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Dixit and Stiglitz model

Dixit and Stiglitz model
Dixit and Stiglitz model

3

Monopolistic Competition

The standard model of economic geography relies on the Dixit–Stiglitz model of monopolistic competition,which makes it possible to inte-grate both increasing returns and imperfect competition in a very simple and elegant way.This combination is crucial for economic geography. Indeed,as seen in chapter2,the presence of increasing returns is neces-sary to explain the agglomeration of activities in a homogeneous space. Under increasing returns at the?rm level,the assumption of perfect competition becomes untenable,as marginal cost pricing results in neg-ative pro?ts.We therefore need a setting that integrates both increasing returns and imperfect competition to analyze the formation of economic agglomeration.Moreover,this setting must be of the general-equilibrium type as it has to include the interactions between product and labor markets.Finally,trade must arise in equilibrium if we want to avoid regions or countries ending up in autarky(see chapter2).Of all the market structures studied in industrial organization,economic geogra-phy has focused on monopolistic competition,even though alternative approaches have been followed(see chapter9).The aim of this chapter is to explain the reasons for this choice and to survey this family of mod-els.We focus here on a closed economy;the case of an open economy will be dealt with in chapter4.

The concept of monopolistic competition goes back to Chamberlin (1933).1It can be described by means of the following four assumptions:

(i)Firms sell products of the same nature but they are not perfect

substitutes—we refer to them as the varieties of a di?erentiated good.

(ii)Every?rm produces a single variety under increasing returns and chooses its price.

1The interested reader will?nd a more concise and,above all,clearer presentation of his ideas in Chamberlin(1951)than in the various editions of his earlier work.In this chapter,we shall focus solely on the formulations used in international economics and economic geography.

54 3.Monopolistic Competition

(iii)The number of?rms belonging to the industry is su?ciently large for each of them to be negligible with respect to the whole group of?rms.

(iv)Finally,there is free entry and exit,so pro?ts are zero.

These assumptions bear a strong resemblance with those of perfect com-petition,the main di?erence being the fact that here each?rm sells a speci?c product and chooses its own price.This endows each?rm with a speci?c market,in which the?rm has some monopoly power.How-ever,the existence of other varieties implies that the size of this market depends on the behavior of the other?rms,thus constraining each pro-ducer in their price choice.In other words,although the?rm is not in a situation of perfect competition,neither is it in a situation of monopoly. Finally,since?rms operate under increasing returns,the resources avail-able in the economy impose a limit on the number of varieties capable of being produced.In general,this number depends on the entry barriers that?rms face.In the case of monopolistic competition,it is assumed that the?xed cost associated with the launching of a new variety is the only e?ective barrier.Such an entry barrier is nonstrategic because it cannot be manipulated by the?rms.

After having attracted a great deal of attention in the1930s,Cham-berlin’s ideas lost most of their appeal until Spence(1976)and,above all,Dixit and Stiglitz(1977)brought them back onto the scienti?c stage by proposing a model capable of being used in various economic?elds. Spence has developed a partial-equilibrium setting,whereas the Dixit–Stiglitz model places itself in a general-equilibrium context.We will therefore focus on the Dixit and Stiglitz model here.The main purpose of this chapter is to present an up-to-date discussion of this model and to study its principal properties.We will place special emphasis on the role played by the assumption of a continuum of?rms.Hotelling(1929) and Aumann(1964)have shown that the idea than an agent(whether a consumer or a?rm)has no in?uence on a market can only be captured by means of a continuum of agents,which are all negligible in the sense of measure theory.2Despite its appropriateness to describe Chamberlin’s intuition,it took a long time for this idea to be integrated into the frame-work of monopolistic competition.In the second section of this chapter we will present a linear model of monopolistic competition recently put forward in economic geography to remedy certain weaknesses of the Dixit–Stiglitz model.

2Since then,the same assumption has been made in several economic?elds.

3.1.The Dixit–Stiglitz Approach55 3.1The Dixit–Stiglitz Approach

The economy is made up of two sectors called agriculture(or the tradi-tional sector)and industry(or the modern sector).The denomination of these two sectors is conventional and can vary according to the histori-cal period under consideration.For example,for a long time the textile industry was the industry of reference,while this role was later taken by steel and,after that,by the automobile sector.What really matters for our purposes are the market and technological properties of these two sectors:in agriculture,a homogeneous good is produced under constant returns and is sold in a perfectly competitive market;in the manufactur-ing sector,?rms produce a di?erentiated good under increasing returns and compete in a monopolistic competition setting.

3.1.1Consumption and Production

3.1.1.1Preferences and Demand Functions

The economy involves L consumers whose preferences are identical and are given by a Cobb–Douglas utility function:

U=CMμA1?μ,0<μ<1,(3.1)

where C is a positive constant chosen so that the coe?cient of indi-rect utility is normalized to1.3In this expression,A denotes the quan-tity of the agricultural good and M denotes the quantity of a composite di?erentiated good de?ned by a CES-type index:

M≡

n

i=1qρ

i

1/ρ

,0<ρ<1,

where q i is the quantity of variety i consumed,n is the total number of varieties available,andρis a parameter that,as we will see later, is an inverse measure of the degree of di?erentiation across varieties. We therefore assume that the varieties are di?erentiated and that they a?ect the value of M in a symmetric way.4Instead of usingρ,it will often prove convenient to use the parameterσ,the elasticity of substitution between any two varieties.These two parameters are related through the following expressions:

σ=

1

1?ρ

orρ=

σ?1

σ

.

3It is readily veri?ed that C is such that C?1≡μμ(1?μ)1?μ.

4The assumption that?rms sell di?erentiated goods is in itself justi?ed by the“princi-ple of di?erentiation,”which says that?rms soften competition by selling di?erentiated goods(Tirole1988).

