The Simple New Keynesian Model

Graduate Macro II, Spring 2010The University of Notre Dame

Professor Sims

1 Introduction

This document lays out the standard New Keynesian model based on Calvo (1983) staggeredprice-setting. The basic model is usually cast in a setting without physical capital, whichmeans that there is no way in equilibrium to transfer resources across time (i.e. in equilibriumaggregate consumption is equal to output). Some argue that this isn�t a problem, but Ithink it makes the model behave very di¤erently. Aside from lacking physical capital, themodel also di¤ers from our benchmark in that it assumes imperfect competition (in particularmonopolistic competition) on the �rm side of the model. To think about price-stickinessyou have to think about price-setting, and to think about price-setting you need some degreeof pricing power. The household side of the model is basically identical to what we�ve seenbefore.I�ll begin with a model of imperfect competition with no price stickiness. Then we�ll

move to a model with price-stickiness.

2 The Model with No Price Stickiness

2.1 Households

The household side of the model is very standard and is similar to setups we have alreadyseen. We assume that money enters the utility function in order to get households to holdmoney. The household�s problem can be written as follows:

maxct;nt;bt;mt

E0

1Xt=0

�t

c1��t � 11� �

+ �(1� nt)

1�� � 11� �

+ m1�vt � 11� v

!

s.t.

ct + bt +mt � wtnt +�t + (1 + it�1)bt�11 + �t

+mt�1

1 + �t

The Lagrangian for the problem can be written:

L = E0

1Xt=0

�t

8<:c1��t �11�� + � (1�nt)

1���11�� +

m1�vt �11�v + :::

:::+ �t

�wtnt +�t + (1 + it�1)

bt�11+�t

+ mt�11+�t

� ct � bt �mt

� 9=;1

The �rst order conditions can be written:

@L@ct

= 0, c��t = �t

@L@nt

= 0, �(1� nt)�� = �twt

@L@bt

= 0, �t = ��t+11 + it1 + �t+1

@L@mt

= 0, �t m�vt � �t�t + �t+1�t+1

1

1 + �t+1= 0

) m�vt = �t � ��t+1

1

1 + �t+1

We can simplify these using the Fisher relationship and simplifying:

(1� nt)�� = c��t wt

c��t = �c��t+1(1 + rt)

m�vt = c��t

�it

1 + it

�2.2 Production

Production in these models is split into two stages �intermediate and �nal goods. The �nalgoods production �technology�is simply a constant elasticity (CES) bundler of intermediategoods �there are no factors (i.e. labor) used to produce �nal goods. Pro�t maximizationin the �nal goods sector (which is competitive) yields a downard sloping demand curve forintermediate goods producers, which gives them some pricing power. It is in the intermediategoods sector that we will assume some nominal rigidity (i.e. price-stickiness), which is inturn capable of generating meaningful non-neutralities.What di¤erentiates monopolistic competition from perfect competition is that a large

number �rms sell di¤erentiated products and have some pricing power. Because of entryand exit, they earn no economic pro�ts in the long run, however.

2.2.1 Final Goods

There is one �nal goods �rm and a continuum (i.e. in�nity) of intermediate goods �rms.These �rms are indexed along the unit interval. The �production function� for the �nalgood is:

yt =

24 1Z0

yt(j)"�1" dj

35"

"�1

We require that " > 1; this is the elasticity of substitution among the di¤erent intermediategoods. As long as " <1, the intermediate goods are imperfect substitutes in consumption;

2

this is what gives them market power. Note that an integral is really just a sum �we�rejust taking a weighted sum of intermediate goods and then raising them all to a power. Itis straightforward to verify that this �production function�has constant returns to scale �if you double all intermediate inputs, you double output.The �nal goods �rm wants to maximize pro�ts (more generally they�d want to maxi-

mize the present discounted value of pro�ts, but there is nothing that makes the probleminteresting in a dynamic sense as they just buy the intermediate goods period by period,so maximizing value is equivalent to maximizing pro�ts period by period). The objectivefunction is written in nominal terms is:

maxyt(j)

ptyt �1Z0

pt(j)yt(j)dj

Total revenue is the �nal goods price times the amount of �nal good. Total cost is the sumover all intermediate goods of the price times quantity. Plug in the production function:

maxyt(j)

pt

24 1Z0

yt(j)"�1" dj

35"

"�1

�1Z0

pt(j)yt(j)dj

The �rst order conditions come from taking the derivative with respect to each yt(j) andsetting it equal to zero. Remember to treat the integral just like a sum in taking thederivatives.

pt"

"� 1

24 1Z0

yt(j)"�1" dj

35"

"�1�1"� 1"

yt(j)"�1"�1 = pt(j) 8 j

Now play around with this and simplify and solve for the demand for each intermediategood:

3

pt

24 1Z0

yt(j)"�1" dj

35"

"�1�"�1"�1

yt(j)"�1"� "" = pt(j) 8 j

pt

24 1Z0

yt(j)"�1" dj

351

"�1

yt(j)� 1" = pt(j) 8 j

yt(j)� 1" =

�pt(j)

pt

�24 1Z0

yt(j)"�1" dj

35�1

"�1

yt(j) =

�pt(j)

pt

��" 24 1Z0

yt(j)"�1" dj

35"

