Melitz’ heterogeneous firm trade model with Pareto ... · Melitz (2003) examines the effects of trade on productivity and welfare in a heterogeneous firm framework. However, Melitz

Melitz’ heterogeneous firm trade model with Pareto−distributed productivity

Author: Richard Foltyn; Date: 2009−12−07

1 IntroductionMelitz (2003) examines the effects of trade on productivity and welfare in a heterogeneous firm framework. However, Melitzdoes not specify a distribution for firm−level heterogeneity, but only presents general results applicable to several families ofdistributions. This Mathematica notebook derives closed−form solutions for this model when firm heterogeneity is represented

by a productivity parameter Φ drawn from the Pareto distribution. The Pareto distribution was chosen as closed−form expres−

sions for all equilibrium variables can be derived, which is not the case for all distributions (such as the exponential distribu−tion). It is furthermore employed in other models with firm−level heterogeneity, such as Helpman/Melitz/Yeaple (2004) andvarious papers by Baldwin et al.

1.1 General model assumptions

The CDF of a Pareto−distributed random variable X is defined as

(1)FXHxL = 1 - Jb

xNk x ³ b

0 otherwise

with location parameter b and shape parameter k (see Evans(1993)). (Mathematica restricts the random variable to x > b, so thebuilt−in Pareto distribution is not used.)

In[1]:= ClearAll@Φ, b, k, Σ, fe, fc, ∆D

In[2]:= phicdf@Φ_D := PiecewiseB::1 -b

Φ

k

, Φ ³ b>>F

In[3]:= phipdf@Φ_D := Evaluate@D@phicdf@ΦD, ΦDD

In[4]:= TraditionalForm 8phicdf@ΦD, phipdf@ΦD<

Out[4]= : 1 - Ib

ΦMk Φ ³ b

0 True

,

b k J bΦNk-1

Φ2b - Φ £ 0

0 True

>

Using the Pareto distribution, the model exhibits most of the characteristics described in Melitz (2003). The notable exception isthat the zero cutoff profit (ZCP) condition does not result in a downward−sloping curve, as the resulting average profit given bythis condition is constant. As a sufficient condition for a downward−sloping ZCP curve, Melitz (see footnote 15) requires the

following expression be increasing in Φ on H0, ¥L. Here, however, this does not hold for all Φ:

In[5]:= AssumingB8Φ ³ b, b > 0, k > 0 0, 8b, k 1, to ensure that varieties actually are substitutes, but not perfectly so.

2.

Printed by Mathematica for Students

2.

Furthermore, k > 2 is required for the Pareto distribution to have a well−defined variance. Additionally, k > Σ - 1 is assumed to hold in order to avoid divisions by zero and to ensure that integrals converge (Helpman/Melitz/Yeaple (2004) use k > Σ + 1 for their Pareto−distributed firm productivity model, but actually only the weaker condition k > Σ - 1 is required. I thank Jonathan Dingel for point this out.)

3. The definition of the Pareto distribution requires that Φ ³ b > 0.

4. Finally, all fixed costs are assumed to be non−negative, i.e. f > 0, fx > 0, fe > 0.

In[6]:= defaultAssump = k > Σ - 1 && Σ > 1 && k > 2 && b > 0;

allAssump = defaultAssump && 0 < ∆ < 1 && fe > 0 && fc > 0;

(Limits for interactive graphs resulting from assumptions:)

In[8]:= kMin@Σ_D := [email protected], Σ - .999D; sMax@k_D := k + .999;Comment on notation: In the Mathematica expressions, Melitz’ notation is slightly modified to ensure that the resulting

Mathematica syntax is legal: Φ* = phistar, L = lsize, f = fc, etc.

2 Closed economy modelTo solve the Melitz (2003) model, one has to first determine the equilibrium distribution of Φ, which is done in section 3 of the

Melitz paper. Section 2.1 contains standard Dixit/Stiglitz results for a continuous spectrum of varieties and thus no detailedtreatment is required.

