The Melitz-Ottaviano Model: Slidesthe Melitz model: only variety effect; here we also observe the "competition" effect. Firms have to reduce their prices and, therefore, consumers

The Melitz-Ottaviano Model: Slides

Alexander Tarasov

University of Munich

Summer 2010

Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 1 / 29

Motivation and Comparison with Melitz (2003)

The linear demand system (no CES preferences).

variable elasticity of demand

variable (endogenous) mark-ups that are affected by the intensity of competition

(the number and average productivity of competing �rms in the market)

Market size does affect the equilibrium distribution of �rms and their

performance.

bigger markets exhibit higher levels of product varieties and host more

productive �rms that set lower markups (lower prices).

The effects of trade and different trade liberalization policies.


Model: Closed EconomyDemand

An individual utility function over a continuum of goods indexed by i and a

homogenous good chosen as numeraire:

U = qc0 + αZi2Ω

qci di �12

γZi2Ω

(qci )2 di � 1

2η

�Zi2Ω

qci di�2,

where qc0 and qci represent the individual consumption levels of the

numeraire and each variety i .

The parameters α and η represent the substitution pattern between the

numeraire and the differentiated varieties.

The parameter γ indexes the degree of product differentiation between the

varieties.



Demand for a certain variety (if it is positive) is given by

pi = α� γqci � ηQc .

Ω� � Ω is the subset of varieties that are consumed (qci > 0).

Then, it can be shown that, for any i 2 Ω�,

qci =α

γ+ ηN� pi

γ+

ηNγ+ ηN

p̄γ,

where N is the measure of Ω�, p̄ =Ri2Ω� pidiN is the average price.



The total demand:

qi =αL

γ+ ηN� L

γpi +

ηNγ+ ηN

Lγp̄.

Therefore, qi > 0 if and only if

pi <γα+ ηNp̄

γ+ ηN� pmax

Notice that pmax is endogenous and pmax � α. If η = 0, then pmax = α (noexit in this case).

A tougher competitive environment (p̄ is lower or N is higher):

pmax decreases: �rms cannot charge so high prices as before.



The indirect utility function:

U = Ic +12(η +

γ

N)�1(α� p̄)2 + 1

2Nγ

σ2p

where σ2p =Ri2Ω� (pi�p̄)

2diN is the variance of prices.

Ic " =) U ": the income effect (through the numeraire)

N " =) U ": the variety effect ("love for variety")

p̄ # =) U ": the price effect

σ2p " =) U ": consumers re-optimize their purchases by shiftingexpenditures towards lower prices varieties and the numeraire.


Model: Closed EconomyProduction and �rm behavior

Labor is the only factor of production

Numeraire good: perfect competition and one-to-one technology =)w = p0 = 1.

The cost of entry into the industry: fe. Then, the cost of production is

realized: c � G(c) with the support on [0, cM ] .

Firms then decide whether to produce or to exit. They maximize their pro�ts

taken N and p̄ as given.



Pro�ts:

π(c) = (p(c)� c)q(p(c))

It can be shown that the optimal price

p(c) =pmax + c

2

New notation: pmax = cD is the cutoff level. Firms with c > cD exit, as

p(c) > pmax, which results in zero demand.



Then, pro�ts are given by

π(c) =L4γ(cD � c)2

Output:

q(c) =L2γ(cD � c)

Revenues:

r(c) =L4γ(c2D � c2)


Model: Closed EconomyFree entry equilibrium

The net value of entry:Z cD0

π(c)dG(c)� fe = 0 ()

L4γ

Z cD0(cD � c)2dG(c) = fe.

Therefore, we can �nd cD as the solution of the last equation.


Model: Closed EconomyFree entry equilibrium

To �nd N, we useγα+ ηNp̄

γ+ ηN= cD

It is equivalent to

N =2γη

α� cDcD � c̄

where c̄ is the average cost of surviving �rms:

c̄ =

R cD0 cdG(c)G(cD)


Model: Closed EconomyBrief Analysis

In larger markets (higher L):

A rise in L immediately implies that cD falls

it can be shown that Ne = NG(cD) increases (there is more entry in bigger

markets)

as a result, the competition becomes tougher and �rms have to reduce their

prices (to set lower markups) and some �rms exit (because of negative pro�ts)

It can be shown that a rise in L leads to a decrease in average price p̄:

p̄ =cD + c̄2

.

Notice that the impact on N is in general ambiguous (we need to make some

assumptions about G(c)).


Model: Closed EconomyParametrization

We assume that

G(c) =�ccM

�k, c 2 [0, cM ].

The shape parameter k indexes the dispersion of cost draws. If k = 1, then

the distribution is uniform. As k increases, the relative number of high-cost

�rms increases, and the cost distribution is more concentrated at these

higher costs levels.

The productivity distribution of surviving �rms is also Pareto:

GD(c) =G(c)G(cD)

=

�ccD

�k.



Given the parametrization,

cD =�2(k + 1)(k + 2)γ(cM)k fe

L

� 1k+2

.

The number of surviving �rms:

N =2(k + 1)γ

η

α� cDcD

Under Pareto distribution, there are more active �rms in bigger markets.



Welfare:

U = Ic +12(η +

γ

N)�1(α� p̄)2 + 1

2Nγ

σ2p

We have shown that if L rises, then p̄ decreases and N rises, resulting in

greater welfare. That is, U is higher in bigger markets.

the Melitz model: only variety effect; here we also observe the "competition"

effect. Firms have to reduce their prices and, therefore, consumers gain.

formally, a rise in L leads to lower σ2p (which reduces welfare), but this effect is

dominated by the effects of p̄ and N.


