GEOGRAPHICAL ECONOMICS

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GEOGRAPHICAL ECONOMICS

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ELTE Faculty of Social Sciences, Department of Economics

Geographical Economics

week 6

MONOPOLISTIC COMPETITION AND THE DIXIT-STIGLITZ MODEL Authors: Gábor Békés, Sarolta Rózsás

Supervised by Gábor Békés

June 2011

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week 6 Gábor Békés

Monopolistic competition:

Introduction Dixit-Stiglitz model: demand side Monopolistic competition II - Supply

Production structure Price setting and prot

Outline

1 Monopolistic competition: Introduction

2 Dixit-Stiglitz model: demand side

3 Monopolistic competition II - Supply Production structure

Price setting and prot

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NEG market structure

The nuts and bolts of Geographical Economics

Market structure - monopolistic competition (mixing the elements of perfect competition and monopoly)

Dixit, A - J. Stiglitz 1977 Monopolistic competition, and the optimum product diversity, AER (top 20)

Necessary to understand the (later) core model Topics for today

Basics: theory and reality Model: demand

Model: supply

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Industrial Organization

Market structure - hing on the market power of the rms Perfect competition

Oligopoly (Bertrand / Cournot) Monopoly

Monopolistic competition (mixing the elements of perfect competition and monopoly)

Firms determine the prices of their products in part as a monopoly,

but the competition is close to the perfectly competitive model

There may arise economic prot (disappear when there are plenty of rms)

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Product varieties

(Product) dierentiation each rm produces a variety which is dierent in some aspects from the products of the other rms

Product varieties are near but imperfect substitutes varieties Price-elastic demand: when price goes up, the quantity demanded decreases

Love-of-variety

Rational for competition dierence, quality

= design, reliability, services, marketing, etc.

Dierentiating products decrease the price elasticity

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Product dierentiation

The features of product dierentiation Material dierence

Convenience Feeling Reputation Vanity, snobbery Fear and desire Private services

The place and circumstances of shopping

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Competition

Monopolistic competition in reality A lot but not too much competitors Innovation dierentiating products Low concentration level

Concentration is measurable

Market share of the rst three or four largest companies Hirschmann-Herndahl index (0-100)

: HHI:

∑

ⁿ

i=1

sh²/100

USA: lower than 10 = competitive market; higher than 18 = enquiry of the Competition Authority

Distinguishable

Almost competitive market Oligopoly

Monopolist with Competitive Fringe

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Demand side: Introduction

BGM Chapter 3.4

The economy has two good sectors: agriculture (F )

(producing food) and manufacturing industry (M) (producing manufacturing varieties)

Firms produce plenty of varieties (N) in the manufacturing industry

Consumers have Cobb-Douglas utility function

U =_F¹⁻^δ_M^δ_, ₀<δ<₁ ₍₁₎ Let the price of food be equal to one, that is all other products are expressed relative to this (numèraire).

Let the price index of manufactures be I (it will be dened later)

The income of the consumers: Y ; the budget constraint:

F+IM =Y (2)

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Demand side: Budget constraint

Optimal spending on food and manufactures?

U =_F¹^−δ_M^δ_{, F}+_IM =_Y L=F¹⁻^δM^δ+κ(Y−(F+IM))

First-order Conditions (FOC):∂L/∂F,∂L/∂M (₁−δ)_F^−δ_M^δ=κandδF¹^−δM^δ−¹=_κI

Taking the ratio of the two rst-order conditions: IM = ₁_−δ^δ F Substituting this in budget constraint:

Y =_F+_IM =_F+₁₋^δ

δF thus F = (1−δ)Y and IM =δY

Consumers spend a fraction, δ, of income on manufactures and a fraction, 1−δ, of income on food

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Demand side: Utility

Dixit, A - J. Stiglitz 1977 Monopolistic competition, and the optimum product diversity, AER (top 20),

CES preferences

i =1...N number of available varieties, where N is a very big number (or in continuous setting the set of varieties has measure N)Consumption of all varieties are symmetrical: c_i

The only arguments of the utility function are the consumption of the N varieties:

M=

∑

^N

i=1c_i^ρ

!¹

ρ

0<ρ<1 (3) Love-of-variety: ρ

Ifρ'1, the varieties are perfect substitutes and only the total amount of consumption matters.

