3 The New Neoclassical Synthesis and 3.1 Monopolistic Competition and Externalities

Up to now we considered dynamic macro models with flexible prices. 1) Standard Growth Models: Consumption smoothing; capital accumulation 2) Models with Money: In that model, tax distortion ( tax) creates inefficiency. Friedman rule as solution: zero nominal rate of interest or payment of interest on money The welfare cost of inflation in these models (welfare triangle under the money demand function) is rather small (empirically: money balances small)

Monetary policy is not only concerned with long run aspects, but also with short run fluctuations. The standard Keynesian view considered Monetary and fiscal policy as instruments to stabilise the economy [IS/LM model]. This view came under attack with the rational expectation revolution by Lucas /Sargent/ Wal- lace: They are argued that systematic stabilisation policy is ineffective. In this Real view, fluctuations are seen as an efficient response to real shocks. In contrast, the New Keynesian re-established a role for monetary policy in the presence of coordination failures/sticky prices in models with rational expectations.

In order to try to understand the role of price rigidities, we need a model which allows for explicit price setting. The first generation of models to introduce mo- nopolistic competition was the static model of Blanchard /Kiyotaki. We first look at this setting in the absence of nominal rigidities and show how imperfect competition creates aggregate demand externalities on the macro level. When introducing sticky prices, we can look at the effects of nominal money and shocks, to understand the effects on output and welfare. If firms have monopoly power, they may want to accommodate shifts in demand as long as price exceeds marginal cost. So movements in demand, both positive or negative, will have an effect on output, at least within some range (as long as MC < P). Section 3.3, looks at the implications of fluctuations for monetary policy in a simple stylized setup. Section 3.4 discusses designs for credible monetary policy.

Much of the macro research during the last 10 years has integrated key elements of these Keynesian features into the dynamic model we have developed until now, with explicit modelling choices for consumption, saving, money holding and leisure in a dynamic stochastic environment with sticky price setting. In the last part of the lecture (section 3.5), we will derive the current workhorse if macro, which incorporates all these aspects, known as the “New Keynesian" model or The New Neoclassical Synthesis. In that context, we will analyse the effects of dynamic price staggering and reexamine implications for monetary and fiscal policy.

88

Note: The intertemporal versions of IS/AS Model based on explicit microfoun- dations with dynamic optimisation are fairly sophisticated models with forward looking behaviour. At the same time, however, they are drastically simplified to allow for explicit solutions (such as: assume Constant Elasticity of Substitution; abstract from capital formation; focus on specific price setting assumptions, rep- resentative agent type models (despite heterogeneity). General equilibrium with specific preferences, designed to give explicit solu- tions (allow for approximation to linear functions) Key features: Monopolistic competition (price setting); heterogenous goods; representative agent models with aggregate externalities (aggregation)

This approach provides some important general insights: a) Forward looking behaviour allows to analyse impact of change in policies b) Captures realistic features of current monetary policy: Under what condi- tions yield interest rate rules stable solutions? c) Helps to understand the role of institutions (independent, credible central bank) d) Allows for welfare analysis e) Empirical analysis: Allows to confront theory with data

Some key features of the new generation dynamic models can be more easily captured in the static set up of the first generation of New Keynesian models. Therefore we start first with a simpler (static) approach to gain intuition for some insights:

A) Systematic, anticipated stabilisation policy can be quite effective in the presence of price rigidities B) When market equilibrium is inefficient, a surprise in monetary policy may be welfare improving (nature of aggregate demand externalities) C) But it is not possible to implement a policy of systematic surprise → there is the problem of dynamic inconsistency, calling for credible commitment mechanisms.

89 Rational expectation equilibrium revolution:

Lucas-Critique of Keynesian Approach:

You may fool all of the people some time, and you can fool some people all of the time, but you can't fool all of the people all of the time.

Key Argument: Private Agents react consciously to changes in policy – seem- ingly stable relations may break down → Proper modelling of economic policy has to be based on sound microeconomic foundations

Policy Conclusions in Lucas type models:

1) Ineffectiveness of a policy of systematic stabilisation Monetary policy can have real effects only if it tries to fool

2) Fooling is welfare reducing, since it distorts optimal rational choice.

But these policy conclusions are based on two special conditions:

1) In the absence of intervention, market will reach equilibrium with flexi- ble prices (no price rigidities; no disequilibrium)

2) Market equilibrium is socially efficient (no externalities))

(1) does not hold in the presence of price rigidities [see Stanley Fisher (1977) and John Taylor (1980)]; Monetary policy as a public good in the presence of coordination failure, if nominal contracts have to be arranged before ob- serving shocks): after the realisation of shocks, monetary policy as central coordination mechanism allows for a smoother reversion to the flexible price outcome, saving on private adjustment and coordination costs (Summer time analogy).

(2) In the presence of aggregate demand externalities, market equilibrium is at best constrained efficient. Due to structural inefficiencies, the market out- come is below the efficient level.

