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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 isnt a problem, but Ithink it makes the model behave very dierently. Aside from lacking physical capital, themodel also diers from our benchmark in that it assumes imperfect competition (in particular

monopolistic 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 weve seenbefore.

Ill begin with a model of imperfect competition with no price stickiness. Then wellmove 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 households problem can be written as follows:

maxct;nt;bt;mt

E0

1Xt=0

t

c1t 1

1 +

(1 nt)1

1

1 +

m1vt 1

1 v

!

s.t.

ct+bt+mt wtnt+ t+ (1 +it1) bt

11 +t

+ mt

11 +t

The Lagrangian for the problem can be written:

L =E0

1Xt=0

t

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The rst order conditions can be written:

@L

@ct= 0 , ct =t

@L

@nt = 0 , (1 nt)

=twt

@L

@bt= 0 , t = t+1

1 +it1 +t+1

@L

@mt= 0 , tmvt

tt+t+1t+1

1

1 +t+1= 0

) mvt =t t+11

1 +t+1

We can simplify these using the Fisher relationship and simplifying:

(1 nt) = ct wt

ct = ct+1(1 +rt)

mvt = ct

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. Prot maximization

in 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 dierentiates monopolistic competition from perfect competition is that a largenumber rms sell dierentiated products and have some pricing power. Because of entryand exit, they earn no economic prots in the long run, however.

2.2.1 Final Goods

There is one nal goods rm and a continuum (i.e. innity) 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 dierent intermediategoods. As long as" < 1, the intermediate goods are imperfect substitutes in consumption;

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this is what gives them market power. Note that an integral is really just a sum werejust 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 prots (more generally theyd want to maxi-

mize the present discounted value of prots, 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 prots 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:

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pt

2

41Z

0

yt(j)"1" dj

3

5

""1

"1"1

yt(j)"1"

"" = pt(j) 8 j

pt

24 1Z0

yt(j)"1" dj

35 1"1 yt(j)1" = pt(j) 8 jyt(j)

1" =

pt(j)

pt

24 1Z0

yt(j)"1" dj

35 1"1

yt(j) =

pt(j)

pt

"

2

41Z

0

yt(j)"1" dj

3

5

""1

yt(j) =pt(j)

pt

"yt

The last line follows from the denition 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, prots 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:

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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 innite 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 a

given 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. rms dont havefactor attachment), so maximizing value is equivalent to maximizing prots 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. Prots 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

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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 byj 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 ='tpt

at

Herewt Wtpt

; i.e. the real wage. If markets were perfectly competitive, price would alwaysbe 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 static

problem.

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 prots,which would entail dividng by the aggregate price level, but given the static nature of theproblem, doing so would not aect 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

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Since the rm is small (i.e. there are an innite 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 therms demand curve is. As" ! 1, the rm faces a horizontal demand curve, "

"1 ! 1, and

price is equal to marginal cost, and were 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 ""1Let yt(j) be the amount of output produced by the typical intermediate goods rm.

Since its the same across allj , 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)

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Note also that, since were 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 =

1

Z0

atnt(j)dj = at

1

Z0

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 rms 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 ='tpt

at

Now use the pricing condition:

wt =" 1

" at

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mvt =ct

it

1 +it

(6)

1 +rt = 1 +it1 +t+1

(7)

dmt+t = (1 m) +mdmt1+mt1+em;t (8)

dmt= ln mt ln mt1 (9)

ln at = ln at1+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) denes 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.Lets 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 prots, since it will, in expectation, be stuck withthis price for more than just the current period. The rm discounts future prots by thegross real interest rate between today and future dates . . . i.e. (1 + rt;t+s)

1 fors = 0;:::1.From the households Euler equation, we can solve for this long real interest rate as:

(1 +rt;t+s)1 =sEt

ct+s

ct

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 prot ows by the probability thatit will be stuck with the price it chooses today. This probability is. If is small, then therms get to update their prices frequently, and thus will heavily discount future prot 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:

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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 prots discounted by the stochastic

discount factor as well as the probability of being able to make price changes. For simplicity,I writet;t+s=

ct+sct

i.e. the ratio of marginal utility between period t + sand 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 currentprots will factor into the pricing decision. Note that the rms price isnt indexed by s,since it is choosing a price today that it wont be able to change in the future. The rstorder condition for this problem is:

Et

1

Xs=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

Lets 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)"

Et

1

Xs=0 ()

s

t;t+s p(1")t+s yt+s= "pt(j)"1Et1

Xs=0 ()

s

t;t+s 't+sp(1")t+s yt+sSimplify:

p#t = "

" 1

Et

1Xs=0

()s t;t+s

't+sp(1")t+s yt+s

Et

1Xs=0

()s t;t+sp(1")t+s yt+s

Above, I replace thept(j)withp

#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, lets write this expression as:

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p#t = "

" 1

AtBt

At = Et

1

Xs=0 ()s t;t+s 't+sp(1")t+s yt+sBt = Et

1Xs=0

()s t;t+sp(1")t+s yt+s

Now, when we go to the computer to solve this, the computer isnt going to like an innite

sum. Fortunately, we can write the expression forAt andBt as follows:

At = 'tp(1")t yt+t;t+1EtAt+1

Bt = p(1")t yt+t;t+1EtBt+1

Recall the denition 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 +p

1"t1

dj

351

1"

pt =

24 1Z0

p#1"t dj+

1Z1

p1"t1dj

351

1"

pt =

h(1 )p#1"t +p

1"t1i

11"

As a general matter we want to allow for the existence of steady state ination (thoughin most linearizations it is assumed that there is zero steady state ination), so we need towrite this such that there is a well-dened steady state. To do this divide both sides by

pt1:

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ptpt1

= p1t1

h(1 )p#1"t +p

1"t1

i 11"

pt

pt1= hp(1")t1 ((1 )p#1"t +p1"t1)i

11"

ptpt1

=

24(1 ) p#tpt1

!1"+

pt1pt1

1"35 11"

Dening1 +t = ptpt1

, we can write this:

1 +t =

2

4(1 )

p#tpt1

!1"+

3

5

11"

Thus, to get an expression for current ination, we need to nd an expression for resetprice ination, which Ill call

p#t

pt1. Go back to the expression for the rest price:

p#t = "

" 1

AtBt

Divide both sides by pt1:

p#tpt1

= "

" 1

1

pt1

AtBt

Lets deal with this part by part. Note that:

Atpt1

= 1

pt1

'tp

(1")t yt+t;t+1EtAt+1

At

pt1=

'tp(1")t ytpt1

+t;t+1EtAt+1

pt1

Deningmct = 'tpt

as real marginal cost, we can write this as:

Atpt1

=mct

pt

pt1

p(1")t yt+

t;t+1EtAt+1pt1

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

Atpt1

=mct

pt

pt1

p(1")t yt+t;t+1

pt

pt1

Et

At+1pt

To save on notation, lets go ahead and call Atpt1

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

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Given this, we can write reset price ination as:

p#tpt1

= "

" 1

bAtBt

Now, were not yet done because bothbAt 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. Denebat =bAt=p(1")t andbbt = Bt=p(1")t :

p#tpt1

= "

" 1

batbbtNow we need to nd expression forbat andbbt:

bat =

bAt

p(1")t

= (1 +t)

mctyt+Ett;t+1

bAt+1

p(1")t

!

bat = bAtp(1")t

= (1 +t)

mctyt+Ett;t+1pt+1

pt

(1") bAt+1p(1")t+1

!bat = (1 +t)mctyt+Ett;t+1(1 +t+1)(1")bat+1

bbt = Btp(1")t

= 1

p(1")t

p(1")t yt+Ett;t+1Bt+1

bbt = yt+Ett;t+1

Bt+1

p(1")tbbt = yt+Ett;t+1pt+1

pt

(1") Bt+1p(1")t+1bbt = yt+Ett;t+1(1 +t+1)(1")bbt+1

A small technical point is that, for this trick to work (i.e. writtingbat andbbt not asinninite sums but rather as as current plus continuation values) it must be the case thatthe eective discount factor be less than one in the steady state. Since = 1, this meansthat(1 +)(1")


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

ct =Etct+1(1 +rt) (11)

ct = yt (12)

yt = atnt (13)

(1 nt) =ct wt (14)

wt = mctat (15)

1 +t= (1 )1 +#t 1" + 11" (16)1 +#t =

"

" 1

batbbt (17)bat= (1 +t)mctyt+Ett;t+1(1 +t+1)(1")bat+1 (18)

bbt = yt+Ett;t+1(1 +t+1)(1")bbt+1 (19)t;t+1= ct+1ct

(20)

mvt =ct

it

1 +it

(21)

1 +rt = 1 +it1 +t+1

(22)

dmt+t = (1 m) +mdmt1+mt1+em;t (23)

dmt= ln mt ln mt1 (24)

ln at= ln at1+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 0.9m 0.5 1" 11 0.01e 0.007em 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 ination of 0.01, this means that the steady statenominal interest rate is about 0.02. The power coecients in preferences being all equal toone means that the within period utility function is log-log-log. = 3:5means that steadystate hours per capita will be roughly 0.2. The shock standard deviations and autoregressivecoecients in the technology and money growth specications are similar to what weve 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 (or

average) 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 is1 .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 )j1

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

Expected Duration between Price Changes =

1

Xj=1(1 )j1j= (1 )

1Xj=1

j1j

We can write the part inside the summation as:

S= 1 + 2+ 32 + 43 + 54 +:::=

1Xj=1

j1j

Multiply everything by:

S = + 22 + 33 + 44 + 55 +:::

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 its between 6 months and one year. Well gowith the long end of that range (four quarters), which suggests that = 0:75.