56 3.Monopolistic Competition The elasticity of substitution therefore ranges from 1to ∞.The index M may then be rewritten as follows:

M = n i =1

q (σ?1)/σi σ/(σ?1).(3.2)

When σtends to ∞(ρ=1),varieties are perfect substitutes as

M =n

i =1q i .

By contrast,they are totally independent when σ=1(ρ=0),as the index M boils down to a Cobb–Douglas subutility function with M = n i =1q i .For all intermediate values of σ,varieties are imperfect substitutes,ρand σbeing inverse measures of the degree of product di?erentiation across varieties.

Assume that a consumer has a quantity ˉM

of the composite good that is uniformly distributed among a limited number k

k i =1

ˉM k (σ?1)/σ μ(σ/(σ?1))A 1?μ=k μ/(σ?1)A 1?μˉM μ,which is an increasing function of k since σ>1.Consequently,rather than concentrating her consumption over a small number of varieties,every consumer prefers to spread it over a larger number of varieties until k is equal to the total n of varieties available.This property means that the CES index M incorporates what is called a preference for diver-sity —a preference that pushes consumers to purchase all the available varieties at an intensity that varies with the parameter σ(Bénassy 1996).This is because individuals want to avoid the boredom generated by the repeated consumption of the same variety:they prefer consuming dif-ferent varieties,either one by one or simultaneously.Such an assump-tion also captures the basic idea,mentioned in the foreword,that a greater range of choices makes large urban regions more attractive to consumers.5As will be seen below,such preferences imply that the intro-duction of a new variety does not lead to the disappearance of existing varieties but rather to a reduction in their consumption.

If p a denotes the price of the agricultural good and P the price index of the manufactured good (which will be de?ned in section 3.1.1.3),the budget constraint of a consumer,whose revenue is y ,is

PM +p a A y.

5Remember

that the preference for diversity is formally identical to the convexity of

indi?erence curves and that this,in turn,is equivalent to the assumption of a quasi-concave utility function.

3.1.The Dixit–Stiglitz Approach 57In this case,it is well-known that the aggregate demand functions take the form M =E P and A =E a p a

,(3.3)where E ≡μLy and E a ≡(1?μ)Ly are the expenditures of all consumers on the manufactured good and the agricultural good,respectively.

Expenditure E being given,the consumption of each variety is then obtained by maximizing (3.2)under the constraint n j =1p j q j E .The Lagrangian of this problem is

L =M +λ E ?

n j =1

p j q j ,and the ?rst-order conditions are

?L ?q i =?M ?q i

?λp i =0,i =1,...,n,(3.4)?L ?λ=E ?n j =1

p j q j =0.(3.5)Conditions (3.4)are equivalent to

M 1/σq ?1/σi =λp i ,i =1,...,n,

which can be rewritten as follows:

q i =Mλ?σp ?σi .(3.6)

If we multiply (3.6)by p i ,add the so-obtained terms over i ,and substitute this sum in (3.5),we get

Mλ?σ=E

j p ?(σ?1)j ,

which gives us,after substitution into (3.6),the aggregate demand function for variety i :

q i =p ?σi

j p ?(σ?1)j E,i =1,...,n.(3.7)

The demand for a variety is,therefore,a function of the prices of all varieties,which is in contrast to what we will see later with models of spatial competition,in which each ?rm is in direct competition only with its immediate neighbors in space (see chapter 9for further details).If one ?rm sets a higher price than its competitors,then consumers reduce their consumption of the corresponding variety,but their preference for variety has the e?ect of keeping the demand for this ?rm positive.

58 3.Monopolistic Competition

Expression(3.7)implies,moreover,that the introduction of a new vari-ety increases the denominator and,consequently,leads to a reduction in the demand for the existing varieties i=1,...,n so long as their prices remain unchanged.In other words,the entry of new varieties triggers the fragmentation of demand over more varieties.This is known as the “market-crowding”(or fragmentation)e?ect.

Finally,the relative demand of two varieties is given by

q i q j =

p i

p j

,

which is independent of the price of other varieties.This property, although restrictive,will prove very useful in various empirical appli-cations in which the denominator of the CES demand function(3.7)is di?cult to estimate.Furthermore,because the elasticity of substitution, which is given by

?ln(q i/q j)

?ln(p i/p j)

=?σ,

is constant,it becomes clear why the term CES stands for“constant elasticity of substitution.”

3.1.1.2Preference for Diversity and Heterogeneous Consumers

The assumption of identical consumers is similar to that of the represen-tative consumer used by Dixit and Stiglitz,which has long been known to have severe weaknesses(Kirman1992).Furthermore,the fact that con-sumers have a taste for diversity implies that all individuals consume the whole array of varieties.Such behavior may seem unrealistic,but it is in fact less restrictive than it seems at?rst glance.Anderson et al. (1992,chapter3)have demonstrated that the same demand functions can be obtained from a population of heterogeneous consumers who buy a single variety,while it was assumed above that consumers are identical and consume all varieties.

To show this,consider a situation in which each individual chooses a single variety i,which she consumes in quantity q i;in this case,we have in mind mutually exclusive or discrete choices.In addition,each individual spends an amount E/L on the manufactured good,where L has been de?ned as the number of consumers.The utility function asso-ciated with the consumption of variety i,ignoring the agricultural good, is assumed to be given by

?U

i=ln q i+εi,i=1,...,n,

whereεi is a random variable whose realization measures the quality of the match between this consumer and variety i.At identical prices,a

3.1.The Dixit–Stiglitz Approach59

speci?c consumer’s ideal variety is the one for which the realization of the random variable is highest.Because they face di?erent realizations of εi,di?erent consumers have di?erent matches with variety i.Given her own set of matches,each consumer chooses the variety i that provides her with the highest utility.The corresponding indirect utility is thus given by

?V

i=ln(E/L)?ln p i+εi,

since her consumption of variety i is equal to q i=E/Lp i,where p i is the price of variety i.In this context,the distribution function ofεi (i=1,...,n),assumed to be the same for each variety,re?ects the hetero-geneity of consumers’tastes with respect to the varieties o?ered because the realizations ofεi vary across consumers.

