Notas Romer

7/31/2019 Notas Romer

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Romer01a.doc

The Solow Growth Model

Set-up

The Production Function

Assume an aggregate production function:

(( )) (( )) (( )) (( ))[[ ]]tLtAtKFtY ,== (1.1)

Notation:

Y output

K capital

L labor

A effectiveness of labor (productivity)

Technical change is labor-augmenting (also known as Harrod neutral).

The production function exhibits constant returns to scale:

(( ))ALKcFcALcKF ,),( == for all 0c . (1.2)

Setting ALc /1== in (1.2) yields the production function in intensive form:

(( ))ALKFALAL

KF ,

1)1,( == (1.3)

Now define:

AL

Kk==

AL

Yy ==

(( )) (( ))1,kFkf ==


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2

Then (1.3) becomes:

(( ))kfy == (1.4)

So output per effective unit of labor is a function of capital per effective unit of labor.

We further assume that:

(( )) 00 ==f

(( )) 0' >>kf

(( )) 0''


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where the dot notation refers to a time derivative: (( ))(( ))

dt

tdXtX ==& .

The constant growth assumption also permits us to describe paths for L andA by:

(( )) (( )) nteLtL 0==

(( )) (( )) gteAtA 0== .

By definition, output can be divided into consumption and investment. The fraction of

output going to investment is s, which is assumed to be a constant. The constant rate of

saving is a key feature of this model. Capital also depreciates at the rate . Thus the path

of capital must satisfy the equation:

(( )) (( )) (( ))tKtsYtK ==& (1.10)

It is assumed that 0>>++++ gn .

Analysis of the Model

Since ,/ALKk==

(( ))dt

AL

Kd

tk

==&

Using both quotient and product rules for derivatives:

(( ))(( )) (( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( )) (( ))[[ ]]2tLtA

tAdt

tdLtL

dt

tdAtK

dt

tdKtLtA

tk

++

==& (1.11)

(( ))(( ))

(( )) (( ))(( ))

(( )) (( ))(( ))(( ))

(( ))(( )) (( ))

(( ))(( ))tL

tL

tLtA

tK

tA

tA

tLtA

tK

tLtA

tKtk

&&&& == (1.11a)1

Now recall (1.10):

1

This can also be written as:

AA

kL

L

kK

K

kk &&&&

++

++

== .


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(( )) (( )) (( ))tKtsYtK ==& (1.10)

Substituting (1.10) and the given growth rates forA andL into (1.11a):

(( ))(( )) (( ))

(( )) (( )) (( ))tgktnktktLtA

tYstk == )(& (1.12)

(( ))(( )) (( )) (( ))tkgntksftk ++++==)(& (1.13)

Equation (1.13) is the key equation in the Solow model. It describes how the capital stockevolves over time. The first term on the RHS is the amount of investment per worker.

The second term is the amount of investment that would be needed to keep kconstant.

Two diagrams are useful here: Figure 1.2 and Figure 1.3 in Romer. (Note that thesediagrams are drawn to satisfy the Inada conditions).

Balanced Growth Path

If *kk>k& and if *kk>> , then 0


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We will consider changes in the savings rate, the key model parameter. Policymakers

might be able to influence this parameter by changing tax rates or the amount and/orcomposition of government spending.

Suppose the savings rate increases. This causes the (( ))ksf line to shift upward.

Investment exceeds its break-even level, so kbegins to increase and does so until itreaches a new higher level of *k . During the transition, output per worker grows faster

thanA, because of the rise in k. So a permanent increase in the savings rate produces a

temporary increase in the growth of output per worker.

The initial impact of the increase in s is to reduce c. Since output is not initially change,the added saving must reduce consumption. As kgrows,y grows, and c grows. However,

when the new balanced growth path is reached, it is questionable whether c is higherthan before (illustrate via diagram).

The golden-rule level of *k occurs at the savings rate that maximizes c (illustrate via

diagram).

Calibration Experiments

When the savings rate changes, the economy moves to a new balanced growth path. But

how much does a change in s affecty, and how quickly?

With plausible functional forms and parameter values, Romer concludes that:

A significant (10%) increase in the saving rate has a modest (5%) effect on outputalong the balanced growth path.

Following a change in the savings rate, convergence to a new balanced growth

path is slow. Half of the movement toward the new growth path is accomplishedin 18 years.

These results seem to imply that it is difficult to increase an economys standard of living

by way of higher saving.

Implications

The results noted above suggest that variations in s and kwill probably account for little

of the variation in growth and output across countries. Instead, variations in theproductivity parameter,A, must account for most of the variation in output. In the Solow

model, variations inA are not explained.

What is A?

The stock of knowledge?Education of the labor force (human capital)?


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Quality of infrastructure (public capital)?

Institutions regulating and enforcing property rights?

Solow on Growth Accounting

Also see the Solow Handout (in lieu of the discussion on pp. 26-27 in Romer).

Recall the production function:

(( )) (( )) (( )) (( ))[[ ]]tLtAtKFtY ,== (1.1)

Differentiate with respect to time:

(( ))(( ))(( ))

(( ))(( ))(( ))

(( ))(( ))(( ))

(( ))tAtA

tYtL

tL

tYtK

tK

tYtY &&&&

++

++

== , (1.28)

where:

AAL

Y

L

Y

==

and LAL

Y

A

Y

==

.

Divide on both sides by (( ))tY and rearrange to get:

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))

(( ))(( ))tA

tA

tA

tY

tY

tA

tL

tL

tL

tY

tY

tL

tK

tK

tK

tY

tY

tK

tY

tY &&&&

++

++

== (1.29a)

(( ))(( ))

(( ))(( ))(( ))

(( ))(( ))(( ))

(( ))tRtL

tLt

tK

tKt

tY

tYLK ++++==

&&& (1.29b)

Note that the s are elasticities of output with respect to the indicated inputs.

Subtracting (( )) (( ))tLtL /& from each side, and noting that 1==++ LK , we obtain:

(( ))(( ))

(( ))(( ))

(( ))(( ))(( ))

(( ))(( ))

(( ))tRtL

tL

tK

tKt

tL

tL

tY

tYK

++

==

&&&&

The rate of growth of the output/labor ratio depends on the rate of growth in the

capital/labor ratio and a residual, the Solow residual. Everything in the equation above is

fairly easily measured, except for (( ))tR . But that means that (( ))tR can be determined as aresidual. One can use this equation to decompose growth into portions due to changes in

capital per worker and changes due to technical change.


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Empirical Results from Solows paper:

For the period 1909-29: Technical change accounts for 0.90 percentage points of per

capita growth per year.

1930-1949: Technical change accounts for 2.25 percentage points of per capita growthper year.

Technical change accounts for 7/8 of per capita growth; increased capital per worker

accounts for just 1/8.

Technical change appears to be highly variable from year to year.

Economic Convergence

Why should we expect convergence across economies (i.e. economies should besimilar in terms of income, capital, etc.)?

Each country should approach its balanced growth path.Countries with lower capital stocks will have a higher marginal product of capital,

and will attract investment.The ability to emulate best technology should allow convergence toward a

common level for A.

Empirical evidence on the hypothesis of convergence is partly contradictory, but it seemsclear that not all poor countries are in a process of catching up.


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Romer02a.doc

Introduction

We next consider a growth model that differs from that of Solow in an important way.

The savings rate is now endogenously determined as the result of choices made bymaximizing households, rather than exogenously imposed. The savings rate also neednot be constant.

Assumptions

Assumptions About Firms

There are many identical firms.

Each firm produces subject to the CRS production function: (( ))ALKFY ,== .

Firms hire workers and rent capital in competitive markets; they also sell output in acompetitive market.

Firms are profit maximizers.

Profits accrue to households as income.

A grows exogenously at rate g.

Households

There are many households.

The size of each household grows at rate n.

Each member of the household supplies one unit of labor at every point in time. (Thenumber of people is equal to the quantity of labor).

Households rent all capital they own to firms.

Each household has initial capital holdings (( )) HK /0 , where (( ))0K is the initial amount of

capital in the economy andHis the number of households.

There is no depreciation.

Households divide income between consumption and saving in order to maximizelifetime utility, given below:


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(( ))(( ))(( ))

dtH

tLtCueU

t

t

==

==0

(2.1)

Here is an individuals discount rate. Note that Cindicates consumption per member

of the household, while HL/ is the number of members (laborers) per household (i.e.

total labor divided by the number of households). So household utility increases withconsumption per member andwith the number of members.

We assume a particular for the instantaneous utility function u:

(( ))(( ))(( ))

==

1

1tC

tCu , 0>> , (( )) 01 >> gn .

(This is a constant relative risk aversion utility function).

Parameter determines the willingness of a household to shift consumption betweenperiods: a small means a household is more willing to tolerate shifts in consumption

between periods. The assumption that (( )) 01 >> gn insures that the householdcannot obtain infinite lifetime utility.