Below are impulse responses to technology shock:

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

mvt =ct

it1 +itRewrite this in terms of the nominal money supply, the price level, and output (since

consumption is equal to output in equilibrium):Mtpt

= y

v

t 1v

1 +it

it

1v

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

v 1and 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.

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These responses look reasonably intuitive. An increase in money growth raises output,ination, 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 can

be 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 ination steady state, so that = 0. AS short hand, lets

call p

#t

pt1= 1 +#t . Log-linearize the ination 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#

= 0aswell:

ln(1 + 0) + dt1 + 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:

dt = (1 )d#t

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Since ination is already in a percentage rate, we want to leave it as an absolute ratherthan percentage deviation. Therefore, letet = dt ande#t =d#t :

et = (1 )

e#t

Quite naturally, then, this says that deviation of ination from 0 is equal to the fractionof rms changing prices times the amount by which they are changing prices. To close thisout, we now need an expression fore#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 ination steady state:

ln(1 + 0) +d#t = ln " ln(" 1) + ln

ba ln

bb +

dbatb

a

dbbtb

b

e#t = ln " ln(" 1) + lnbabb +ebat ebbtWheree#t =d#t . Now, what is babb ? Note thatt;t+1= 1. Solve for them individually

using the denitions:

bat = (1 +t)mctyt+Ett;t+1(1 +t+1)(1")bat+1ba = mcy +bab

a = mcy

1

bbt = yt+Ett;t+1(1 +t+1)(1")bbt+1bb = y +bbbb = y1

To derive the above Im using the assumption that ination is zero in the steady state.Thus, I have:

babb =mcFrom above, we know that price is equal to a markup over nomina marginal cost. Thus

real marginal cost is equal to the inverse of that markup, or, in the steady state:

mc =" 1

"

This means that:

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ln

babb

= ln(" 1) ln "

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

e#t =ebat ebbtSo now we needebat andebbt. Begin with the rst by rst taking logs:

bat = (1 +t)mctyt+Ett;t+1(1 +t+1)(1")bat+1lnbat = ln(1 +t) + lnmctyt+Ett;t+1(1 +t+1)(1")bat+1

Now do the Taylor series expansion evaluated at the steady state. Before proceeding,note that mcy +Etba

=ba since steady state ination is 0:

lnba +dbatba = ln(1 + 0) +dt+ lnba +dmctyba +dytmcba +::::::+

dt;t+1baba (1 ")dt+1baba +dbat+1baSimplifying, we have:

ebat = dt+ dmcty

ba

+dytmc

ba

+dt;t+1 (1 ")dt+1+ebat+1Leave this alone for a minute. Now go toebbt:

lnbbt = lnyt+Ett;t+1(1 +t+1)(1")bbt+1As above, note that y +Etbb =bb. Proceed with the rst order Taylor series

expansion:

lnbb +dbbtbb = lnbb +dytbb + + dt;t+1bbbb (1 ")dt+1bb

bb +dbbt+1bbNow simplify some:

ebbt = dytbb +dt;t+1 (1 ")dt+1+Qebbt+1We can now rewrite part of this as:

ebat = dt+ ybadmct+ mca dyt+dt;t+1 (1 ")dt+1+ebat+1ebbt = dytbb +dt;t+1 (1 ")dt+1+ebbt+120


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Now subtract the latter from the former:

ebat ebbt = dt+ y

ba

dmct+mc

ba

dyt dyt

bb

+

ebat+1 ebbt+1Note that y

ba

= (1)

mc

and mc

ba

= 1

b

b

. Using these facts, we can write:

ebat ebbt =et+ (1 )fmct+ebat+1 ebbt+1Now note thatet= (1 )ebat ebbt and ebat+1 ebbt+1= Etet+11 :

et = (1 )et+ (1 )(1 )fmct+Etet+1Now solve foret:

et = (1 )(1 )fmct+Etet+1et = (1 )(1 )

fmct+Etet+1The 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 thatspecication, lets start with what denes real marginal cost and then go from there:

mct =wtat

We can substitute out for the wage using the households rst order condition for laborsupply:

wt = ct(1 nt)

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

mct =yt(1 nt)

at

Now lets log-linearize this expression. Begin by taking logs:

ln mct= ln yt+ ln ln(1 nt) ln at

Now do the rst order Taylor series expansion about the steady state:

ln mc +dmct

mc= ln mc +

dyty

+ dnt1 n

dat

a

fmct = eyt+ n1 n

ent eat21


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

1 n(eyt eat) eat

Simplifying:

fmct = + n1 n

eyt 1 + n1 n

eatThe output gap is dened 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 thatfmct= 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 + n1 neat

eyft = 1 + n1n+ n1n eatNote that, if we have log utility over consumption (i.e. = 1), theneyft =eat (i.e.

employment is constant in the exible price equilibrium. Using the above, we can eliminate

eat from the expression for the log deviation of real marginal cost:

fmct = + n1 n

eyt + n1 n

eyftfmct = + n

1 n

eyt eyft Letting =

+ n

1n

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

as:

et =

(1 )(1 )

eyt

eyft

+Et

et+1

Holding expected ination xed, we see that positive output gaps put upward pressureon current ination.

