New Keynesian Macroeconomics

Sources: C. Walsh Monetary Theory and Policy Ch. 8 (derivation follows him) J. Gali Monetary Policy, Inflation and Business Cycles Ch. 3 D. Romer Advanced Macroeconomics Ch. 7 (a good general discussion) Sbordone et al (2010) ‘Policy Analysis using DSGE Models’ FRBNY

Economic Policy Review (a good discussion of the basics)

What is a New Keynesian (NK) Model?

- A modern macroeconomic model: derived from microfoundationsa DSGE model

- This is also true of RBC models: what’s new in NK Models?

- Different microfoundations:

RBC: Perfectly competitive markets (businesses price takers)Prices are flexibleShocks: productivity, preferences, governmentMoney and monetary policy: minor extensions; monetary

policy has little effect.

NK: Businesses in monopolistically competitive marketsPrices are not perfectly flexible (like old macro!)Monetary policy is an important part of the model:

monetary policy has important real effects.Shocks: same shocks as RBC plus monetary policy and

markup (price setting) shocks.

- These differences give rise to quite different policy implications.

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- Differences in policy implications: RBC vs NK

RBC economies: Business cycles are economies responding efficiently to shocks

No role for stabilization policy.

NK economies: Economies are not efficient (reflects non-competitive market structure

and price rigidity)

Business cycles reflect responses to shocks but are not efficient.

Policy can improve well-being: offset the effects of price rigidity and imperfect competition.

- NK models attempt to bridge the gap between new macro and old macro.

- Old macroeconomics is still the main approach of forecasters and policy-makers like central banks.

- NK approach develops microfoundations-based models that could be used in policy analysis and possibly forecasting.

- Advantage: with microfoundations immune from Lucas critique.

- Walsh Ch. 8 p.327:‘In the 1970s, 1980s and early 1990s, models used for monetary

policy analysis combined the assumption of nominal rigidity with a simple structure that linked the quantity of money to aggregate spending. … Today the standard approach in monetary economics and monetary policy analysis incorporate nominal wage or price rigidity into a DSGE framework that is based on optimizing behavior

…’

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- NK models also seek to address the short-comings of the RBC approach.

- RBC approach: some empirical successes but failed to capture other features of the economy (see RBC notes)

- RBC models have little role for money and monetary policy.

- RBC world is not really a monetary economy.

- Perfectly competitive markets with flexible prices: - markets clear, relative prices matter. - Money affects the levels of monetary variables (prices;

nominal interest rates; inflation) but has no effects on real variables (real GDP; employment, etc.).

- Example: Money-in-Utility function approach. - Utility depends on consumption, leisure and money balances.- Idea that money is useful: implies demand for money. - Adding this to the RBC model has no effect on real variables

or how the economy responds to shocks.- See for example McCandless Fig. 8.1 and Fig. 9.2 : identical

results in the RBC model with and without money.

- Empirical evidence suggests that money has real effects (Romer discussion of monetary effects and the shortcomings of the RBC model Ch. 5 and the end of last set of course notes)

e.g. Christiano, Eichenbaum and Trabandt (2018): US data

“ expansionary monetary policy … leads to hump-shaped responses in consumption, investment and output, as well

as relatively small rises in real wages and inflation”

(diagram: next page)

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4

Figures 8.1 and 9.2 from McCandless (Fig. 9.2 has money in the model, Fig. 8.1 does not):

Source: G. McCandless (2008) The ABCs of RBCs. Harvard University Press.

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New Keynesian Models: Basic Structure and Comparison to Older Models

- Sbordone at al (2010): a useful diagram and overview of NK model structure (pp.24-26)

- Three blocks of the model (rectangles):

(1) Monetary policy block: captures behavior of the central bank (typically sets ‘i’ some policy interest rate)

- monetary policy is set in response to what is happening on the demand and supply side of the economies (dotted

arrows)

- monetary policy affects aggregate demand via its effect on ‘i’ and on expectations (solid arrows)

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(2) Aggregate Demand block:

- aggregate demand: reflects actions of consumers; businesses, governments.