"�1

yt(j) =

�pt(j)

pt

��"yt

The last line follows from the de�nition of the �nal goods CES aggregator. This says thatthe demand for each intermediate good depends negatively on its relative price and positivelyon total production. We can interpret " as the elasticity of demand �the requirement that" > 1 is just say that monopolists produce on the elastic portion of the demand curve. As" ! 1 demand becomes perfectly elastic (equivalently, the intermediate goods are perfectsubstitutes), which will end up putting us back in the case of perfect competition.Since the �nal good �rm is competitive, pro�ts are zero, which implies that:

ptyt =

1Z0

pt(j)yt(j)dj

Plug in the demand functions for intermediate goods and solve for the price of the �nal good:

ptyt =

1Z0

pt(j)

�pt(j)

pt

��"ytdj

We can �take out of the integral�(i.e. sum) the variables not indexed by j on the righthand side, leaving:

4

ptyt = p"tyt

1Z0

pt(j)1�"dj

p1�"t =

1Z0

pt(j)1�"dj

pt =

24 1Z0

pt(j)1�"dj

351

1�"

This can be thought of as the aggregate price index.

2.2.2 Intermediate Goods

Intermediate goods (remember, there an in�nite number of them populated along the unitinterval) produce output using a production function using labor and TFP. The level ofTFP is common to all of them. Assume that this production function takes the form:

yt(j) = atnt(j)

Hence, production is linear in labor given TFP.The typical intermediate goods �rm optimizes along two dimensions �it must choose its

employment and its price. We consider these problems sequentially.Intermediate goods �rms are price takers in factor markets (i.e. they take the wage as

given). The market structure requires them to produce as much output as is demanded at agiven price (they will be willing to do this since price, as we will show, will be above marginalcost). Nothing makes the value of the �rm explicitly time dependent (i.e. �rm�s don�t havefactor attachment), so maximizing value is equivalent to maximizing pro�ts period by period,which is in turn equivalent to minimizing costs period by period. It is easiest to think aboutthe choice over labor as a cost minimization problem as follows:

minnt(j)

Wtnt(j)

s.t.

yt(j) ��pt(j)

pt

��"yt

yt(j) = atnt(j)

Here, Wt is the nominal wage, which is common to all �rms since they are competitive infactor markets. Pro�ts are maximized when costs are minimized subject to two constraints�production is at least as much as demand and production is governed by the production

5

�technology�given above. Minimzing a function is the same as maximizing the negative ofthe same function, so we can write the problem out as a standard Lagrangian:

L = �Wtnt(j) + 't

atnt(j)�

�pt(j)

pt

��"yt

!The �rst order condition is:

Wt = 'tat

The Lagrange multiplier, 't, has the interpretation of nominal marginal cost �how muchnominal costs change (the objective function) if the constraint is relaxed (i.e. if the �rmhas to produce one more unit of its good). note that marginal cost is not indexed by j �constant returns to scale plus competitive factor markets insure that marginal cost is thesame for all �rms. Divide both sides of this expression by the aggregate price level (thisputs this in terms of the consumption wage, which is what households care about). Thisleaves a relationship between the real wage, real marginal cost, and the marginal product oflabor.

wt ='tptat

Here wt � Wt

pt; i.e. the real wage. If markets were perfectly competitive, price would always

be equal to marginal cost, and so real marginal cost would always be one, and the labordemand condition would be the familiar wage equals marginal product. More generally, realmarginal cost will equal the real wage divided by the marginal product of labor.Now consider the choice of the optimal price conditional on the optimal choice of labor.

Again, since the �rm can choose its price each and every period, we can write this as a staticproblem.

maxpt(j)

pt(j)yt(j)�Wtnt

s.t.

yt(j) =

�pt(j)

pt

��"yt

Wt = 'tat

In other words, the optimization is done subject to the demand function and the require-ment that labor is chosen optimally. Techinically, we should be maximizing real pro�ts,which would entail dividng by the aggregate price level, but given the static nature of theproblem, doing so would not a¤ect the optimal decision rule. We can plug these constraintsin to write the problem as:

maxpt(j)

pt(j)

�pt(j)

pt

��"yt � 'tatnt = pt(j)

�pt(j)

pt

��"yt � 't

�pt(j)

pt

��"yt

6

Since the �rm is small (i.e. there are an in�nite number of them), it takes aggregateoutput, yt, and the aggregate price level as given. Take the FOC:

(1� ")pt(j)�"p�"t yt + "'tpt(j)

�"�1p�"t yt = 0

Simplifying:

("� 1)pt(j)�" = "'tpt(j)�"�1

pt(j) ="

"� 1't

Since " > 1, ""�1 > 1. This means that the optimal price is a markup over marginal cost

(i.e. price exceeds marginal cost). The extent of the markup depends on how �steep�the�rm�s demand curve is. As "!1, the �rm faces a horizontal demand curve, "

"�1 ! 1, andprice is equal to marginal cost, and we�re back in the perfectly competitive case.