90 Intuition behind aggregate demand externality: Blanchard /Kiyotaki (1987) We first analyse a one-period problem without uncertainty. [Later, we will add (a) uncertainty (shocks) and (b) will consider a dynamic ver- sion with bonds and money]. For the moment, we also abstract from nominal rigidities. The objective of the representative household is to M Max = CuU j − Nv )();( jjj P j st. per period budget constraint: M +T + PC = Y jjjj where nominal 1 income is ∫ jjj )( +∏+= MdiiNwY j . 0 This budget constraint is a short cut for a dynamic budget constraint. 1 α 1−α ⎛ j ⎞ ⎛ j / PMC ⎞ 1 1+γ For U we use the specification: ⎜ ⎟ ⎜ ⎟ − N n j ⎜ ⎟ ⎜ ⎟ j ⎝ ⎠ ⎝ 1−αα ⎠ 1+ γ n Note: This model specification is designed to separate intratemporal problem and intertemporal problem to allow for straightforward ag- gregation for demand for composite goods. Among the advantages of this specification will be a very simple relation between consumption and real money balances, and constant marginal utility of income.

Each household gets utility out of consuming a consumption basket, composed of all goods produced (Dixit Stiglitz version of monopolis- tic competition – heterogenous producers with market power, derived from preferences with constant elasticity of substitution). Here, we consider the continuous case (see Blanchard/Kiyotaki for discrete ver- sion): θ θ −1 ⎡1 ⎤θ −1 θ = ⎢∫ jj )( diicC ⎥ ⎣⎢0 ⎦⎥ 1 1 ⎡ 1−θ ⎤1−θ with Dixit Stiglitz price index: = ∫ )( diipP . ⎢⎣0 ⎦⎥

1 Note: In a dynamic setting, we should only include the opportunity cost for real money bal- ances in the per period budget constraint (see Woodford)! 91 Production: Each firm produces a differentiated good using labor with a constant returns technology iy = A N i)()( . A is the level of technology. We can think of movements in A as technological shocks.

We have to derive the demand curve each firm is facing for its prod- uct from the demand for the good by all consumers. To characterize the equilibrium, proceed in 4 steps: 1) Given spending on consumption, derive consumption de- mands for each good by each household. 2) Derive of the relation between aggregate consumption and aggregate real money balances. 3) Derive labour supply 4) Derive the demand curve facing each firm, and its pricing decision 5) Characterise the general equilibrium

The households problem is α 1−α ⎛ ⎞ ⎛ / PMC ⎞ 1 U = ⎜ j ⎟ ⎜ j ⎟ − N1+γ with j ⎜ ⎟ ⎜ ⎟ j ⎝ ⎠ ⎝ 1−αα ⎠ 1+ γ θ θ −1 ⎡1 ⎤θ −1 θ = ⎢∫ jj )( diicC ⎥ ⎣⎢0 ⎦⎥ 1 subject to ∫ j )()( NwYMdiicip +Π+==+ M jjjjj 0 We solve the problem step by step: 1) Choose optimal j ic )( for a given budget X j spent on the consumption −θ −θ ⎛ ip )( ⎞ X j ⎛ (ip ) ⎞ basket: j ic )( = ⎜ ⎟ = ⎜ ⎟ C j with P C = X jj ⎝ p ⎠ p ⎝ p ⎠

2) Choose optimal mix between C j and M j for given total income Yj: α 1−α ⎛ j ⎞ ⎛ j / PMC ⎞ 1 1+γ U = ⎜ ⎟ ⎜ ⎟ − N s.t. + += MYMCP j ⎜ ⎟ ⎜ ⎟ j jjjj ⎝ α ⎠ ⎝ 1−α ⎠ 1+ γ

92 M j 1−α This gives as FOC: = C , so P α j ++Π MNw jjj M j Π + + MNw jjj C = α [ ]; −= α [)1( ] j P P P

−θ 1 ⎛ ip )( ⎞ t and Note that = ∫ tttt )()( diicipCP since )( = Cic jj ⎜ ⎟ 0 ⎝ Pt ⎠ 1−θ 1−θ pt = ∫ ( t ))( diip

From the FOC we get as indirect utility: 1 Y j 1 1+γ w 1 1+γ 1 ⎡ ⎤ U j −= N j j −= NN j ⎢∫ ∏+ j )( + Mdii j ⎥ P 1+ γ P 1+ γ P ⎣0 ⎦

3) So it is straightforward to characterise labour supply as an increas- ing function of the real wage w/p.: 1 w γ ⎛ w⎞γ = N j or N j = ⎜ ⎟ P ⎝ P ⎠

1 1 4) Total aggregate income is: = ∫ ttt ∫ t )()()( +∏+ MdiidiiNiwY t 0 0 1 with = ∫ tttt )()( diicipcP 0

Distortion due to monopolistic price setting: The firm’s optimization problem (for the case of linear production): Max profit: i =Π p − wiyi N i)()()()( with iy = A N i)()( p i)( Y Since demand for good i is: d iy = α ()( )−θ t , we get: Pt Pt ⎡ 1−θ −θ ⎤ Πt i ⎛ t ip ⎞ ⎛ t ip ⎞ t iw )()()()( = α ⎢⎜ ⎟ − ⎜ ⎟ ⎥ Yt p ⎢⎜ p ⎟ ⎜ p ⎟ A ⎥ t ⎣⎝ t ⎠ ⎝ t ⎠ t ⎦

93 1 FOC: p i = + μ MC i)()1()( with mark up μ = t t θ −1 v NtN i))(( t iw )( 1 1 For )( = Pip tt : == At 1( ) <−= AA tt mcu ttc );( t ip )( 1+ θμ

So we have derived the price setting curve

We could derive a corresponding wage setting curve if we model workers as monopolistic suppliers of specific inputs in a symmetric way (Blanchard/Kiyotaki).

