Three Sources of Increasing Returns to Scale

Home , Returns to scale

Jinill Kim

First draft: March 1996

This draft: April 3, 1997

Abstract

This pap er reviews various typ es of increasing returns from a critical p er-

sp ective. Increasing returns have b een intro duced b oth at the rm level and

at the aggregate level in a monop olistic-comp etition mo del. We show that

the degree of the aggregate returns to scale is a linear combination of three

return parameters, with the weights determined by the sp eci cation of a zero-

pro t condition. Identi cation issues are discussed with an emphasis on recent

macro literature. We argue that disaggregate data give information on the

market structure rather than the technology. Welfare implications explain

why it is imp ortant to identify various increasing returns.

Key words : Increasing Returns; Monop olistic Comp etition; Returns to Vari-

ety

JEL classi cation : E32

Federal Reserve Board, Mail Stop 61, Washington, D.C. 20551. Telephone: (202) 452-2715.

E-mail: [email protected]. This is a revised version of a chapter in my dissertation at Yale University.

Sp ecial thanks to Christopher Sims for his guidance and supp ort. Thanks also to William Brainard,

John Fernald, Rob ert Shiller, Steve Sumner, Michael Wo o dford, and seminar participants at the

Universities of Maryland and Virginia, and Federal Reserve Board for their valuable comments.

This pap er represents the view of the author and should not b e interpreted as re ecting the views

of the Board of Governors of the Federal Reserve System or other memb ers of its sta . 1

Contents

1 Intro duction 2

2 The Mo del 4

2.1 Firms ...... 5

2.2 Aggregation ...... 8

2.3 Returns to Scale ...... 10

2.4 A Dynamic Mo del ...... 15

3 Implications 19

3.1 Identi cation with Aggregate Data ...... 19

3.2 Interpretation of Disaggregate Data ...... 21

3.3 Comparison with a So cial Planner ...... 24

4 Further research 26

A External Increasing Returns 28

B Input Fixed Cost 29

C Cost Minimization 31

D Fixed Cost Externalities 33

1 Intro duction

The hyp othesis of noncomp etitive markets and/or increasing returns to scale has

recently b een used in dynamic sto chastic general-equilibrium (DSGE), more often

called real-business-cycle, mo dels. Using the Solow residual as a measure of pro duc-

tivitychanges is appropriate only under the jointhyp othesis of p erfectly comp etitive

markets and constant returns to scale. In a series of pap ers, Hall (1986, 1988, 1990)

argues that evidence from the Solow residual is not consistent with this maintained

hyp othesis but with the alternativehyp othesis of noncomp etitive markets and/or in-

creasing returns to scale. Under this alternativehyp othesis, the Solow residual has

Imp erfect comp etition makes equilibrium p ossible in the presence of increasing returns. In-

creasing returns are compatible with comp etitive rms if the increasing returns are external to the

rms. Internal returns may b e motivated as a representation of external ones, as in Beaudry and

Devereux (1995a). The twotyp es are compared using mo dels with b oth typ es of increasing returns

in App endix A. 2

endogenous comp onents which cause it to over-represent the contribution of pro duc-

tivity sho cks. Furthermore, this alternative hyp othesis helps explain some puzzles

in the DSGE literature, e.g. little correlation b etween employment and pro ductivity.

Following Dixit and Stiglitz (1977) and Blanchard and Kiyotaki (1987), the

monop olistic-comp etition framework has b een widely used in macro economics. The

assumption of unrestricted entry and exit implies that pro ts are zero in equilib-

rium. In a monop olistically comp etitive market, the technology of constant returns

to scale lets rms pro duce p ositive pro ts regardless of their size. Intro ducing in-

creasing returns at the rm level leaves ro om for reducing pro ts to zero. The

ob jective of this pap er is to discuss three di erent typ es of increasing returns in a

monop olistic-comp etition mo del and to derive implications for the related literature.

There are two ways of intro ducing increasing returns at the rm level. The

more conventional way is including xed costs as part of a rm's technology. This

way has b een followed whenever a zero-pro t condition is imp osed. An alternate

way is amplifying the constant-returns-to-scale term by a power larger than one,

which amounts to diminishing marginal cost. When we incorp orate b oth sources

of increasing returns simultaneously, as in Hornstein (1993), their e ect on the ag-

gregate returns to scale is di erent from each other. Increasing returns due to the

third source o ccurs only at the aggregate level. It involves a technology or a pref-

erence for the variety of go o ds. The intro duction of a new go o d might enhance the

pro duction eciency and the consumption convenience. Romer (1987) fo cuses on

this as an engine of growth and Matsuyama (1995) relates this to complementarities

and cumulative pro cesses of macro economics. The mo del in Devereux, Head and

Lapham (1996a), even without pro ductivity sho cks, generates business cycle uc-

tuations of real variables from government sp ending sho cks since these a ect the

varietyofgoods.

This pap er shows that, in a static mo del, the resulting degree of aggregate re-

turns to scale is the average of the second and the third sources of increasing returns,

without any in uence of p ositive xed costs. The derivation of aggregate returns

from a rm's technology involves two steps. First, the di erentiated outputs are

aggregated to pro duce a measure of aggregate output. Second, a zero-pro t con-

See Benassy (1991) for quali cations of zero-pro t conditions. The assumption of zero pro ts

matches the observation in Hall (1990) and Rotemb erg and Wo o dford (1995) that there are no

signi cant pure pro ts in the United States.

Not that all pap ers in DSGE literature imp ose a zero-pro t condition. Hairault and

Portier (1993) and Beaudry and Devereux (1995b) do not imp ose a zero-pro t condition and

so parameterize b oth xed cost and the numb er of rms. In such mo dels, the rm-level returns

to scale are the aggregate returns to scale and the p ermanent presence of p ositive pro ts remain

unexplained. For example, the steady-state pro t rate is 17% in the b enchmark mo del of Hairault

and Portier (1993). 3

dition is imp osed up on the aggregate version of a rm's technology. Sp eci cation

of a zero-pro t condition determines the weights of the averaging. In a dynamic

mo del where adjustments to zero pro t are not instantaneous, the market structure

of monop olistic comp etition plays a role|the slower the adjustments, the larger the

role. Even if market structure do es not directly a ect the technology, this source

in uences the resp onse of output in a way indistinguishable from the previous two

ways.

The aggregate dynamics of a mo del which combines various sources of increasing

returns to scale show that there are identi cation problems in the recent macro e-

conomics literature using the framework of monop olistic comp etition. We compare

various pap ers to see how they sp ecify a zero-pro t condition and what the resulting

degree of returns to scale is. We also argue that the literature using disaggregate

data provides information di erent from what it intends to provide: on the mar-

ket structure rather the technology. Lastly, welfare implications are discussed from

the p ersp ective of a so cial planner who do es not need to satisfy zero-pro t condi-

tions. While having similar p ositive implications for the aggregate returns, various

increasing returns have di erent normative implications.

2 The Mo del

To illustrate the p oints in as simple a structure as p ossible, we analyze only the

pro duction side of the economy. This analysis is tractable and gives much insight

on how di erent returns to scale interact with one another. Most pap ers on monop-

olistic comp etition deal with b oth the pro duction and the consumption side of an

economy. However, intro ducing a utility function complicates the mo del so that it is

dicult to disentangle the pro duction features from consumer b ehavior. Our mo del

is a partial-equilibrium mo del, since the pro duction side generates the demand for

inputs. The transformation of this mo del into a general-equilibrium framework is

straightforward by stacking it with a consumer problem and, if needed, a government

problem. The consumer problem would generate the supply function of aggregate

inputs through a lab or-leisure choice and capital accumulation. Therefore, through-

out this pap er, we may consider the aggregate inputs as exogenous variables.

