29
Background Model Analysis, in closed economy Open Economy Analysis, in open economy The Melitz Ottaviano (2008) Model Alexander Tarasov 1 Davide Suverato 2 1 LMU University of Munich 2 LMU University of Munich Topics in International Trade, 2 June 2015 Davide Suverato, LMU The Melitz Ottaviano (2008) Model 1 / 29

The Melitz Ottaviano (2008) Model - LMU

  • Upload
    others

  • View
    4

  • Download
    0

Embed Size (px)

Citation preview

Page 1: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

The Melitz Ottaviano (2008) Model

Alexander Tarasov1 Davide Suverato2

1LMU University of Munich

2LMU University of Munich

Topics in International Trade, 2 June 2015

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 1 / 29

Page 2: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

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 productivityof competing firms in the market)

• Market size does affect the equilibrium distribution of firmsand their performance.

• bigger markets exhibit higher levels of product varieties andhost more productive firms that set lower markups (lowerprices).

• The effects of trade and different trade liberalization policies.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 2 / 29

Page 3: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Preferences

• An individual utility function over a continuum of goodsindexed by i and a homogenous good chosen as numeraire:

U = qc0 + α

∫i∈Ω

qci di −1

∫i∈Ω

(qci )2 di − 1

(∫i∈Ω

qci di

)2

,

where qc0 and qci represent the individual consumption levelsof the numeraire and each variety i .

• The parameters α and η represent the substitution patternbetween the numeraire and the differentiated varieties.

• The parameter γ indexes the degree of product differentiationbetween the varieties.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 3 / 29

Page 4: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Demand

• 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 ∈ Ω∗,

qci =α

γ + ηN− piγ

+ηN

γ + ηN

p

γ,

where N is the measure of Ω∗, p =∫i∈Ω∗ pidi

N is the averageprice.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 4 / 29

Page 5: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Aggregate demand

• The total demand:

qi =αL

γ + ηN− L

γpi +

ηN

γ + ηN

L

γp.

• Therefore, qi > 0 if and only if

pi <γα + ηNp

γ + ηN≡ pmax

• Notice that pmax is endogenous and pmax ≤ α. If η = 0, thenpmax = α (no exit in this case).

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

• pmax decreases: firms cannot charge so high prices as before.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 5 / 29

Page 6: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Welfare

• The indirect utility function:

U = I c +1

2(η +

γ

N)−1(α− p)2 +

1

2

N

γσ2p

where σ2p =

∫i∈Ω∗ (pi−p)2di

N is the variance of prices.

• I c ↑ =⇒ U ↑: the income effect (through the numeraire)

• N ↑ =⇒ U ↑: the variety effect (”love for variety”)

• p ↓ =⇒ U ↑: the price effect

• σ2p ↑ =⇒ U ↑: consumers re-optimize their purchases by

shifting expenditures towards lower prices varieties and thenumeraire.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 6 / 29

Page 7: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Production

• Labor is the only factor of production

• Numeraire good: perfect competition and one-to-onetechnology =⇒ w = p0 = 1.

• Sunk cost of entry into the industry: fe . Then, the cost ofproduction is realized: c ∼ G (c) with the support on [0, cM ] .

• Firms then decide whether to produce or to exit. Theymaximize their profits taking N and p as given.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 7 / 29

Page 8: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Firm behavior

• Profits: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 withc > cD exit, as p(c) > pmax, which results in zero demand.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 8 / 29

Page 9: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Firm behavior

• Then, profits are given by

π(c) =L

4γ(cD − c)2

• Output:

q(c) =L

2γ(cD − c)

• Revenues:

r(c) =L

4γ(c2

D − c2)

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 9 / 29

Page 10: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: free entry

• The net value of entry:∫ cD

0π(c)dG (c)− fe = 0 ⇐⇒

L

∫ cD

0(cD − c)2dG (c) = fe .

• Therefore, we can find cD as the solution of the last equation.Notice that the l.h.s. is increasing in cD =⇒ larger marketsize leads to lower cutoff.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 10 / 29

Page 11: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: free entry

• To find N, we use the equation for pmax

γα + ηNp

γ + ηN= cD

• It is equivalent to

N =2γ

η

α− cDcD − c

where c is the average cost of surviving firms:

c =

∫ cD0 c dG (c)

G (cD)

which is increasing in cD , because of the properties of thedistribution we are assuming.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 11 / 29

Page 12: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Market SizeIn 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 firms have

to reduce their prices (to set lower markups) and some firmsexit (because of negative profits)

• It can be shown that a rise in L leads to a decrease in averageprice p:

p =cD + c

2.

increasing in cD .

• Notice that the impact on N is in general ambiguous (we needto make some assumptions about G (c)).

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 12 / 29

Page 13: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Parametrization of the technology

• We assume that

G (c) =

(c

cM

)k

, c ∈ [0, cM ].

• The shape parameter k indexes the dispersion of cost draws.If k = 1, then the distribution is uniform. As k increases, therelative number of high-cost firms increases, and the costdistribution is more concentrated at these higher costs levels.

• The productivity distribution of surviving firms is also Pareto:

GD(c) =G (c)

G (cD)=

(c

cD

)k

.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 13 / 29

Page 14: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Parametrization of the technology

• Given the parametrization, the solution to the free entrycondition is:

cD =

[2(k + 1)(k + 2)γ(cM)k fe

L

] 1k+2

.