94 3.2 Price Rigidities and Monopolistic Competition:

Menu Costs: Macroeconomic Externalities

Intuition: Due to monopolistic distortion, aggregate price level is inefficiently high. Thus, aggregate demand out of real money balances is inefficiently low.

Welfare Gains if each firms would lower its price. But no Nash-equilibrium (other firms gain from individual price reduction: aggregate externality!

In the presence of price adjustment costs, surprise increase in money supply may result in higher profits per firm (stay in B). But: multiple equilibria likely

Gd i 1 ⎡∂ Gi ∂ Gi ∂ Pi ⎤ 1 ∂ Gi ∂ Gi Key argument: Envelope theorem: ⎢ += ⎥ = as = 0 PMd ⎣∂ M ∂ Pi ∂ ⎦ PM ∂ M ∂ Pi

⎛ M ⎞ G ⎜ 1 ⎟ 1⎝ p ⎠

⎛ M0 ⎞ G0 ⎜ ⎟ ⎛pi M1⎞ ⎝ p ⎠ G1⎜ , ⎟ ⎝p0 p0 ⎠

⎛pi M1⎞ G1⎜ , ⎟ ⎝p1 p1 ⎠

95 3.3 Monetary Policy as an Optimal Control Problem

The issues: Design the optimal mechanism at the stage of constitution: What should be the rules of the game? [See: Illing 1998]

Principal Agent Problem:

Society (representative agent) cares both about price stability and output fluctuations ↓ Government ↓ Central Bank

A) To whom should monetary policy be delegated? B) Accountability of Monetary Authorities (Public/ Political)

Problems involved: a) Dynamic Incentive Constraints b) Political Incentive Constraints c) Flexibility and Robustness against Uncertainty d) Credibility of Specific Mechanisms

Two competing popular mechanisms: 1) Delegation to a Conservative Central Bank: Bundesbank Model 2) Optimal Central Bank Contract: The New Zealand/UK Model

96 The basic model: Three Stage Game – Basic Set up: Short run supply function:

s e 2 yy +−=− εππ with EVar(εε) ==0 ; ( ) σε ; Δ =−yy*

Aggregate Demand function:

d 2 y +−= ηπμ with E(η) = ;0 Var( ) = ση η ;

Central Bank pursues policy μ η ε );( such as to minimize:

22 Loss Function: Lb=−(*)(ππ +y −y*)

π * inflation target; y * output growth target y *−y =Δ ≥0

Private Agents: try to minimize prediction error E − ππ e )( 2

→ Rational Expectations: Eπ − π e = 0

The stages of the game: Stage 1: Central Bank announces some policy Stage 2: Private agents sign contracts, based on expectations π e After contracts are settled, shocks η;ε occur Stage 3: Central Bank responds with policy μ η ε );(

97 A) Discretionary (subgame perfect) solution:

Solve backwards - First: Optimal policy at stage 3:

Central bank minimizes losses, given expectation π e and realisation ε

Min L 2 b(*)( e εππππ Δ−+−+−= )2

1 b →Reaction function: = *+ [ e επππ Δ+− ] 1+ b 1+ b Stage 2: Private agents rationally anticipate the reaction in stage 3, so they calculate: 1 b E = *+ [πππ e Δ+ ] 1+ b 1+ b

→ Rational Expectations: π e = π * +b ⋅ Δ Summary: Discretionary Solution: b 1 ππe =+⋅*,bbΔΔ ππ −=⋅− * ε ,yy−= ε D DD1+ b 1+ b b L = σ 2 )1( bb Δ++ 2 D 1+ b ε B) Commitment Solution:

Central bank announces a credible policy in stage 1:

Min Loss Function s. t. expectation constraint: Eπ − π e = 0 2 2 e E ππ yyEb −+−+−=Λ E ππλ )]([*)(*)(

2 2 e ∫ E{ ππ yyb −+−+−= } )(][*)(*)( df εεππλ

with s yy * e+−+Δ=− εππ and the density function f(ε).

98 FOC for commitment solution: d ∂ Note: Use Leibniz rule ∫ ),( dzzxf = ∫ ),( dzzxf xd ∂ x a) Choose, after observing state ε , the optimal rate of inflation: ∂Λ ∂π = 0/ : λ ↔ * b( e εππππ )=Δ−+−+− for each ε 2 λ thus: E )(*)( bEb ππππ e =Δ−−+− 2 b) Determine optimally π e before knowing state ε : ∂ π e =∂Λ 0/ λ λ ↔ bE()e εππ =Δ−+− → b Δ−= 2 2

e → Central bank directs expected inflation such that: ππC = * Thus, for each state ε: π + b = π + )1(*)1( − bb ε

Summary: Commitment Solution: b 1 ππe = *, ππ−=−*; εyy−= ε C CC1+ b 1+ b

b 2 L = σ 2 + bΔ C 1+ b ε b L = σ 22++()1 bbΔΔ = L + b22 DC1+ b ε C) Cheating Strategy: b b 1 e ππππ **)( +== []−Δ ε ; yy * =− +Δ ε S 1+ b S 1+ 1+ bb b b 2 b2 2 L = σ 2 + L −=Δ Δ S + b ε 11 + b C 1+ b

99 Implementation of optimal policy: Choose (π, y) optimally.

Assume first Δ =−y* y=0 (no credibility problem). b 1 b Optimal policy: ππ * −=− ε; yy =− ε ; LE )( = σ 2 1+ b 1+ b 1+ b ε

When π, y can be controlled perfectly, all policies are equivalent.

