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Monopolistic Competition in Trade

• The Dixit-Stiglitz Model• Monopolistic Competition and Trade in a One-Sector Model• Monopolistic Competition and Trade in a Two-Sector HO

Model• Transport Costs and the Home-Market Effect• Economic Geography

© J.P. Neary Wednesday 19 April 2023

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Monopolistic Competition

Due to Chamberlin; key features:

1. Differentiated products: reflecting a “taste for variety”

– Hotelling approach (used by Helpman, JIE 1981): each consumer has an “ideal type” - difficult!

– Dixit-Stiglitz “taste for variety” approach is now standard

– Both approaches have identical implications for positive questions (but not for normative ones)

2. Increasing returns (due to fixed costs perhaps)

– Otherwise, every conceivable variety could always be produced, in tiny amounts

3. Free Entry => No long-run profits

– Just like perfect competition

4. No strategic behaviour: Firms ignore their interdependence when taking their decisions.

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1. Tastes: Dixit-Stiglitz Utility Function

• A symmetric CES function

• xi is the consumption of variety i

• n, the number of varieties, is given to consumers, but endogenous in equilibrium

• The index is a measure of substitutability, and must lie in [0,1]

• As we will show, it is related to the elasticity of substitution :{0 << 1} <=> {1 < <

A. Preference for Variety/Diversity

Proof: Assume all varieties have the same price p and are consumed in equal amounts, so total expenditure is I = npx

(This is the indirect utility function in symmetric equilibria)

Logarithmically differentiating, with I and p fixed:

i.e., utility rises with diversity, and by more so the lower is QED

u n 1

1

u xii

n ( ) / 1

1

x x u nx n x n I pi ( ) // / /( ) 1 1 1 1

1

110

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The indirect utility function can be inverted to get the expenditure function in symmetric equilibria:

The unit expenditure function P is a true price index for the industry.

It is decreasing in n (again, because consumers like variety) and to a greater extent the lower is

e p u Pu P pn( , ) /( ) where: 1 1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50 60 70 80 90 100

n

P

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B. Derive the Elasticity of Substitution

Rewrite utility function as u and differentiate w.r.t. pi:

(from consumer’s FOC)

Now, take ratio for goods i and j:

{0<<1} <=> {1<<

C. Marshallian Demands

Solve for xj, multiply by pj, sum over j and substitute into the budget constraint, pjxj = I, to obtain:

D. Industry Price Index

Substitute into u(x) to get the indirect utility function V(p,I):

So:

( ) u u x pi i i

1 1 =

xx

pp

i

j

i

j

1

110

x Iip

pi

j j

1

u x I p Ii ip

p j ji i

j j

1

1 1

u p Ij j 1 1 1 /( )

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Finally, as in the symmetric case, this can be inverted to get the expenditure function:

The demand functions can be expressed more simply in terms of the unit expenditure function P :

(The Marshallian demand function is log-linear in relative price and real income, both defined

with respect to P)

e p u Pu P pj j( , )/( )

where: 1 1 1

xp

P

I

Px

p

Pui

iic i

;

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Production: Profit Maximisation by Firms

Firms maximise profits given the (perceived) demand curve (i.e., they take income and the industry price index as given):

p Ax A P Ii i 1 1 1/ ( )/ / where:

TR Ax MR Ax p ( )/ /

1 1 1and

Hence their total and marginal revenue curves are (suppressing i):

So the demand and MR curves are iso-elastic, with the latter a fraction of the former.

CostsHomotheticity: Production uses a composite input, at unit cost W Overheads require F units; and production c units per unit output: TC = (F+cx)W (Set W=1 for now)Hence: MC = c

AC = c+F/x (a rectangular hyperbola)

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MR

p

x

D

Dixit-Stiglitz (symmetric CES) preferences:

==> Demand: p = Ax Marginal Revenue: p

-1/

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p

x

MC

AC

Total Costs: C = F+cx

==> Marginal Costs: c Average Costs: c+F/x

c

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Profit Maximisation

MC=MR => c =p i.e., p = c / (p independent of A, F)

Alternatively: p /c = 1/ = /( -1) (the price-cost margin is decreasing in )

Free Entry

AC=AR => c + F/x = p

+ {MC=MR} => c + F/x = c / => x = (-1) F/c

In equilibrium: p Ax A F c px 1/ 1/ ( , , )

i.e., Equilibrium A is also independent of P, I, and therefore n

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MR

p

F/c

c

c/

x

MC

ACD

Equilibrium: Chamberlinian Tangency• Profit maximisation: MR=MC • Free entry: =0 ==>

• Firm output depends only on F, c, • Industry output adjusts to demand shocks via changes in n only

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Technical Digression

As drawn, the AC curve is more convex than the demand curve.