Behavior

Behavior of Firms

Firms employ capital and labor, pay them their marginal products, and earn zero profits(under perfect competition).

The marginal product of capital is given by (( ))kf' , where (( ))kf is the intensiveproduction function. With no depreciation, the real rate of return on capital (real rate ofinterest) is given by

(( )) (( ))(( ))tkftr '== (2.3)

We can also show that the real wage per effective unit of labor is:

(( )) (( ))(( )) (( )) (( ))(( ))tkftktkftw '== .

To show the last result:

(( ))ALKFY ,==

==AL

KALfY


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Now let *LAL ==

==

*

*

L

KfLY

(( ))2***

*'

L

K

L

KfL

L

Kf

AL

Y

==

(( )) (( ))kkfkfAL

Y'==

Since the marginal product of labor (not effective labor) is (( )) ALALKFA /, , a workerslabor income at time tis (( )) (( ))twtA .

The Households Maximization Problem

Each household takes rand w as given.

Define (( ))tR :

(( )) (( ))

drtRt

==== 0( )tRe serves to discount future flows when the interest rate is not constant over time. This

is a generalization of the case of a constant interest rate, r. In that case multiplication byrt

e

accomplishes the discounting.

To consider why the generalization works, consider a series of short (1-period)intervals. Within each of these short intervals, the interest rate is constant.Repeatedly apply the constant interest rate formula.

Example: At time zero you have $1. At the end of period 1, you havetr

e 11$ . At

the end of two periods you have 211$ rree . At the end of three periods, you have3213211$

rrrrrreeee

++= , etc. For very short periods, the summation is replaced by theintegral.

Since the household has ( ) HtL / members, its labor income is ( ) ( ) ( ) HtLtwtA / and

consumption is ( ) ( ) HtLtC / . The households budget constraint is therefore:

( ) ( )( ) ( ) ( ) ( ) ( )

( )

=

=

+00

0t

tR

t

tRdt

H

tLtwtAe

H

Kdt

H

tLtCe (2.5)


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Define (( ))tc as consumption per effective unit of labor. This can be written as

consumption per person, (( ))tC , divided by the amount of effective labor per person, (( ))tA .

So (( )) (( )) (( ))tAtCtc /== . Similarly, (( ))0k is the initial capital stock per effective unit of labor,so that (( )) (( )) (( )) (( ))00/00 LAKk == .

Using the relationships described above, rewrite (2.5) in terms of consumption and laborincome per effective unit of labor:

(( )) (( ))(( )) (( ))

(( ))(( )) (( )) (( )) (( )) (( ))

(( ))

==

==

++00

000

t

tR

t

tRdt

H

tLtwtAe

H

LAkdt

H

tLtAtce (2.6)

Next, substitute (( )) (( )) (( )) (( )) (( ))tgneLAtLtA ++== 00 into (2.6):

(( )) (( ))(( )) (( )) (( ))

(( ))(( )) (( )) (( )) (( ))

(( )) (( )) (( ))

==

++

++

==

++00

00000

00t

tgntR

tgn

t

tRdt

H

eLAtwe

H

LAkdt

H

eLAtce ,

then divide each side by (( )) (( )) :/00 HLA

(( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

++++

==

++00

0t

tgntRtgn

t

tRdtetwekdtetce . (2.7)

Romer also shows that the budget constraint can be written in an alternative way:

(( )) (( )) .0lim H

sKe

sR

s

It is also useful to rewrite the households objective function in terms of consumption per

effective unit of labor. Recall that (( )) (( )) (( ))tAtCtc /== . We can rewrite the utility function:

(( )) (( )) (( ))[[ ]]

==

11

11tctAtC

(( )) (( )) (( ))[[ ]]

==

1

0

1

11tceAtC

gt

(( )) (( )) (( )) (( ))

==

10

1

111

1 tceA

tC gt(2.13)

Recall the households objective function:

(( ))(( ))(( ))

dtH

tLtCueU

t

t

==

==0

(2.1)


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Substitute (2.13) and (( )) (( )) nteLtL 0== into (2.1):

(( )) (( ))(( )) (( ))

dtH

eLtceAeU

ntgt

t

t 0

10

1

11

0

==

==

(( ))(( )) (( )) (( ))

dttc

eeeH

LAU

ntgt

t

t

==

==

10

01

1

0

1

(( ))dt

tceBU

t

t

==

==

11

0(2.14)

where:

(( )) (( ))H

LAB 00 1 ==

and

(( ))gn == 1 .


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Romer02b.doc

Household Behavior: the Maximization Problem

The households problem is to choose a path for (( ))tc to maximize lifetime utility subject

to the budget constraint:

Recall the objective function:

(( ))dt

tceBU

t

t

==

==

11

0(2.14)

Also recall the budget constraint:

(( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

++++

==

++00

0t

tgntRtgn

t

tRdtetwekdtetce . (2.7)

Form the Lagrangean:

(( ))(( )) (( )) (( )) (( )) (( )) (( )) (( ))

++++

== ++

==

==

++

==

dtetcedtetwekdttc

eBLtgn

t

tR

t

tgntR

t

t

00

1

00

1*

The first order condition for an individual (( ))tc is:1

(( )) (( )) (( ))tgntRt eetcBe ++ == (2.16)

Take logs on each side:

(( )) (( )) (( ))tgntRtctB ++++== lnlnln

Differentiate with respect to ton each side:

(( ))(( )) (( )) (( ))gntrtctc

++++==

&

(In taking the derivative above, recall that (( )) (( ))

drtRt

==== 0 0)).

1

As Romer notes, this step is somewhat informal. See Romer, p. 44, footnote 7.


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Finally, solve for(( ))(( ))tctc&

to get:

(( ))(( ))

(( ))

==

gntr

tc

tc&

or, using (( ))gn == 1 :

(( ))(( ))

(( ))

gtr

tc

tc ==

&(2.19)

Equation (2.19) is theEuler equation for this problem.

The Euler equation describes how consumption must behave over time for a given (( ))0c ;(( ))0c must be chosen so as to satisfy the budget constraint.

Essentially, optimality requires that it not be possible for a consumer to gain by way of

making a small shift of consumption from one time period to another, while satisfying the

budget constraint. The Euler equation is an implication of that optimality requirement

Dynamics of the Economy

Dynamics for c

Since all households are the same, condition 2.19 holds for the entire economy, not just a

single household. All firms are identical also under our assumptions the intensive

production function for a single firm is identical to that for the entire economy.

Recall that (( )) (( ))(( ))tkftr '== and rewrite (2.19):

(( ))(( ))

(( ))(( ))

gtkf

tc

tc ==

'&(2.22)

By (2.22). we know that 0==c& when (( )) gkf ++==' . Let *k denote the capital stocksuch that the latter condition holds.

When *kk>> , (( )) gkf ++


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Dynamics for K

Capital accumulation is governed by an equation similar to that in the Solow model

(except that the savings rate need not be constant and depreciation has been assumed to

equal zero):

(( )) (( ))(( )) (( )) (( )) (( ))tkgntctkftk ++==& (2.23)

Recall Figure 1.6, Romer, p. 19. For any value ofk, k& will be equal to zero so long as

consumption is equal to the vertical distance between the (( ))kf curve and the (( ))kgn ++line (remember that deprecation is zero). This is so because actual saving is equal to

break-even saving.

So the 0==k& locus will have an appearance like that shown in figure 2.2, Romer, p. 47.

Above this curve, consumption is higher, hence saving is lower, and .0k&

See Figure 2.3, Romer p. 48, for a diagram summarizing the dynamics of both c and k.

Point E indicates a balanced growth path, with both c& and k& equal to zero.

Note that the balanced growth path is to the left of the golden rule level for k. This

must be true, as is argued below:

Recall that in this model 0==c& when (( )) gkf ++==*' (again see (2.22)). At thegolden rule point, (( )) gnkf

GR++==' . Concavity of the intensive production

function implies that if GRkk >>* , then

gng ++


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We conclude that it is not the case thatGR

kk >>* ; insteadGR

kk there will similarly be a unique

path leading back to the balance growth path.


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Romer02c.doc

Balanced Growth in the Ramsey Neoclassical Growth Model

Welfare

Diagram 2.5 in Romer illustrates movement toward the balanced growth path.

Convergence to this point and the subsequent balanced growth path are optimal. A socialplanner would choose the same path as utility maximizing individuals competitivemarket outcomes are optimal in the absence of externalities, etc.

Properties of the Balanced Growth Path

Along the balanced growth path, consumption and capital per effective unit of labor areconstant (as the diagram directly shows). If the capital to effective labor ratio is constant,

then output per effective unit of labor must also be constant. Sincey and c are constant,the savings rate (( )) ycys /== will also be constant. This implies that on the balancedgrowth path, this model economy looks no different than the Solow economy.Approaching the balanced growth path, the savings rate is not constant, however.