Since a zero-pro t condition is crucial in deriving the economy-wide returns to

scale, we will be very careful in discriminating two meanings of `pro duction func-

tion.' A structural pro duction function is a purely technological relation without

reference to the equilibrium condition of zero pro ts. However, a pro duction func-

tion in a reduced form, whether a rm's or an aggregate one, is a combination of

the appropriate technology and a zero-pro t condition. That is, a reduced-form 4

pro duction function is a structural pro duction function with a zero-pro t condition

imp osed.

Now our ob jective of this section is to transform a rm-level structural pro-

duction function, Eq. (1), into an aggregate reduced-form pro duction function, e.g.

Eqs. (16) and (24). This transformation is a contribution to the literature since it

simpli es one step of complex DSGE mo dels and so makes it easy to understand

their pro duction features. We start with a static mo del since it is a sp ecial case

of a dynamic mo del. The static mo del serves as a steady-state, or low-frequency

in general, feature of the dynamic mo del. As a preparation for the analysis of the

aggregate variables, we analyze the b ehavior of rms.

2.1 Firms

Firms are identical except for the heterogeneity of outputs. Firm i pro duces y units

of net output under a technology of increasing returns to scale:

i i

y = A k l ; (1)

i i i

i i

with the restrictions that 0 1, > 0, and > 0. A denotes the pro ductivity

i i i i

sho ck, k is the capital sto ck and l is the quantity of lab or. Since pro ductivity sho cks

i i

are not crucial in deriving the implications on returns to scale, they are normalized

to 1 except when necessary for discussing econometric issues.

The parameter represents what rm i should pay at each p erio d regardless of

its activity level. For example, a rm advertises its go o d each p erio d to maintain its

market share. Note that the xed cost is measured in units of its own output, not

its inputs. Additionally, this pap er follows the convention that a rm's xed cost

is exogenous to the rm. The presence of a xed cost makes it p ossible to imp ose

a zero-pro t condition, as in Hornstein (1993), Rotemb erg and Wo o dford (1995),

Beaudry and Devereux (1995a), and Devereux, Head and Lapham (1996a,b), and

is a source of increasing returns to a rm's technology. However, in a static mo del,

rm-level increasing returns due to xed costs are not transmitted to the increasing

returns of an aggregate pro duction function in a reduced form, Eq. (16).

If is greater than 1, a rm's gross output features additional increasing returns

to scale. This source of increasing returns has not b een p opular in the literature.

Actually, most pap ers in the DSGE literature restrict to be exactly equal to 1,

except for Hornstein (1993) and Benhabib and Farmer (1994). This pap er names

Chatterjee and Co op er (1993) and Yun (1996) assume that xed costs are a part of inputs.

The mo dels with input xed costs b ehave in a similar way. See App endix B for a mo del with xed

costs as part of its inputs. 5

this source `diminishing marginal cost.' App endix C on cost minimization shows

that the bigger is, the smaller the slop e of marginal cost is. Hornstein (1993)

call this `the scale co ecient.' However, we will show that the degree of returns

to scale dep ends also on other parameters. Furthermore, diminishing marginal cost

may not a ect the resp onse of aggregate output, dep ending on the sp eci cation of

a zero-pro t condition.

These two sources of increasing returns to scale at the rm level are added up

in overall returns to scale of a rm's technology. From the p ersp ective of a rm,

to whom the xed cost is exogenous, the log-linearized reduced-form pro duction

function is:

h i

y +

i i

y ' k l :

i i

That is, the degree of returns p erceived by a rm is the pro duct of the degree of

diminishing marginal cost and the ratio b etween the gross and the net output. The

ratio turns out to be , where is the degree of market power. This degree

is assumed to be greater than and de ned in Eqs. (5) and (6). So the degree

of returns to scale of a rm's technology is equal to the degree of market power.

However, this exercise is meaningless in that wehave not incorp orated a zero-pro t

condition into a rm's technology. Actually, we will see that the presence of xed

costs by itself do es not imply increasing returns in a static mo del. However, the

degree of market power turns out be the right measure in a dynamic case where

there are only steady-state adjustments to zero pro t, without any p erio d-by-p erio d

adjustments.

A rm maximizes its pro t:

= p y PZk PWl ;

i i i i i

where p is the price of rm i's pro duct, P is the general price level, Z is the

real rental rate of capital, and W is the real wage. In mo dels with monop olistic

comp etition, the price of outputs, p , dep ends on the amount pro duced, y . The

i i

inverse demand for the output of the ith rm will be derived later in Eq. (8),

which involves the parameter representing the market structure of monop olistic

comp etition, . The assumption that rms rent capital rather than holding and

accumulating it simpli es the analysis. Under this assumption, we do not need

to explain where entering rms buy capital or where exiting rms sell remaining

An alternative, and probably more conventional way of analyzing the rm b ehavior is the

framework of cost minimization in App endix C. However, the pro t-maximization framework is

b etter for understanding the mechanics of the zero-pro t condition since it expresses the endoge-

nous variables as a function of aggregate inputs, considered exogenous in this pap er. Furthermore,

extension to a dynamic mo del b ecomes more complicated in the framework of cost minimization. 6

capital. Capital and lab or are homogeneous and a rm b ehaves as a price-taker in

the input markets.

Noting that y is a function of (k ;l ), the rst order conditions with resp ect to

i i i

k and l are:

i i

! !

y + p

i i i

= PZ; (2)

i i

p y +

i i i

(1 ) = PW: (3)

i i

Input prices determine the ratio b etween gross output |not net output| and each

input. This p oint will b e imp ortant later in comparing the dynamics of output with

those of input prices. The rst order conditions, Eqs. (2) and (3), determine the

optimal capital and lab or, dep ending on the demand curve and the size of the xed

cost. A rm's maximized pro t is:

! # "

i i

k l : = p 1

i i

This derivation shows that neither form of increasing returns to scale is allowed in

a p erfectly comp etitive economy, i.e. when = 1. Intro ducing increasing returns

would generate corner solutions. Therefore, the discussion of increasing returns in

a comp etitive economy in Hall (1988) is futile. It is also interesting to note that

marginal costs decreasing at a faster rate do not always b ene t rms. Other things

b eing equal, a rm would pro duce more output, which would decrease its pro t by

i i

lowering its output price. If k l is b elow a certain level, a faster rate means

i i

lower pro ts. Otherwise, pro ts b ecome higher as increases only when is close

i i

to 1.

The zero-pro t condition which should hold in equilibrium implies the following

restriction:

i i

= 1 : (4) k l

i i

The assumption of p ositive xed costs is equivalent to the assumption that the

degree of diminishing marginal cost ( )may not b e larger than the degree of market

Here we neglect the price-index e ect of individual pricing decisions. See d'Aspremont, Ferreira

and Gerard-Varet (1996) for this issue.

If marginal cost is decreasing, the solution to (2) and (3) might not be a pro t-maximizing

choice. The restriction that is nonnegative is sucient to guarantee that the choice de ned

by (2) and (3) maximizes pro ts in equilibrium.

This is corrected in Hall (1990), where rms have market p ower under increasing returns. 7

power (). This equation may be interpreted as determining the size of the xed

cost. However, it may also be interpreted as a condition deciding the number of

rms, since k and l dep end on the number of rms. Since aggregate inputs are

i i

considered exogenous in this pap er, we should transform the zero-pro t condition

into a relation among aggregate variables. Before discussing aggregate implications

of two rm-level increasing returns, we need to sp ecify how the economy values

the variety of go o ds in aggregation, which intro duces the third source of increasing

returns.

2.2 Aggregation

From a macro economic viewp oint, heterogeneous outputs need to b e aggregated. A

convention is intro ducing an additional agent in the economy, called an aggrega-

tor. The aggregator is equivalent to a rm pro ducing a nal good. The presence

of the aggregator can be avoided, when every agent cho oses the goods and lab or

index comp osition optimally, as in Blanchard and Kiyotaki (1987) and Hairault and

Portier (1993). In this case, aggregation is a matter of preference as well as tech-

nology. Since this choice is static and the same for all agents, notation is simpli ed

when the aggregator solves the problem instead.