- influenced by the interest rate set by the central bank (i); and depends on decision-maker expectations about future (Ye, e)

(3) Aggregate Supply block:

- determines inflation (). Depends on current real GDP (Y) and expectations of future inflation.

- assumptions about price stickiness and market structure underlie this relationship.

- Imperfectly competitive models give a ‘markup’ result i.e. prices are set a % markup over marginal costs.

- Sticky prices make price setting forward-looking.

- Expectations affect both aggregate demand (Y) and inflation ().

- Expectations are rational: reflect monetary policy behavior and the state and structure of the economy.

- The economy is subject to shocks to demand; supply (productivity or markup) and policy changes.

- As in RBC models, business cycles reflect adjustment to these shocks.

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- The basic structure is similar to old-style macroeconomic models.

Old Macro:

Demand block: An IS curve story about spending determination (Y=C+I+G+NX)

Monetary policy: Affects aggregate demand through interest rate effects.

Supply block: Some kind of Phillips curve relationship linking the current state of the economy to the inflation

rate.

Shocks: Exogenous changes in spending components. Supply shocks (often due to input price changes)Policy changes.

- Differences from old-style models:

- Microfoundations underlie the demand and supply blocks in NK models.

- Lucas Critique does not apply: can be used for policy analysis.

- We will derive the most basic NK model which goes by the name ‘Canonical NK Model’ in the literature.

- emphasizes what’s new about the approach.

- what’s missing in the Canonical NK Model?

- No capital accumulation or investment.

- Policy versions of the model tend to add this to the model, see for example Heijdra Ch. 19 (a version with capital accumulation).

- Walsh p. 330 notes arguments why this omission may not be critical.

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Canonical New Keynesian (NK) Model: the Three Main Relationships

- These relationships correspond to the three blocks in Sbordone et al’s diagram.

NK IS curve (Demand): y t=−1θ

(it−π t+1)+ y t+1

NK Phillips Curve (Supply): π t=κ (ϕ+θ)[ y t− y tf ]+b π t+1

Monetary policy (Taylor rule): it=δ1 π t+δ 2( y¿¿ t− y tf )+vt ¿

Where (a ^ denotes log-deviations from the s.s.):

y is a measure of real GDP;

is inflation

yf is a measure of the efficient level of output;

i is a measure of the policy interest rate;

Future variables yt+1, t+1 are expected values.

and b are all exogenous parameters

vt is a monetary policy shock (shocks can be added to the other two equations as well).

- We will derive a version of each and discuss their implications.

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New Keynesian IS Curve:

- It links interest rates to aggregate demand: like the IS curve in Old-macro.

- It comes from the intertemporal consumption model’s Euler equation.

- consumption spending is part of a life-time plan.

- how consumption spending changes over time depends on the interest rate.

- in the Canonical NK Model there is no investment spending or government and the economy is closed so consumption spending is the same as

aggregate spending.

- Presentation here is like Romer Ch. 7 or Walsh Ch. 8: Money-in-Utility function; labour supply decision is part of the model.

- Household problem: representative-agent model, like RBC.

Household Utility: U=∑t=0

∞

β t[U (Ct )+F (M t

Pt )−V (Lt)]C = consumption M = money P=price level L = labour supplied

- C and real money supply (M/P) are goods (more is better).

- Labour supply (L) is negative leisure so more L is bad (negative sign).

- Diminishing marginal utility assumed for C and M/P; increasing disutility

of work (∂2V ( L)∂ L2 >0).

- Money contributes to utility: captures the idea that money is useful(can be used as a medium of exchange or a store of value)

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- CRRA style utility is typical in empirical versions of the NK model:

U=∑t=0

∞

β t[ Ct1−θ

1−θ+h

( M t /Pt )1−ω

1−ω− χ

Lt1+ϕ

1+ϕ ] with ,ϕ, h>0

- The household faces a period-specific budget constraint:

Bt+PtCt +Mt = (1+it-1)Bt-1+Mt-1+WtLt

with: B = financial asset (bonds); i=interest rate; W=wage

RHS: financial resources available in period t (bonds and interest earned; money balances from last period and wage income)