Our objective is to determine the aggregate demand for each variety. To do this,it is necessary to make certain assumptions about the het-erogeneity of consumers,i.e.,about the distribution ofεi.Ever since the work of McFadden(1974),it has been known that the probability of a consumer choosing variety i is given by the multinomial logit(MNL),

P i=

exp(?(1/ν)ln p i)

n

j=1

exp(?(1/ν)ln p j)

=

p?1/ν

i

n

j=1

p?1/ν

j

,

if and only if theεi are independent and distributed according to the Gumbel distribution(the parameterνrepresents the degree of dis-persion of consumers’preferences).6Moreover,if consumers’individ-ual choices are independent,the expected aggregate demand is equal to L times the individual probability of choosing i,multiplied by the individual consumption of this variety:

D i=L P i q i=

p?1/ν

i

n

j=1

p?1/ν

j

E

p i

.

Ifν=1/(σ?1)>0,we fall back on the CES-type demand(3.7).In other words,the assumption that consumers have a preference for diversity is equivalent to assuming the existence of a heterogeneous population in which each individual consumes a single variety chosen from the array of available varieties.Consequently,we can focus on the CES formulation

6The cumulative function of a random variableε

i distributed according to the Gumbel law is given by F(εi)=exp[?exp(?(εi/ν+γ))],whereγ≈0.577is Euler’s constant.Its mean isγνand its varianceν2π2/6.The Gumbel law provides a fairly good approxima-tion of the normal law and has the further advantage of having an explicit form for its cumulative function.It also has several appealing properties in discrete choice theory. The reader is referred to Anderson et al.(1992,chapter2)for a detailed discussion of this distribution.

60 3.Monopolistic Competition

presented above without worrying too much about the assumption that all individuals consume all varieties.

Clearly,the expression

s i≡p i q i

E

=

p?(σ?1)

i

n

j=1

p?(σ?1)

j

denotes the market share of variety i.It is therefore equal to a logit-type probability,the interpretation of which is fairly straightforward:the probability that a variety is chosen depends negatively on its price and positively on the price of the competing varieties.This property has two interesting implications.First,it highlights some of the links between CES and MNL.In both cases,the probability of choosing a variety is the same.It is the quantity of the chosen variety that is consumed which distinguishes these two models.In the CES case,this quantity is equal to the income spent on the manufactured good divided by the price of the variety,while in the MNL the demand is totally inelastic and is equal to 1.Furthermore,as we have seen,the ratio of the CES demands for two varieties is independent of the price of the other varieties—a property also shared by the MNL case.

3.1.1.3Price Index

We?rst show that the denominator of the demand function(3.7)is directly related to the price index of the manufactured good.Introduc-ing the equilibrium consumption of each variety into the de?nition of the composite good(3.2)yields

M=

n

i=1

p?σ

i

j

p

j

E

(σ?1)/σ σ/(σ?1)

=E

i

p?(σ?1)

i

(

j

p?(σ?1)

j

)(σ?1)/σ

σ/(σ?1)

=E

n

i=1p?(σ?1)

i

1?((σ?1)/σ) σ/(σ?1)

=E

n

i?1p?(σ?1)

i

1/(σ?1)

,

so the total expenditure E on the manufactured good,which is a fraction of the total income,is given by

E=M

n

i=1p?(σ?1)

i

?1/(σ?1)

.

3.1.The Dixit–Stiglitz Approach 61This expression says that the level of expenditure on the manufactured good is equal to the CES aggregate of the quantities consumed,M ,times a term that may be interpreted as the price index of the manufactured good:P ≡

n i =1

p ?(σ?1)i ?1/(σ?1).One important property of this index is that it decreases with the num-ber of varieties available .Indeed,if all the prices are identical and equal to p ,we come up with

P =

n i =1

p ?(σ?1)i ?1/(σ?1)=pn ?1/(σ?1),(3.8)which is a decreasing function of the number n of varieties because σ>1.This property is nothing but the counterpart in the price space of the preference for diversity in the variety space.Furthermore,the less di?erentiated the varieties,the lower the price index.This accounts for the fact that a larger elasticity of substitution makes the di?erentiated product less attractive to consumers.A lower price index may then be viewed as a compensation.

The demand functions (3.7)may then be rewritten as follows:q i =p ?σi P σ?1E = p i P ?σE P and A =E a p a

.(3.9)Hence,the demand for a variety increases with the price index so that a low (respectively,high)price index means that the product market is more (respectively,less)competitive.In other words,a ?rm’s demand accounts for the aggregate behavior of its competitors via the price index .This demand may thus be interpreted as the result of a two-stage pro-cess:consumers ?rst choose the amount to spend on the manufactured good according to the index P ,before sharing it among the varieties available according to their speci?c price.In other words,the term p i /P encapsulates the competition e?ect between variety i and the other vari-eties,while the second term E/P denotes the aggregate demand for the manufactured good.

Introducing the demands (3.9)into the utility function (3.1)yields a consumer’s indirect utility,which evaluates her well-being in terms of her income y and prices,

V =y P μp 1?μa

≡ω,(3.10)which is here equal to her real income ,the nominal income being divided by the price P μp 1?μa

of the batch consumed.As will be shown later on,

62 3.Monopolistic Competition

in the standard model of economic geography with labor mobility,the consumer chooses her residence and her workplace by comparing the real incomes she can earn in the various regions.

3.1.1.4Technologies

Agriculture uses only unskilled labor.We assume that there is perfect competition and constant returns in this sector,so that the price of the agricultural good(p a)is equal to its marginal cost,which is equal to the marginal labor requirement(m a)times the agricultural wage(w a): p a=m a w a.Without any loss of generality,we assume that one worker produces one unit of agricultural good,which means that m a=1.Conse-quently,the price of this good is equal to the wage of unskilled workers (p a=w a).The agricultural good being the numéraire(p a=1),we have

p a=w a=1.