The aggregator purchases di erentiated outputs from rms, which are describ ed

byanN-dimensional vector, (y ;y ;:::;y ), and transforms them into Y units of a

1 2 N

nal good. In the literature on monop olistic comp etition, the aggregating function

follows b oth constant returns to scale and constant elasticity of substitution. The

sp eci cation used by Dixit and Stiglitz (1977), and thenceforth conventional, is:

Y = y ; (5)

i=1

with > 1. The elasticity of substitution between two di erentiated outputs is

constant at . Since is the markup in equilibrium, it is de ned as the degree

of market power. Besides determining the elasticity of substitution, the parameter

plays an additional role involving the varietyofgoods.

If all go o ds are hired in the same quantity, y , then aggregate output is:

Y = N y:

Thus, there are increasing returns to variety (N ), together with constant returns

to quantity (y ). If the number of rms is constant in the mo del, this typ e of in-

creasing returns do es not pro duce aggregate increasing returns. Hornstein (1993)

and Beaudry and Devereux (1995a) fall into this category. However, in a mo del 8

where the number of rms is endogenous, increasing returns to variety do generate

aggregate increasing returns, as in Devereux, Head and Lapham (1996a,b). They

argue that increasing returns to variety capture the spirit of a thick-market e ect

and also explain the pro cyclical b ehavior of job creation and the entry of new rms.

In the conventional sp eci cation of Eq. (5), the degree of increasing returns to

variety is linked to the degree of market power. Theoretically sp eaking, there is

no a-priori reason to prefer the presence or the absence of increasing returns to

variety. Furthermore, with all the agreement on the presence of increasing returns

to variety, the degree do es not need to b e related to the degree of market p ower, as

in Eq. (5). This pap er parameterizes, and so disentangles, the degree of returns to

variety separately from that of market p ower. Dixit and Stiglitz (1975)|a working-

pap er version of Dixit and Stiglitz (1977)| and Benassy (1996) also disentangle the

two parameters, but this is not related to the degree of returns to scale.

Disentangling the two parameters is not just a theoretical curiosity. We will

show that input prices, as a function of aggregate variables in Eqs. (26) and (27),

are a function of the variety parameter but not of the market structure parameter.

Therefore, only the former contributes to the existence of equilibrium indeterminacy.

Besides, comparing the economy of monop olistic comp etition with that of a so cial

planner, we will show that the two parameters have di erent welfare implications.

These two p oints will b e discussed after the algebra of the mo del.

This pap er parameterizes the aggregator as follows:

y Y = N ; (6)

i=1

which results in the following equation for aggregate output at symmetry:

Y = N y: (7)

The new parameter, , represents the returns to variety. If it is greater than 1, there

are increasing returns to variety. Note that Eq. (5) corresp onds to the case when

is equal to . If it is equal to 1; the aggregator implies constant returns to variety

Romer (1987) and Devereux, Head and Lapham (1996a,b) call this a return to `sp ecialization'

rather than `variety.' Following the title of Dixit and Stiglitz (1977), Chatterjee and Co op er (1993)

use the terminology of `pro duct diversity.'

A new multiplicative term, N , is called the \public-go o d feature of diversity" in Dixit and

P P

N N

( )

Stiglitz (1975). Two equivalent representations are N y and (N y ) .

i=1 i=1

This function is discontinuous with resp ect to the intro duction of a new good. However, the

discontinuity disapp ears if we assume a continuous go o ds space. 9

as assumed in Rotemb erg and Wo o dford (1995). They motivate constant returns as

a normalization, but this assumption is a restriction rather than a normalization.

The aggregator is assumed to maximize its pro t:

=PY p y ;

i i

i=1

where all price variables are exogenous to the aggregator and the aggregate price

index is de ned as:

( 1)

( )

p P = N :

i=1

The rst order condition with resp ect to y reduces to a constant-elasticity inverse

demand function:

) (

: (8) p = PN

Note that the new parameter, ; a ects only the level of the demand without a ect-

ing its elasticity. Hence the rst order conditions of the rms, Eqs. (2) and (3), are

not sensitive to the parameterization of returns to variety.

Besides aggregating di erentiated go o ds, macro economics has also made it a

convention to scrutinize a symmetric equilibrium under identical technologies of

the rms. The homogeneity of capital and lab or implies the following relations in

equilibrium:

K = Nk;L = Nl; (9)

where K and L denote aggregate capital and aggregate lab or, resp ectively.

2.3 Returns to Scale

Based up on previous discussions on rms' b ehavior and aggregation, wenow derive

the aggregate reduced-form pro duction function, i.e. an aggregate version of rms'

technology with a zero-pro t condition imp osed. Aggregation of homogeneous in-

puts in Eq. (9), together with aggregation of outputs as shown in Eq. (7), transforms

the rms' technologies, Eq. (1), as follows:

( ) 1

Y = N K L N : (10)

At symmetry, P = p. Increasing returns to variety imply that an increase in the number

(1)

of rms leads to a decrease in the price index due to an eciency gain. See Feenstra (1994) for an

application to imp ort go o ds. Existing price indices do not adjust b ene ts to variety: they assume

=1. 10

This is the aggregate structural pro duction function. Both the aggregate gross out-

put and the aggregate xed cost feature an element of returns to variety due to

aggregation. Note that we have not yet imp osed a zero-pro t condition, which is a

part of the equilibrium conditions. That is, the aggregate pro duction function in a

reduced form is derived only after the zero-pro t condition is imp osed.

We now derive what the zero-pro t condition, Eq. (4), implies for the aggre-

gate reduced-form pro duction function. The relationship between individual and

aggregate inputs transforms the zero-pro t condition as follows:

N = 1 K L : (11)

Note that this do es not involve the degree of returns to variety, . This aggregate

version of the zero-pro t condition has b een interpreted in two di erentways in the

literature.

The rst interpretation endogenizes the numb er of rms, N , as a function of the

aggregate variables as follows:

0 1

@ A

N = K L : (12)

If there is a change in the economy causing a p ositive pro t, e.g. a p ositive pro duc-

tivity sho ck, the number of rms increases to take advantage of this change, thus

reducing pro ts to zero. The xed cost of a rm is assumed to remain constant.

Rotemb erg and Wo o dford (1995) and Devereux, Head and Lapham (1996a,b) adopt

this interpretation. This interpretation has a conceptual problem asso ciated with

the de nition of the equilibrium, since an equilibrium with entry and exit involves

achange of the go o ds space.

Conversely, the second interpretation considers the xed cost as an endogenous

variable, that is:

= N 1 K L : (13)

Taken literally, a rm's xed cost increases with aggregate inputs and is also a ected

by the market structure represented by the degree of market power, . This inter-

pretation has b een less attractive than the former, since the amount of the xed cost

is not a technological parameter but a function of other parameters and exogenous

variables.

See Benassy (1991) for the details.

The endogeneity with resp ect to the degree of market power might be rationalized by the

argument that a larger xed cost (e.g. advertising) should b e paid at each p erio d in an economy

with a larger degree of market p ower. 11

However, an increase in the xed cost per rm is equivalent to job creation on

the intensive margin. That is, output per rm is prop ortional to the xed cost.

Furthermore, over low frequencies of a growing economy, it is reasonable to assume

that xed costs grow as rm size grows over time. No pap ers have argued for this

interpretation seriously, but some follow this interpretation implicitly by assuming

that the number of rms is constant. If the number of rms is assumed to be

constant, as in Hornstein (1993), Yun (1996), and Beaudry and Devereux (1995a),

the only way to achieve the zero-pro t condition is by endogenizing the xed cost

of a rm.