In the Ramsey model, the capital stock is less than the golden rule capital stock (thatwould maximize consumption per effective unit of labor). Recall that we earlier showed

that GRkk


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Skip the section on the Rate of Adjustment and the Slope of the Saddle Path

Government Purchases

Assume that the government buys goods at the rate ( )tG per unit of effective labor, and

that government purchases do not affect the utility of private good consumption or futureoutput. Government spending is financed by lump-sum taxes. Initially assume that

current spending must be financed by current taxes,; i.e., (( )) (( ))tGtT == . Here (( ))tT is taxesper effective unit of labor.

Equation (2.23) now becomes:

( ) ( )( ) ( ) ( ) ( ) ( )tkgntGtctkftk +=& (2.38)

The Euler equation will hold as it did before (it was derived without using either 2.23 or

the budget constraint). The budget constraint (see equation (2.9)) is changed however. Itbecomes:

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )

=

++

=

+00

0t

tgntRtgn

t

tRdtetGtwekdtetce . (2.39)

Suppose that the economy initially is on a balanced growth path with ( )L

GtG = . Then

suppose that ( )tG suddenly and permanently increases to ( ) HGtG = . Governmentspending does not affect the Euler equation, from which the 0=c& locus was derived. Sothe 0=c& locus is unchanged. At any level ofk, c is now lower by exactly the amount of

the increase in G on the 0=k& locus. Initial consumption must adjust to get the economy

back on a path towards balanced growth in this case, consumption simply falls by theamount of the increase in G, and we are immediately back on the balanced growth path.Since kis unchanged, the marginal product of capital and the real interest rate are alsounchanged.

Now suppose that instead of a permanent increase in G, there is a sudden temporaryincrease in G (of known duration). As we have noted before, the 0=c& locus is

unchanged, but the 0=k& locus falls. Ifc fell by the full amount of the change in G, thenwe would stay at that point until G reverted to its lower level, at which time it would haveto shift back up. But the discontinuous upward shift cannot be optimal there can be noanticipated discontinuity in consumption because it would imply a discontinuity inmarginal utilities (which would lead to a desired reallocation of consumption from oneperiod to the next).

Instead, c will fall by less than the full amount of the change in G. At this point, the

economy is above the 0=k& line, so kfalls. As kfalls, we move to the left of the 0=c&locus, so c rises (we are moving northwest in Fig. 2.9a). When G returns to its normal,


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low level we must be back on the path of convergence to the original balanced growthpath.

So, a temporary increase in G causes a downward jump in consumption (less than theincrease in G). While the higher G is in effect, capital is decumulating. Consumption

immediately begins a gradual return to its original level, however. When G reverts to itsnormal level, consumption is still below its balanced growth path level, and capitalbegins to accumulate.

During the period of high G, kis declining, so the marginal product of capital isincreasing, as is the interest rate. Once G returns to normal, kbegins to increase, and theinterest rate falls. We would therefore expect high interest rates in wartime (whengovernment spending is temporarily higher than normal).

Bond and Tax Finance

The Government Budget Constraint

We will now permit the possibility that a government might finance current spendingeither with taxes or bonds (borrowing). The governments budget constraint says that thepresent value of its spending stream must be less than or equal to initial governmentwealth plus the present value of its tax revenues. We will assume that the constraint holdsas an equality. So the budget constraint can be written as:

(( )) (( )) (( )) (( )) (( ))[[ ]] (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

==

++++ ++==0 0

0000000t t

tgntRtgntRdtLAetTeLAbdtLAetGe

In the above equation, (( ))0b is the initital level of the outstanding stock of governmentdebt per effective unit of labor. This represents negative wealth for the government.

Dividing by (( )) (( ))00 LA on each side yields:

(( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

==

++++ ++==0 0

0t t

tgntRtgntR dtetTebdtetGe (2.42)

Romer notes that the budget constraint (as an equality) could also be written as:

(( )) (( ))

(( )) 0lim==++

sbee

sgnsR

s

This says that in the limit, the present value of the governments outstanding debt mustapproach zero. (The debt itself could converge to a positive number, but the present valuemust go to zero). This implies that if I buy a bond, I expect a stream of future paymentsfrom the government equal in present value to what I paid for the bond. (If this were nottrue, I would not buy the bond).


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Households also have a budget constraint. Modify equation (2.7) to include taxes:

(( )) (( )) (( )) (( )) (( )) (( )) (( )) (( ))[[ ]] (( ))

==

++++

==

++++00

00t

tgntRtgn

t

tRdtetTtwebkdtetce (2.44)

Now rewrite (2.42), isolating the present value of taxes on one side:

(( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

++

==

++ ++==00

0t

tgntR

t

tgntRdtetGebdtetTe (2.42)

Substitute into (2.44) to get:

(( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

++

==

++++

==

++++0

00000

t

tgntR

t

tgntRtgn

t

tR dtetGebdtetwebkdtetce

or, canceling the (( ))0b terms,

(( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( )) (( ))

==

++

==

++++

==

++0

000

t

tgntR

t

tgntRtgn

t

tRdtetGedtetwekdtetce (2.45)

Notice that government spending enters the budget constraint above, but the quantity ofbonds at any time does not. In fact, the latter equation is identical to (2.39), which wasthe budget constraint in the case where the government never issued bonds.

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )

=

++

=

+00

0t

tgntRtgn

t

tRdtetGtwekdtetce . (2.39)

The implication is that only the path of government purchases, and not the path of thetaxes that finance those purchases, affects the economy. This is theRicardianEquivalence Theorem.

Ricardian Equivalence

Consider the intuition for the result noted above. For a given government spending plan,a government can finance todays spending (say $100) by collecting taxes or issuingbonds. Suppose that the government levies no current tax, but instead issues $100 of

bonds. This implies that households now face a future tax liability which has a presentvalue equal to $100 -- bondholders must be paid a future stream equal in present value tothe amount they pay for the bonds. Households see no change in wealth they are freedof the need to pay $100 in taxes, but they are simultaneously burdened with a futureliability of equal present value. Households also have no reason to alter theirconsumption plan when a tax is replaced by bond financing a household can maintainconsumption as before. The funds freed up by the reduction in taxes can be used to buy abond. When higher taxes must be paid in the future, returns from that bond will provide


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exactly the funds required to pay the taxes. So the choice of bond versus tax finance hasno affect on the real economy.

Clearly, this result is contrary to the implications of conventional Keynesian models,which imply that the decisions to alter taxes will affect aggregated demand, output

consumption, interest rates, etc.

Does Ricardian Equivalence Hold?

Our model has made some strong assumptions, which might affect the Ricardianequivalence result:

Infinitely lived individuals. If individuals can die, they can avoid future taxes.However, if individuals care enough about descendants to leave bequests, thegovernments financing decisions are again irrelevant. The government may cutmy taxes, which apparently redistributes from my child to me. But if I am

planning to leave a given bequest to my child, I can still do so. I can just by abond with my current tax savings, and leave the bond to my child when I die.Proceeds from the bond will just pay his added future tax liability.

Even if individuals are not so calculating, it will often be the case that quantitativedeviations from Ricardian equivalence will be small. Suppose that thegovernment gives me a one-time tax break of $20,000. Even if I believe that thisis wealth (I dont worry about the future tax burden), I am not likely to spend it allat once. The optimal consumption plan will typically result that I increaseconsumption slightly in many future periods. So a big tax cut will not have bigimpacts on current spending.

Liquidity Constraints.

When the government cuts your taxes today (which you must pay back in highertaxes in the future), they are giving you a loan. Our model supposes thatindividuals already had the option to borrow at the prevailing interest rate, so thegovernments beneficence is not appreciated.

In the real world, most individuals cannot borrow at the same interest rate as thegovernment. Given an opportunity to borrow at a lower rate, I might borrowmore, so my consumption is affected by the governments decision to tax orborrow for me.

There are two counter-arguments to favor the Ricardian position. First, if myborrowing rate were lower, it is not obvious that I would borrow a lot more(enough to give a notable Keynesian kick to the economy).

Second, liquidity constraints are not exogenous. When the government borrowsmore on your behalf, private lenders may become less likely to lend to you


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6

(because of your future tax liability you are seen as a riskier prospect) so that theimpacts of the initial government action are reduced.

Taxes are Not Lump-Sum

When taxes are not lump-sum, there will clearly be some real world effectsassociated with changing the time path of tax collections. E.g., people willreallocate work effort to avoid taxes.

One effect of imposing income taxes instead of lump sum taxes is that uncertaintyabout future (after-tax) income is reduced, which may lead to higher currentconsumption.

Imposing an income tax may also affect the importance of liquidity constrainteffects. My riskiness as a potential borrower is lessened under an income tax. I.e.,if my income turns out to be low, my future tax burden will also be low so the

future tax liability will not have a big negative impact on my ability to borrow inthe private sector today.

Non-Optimizing Behavior

Traditional Keynesian models did not posit optimizing behavior theconsumption function could instead reflect rule-of-thumb behavior. Individualsnaively ignore future tax liabilities and increase consumption in response to a taxcut.