In the manufacturing sector,there are increasing returns at the?rm level,but no scope economies that would induce a?rm to produce sev-eral varieties.Each?rm therefore produces a single variety.Furthermore, no two?rms sell the same variety because this would allow them to relax price competition(see also chapter9).Consequently,the n varieties of the manufactured good are produced by n di?erent?rms.We can there-fore identify the set of varieties with the set of?rms.The technology is identical in all locations—there are no comparative advantages—and for all the varieties—there are no speci?c advantages—so that space is homogeneous in the sense of chapter2.

Three strategies for the modeling of technologies are found in the literature.They make use of various assumptions regarding production factors.In particular,labor is either homogeneous or heterogeneous,in which case we distinguish between skilled and unskilled workers.Each worker supplies one unit of her type of labor.

In the?rst modeling strategy,labor is homogeneous,and therefore perfectly mobile between sectors,which implies that the wage prevailing in the manufacturing sector is equal to w a.The amount f denotes the ?xed requirement and m denotes the marginal requirement of labor in each?rm.The production of q i units of variety i thus requires a total quantity of labor equal to l=f+mq i.In this case,the production cost is given by

C(q i)=f w a+mw a q i=f+mq i.(3.11)

This cost function is therefore of the unique-factor type.

In the second strategy,labor is heterogeneous and speci?c to each sector.There are L a unskilled workers employed in agriculture and L

3.1.The Dixit–Stiglitz Approach63

skilled workers in the manufacturing sector.The cost of producing q i units of variety i is now given by

C(q i)=f w+mwq i,(3.12)

where w is the wage of skilled workers,which typically di?ers from w a. This cost function is of the speci?c-factor type.

Finally,in the third strategy,labor and capital are the production https://www.wendangku.net/doc/ab9961975.html,bor is homogeneous and a worker’s wage is equal to w a,regard-less of the sector.Each?rm uses a?xed requirement f of capital and a variable quantity of labor mq i.The production cost is therefore given by

C(q i)=f r+mw a q i=f r+mq i,(3.13)

where r is the return on capital.In this case,the cost function is of the crossed-factor type.

We will use(3.12)later on in this chapter,while the other two speci?cations will be considered elsewhere in the book.

3.1.2Market Equilibrium

3.1.2.1Price

Let w be the wage of skilled workers.Firm i’s pro?t is then

πi=p i q i?C(q i)=(p i?mw)q i?f w,

where the demand q i is given by(3.9).Firm i’s equilibrium price is deter-mined by maximizing the pro?tπi with respect to p i.Letting the price elasticity of the demand(3.9)for variety i,denoted by i,be given by

i=??q i

?p i

p i

q i

,

the?rst-order condition gives the following classical result:

p i

1?

1

i

=mw,

provided i>1,otherwise the left-hand side of this expression would be negative.Here lies one of the key assumptions of the Dixit–Stiglitz model: the absence of strategic interactions between?rms—an assumption that explains both its simplicity and its success in various applications.This point deserves further elaboration.

64 3.Monopolistic Competition

First,if we assume that the?rm takes expenditure E as constant,we have

?q i ?p i =

?σp?σ?1

i

j

p?(σ?1)

j

+p?σi(σ?1)p?σi

j

p?(σ?1)

j

2E

=?σ

p?σ?1

i

j

p?(σ?1)

j

E+

p?σ

i

j

p?(σ?1)

j

2

(σ?1)E

=?σq i

p i

+q2iσ?1

E

,

so

i=σ?(σ?1)q i p i

E

=σ?(σ?1)s i.(3.14)

In a symmetric market,when the number of?rms increases,each of them experiences a drop in its market share,s i.Ultimately,when n tends to ∞,s i tends to0,so that the price elasticity is such that

i=σ.

Therefore,if we assume that the number of?rms is large,the?rst-order condition determining the equilibrium price boils down to a very simple

expression:

p?=

σ

σ?1

mw,(3.15)

so the relative markup is constant and equal toσ/(σ?1),withσ>1. The second-order condition is also satis?ed.For a given wage,the Lerner index is independent of the number of?rms and is equal to

p??mw

p =1

σ

,

which increases with the degree of di?erentiation across varieties.The expression(3.15)agrees with what we know from industrial economics: the equilibrium price exceeds the marginal cost(mw)as soon as vari-eties are di?erentiated,i.e.,as soon asσtakes on a?nite value greater than one,and the pro?t margin increases with the degree of di?erentia-tion,i.e.,as soon asσdecreases.Consumers being less sensitive to price considerations,?rms are then able to charge higher prices.Conversely, in the special case of homogeneous varieties(σtends to∞),we fall back on Bertrand’s solution:the equilibrium price is equal to the marginal cost.7

7Even though the number of?rms is arbitrarily large,a?nite value forσimplies that the equilibrium price remains greater than the marginal production cost.This result runs against the conventional wisdom that a very large number of producers is equivalent to perfect competition.

3.1.The Dixit–Stiglitz Approach 65The above expressions have been obtained under price competi-tion (Bertrand).Let us now assume that there is quantity competition (Cournot).Applying the ?rst-order condition yields p i 1?1 C i

=mw,where 1 C i

≡??p i ?q i q i p i .Using the inverse demand function p i =q ?1/σi

j q 1?1/σj E,

we can determine C i as follows:1 C i =1σ+ 1?1σ s i .(3.16)

When the number of ?rms tends to in?nity,we again ?nd that C i =σ.

To sum up,when there are many ?rms,everything works as if each ?rm had a zero market share.Consequently,when it chooses its strategy,a ?rm anticipates that its decision has no signi?cant impact on the mar-ket,so that it does not a?ect its competitors’own choices of strategies.In other words,everything works as if the best-reply functions were hor-izontal .Unlike in the case of oligopolistic competition,therefore,there are no strategic interactions because a ?rm’s pro?t-maximizing strategy does not depend on the strategies of the other ?rms.