1− b Money growth targeting: y y * ++=−+= −ηεπηπμ 1+ b

Interest rate policy: Set real interest rate such that desired combina- tion (π, y) will be realized (requires specification of link between real interest rate and aggregate demand.

In reality: Control errors; incomplete information (identification); uncertainty about transmission mechanism

Strict Rules as Second Best? a) Strict Inflation Targeting: Fix inflation rate: π=π* π e * ==⇒ ππ

2 π F − π*,= 0 yyF − = ε. Expected loss F = bEL σε b) Strict Monetary Targeting: Fix money growth μ = μ π += y **

Demand: π y μ η =−=+ π *+ y −η Supply: yy *+−+= εππ

1 1 π MM−=π*(2 η − ε ),yy− = 2 (ε + η).

1+ b Expected loss EL = +σσ 22 )( . 4 ηε

100 Stochastic Control Errors (Brainard Uncertainty)

Uncertainty about instruments/ precision of information variable

2 Simplify: = ELMin π )( (note: certainty equivalence)

Impact on target Inflation Rate π by the instrument Z: π zZ += η

2 η random variable with normal distribution: η N ση ),0(~ .

2 A) Uncertainty about transmission mechanism z: zNz σ z ),(~

22 B) Noisy Signal ψ about η : ψ = ξ +η with ψ N ++ σσξη ξη ),(~ .

Minimise expected losses: ZzE +η)( 2

22 + + EEzEZzEZ ψηψη )()()(2 2

EzE ηψ )()( FOC: Z −= zE )( 2

A) Complete Information and perfect control:

First order condition: zz Z +η = 0)(2 .

η Stabilise price level perfectly: Z −= z

101 B) Multiplicative uncertainty about transmission mechanism:

Stochastic fluctuations of parameter z (with normal distribution 2 zNz σ z ),(~ .

zη 22 + zEZzEZ )(2 +ηη 2 FOC: Z −= . zE 2

22 2 22 222 Since )( σ z ( )( ) −=−== zzEzEzEzVar , σ z += zzE z Thus Z −= 22 η . σ z + z

C) Imperfect Observability of Shocks:

Instead of η observe only noisy signal ψ = ξ +η

Note: If both η and ξ are normally distributed independent ran- dom variable, then ψ is normally distributed with

2 2 ψ N(~ η + ,σξ η +σξ ).

2 ση For η = ξ = 0 conditional expectation E ψη )( = 22 ψ . +σσ ξη

2 2 ση 1 FOC: : + EzzZ ψη = 0)(22 , thus: Z −= 22 ψ +σσ ξη z

102 Interest rate vs. monetary targeting with unobservable shocks (Poole):

LM-curve μ − π = y − id + ζ

e e IS-Curve ˆ ibcy t+1)( ibc +−=+−−= ηηπ with ˆ += bcc π t+1 Shocks η and ζ disturbe goods/money market: E η = 0)( E ζ = 0)( . b ⎡ dc d ⎤ Aggregate Demand: y = −+−+ ζηπμ + db ⎣⎢ b b ⎦⎥ In the absence of nominal rigidities: y = y

Assume now that short run supply is perfectly elastic at πe

dc + db 1 πe determined by e μππ −+== y with i ()−= yc b b b

Real shocks:

y − y −= − iib )( +η

μ μ −−=− − iidyy )( + ζ

Interest rate policy: y y =− − − iib )( +η

b d b Money growth targeting: yy =− )( +− − ζημμ + db + db + db

Stabilising interest rate ( = ii ) → Strong fluctuations if η volatile

Stabilising money growth (μ = μ ) → Strong fluctuations if ζ volatile

103 3.4 Designs for Commitment in Monetary Policy (How to cope with dynamic consistency when Δ>0?)

What mechanisms can implement commitment solution? b L = σ 22++()1 bbΔΔ = L + b22 DC1+ b ε Reputation approach: Repeated Games Institutional approach: Design specific rules (different institu- tional arrangements)

3.4.1 Strict Rules a) Strict Inflation Targeting:

22 22 LbF =+σε bΔ . LLD > F if Δ ()1+>b σ ε . b) Strict Monetary Targeting μ = π *.+ y + η

1 1 π M π 2 η −=− ε ,)(* M − yy = 2 ε +η).(

1+ b L = σσ22++bΔ2. M 4 ( εη)

104 3.4.2 Delegation to a Conservative Central Banker (Rogoff)