As with the sp eci cation of the aggregator, there is no a priori reason to prefer

either interpretation. In general, b oth the xed cost and the number of rms may

change. This pap er, rst in the literature, incorp orates this generality by de ning

an additional parameter, " 2 [0; 1], which represents the ratio of the intensive and

extensive margins. Our parameterization p ostulates the xed cost and the number

of rms as follows:

# " !

= ; (14) 1 K L

" # !

N = 1 ; (15) K L

where is an arbitrary constant. This constant a ects only the level of variables,

but not their p ercentage deviation. We can interpret the parameter " as one related

with the elasticity of supplying new rms. Manipulating Eqs. (14) and (15), wehave

a constant-elasticity supply schedule,

" 1"

N = ;

where is interpreted as the marginal cost of some sp ecialized resource, e.g. en-

trepreneurial ability, required to create a new rm. For example, when " = 0, the

supply of entrepreneurial ability is in nitely elastic at a price of per unit.

Note that the extreme cases of " = 0 and " = 1 corresp ond to the twointerpreta-

tions in the literature, Eqs. (12) and (13). This pap er shows that our generalization

is analytically tractable and gives intuition for the interaction among the various

increasing returns. Before discussing the general case containing the parameter ",

we consider the two extreme cases.

This parameterization of endogenous xed costs can b e reconciled with that of exogenous xed

costs, when negative externalities are intro duced via xed costs as in App endix D. Optimal decision

of xed cost in industrial-organization literature endogenizes the " parameter, but constant " is

not a bad approximation when time unit if constant. 12

Supp ose that the number of rms is endogenous, i.e. " =0, then the aggregate

output in equilibrium is derived by substituting Eq. (12) into Eq. (10):

( )

Y = 1 K L :

In this case, , the degree of returns to variety, determines the degree of aggregate

returns to scale. The intuition b ehind this is that the rm size is constant while the

number of rms is variable. Since the endogenous number of rms is prop ortional

to the constant-returns-to-scale term (K L ) in Eq. (12), the channel of returns

to variety works while overriding that of diminishing marginal cost. Note that the

economy may pro duce more output with a larger xed cost. If is smaller than ,

output increases as the xed cost increases. This apparently abnormal result comes

from the following intuition: since the degree of diminishing marginal cost is higher

than that of returns to variety, it is b etter to have larger rms. The gains from

larger rms o set the losses from a larger aggregate xed cost.

However, if we assume an endogenous xed cost, i.e. " =1, then the aggregate

output b ehaves di erently. Using Eq. (13), the aggregate technology of Eq. (10) is

transformed as follows:

( ) 1

Y = N : K L

Note that the degree of returns to scale is , the degree of diminishing marginal cost.

Opp osite from the previous case of an endogenous number of rms, the channel of

diminishing marginal cost is in e ect while there is no e ect of the returns to variety.

In a sense, this is trivial since the numb er of rms is constant. A rm's output and

xed cost move prop ortionally with the aggregate output. Inputs per rm move

also prop ortionally with the aggregate inputs. Note that an increase in the number

of rms may increase or decrease the aggregate output, dep ending on the sign of

( ). If the degree of returns to variety is higher than that of diminishing marginal

cost, it is b etter to have more rms in the economy.

Now let's consider our general case where b oth the xed cost and the number

of rms change according to Eqs. (14) and (15). It is natural to sp eculate that the

degree of returns to scale is between the two extreme cases. Substituting Eqs. (14)

and (15) into Eq. (10), the aggregate pro duction function in a reduced form is

simpli ed as follows:

(1") 1

( )

" +(1")

( )

Y = 1 K L : (16)

The degree of returns to scale in this general case is [" +(1 ")], a convex combi-

nation of and . The degree of returns to scale in the aggregate economy dep ends 13

on the choice of three parameters representing diminishing marginal cost ( ), returns

to variety(), and the ratio of the two endogenous changes ("). Note that increasing

returns at a rm level due to p ositive xed costs do es not generate increasing returns

at an aggregate level.

The e ect of monop olistic comp etition on the aggregate output is now clear.

Both the xed cost and the numb er of rms are an increasing function of the degree

of market power, . Since a larger degree of market power means a larger pro t,

we need more rms and a larger xed cost per rm to reduce the pro t to zero.

Given a certain amount of the aggregate inputs, a higher degree of market power

decreases the amount of di erentiated output due to an increase in the xed cost and

a decrease in inputs p er rm. However, if the numb er of rms is endogenous ("<1)

and the degree of returns to variety is larger than that of diminishing marginal

cost ( > ), an o setting mechanism is at work, since a higher degree of market

power increases the number of rms. This mechanism overturns the decrease in

a rm's output if the degree of market power is lower than the degree of aggregate

returns to scale, [" +(1 ")]. In other words, an economy with monop olistic

comp etition would pro duce maximum output if the degree of market power were

equal to the degree of returns to scale. The -shap ed graph in the rst plot of

Figure 1 on page 27, denoted as Y , shows this prop erty. For example, in a mo del

with a Dixit-Stiglitz aggregator ( = ) and an endogenous numb er of rms (" = 0),

the monop olistic-comp etition economy is optimal in the sense that it pro duces the

most aggregate output. This optimality p ertains to the comparison among di erent

market economies and has nothing to do with the comparison with a so cial planner,

which will b e discussed later.

Before moving on to a dynamic mo del, we analyze the mechanics of input prices.

The pro t maximizations of the rms have implications for the rental rate and the

wage, Eqs. (2) and (3), which result in the following aggregate relations:

Z = ; (17)

W = (1 ) : (18)

Although the dynamics of input prices dep end on gross output, they mimic those of

net output b ecause the aggregate xed cost is prop ortional to the aggregate gross

output, and so the net output. However, this prop erty do es not extend to a dynamic

case. This p oint turns out to b e imp ortant in the discussion of identi cation issues.

It is easy to check that the aggregate output is decreasing over the whole range of the degree

of market p ower if b oth conditions do not hold. 14

2.4 A Dynamic Mo del

Up to now, neither the size of the xed cost nor the degree of market p ower a ects

the degree of returns to scale. Then, is there no p oint of intro ducing monop olistic

comp etition and the xed cost in the b ehavior of returns to scale? Yes, the presence

of the xed cost do es a ect returns to scale in a dynamic mo del where adjustments

to zero pro t are not instantaneous. Note that the static case considered ab ove

is analogous to the Marshallian long run, in that no sp ecialized resource prevents

zero pro ts from b eing achieved. Following the same spirit, we interpret a dynamic

case considered b elow as the Marshallian short run since existing rms may earn

quasi-rents.

To consider a dynamic case, the aggregate technology without the zero-pro t

condition imp osed, Eq. (10), is rewritten with time subscripts for the variables.

( )

K L N Y = N : (19)

t t

This structural pro duction function do es not directly involve the degree of market

power, . However, this degree representing the market structure may a ect the

degree of returns to scale of the aggregate reduced-form pro duction function in a

dynamic mo del.

Apart from the choice of endogenous variables in a static mo del, an additional

consideration arises in a dynamic mo del: how fast do es the economy move towards

the state of zero pro ts? Two extreme cases have b een considered in the literature.

In the case of full adjustment, pro t is zero at every p erio d. This sp eci cation is

used in Devereux, Head and Lapham (1996a,b). The short-run dynamics are the

same as that of the static mo del describ ed in Eqs. (16){(18). The other extreme

is the case of no short-run adjustment. Pro t is zero only at the steady state, so

the short-run dynamics of the xed cost and the number of rms are not a ected

even if pro t is not zero in a particular p erio d. Hornstein (1993) and Beaudry and

Devereux (1995a) adopt this sp eci cation. In this case, returns to scale are governed

by neither diminishing marginal cost nor returns to variety. They are governed by

the degree of market p ower, which is a function of the degree of substitution among

di erentiated go o ds. These two extreme cases are compared in Chatterjee and

Co op er (1993).