• The number of surviving firms:

N =2(k + 1)γ

η

α− cDcD

• Under Pareto distribution, there are more active firms inbigger markets.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 14 / 29

Page 15: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Welfare implications

since the industry profit net of entry cost is zero, individual incomeis only labor income, then:

• Welfare:

U = 1 +1

2η(α− cD)

(α− k + 1

k + 2cD

)• We have shown that if L rises, then p decreases and N rises,

resulting in greater welfare. That is, U is higher in biggermarkets.

• 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 p and N.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 15 / 29

Page 16: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Some 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

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 16 / 29

Page 17: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Open economy

• If there are no trade costs, then trade is equivalent to a rise inmarket size.

• an increase in average productivity and product variety, and adecrease 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 inboth countries.

• The markets are segmented: firms choose different prices fordifferent markets.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 17 / 29

Page 18: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Trade costs

• The delivered cost of a unit with cost c to country l ∈ H,Fis τ lc (the analogue of iceberg transport cost).

• Thus, countries are different in two dimensions: market size Ll

and barriers to imports τ l .

• Let plmax denote the price threshold for positive demand inmarket l . Then,

plmax =αη + ηN l pl

ηN l + γ

where N l is the total number of firms selling in country l (thetotal number of varieties), pl is the average price (across bothlocal and exporting firms).

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 18 / 29

Page 19: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Firm behavior

• Let plD(c) and qlD(c) represent the domestic levels of theprofit maximization price and quantity sold for a firmproducing in country l with cost c.

• Such a firm may also decide to produce some output qlX (c)that it exports at a delivered price plX (c).

• Since markets are segmented, firms independently maximizetheir profits earned from domestic and export sales.

• Let πlD(c) be the maximized profits from selling domestically,then

πlD(c) =(plD(c)− c

)qlD(c)

• Similarly, maximized export profits are given by

πlX (c) =(plX (c)− τhc

)qlX (c)

where h 6= l .

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 19 / 29

Page 20: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Cutoffs

• Let c lD denote the upper bound cost for firms selling in theirdomestic market and c lX denote the upper bound cost forexporters from l to h.

• These cutoffs satisfy

c lD = supc : πlD(c) > 0

= plmax

c lX = supc : πlX (c) > 0

=

phmax

τh

• That is,

chX =c lDτ l.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 20 / 29

Page 21: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Prices and profits

• Similar to the closed economy case:

plD(c) =c lD + c

2,

plX (c) = τhc lX + c

2,

πlD(c) =Ll

(c lD − c

)2,

πlX (c) =Lh

(τh)2 (

c lX − c)2.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 21 / 29

Page 22: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: free entry

• The free entry condition in country l means that∫ c lD

0πlD(c)dG (c) +

∫ c lX

0πlX (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,

whereφ ≡ 2(k + 1)(k + 2) (cM)k fe .

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 22 / 29

Page 23: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: free entry

• Notice that the free entry condition holds so long as there is apositive mass of domestic entrants N l

e > 0, otherwise countryl is specialized in the numeraire!!

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

• Then, taking into account that chX =c lDτ l

, we can rewrite thefree entry condition in the following way:

Ll(c lD

)k+2+ Lhρh

(chD

)k+2= γφ,

where ρh ≡(τh)−k

.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 23 / 29

Page 24: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: market clearing

Equilibrium equations:

LH(cHD

)k+2+ LFρF

(cFD

)k+2= γφ

LF(cFD

)k+2+ LHρH

(cHD

)k+2= γφ.

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

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 24 / 29

Page 25: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: cutoff

It can be shown that

c lD =

[γφ

Ll1− ρh

1− ρhρl

] 1k+2

.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 25 / 29

Page 26: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Equilibrium: price and mass of firms

• It also can be shown that

pl =2k + 1

2k + 2c lD

N l =2(k + 1)γ

η

α− c lDc lD

• Finally, welfare is a decreasing function of c lD . This capturesthe effects of product variety and average prices (see theclosed economy case).

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 26 / 29

Page 27: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Number of varieties

• The number of sellers in country l consists of domesticproducers and exporters from h.

• Given a positive mass of entrants, G (c lD)N le represents

domestic producers, while G (chX )Nhe represents exporters

selling in l .

• Hence,G (c lD)N l

e + G (chX )Nhe = N l ,

and we can find N le for l = H,F.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 27 / 29

Page 28: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Efficiency

• 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 > c lD .

• Therefore, trade

• increases aggregate productivity by forcing the least productivefirms to exit (a la Melitz (2003)).

• decreases average price and markups (the competition effect).• increases the number of available varieties• increases, thereby, welfare!!

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 28 / 29

Page 29: The Melitz Ottaviano (2008) Model - LMU

Background Model Analysis, in closed economy Open Economy Analysis, in open economy

Mechanism

Intuition:

• Recall that in Melitz (2003), trade induces increasedcompetition for scarce labor resources. As a result, real wagerises and the least productive firms exit.

• Here, the intuition behind exit of least productive firms isdifferent. In the current model, increased factor marketcompetition plays no role, as the supply of labor to thedifferentiated sector is perfectly elastic (wage is determined bythe price of the numeraire).

• Firms exit only because of ”tougher” competition that affectsdemand elasticities.

Davide Suverato, LMU The Melitz Ottaviano (2008) Model 29 / 29