2.3 Aggregation

We will restrict attention to a situation in which all �rms behave identically (i.e. a �symmet-ric equilibrium�). This is not without loss of generality. Since �rms operate in competitivefactor markets, they all have the same marginal cost of production, 't. Since they all facethe same demand elasticity, from above, we see that they will all choose the same price. Butif they all choose the same price, they face the exact same demand. This in turn meansthat they will each produce an equal amount and will hire an equal amount of labor (sincethey all face the same aggregate TFP). Starting with the aggregate production function, wehave:

yt =

24 1Z0

yt(j)"�1" dj

35"

"�1

Let yt(j) be the amount of output produced by the typical intermediate goods �rm.Since it�s the same across all j, we have can take it out of the integral and get:

yt = yt(j)

24 1Z0

dj

35"

"�1

= yt(j)

In other words, output of the �nal good is equal to output of the intermediate goods(or, more correctly, the production of the �nal good is equal to the sum of production ofintermediate goods in the symetric equilibrium . . . since we are summing across the unitinterval, the sum is equal to the amount produced by any one �rm on the unit interval).Taking note of this fact, and using the intermediate goods production function, we have

yt = yt(j) = atnt(j)

7

Note also that, since we�re integrating over the unit interval and every �rm produces the

same amount, yt(j) =

1Z0

yt(j)dj. Hence we can apply an integral above and get:

yt =

1Z0

atnt(j)dj = at

1Z0

nt(j)dj = atnt

This follows from the fact that employment supplied by the houshold is split amount the

�rms along the unit interval (i.e. nt =

1Z0

nt(j)dj).

Since all intermediate goods �rms are behaving the same, we get the same result thatthe aggregate price level is equal to the price level of the intermediate goods �rm:

pt = pt(j)

From above, we know what each �rm�s price will be:

pt(j) ="

"� 1't

The labor demand condition for each intermediate goods �rm is:

Wt = 'tat

Divide both sides by the price level:

wt ='tptat

Now use the pricing condition:

wt ="� 1"

at

"�1"< 1, so the real wage is less than the marginal product.

We can summarize the entire model with the following equations:

c��t = �Etc��t+1(1 + rt) (1)

ct = yt (2)

yt = atnt (3)

�(1� nt)�� = c��t wt (4)

wt ="� 1"

at (5)

8

m�vt = c��t

�it

1 + it

�(6)

1 + rt =1 + it1 + �t+1

(7)

dmt + �t = (1� �m)�� + �mdmt�1 + �m�t�1 + em;t (8)

dmt = lnmt � lnmt�1 (9)

ln at = � ln at�1 + ea;t (10)

(1) is the Euler equation; (2) is the aggregate accounting identity; (3) is the productionfunction; (4) is labor supply; (5) is labor demand; (6) is demand for real balances; (7) is theFisher relationship; (8) is the exogenous process governing the growth rate of real balances;(9) de�nes the growth rate of real balances; and (10) is the familiar process for log technology.

3 The Model with Calvo Price Stickiness

Above �rms could change their prices each period; each period, they would set prices as aconstant markup over marginal cost, with the size of the markup related to the slope of thedemand curve for their good. Now we assume that �rms cannot change their prices freelyeach period. In particular, �rms face a constant probability, 1� �, of being able to adjusttheir price in any period. This hazard rate is constant across time.The household side of the model is identical to above; the �nal goods production is also

identical to above. The pricing decision is simlar but cannot be undertaken every period.Let�s consider a �rm who, at time t, is given the ability to adjust its price. It will do so tomaximize the expected discounted value of pro�ts, since it will, in expectation, be stuck withthis price for more than just the current period. The �rm discounts future pro�ts by thegross real interest rate between today and future dates . . . i.e. (1+ rt;t+s)�1 for s = 0; :::1.From the households Euler equation, we can solve for this �long�real interest rate as:

(1 + rt;t+s)�1 = �sEt

�ct+sct

���This is often called the stochastic discount factor and is frequently used in the asset pricing

literature. In addition, the �rm will also discount future pro�t �ows by the probability thatit will be stuck with the price it chooses today. This probability is �. If � is small, then the�rms get to update their prices frequently, and thus will heavily discount future pro�t �owswhen making current pricing decisions. On the other hand, if � is large, it is very likely thata �rm will be �stuck�with whatever price it chooses today for a long time, and will thus berelatively more concerned about the future when making its current pricing decisions.Similarly to above but now taking account of the possibility of being stuck with a price,

we can write the �rm with the opportunity to change its price solves the following problem:

9

maxpt(j)

Et

1Xs=0

(��)s �t;t+s

pt(j)

pt+s

�pt(j)

pt+s

��"yt+s �

't+spt+s

�pt(j)

pt+s

��"yt+s

!Here the problem is written as maximizing real pro�ts discounted by the stochastic

discount factor as well as the probability of being able to make price changes. For simplicity,

I write �t;t+s =�ct+sct

����i.e. the ratio of marginal utility between period t+ s and period

t. When � = 0, so that there is no price stickiness, it is straightforward to verify that theproblem reduces to what we had above (because (��)s = 0 for every s > 0, so only currentpro�ts will factor into the pricing decision. Note that the �rm�s price isn�t indexed by s,since it is choosing a price today that it won�t be able to change in the future. The �rstorder condition for this problem is:

Et

1Xs=0

(��)s �t;t+s

�(1� ")pt(j)

�"p�(1�")t+s yt+s + "pt(j)

�"�1't+sp�(1�")t+s yt+s

�= 0

Let�s simplify:

Et

1Xs=0

(��)s �t;t+s

�("� 1)pt(j)�"p�(1�")t+s yt+s

�= Et

1Xs=0

(��)s �t;t+s

�"pt(j)

�"�1't+sp�(1�")t+s yt+s

�Since the price they choose does not depend upon s, we can pull it out of the sums:

("�1)pt(j)�"Et1Xs=0

(��)s �t;t+s

�p�(1�")t+s yt+s

�= "pt(j)

�"�1Et

1Xs=0

(��)s �t;t+s

�'t+sp

�(1�")t+s yt+s

�Simplify:

p#t ="

"� 1

Et

1Xs=0

(��)s �t;t+s

�'t+sp

�(1�")t+s yt+s

�Et

1Xs=0

(��)s �t;t+s

�p�(1�")t+s yt+s

�Above, I replace the pt(j) with p

#t , which is called the optimal reset price. Since �rms face

the same marginal cost and take aggregate variables as given, any �rm that gets to update itsprice will choose the same price. Essentially, the current price that price-changing �rms willchoose is a present discount value of marginal costs. As noted, if there is no price-stickiness,so that � = 0, then the solution is the same as above, with p#t =

""�1't.

For ease of notation, let�s write this expression as:

10

p#t ="

"� 1AtBt

At = Et

1Xs=0

(��)s �t;t+s

�'t+sp

�(1�")t+s yt+s

�Bt = Et

1Xs=0

(��)s �t;t+s

�p�(1�")t+s yt+s

�Now, when we go to the computer to solve this, the computer isn�t going to like an in�nite

sum. Fortunately, we can write the expression for At and Bt as follows:

At = 'tp�(1�")t yt + ���t;t+1EtAt+1

Bt = p�(1�")t yt + ���t;t+1EtBt+1

Recall the de�nition of the aggregate price level:

pt =

24 1Z0

pt(j)1�"dj

351

1�"

We can split this intergral into a convex combination of two things �the optimal resetprice and the previous price. This is because all �rms that can reset will choose the samereset price, and the �average�price of the �rms that cannot reset will equal the previousaggregate price level:

pt =

24 1Z0

�(1� �)p#1�"t + �p1�"t�1

�dj

351

1�"

pt =

24 1��Z0

p#1�"t dj +

1Z1��

p1�"t�1dj

351

1�"

pt =h(1� �)p#1�"t + �p1�"t�1

i 11�"

As a general matter we want to allow for the existence of steady state in�ation (thoughin most linearizations it is assumed that there is zero steady state in�ation), so we need towrite this such that there is a well-de�ned steady state. To do this divide both sides bypt�1:

11

ptpt�1

= p�1t�1

h(1� �)p#1�"t + �p1�"t�1

i 11�"

ptpt�1

=hp�(1�")t�1 ((1� �)p#1�"t + �p1�"t�1)

i 11�"

ptpt�1

=

24(1� �)

p#tpt�1

!1�"+ �

�pt�1pt�1

�1�"35 11�"

De�ning 1 + �t =ptpt�1

, we can write this:

1 + �t =

24(1� �)

p#tpt�1

!1�"+ �

35 11�"

Thus, to get an expression for current in�ation, we need to �nd an expression for �reset

price in�ation�, which I�ll call p#tpt�1

. Go back to the expression for the rest price:

p#t ="

"� 1AtBt

Divide both sides by pt�1:

p#tpt�1

="

"� 11

pt�1

AtBt

Let�s deal with this part by part. Note that:

Atpt�1

=1

pt�1

�'tp

�(1�")t yt + ���t;t+1EtAt+1

�Atpt�1

='tp

�(1�")t ytpt�1

+���t;t+1EtAt+1

pt�1

De�ning mct ='tptas real marginal cost, we can write this as:

Atpt�1

= mct

�ptpt�1

�p�(1�")t yt +

���t;t+1EtAt+1pt�1

We need to play around further with the dates on the very end of the expression on theright hand side:

Atpt�1

= mct

�ptpt�1

�p�(1�")t yt + ���t;t+1

�ptpt�1

�EtAt+1pt

To save on notation, let�s go ahead and call Atpt�1

= bAt. Thus, we can write this as:bAt = (1 + �t)�mctp�(1�")t yt + ���t;t+1 bAt+1�

12

Given this, we can write reset price in�ation as:

p#tpt�1

="

"� 1bAtBt

Now, we�re not yet done because both bAt and Bt have a p�(1�")t component in them.Fortunately, we can divide both numerator and denominator by p�(1�")t without changingthe equality. De�ne bat = bAt=p�(1�")t and bbt = Bt=p

�(1�")t :

p#tpt�1

="

"� 1batbbt

Now we need to �nd expression for bat and bbt:bat =

bAtp�(1�")t

= (1 + �t)

mctyt + Et���t;t+1

bAt+1p�(1�")t

!

bat =bAt

p�(1�")t

= (1 + �t)

mctyt + Et���t;t+1

�pt+1pt

��(1�") bAt+1p�(1�")t+1

!bat = (1 + �t)

�mctyt + Et���t;t+1 (1 + �t+1)

�(1�") bat+1�

bbt =Bt

p�(1�")t

=1

p�(1�")t

�p�(1�")t yt + Et���t;t+1Bt+1

�bbt = yt + Et���t;t+1

Bt+1

p�(1�")tbbt = yt + Et���t;t+1

�pt+1pt

��(1�")Bt+1

p�(1�")t+1bbt = yt + Et���t;t+1 (1 + �t+1)