Another contribution in this pap er is to generalize the sp eci cation of zero pro ts

byintro ducing an additional parameter, 2 [0; 1], representing the sp eed of adjust-

ments. This sp eed is inversely related to cost of adjustments, e.g. entry and exit.

Rotemb erg and Wo o dford (1995) is the only example which do es not follow one of the two

extreme cases. The calibration corresp onds to a very small of this pap er. Optimal decision of 15

We assume that the xed cost and the numb er of rms adjust p erio d by p erio d, but

not fast enough to achieve zero pro t at every p erio d:

(1 )

= ;

(1 )

N = N N ;

where ; N guarantees zero pro t at every p erio d and ; N guarantees zero

pro t at the exogenous steady state. Note that no adjustment corresp onds to the

case when = 0, and full adjustment, when =1.

Sp eci cally, the four variables are de ned as follows:

" ! #

= 1 K L ; (20)

t t

# " !

; (21) = 1 K L

" ! #

1"~

N = ~ 1 K L ; (22)

t t

# " !

: (23) N = 1 K L

We will see that only "~, a parameter regarding zero pro t at every p erio d, matters

for the dynamics of the mo del. The other parameter, ", a ects only the steady-state

prop erties. Alternatively, the sp eed of adjustments can vary b etween the xed cost

and the number of rms with the assumption that "~ = . In this case, the three

parameters are ("; ; ). However, this new mo del is equivalent to our mo del with

the parameters de ned as follows:

= " +(1 ") ;"~= "; " = ":

Substituting Eqs. (20){(23) into Eq. (19), the aggregate reduced-form pro duction

function in the dynamic mo del is as follows:

i i h h

(~" +( 1"~) ) (~" +(1"~)) +(1 )

1 1

; (24) K L Y = K L

2 t 1

t t t t

entry and exit in industrial-organization literature endogenizes the parameter, but constant is

an approximation under constant time unit. A higher frequency would imply a lower .

Our sp eci cation with an additional parameter preserves the simple one-p erio d nature of the

mo del. The mo del b ecomes intertemp oral if we intro duce partial adjustments by assuming the

xed cost and the numb er of rms predetermined, as in Ambler and Cardia (1996). 16

where

[ (1"~) +(1 )( 1")] 1

)

(

h i

1 (1 )( 1")( )

( )

1 1

= ~ 1 K L ;

" ~ +( 1"~) " +( 1")

+(1 )

h i

1 (1 )(" +(1") )

)

(

1 1

= ~ 1 K L :

Note that the two exp onent terms in Eq. (24) are di erent from each other and that

neither of them involves the degree of market power, . Gross output uctuates

strictly more than the aggregate xed cost, unless adjustments are instantaneous.

This nonlinear reduced-form pro duction function has such complicated dynamics

that it cannot b e compared with the static case. However, in a linearized version of

the dynamic mo del, the b ehavior of the variables is comparable to the static mo del.

In practice, most DSGE pap ers use a linearized version.

The log-linearized version of Eq. (24) is:

^ ^ ^

Y =[ (~" +(1 "~))+(1 )] K +(1 )L ; (25)

t t t

where x^ is the p ercentage deviation of x from its steady state. This clearly shows

t t

that market structure a ects output uctuations in a way indistinguishable from the

two previous sources of increasing returns, diminishing marginal cost and increasing

returns to variety. The degree in the dynamic case is a convex combination of

the degree of the static case in Eq. (16), [~" +(1"~)], and the degree of market

power, . Equivalently, the degree of returns to scale is a convex combination of

the three parameters: the degree of diminishing marginal cost ( ), the degree of

returns to variety(), and the degree of market p ower (). Note that the degree of

returns to scale in the long run, i.e. at the steady state, would be [" +(1")]

which is di erent from that of the short run in Eq. (25). This helps explain the

di erence b etween low- and high-frequency uctuations, b oth of which are imp ortant

in macro economics.

In the extreme case where there is no p erio d-by-p erio d adjustment, the degree

of short-run returns to scale is the degree of market power; the other two param-

eters do not enter at all. The intuition of this extreme case is as follows. Since

aggregate xed cost do es not resp ond to the change of aggregate inputs, net aggre-

gate output uctuates more than gross aggregate output. Furthermore, the amount

rms pro duce relativetothe size of the xed cost dep ends on the degree of market

power. The result of this extreme case shows that the intro duction of diminish-

ing marginal cost in Hornstein (1993) and Beaudry and Devereux (1995a) do es not

a ect the dynamics of the aggregate reduced-form pro duction function. However, 17

this do es not mean that the intro duction has no in uence on the dynamics of a

general-equilibrium mo del at all. The degree of diminishing marginal cost a ects

the dynamics of input prices as follows.

In a dynamic case, the dynamics of the rental rate and the wage are as follows:

h i

(~" +(1"~)) +(1 )

K L

; (26) Z =

h i

(~" +(1"~)) +(1 )

K L

W = (1 ) : (27)

Regardless of the assumption ab out adjustments, the degree of market power do es

not a ect the dynamics of the input prices. The dynamic prop erties of input prices

are governed by those of gross output rather than net output, since xed costs and

the number of rms are exogenous to the rm's input decision. For example, in

mo dels where the presence of xed costs is the only source of increasing returns

as in Hornstein (1993) and Rotemb erg and Wo o dford (1995), the intro duction of

monop olistic comp etition and increasing returns do es not a ect the dynamics of

input prices as a function of aggregate inputs.

Note that the exp onent term relevant for the input-price dynamics is strictly

smaller than that of the aggregate returns to scale, unless the adjustments are

instantaneous. This di erence has an implication for deriving indeterminacy from

increasing returns to scale, as in Benhabib and Farmer (1994). Since the dynamics

of the input prices are critical for the existence of indeterminacy, all sources of

increasing returns do not contribute to its existence. Of the three sources, the

degree of market power has nothing to do with the existence of indeterminacy.

The b ottom line of the mo del is as follows. The degree of returns to scale

of the aggregate reduced-form pro duction function is a convex combination of three

parameters: the degree of diminishing marginal cost, the degree of returns to variety,

and the degree of market p ower. The weights dep end on the sp eci cation of a zero-

pro t condition. Unless zero pro t is imp osed p erio d by p erio d, the dynamics of

input prices have information indep endent of the aggregate reduced-form pro duction

function.

In other words, even if wehave the dynamics of output and each input, we do not derive the

dynamics of the input prices. The dynamics of the input prices have some indep endent implications. 18

3 Implications

This section derives implications of the mo del and, based up on them, reviews and

critiques existing literature. The mo del is related to two branches of empirical

literature. A direct implication comes from the b ehavior of the aggregate reduced-

form pro duction function. The DSGE literature on monop olistic comp etition uses

a sp eci c parameterization of this pap er. Our general mo del in this pap er gives a

warning sign to b oth calibration and estimation approaches. Another implication of

the mo del involves the works using disaggregate data to identify some parameters.

This pap er gives interpretations for the estimates this literature have found, di erent

from how they used to be interpreted. In addition to these empirical implications,

this pap er derives welfare implications. We compare an economy of monop olistic

comp etition with a so cial planner's economy and nd that some previous welfare

results are due to restricted sp eci cations.

3.1 Identi cation with Aggregate Data

To facilitate the discussion of identi cation issues, we b egin by comparing the sp eci -

cations of how the existing literature parameterizes the sp eed of adjustments ( ) and

the p erio d-by-p erio d endogeneitybetween the xed cost and the numb er of rms (" ~).

For convenience, recall that the degree of returns to scale of the aggregate reduced-

form pro duction function is [ (~" +(1"~))+(1 )]. Therefore, this value is

of sole imp ortance in determining the aggregate dynamics of the pro duction func-

tion. The dynamics of input prices are governed by [ (~" +(1 "~))+(1 ) ],

the degree of returns to scale of aggregate gross output.