LHS: uses for financial resources (bonds and money to carry over to next period; consumption spending)

- So the households problem is to maximize the Lagrangian:

L=∑t=0

∞

βt [ C t1−θ

1−θ+h

( M t

P t )1−ω

1−ω− χ

Lt1+ϕ

1+ϕ ]+∑t=0

∞

λt ((1+it−1) Bt−1+W t Lt+M t−1−B t−Pt C t−M t )

- First order conditions for some typical period (t) Ct, Ct+1,Bt ,Lt and Mt give:

Ct: β t Ct−θ−λ t P t=0

Ct+1: β t+1C t+1−θ −λt+1 Pt+1=0

Bt: −λ t+λ t+1 (1+it )=0Lt: -β t χLt

ϕ+λ t W t=0

Mt: β t h( M t

Pt)−ω

( 1P t )− λt+ λt+1=0

- Labour supply is governed by the following (i.e. combine f.o.c’s for Lt and Ct):χLt

ϕ

Ct−θ =W t / Pt

- Money demand is governed by (i.e. combines f.o.c.’s for Mt, Bt and C):

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h( M t

Pt)−ω

Ct−θ =

it

1+it

- Combining the conditions for Ct, Ct+1 and Bt and using Pt+1/Pt=(1+t+1) gives an Euler equation:

C t−θ=β

(1+it )(1+π t+ 1)

C t+1−θ

- Taking logs and dividing through by : and use the fact that Yt=Ct in this model:

ln Y t=−1θ

lnβ−1θ

ln( 1+it

1+π t+1)+ ln Y t+1

Using ln ( 1+it

1+π t+1)≅ it−π t+1 ≡rt (real interest rate) gives:

ln Y t=−1θ

lnβ−1θ

rt+ lnY t+1

Let lowercase denote logs (so yt=lnYt) then the NK IS can be written as:y t=

−1θ

lnβ−1θ

r t+ y t+1

- It is often written in terms of deviations from the steady state:

- In a steady state, y=−1θ

lnβ−1θ

r+ y (a ‘bar’ denotes steady state value). Subtract this from the NK IS above to to get:

y t=−1θ

rt+ y t+1

or y t=−1θ

( it−π t+1)+ y t+1

where a hat (^) denotes a deviation from the steady state.

(Note: could have used Uhlig’s log-linearization method to obtain this)

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- Consequences of the NK IS curve:

- The NK IS curve relationship relates Y to the real interest rate: much like the IS curve in IS-LM model.

- It is based on an intertemporal consumption story. i.e. substitution: consume less save more when interest rates are high.

- Note that yt depends on future y (yt+1) which in turn depends on yt+2 etc,

- they are all part of a forward-looking intertemporal plan.

i.e. an income or wealth effect style story: if future income and consumption is high so is current income and

consumption.

- The NK IS curve is a linear, first-order difference equation and can be solved forward, i.e. substitute for y t+1=

−1θ

rt+1+ y t+2 then do the same for y t+2 then y t+3 etc.).

Result: y t=−1θ ∑

j=0

∞

r t+ j

- So current y t depends on the future path of interest rates.

- Consequences for monetary policy:

- expectations of future interest rates matter for real GDP now,

- central banks can conduct monetary policy both by changing the current interest rate and by affecting

expectations of future interest rates (‘forward guidance’)

- changes in interest rates that persist will have larger effects on current real GDP.

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- Extended versions of NK IS have Investment and Net Export spending and givesadditional reasons for a negative relationship between r and Y.

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NK Phillips Curve:

- NK IS curve is familiar from intertemporal consumption models.

- Phillips curve in Old-Macro: concerned with inflation determination, same here.

- It is a Supply-side relationship: reflects price-setting behavior by firms, relation between production costs and state-of-the-economy.

- NK Phillips curve is a new story (not in RBC models)

- Hinges on:

(1) Imperfectly competitive firms: price setters (monopolistic competition)

(2) Sticky prices: each period some firms can’t change price (Calvo Pricing). - Sticky prices are familiar from older Keynesian macro models.