�(1�")bbt+1A small technical point is that, for this �trick� to work (i.e. writting bat and bbt not as

in�ninite sums but rather as as current plus �continuation values�) it must be the case thatthe e¤ective discount factor be less than one in the steady state. Since �� = 1, this meansthat �� (1 + ��)�(1�") < 1. �� < 1, so if steady state in�ation is zero, this is never an issue.But if steady state in�ation is very high, or " is very large, then this may not hold. Giventhe auxilliary variables bat and bbt, which, subject to the caveat above, have been renderedstationary, we can write actual price in�ation as:

1 + �t =

24(1� �)

p#tpt�1

!1�"+ �

35 11�"

p#tpt�1

="

"� 1batbbt13

Given this, we can write down the equations characterizing equilibrium of the model withprice stickiness as follows:

c��t = �Etc��t+1(1 + rt) (11)

ct = yt (12)

yt = atnt (13)

�(1� nt)�� = c��t wt (14)

wt = mctat (15)

1 + �t =

�(1� �)

�1 + �#t

�1�"+ �

� 11�"

(16)

1 + �#t ="

"� 1batbbt (17)

bat = (1 + �t)�mctyt + Et���t;t+1 (1 + �t+1)

�(1�") bat+1� (18)

bbt = yt + Et���t;t+1 (1 + �t+1)�(1�")bbt+1 (19)

�t;t+1 =

�ct+1ct

���(20)

m�vt = c��t

�it

1 + it

�(21)

1 + rt =1 + it1 + �t+1

(22)

dmt + �t = (1� �m)�� + �mdmt�1 + �m�t�1 + em;t (23)

dmt = lnmt � lnmt�1 (24)

ln at = � ln at�1 + et (25)

Note that I have �fteen equations and �fteen variables. Some of these (in fact many ofthese) variables can be eliminated from the solution.I calibrate the parameters of the model as follows:

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Parameter Value� 0.99� 1� 1v 1� 3.5� 0.75� 0.9�m 0.5 1" 11�� 0.01�e 0.007�em 0.002

How can these parameters be interpreted? The discount factor of 0.99 implies a steadystate real interest rate of about one percent (or about four percent expressed at an annualfrequency). Coupled with steady state in�ation of 0.01, this means that the steady statenominal interest rate is about 0.02. The power coe¢ cients in preferences being all equal toone means that the within period utility function is �log-log-log�. � = 3:5 means that steadystate hours per capita will be roughly 0.2. The shock standard deviations and autoregressivecoe¢ cients in the technology and money growth speci�cations are similar to what we�ve beenusing.The two new parameters here that need some discussion are �, which governs price-

stickiness and is often called the �Calvo parameter�, and ", which controls market power. "is easier to deal with, so we begin there. Recall from our derivation that the steady state (oraverage) markup of price over marginal cost is equal to "

"�1 . In the data, average markupsappear to be about 10% (Basu and Fernald (1997)). This means that "

"�1 = 1:1, or " = 11.The Calvo parameter will govern the average duration between price changes. Condi-

tional on changing a price in the current period, what is the expected duration until yournext price change? Well, the probability of getting to change prices next period is 1 � �.The probability of getting to change prices in two periods is 1 � � times the probability ofnot changing prices after one period, or (1 � �)�. The probability of getting to changeprices in three periods is 1� � times the probability of not getting to change prices for twoconsecutive periods, or (1� �)�2. More compactly:

Duration Probability1 1� �2 (1� �)�

3 (1� �)�2

4 (1� �)�3

......

j (1� �)�j�1

15

The expected duration between price changes is then just the sum of probabilities timesduration:

Expected Duration between Price Changes =1Xj=1

(1� �)�j�1j

= (1� �)1Xj=1

�j�1j

We can write the part inside the summation as:

S = 1 + 2�+ 3�2 + 4�3 + 5�4 + ::: =

1Xj=1

�j�1j

Multiply everything by �:

S� = �+ 2�2 + 3�3 + 4�4 + 5�5 + :::

Subtract the former from the latter:

S � S� = 1 + (2� 1)�+ (3� 2)�2 + (4� 3)�3 + (5� 4)�4 + ::::

(1� �)S = 1 + �+ �2 + �3 + �4 + ::::

Now multiply this expression by �:

(1� �)�S = �+ �2 + �3 + �4 + ::::

Now subtract this from the former:

(1� �)S � (1� �)�S = 1

This follows from the fact that, as j !1, �j+1 = �j = 0. Simplifying:

(1� �)2S = 1

S =1

(1� �)2

Now plugging this back in to the original expression, we have:

Expected Duration between Price Changes = (1� �)1

(1� �)2=

1

(1� �)

Thus, we can calibrate � by looking at data on the average duration between pricechanges. Bils and Klenow (2004) �nd that it�s between 6 months and one year. We�ll gowith the long end of that range (four quarters), which suggests that � = 0:75.Below are impulse responses to technology shock:

16

We see that a technology shock leads to an increase in output and the real interest rateon impact, with decreases in in�ation, hours, and the price level. The fall in hours mayseem non-intuitive at �rst. To see why hours fall, look at the money demand speci�cation:

m�vt = c��t

�it

1 + it

�Rewrite this in terms of the nominal money supply, the price level, and output (since

consumption is equal to output in equilibrium):�Mt

pt

�= y

�vt

1v

�1 + itit

� 1v

To make this as simple to see as possible, suppose that both v and � are very big, sothat �

v� 1 and 1

v� 0. Then we recover exactly the simple quantity equation:

Mt = ptyt

If prices were fully �exible, when technology increases prices would fall by the amount ofthe increase in output. But because we have here assumed price stickiness, prices cannotfall by that much, so output cannot rise by as much as it would if prices were fully �exible.This means that hours cannot rises by as much as they do when prices are fully �exible; sincein the way I wrote down preferences hours actually do not respond at all to a technologyshock when prices are perfectly �exible, this necessitates a decrease in hours on impact.Next, consider the responses to a money growth shock.