Hornstein (1993) and Beaudry and Devereux (1995a) assume that only the xed

cost is endogenous and that the adjustment to zero pro t o ccurs only at the steady

state. Since the resulting degree of aggregate returns is , diminishing marginal

cost and increasing returns to variety do not a ect the dynamics of the aggregate

reduced-form pro duction function. However, this do es not mean that, in a particular

general-equilibrium mo del, the dynamics of output dep end only on the degree of

market power. Since the dynamics of input prices dep end on diminishing marginal

cost and returns to variety, they also a ect output in a general-equilibrium mo del.

In Rotemb erg and Wo o dford (1995), the endogenous variable is not the xed cost

but the numb er of rms. However, this do es not makemuch di erence b ecause their

calibrated sp eed of adjustments is very low, a small . If the resp onses of the number

For example, the second and the third cases of Hornstein (1993) do not pro duce the same

results even if the only di erence b etween the two cases lies in the degree of diminishing marginal

cost. 19

"~n 0 1

0 Rotemb erg and Wo o dford (1995) Devereux, Head and Lapham (1996a,b,c)

Hornstein (1993)

Beaudry and Devereux (1995a)

Table 1: Examples of Sp eci cation

of rms happ en only at the steady state, a zero , then di erent assumptions on

endogeneity do not make any di erence at all.

The endogeneity of the number of rms matters if the zero-pro t condition is

imp osed p erio d by p erio d, as in Devereux, Head and Lapham (1996a,b,c). The

dynamics of b oth output and input prices dep end on the degree of returns to variety.

The degree of diminishing marginal cost and that of market power do not a ect

the prop erties of the variables, as far as the p ercentage deviations are concerned.

Table 1 summarizes the sp eci cations. The rst column corresp onds to a mo del

where adjustments to zero pro t o ccur only at the steady state. The second

column contains the static version of this pap er, where returns to variety matters.

The rst and the second rows corresp ond to mo dels of endogenous numb er of rms

and endogenous xed cost, resp ectively.

Discussion of econometric issues such as identi cation naturally involves the

sp eci cation of an error structure. A convention is to interpret the pro ductivity

sho cks as a random variable. Accordingly, this section restores the notation for

pro ductivity sho cks, A . Note that identi cation issues can be discussed only in

the context of a sp eci c mo del and available data. From the calibration p oint of

view, identi cation issues are interpreted as follows. If the mo del is not identi ed,

di erent parameter calibrations may result in the same mo del.

Supp ose that our mo del is the aggregate reduced-form pro duction function with

exogenous steady states and that wehave aggregate data on output, capital and la-

b or. Augmenting the linearized aggregate reduced-form pro duction function, Eq. (25),

with pro ductivity sho cks as an error structure, we have:

^ ^ ^ ^

Y =[ (~" +(1 "~))+(1 )] K +(1 )L + A :

t t t t

Using the aggregate data, we can draw inferences ab out the share parameter, , and

the aggregate returns to scale, [ (~" +(1"~))+(1 )]. The three parameters

representing the degrees are not identi ed separately. From the calibration p oint

Rotemb erg and Wo o dford (1995) is included in this category since is very small, even if not

exactly zero. 20

of view, di erent parameter calibrations may result in the same aggregate returns

to scale. For example, the dynamic prop erties of Rotemb erg and Wo o dford (1995)

with slow p erio d-by-p erio d adjustments, i.e. a small nonzero , can be replicated

by another economy where adjustments o ccur only at the steady state, i.e. zero

. Since the xed cost is exogenous (~" = 0), the degree of aggregate returns to

scale is [ +(1 )]. Because the degree of returns to variety is normalized to

1, an appropriate decrease in the degree of market power by ( 1) is the only

mo di cation necessary in the new economy without any short-run adjustments. The

dynamic b ehavior of all the aggregate variables, except for the gross output and the

number of rms, is the same as that of the original mo del.

More aggregate data may solve the identi cation problem. Data on the xed cost

or the number of rms would be helpful. However, it is not likely that we can get

measures of these two variables, consistent with this pap er. Now supp ose that we

have additional data on the rental rate or the wage and that the mo del also includes

the appropriate input-price equation, Eq. (26) or (27). The new mo del identi es

another linear combination of parameters, [ (~" +(1"~))+(1 ) ]. This

is di erent from the degree of aggregate returns and so the mo del identi es two

parameters, unless adjustments are instantaneous. For example, the two free pa-

rameters estimated in Kim (1996) are and , b oth of which are identi ed for the

following reason. Since adjustments are assumed to o ccur only at the steady state,

the degree of aggregate returns is and the dynamics of input prices are governed

by . Additional data on the interest rate provide information on the rental rate

and so is also identi ed.

Considering the identi cation problems, one may wonder why we care ab out

various increasing returns separately. This question will be answered when the

market economy is compared with a so cial planner. We will show that di erent

increasing returns have di erent normative implications. Before comparing with a

so cial planner, we review the literature using disaggregate data to identify some

parameters of the mo del.

3.2 Interpretation of Disaggregate Data

Recall that the aggregate economy has b een derived from a rm's problem. If we

interpret a rm as a particular sector of the economy, the data disaggregated to the

sectoral level have implications on the state of the economy. Two related literatures

apply this interpretation. Here, we evaluate the implications of these literatures by

Note that the replication of Rotemb erg and Wo o dford (1995) in the previous paragraph do es

not change input-price dynamics. Since b oth and are equal to 1, the change in do es not

change the dynamics. 21

using our general sp eci cation of increasing returns in a monop olistic-comp etition

mo del.

Firm's rst order conditions are the starting p oint of testing the jointhyp othesis

of p erfectly comp etitive markets and constant returns to scale, as in Hall (1996, 1998,

1990). However, compared with the mo del in this pap er, they do not incorp orate

a zero-pro t condition. Since it is assumed that there are no xed costs, there is

no di erence b etween the structural pro duction function and its reduced form. The

log-linearized pro duction function is:

dk dy dl dA

it it it it

= +(1 ) + :

i i i

y k l A

it it it it

The analysis of Hall (1986, 1988, 1990) is based on the measurement of pro ductivity

growth in Solow (1957). Calculation of the Solow residual requires the revenue share:

W l

t it

s = : (28)

p y

it it

(1 ). Without zero-pro t conditions or xed costs, the revenue share is equal to

Therefore, the Solow residual is:

dy dk dk dk dA dl dl

it it it it it it it

+( 1) s (1 s ) =(1)s + :

i R R R

y l k l k k A

it it it it it it it

This is the basis of testing for the joint hyp othesis of p erfect comp etition ( = 1)

and constant returns to scale ( = 1).

Hall (1990) prop oses a way to di erentiate diminishing marginal cost from market

power. He de nes a new share as follows:

W l

t it

s = :

Z k + W l

t it t it

This is called the cost share, since it is the share of lab or input in total cost, rather

than in total revenue as in Eq. (28). Note that it is equal to the share parameter,

(1 ), regardless of increasing returns and market p ower. The cost-based residual

is as follows:

dl dk dl dk dy dA

it it it it it it

s (1 s ) =( 1) s +(1s ) : +

C C i C C

y l k l k A

it it it it it it

Unlike the Solow residual, the cost-based residual can provide information only on

increasing returns. This algebra lets Hall (1990) conclude that his evidence p oints

in the direction of increasing returns, presumably coupled with market power.

The similar results for the Solow residual and the cost-based residual supp ort the absence of

pro ts in Hall (1990). Note that a zero-pro t condition is not imp osed. 22

However, this di erence between the Solow residual and the cost-based residual

disapp ears under the sp eci cation of this pap er which incorp orates xed costs and

zero-pro t conditions. First order conditions imply that the steady state of the rev-

enue share is the share parameter. The cost share is the share parameter regardless

of the sp eci cation. Considering that the xed cost do es not adjust instantaneously,

the rm-level reduced-form pro duction function is log-linearized as follows:

dk dy dl

it it it

=[ "~e +(1 "~e)] +(1 ) + error; (29)

i i i

y k l

it it it

where e represents the elasticity of aggregate inputs with resp ect to rm inputs.