Ifρdecreases, the utility, arising from the ability of consuming more varieties, will increase.

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Demand side: CES

CES preferences

If the level of consumption of each variety is the same:

M =

∑

N i=1c^ρ

!¹_ρ

= (_Nc^ρ)¹^ρ = (_N)¹^ρ _c= (_N)¹^ρ⁻¹(_Nc) Nc - total amount of product produced

(N)¹^ρ⁻¹ - externality, which arises because of more varieties

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Demand side: derivation (1)

Manufacturing market, budget constraint, p_i = the price of variety i

∑

N

i=1p_ic_i =δY (4)

Optimally allocate spending among the dierent varieties of manufactures

L=

∑

^N

i=1c_i^ρ

!¹

ρ

+κ δY−

∑

^N

i=1p_ic_i

!

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FOC:∂L/∂c_j ⇒

∑

^N

i=1

c_i^ρ

!¹_p₋₁

c_j^ρ⁻¹=κp_j for each j=1...N Let us choose two varieties i=1 and i =j, then:

c_j^ρ⁻¹/c₁^ρ⁻¹=p_j/p₁ e:=1/(1−ρ) c_j =p_j⁻^ep₁^ec₁

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Demand side: derivation (2)

Back to the budget constraint:

δY =

∑

^N

i=1p_ic_i =

∑

^N

i=1p_i(_p^−e_i _p^e₁_c₁) =_p₁^e_c₁

∑

^N

i=1p_i¹^−e Let the price index be

I =

∑

^N

i=1p_i¹⁻^e

!₁/(1−e)

then p₁^ec₁

∑

^N

i=1

p_i¹^−e =pê₁c₁I¹⁻ê=δY ⇒c₁ =p₁^−eIê⁻¹δY The demand for variety i is derived analogously:

c_i =p_i^−eI^e⁻¹δY

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Demand side: I

Why the denition of price index I is good for us? Substitute it in the CES utility function:

M =

∑

N i=1c_i^ρ

!¹_ρ

=

∑

N i=1

(_p_i^−e_I^e−¹_δY)^ρ

!¹_ρ

=

δYI^e−¹

∑

^N

i=1p_i^−eρ

!¹_ρ

e=1/(1−ρ) M =_δY_/I

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Demand eects:

The demand for variety i is given by: c_i =_p⁻_i ^e_I^e−¹_{δY , which} appears to be inuenced by:

(1) the incomeδY spent on manufactures (proportional), (2) the price p_i of good i, (3) some parametere, (4) the price index I

What is the connection between the quantity demanded and the price?

We know, that I^e−¹δY =_const

There appears p_i in I , however, if N is big enough, the eect of it is negligible

As a result of optimization: constant elasticity of substitution (CES) - (−_∂c_i_/∂p_i)(_p₁_/c₁) =e

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Demand function and e Figure 1

The higher the e, the more rapidly falls the demand for a variety as a result of a small price increase

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Demand function and e Figure 2

The limitation of the Dixit-Stiglitz model: there is a relationship between the price elasticity (e) and the `substitution parameter' (ρ)

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Price index

c_i =_p_i^−e_I^e⁻¹_δY = ^p_Iⁱ⁻^{e δY}_I

The demand for a given variety depends on the average price level, that is on the average of the prices of the other varieties = substitutes

In other words, the quantity demanded hinges on the relative price and the relative income.