From a more formal point of view,this amounts to saying that,when it maximizes its pro?t,a ?rm assumes that its price choice has no impact on the price index P or on consumers’expenditure E ,which both appear

in the demand (3.9).We thus obtain i = C i

=σ.We will see in sec-tion 3.1.2.2how an alternative model makes it possible to justify such a behavioral assumption.Furthermore,competition on price or on quan-tity leads to the same equilibrium in monopolistic competition,while these two forms of competition give di?erent results in oligopolistic competition.Finally,the assumption of symmetry between varieties,with respect to both consumer preferences and ?rms’technologies,is re?ected in the same equilibrium price for all varieties.

3.1.2.2Quantity

We can substitute the equilibrium price (3.15)into the demand function (3.9)to determine the quantity produced by each ?rm as a function of the

66 3.Monopolistic Competition

number of varieties(and wages).Then,by reintroducing prices and quan-tities into the free-entry condition,πi=0,we can obtain the number of ?rms,and therefore the number of varieties,existing in equilibrium.

It is in fact simpler,but strictly equivalent,to proceed in the reverse order by?rst determining the volume of production thanks to the free-entry condition given by

πi=(p?mw)q?f w

=mw

σ?1

q?f w=0,

which makes it possible to obtain the equilibrium production of each

?rm:

q?=(σ?1)f

m

.(3.17)

Regardless of the total number of?rms,they all have the same size.This result is a direct consequence of the fact that the markup is constant, and it is one of the major weaknesses of the Dixit–Stiglitz model.More generally,in this model,the entry of new?rms does not generate any procompetitive e?ect:the markup is independent of the number of?rms n,while industrial economics suggests that it decreases with n.8Second, there is no scale e?ect,as q?is independent of both the share of the manufactured goodμin consumption and the number of consumers L. It is important to keep these two limitations in mind.Although the Dixit–Stiglitz model is very useful for its great simplicity,it fails to capture some important e?ects.

3.1.2.3The Number of Firms

As the quantity of skilled labor used by a?rm is equal to l?=f+mq?= fσ,the number of?rms is thus determined by the full-employment condition:

L=nl?=n(f+mq?),

from which it immediately follows that

n?=

L

σf

.(3.18)

One di?culty should be pointed out right away:n?is not necessarily an integer,so the number of?rms in equilibrium is given by the largest

8This observation needs quali?cation,however.Indeed,even if the equilibrium price remains unchanged when the number of?rms increases,the consumption of the man-ufactured good is fragmented over a greater number of varieties.This in turn implies that each?rm’s pro?ts go down.In other words,we come back,albeit very indirectly,to a kind of competitive e?ect(which has been called the market-crowding e?ect),as the entry of new?rms has a negative e?ect on the pro?tability of the incumbents.

3.1.The Dixit–Stiglitz Approach67

of the integers lower than or equal to n?.Such an approximation only makes sense when n?is large.Having said that,the zero-pro?t condi-tion is equivalent to the famous Chamberlinian condition of tangency between the?rm’s demand and its average cost at the free-entry equi-librium.This implies that increasing returns are not totally exploited in equilibrium,as the average cost is not minimized,which is explained by the fact that consumers value diversity in their consumption of the manufactured good.

Although?rms’sizes are una?ected when the economy gets larger, there is a scale e?ect at the market level.It takes the speci?c form of a growth in the number of varieties.The greater the increase in the num-ber of workers/consumers L,the greater the increase in the number of ?rms—and consequently in the number of varieties.The price index then decreases,as seen above,thus contributing to making all consumers better-o?.Likewise,the greater a drop in?xed costs f is,the smaller the ?rms and the higher their number in equilibrium,with the same conse-quences on the price index and on individual welfare.We should note, however,that,so long as the?xed costs are positive,the number of?rms and varieties is?nite.Indeed,if the demand for a new variety proves to always be positive,consumers are not numerous enough for the pro?ts that they create to cover the?xed cost associated with the launching of an additional variety.The entry process must therefore come to a halt. The number of varieties obtained in this way has little chance of being the socially desirable number.Indeed,when a new?rm enters the mar-ket,it ignores the fact that its entry triggers a loss in earnings for its competitors.This force favors an excessive number of varieties.On the other hand,due to the absence of price discrimination,no single?rm can capture the whole social surplus created by the introduction of its variety.This force,in contrast,favors an insu?cient number of varieties. Consequently,the equilibrium and optimal outcomes are generally dif-ferent.In addition,there is a priori no reason to expect the market to supply too many or too few varieties.

3.1.2.4Wages

It remains for us to determine the wage,w,of the skilled workers in the manufacturing sector.We do this by using the equilibrium conditions of the product market.Indeed,the equilibrium output q?depends only on the exogenous parameters of the model,while the demand(3.7)varies with the price set by?rm i and the number of varieties n?.As this num-ber depends only on the exogenous parameters,and since the markup is constant,the equilibrium price varies only with w.Wages are indeed

68 3.Monopolistic Competition the only parameter of adjustment left.More precisely,we have

q ?= p ?P ?σμ(L a +wL)P

,where yL =L a +wL is the total income in the https://www.wendangku.net/doc/ab9961975.html,bining (3.8),(3.15),(3.17),and (3.18),it is readily veri?ed that the equilibrium wage is given by w ?=μ1?μL a L

.(3.19)Because each ?rm takes the wage level as given,this wage is similar to the equilibrium wage of a competitive labor market.This also implies that all the operating pro?ts are redistributed to the skilled workers.The equilibrium wage increases with the consumption share of the man-ufactured good because the demand for this good increases.It decreases with the number of skilled workers because there is more competition on the skilled labor market.Furthermore,the zero-pro?t condition has another important implication:a consumer’s income is equal to her wage (y =w a for agricultural workers and y =w ?for industrial workers)and the overall income is given by the total wage bill.The free-entry assump-tion thus enables us to avoid having to tackle the question of how pro?ts are distributed.