- Derivation of the NK Phillips Curve: two common set-ups (same result)

(1) Treat Consumption as an aggregate of consumption of ‘i’ goods produced by monopolistically competitive firms;

- See for example Romer’s chapter or Walsh’s presentation.

(2) One Final Good which is produced using ‘i’ intermediate goods as inputs where the intermediate goods are produced by monopolistically

competitive firms.

- McCandless follows this route.

- These two setups both give a NK Phillips Curve. We use the first.

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- Derivation is based on Romer Section 8.5 and Walsh Chapter 8.

- Outline of steps;

(1) Derive consumer demand functions for each consumer good (Ci);

(2) Solve the price-setting problem of a profit-maximizing monopolistically competitive firm under the assumption of Calvo pricing (sticky

prices) where demand for the firm’s product comes from step (1);

(3) Use the price setting solution from (2), the expression for the aggregate price level Appendix 1) and relation between costs

and state of the economy to characterize how inflation is determined (this is the NK Phillips Curve).

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(1) Consumer Demand Functions:

- Ct is the representative person’s consumption in period t.

- Ct is a composite consumption good ‘summed’ over a continuum of goods:

C t=[∫i=0

1

C¿(η−1)/η di]

η/(η−1)

with η>1

i.e. an aggregate of individual consumer goods Ci (CES Aggregator)

- Person chooses the mix of Cit to minimize the cost of a level of Ct:

min ∫i=0

1

P¿C ¿di s . t .Ct=[∫i=0

1

C¿(η−1 )/ηdi ]

η/(η−1)

- Let ‘’ be the Lagrange multiplier (marginal cost or price of a unit of Ct) then:

f.o.c. for Cit: P¿−μt [∫i=0

1

C ¿

η−1η di ]

1η−1

C¿(−1

η )=0

This can be written in terms of aggregate Ct as:

P¿−μt C t

1η C¿

−1η =0

Solving for Cit gives the conditional demand function:

C ¿=¿¿

- Conditional demand function can be written in terms of the price level, Pt:C ¿=¿¿where t=Pt

- Why? measures the cost of an extra unit of aggregate consumption (Ct), i.e. Pt.

(Appendix 1 Shows how relates to individual prices)

- More demand for Ci if: Pit/Pt is low (substitution) and if overall demand (Ct) is high.

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(2) Price Setting Problem of a Monopolistically Competitive Firm:

- Each good ‘Ci’ is produced by a monopolistically competitive firm.

- Each firm has monopoly power over its own specific good.

- Each variant of the consumer good is a substitute for others (competition)(implied by the expression for Cit (last page))

- Firm’s problem: choose the price of its variant of the good (Pi) given that it knows that the consumer demand curve for its good is:

C ¿=( P¿

Pt )−η

C t

- Follow Walsh and Romer and assume a linear production function:

Yit=ZtLit

Yit = output of good i in period t. Lit = labour used to produce i in period t.Zt = productivity term (time but not firm specific; changes to Zt are

economy-wide)

- With linear production the real marginal cost of output is:

mt=(Wt/Pt)/Zt (takes 1/Z labour to produce 1 unit output)

- this doesn’t vary with output of ‘i’ and is the same for all firms (all firms hire workers at the real wage W/P and the

productivity parameter is common to all firms)

- Real Profits in period t are then: (Pit/Pt) Yit – mt Yit

with: Yit = Cit ¿( P¿

Pt )−η

Ct each firm produces enough

to satisfy demand for Cit So profits per period are:

( P¿

Pt ) ( P¿

Pt )−η

C t-mt( P¿

P t )−η

Ct

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- Flexible price solution:

- Say the firm can choose a different price each period if it wishes.

- The firm then chooses Pit each period to maximize per period profits:

( P¿

Pt ) ( P¿

Pt )−η

C t-mt( P¿

P t )−η

Ct

f.o.c. for Pit:

(1−η )( P¿

Pt)−η

( 1P ¿¿ t)C t−mt (−η )( P¿

P t)−η−1

( 1P ¿¿ t)C

t¿¿= 0

- This gives a mark-up condition:

( P¿

Pt )= ηη−1

mt

- The price of good ‘i’ is a constant markup over marginal cost.