Appoint agent with weight bk as central banker:

bk 1 ππkk−=*;b Δ − εyyk −= ε.; E()π k = π *+Δbk . 1+ bk 1+ bk 2 ⎛ b ⎞ b Lb()=+ b22ΔΔk σσ2 + m 22+ b kk ⎜ ⎟ εε2 m ⎝1+ bk ⎠ ()1+ bk

Choose optimal type bk (0 < bbk < m ): bb− Δ2 mk=>0. 3 2 bbkk()1+ σ ε Rogoff: Trade off between inflation bias and stabilisation bias

3.4.3 The optimal central bank contract (Walsh) Transfer Payments to the central bank: 22 Lbii=−(*)(ππ +y −−yΔ )(, −Tπε) Optimal transfer payments T(,)π ε = cb− 22Δ π = bΔ (*π −π) 1 b 1 ∂ T Policy: ππ= * + πεe −+Δ +1 11+ b + bb[ ] 2 1+ ∂π ∂ ET Thus, Eb()ππ=++ * Δ 1 2 ∂π ∂ ET No inflation bias if =−2 b Δ . ∂π

Walsh: Trade off between inflation bias and stabilisation bias can be completely eliminated by choosing the appropriate contract

105 3. 5 The New Neoclassical Synthesis Optimising Model with Nominal Rigidities (Dynamic IS AS Model)

Recent research focused on dynamic models with micro-foundations, building a bridge between academics and practitioners (empirically oriented, dynamic structure). This research has had strong influence on monetary policymaking. New issues have been discovered and old results re-emerge in new setting. New Neoclassical Synthesis: simple, widely applied small-scale macro model Woodford Chapter 3/ 7; Clarida/Gali/Gertler (1999), Gali Chapter 4/5

Key ingredients: A) Dynamic IS-Curve: derived from Euler equation ˆ n ttt +1 [( πσ +1) −−−= rEixEx ˆtttt ] ˆˆ n with output gap: −≡ YYx ttt (log deviations from steady state) ˆ n Yt : stochastic, hypothetical flex-price output (natural rate of output in a real business cycle model). B) Forward looking Phillips-Curve

Derived from dynamic price setting

π = κx + βE πtttt +1 ˆˆ n ˆˆ n with output gap: −≡ YYx ttt : ( κπ ( ttt ) +−= EYY πβ t+1 ) κ depends on proportion of firms that prefix prices

C) Monetary Policy Rule:

C1) Interest rate rule (Taylor rule): * * * ii tt π t φππφ tx −+−+= xx )()( Alternatively: C2) Money supply rule: (based on equilibrium on money market – requires knowledge about money demand function) (LM curve)

106 3.5.1) Derivation of the dynamic IS-Curve Solve stochastic intertemporal optimisation problem (rerun)

∞ 1 ⎡ t ⎡ M t ⎤⎤ Max = 00 ⎢ ∑ β ⎢ cuEU t ξt − ∫ ξtt ));(();;( diihv ⎥⎥ ⎣t=0 ⎣ Pt 0 ⎦⎦ subject to per period budget constraint M B W P Y −+=+ T − P Ctttttttt θ ⎡1 θ −1 ⎤θ −1 θ with consumption bundle: = ⎢∫ tt )( diicC ⎥ ⎣⎢0 ⎦⎥ 1 1 ⎡ 1−θ ⎤1−θ Dixit Stiglitz price index: = ⎢∫ tt )( diipP ⎥ ⎣0 ⎦

1 1 Total income: = ∫ tttt ∫∏+ t )()()( diidiihiwYP 0 0

1 = ∫ tttt )()( diicipCP 0

Shocks ξt disturb allocation around steady state: they affect productivity; labour supply (leisure); government spending,… (contains suppy side element, even though considered as demand-side relationship

Note: Model specification designed to separate intratemporal prob- lem and intertemporal problem to allow for straightforward aggregation for demand for composite goods

107 Recall: Intertemporal Portfolio Allocation with Complete contingent claims markets

Value of a portfolio which yields payoff Ds ξt+1)( in state ξt+1: N i t = ∑ qq +1 Dstts ξξξξ t+1)(),()( s=1 i q ξt )( : price of a portfolio of contingent claims; with ξt+1 denoting states of nature, q ξ ξtts +1),( is the price to be paid in state ξt for a claim on one Euro received in t+1 contingent on the state ξt+1 occurring: prob ξ ξtt +1);( - probability of event ξt+1, given state ξt q ξ ξtts +1),( Q +1, ξξ ttss +1),( = stochastic discount factor prob ξξ tt +1);( 1 = QE ttt +1, )( Interest coefficient for a riskless asset 1+ it

Intertemporal Budget Constraint: M − ii tt cp tt + + ++ 1,1, )( −+≤ TypWWQEM tttttttttt or: 1+ it p ctt −+ E Q +1, M + E ++ 1,1, )())(1( ≤ WWQ + p y −Tttttttttttttt

Complete Markets: All households face the same asset prices and have the same subjective probabilities; so, all households face the same discount factors Q tt +1,

Aggregating across all period budget constraints, we get: ∞ ∞ it 0 + ∑ ,00 ttt =∑ ,00 ([)( cpQEypQEW ttt + +TM tt ]) t=0 t=0 1+ it with Q ,0 t as stochastic discount factor at time 0 T with Q ,0 t = ∏t=1Q − ,1 tt Q 0,0 = 1