Proof that this must be so:

p Ax p Ax

p Axp

x

1 1

1 1

1

2

2 1

2 2

/

AC c AC

AC

Fx

Fx

Fx

2

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AC p F px

F px F

F

2 1

2 1

1

2

( )

[ ( )] ( )

( )

using

So, > 1 is necessary and sufficient for AC to be more convex.

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Equilibrium Firm Size

In equilibrium, x depends only on c, F and : all adjustment to changes in other exogenous variables is via changes in n

How to avoid this implausible property?

1. Relax CES assumption.[Krugman, JIE 1979]

2. Assume more than one factor with non-homothetic costs:[Lawrence/Spiller, QJE 1983; Flam/Helpman, JIE 1987; Forslid/Ottaviano, JEG 2003]

TC = rf + wx x = (-1)rf/w3. Assume heterogeneous firms:

[Melitz, Em 2003]

xi = (-1)Fi/ci

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Role of

High :

• different varieties are close substitutes for each other(preference for diversity is not so strong)

• p close to c : so p and MR curves are flat and close together

• x large: economies of scale are highly exploited

• fewer varieties, higher output of each

Low :

• different varieties are less close substitutes (greater preference for diversity)

• p >> c : so p and MR curves are steep and far apart

• x small: economies of scale are not highly exploited

• more varieties, lower output of each

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MR(high )

p

x

MC

AC

B

Effects of Changes in the Elasticity of Substitution

MR(low )

A

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Monopolistic Competition and Trade in a One-Sector Model[Krugman, JIE 1979]

2 countries; 1 sector; labour the only factor of production (W=w=1); identical technology and tastes in both countries.

Full employment: L = nLi = n(F+cx) => n = L /(F+cx) But:

i.e., number of varieties is linear in the size of the economy

Autarky: L, L* => nA , nA*, xTrade: • All trade is intra-industry• Trade is unrelated to comparative advantage

(Both countries have the same autarky prices, since they are identical except for size, which has no effect given identical homothetic tastes)

• Trade is welfare improving (since it increases the number of varieties available)

• Volume of trade is maximised when countries are of equal size

x nLF

Fc ( )

1

L L L n *

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Monopolistic Competition and Trade in the Two-Sector Heckscher-Ohlin Model

We extend the two-sector HO model by assuming that one sector has a Dixit-Stiglitz monopolistically competitive structure.

2 sectors:

• X1 “Food”: perfectly competitive, output homogeneous, p=1

• X2 “Manufactures”: monopolistically competitive

Tastes:

An example of “two-stage budgeting”: • the utility function is Cobb-Douglas in food and manufacturing;

• the manufacturing sub-utility function in turn has the Dixit-Stiglitz form.

Expenditure function:

U X X X xii

n

1 21

2 1

1 , ( ) /

e P u P np n p(.) [ ( ) ]* * /( ) 1 1 1 1 1 where:

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Economic GeographyStandard features of Dixit-Stiglitz type models:

1. Demand intercept A depends:

• positively on industry price index P

• positively on expenditure I

2. Industry price index depends:

• negatively on number of firms at home and abroad

• positively on trade costs

Additional feature of Venables model: Each firm uses the output of every other as an input. So:

• Expenditure I depends positively on n

• Input costs depend positively on P[Now we need to make W explicit: replace F and c by FW and cW]

Implications for stability of diversified equilibria:

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MR

p

F/c

cW

cW/

x

MC

ACD

Effects of entry by one new firm: 1. P, P* fall => fall 2. Cost linkage: P falls => W falls => rise 3. Demand linkage: Demand rises => rise

3

2

1

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T

Dispersal

Figure 2: Agglomerated and Dispersed Equilibriaas a Function of Trade Costs

Core

TSTB

Periphery

0.5

1

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Heterogeneous Firms

Firm heterogeneity in monopolistic competition:[Melitz (Em 2003), Helpman-Melitz-Yeaple (AER 2004)]

• Firms pay a sunk cost to reveal their productivityDraw c from g(c) with positive support over (0,)

• Given their productivity, they calculate their expected profits and choose to produce or exit

Exit if c < ce where (ce) = 0 or r(ce) = f .

• If exporting and/or FDI require an additional fixed cost, only high-productivity firms will engage in them

• Predictions are consistent with micro-empirical evidence