- New Keynesian models assume sticky prices and/or sticky wages.

- Rooted in the idea that changing prices is costly so firms won’t change them constantly.

e.g. costs of determining a new price; menu costs of changing prices.

- Empirical studies: suggest that prices are somewhat sticky

- Romer reviews some US studies (Section 7.6 pp. 335-338).

- Frequency of price changes: Intermediate goods: once a year; Final goods: every four months.

- These are averages: lots of heterogeneity among firms.

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- How to introduce price stickiness?

- Contract models: prices are set for the duration of a fixed-length contract.

e.g. Taylor – the price stipulated at the start of the contract is in effect until the contract expires.

Fischer – the contract stipulates prices in each year of the contract.

Calvo pricing: firm sets a price now knowing that it can change that price in a given future period

with probability (1-).

- Calvo pricing is the most common assumption in NK models:

-gives stickiness;

- computationally the easiest to deal with (aggregates out price heterogeneity)

- easy to experiment with the degree of stickiness (can try different ’s)

- relatively simple inflation dynamics (NK Phillips curve)

- But arbitrary, e.g. why wouldn’t probability of changing a price rise over time?

- We look at stickiness in output prices using the Calvo pricing assumption.

- Comment: from a microfoundations point of view this assumption seems arbitrary.

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- Sticky Price Solution with Calvo pricing and monopolistic competition:

- Firm knows that the price it sets now could be in place for some time.

- Profit maximization becomes an intertemporal problem (unlike the 1-period problem when prices were flexible)

- Say the firm sets a price in time t and that the probability it can change its price in any specific future period is 1-.

- Probability that the price can’t be changed between two periods is .

- Probability that a price set in t is still in place in period t+k is k.

- So a firm chooses its price in period t (Pit) to maximize the present value of expected profits resulting from that choice:

∑k=0

∞

αk bk [( P¿

P t+ k)( P¿

Pt+k)−η

−mt+ k( P¿

Pt+k)−η]Ct+k

k is the probability that the price chosen in t is unchanged in period t+k and bk is a discount factor (0<b<1 i.e. to calculate present value).

Term in square brackets times Ct+k is the profit made in period t+k if the price set in period t is still in effect.

f.o.c. for Pit:

∑k=0

∞

αk bk [ (1−η )( P¿

Pt+k)−η 1

P t+ k+η mt+k ( P¿

Pt+k)−η−1 1

Pt+k ]Ct+k=0

Manipulating the f.o.c. gives:

(1−η ) P¿−η∑

k=0

∞

αk bk C t+ k( 1P t+ k )

1−η

+η P¿−η−1∑

k=0

∞

αk bk mt+k C t+k ( 1Pt+k )

−η

=0

Then solving for Pit:

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P¿=η

η−1

∑k=0

∞

α k bk mt+ k C t+k ( 1Pt +k )

−η

∑k =0

∞

α k bk C t+k ( 1P t+ k )

1−η

To express this in real terms divide by Pt and invert the 1/Pt+k terms to give:

P¿

Pt= η

η−1

∑k=0

∞

αk bk mt+k C t+k ( Pt+k

Pt)

η

∑k=0

∞

α k bk Ct+k ( Pt+k

Pt )η−1

- if =0 this collapses to flexible price result (to see this separate out period k=0 from the sum terms so that there is a separate

term for the period when the price is first set then note 0=1)

- What does this suggest more generally?

- Price setting is forward looking: - depends on expected future marginal costs- future demand level.

- Uhlig’s method from the RBC notes can be used to derive a log-linearized version of the Sticky price solution (see Appendix 2):

P¿−Pt=(1−αb )∑k=0

∞

α k bk ¿¿

(a ‘hat’ denotes a log-deviation of the variable from its steady state value)

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(3) Inflation Determination with Sticky Prices

- Last step is to turn the result for price setting into a story about inflation.