108 Dynamic Lagrangian approach:

∞ 1 ⎡ t ⎡ M t ⎤⎤ = 00 ⎢ ∑ β ⎢ cuELMax t ξt − ∫ ξtt ));(();;( diihv ⎥⎥ ⎣t=0 ⎣ Pt 0 ⎦⎦ ∞ ∞ M − ii tt λ 0 ++ ∑ ,00 tttt −− ∑ ,00 (()(([ cpQETypQEW ttt + M t ]) t=0 t=0 1+ it

First Order Conditions:

mcu ξ );;( − ii M (1) tttm = tt mcu ξtttc 1);;( + it

mcu ξ );;( β P (2) tttc = t mcu ξ+++ );;( Q tttttc Pt++ 11,111

⎡ mcu ξtttc );;( Pt+1 ⎤ ⎡ β ⎤ or: (2a) Et ⎢ ⎥ = Et ⎢ ⎥ += it )1( β ⎣ mcu ξ+++ 111 tttc );;( Pt ⎦ ⎣Q tt +1, ⎦

ihv ξ ));(( iw )( (3) tth = t mcu ξtttc );;( Pt

Specific Preferences:

a) Additive separability between consumption and real money → M Money demand function ;(/ −= iicLpM ttttt ξt ); 1 b) CES utility for consumption bundle: ))(( = CtCu 1−γ 1−γ t 1 c) CES utility for labour ))(( = hHthu 1+ν 1+ν tt

109 A) For linear production iy = A ttt ih )()( , only the combination of the “supply” shocks (labour supply shocks Ht and productivity shocks At) matter: 1 1 hH 1+ν = AH +− ν )1( iy )( 1+ν 1+ν tt tt 1+ν t − +ν )1( Both shock can be combined to: AH tt

B) Distortion due to monopolistic price setting: The firm’s optimization problem (case of linear production): Max profit: i =Π p − iwiyi A tttttt ih )()()()()( d t ip )( −θ Since demand for good i is: t iy = ()( ) Yt , we get: Pt ⎡ 1−θ −θ ⎤ ()t ip ⎛ t ip ⎞ t iw )()()( t i)( =Π ⎢ − ⎜ ⎟ ⎥ Yt ⎢ p −θ ⎜ p ⎟ A ⎥ ⎣ t ⎝ t ⎠ t ⎦ 1 FOC: p i = + μ MC i)()1()( with mark up μ = t t θ −1 For p i)( = Ptt : vh ih ξtt ));(( t iw )( 1 1 == At 1( ) <−= AA tt uc ( mc ξttt );; t ip )( 1+ θμ

B1) Additional distortion with monopolistic wage setting: Composite labor input bundle: φ φ−1 1 ⎡1 ⎤φ−1 1 φ ⎡ 1−φ ⎤1−φ = ⎢∫ tt )( djjhH ⎥ with wage index: = ⎢∫ tt )( djjww ⎥ ⎣⎢0 ⎦⎥ ⎣0 ⎦ φ ⎛ wt ⎞ Demand for household j’s labor: t jh )( = ⎜ ⎟ Ht , so ⎝ t jw )( ⎠ ihv ξ ));(( φ − iw iw )()(1 tth = t < t mcu ξtttc );;( φ Pt Pt Monopolistic labor supply charges a mark up μh = φ − )1(/1

110 IS-Curve: Intertemporal Equilibrium condition: Euler equation: −1 −1⎛ Cu ξttc ++ 11 );( −1 ⎞ 1 t =+ β ⎜ Ei t ∏t+1 ⎟ ⎝ Cu ξttc );( ⎠ Pt Ct aggregate consumption bundle t+1 =∏ gross inflation rate of Pt−1 the Dixit-Stiglitz price index Pt

1 Log linear approximation with ρ = − β )ln( or β = : 1+ ρ

ln tttt +1 −== σ ()( − EicEcC π ttt +1 − ρ) Optimal intertemporal consumption path. Aggregating across all agents, in general equilibrium + = YGC ttt for all t. Log-linearizing, we get:2 ( −=− ttttt ++ 11 −σ () − EigyEgy πttt +1 − ρ) If government spending is proportional to consumption bundle, the Euler relation also holds for intertemporal production, except for a crowding out term due to fiscal shocks (for ≠ gEg ttt +1 )(

Assume gt+1 λ g += ηtt +1, so E gtt +1)( = λ gt − λ)1( gt = crowding out term due to fiscal shock

Intertemporal IS Curve: ˆ ˆ (YEY ttt +1 ()1() Eig tttt +1 −−−−+= ρπσλ ) Real interest rate has to adjust such that IS curve holds With flexible prices: ˆˆ n ˆ n = YY tt as natural output at t (Yt fluctuates with supply shocks) n (RBC models) and rˆt as natural real rate of interest ˆ With sticky prices, effective production Yt is demand determined (de- viates from natural output due to price stickiness)

C G G G 2 t t t t Use: 1−= and define gt )1(ln ≈−−= Yt Yt Yt Yt 111 ˆˆ n Output gap: −≡ YYx ttt , so ˆ n ttt +1 σ[( −−= EixEx π +1) − rˆtttt ] Solving the IS equation forward, we get: ∞ n xt −= σ ∑k=0 t[(iE π 1) −− rˆ++++ ktktkt ]