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1

Romer04_Notation.doc

Y Aggregate Output

C,I, G, Aggregate Consumption, Investment, Government Spending

Depreciation rate

U Utility (grand utility function)

w Real wage

r Real rate of interest

Utility function discounting parameter

N Population (grows at rate n)

H Number of Households (constant)

cN

Cc = (Not

AN

C)

lN

Ll =


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2

Romer04a.doc

Business Cycle Facts

Business cycles do not exhibit a simple, regular, cyclical pattern. Both amplitude and

duration of cycles are irregular.

Fluctuations in output are distributed unevenly over its components.

Inventory investment is a very small fraction of total output, but it accounts for1/3 of the shortfall in output in recessions. Investment in general is has highamplitude and is procyclical.

Over rather long periods, output tends to be slightly above its trend path; over rather short

periods, output tends to be sharply below its trend path.

Fluctuations before the Great Depression and after WWII are similar in character.

Cycles of the magnitude experienced in the Great Depression in the U.S. are unparalleled

in the remainder of the historical record.

Some evidence on co-movements. In a recession:

Employment falls.

Unemployment rises.Length of average workweek falls.Declines in employment and hours are small relative to the decline in output (so

measured productivity falls).There is disagreement about the behavior of inflation over the business cycle.

The real wage falls slightly.Nominal interest rates and the real money supply fall in recessions.

Theories of Fluctuations

We have already studied Keynesian, New Classical, and New Keynesian Models ofFluctuations.

Oddly, these models do not seem to depart directly from the neoclassical growth model(in which individuals maximize utility over long time horizons, firms maximize profits,

markets are competitive, and market imperfections are absent).

We will now study real business cycle models, which do take neoclassical growth theory

as a point of departure.

How will the Ramsey model be modified?


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3

We must have random shocks hit the economy (or else we will approach abalanced growth path). We will emphasize real shocks: shifts to technology

(hence real business cycle models).

We must also permit labor supply to be endogenous (to reflect the fact that labor

is procyclical).

Although real shocks may not turn out to provide the best explanation of business cycles,we should certainly consider the possibility that they do (before assuming market

imperfections, incomplete nominal adjustment, etc.).

A Baseline RBC Model

Assumptions

Time is discrete.

There are many identical price-taking firms and many identical price-taking households.

Households are infinitely lived.

Inputs into production are capital and labor.

The production function is Cobb-Douglas:

( ) = 1tttt LAKY , .10


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4

( ) ttt

t

t ALA

Kw

= 1 (4.3)

and

=

1

t

tt

tK

LAr (4.4)

A representative household maximizes the expected value of :

( )

=

=0

1,t

ttt

t

H

NlcueU

(4.5)

Here, tN is the population, NCc /= and NLl /= .L represents labor supplied. Eachindividual is endowed with 1 unit of time per period, so l1 is leisure per person.

Population grows exogeously at rate b . (4.7)

Finally, we need to describe processes for two shock variables, G andA:

tt AgtAA~

ln ++= (4.8)

tAtAt AA ,1~~

+= 11


27/72

5

Begin at Romer p. 156 on Household Optimization under Uncertainty

Household Optimization under Uncertainty

In our model, a household does not have information about future rates of return or future

wages. At any point in time, an individual makes optimal decisions given informationcurrently available.

We follow Romers informal approach to derive first order conditions for our

optimization problem.

Consumption

Consider a household in period t. Suppose that the household reduces consumption per

member by a small amount c and uses the added wealth to increase consumption permember in period 1+t . Optimality requires that such a change leave expected utility

unchanged. Recall the utility function described earlier in equations (4.5) and (4.7):

( )

=

=0

1,t

ttt

t

H

NlcueU

(4.5)

( )ttt lbcu += 1lnln , 0>b . (4.7)

In the equation above, c is defined by NCc /= and l is defined by NLl /= .

From these equations, we find that the marginal utility of tc is:

( )( )ttt

t cHNecU /1//=

The utility cost of a small discrete change c is then ( )( )ttt

ccHNe // .

The total consumption given up by the household in period tis ( )HNc t/ . This permitsan increase in total consumption for the household next period of )( )11/ ++ tt rHNc

1.

The added consumption per member next period is then

( )( ) ( ) ( ) )( ) ( ) ( ) ) ( ) creeNreNcHNrHNc tntn

t

nt

ttt +=+=+ ++

+++ 11

111 10/10//1/ .

From (4.5) The marginal utility of in consumption per member in period 1+t is:

( ) ( )( )111

1 /1// +++

+ = ttt

t cHNecU .

1 If I save today, my return comes via the marginal product of capital next period,which

must equal the interest rate over that period, which is denoted 1+tr .


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6

So, the expected utility gain of the added consumption per member is:

( ) ( )( ) ( ) crecHNeE tn

tt

t

t + +

+++

111

1 1/1/

or

( ) ( ) ( )[ ] ccreHNeE ttn

t

t

t + ++

++

111

1/1/

.

Equating the marginal gains and losses from this shift of c yields:

( ) ( ) crc

eH

NeE

c

c

H

Ne t

t

ntt

t

t

tt

+

=

+

+

++1

1

11 11

(4.22)

or, since ( ) ( ) ntt

eHNe

++ /1

1 is not random and since ntt eNN =+1

( )

+= +

+

1

1

111

t

t

t

t

rc

Eec

(4.23)

This is analogous to the Euler equation in the Ramsey model.

Equation (4.23) can again be rewritten, using the formula for the expected value of aproduct:

[ ]

+++=

++

++

11

1

1

1,1Cov111 tt

tt

t

t

t

rc

rEc

Eec

(4.24)

Labor Supply

Each household must also make a labor supply decision in each period.

Optimality will require that the marginal utility of working (to gain additionalconsumption) be equal to the marginal utility from leisure. Equivalently, a small

increase in labor supplied should leave utility unchanged.

Again recall the utility function:

( )

=

=0

1,t

ttt

t

H

NlcueU

(4.5)


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7

( )ttt lbcu += 1lnln , 0>b . (4.7)

The marginal disutility of working is given by:

t

tt

l

b

H

Nel

U

=

1

,

and the loss associated with a small increase l is then:

ll

b

H

Ne

t

tt

1

.

By working more, one gains additional income and consumption. The added

consumption per worker is given by tt lw , and gives an added utility gain of:

( )( ) ttttt

lwcHNe /1/

Equating the marginal utility gain with the marginal utility loss yields:

tt

t

tt

t

tt lwcH

Nel

l

b

H

Ne =

1

1

(4.25)

or

b

w

l

c t

t

t =1

(4.26)

Equations (4.23) and (4.26) are the key equations of the model.

Skip to section 4.6

Romer04c.doc

Start on Romer p. 164

Solving the Model

The RBC model we have been developing, like most RBC models, cannot be solved

analytically.

We will instead describe the solution to a log- linear approximation of the model.


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8

The Log-Linearized Model

In any period, the state of the economy can be described by level of the inherited capitalstock and current values of government spending (G) and technology (A).

If we log- linearize around around the non-stochastic blanced growth path, solutions forconsumption and labor supply must be given by:

tCGtCAtCKt GaAaKaC~~~~

++ (4.43)

and

tLGtLAtLKt GaAaKaL~~~~

++ (4.44)

where X~

represents the difference between the log ofXand the log of its balanced

growth path level.

To solve the model, we must find values for the as.

It turns out that the key equations, 4.23 and 4.26 impose restrictions that permit us toidentify the a's, hence solve the model.

Intratemporal (Current Consumpt ion vs. Current Leisure) First Order Condition

Recall (4.26):

bw

lc t

t

t =1(4.26)

Also recall (4.3):

( ) ttt

t

t ALA

Kw

= 1 (4.3)

Substitute (4.3) into (4.26) and take logs:

( )

b

ALA

K

l

ct

tt

t

t

t

=

1

1

( ) ( ) ttttt LKAb

lc lnlnln11

ln1lnln

++

= (4.45)


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9

The deviation of the actual value of the RHS of this equation from its balanced growth

path level is simply ( ) ttt LKA~~~

1 + .