The formula for b oth the Solow residual and the cost-based residual is:

dy dl dk dl dk

it it it it it

+ error; s (1 s) =[ "e ~ +(1 "~e) 1] s +(1 s)

y l k l k

it it it it it

where s represents b oth s and s . Since the revenue share is equal to the share

R C

parameter only at the steady state, this formula holds only approximately for the

cost-based residual. Since the cost-based residual provides the same information as

the Solow residual, the way prop osed in Hall (1990) cannot identify the degree of

diminishing marginal cost separately from that of market power.

While Hall (1986, 1988, 1980) incorp orates rst order conditions in his analy-

sis, others study increasing returns directly by regressing outputs on inputs, using

Eq. (29) as a regression equation. It would b e realistic to assume that "~e is close

to zero, since all three terms are likely to b e small. Furthermore, this assumption is

true for all existing sp eci cations summarized in Table 1. Under this assumption,

the literature can give information only on the market structure. In equilibrium, the

degree of market power is a mixture of two sources of increasing returns at a rm

level: diminishing marginal cost and the presence of xed costs. The two sources

cannot be separately identi ed by pro duction function regressions which use a re-

duced form. However, most DSGE pap ers arguing for or against the existence of

indeterminacy draw information on diminishing marginal cost from this literature.

This is valid only when there is no xed cost in the economy. We can discuss the

existence of indeterminacy only after identifying the degree of diminishing marginal

cost separately from that of market power. Furthermore, as we will show shortly,

this separate identi cation is imp ortant from a welfare p ersp ective.

dK k dL l

t it t it

The elasticity is: e = +(1 ) .

dk K dl L

it t it t

For a detailed discussion and the references, see Basu and Fernald (1997). 23

3.3 Comparison with a So cial Planner

A basic issue of welfare economics is whether a market solution will yield the so cial

optimum or not. Unlike the case of monop olistic comp etition, the zero-pro t con-

dition is not binding for a so cial planner. So the planner's problem is static in the

sense that there is no concern ab out the sp eed of adjustments. For ease of compar-

ison, adjustments to zero pro ts are assumed to be instantaneous in the economy

of monop olistic comp etition, i.e. = 1, and so time subscripts are suppressed. The

b ehavior of the market economy is summarized as follows:

market 1

= 1 K L ;

1 market

; K L N = 1

(1") 1

( )

" +(1")

market 1

; Y = K L 1

where the level parameter, , is normalized to 1 in the original equations, Eq. (14),

(15) and (16). The sup erscripts of `market' denote the market economy of monop-

olistic comp etition.

Note that the planner's optimization problem with resp ect to the xed cost is

not well-de ned. For the planner, it is optimal to decrease the xed cost as close

to zero as p ossible. So the welfare issues are considered only under the assumption

that the so cial planner is not allowed to control the xed cost. Given a pro cess of

the xed cost, the number of rms in an economy is chosen by the so cial planner

whose ob jective is:

i h

market 1 ( )

: N K L max N

Due to the partial-equilibrium setup of this pap er, it is natural for the so cial plan-

ner to consider aggregate capital and lab or as exogenous. Note that the market

structure a ects the planner's problem only through the exogenous pro cess of the

xed cost. For this problem to be well de ned, we assume that > . Under this

assumption, the number of rms has two o setting e ects on the aggregate out-

put. An increase in the number of rms increases b oth aggregate gross output and

aggregate xed cost.

This problem is di erent from that of Dixit and Stiglitz (1977) in that the cost function is

parameterized and that the utility function is not intro duced. This makes it easy to consider

monop olistic comp etition and increasing returns simultaneously. 24

The solution of this problem gives the optimal numb er of rms and the optimal

output:

! !

planner 1

N = 1 1 K L ;

! !

1 " 1

( ) ( )

" +(1")

planner 1

Y = 1 1 K L :

planner market

By construction Y Y , where the equality holds when = : A larger

degree of returns to variety is an incentive for the so cial planner to create more

planner

rms, i.e. N > 0.

Noting the similarity between the planner's economy and the market economy,

we have the following relation:

8 9 8 9 8 9

planner market planner market

> > > > > >

< N >N y

< = < = < =

planner market planner market

= N = N y = y

() () :

> > > > > >

: ; : ; : ;

planner market planner market

> N y

Figure 1 on page 27 compares the monop olistic-comp etition economy with the so cial

planner's. The horizontal axis is the degree of market power (). It is easy to see

that government intervention has ro om for welfare improvement by a ecting the

numb er of rms, unless the degree of market p ower is equal to the degree of returns

to variety.

When the two degrees match as in the conventional aggregator sp eci cation of

Eq. (5), the so cial planner has no ro om for improvement. With an endogenous

number of rms (" = 0), the graph of Y in the rst plot would be horizontal and

tangent to Y at its maximum, where the economy is lo cated. In other words, an

economy with the sp eci cation of Eq. (5) following Dixit and Stiglitz (1977) is not

only the b est in the class of monop olistic-comp etition economies but also as go o d as

the so cial planner's economy. It has b een argued that this optimalityis due to the

feature of constant elasticity of substitution. However, the more relevant reason

consists in the sp eci cation that the degree of returns to variety is equal to the degree

of market power. Note also that this particular optimality result relies on the

partial-equilibrium setup of this pap er. A general-equilibrium framework featuring

endogenous aggregate inputs may break the optimality of the partial-equilibrium

setup. For example, the equality between the outputs per rm do es not imply the

equalitybetween the numb ers of rms in Dixit and Stiglitz (1977), and government

exp enditure is welfare improving in Devereux, Head and Lapham (1996c).

See Romer (1987) for an example.

Benassy (1996) also p oints out that this match \is purely owing to an accidental implicit choice

of " the degree of returns to variety. 25

4 Further research

Based on the critical review of the literature, empirical work to identify the parame-

ters needs to b e done. Enough data b oth at aggregate and disaggregate levels might

enable the identi cation of all the parameters in this pap er. The issue is howto nd

a measure, say of the numb er of rms, relevant to this pap er. As to the mo del, the

sp eci cation of endogenous xed costs needs further analysis. It is more app ealing if

wehave a mo del which explains how the xed cost changes endogenously in resp onse

to the change in exogenous variables. The literature on R&D with entry and exit

can b e a starting p oint. Another topic that deserves attention is to include materials

as an input to the technology. Such extension makes the aggregation in an imp er-

fectly comp etitive economy less straightforward, since it involves value added rather

than gross output. Furthermore, it mightchange some results regarding returns to

scale.

See, for example, Hall (1986, 1988, 1990), Rotemb erg and Wo o dford (1995), and Basu and

Fernald (1997). 26 Planner So cial a with 27 Comparison

Output per Firm Number of Firms Aggregate Output γ γ γ Figure 1: γ(−) ρ εγ+(1−ε)ρ Degree ofMarketPower ρ ρ (µ) Y Y y y N N m p m p p m

A External Increasing Returns

Here we supp ose that externalities come from inputs, rather than outputs. The

technology of the rms without externalities, Eq. (1), is rewritten for ease of com-

parison:

y = k l :

Input externalities have b een parameterized in two di erent ways.