Under the expectations hypothesis of the term structure, the long term interest rate is equal to the sum of short term interest r ates: long T rt = ∑k =0 Et[(i − π ktkt +++ 1)] Thus, we get as long term relation:

longT T nlong t σ [ ˆtt ] +−−= xErrx tt +T

Current output gap depends on deviation of long term real interest rate from natural long term real interest rate

112 3.5.2) The Forward looking Phillips-Curve We follow Calvo for modelling the staggered price setting process - a very stylised approach to represent sticky prices. It is designed to en- sure simple aggregation, independent of the history. In each period, each firm faces a state-independent probability α of being stuck with its old price (Poisson process). α is independent of when the firm last changed its price. The Poisson process is extremely artificial, but it is allows for a stationary aggregate structure. 1−α Probability that a firm gets the chance to revise its own price, independent of other firms and of its own history α t Probability that firm’s price cannot be changed until period t Expected duration of price stickiness (expected length of fixed prices): ∞ 1 2 ααααα )1(...)1(3)1(2)1(1 ∑ s ⋅−=+−⋅+−⋅+−⋅ αα s−1 = s=1 1−α The price level evolves according to * (1) τ α pp τ −1 −+= α)1( pτ * → Partial price adjustment τ pp τ −1 α τ −−=− pp τ −1][)1( Intuition: Aggregate prices today depend on a) Prices yesterday (because a share α of prices is sticky) and on * b) The optimal flexible price pτ which is set knowing that the price may stay fixed for a long period. So it also depends on the expected optimal prices in the future (forward looking agents care about prices even in distant future) ∞ t * * 2* * a) pτ −= )1( ∑t=0αα pτ −t )[1( τ τ −1 ααα ppp τ −2 +++−= ... * 2* * Since: pτ −1 )[1( τ −1 τ −2 ααα ppp τ −3 +++−= .... 2* 3* * Thus: α pτ −1 )[1( τ −1 τ −2 αααα ppp τ −3 +++−= ... * So pτ τ +−= αα pp τ −1.)1( b) Forward looking price setting: Those firms lucky to get the chance * to adjust set their new prices pτ as a weighted average of current and

113 expected future nominal marginal cost: mcτ +t (apart from some cost push shock uτ ):

(2) pτ = 1( − β α)* mcτ + β α ( pE τ +1*) + uτ ∞ t pτ = 1( − ∑t=0 αβαβ τ ()()* mcE τ +t ) + uτ

Up to now, we have: * (1) τ α pp τ −1 1( −+= α) pτ * (2) pτ * −= )1( mcτ + αβαβ τ ( pE τ +1) + uτ

Since (1) holds also for τ +1; multiply by β α and subtract from (1): * * pτ − βα τ pE τ +1 ( τ −1 −= βαα pp τ 1() −+ () τ − βαα τ pEp τ +1)

Using 2) this gives (with mcτ = pt + rmcτ ): pτ −α β τ pE τ +1 =

( pτ −1 βαα pτ −+− α − αβ t rmcp τ −++ α)1(][)1)(1() uτ Rearranging gives: τ pp τ −1 =− β [Eτ τ +1 − pp τ + −α 1)(1(] − β α /) α rmcτ + −α /)1( α uτ or πτ = β Eτ πτ +1 −+ α − β α /)1)(1( α rmcτ + 1( −α /) α uτ

If we assume that real marginal cost rise linear with the output gap rmcτ = γ xt , we get as aggregate supply curve

~ π = κ + β Ex ttt π t +1 + ut 1−α with output gap: −≡ YYx ˆˆ n and k = 1( − ) γαβ ttt α k is an (inverse) measure of nominal rigidity: The higher α, the lower k (for α →0, k→∞) ~ ut : captures variation in prices not related to output gap (e.g. fluctuations in firm’s mark up; cost push shocks) Key result: no stickiness in inflation, despite price stickiness (π is fully forward looking) →Disinflation can be achieved without any cost in output!

114 Transmission mechanism for monetary policy:

ˆ Due to price stickiness, effective production Yt is demand determined Calvo price adjustment process gives as forward looking Phillips-Curve π = κx + βE πtttt +1

Current inflation is determined by expected future output gaps (infla- tion as a forward looking phenomenon): ∞ k t = ∑k =0 βκπ xE tt +k

3.5.3) Optimal Monetary Policy for New Neoclassical Synthesis Set interest rates so as to eliminate the output gap – in the presence of price rigidities only, this is equivalent to price stability.

Set E π tt +1 = 0. π t = 0 , when there is no output gap: xt = 0 → sta- bilise mark-up (unless there is a cost push shock creating a trade off)! Neutral interest rate policy: Set it such that the natural interest rate is realized: n For tt += Eri π tt +1, we get: 1 r n ρ =− ([ ˆ n ) −YYE ˆ n ] t σ tt +1 t

By changing the short term interest rate, monetary policy affects the long term real rate – impact via intertemporal substitution effect:

An increase in the real interest rate dampens current aggregate de- mand due to the intertemporal substitution effect. ”Standard” transmission mechanism: Short term interest rate af- fects real interest rate and thus the output gap. Inflation is then af- fected by the output gap! Note: Current values π , xtt depend on their expected future values and thus on expected future monetary policy. Credibility of central bank is crucial - but more complex than in Barro/Gordon!