- Repeating the result from last page:

P¿−Pt=(1−αb)∑k=0

∞

α k bk ¿¿

Break Pt out of the sum (note: (Pt/(1-ab)) x (1-ab) = Pt)

P¿−Pt=(1−αb )∑k=0

∞

α k bk (mt+k+ Pt+k )−P t

Then separate out the k=0 terms from the sum:

P¿−Pt=(1−αb ) (mt+ Pt )+ (1−αb )∑k=1

∞

α k bk (mt+k+ Pt+k )−Pt

The second RHS term can be written as: αb ( P¿+1−Pt+1 )+ Pt+1 (this uses the expression for P¿− Pt on previous line). Then we have:

P¿−Pt=(1−αb ) mt+αb ( ( P¿+1−P t+1)+ ( Pt+1−Pt ))

- The price level is an average of the Pit: Pt=[∫i=0

1

P¿1−η di ]

1/(1−η)

- With Calvo pricing (1-) firms set their price at Pit in period t while the remaining firms still have prices equal to their t-1 values.

Using this and the expression for Pt gives:

Pt1−η=(1−α ) P¿

1−η+α Pi , t−11−η

Since the firms who have sticky prices are a random sample of all firms Pi,t-1=Pt-1

Pt1−η=(1−α ) P¿

1−η+α Pt−11−η

- Dividing through byPt1−η gives a version in terms of the inflation rate:

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1 ¿ (1−α ) ( P¿ /Pt )1−η+α ( Pt−1/Pt )1−η

- Do a log-approximation around the steady state (use Uhlig to replace P terms) and use Pit=Pt=Pi,t-1 in the s.s to get (Walsh p. 379):

α ( Pt−P t−1 )=¿ (1−α ) ( P¿−Pt )

Or: ( P¿−Pt )=( α1−α

)( Pt−Pt−1)

- Use this last result to replace the( P¿−Pt ) terms in the optimal price setting result:

P¿−Pt=(1−αb ) mt+αb ( ( P¿+1−P t+1)+ ( Pt+1−Pt ))

And use Pt−Pt−1=π t ( is inflation)

α1−α

π t=(1−αb ) mt+αb ( α1−α

π t+1+π t+ 1)

π t=[ (1−α ) (1−αb )α ]mt+b π t+1=κ mt+b π t+1

Where : κ ≡[ (1−α ) (1−αb )α ]> 0

- This is the NK Phillips curve.

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- Looking at the result:π t=κ mt+b π t+1

- Inflation rate is high: - if current marginal costs are high;

- if expected future inflation is high.

- Effect of current marginal cost is larger the smaller is (less sticky are prices), i.e. small means larger .

- In this form inflation does not depend on output or the state of the economy

- Old macro Phillips curves: inflation depends on output.

- The NK Phillips curve can also be written in these terms.

- this works through the effect of real GDP on marginal cost.

- How might this work?

- If the production function had diminishing returns then ‘m’ will rise with firm-level output and y.

- We assumed linear production, so: m=(W/P)/Z

- this doesn’t vary with the firm’s own output level

- it will rise with aggregate output via rising wages.

i.e. more output requires more labour; workers economy-wide supply more labour if wages are

higher.

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- Formally, with linear production (as in Walsh):

Production: Yt=ZtNt gives y t= z t+nt

Marginal cost: mt =(Wt/Pt)/Zt gives mt=wt− pt− zt

Labour supply:χN t

ϕ

Y t−θ =W t / Pt gives w t− p t=θ y t +ϕ nt

Labour supply condition is from the Household problem (p.10).

- have set L (labour supply) equal to N (labour demand) and output (Y) equal to consumption (C).

Substitute labour supply into ‘m’ definition (replace w t− p t) then use the linearized production function to eliminate nt= y t− z t.

This gives:mt=¿

So marginal cost depends positively on output and negatively on productivity.

Replacing m in the NK Phillips curve gives:

π t=κ (ϕ+θ)[ y t−(ϕ+1)(ϕ+θ)

z t]+b π t+1

Or in terms of an output gap measure:

π t=κ (ϕ+θ)[ y t− y tf ]+b π t+1

Where: y tf =

(ϕ+1)(ϕ+θ)

zt is the flexible-price output

level.