If average inputs a ect the output of rms, their technology is:

1 1

y = k l k l ;

where k and l are average inputs and denotes the degree of externalities. Following

the same pro cedure of the aggregation and the zero-pro t condition, the degree of

aggregate returns to scale in a static mo del is [~" ( + )+(1 "~)]. Note that

externalities via average inputs change the degree of diminishing marginal cost from

to ( + ), without any change in the degree of returns to variety. In a dynamic

case, since the relative size of xed costs dep ends on the internal increasing returns

only, the e ect of market structure is magni ed by a factor of . That is, the

returns to scale in our most general sp eci cation is,

" ! #

[~" ( + )+(1 "~)]+(1 ) :

Pro duction externalities may come through total inputs as follows:

1 1

; K L y = k l

where K and L denote total capital and lab or in the economy. Such externalities

magnify not only the degree of diminishing marginal cost but also that of returns

to variety by a factor of . The degree of market power is magni ed, to o. The

degree of returns to scale is,

[ (~" +(1 "~))+(1 )]:

The ab ove algebra shows that a mo del with external increasing returns and one

with internal increasing returns can replicate the dynamic prop erties of each other,

sub ject to the following quali cation. The degree of returns to variety and the

degree of market power should be changed appropriately. For example, if internal

returns are replaced with external returns, the degree of market power in a new

), the old degree of market power divided by the mo del should be reduced to (

degree of internal returns.

In the App endix, all subscripts denoting rm and time are omitted.

This equivalence is also explained in Benhabib and Farmer (1994). 28

B Input Fixed Cost

Supp ose that the rms pay xed costs as part of their inputs as well as their output.

Then the technology of the rms is:

y = (k k ) (l l ) ;

0 0

where k and l are input xed costs. The aggregate version of the zero-pro t

0 0

condition is:

" #

K L

N = 1 +(1 ) (K Nk ) (L Nl ) :

0 0

K Nk L Nl

0 0

Unless there is no output xed cost ( = 0), this condition is not analytically

tractable. Assuming that there is no output xed cost, the aggregate technology

and the zero-pro t condition are simpli ed as follows:

; (K Nk ) (L Nl ) Y = N

0 0

K L

= +(1 ) :

K Nk L Nl

0 0

With b oth input xed costs, the zero-pro t condition has a conceptual problem.

Interpreted as an equation determining N , it pro duces two discrete solutions. If k

and l are endogenous, it has a continuum of solutions. Therefore, we need to pin

L K

or . With the two ratios de ned as R and R , the down either

K L

K Nk LNl

0 0

linearized version is:

dY dN

= ( )

Y N

" ! !#

dK d(Nk ) d(Nl ) dL

0 0

+ R (R 1) (R 1) +(1 ) R ;

K K L L

K Nk L Nl

0 0

" # " #

dK d(Nk ) dL d(Nl )

0 0

0 = R (R 1) +(1 )R (R 1) :

K K L L

K Nk L Nl

0 0

If the input xed costs are exogenous and the number of rms resp onds instan-

taneously, the linearized aggregate reduced-form pro duction function is:

dK R 1 dY

= ( )+ +(1 )(R R +1)

L K

Y R 1 K

R 1 dL

+(1 ) ( )+ (1 )+ (R R +1) ;

K L

R 1 L

L 29

R ( R 1)

K K

where = . With all the complexity of the functional form,

R (R 1) +(1 )R ( R 1)

K K L L

the degree of returns to scale is simply . However, we may assume that more

unpro ductive inputs are necessary in a b o om. If the aggregate input xed costs are

endogenous, the aggregate pro duction function in a reduced form is:

dK dL dY

= +(1 ) :

Y K L

The degree of returns to scale is .

In a dynamic case when the adjustments o ccur only at the steady state, the

linearized aggregate reduced-form pro duction function is:

dY dK dL

= R +(1 )R :

K L

Y K L

The zero-pro t condition implies that the degree of returns to scale is . The mo del

of Yun (1996) intro duces only lab or xed cost, i.e. k = 0, and so magni es only

lab or uctuations in output dynamics.

The b ottom line of this app endix is that the mo del with b oth input xed costs

has richer dynamics in the sense that capital and lab or uctuations may b e magni ed

to a di erent degree. However, this comes at a cost of one additional restriction.

Three extreme cases show that the degree of returns to scale is the same as that of

the mo del with only the output xed cost. 30

C Cost Minimization

The framework of cost minimization has b een used widely, for example in Horn-

stein (1993) and Devereux, Head and Lapham (1996a). The cost function of a rm

is de ned with rescaled input prices:

c (y ) = min Z k +(1 )Wl s.t. k l =(y+) ;

( k;l)

where the general price level, P , is normalized to 1. The rst order conditions

pro duce the conditional demands:

(y + ) k = ;

l = (y + ) :

So the cost function is:

c (y )= Z W (y+) :

Pro t maximization sub ject to the output demand of the aggregator, Eq. (8),

implies the constant markup rule:

= ;

where mc = Z W : The marginal cost is decreasing if > 1. How- (y + )

ever, the second-order sucient condition is satis ed in equilibrium, since xed cost

is assumed to b e p ositive. The maximized pro t is:

" #

= Z W (y + ) y (y + ) :

Therefore, the condition of zero pro ts at each p erio d results in the following rela-

tion:

= 1 y;

which involves output rather than inputs as in Eqs. (4) and (11). This relation

shows that output p er rm is prop ortional to the xed cost, whether it is exogenous

or not.

Interpreting the zero-pro t condition as follows:

! # "

(1)

; 1 Y N =

" ! #

= 1 Y ;

we transform the conditional demands and the cost function into an aggregate ver-

sion as follows:

K = ; (30) Y

L = ; (31) Y

C (Y ; N (Y );(N)) = Z W Y ;

( )

where = . The constant markup rule is expressed as follows:

1 1

(1)

( )

= ; (32)

h i

(1)

( )

. MC is not where p = 1 Y and MC = Z W Y

(1)

( )

the aggregate marginal cost but just a name for the aggregate counterpart of the

rm's marginal cost. Note that the inverse of the markup is equal to C (Y ; N; )

evaluated at the zero-pro t condition, not to C (Y ; N (Y );(Y )).

Eqs. (30){(32) are a complete description of the economy. Solving for Y as a

function of K and L, we have:

)

(

1 1

: Y = K L

The degree of returns to scale is the weighted harmonic average of the degree of

diminishing marginal cost and that of returns to variety. The input prices are:

Z = ;

W = :

This shows that our results on input prices in a static mo del are robust to the choice

of the framework. 32

D Fixed Cost Externalities

In the static case, b oth the xed cost and the number of rms are endogenous and

so the degree of returns to scale is a convex combination of the degree of diminishing

marginal cost and that of returns to variety. This app endix shows that the static

mo del is equivalent to two mo dels where the xed cost is exogenous and only the

numb er of rms is endogenous, i.e. " =0. That is, aggregate output and the number

of rms follow Eqs. (10) and (12), rewritten here for convenience.

( ) 1

Y = N K L N ;

0 1

@ A

N = K L :

Instead of endogenous xed costs, each mo del intro duces an assumption ab out xed

cost externalities. The new parameters, and , represent the external diseconomies

such as congestion e ects.

First, external diseconomies are assumed to a ect the xed cost as follows:

= Y ; (33)

where 2 [0; 1]. An increase in aggregate output would increase the xed cost per

rm, less than prop ortionately. Combining Eq. (33) with the ab ove two equations,

the equilibrium aggregate output is:

)

(

Y = K L ;

( )

where = 1 . The degree of returns to scale is a harmonic

average of the degree of diminishing marginal cost and that of returns to variety.

Alternatively, negative externalities is assumed with resp ect to the number of

rms as follows:

= N ;

where 2 [0; 1). An increase in the number of rms would increase the xed cost

per rm, p ossibly more than prop ortionately. The same algebra shows that the

equilibrium output is:

( ) ]

) [(

+ +

; Y = K L

( )

+ +

where = 1 . As the degree of external- 1 1

2 2

ity ( ) grows from 0 to 1, the degree of returns to scale moves from the degree of

returns to variety() to the degree of diminishing marginal cost ( ). 33

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