Price index I - utility derived from the consumption of manufactures (one unit of consumption bundle) = consumption-based price index

I , i.e. the utility depends positively on the number of varieties:

Proof: Suppose that every variety has the same price, p₀ I =

∑

^N

i=1p_i¹^−e

!₁/(1−e)

=_p₀_N¹^/(¹^−e)

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Price index

Indirect utility can be derived from the budget constraint and the utility function

What is the minimum amount of expenditure required to buy one unit of utility?

The price of food = 1

The utility rises if Y/I^δ increases therefore the real income: y =_YI^−δ

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Key terms

Monopolistic competition

Marginal rate of substitution between products CES utility

Love-of-variety

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Basics

BGM Chapter 3.5 Supply side: two sectors Total labor force: L

Manufacturing industry: γL , Food sector: (₁−γ)_{L s.t.}

0<γ<₁

Agriculture: CRS, competitive market numèraire

Production function: F = (₁−γ)_L as p_F =1 and MPL=1⇒w_F =1

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Industry

Industry: increasing return to scale, imperfect competition Manufactures are symmetrical:

Each variety has the same technology

Dierent varieties are produced by dierent rms (the rm with the largest sales can always outbid a potential competitor)

Economies of scale (l_i is the amount of labor necessary to produce x_i of variety i)

l_i =α+βx_i (6)

FC:αand MC: β

= Increasing internal economies of scale

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Industry: products

Each variety is produced by a single rm - monopolist behavior

But: each rm takes the price-setting behavior of other rms as given

Thus there is no strategic behavior: if rm i increases its price, it does not assume that the other rms react.

The rm also ignores the eect of changing its own price on the price index I of manufactures

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Prot

Symmetrical rms that produce x unit of output, using l unit of labor and paying wage rate W will earn protsπ:

π=px−Wl=px−W(α+βx) (7) Recall that the demand for x: x =p^−econ where

con=_I^−e_δY

Then π=p¹⁻^econ−W(α+βp⁻^econ)

∂π/∂p and FOC=₀

p=βW/(1−1/e) =βW/ρ (8)

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Price setting (2)

p=_βW/(₁−_1/e) =_βW/ρ (9) constant margin (price marginal cost = mark-up), asβW denotes the marginal cost

Ife=5 the mark-up is 20%

Why the mark-up is `necessary'?

In order to recuperate the xed costs of labor

Constante →constant mark-up (does not depend on the quantity produced)

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Price setting (2a)

Recall that in case of DS: constant mark-up - p(1−¹_e) =βW

In opposition to this, oligopoly, e.g. Cournot is somewhat dierent

p(₁− _e(^s_Y₎) =_{MC, where}e(_Y)is the demand elasticity and MC is equal toβW

The mark-up depends on the market share. If s →0 it is the same.

If N is large (s is small), equilibrium price does not depend on the type of competition.

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Prot and the equilibrium output

Suppose, that the prots are positive (economic prot). Then it is worth setting up a rm and beginning to produce a new variety.

Consumers: if N ↑_{then x}_i ↓_andπ_i ↓_{- lim}_N_=∞(π_i) =_{0, i.e.}

if N is large enough,π_i =0

π=₀⇒_px=_W(α+_βx)_{and p}=_βW/(₁−_1/e) (_e−^e₁)βWx =Wα+Wβx ⇒(_e−^e₁−1)βWx =Wα

x = ^α(e−₁)

β (10)

The output per rm is xed in equilibrium - it depends only on exogenous parameters

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Prot and the equilibrium output

How much labor is required?

l =α+βx =α+β^α(e−_β¹⁾ =αe

How much is N? total labor force/the amount of labor required to produce one unit of output

N=_γL/l=_γL/αe

Size of the economy = number of varieties (as x_i is xed) Cheating: N is nite and can be determined. However, we have repeatedly supposed that N is almost innite.

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Economies of scale

Does the economies of scale matter?

AC/MC(l):

AC = _l/x =αe/^α(e_β⁻¹⁾ =βe/(e−₁)_, MC =β

Economies of scale = _e₋^e₁

For a high value ofe(similar products, substitutes) this measure of scale economies is low.