17

These responses look reasonably intuitive. An increase in money growth raises output,in�ation, and the price level, while lowering nominal and real interest rates. The intuitionfor why this happens can be gained from the quantity theoretic equation above as well. Theprice level cannot adjust upward the same amount it would if prices were �exible when themoney supply increases �therefore, output must rise to make the money market clear. Notethat the Matlab �led used to produce these �gures is titled nk_basic_notzero.mod and canbe run from new_keynesian.m.

3.0.1 Log-Linearizing

Suppose that we want to log-linearize this expression about a steady state. The conventionallinearization is about the zero in�ation steady state, so that �� = 0. AS short hand, let�s

call p#tpt�1

= 1 + �#t . Log-linearize the in�ation equation by �rst taking logs of both sides:

ln(1 + �t) =1

1� "ln

�(1� �)

�1 + �#t

�1�"+ �

�Now do the Taylor series expansion about the point �� = 0, which will mean ��# = 0 as

well:

ln(1 + 0) +d�t1 + 0

=1

1� "ln(1) +

1

1� "

1

1(1� ")(1� �)(1 + 0)�"d�#t

Some of this follows from the fact that (1� �) (1 + 0)1�" + � = 1. Simplify, we have:

d�t = (1� �)d�#t

18

Since in�ation is already in a percentage rate, we want to leave it as an absolute ratherthan percentage deviation. Therefore, let e�t = d�t and e�#t = d�#t :

e�t = (1� �)e�#tQuite naturally, then, this says that deviation of in�ation from 0 is equal to the fraction

of �rms changing prices times the amount by which they are changing prices. To close thisout, we now need an expression for e�#t . Log-linearize that expression by �rst taking logs:

ln(1 + �#t ) = ln "� ln("� 1) + lnbat � lnbbtNow do a Taylor series expansion about the zero in�ation steady state:

ln(1 + 0) + d�#t = ln "� ln("� 1) + lnba� � lnbb� + dbatba� � dbbtbb�e�#t = ln "� ln("� 1) + ln�ba�bb�

�+ ebat � ebbt

Where e�#t = d�#t . Now, what isba�bb� ? Note that ��t;t+1 = 1. Solve for them individually

using the de�nitions:

bat = (1 + �t)�mctyt + Et���t;t+1 (1 + �t+1)

�(1�") bat+1�ba� = mc�y� + ��ba�ba� =

mc�y�

1� ��

bbt =�yt + Et���t;t+1 (1 + �t+1)

�(1�")bbt+1�bb� = y� + ��bb�bb� =y�

1� ��

To derive the above I�m using the assumption that in�ation is zero in the steady state.Thus, I have:

ba�bb� = mc�

From above, we know that price is equal to a markup over nomina marginal cost. Thusreal marginal cost is equal to the inverse of that markup, or, in the steady state:

mc� ="� 1"

This means that:

19

ln

�ba�bb��= ln("� 1)� ln "

Now plugging this in above, we see that the "s disappear, leaving:

e�#t = ebat � ebbtSo now we need ebat and ebbt. Begin with the �rst by �rst taking logs:

bat = (1 + �t)�mctyt + Et���t;t+1 (1 + �t+1)

�(1�") bat+1�lnbat = ln(1 + �t) + ln

�mctyt + Et���t;t+1 (1 + �t+1)

�(1�") bat+1�Now do the Taylor series expansion evaluated at the steady state. Before proceeding,

note that mc�y� + Et��ba� = ba� since steady state in�ation is 0:lnba� + dbatba� = ln(1 + 0) + d�t + lnba� + dmcty

�ba� +dytmc

�ba� + :::

:::+��d�t;t+1ba�ba� � (1� ")��d�t+1ba�ba� +

��dbat+1ba�Simplifying, we have:

ebat = d�t +dmcty

�ba� +dytmc

�ba� + �d�t;t+1 � (1� ")��d�t+1 + ��ebat+1Leave this alone for a minute. Now go to ebbt:

lnbbt = ln�yt + Et���t;t+1 (1 + �t+1)�(1�")bbt+1�

As above, note that y� + Et��bb� = bb�. Proceed with the �rst order Taylor seriesexpansion:

lnbb� + dbbtbb� = lnbb� + dytbb� ++��d�t;t+1bb�bb� � (1� ")��d�t+1bb�bb� +

��dbbt+1bb�Now simplify some:

ebbt = dytbb� + �d�t;t+1 � (1� ")��d�t+1 + �Q�ebbt+1

We can now rewrite part of this as:

ebat = d�t +y�ba�dmct + mc�

a�dyt + �d�t;t+1 � (1� ")��d�t+1 + ��ebat+1ebbt =

dytbb� + �d�t;t+1 � (1� ")��d�t+1 + ��ebbt+1

20

Now subtract the latter from the former:

ebat �ebbt = d�t +y�ba�dmct + mc�ba� dyt � dytbb� + ��

�ebat+1 �ebbt+1�Note that y

�ba� = (1���)mc� and mc�ba� = 1bb� . Using these facts, we can write:ebat �ebbt = e�t + (1� ��)fmct + ��

�ebat+1 �ebbt+1�Now note that e�t = (1� �)

�ebat �ebbt� and �ebat+1 �ebbt+1� = Ete�t+11�� :

e�t = (1� �)e�t + (1� �)(1� ��)fmct + ��Ete�t+1Now solve for e�t:

�e�t = (1� �)(1� ��)fmct + ��Ete�t+1e�t =(1� �)(1� ��)

�fmct + �Ete�t+1

The above relationship is what is often called the New Keynesian Phillips Curve.It is actually quite common to see the Phillips Curve expressed not in terms of the log-

deviation of real marginal cost, but rather in terms of an �output gap�. To get to thatspeci�cation, let�s start with what de�nes real marginal cost and then go from there:

mct =wtat

We can substitute out for the wage using the household�s �rst order condition for laborsupply:

wt = c�t �(1� nt)��

Now use the accounting identity fact that consumption equals income to get:

mct =y�t �(1� nt)

��

at

Now let�s log-linearize this expression. Begin by taking logs:

lnmct = � ln yt + ln � � � ln(1� nt)� ln atNow do the �rst order Taylor series expansion about the steady state:

lnmc� +dmctmc�

= lnmc� + �dyty�+ �

dnt1� n�

� data�

fmct = �eyt + �n�

1� n�ent � eat

21

Now, note that, from the aggregate production function, ent = eyt � eat:fmct = �eyt + �

n�

1� n�(eyt � eat)� eat

Simplifying:

fmct = �� + �n�

1� n�

� eyt � �1 + � n�

1� n�

�eatThe output gap is de�ned as the deviation between the actual level of output and the

��exible price� level of output, eyft which is the level of output which would obtain in theabsence of price stickiness. If prices are not sticky, price is a constant markup over nominalmarginal cost, which implies that real marginal cost is constant, or equivalently that fmct = 0(i.e. the log deviation of a constant is zero). We can then solve for the �exible priceequilibrium level of output in terms of the exogenous driving variable using this fact and theabove expression:

0 =

�� + �

n�

1� n�

� eyft � �1 + � n�

1� n�

�eateyft =

1 + � n�

1�n�

� + � n�

1�n�eat

Note that, if we have log utility over consumption (i.e. � = 1), then eyft = eat (i.e.employment is constant in the �exible price equilibrium. Using the above, we can eliminateeat from the expression for the log deviation of real marginal cost:

fmct =

�� + �

n�

1� n�

� eyt � �� + �n�

1� n�

� eyftfmct =

�� + �

n�

1� n�

��eyt � eyft �Letting � =

�� + � n�

1�n��, we can re-write the Phillips Curve in terms of the output gap

as:

e�t = (1� �)(1� ��)

���eyt � eyft �+ �Ete�t+1

Holding expected in�ation �xed, we see that positive output gaps put upward pressureon current in�ation.We can also log-linearize the rest of the model. Start with the Euler equation, after

having already imposed the accounting identity:

y��t = �Et(y��t+1(1 + rt))

Take logs:

22

�� ln yt = ln � � � ln yt+1 + rt

Above I have imposed the approximation that ln(1 + rt) � rt. Now do the �rst orderTaylor series expansion:

�� ln y� � �dyty�= ln � � � ln y� + r� � �

dyt+1y�

+ drt

De�ning eyt = dyty� and ert = drt, we have:

��eyt = ��eyt+1 + erteyt = eyt+1 � 1

�ert

The log-linearized Euler equation is often referred to as the �New Keynesian IS�curve,as it shows a negative relationship between current spending and the current real interestrate, holding �xed expected future spending.Now let�s log-linearize the money supply curve (written in terms of real balances). It

can be written out as follows:

lnmt � lnmt�1 + �t = (1� �m)�� + �m(lnmt�1 � lnmt�2) + �m�t�1 + em

Since this equation is already in logs and already linear, we can write it exactly the sameway but interpreting the variables as log deviations emt =

dmt

m� and e�t = d�t:

emt = (1� �m)�� + emt�1 + �m(emt�1 � emt�2)� e�t + �me�t�1 + em

Now let�s log-linearize the money demand function. First take logs:

ln � v lnmt = �� ln yt + ln it � ln(1 + it)Do the �rst order Taylor series expansion:

ln � v lnm� � vdmt

m� = �� ln y� + ln i� � ln(1 + i�)� �

dyty�+diti�� dit1 + i�

Simplifying and use the tilde notation:

�v emt = ��eyt + � 1i�� 1

1 + i�

�eitSimplifying further:

emt =�

veyt � � 1

vi�(1 + i�)

�eitEquilibrium requires that money demand be equal to money supply, so we can eliminate

money altogether from the set of equations by equating demand with supply:

23

�

veyt � � 1

vi�(1 + i�)

�eit = (1� �m)�� + emt�1 + �m(emt�1 � emt�2)� e�t + �me�t�1 + em

Simplify by solving for the current log deviation of output:

eyt = � 1

�i�(1 + i�)

�eit+ v

�(1� �m)��+

v

�emt�1+

v

��m(emt�1� emt�2)�

v

�e�t+ v

��me�t�1+ v

�em

We can write this in terms of the real interest rate by using the linearized Fisher rela-tionship (eit = ert + e�t+1):eyt = � 1

�i�(1 + i�)