To sum up,the monopolistic competition equilibrium is unique and is described by the expressions (3.15),(3.17),(3.18),and (3.19).The welfare of an industrial or agricultural worker is given by her indirect utility (3.10)evaluated at equilibrium prices and wages (recall that p a =1):

V =w ?[(n ?)?1/(σ?1)p ?]?μ= σm σ?1 ?μ μL a (1?μ)L 1?μ L σf μ/(σ?1)V a =[(n ?)?1/(σ?1)p ?]?μ= (σ?1)(1?μ)L σμmL a μ L σf

μ/(σ?1).All else being equal,a large labor force (L a +L )favors both industrial and agricultural workers provided that the relative sizes of the two groups remain the same.On the other hand,an increase in the size of one of the two groups has contrasting e?ects on their welfare.A larger num-ber of agricultural workers proves favorable to the industrial workers,by increasing their relative wage,but it is obviously unfavorable to the agricultural workers.A larger number of industrial workers enhances the welfare of agricultural workers,who have access to a greater num-ber of varieties sold at lower prices.Likewise,more industrial workers leads to a higher number of varieties;however,this also makes compe-tition in the skilled-labor market more ?erce.Inspecting V tells us that the net impact is positive if and only if 1>σ(1?μ),i.e.,when the size

3.1.The Dixit–Stiglitz Approach 69of the manufacturing sector is large and the manufactured good is very di?erentiated.

3.1.2.5The Continuum of Firms

The assumption of nonstrategic behavior has generated considerable controversy.So long as the number n of ?rms is described by an integer,the elasticity (3.14)varies with s i and,therefore,with the prices chosen by the other ?rms.As a result,a ?rm’s equilibrium price is never equal to mwσ/(σ?1).More precisely,when n is an integer and a ?nite number,it must be that ?P/?p j >0for every j ≠i ,as the demand for i depends on the price set by each of the other producers.In such a context,the price game among ?rms has a unique symmetric Nash equilibrium given by (see Anderson et al.1992,chapter 7):p ?=mw 1+n n ?11σ?1 >σσ?1

mw.To obtain the results derived above,we must therefore assume that the number of ?rms is in?nitely large,in which case we come up with p ?=mwσ/(σ?1)>mw .Nevertheless,this assumption contradicts the endogenous determination of the number of ?rms.Moreover,the number n ?is small when f or σis large,which automatically rules out the absence of nonstrategic behavior.Finally,in a general-equilibrium context,a ?rm’s pricing strategy also in?uences,albeit only slightly,con-sumers’income and therefore their demand for the variety produced by this ?rm.The same is true of the wage rate,which depends,if only to a minor degree,on every ?rm’s hiring policy.

It is possible,however,to resolve these contradictions in a rigorous and elegant manner by assuming the existence of a continuum of ?rms whose total mass is N .This leads us to assume that the composite good entering the utility function takes the form

M = N 0

q(i)(σ?1)/σd i σ/(σ?1),(3.20)where q(i)is the quantity of variety i consumed.Intuitively,as each variety is assumed to be in?nitely close to its neighboring varieties in

[0,N],we may treat q(·)as a continuous density function.9

This assumption implies that each ?rm is negligible .Indeed,it is easy to show that the price index,in the demand function (3.9),may be

9Note

that in the ?rst version of their working paper,which has been republished in Brakman and Heijdra (2004),Dixit and Stiglitz describe the manufacturing sector by means of a continuum of ?rms.Such a formulation does not appear in the subsequent versions of their paper,where they assume a ?nite number of ?rms.

70 3.Monopolistic Competition rewritten as follows:

P≡ N

p(i)?(σ?1)d i

?1/(σ?1)

.

Consequently,a?rm’s price choice has no impact on either the value of P(i.e.,?P/?p(i)=0)or on the level of income y(i.e.,?y/?p(i)=0), as each?rm is equally negligible in both the product and labor markets. Such a modeling strategy captures in a very neat way the essence of the Chamberlinian idea of monopolistic competition as summarized in the following quote:

A price cut,for instance,which increases the sales of him who made

it,draws inappreciable amounts from the markets of each of his many

competitors,achieving a considerable result for the one who cut,but

without making incursions upon the market of any single competitor

su?cient to cause him to do anything he would not have done anyway.

Chamberlin(1933,p.83)

The demand elasticity that any?rm faces is therefore constant and equal toσ.In this case,the equilibrium price is given exactly by(3.15), the other equilibrium magnitudes being unchanged.Furthermore,the assumption of the existence of a continuum of?rms does not contra-dict what we have seen regarding the equilibrium values of the other variables.The only di?erence is that all magnitudes related to?rms and varieties are now described by continuous densities over the inter-val of varieties[0,N].Once we have derived?rms’equilibrium prices, the free-entry condition and the full-employment constraint allow us to determine the output of each?rm and the mass N?of varieties/?rms. The assumption of a continuum enables us to grasp the simple,but fundamental,idea that a?rm can be negligible with respect to the econ-omy as a whole while having some monopoly power on its market.And indeed,each?rm faces a downward-sloping demand for its product.Fur-thermore,with a continuum of?rms,the di?erence between price com-petition(Bertrand)and quantity competition(Cournot)disappears—a distinction that plagues oligopoly theory.

The assumption of a continuum presents another advantage that is lit-tle known but nevertheless fundamental.In a general-equilibrium model with imperfect competition,the choice of the numéraire matters for the equilibrium.10If oligopolistic competition prevails in the product mar-ket,taking variety i as a numéraire changes the behavior of its producer,

10See Bonanno(1990)for a detailed discussion of this problem.Dierker et al.(2003) have recently shed light on the di?culties associated with the choice of numéraire in some models of international trade with imperfect competition.