On the LHS we will approximate using a first-order Taylors series approximation. 2Think of the LHS as a function of tcln and tlln ; i.e.:

( )tt lcz = 1lnln

( )tlt eczln

1lnln =

Taking the Taylors series approximation around the balanced growth path values of tcln

and tlln :

tt ll

lczz

~

1

~*

*

0

+

Because population is not affected by shocks, tt cC~~ = and tt Ll

~~= and we can rewrite the

ezpression for the LHS of (4.45) as a deviation from its balanced growth path level as:

tt Ll

lCzz

~

1

~*

*

0

+

To see that tt cC~~ = recall:

N

Cc =

so:

NCc lnlnln =

2

For ( )yxfz ,= ,

( )( )

( )( )

( )000

000

00

,,, yy

y

yxfxx

x

yxfyxfz

+

+

or

( )( )

( )( )0

000

000

,,yy

y

yxfxx

x

yxfzz

+

=


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10

Letting asterisks indicated balanced growth path levels:

*** lnlnln NCc =

Differencing the last two equations and noting that *NN= :

Cc~~ =

By equating the expressions derived for the LHS and RHS of (4.45) as deviations frombalanced growth path values, we have:

( )ttttt LKAL

l

lC

~ln

~ln

~ln1

~

1

~*

*

+=

+ (4.46)

romer04d.doc

Recall

( )ttttt LKAL

l

lC

~ln

~ln

~ln1

~

1

~*

*

+=

+ (4.46)

Furthermore, recall that we have conjectured solutions:

tCGtCAtCKt GaAaKaC~~~~ ++ (4.43)

and

tLGtLAtLKt GaAaKaL~~~~ ++ (4.44)

Substituting (4.43) and (4.44) into (4.46) yields:

( )GaAaKal

lGaAaKa LGtLAtLKtCGtCAtCK

~~~

1

~~~*

*

++

+

+++

( ) tt KA~

ln~

ln1 += (4.47)

Equation (4.470 must hold for all values of K~

, A~

, and G~

. This implies that coefficientsfor each of these variables must be identical on the two sides of this equation, leading to

three equations which impose restrictions on the s' . See equations (4.48)-(4.50) inRomer.


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11

The Intertemporal First-Order Condition

Our strategy here will be to use the intermporal first order condition to get three morerestrictions on the s. We outline the procedure for doing so.

Recall the intertemporal first -order condition:

( )

+= +

+

1

1

111

t

t

t

t

rc

Eec

(4.23)

Define the bracketed expression above as 1+tZ , i.e.:

( )

+= +

++ 1

1

1 11

t

t

t rc

Z

Let 1~

+tZ be the difference between the log of 1+tZ and the log of its balanced growth path

value.

Recall our conjectured solution:

tCGtCAtCKt GaAaKaC~~~~

++ (4.43)

Update this equation to time t+1:

1111

~~~~

++++ ++ tCGtCAtCKt GaAaKaC (4.51)

Recall (4.4), now updated one period:

=

+

+++

1

1

11

1

t

tt

tK

LAr (4.4)

We can substitute (4.4) into the definition of 1+tZ given above. Also substitute the

definition 111 / +++ = ttt NCc .

This would give us 1+tZ as a function of 1+tA , 1+tK , 1+tC , and 1+tL ( 1+tN is

predetermined).

Take logs and use a Taylors series expansion to express 1~

+tZ as a function of 1~

+tK , 1~

+tA ,

and 1~

+tC and 1~

+tL .


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12

Substitute the conjectured solutions (4.43 and 4.44) for 1~

+tL and 1~

+tC .

Recall (4.43) and (4.44):

tCGtCAtCKt GaAaKaC ~~~~ ++ (4.43)

and

tLGtLAtLKt GaAaKaL~~~~

++ (4.44)

Substituting updated versions of (4.43) and (4.44) would give 1~

+tZ as a function of 1~

+tK ,

1

~+tA , and 1

~+tG .

To get rid of the endogenous 1~

+tK term, recall equation (4.2):

tttttt KGCYKK +=+1 (4.2)

Use the production function to eliminate tY , then log- linearize equation (4.2) and write

1

~+tK as a function of tK

~, tA

~, tG

~, tL

~and tC

~. Use (4.43) and (4.44) to substitute for tL

~

and tC~

.

We then have 1~

+tK as a function of tA~

, tG~

, and tK~

:

tKGtKAtKKt GbAbKbK~~~~

1 +++ (4.52)

Next, we can then get rid of the 1~

+tK term in the 1~

+tZ expression, so 1~

+tZ is now a

function of tK~

, tA~

, tG~

, 1~

+tA , and 1~

+tG .

Finally, use this expression to calculate [ ]1~

+tt ZE . The 1~

+tA and 1~

+tG terms drop out when

we take expectations, so we will have [ ]1~

+tt ZE as a function of tK~

, tA~

, and tG~

.

Finally, recall the intertemporal first order condition, (4.23):

[ ]11

+= tt

t

ZEec

(4.23)


35/72

13

On each side, we wish to take logs, and then and express as deviations from balanced

growth path levels. First substituting ttt NCc /= , the LHS will become a linear function

of tC~

, which we will replace with our conjectured solution, tCGtCAtCKt GaAaKaC~~~~

++ .

On the RHS assume that [ ] BZEZEtttt += ++ 11

lnln , whereB is a constant. Romer

provides assumptions for which this will be true. 3 Given this, when we express the RHS

as a deviation from the balanced growth path level, we get [ ]1~

+tt ZE , which we have

already found is a function of tK~

, tA~

, and tG~

.

Equating coefficients on the LHS and RHS of the modified (4.23) will provide 3additional restrictions on the s. We now have 6 linear restrictions (we had three othersfrom the other first order condition) and can solve for the 6 s, giving us anapproximate solution to the model. Thereafter we can use our approximate solution toinvestigate the properties of the model.

romer04e.doc

Start on Romer p. 168 on Implications

Introduction

Given our approximate solution for the model, we can calculate tC~

and tL~

(and other

variables) when given values for tK~

, tA~

, and tG~

. This permits us to calculate paths for

the models variables following shocks to technology or government spending.

One generally begins by selecting baseline values for the model parameters, solving forthe s in our solution, and then tracing the impacts of shocks. Romer selects plausibleparameter values based on empirical evidence.

Technology Shock

Let 95.0== GA . Then consider the impact of a positive 1% shock to technology.The qualitative effects of the shock are:

Capital gradually accumulates and then returns to normal:

The marginal product of capital is higher because of the shock.

Higher output and a desire to spread out consumption leads to more saving (andinvestment) initially.Eventually the shock dissipates, and we must return to the balanced growth path.

3For more on this see Lindgren, Statistical Theory, p. 176.


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14

Labor supply jumps, then gradually falls, goes below the balanced growth path, and thenreturns to the balanced growth path:

High marginal product of capital (hence interest rate) and high marginal product

of labor both induce high current labor supply after the shock (the interest rate

works via an intertemporal substitution effect).As the shock dissipates, we are left with an above balanced growth path capital

stock (i.e. higher wealth than on the balanced growth path).As the capital stock is permitted to return to normal, we enjoy both more

consumption and leisure (relative to the balance growth path), hence below pathlabor supply.

Output increases in the period of the shock, then returns gradually to the balanced growthpath.

The shock and intertemporal work effort effects both lead to an initial increase in

output.The effect persists because of the persistence of the original shock and theaccumulation of capital in the initial periods (higher consumption is spread over a

long horizon).

Consumption rises more slowly than output, then gradually returns to normal:

This reflects the consumption smoothing motive.

The wage rises and then gradually returns to normal:

The shock directly increases the marginal product of labor and then dissipates.Capital accumulation also contributes to the rising marginal product of labor

initially.

The interest rate immediately rises, then gradually falls below the balanced growth path

level, before returning to it:

Initially, the shock increases the marginal product of capital. When the shockbegins to dissipate and labor supplied is reduced, the capital stock is high relativeto the amount of effective labor, and the marginal product of capital is low. As the

capital stock returns to its balanced growth path level, so does the marginalproduct of capital (and the interest rate).

A Less Persistent Technology Shock

If the A parameter is smaller, technology shocks are less persistent. This implies that

wealth effects of the shocks will be smaller, and substitution effects larger. We have

shorter, sharper output fluctuations (the period of the shock is now an especially good


37/72

15

time to work, but consumption will not be affected greatly). In contrast, if 1=A ,technology shocks are permanent; and the output burst is initially smaller but sustained.

Changes in Government Spending

Other things equal, a positive government spending shock reduces output available forother uses. Output is scarce in the period of the shock. The desire to maintainconsumption (spread out the effects of the shock) leads to capital decumulation and a risein the interest rate. The increase in the interest rate induces more work effort in the initial

periods of the shock. The wealth effect also induces higher work effort and lowerconsumption (i.e., lower wealth reduces both consumption and leisure). The wage

declines at the time of the shock (with more labor working, the marginal product islower).


38/72

1

Romer Chapter 6 on the Lucas Model

The Case of Perfect Information

Producer Behavior

There are many different goods.

A representative producer of a typical good, good i, produces according to the productionfunction:

iiLQ == (6.1)

where iL is the amount the individual works and iQ is the amount he produces.

Real consumption, is nominal income divided by a price index:

P

QP

Cii

i == .

Utility depends positively on consumption and negatively on hours worked:

1,1

>>==

iii LCU (6.2)

Substituting the previous equations into (6.2) gives:

i

ii

i

LP

LPU

1

==. (6.3)

Taking prices as given, an individual maximizes utility by selecting iL to satisfy the first

order condition:

01 == i

i LP

P. (6.4)

Rearranging, we get:

11

==

P

PL ii (6.5)

Letting lowercase letters denote logs:

(( ))pplii

==

1

1

(6.6)


39/72

2

This is a labor supply function (and, indirectly, an output supply function) in which anindividuals hours depend on the relative price of the individuals output price. Note that

this supply function does not include inertial effects like those in the Lucas paper we readearlier.

Note that if PPi == , then 0,0,1,1 ======== iiii qlQL (the utility function was designed sothat this would be the result).

Demand

We assume that the demand for good i has the following form (note: this is NOT derived

from a utility maximization problem):

(( )) 0, >>++== ppzyq iii , (6.7)

wherey is the log of a measure of aggregate income, iz is a shock to demand for good i

(with mean zero across goods), and is the demand elasticity. More specifically,y is

defined to be the average of the sqi' across goods, andp is defined to be the average of

the spi' across goods:

iqy == (6.8)

and

ipp == . (6.9)

Aggregate demand is given by:

pmy == (6.10)

This is just a simple way of modeling aggregate demand; the essential property is that the

price level and output are inversely related. While m can be literally interpreted as the logmoney supply, it might be thought of more generally as any aggregate demand shifter.

Equilibrium

We require that quantities demanded equal quantities supplied in each market i. From

(6.6) and (6.7) we obtain:

(( )) (( ))ppzyp iii ++==

1

1. (6.11)


40/72

3

Solving fori

p yields:

(( )) pzyp ii ++++++

==

1

1(6.12)

Next, average the left- and right-hand sides of (6.12):

pyp ++++

==

1

1(6.13)

Solve fory:

.0==y (6.14)

Recall equation (6.10):

pmy == (6.10)

If 0==y , then (6.10) implies:

mp == (6.15)

Thus money is neutral in this model. An increase in m leads to a proportional increase inp. Also, sincep is observable and markets clear, the average level of log output is zero.

An increase in aggregate demand does NOT lead to higher aggregate output in the perfectinformation version of the model.

Imperfect Information

Producer Behavior

We now consider the case where producers observe the price of the good they produce,

but not the aggregate price level.

Define the relative price of good i as ppr ii == , we get:

ii

ii

rpppppp

++== ++==)( (6.16)

Individuals supply choices are motivated by relative prices, but relative prices are not

observed; ip is observed, but the individual must make a forecast regarding ir.


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4

We assume that individuals calculate the expectation ofi

r giveni

p , and then act as if

this expected value were known with certainty (we implicitly have been making this

assumption ofcertainty equivalence in all of our work with rational expectations). Thisimplies that with uncertainty, equation (6.6) is modified to give:

[[ ]]iii prEl |

11

==

(6.17)

We must next describe the process generating values for m. We assume that m is

normally distributed with mean [[ ]]mE and variance mV (this is a bit different and moregeneral than the demand process specified in the paper by Lucas that was assigned

earlier). We also assume that the szi' , which we earlier assumed had mean 0, are

normally distributed, and that thei

z and m shocks are independent.

We will next invoke our solution to the signal extraction problem. We wish to forecasti

r

using our knowledge ofi

p . Recall thatii

rpp ++== ; i.e. we observe a sum, but wish toforecast a component of the sum. Writing our synthetic regression forecast in a formwhere variables are expressed as deviations from means we get:

[[ ]] [[ ]](( ))pEpVV

VprE

i

pr

r

ii

++==| , (6.19)

wherer

V is the variance ofi

r and pV is the variance ofp . Use of this formula requires

that ir andp be independent normal variables. At the moment we have not derived

expressions for rV and pV , but we will be able to do so eventually.

Note here that the sri' have unconditional mean zero, and the sp

i' have unconditional

mean [[ ]]pE .


[[ ]]iii

prEl |1

1

==

(6.17)

Substituting (6.19) into (6.17) yields:

[[ ]](( ))pEpVV

Vl

i

pr

ri

++

==1

1

(6.20a)

or


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5

[[ ]](( ))pEpblii . (6.20b)

Averaging across producers yields:

[[ ]](( ))pEpby == (6.21)

This is the Lucas supply function.

Equilibrium

Now combine aggregate demand, equation (6.10) and the Lucas aggregate supply curve,

(6.21):

[[ ]](( ))pEpbpm == .

Solve forp:

[[ ]]pEb

bm

bp

++++

++==

11

1. (6.22)

We know how to solve from this point on using the method of undetermined coefficients.However, Romer uses a trick that often (but not always) lets us solve rational

expectations models more quickly. Take expectations on both sides of (6.22):

[[ ]] [[ ]] [[ ]]pEb

bmE

bpE

++++

++==

11

1(6.24)

This equation can be solved for [[ ]]pE :

[[ ]] [[ ]]mEpE == . (6.25)

Substituting into (6.22):

[[ ]]mEb

bm

bp

++++

