the melitz-ottaviano model: slidesthe melitz model: only variety effect; here we also observe the...
TRANSCRIPT
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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
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 2 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 3 / 29
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Model: Closed EconomyDemand
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 4 / 29
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Model: Closed EconomyDemand
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 5 / 29
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Model: Closed EconomyDemand
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 6 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 7 / 29
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Model: Closed EconomyProduction and �rm behavior
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 8 / 29
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Model: Closed EconomyProduction and �rm behavior
Then, pro�ts are given by
π(c) =L4γ(cD � c)2
Output:
q(c) =L2γ(cD � c)
Revenues:
r(c) =L4γ(c2D � c2)
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 9 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 10 / 29
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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)
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 11 / 29
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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)).
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 12 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 13 / 29
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Model: Closed EconomyParametrization
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 14 / 29
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Model: Closed EconomyParametrization
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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 15 / 29
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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
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 16 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 17 / 29
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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).
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 18 / 29
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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
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 20 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 21 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 22 / 29
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Model: Open EconomyFree entry equilibrium
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
.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 23 / 29
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Model: Open EconomyFree entry equilibrium
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 .
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 24 / 29
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Model: Open EconomyFree entry equilibrium
It can be shown that
c lD =�
γφ
Ll1� ρh1� ρhρl
� 1k+2
.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 25 / 29
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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).
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 26 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 27 / 29
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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!!
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 28 / 29
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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.
Alexander Tarasov (University of Munich) The Melitz-Ottaviano Model Summer 2010 29 / 29