- Flexible-price or Efficient output: yf

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- When price are flexible (p.17): ( P¿

Pt )= ηη−1

mt

- Each firm is basically the same so Pit/Pt=1 so 1¿ ηη−1

mt

- Rewrite: m=(W/P)/Z as η−1η =(W/P)/Z

- Then we can write W/P=Zη−1η

- Use this to replace W/P in the labour supply condition: χN t

ϕ

Y t−θ =W t / Pt

and use Y=ZN to eliminate N from the labour supply condition then solve for Y (this gives flexible price Y or Yf):

Y tf =[ 1

χη−1

η ]1

θ+ϕ Z t

1+ϕθ+ϕ

- Log-linearizing the results gives:

y tf =

(ϕ+1)(ϕ+θ)

z t (see Walsh p.335,337).

- So: π t=κ mt+b π t+1

π t=κ (ϕ+θ)[ y t− y tf ]+b π t+1

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- Comments on the NK Phillips curve:

- Based on microfoundations for a specific economic environment: monopolistic competition; constant elasticity demand curves;

Calvo pricing. (Walsh p. 337)

-Real marginal cost (m) drives inflation.

We can solve the NK Phillips curve forward (i.e. use π t=κ mt+bπ t+1 to infer π t+1=κ mt+1+b π t+2 and substitute this into the NK Phillips curve

then repeat for other future inflation rates).

This shows that current inflation is a function of current and future m:

π t=κ mt+b κ mt+1+b2 κ mt+2+¿ … = κ∑k=0

∞

bk mt+k

Future costs matter given that a price chosen now may be sticky.

- Future inflation is more important to current inflation the more the future is valued (high b).

- Output gap version: stabilization – a central bank can stabilize inflation by stabilizing the output gap

i.e. setting output at its flexible price value gives mt=0 each period.

- Differs from Old-macro’s ‘Expectations Augmented Phillips Curve’ in that it is not backward-looking. No inertia to in NK Phillips curve.

- No shock term (vs. older Phillips curves). Common extension is to allow for a ‘cost-push’ shock, e.g. mark-up shock or firm-

specific MC

30

shocks. A productivity shock works through y tf .

31

Monetary Policy Rule:

- Final piece of the NK model is concerned with interest rate (i) determination.

- Central bank is typically treated as an interest rate setter.

- Why? Many central banks often conduct policy this way.

e.g. Bank of Canada: targets the ‘overnight rate’; US Federal Reserve targets the Federal Funds Rate.

- Theories of interest rate structure suggest that altering the policy rate the central bank can influence the entire structure of interest rates.

e.g. higher policy rate, raises other short-term rates (substitutes) and term structure models suggest it pushes up longer-term rates

too.

- Stabilization goals underlie the Central Bank’s interest rate choice:

- Stabilize what?

1) Inflation: try to keep inflation within some target range.

- Why target inflation?

- Avoid the costs of inflation (resource costs in forecasting and changing prices; uncertainty; distributional

effects);

- Many models suggest that in the long-run monetary policy can only affect inflation and other nominal variables.

2) Output and Employment: stay near full employment

e.g. US Federal Reserve: targets both inflation and output and employment (‘dual mandate’)

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- Why? Avoid the costs of lost output and incomes; economic and personal costs of unemployment etc.

- How does stabilization work?

- The interest rate affects current output via the NK IS curve, i.e. by changing aggregate demand.

- Output affects inflation through the NK Phillips curve.

- Monetary policy can be represented via a rate setting rule: Taylor Rule

- Interest rate depends on inflation rate and possibly the output gap:

it=δ1 π t+δ 2( y¿¿ t− y tf )+v t ¿ with 1>0, 2≥0

- where itis the deviation from the steady state i.

- If 2=0 then the central bank only targets inflation.

- Error term (vt): what does it represent? Shocks to i.

(changes in relationship between policy rate and structure of interest rates; politics; goal changes)

- Central bank raises interest rates if inflation rises or the output gap rises.

- rise in i will reduce demand (NK IS), lower the output gap and cause inflation to fall (via NK Phillips Curve).