�(ert + e�t+1)+v

�(1��m)��+

v

�emt�1+

v

��m(emt�1�emt�2)�

v

�e�t+v

��me�t�1+v�em

The expression above can be interpreted as an LM curve from intermediate macro �itis the set of points in (ert; eyt) space consistent with the money market clearing. The IScurve is the set (ert; eyt) pairs consistent with the �goods market�clearing, which means thatconsumption is equal to income and the Euler equation holds. The IS equation is downwardsloping, while the LM curve is upward sloping.Above we derived an expression for the �exible price equilibrium level of output as:

eyft = 1 + � n�

1�n�

� + � n�

1�n�eat

For notational ease, call =1+� n�

1�n�

�+� n�1�n�

, so:

eyft = eatNow plug in this process for technology:

eyft = �eat�1 + et

Now we know that eat�1 = 1 eyft�1, so we can write this as:

eyft = �eyft�1 + et

The full set of log-linearized equations which allow us to solve the model are then:

eyt = eyt+1 � 1

�ert (26)

eyt = � 1

�i�(1 + i�)

�(ert + e�t+1)+v

�(1��m)��+

v

�emt�1+

v

��m(emt�1�emt�2)�

v

�e�t+v

��me�t�1+v�em

(27)

24

eyft = �eyft�1 + et (28)

e�t = (1� �)(1� ��)

���eyt � eyft �+ �Ete�t+1 (29)

Equation (26) is the IS curve, (27) is the LM curve, (28) is the process for the �supplyshock�, and (29) is the Phillips Curve. There are four equations and four variables (output,real interest rate, the �exible price level of output, and in�ation).It turns out there is a graphical interpretation of this model that is is visually similar to

what one sees in intermediate macro. Holding the values of all future and past variables�xed, as well as the value of current in�ation, we can plot out the IS and LM curves asfollows:

Recall that the LM curve is drawn holding current in�ation �xed (the IS curve does notdepend on current in�ation). E¤ectively what this does is de�ne an equilibrium level ofoutput and the interest rate for each level of current in�ation possible. If in�ation goes up,the LM curve shifts horizontally to the left (i.e. holding the real interest rate �xed outputmust fall when in�ation goes up). The opposite holds when in�ation goes down. We canthen trace out an aggregate demand curve in (e�t; eyt) space as follows:

25

When in�ation is relative high, the LM curve is relatively far in, and so output is relativelylow, and vice versa. Tracing out the points, the AD curve is downward sloping. Wecan complete the model by adding in the Phillips curve, which is an upward sloping ASrelationship, de�ned for a give value of the �exible price level of output and a given expectedfuture in�ation:

26

Given this framework, we can graphically conduct comparative statics exercises. I shouldbe very upfront that this exercise is frought with hazards �there are lots of expected futureendogenous variables in these equations, all of which will, in general, move when exogenousvariables change. This means that shifting curves holding expectations of future endogenousvariables constant really isn�t correct. Nevertheless, if shocks are transitory enough, thiswill provide a very good approximation.Let�s �rst consider a monetary policy shock �this will cause the LM curve to shift right

(i.e. a positive innovation to em raises output for a given interest rate).

27

The increase in money supply shifts the LM curve out �this raises the equilibrium levelof output for a given level of in�ation, shifting the AD curve horizontally. In order to alsobe on the Phillips Curve/AS relationship, in�ation rises. This means that output rises byless than the horizontal shift in the AD curve. The rise in in�ation causes the LM curveto shift back in some, so as to intersect the IS curve at the same level of output. We seethat, in equilibrium the real interest rate is lower, output is higher, and in�ation is lower �in other words, more or less exactly what our undergraduate intuition is. Furthermore, wesee that the increase in output due to monetary shocks is increasing in the �atness of thePhillips Curve. When is the Phillips Curve �at? When �, the probability of not beingable to adjust one�s price, is big. In other words, money supply shocks have a bigger e¤ecton output (and a smaller e¤ect on in�ation), the stickier are prices. If prices are �exible,so that � = 0, then the Phillips Curve is vertical at the �exible price level of output, whichmeans that monetary shocks have no real e¤ect and just lead to in�ation.Now let�s consider a �supply shock��i.e. a shock to the �exible price level of output.

From inspection of the Phillips Curve, this leads to an outward shift of the AS relationship.Graphically:

28

The outward shift in the AS relationship raises output and lowers in�ation. The lowerin�ation forces the LM curve outward. At the end of the day, the supply shock leads tohigher output, lower in�ation, and a lower real interest rate. Note that the increase in outputis smaller than if the AS/Phillips Curve were perfectly vertical. This is what necessitatesthe reduction in hours on impact in response to a technology shock in the model.Finally, consider an �IS Shock�. We don�t formally have that in the model as speci�ed,

but would could think of it as a shock to expected future output. We will ignore the factthat this would in�uence expected in�ation in equilibrium, which would in turn shift thePhillips Curve:

29

Here the outward shift of the IS curve shifts the AD curve out, which raises both outputand in�ation. The increase in in�ation leads to the LM curve shifting back in some so asto restore equilibrium. At the end of the day, output, in�ation, and the real interest rateare all higher.The above exercise shows that this dynamic, optimizing model can be thought of in terms

very similar to what one learns in a typical intermdiate micro course. Of course, this is allapproximate. Nevertheless, it restores a lot of the Keynesian intuition.

30