3.2.Monopolistic Competition:A Linear Setting71

as its pro?t function is no longer the same,thus also changing the behav-ior of the other producers.This results in the emergence of a new market equilibrium.To put it another way,the market solution changes with the choice of numéraire.By contrast,in a situation of monopolistic competi-tion with a continuum of?rms,the behavior of producer i has no impact on the other agents because it is negligible.In consequence,the equilib-rium remains the same,regardless of the choice of numéraire(Neary 2003).

Finally,this assumption greatly simpli?es the analysis of the distribu-tion of?rms between regions that will occupy us later,as all the variables are continuous.With a discrete number of?rms,a?rm’s relocation from one region to another always has a nonnegligible impact on the regions of origin and destination,which in turn implies discrete jumps in the vari-ables.The conditions of spatial equilibrium must therefore be described by inequalities.With a continuum,the move of a single?rm is negligible and we can work with equalities.Furthermore,the question of whether or not n?is an integer no longer needs to be asked.

It is important to understand the precise meaning of the continuum assumption.If we want to work with a model in which agents are negligi-ble,this approach is formally the right one,even though it does not seem realistic.In addition,it allow us to avoid the formidable di?culties posed by the existence of a market equilibrium,once we account for strategic behavior in a general-equilibrium setting.Finally,the monopolistic com-petition model with a continuum of?rms allows for a combination of increasing returns and imperfect competition,something that has not been achieved in other contexts.Wether or not using such a setting is reasonable must,therefore,be evaluated on the basis of its overall impli-cations,and not just its realism.In this respect,it can hardly be denied that monopolistic competition presents real advantages over oligopolis-tic competition,even though,as we will see in chapter9,the latter brings new e?ects to light.

3.2Monopolistic Competition:A Linear Setting

We have just seen that the Dixit–Stiglitz model takes away all forms of strategic interaction among?rms.It is therefore legitimate to won-der whether we can keep the?exibility inherent to a model of monop-olistic competition while introducing certain forms of interaction that agree with what we know from industrial economics.This is precisely the objective of the linear model,introduced into economic geography by Ottaviano et al.(2002),which we examine in this section.In addition,

72 3.Monopolistic Competition

this setting,unlike the Dixit–Stiglitz model,allows us to account for a particularly robust empirical fact:namely,that larger markets have lower markups but larger output(see,for example,Campbell and Hopenhayn 2005).11

3.2.1Quadratic Utility with a Continuum of Varieties

In the case of two varieties,the utility function generating a system of lin-ear demands is given by the quasi-linear utility encapsulating a quadratic subutility:

U=α(q1+q2)?12β(q21+q22)?γq1q2+A,(3.21) whereα,β,andγare three positive parameters.For the utility function U to be quasi-concave,it must be thatβ>γ.As the function U is linear in the numéraire A,income e?ects are absent from individual consump-tion:a variation in income only a?ects the demand for the numéraire, not the demand for varieties.

The demand function for variety i,obtained by maximizing(3.21) under the budget constraint,takes the following form:

D i(p1,p2)=a?bp i+c(p j?p i),i,j=1,2,i≠j.(3.22) The parameter a≡α/(β+γ)expresses the desirability of the di?er-entiated product with respect to the numéraire and may,therefore,be viewed as a measure of the size of this market;b≡1/(β+γ)gives the link between individual and industry demands:when b rises,consumers become more sensitive to price di?erences.Finally,c≡γ/[(β?γ)(β+γ)] is an inverse measure of the degree of product di?erentiation between varieties:when c→∞,varieties are perfect substitutes,whereas they are independent for c=0.

In the case of n>2varieties,the utility function(3.21)can be generalized as follows:

U=α

n

i=1

q i?12β

n

i=1

q2

i

?1

2

γ

n

i=1

j≠i

q i q j+A

n

i=1

q i?12(β?γ)

n

i=1

q2

i

?1

2

γ

n

i=1

n

j=1

q i q j+A

n

i=1

q i?12(β?γ)

n

i=1

q2

i

?1

2

γ

n

i=1

q i

2

+A.

11Also note that the Dixit–Stiglitz model presupposes homothetic preferences—an assumption that is rarely validated by empirical consumption studies.We assume here quasi-linear preferences.Although they rank far behind homothetic preferences in general-equilibrium models of trade and geography,Dinopoulos et al.(2007)show that “quasi-linear preferences behave reasonably well in general-equilibrium settings.”

等效电路模型参数在线辨识

第四章 等效电路模型参数在线辨识 通过第三章函数拟合的方法可以确定钒电池等效电路模型中的参数,但是在实际运行过程中模型参数随着工作环境温度、充放电循环次数、SOC 等因素发生变化,根据离线试验数据计算得到的参数值估算电池SOC 可能会造成较大的估计误差。因此,在实际运行时,应对钒电池等效电路模型参数进行在线辨识,做出实时修正,提高基于模型估算SOC 的精度。 4.1 基于遗忘因子的最小二乘算法 参数辨识是根据被测系统的输入输出来,通过一定的算法,获得让模型输出值尽量接近系统实际输出值的模型参数估计值。根据能否实时辨识系统的模型参数,可以将常用的参数辨识方法分为离线和在线两类,离线辨识只能在数据采集完成后进行,不能对系统模型实时地在线调整参数,对于具有非线性特性的电池系统往往不能得到满意的辨识结果;在线辨识方法一般能够根据实时采集到的数据对系统模型进行辨识,在线调整系统模型参数。常用的辨识方法有最小二乘法、极大似然估计法和Kalman 滤波法等。因最小二乘法原理简明、收敛较快、容易理解和掌握、方便编程实现等特点,在进行电池模型参数辨识时采用了效果较好的含遗忘因子的递推最小二乘法。 4.1.1 批处理最小二乘法简介 假设被辨识的系统模型: 12121212()()()1n n n n b z b z b z y z G z u z a z a z a z ------+++==++++L L (4-1) 其相应的差分方程为: 1 1 ()()()n n i i i i y k a y k i b u k i ===--+-∑∑(4-2) 若考虑被辨识系统或观测信息中含有噪声,则被辨识模型式(4-2)可改写为: 1 1 ()()()()n n i i i i z k a y k i b u k i v k ===--+-+∑∑(4-3) 式中, ()z k 为系统输出量的第k 次观测值;()y k 为系统输出量的第k 次真值,()y k i -为系统输出量的第k i -次真值;()u k 为系统的第k 个输入值,()u k i -为 系统的第k i -个输入值;()v k 为均值为0的随机噪声。