++==

11

1

or , using the fact that [[ ]] [[ ]](( ))mEmmEm++==

:

[[ ]] [[ ]](( ))mEmb

mEp ++

++==1

1. (6.26)


[[ ]](( ))pEpby == (6.21)


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6

Now substituting (6.25) and (6.26) into (6.21) gives a solution for output:

[[ ]] [[ ]](( )) [[ ]]

++++== mEmEm

bmEby

1

1

[[ ]](( ))mEmb

by

++==

1. (6.27)

Equations (6.26) and (6.27) illustrate the basic features of the Lucas model. Expectedvariations in money affect prices in proportion. Money surprises affect both prices and

output, with the division of the impacts depending on underlying variances of relative andgeneral prices.

From equations (6.20), recall that

++

pr

r

VV

Vb

1

1

.Clearly, the larger the variance

in relative prices, the larger is b. With larger b values money shocks have bigger impactson output and small impacts on the aggregate price level.

Finally, to tie up loose ends, we need to go back and figure out the variances ofr

V and

pV .


[[ ]] [[ ]](( ))mEmb

mEp ++

++==1

1. (6.26)

This equation implies that(( ))21 b

VV m

p++

== .

To findr

V , first recall equations (6.7), (6.20b), and (6.21):

(( )) 0, >>++== ppzyq iii , (6.7)

[[ ]](( ))pEpbql iii ==== . (6.20b)

[[ ]](( ))pEpby == (6.21)

Substitute (6.21) into (6.7) to get:

[[ ]](( )) (( ))ppzpEpbq iii ++==


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In (6.20b), add and subtract bp to the right-hand side:

(( )) [[ ]](( ))pEpbppbqii

++== .

Combining the last two equations yields:

[[ ]](( )) (( )) (( )) [[ ]](( ))pEpbppbppzpEpb iii ++==++ (( )) (( ))ppbppz iii ==

(( ))(( ))ppbzii++==

b

zpp ii ++

==

Since pprii== , the variance of

ir is:

(( ))2bVV z

r++

==

.

Again recall from (6.20) that b is given by:

pr

r

VV

Vb

++==

1

1

where

(( ))21 bVV m

p++

==

and

(( ))2b

VV z

r++

==

.

Substituting the last two equations into the definition ofb, one gets:

(( ))

(( ))

++++

++

==

mz

z

Vb

bV

Vb

2

2

1

1

1

,

which implicitly defines b in terms of known parameters.


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8

Finally, note that since:

[[ ]] [[ ]](( ))mEmb

mEp ++

++==1

1. (6.26)

and

b

zppr iii ++

====

,

thatp andi

r are respectively linear functions ofm andi

z . This implies thatp andi

r are

normal and independent, as required when we invoked the solution to the signal

extraction problem.


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9

Discussion of the Lucas Model

Policy shocks affect both output and prices positively, leading to a Phillips curverelation.

There is no exploitable tradeoff between output and inflation only the money supply

process error term has an effect on output.

Changes in policy rules change expectations, which change apparent aggregaterelationships (i.e., observed Phillips curve tradeoff parameters). If policymakers try to

take advantage of statistical relationships, effects operating through expectations maycause those relationships to break down. This is the Lucas critique.

Policy ineffectiveness: Systematic stabilization policy must be ineffective. This is a

general result. [[ ]]mE can be inferred on the basis of a very complicated policy rule, and

still only the error term [[ ]]mEm appears in (6.27).

Suppose that governments observe aggregate demand shocks (from a source other than

money) but the public does not. This would permit the government policymaker topotentially stabilize output. However, this is not a good justification for stabilization

policies. First, most stabilization policy choices are generally based on observableperformance indicators. Second, if the public did not observe those indicators, it would be

easier to simply announce them than to carry out a counter-cyclical stabilization policy.

Empirical evidence on impacts of expected and unexpected money on output: Barro,Romer and Romer.

Is it plausible that there is substantial imperfect information about m andp?

Are labor supply elasticities large enough that we can explain output fluctuations as

responses to relative price misperceptions?


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1

Romer Chapter 6 Part B Staggered Price Adjustment

The Lucas model showed that with rational expectations and market-clearing

assumptions, systematic countercyclical stabilization policy would be ineffective. Earlycriticism from Keynesians focused on the implausibility of the rational expectations

assumption. Over time, the assumption of rational expectations has become moreacceptable (at least as a modeling assumption), and criticism has shifted to the

assumption that markets quickly adjust to market clearing equilibria.

Initially Keynesians responded by assuming some nominal price or wage rigidity.We will follow this approach, and will later come back to the issue of why it is that prices

might be rigid.

It turns out that modern New Keynesian models generally rely on the existenceimperfect competition. This helps out later in explaining why prices might be rigid.

Given that this is so, we will develop a model which incorporates imperfect competition.

A Model of Imperfect Competition and Price Setting

Assumptions

The economy consists of a large number of individuals.

Each individual is the producer of a good and sets the price of that good.

Labor is the only input: the output of good i is equal to the amount of labor employed inits production.

One major difference from the Lucas model: there is a competitive labor market where

individuals may sell and hire labor in order to produce goods (in the Lucas model, onehad to use own labor to produce your good).

Demand equations for each good are as follows (same as the Lucas model except for the

absence of market specific shocks):

(( ))ppyq ii == , 1>> ,

where the lower case letters represent logs,p is the average of the spi ' , andy is the

average of the sqi ' . Converting from logs to levels, the demand equation can be written:

==

P

PYQ ii .

Sellers with market power will sell at a price above marginal cost. If they cannot adjustprice, they will be willing to satisfy small increases in demand.


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2

We employ the same utility function as the Lucas model did:

iii LCU

1

Income now includes profit as well as wage income, so utility can be written as:

(( ))

i

iii

i LP

WLQWPU

1

++== . (6.37)

Once again, aggregate demand is given by:

pmy == .

Unlike the Lucas model, here we assume that the money supply is observed.

Individual Behavior

Substituting the demand equation,

==

P

PYQ ii , into the utility function, yields:

(( ))

i

i

i

i

i LP

WLP

PYWP

U1

++

==

(6.38)

The individual chooses his price,i

P , and his work effort,i

L . The first order conditions

for the utility maximization problem lead to the following results (after some

manipulation):

P

W

P

Pi

1==

(6.40)

1

1

==

P

WLi (6.42)

The first of these is a price mark-up equation; the second is an individuals labor supply

function.


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3

Equilibrium

By symmetry, in equilibrium, each individual works the same amount and produces thesame amount of output. From (6.42) the real wage is:

1== YP

W.

Substituting into (6.40) yields an expression for each producers desired relative price as

a function of aggregate output:

1

*

1

==

Y

P

Pi . (6.44)

In logs:

(( ))yppi 11

ln* ++

==

or

ycppi ++==* . (6.45)

Since producers are symmetric, all prices are the same, and each price is equal to the

average price. We can then use (6.44) to solve for the equilibrium level of output:

1

1

1

==

Y . (6.46)

From the aggregate demand equation, PMY /== , we find the equilibrium price:

1

1

1

==

MP .

Implications

Output is less than 1, which was the competitive market level of output. This reflects the

suboptimality of the monopoly outcome. This implies:


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4

Recessions and booms have asymmetric welfare effects. Booms are good,

recessions are bad.

Pricing decisions have external effects. Suppose that starting from an equilibriumpoint, each producer cuts price by a small amount. The average price declines,

and aggregate output rises, shifting out the demand curve for each product. Thedemand shift makes everyone better off (even though each individual is privately

better off not cutting price).

The existence of imperfect competition is consistent with monetary neutrality.The solution for output in (6.46) does not depend onM.

Predetermined Prices

This is the Fischer model of staggered pricing.

Assumptions

Price setters cannot freely adjust prices in each period. Instead, each price setter sets

prices every other period for each of the next two periods.

Example, at time 0==t , I set a price for time 1==t and a price (possibly a differentone) for time 2==t . At 2==t , I would set prices again, this time for times

3==t and 4==t .

Recall (6.45):

ycppi ++==*

. (6.45)

Since pmy == , this becomes:

(( )) mpcpi

++++== 1* .

For simplicity, normalize the constant c to be zero (one can think of this as simply

redefining the LHS as a deviation from c):

(( ))pmpi ++== 1*

No specific assumptions are made about the process for m.

We will assume that price setters set future log prices at the expected profit maximizing

log prices, given information available at the time prices are set. Expectations areassumed to be rational.


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5

Solution

The average price at time tis given by:

(( ))212

1ttt

ppp ++==

where 1t

p is the price set for time tby those setting prices at time 1t and 2tp is theprice set for time tby those setting prices at time 2t .

Since individuals select expected profit maximizing prices, *11

itttpEp = and

*

2

2

itttpEp = . From these we get:

*

1

1

itttpEp =

( )[ ]tttt pmEp += 111 (6.50)

( ) ( )2111

2

11

tttttppmEp ++=

*

2

2

itttpEp =

( ) ( )21

22

2

2

11 tttttt ppEmEp ++= (6.51)

Solve (6.50) for 1t

p :

2

1

1

1

1

1

2tttt

pmEp

+

++

= . (6.52)

Take expectations at time 2t on both sides of (6.52):

2

2

1

21

1

1

2ttttt pmEpE

+

++= (6.53)

Substitute (6.53) into (6.51)

( )

+

+

++

+= 22

22

2

1

1

1

2

2

11

tttttttppmEmEp

(6.54)


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Solve for 2t

p :

tttmEp

2

2

=

Substitute (6.55) into (6.52) and simplify to get:

( )ttttttt mEmEmEp 2121

1

2 +

+=

Recall that (( ))212

1ttt

ppp ++== and that ttt pmy = , so:

( )ttttttt mEmEmEp 2121

++=

( ) ( )tttttttt mEmmEmEy 1211

1 ++

=

Implications

Aggregate demand shifts have real effects; see the ttt mEm 1 term.

Anticipated demand shifts (that become anticipated after the first prices are set) affect

output; see the tttt mEmE 21 term. Illustrate via numerical example.

It is possible for (some) policy rules to stabilize output. The policymaker can observe the

effect of tttt mEmE 21 when choosing a target for tm .

The impacts of an anticipated demand shock depend on real rigidity, as measured by

parameter . If is low, producers are reluctant to allow relative prices to get out of

line. So when setting the one-period-ahead price, they set it close to those already set for

the next period, which increases overall price rigidity.


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7


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This handout describes a staggered price-setting New Keynesian model, as presented by

David Romer,Advanced Macroeconomics, pp. 265-270.

Set-up

There are two groups of firms (1 and 2). Every two periods, one group sets a price thatthen stays in effect for two periods (the current period and the next one). Groups 1 and 2

alternate in setting prices; i.e., group one may set prices in periods 1, 3, 5 etc., while group

2 sets prices in periods 2, 4, 6, etc. Firms in the two groups are in other respects similar,

but they are not perfect competitors -- different groups may charge different prices

without having sales go to zero. Rather, firms are imperfectly competitive.

There are five basic equations:

ttit ypp ==*

i = 1 2, (6.45)

The equation above is the supply relation. The variable pit

* is the log price that a firm in

group i would ideally like to charge in period t(if it could change prices in every period,

which it cant). The variable pt

is the log aggregate price level in period t. The variable