115 Optimal Monetary Policy

ˆˆ n Min. welfare losses out of inflation π t and output gap −≡ YYx ttt

Second order approximation for losses due to deviations from price ˆ ˆ n stability π = 0* and natural output * = YY tt : (Output target assumed to be equal to the natural rate [which is time varying] – abstracts from monopolistic distortions)

a) Loss function: ∞ τ +t Eτ ∑t=0 []πβ t + xb t )²()²(

b) Aggregate demand curve (IS-Curve): ˆ n ttt +1 [( πσ +1) −−−= rEixEx ˆtttt ]

c) Aggregate supply curve (Forward looking Phillips-Curve): π κx βE π tttt +1 ++= ut 2 Cost push shocks: = ρ uu −1 + ε ttt ≤ ρ ≤ 10; ; E t = ;0)( σε ε

Derive Optimal Policy by minimising (a) s.t. (c) Implementation (Monetary Policy Rule) via aggregate demand (interest rate rule (b) or money supply rule (bb)

In the absence of cost push shocks: Full accommodation of supply shocks ( xt = 0) ensures price stability π t = 0 No trade off between price and output stability: When cost push shocks occur, there is a trade off!

Optimal monetary policy: depends on credibility of central bank Discretionary Solution: central bank optimises, given rational expec- tations of private agents (policy cannot affect expectations) Commitment Solution: central bank optimises, taking its impact on expectations of private agents into account Technique and solution: see Clarida/ Gali/ Gertler, 1999, part 3/4

116 3.5.3.1 Discretionary Solution:

Central bank takes Eτ πτ +t given for all t=1,2,… So at each τ , the central bank’s task is to choose πτ ; xτ optimally in the presence of cost push shocks.

Min Lτ = πτ + xb τ )²()²( πτ κ τ += β τ πτ +1 + uEx τ with Eτ πτ +t ; uτ given and Eτ πτ +t derived from rational expectations about future policy! k FOC: π + xbk = 0 or: x −= π τ τ τ b t Lean against the wind: with inflationary pressure, contract output! Trade-off worsens with increasing nominal rigidity (lower impact on inflation) 1 From FOC, we get π = ( πβ + uE ) τ + k 2 /1 b +1 τττ Rational forward looking agents know that, in all future periods t, the k policy will be x −= π : τ +t b τ +t 1 Thus for all t: E π = ( πβ + uEE ) ττ +t + k 2 /1 b ττ t++ 1 ττ +t At time τ , the best guess for the cost push shock is t τ τ +t = ρ τ uEuE τ +t−1 = ρ uτ Thus, with rational expectations, we get: 2 1 ⎡ βρ ⎛ βρ ⎞ ⎤ π = ⎢u + u + ⎜ ⎟ ( + Eu πβ )⎥ τ + 2 bk τ + 2 bk τ ⎜ + 2 /1/1/1 bk ⎟ τττ +3 ⎣⎢ ⎝ ⎠ ⎦⎥ t 1 ⎛ βρ ⎞ 1 b π = ∞ ⎜ ⎟ u = u = τ 2 ∑t=0 ⎜ 2 ⎟ τ 2 τ 2 + ⎝ + /1/1 bkbk ⎠ /1 bk −+ βρ bk 1( −+ ρ β k xτ −= uτ 2 bk −+ βρ )1( Trade off with Cost Push Shock. Inflation and output gradually moves back The higher b, the higher the inflation variability relative to output variability The more persistent the shock (high ρ), the stronger reacts inflation and output.

117 3.5.3.2 Commitment Solution: see Clarida/ Gali/ Gertler, 1999

∞ τ +t Eτ ∑t=0 []πβ τ +t + xb τ +t )²()²( +λτ + (πτ + −κxτ +ttt − β πτ +t+1)

FOC: π +τ + λ +τ − λttt +τ −1 = 0 xb t+τ )( − k λτ +t = 0 λt−1 = 0 k k so we get: xx tt ττ −++ 1 −=− π t+τ instead of x −= π b τ +t b τ +t k except for x −= π (because λt−1 = 0) τ b τ Optimal response: central bank should adjust the change in output gap to inflation, not the level (as in the discretionary solution)

Lagged dependence in policy rule: Inflation and output gap exhibit history dependence (policy inertia). Key: Central bank can influence future expectations about inflation and thus improve the inflation-output trade off

Intuition: Consider the case of temporary inflation shocks ( ρ =0) Under policy inertia, next-period policy will also be contractive → Next period inflation is dampened → current inflation expectations are dampened → current inflation is dampened → A mild, but prolonged contraction provides better inflation stabili- sation Policy inertia improves predictability of future policy: → current vari- ables can be affected by smaller current policy adjustments

But: Policy should be optimal from a “timeless perspective”: Initial constraint: equivalent to discretionary impulse: λt−1 = 0 (makes use of the fact that the past does not matter) * More sensible instead: − = λλ tt −11 determined by credibility in the past ~ not unique (Jensen/ McCallum)

118