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细菌性病害是植株由于受到细菌侵染而引起的一种病害,一般表现出坏死、腐烂、萎蔫、畸形的特点。 软腐病 您瞧这颗大白菜,叶球直接露出来了,叶柄基部和根茎处的心髓部组织已经完全腐烂,充满了灰黄色的粘稠物,还散发出很大的臭味,这就是软腐病的典型症状。 软腐病又叫做烂疙瘩、烂葫芦、腐烂病、水烂病等,发生极为普遍。它的主要特点是轻轻一掰,植株就倒了,病部呈黏滑软腐状.并伴有恶臭味。小白菜、菜心等白菜类蔬菜发生软腐病时,症状与大白菜基本相似。 拿大白菜来说,大白菜定植后直到形成心叶的这个过程是长外叶的过程,这个过程中软腐病不会发生。而当植株外叶即将罩严地面的时候,大白菜渐渐进入壮心期,这时软腐病开始发生,从壮心开始至收获的整个过程中都有发病的可能,如果在这个时期,一开始植株外围的叶片在烈日下表现出萎蔫,但早晚尚能恢复,慢慢儿地外叶不能恢复的话,那您就要注意了,这有可能是得软腐病的早期症状。 黑腐病 您瞧这棵大白菜,从叶片的边缘往两侧和里边扩展,形成“V”字形黄褐色枯斑,病斑的周边呈淡黄色,这就是黑腐病的症状。以后,病原菌还会沿着叶脉向里扩展,形成大块黄褐色病斑或网状黑脉,并感染叶柄。大白菜一般在莲座期以后容易得这种病,它也是由细菌引起的。

蔬菜病虫害防治

蔬菜——不用农药怎么防止病虫害? ?A+ ?A- 2017-04-05 10:01:47农产信息网关注 说实话,作物病虫害防治不用农药很不现实,比较麻烦且费人力,但要有想学习如何不用农药来防治病虫害的朋友可以来看一下。 蔬菜虫害是蔬菜种植户们非常头疼的问题,若是用传统的农药喷洒方式解决虫害,会因为农药残留影响蔬菜的品质。这里和大家分享一下蔬菜虫害的科学防治方法。

一、伴生植物法: 1.青椒和大蒜间作。由于大蒜有一种特殊气味,能使为害青椒的害虫闻之即逃,避免青椒受害。 2.番茄和甘蓝套种。番茄的叶片会散发一种特殊的气味,可驱赶走为害甘蓝的菜青虫和蚜虫。除此之外,这两种蔬菜吸收的营养有很强的互补性,能充分发挥地力。 3.葱头与胡萝卜间作。它们各自散发的气味能驱走相互间的害虫。若单一种植胡萝卜,为防止虫害,可在地内或四周种上几棵葱头,这也能起到驱虫的作用。 并非所有的蔬菜都可以间作,如甘蓝和芹菜、黄瓜和番茄等不宜间作在一起,因为它们各自的分泌物能抑制对方的生长。 这种方法对适用于种植户和家庭小面积种植者。 二、自然材料治虫

1.草木灰液治虫。草木灰10千克对水50千克浸泡24小时,取滤液喷洒可有效地防治蚜虫、黄守虫。若葱、蒜、韭菜受种蝇、葱蝇的蛆虫危害,每亩沟施或撒施草木灰20~30千克,既治蛆又增产。 2.红糖液防治病。害红糖300克溶于500毫升清水中,加入10克白衣酵母,置于温室或大棚内,每天搅拌1次,发酵15~20天,待其表面出现白膜层为止。然后将此发酵液再加入米醋、烧酒各100克,对入100千克水。每隔10天1次,连喷4~5次,防治黄瓜细菌性斑点病和灰霉病有良好效果。 3.兔粪治地老虎每10千克水加兔粪1千克,装入瓦缸内密封沤15~20天,用时搅拌均匀,浇于瓜菜根部,可防治地老虎。 4.尿洗合剂治菜蚜用洗衣粉、尿素、水按1∶4∶400的比例制成混合液,可防治菜蚜,杀虫率达90%以上。 5.猪胆液治病虫10%浓度的猪胆液加适量小苏打、洗衣粉,能防治茄子立枯病、辣椒炭疽病,能驱赶长豆角、四季豆、瓜类等蔬菜上的蚜虫、菜青虫、蜗牛等多种害虫。稀释液可保持10天有效。 6.大蒜、番茄叶巧杀红蜘蛛用大蒜(捣烂成泥状)2份,水1份混拌均匀,取其滤液喷治。或用新鲜的番茄叶(捣烂成浆)加清水2倍并浸泡5小时然后取滤液喷洒果树、花木或蔬菜,都可有效将红蜘蛛杀死。 7.糖醋、烂果诱捕金龟子选用熟烂酸臭的无花果、烂西瓜等,与糖醋液(红糖、醋、水比为1:3:16),一起放入陶钵,支撑分布在果园或菜园中,每2—3天收集钵中的金龟子即可。 8.三合板涂漆聚捕微型害虫在较大的三合板两面涂上橙黄色油漆,干后再涂一层机油、黄油混合油,分布挂在果园或菜园中,蚜虫、白粉虱、美洲斑潜蝇等害虫就会自投罗网。1周后更换涂刷油漆、混合效果更好。

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