- Romer’s version of the rule is forward looking this version focuses on current inflation and the output gap when setting i.

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- Monetary policy in older models often focuses on money supply

- Central bank chooses size or growth rate the of the money supply.

- Money demand in the NK model is implied by the f.o.c.’s of the household problem.

- Interest rate is at the level where money demand = money supply (see Walsh eqn. 8.30)

- Money supply in our model accommodates the target interest rate.

- Taylor rule suggests the value of i; the money supply is set at the level consistent with the target interest rate.

- Monetary policy here is a rule based on aggregates.

- Justified by appeal to actual policy practice.

- Isn’t this like an old-style macro relationship?

- Optimal monetary policy literature: has the central bank set monetary policy in a way which maximizes household

well-being.

- Woodford shows that in some models this objective can be approximated by a function that depends on inflation and the output gap. (see Walsh Section 8.4.1)

- So some grounding in optimization.

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Appendix 1: t (Nominal Marginal Cost of Ct ) and Prices (see p. 12 of notes)

Substitute C ¿=¿¿ into the CES aggregator:

C t=[∫i=0

1

C¿

η−1η di ]

ηη−1

=[∫i=0

1

[( P¿

μt )−η

C t]η−1

η di ]η

η−1

This gives:C t=μ t

η¿¿Solve for :

μt=[∫i=0

1

P¿1−η di ]

1 /(1−η)

≡ Pt aggregate price level

So ‘μ’ is an average of the prices of the individual goods (Ci): a measure of the price level (makes intuitive sense: interpret ‘μ’ as a Lagrange multiplier).

Appendix 2: Log-linear Approximation of the Sticky Price Solution Around the Steady State

- The solution to the Sticky-Price Firm’s problem on p. 17 was:

P ¿

Pt= η

1−η

∑k=0

∞

αk bk mt+k C t+k ( Pt+k

Pt)

η

∑k=0

∞

α k bk Ct+k ( Pt+k

Pt )η−1

- Multiply both sides by the denominator then the LHS of the new expression is: P¿

Pt∑k=0

∞

αk bk C t +k(P t+ k

Pt)

η−1

Use Uhlig’s method to log-linearize this expression:

P ¿

Pt∑k=0

∞

αk bk C t+k( P t+ k

Pt)

η−1

≅C∑k=0

∞

α k bk ¿¿

¿C ¿

- In Uhlig’s notation a ‘hat’ denotes log-deviation from the steady state and a ‘bar’ is a steady state value. Each of the Pit, Pt, Pt+k, Ct+k terms are replaced by expressions

of the form Pt=P ePt with Pt=ln (Pt

P) and in the steady state

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Pt=Pt+k=Pit) The second step takes the terms that do not depend on k out of the summation and then uses formula for a geometric progression)

- The RHS of the expression can also be approximated as:η

1−η∑k=0

∞

α k bk mt+k Ct+ k( Pt+k

Pt)

η

≅ η1−η

C m ¿

So the full approx. (LHS=RHS):C ¿

This can be simplified (cancel C-bars, use η

1−ηm=1 (holds in steady state – everyone has

adjusted prices so are at the flex price outcome), then the 1/(1-ab) terms and the C t+ kterms wash out and the ( P¿¿ t+k−P t)¿terms can be combined) to give:

P¿−Pt=(1−ab)∑k=0

∞

α k bk¿¿

Break Pt out of the sum (note: (Pt/(1-ab)) x (1-ab) = Pt)

P¿−Pt=(1−ab )∑k=0

∞

α k bk ( mt+ k+ Pt+ k)−Pt

Then separate out the k=0 terms from the sum:

P¿−Pt=(1−ab ) ( mt+ Pt )+ (1−ab )∑k =1

∞

αk bk ( mt+k+ Pt+k )−Pt

Notice that the second RHS term can be written as: ab ( P¿+ 1−Pt+1 )+ Pt+1 (uses expression for P¿−Pton previous line). Then we have:

P¿−Pt=(1−ab )mt+ab (( P¿+1−Pt+1 )+( Pt+1−Pt ))

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