Model: Closed EconomySome Evidence:

Syverson (2004):

higher average productivity in larger markets

the distribution of productivities is less disperse

average prices are lower in bigger markets

higher lower bound for the productivity distribution


Model: Open Economy

If there are no trade costs, then trade is equivalent to a rise in market size.

an increase in average productivity and product variety, and a decrease in prices

(markups)

If there are trade costs, the situation is not so straightforward.

Two countries: H and F ; LH and LF are market sizes.

Preferences are the same, therefore same demand functions in both

countries.

The markets are segmented: �rms choose different prices for different

markets.


Model: Open Economy

The delivered cost of a unit with cost c to country l 2 fH,Fg is τlc (theanalogue of iceberg transport cost).

Thus, countries are different in two dimensions: market size Ll and barriers to

imports τl .

Let plmax denote the price threshold for positive demand in market l . Then,

plmax =αη + ηN l p̄l

ηN l + γ

where N l is the total number of �rms selling in country l (the total number of

varieties), p̄l is the average price (across both local and exporting �rms).


Model: Open Economy

Let plD(c) and qlD(c) represent the domestic levels of the pro�t maximization

price and quantity sold for a �rm producing in country l with cost c.

Such a �rm may also decide to produce some output qlX (c) that it exports at

a delivered price plX (c).

Since markets are segmented, �rms independently maximize their pro�ts

earned from domestic and export sales.

Let πlD(c) be the maximized pro�ts from selling domestically, then

πlD(c) =�plD(c)� c

�qlD(c)

Similarly, maximized export pro�ts are given by

πlX (c) =�plX (c)� τhc

�qlX (c)

where h 6= l .Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 19 / 29

Model: Open EconomyCutoffs

Let c lD denote the upper bound cost for �rms selling in their domestic market

and c lX denote the upper bound cost for exporters from l to h.

These cutoffs satisfy

c lD = supnc : πlD(c) > 0

o= plmax

c lX = supnc : πlX (c) > 0

o=phmaxτh

That is,

chX =c lDτl.


Model: Open EconomyPrices and Pro�ts

Similar to the closed economy case:

plD(c) =c lD + c2

,

plX (c) = τh c

lX + c2

,

πlD(c) =Ll

4γ

�c lD � c

�2,

πlX (c) =Lh

4γ

�τh�2 �

c lX � c�2.


Model: Open EconomyFree entry equilibrium

The free entry condition in country l means that

Z clD0

πlD(c)dG(c) +Z clX0

πlX (c)dG(c)� fe = 0.

Given the Pareto parametrization for G(c) (G(c) =�ccM

�k), the free entry

condition is equivalent to

Ll�c lD�k+2

+ Lh�

τh�2 �

c lX�k+2

= γφ,

where

φ � 2(k + 1)(k + 2) (cM)k fe.



Notice that the free entry condition holds so long as there is a positive mass

of domestic entrants N le > 0, otherwise country l is specialized in the

numeraire!!

We assume that for l = H,F , N le > 0.

Then, taking into account that chX =clDτl, we can rewrite the free entry

condition in the following way:

Ll�c lD�k+2

+ Lhρh�chD�k+2

= γφ,

where ρh ��τh��k

.



Equilibrium equations:

LH�cHD�k+2

+ LF ρF�cFD�k+2

= γφ

LF�cFD�k+2

+ LHρH�cHD�k+2

= γφ.

So we have two equations and two unknowns: cHD and cFD .



It can be shown that

c lD =�

γφ

Ll1� ρh1� ρhρl

� 1k+2

.


Model: Open EconomyPrices, product variety, and welfare.

It also can be shown that

p̄l =2k + 12k + 2

c lD

N l =2(k + 1)γ

η

α� c lDc lD

Finally, welfare is a decreasing function of c lD . This captures the effects of

product variety and average prices (see the closed economy case).


Model: Open EconomyNumber of entrants

The number of sellers in country l is comprised of domestic producers and

exporters from h.

Given a positive mass of entrants, G(c lD)Nle represents domestic producers,

while G(chX )Nhe represents exporters selling in l .

Hence,

G(c lD)Nle +G(c

hX )N

he = N

l ,

and we can �nd N le for l = fH,Fg.


Model: Open EconomyThe impact of trade

Let us denote c laD as the cutoff in autarky. Then,

c laD =�

γφ

Ll

� 1k+2.

As ρh < 1 and ρl < 1, it is straightforward to show that

c laD > clD.

Therefore, trade

increases aggregate productivity by forcing the least productive �rms to exit (a la

Melitz (2003)).

decreases average price and markups (the competition effect).

increases the number of available varieties

increases, thereby, welfare!!


Model: Open EconomyThe impact of trade

Intuition:

Recall that in Melitz (2003), trade induces increased competition for scarce

labor resources. As a result, real wage rises and the least productive �rms

exit.

Here, the intuition behind exit of least productive �rms is different. In the

current model, increased factor market competition plays no role, as the

supply of labor to the differentiated sector is perfectly elastic (wage is

determined by the price of the numeraire).

Firms exit only because of "tougher" competition that affects demand

elasticities.


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The Melitz-Ottaviano Model: Slidesthe Melitz model: only variety effect; here we also observe the "competition" effect. Firms have to reduce their prices and, therefore, consumers