We can also log-linearize the rest of the model. Start with the Euler equation, afterhaving already imposed the accounting identity:

yt =Et(yt+1(1 +rt))

Take logs:

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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 dyt

y= ln ln y +r dy

t+1

y+drt

Deningeyt = dyty andert = 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 lets log-linearize the money supply curve (written in terms of real balances). Itcan be written out as follows:

ln mt ln mt1+t = (1 m) +m(ln mt1 ln mt2) +mt1+em

Since this equation is already in logs and already linear, we can write it exactly the sameway but interpreting the variables as log deviationsemt = dmtm andet = dt:

emt = (1 m)

+

emt1+m(

emt1

emt2)

et+m

et1+em

Now lets log-linearize the money demand function. First take logs:

ln v ln mt = ln yt+ ln it ln(1 +it)

Do the rst order Taylor series expansion:

ln v ln m vdmtm

= ln y + ln i ln(1 +i) dyty

+dit

i

dit1 +i

Simplifying and use the tilde notation:

v emt = eyt+ 1

i

1

1 +ieitSimplifying 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:

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veyt 1

vi(1 +i)

eit = (1 m) +emt1+m(emt1 emt2) et+met1+emSimplify by solving for the current log deviation of output:

eyt = 1i(1 +i)

eit+v

(1 m) +

v

emt1+v

m(emt1 emt2) vet+v met1+v em

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

eyt =

1

i(1 +i)

(

ert+

et+1)+

v

(1m)

+v

emt1+

v

m(

emt1

emt2)

v

et+

v

m

et1+

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

1n

+ n

1n

eat

For notational ease, call= 1+ n

1n

+ n

1n , so:eyft =eatNow plug in this process for technology:

eyft =eat1+etNow we know thateat1 = 1eyft1, so we can write this as:

eyft =

eyft1+et

The full set of log-linearized equations which allow us to solve the model are then:eyt =eyt+1 1ert (26)

eyt = 1i(1 +i)

(ert+ et+1)+ v

(1m)

+v

emt1+ v

m(emt1emt2) vet+ v met1+ v em

(27)

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eyft =eyft1+et (28)

et =

(1 )(1 )

eyt

eyft

+Et

et+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 ination).

It turns out there is a graphical interpretation of this model that is is visually similar towhat one sees in intermediate macro. Holding the values of all future and past variablesxed, as well as the value of current ination, we can plot out the IS and LM curves asfollows:

Recall that the LM curve is drawn holding current ination xed (the IS curve does notdepend on current ination). Eectively what this does is dene an equilibrium level ofoutput and the interest rate for each level of current ination possible. If ination goes up,the LM curve shifts horizontally to the left (i.e. holding the real interest rate xed outputmust fall when ination goes up). The opposite holds when ination goes down. We canthen trace out an aggregate demand curve in (

et;

eyt)space as follows:

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When ination 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, dened for a give value of the exible price level of output and a given expectedfuture ination:

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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 isnt correct. Nevertheless, if shocks are transitory enough, thiswill provide a very good approximation.

Lets 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).

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The increase in money supply shifts the LM curve out this raises the equilibrium levelof output for a given level of ination, shifting the AD curve horizontally. In order to alsobe on the Phillips Curve/AS relationship, ination rises. This means that output rises byless than the horizontal shift in the AD curve. The rise in ination 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 ination 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 ones price, is big. In other words, money supply shocks have a bigger eect

on output (and a smaller eect on ination), 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 eect and just lead to ination.

Now lets 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:

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The outward shift in the AS relationship raises output and lowers ination. The lowerination forces the LM curve outward. At the end of the day, the supply shock leads tohigher output, lower ination, 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 dont formally have that in the model as specied,but would could think of it as a shock to expected future output. We will ignore the factthat this would inuence expected ination in equilibrium, which would in turn shift thePhillips Curve:

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Here the outward shift of the IS curve shifts the AD curve out, which raises both outputand ination. The increase in ination leads to the LM curve shifting back in some so asto restore equilibrium. At the end of the day, output, ination, and the real interest rateare all higher.

The above exercise shows that this dynamic, optimizing model can be thought of in termsvery 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.

Documents

New Keynesian Model[1]