰍 􏰐 φyt +β(1−α)(Etxt+1 −pt) 􏰏 􏰎􏰍 􏰐 Etπt+1/α • Simplifying: πt = α [1−β(1−α)]φyt +βEtπt+1 1−α 􏰏 􏰎􏰍 􏰐 ≡κ This is the new Keynesian Phillips curve. 38/46 New Keynesian Phillips Curve πt =κyt +βEtπt+1 • Higher output (via higher marginal costs) raises inflation, as does expected future inflation. • What is new? 1. It is derived by aggregating the behavior of price-setter firms facing barriers to price adjustment. 2. The inflation term on the right-hand side is the current expectations of next period’s inflation. 39/46 Canonical New Keynesian Model 40/46 The Canonical New Keynesian model • 3-equation model due to Clarida, Galí and Gertler (2000): y=E[y ]−1r+uIS, θ>0 (14)
t t t+1 θt t
πt=βEt[πt+1]+κyt+utπ, 0<β<1,κ>0 (15)
r=φE[π ]+φE[y ]+uMP, φ,φ>0 (16) t π t t+1 y t t+1 t π y
IS curve, Phillips curve, forward-looking interest rate rule. Constant terms set to zero ⇒ interpret variables as departures from steady state
• Adds three shock processes:
uIS=ρuIS +eIS, −1<ρ <1 tISt−1t IS uπ=ρuπ +eπ, −1<ρ<1 tπt−1t π uMP =ρ uMP +eMP, −1<ρ <1 tMPt−1t MP where (eIS,eπ,eMP) are independent white-noise disturbances 41/46 Solving the Model • One can solve the model using the method of undetermined coefficients, like the RBC model. Positthatyt,πt linearfunctionsofdisturbances: y=auIS+auπ+a uMP t ISt πt MPt π=buIS+buπ+b uMP t ISt πt MPt Plug these equations into (14), (15), (16) and solve for the (a, b) such that these equations are always satisfied. • Then, one can calibrate to model to understand its dynamics. Following Galí (2015), we set θ = 1, κ = 0.172, β = 0.99 (one period is a quarter), φπ = 0.5, and φy = 0.125. Then we consider two scenarios for (ρIS,ρπ,ρMP): • ρIS =ρπ =ρMP =0 • ρIS =ρπ =ρMP =0.5 42/46 Effects of a Monetary Policy Shock Output 00 -0.5 -0.2 Inflation -1 -0.4 -1.5 0 5 10 0 5 10 -0.6 Quarters Quarters Interest Rate MP Shock 11 0.5 0.5 00 0 5 10 0 5 10 Quarters Quarters Keyequations: y =E [y ]−r/θ+uIS t t t+1 t t π =βE [π ]+κy +uπ, r =φ E [π ]+φ E [y ]+uMP t tt+1 t t t πtt+1 ytt+1 t 43/46 Pct. Dev. Pct. Dev. Pct. Dev. Pct. Dev. Effects of a Demand Shock Output Inflation 1.5 1 0.5 0.6 0.4 0.2 00 0 5 10 0 5 10 Quarters Interest Rate Quarters IS Shock 0.2 0.1 1 0.5 00 0 5 10 0 5 10 Quarters Quarters Keyequations: y =E [y ]−r/θ+uIS t t t+1 t t π =βE [π ]+κy +uπ, r =φ E [π ]+φ E [y ]+uMP t tt+1 t t t πtt+1 ytt+1 t 44/46 Pct. Dev. Pct. Dev. Pct. Dev. Pct. Dev. Effects of an Inflation Shock Output 02 -0.2 Inflation -0.4 -0.6 1 0 0 5 10 0 5 10 Quarters Interest Rate Quarters Inflation Shock 0.4 0.2 1 0.5 00 0 5 10 0 5 10 Quarters Quarters Keyequations: y =E [y ]−r/θ+uIS t t t+1 t t π =βE [π ]+κy +uπ, r =φ E [π ]+φ E [y ]+uMP t tt+1 t t t πtt+1 ytt+1 t 45/46 Pct. Dev. Pct. Dev. Pct. Dev. Pct. Dev. Discussion • Baseline New Keynesian model appealing because of it’s simplicity, but it is also unrealistic. • Large and active literature engaged in constructing more sophisticated DSGE models but maintain the core elements of the New Keynesian model. • Key extensions: • Allow for inflation inertia • Model wage stickiness as well • Assume habit formation in household utility • Assume some fraction of consumption is determined by rule-of-thumb or liquidity-constrained households who devote all current income to consumption • Allow for credit market imperfections 46/46