yt

is the log of aggregate output. This is equation (6.45) in the text, with the

normalization requiring that 0=c .

The equation says that a firm would desire a higher price for its output when the general

price level is high. One can think of firms as imperfect competitors who are sensitive to

rivals prices. When rivals are charging high prices, I can also charge high prices; when

rivals charge low prices, I must also keep prices low. In other words, my price should not

be too far out of line with others prices. The equation also says that I am willing to seemy relative price be higher when aggregate output is high. I.e., if demand is strong overall,

then I would be willing to charge a somewhat higher price than my rivals, because sales

will remain strong anyway.

Romers book provides a slightly different justification for this equation. Essentially, in an

imperfectly competitive market, firms charge prices that are a markup over costs (with the

markup depending upon the elasticity of demand). In this model, firms must set prices and

then satisfy demand. But if aggregate demand turns out to be unusually high, firms must

then produce more output in the aggregate. But to produce more output, they must hire

more labor. In this model, the supply of labor depends on the real wage, so wages (and

costs) must rise with output. Because of the markup in pricing, with higher costs firms

would wish to be charging a higher price for their product. So, other things equal, firms

prefer to be charging a higher relative price when aggregate output is high.

( )x p E pt it t it = + +1

21

* * (6.60)


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The variable xt is the actual price set by firms in the group that sets a price at time t. This

price must prevail for two periods (i.e. periods tand t+1). It is equal to an average of this

periods optimal price and next periods expected optimal price.

( )p x x

t t t= +

1

2 1p. 266

As noted, pt

is the aggregate price level in period t. In the equation above, it is shown as

an average of the price set last period (by one group) and the price set this period (by the

other group).

y m pt t t= (6.10)

Equation (6.10) is a conventional, simple aggregate demand relation with mt

representing

the money supply.

m m ut t t= +1 (6.59)

Equation (6.59) is a policy rule. It says that the money supply is a random walk.

Analysis

Our strategy will be to obtain a solution for xt, the price set by firms who choose a price

at time t. Clearly, once we have a solution for xt , we will also be able to solve for pt .

Substitute (6.10) into (6.45):

(( ))tttit

pmpp ++== *

(( ))ttit

pmp ++== 1*

Now substitute the equation above into (6.60):

(( ))[[ ]] (( ))[[ ]]{{ }}11 112

1++++ ++++++== ttttttt pEmEpmx

(( )) (( )) (( ))tttttt

tt xxEmxx

mx ++++++

++++== ++

1

1

2

11

2

1

2

1

21

2

1

2

1

(( ))(( ))11214

1++ ++++++== tttttt xExxmx


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Notice that if we regard mt as exogenous, then the preceding equation includes only one

endogenous variable, xt , and also the expectational term E xt t+1 . We now have the model

in a form we know how to handle. Following Romer, I will isolate xt

on the LHS of this

equation before following our usual procedures. After some algebra we obtain:

( ) ( )x A x E x A mt t t t t = + + +1 1 1 2 (6.62)

where

(( ))(( ))

++

==1

1

2

1A .

Now conjecture a solution:

ttt mxx ++++== 1

Update one period:

11 ++++ ++++== ttt mxx

Take expectations:

ttttmxxE ++++==++1

Substituting our conjectured solution for xt into the equation above gives:

(( )) ttttt mmxxE ++++++++== ++ 11 .

Now substitute for xt and E xt t+1 in equation (6.62):

(( ))[[ ]] (( )) ttttttt mAmmxAAxmx 21111 ++++++++++++==++++ .

Now equate coefficients on the two sides of the equation. This yields the following three

equations:

(1) AA ++==

(2) 2 AA ++==

(3) )21( AAA ++++==


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The second equation is quadratic and can be solved for . The solution is:

(( ))(( ))

A

AA

2

21211 ++==

After substituting forA and doing lots of algebra, we obtain:

++

==

1

1or

++==

1

1

Only the first solution is reasonable (because the second solution would give an unstable

solution for xt).

Now that we have a solution for getting solutions for and should be

straightforward. However, in this case there is a complication. In equation (1) it appearsthat any value of will satisfy the equation.1

To find we will take a short-cut. We can pin that parameter down using an analysis of

the steady state of this model when there are no shocks. In such a steady state, we will

have a stable price and a stable level of output. I.e., the sequence of chosen prices will

have identical values in each period. Moreover, the chosen prices will also be the same as

the desired prices in each period. But if desired price equals actual price in each period,

then yt

= 0 by equation (6.45). And ifyt

= 0 , then equation (6.10) implies that p mt t

=

1

For any non-zero value of , the first equation implies that :

AA ++==1

or

(( ))(( ))

++

==

==1

12

1

A

A

However, this value of is not compatible with the value we solved for earlier:

(( ))

(( ))

++

++

1

1

1

12 .

This implies that if we are to satisfy both (1) and (2), then 0== . (Any non-zero valueleads to the contradiction noted). The text gives an alternative route for finding this value

for .


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in each period. But if actual and desired prices are equal in each period, then equation

(6.60) implies that 111 ========== tttttt mpxmpx , etc. Now recall that our conjecturedsolution was:

ttt mxx ++++== 1

The preceding analysis implies that

ttt mmm ++++== .

But this is generally true only if 0== and 1==++ . We now have a solution for .Moreover, the solution for will be given by == 1 .

Given this solution for pt , and given that y m pt t t= , a solution for output can be

found:

ttt uyy2

11

++++==

Implications

The aggregate price level adjusts gradually to a shock. Because price adjusts gradually, the

shock to output is persistent. The intuition is that when there is a demand shock, one

group of firms has preset its price. The group which gets to chose price does not want a

price too different from the other group, for competitive reasons. So price does not fully

and immediately move to the new long-run price level. In the next period the next groupchooses price, but they are similarly constrained by the need to keep their relative price in

line. So prices adjust slowly. With prices adjusting slowly, the demand shock affects

output instead (firms with market power are willing to produce the additional units

demanded because price is in excess of marginal cost). When price adjusts slowly, output

effects will last a long time.


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Romer06D.doc

Introduction

There are problems with the Lucas model and with models where prices are assumed to

be rigid. The money supply and the aggregate price level are easily and quicklyobservable, contrary to the assumption of the Lucas model. Prices can usually be changed

at low cost, apparently undermining the rigid price models.

Agents are presumably interested in real magnitudes. Nominal magnitudes matter only inminor ways to them. If nominal imperfections are macroeconomically important, one

must show that small frictions at a microeconomic level have large impacts at amacroeconomic level.

We will assume the existence of menu costs, small costs associated with changing prices.

We investigate whether such costs might have important macroeconomic consequences.

Market power, Menu Costs, and Price Adjustments

Consider a firm with market power, which currently sets a price at the profit maximizinglevel.

Now suppose that there is a reduction in the demand facing this firm.

Momentarily suppose that the firm does not lower its price to the new lower optimal

level.

Fig. 6.3 in Romer shows that:

The profit consequence of a reduction in demand is large.The profit consequence of failing to cut price is small.

We can illustrate the same point with an alternative diagram (show as a function ofP),or with a Taylors series expansion of the profit function.

If the menu cost (the cost incurred when one changes a price) is larger than the gain fromchanging price, then the firm will leave its price unchanged.

How Aggregate Demand Externalities Arise

For simplicity suppose that AD is given by:

Y = M/P.

If prices are rigid, then a reduction inMreduces real demand for all firms in proportion.


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But for each of many small firms (each imperfectly competitive), the losses from failing to

adjust prices are very small (as shown above).

If all firms did adjust prices, they would all be better off (because their demand curveswould be shifted back out to the right). But no firm can make its demand curve shift back

out by unilaterally being flexible on price.

Thus one can view a recession as a result of market failure -- the failure to overcomeexternalities because of free-rider problems.

Some Welfare Implications.

In some models (Lucas, Fischer, McCallum) booms as well as recessions are associated

with welfare losses.

In this model monopoly equilibrium is suboptimal (with output too low) so booms increase

welfare (by causing higher output). Recessions result in welfare losses, consistent witheveryday views.

It is not obvious, but in such models fluctuations (i.e. demand induced business cycles) areinefficient. There can be a role for stabilization policy.

Some Problems for the Market Power/Price Rigidity Model

The analysis of price rigidity for an individual firm was conducted in a partial

equilibrium environment.

It turns out that a general equilibrium analysis reveals some problems with the precedinganalysis.

Romer Section 6.6 Revisited

First, recall that the first order condition for utility maximization describing labor

supply behavior was given by:

1

1

==

P

WLi (6.42)

Because of symmetry, in equilibrium each individual works and produces the

same amount, so YLi == and:

1

YP

W== (6.42a)


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where1

1

==

. This is the elasticity of labor supply with respect to the real

wage.

Each producer faces a demand function:

==

P

PYQ ii (6.7a)

Each firms (real) profit is given by (again recall that one unit of output requiresthe use of one unit of labor input):

(( ))

P

QWP ii == (6.37a)

or

i

i QP

W

P

P

== (6.37b)

Subsituting (6.7a) and (6.42a) into (6.37b) we get:

==

1

YP

P

P

PY ii

i

or, sinceP

MY==

11

==

P

M

P

P

P

M

P

P

P

M iii

++

==

P

P

P

M

P

P

P

M iii

11

(6.84)

Remember that each firm regardsMand P as exogenous, so 6.84 expresses profits

as a function of iP .

Price Adjustments

In section 6.6, we used this model to solve for an equilibrium under the assumption that

all firms adjusted prices to optimal levels.


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We will now consider the profit consequences of failing to adjust prices when demand

(M) changes.

Profits when Price is not Adjusted

By assumption, in the original flexible price equilibrium other firms are charging P and

firm i is charging PPi == . Now, whenMchanges, if firm i does not change price, then westill have PP

i== and by equation (6.84) profits for firm i are:

++

==

1

P

M

P

MFIXED (6.85)

Profits when Price is Adjusted:

When firm i adjusts its price, holding P fixed, it sets it to the profit maximizing value.

We also recall that the utility maximization problem for this model also involved

choosing price, iP . The first order condition implied this mark-up pricing result:

P

W

P

Pi

1==

(6.40)

Successively substituting (6.42a), 1

YP

W== , and

P

MY== into this equation yields:

1

1

==

P

M

P

Pi (6.40a)

Recall (6.84):

++

==

P

P

P

M

P

P

P

M iii

11

(6.84)

Now substituting

1

1 == PM

P

Pi into (6.84) yields:

++

==

111

1

11 P

M

P

M

P

M

P

MADJ


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which simplifies to:

++

==

1

11

1

P

MADJ (6.86)

We can now compare profits gained by the firm when it does and does not adjust price

whenMchanges.

Notas Romer

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