To determine the equilibrium value of Φ* (and thus the equilibrium distribution of Φ), all that is needed is the expression for

average productivity Φ HΦ*L (see Eq. (9) in the paper), the PDF/CDF of the ex ante distribution of Φ (gΦHΦL and GΦHΦL), and the zero

cutoff profit (ZCP) and free entry (FE) conditions.

2.1 Firm entry and exit

Firm entry and exit in the static equilibrium is governed by two conditions, the zero cutoff profit (ZCP) condition and the freeentry (FE) condition.

The distribution of the productivity of active firms in equilibrium (with PDF ΜHΦL) depends on one exogenous factor: the ex antedistribution of firm productivity (with PDF gΦHxL) (the cutoff productivity Φ* is determined endogenously). Hence, ΜHΦL is theequilibrium PDF conditional on the firm having a sufficiently high productivity to start producing, otherwise the firm exits

immediately after observing its productivity draw. Let Φ* denote this cutoff productivity level; then

(1)ΜHΦL =gΦIΦM

1-GΦIΦ*M Φ ³ Φ*

0 otherwise

where gΦHΦL and GΦHΦL are the PDF and CDF of the ex ante productivity distribution, respectively. Once a firm has secured aproductivity level Φ > Φ*, it earns a positive profit ΠHΦL in every period as productivity stays constant throughout the firm’s lifetime. Consequently, all firms but the marginal firm earn positive profits.

Furthermore, each active firms faces stochastic shocks which force it to exit the market with probability ∆ in each period. As timediscounting is ignored for simplicity, the resulting firm value is defined as follows:

Definition (Firm value): Let ΠHΦL be the per−period profit of a firm with productivity Φ. Then the expected firm value vHΦL is givenby

(2)vHΦL = max :0,ât=0

¥

H1 - ∆Lt ΠHΦL> = max :0, ΠHΦL∆>

(this follows from the summation rule for geometric series).

Definition (cutoff productivity level Φ*): Given the firm value vHΦL defined above, any firm with non−positive firm value willimmediately exit the market. Hence the productivity cutoff level Φ* is defined as

Φ* = inf 8Φ : vHΦL > 0<As ΠHΦL is continuous and increasing in Φ and ΠH0L = -f (see Eq. 5 in the paper), ΠHΦ*L = 0.The resulting average weighted productivity for active firms can be obtained from Eq. (7) in the paper and the definition of ΜHΦLfrom above:

Richard Foltyn melitz_pareto.nb 2


Φ HΦ*L = B 1

1 - GΦHΦ*L àΦ*

¥

ΦΣ-1 gΦHΦL â ΦF1

Σ-1

For the Pareto distribution, this can be calculated with Mathematica:

In[9]:= phiavg@phistar_D :=EvaluateBAssumingB8defaultAssump && phistar >= b 0

v =Π

∆= à

Φ*

¥

vHΦL ΜHΦL â Φ

ve =1 - GΦHΦ*L

∆Π - fe = 0

Π =∆ fe

1 - GΦHΦ*LThe last equation again relates the average profits to the cutoff productivity level Φ*. It can easily be seen that this is non−

decreasing in Φ* as GΦHΦ*L is non−decreasing in Φ* by definition of a CDF (for most distribution this will be strictly increasing).In[13]:= profitavgFE@phistar_D :=

EvaluateBPiecewiseExpandB

PiecewiseB:: ∆ fe1 - phicdf@phistarD

, phistar ³ b>, 8Null, TrueFFF

In[14]:= TraditionalForm@profitavgFE@Φ*DDOut[14]//TraditionalForm=

fe ∆ J bΦ*N-k b - Φ* £ 0

Null True

2.2 Equilibrium in the closed economy

The equilibrium value of Φ* is obtained by equating the ZCP and FE conditions.

In[15]:= eqn = Assuming@8allAssump, phistar ³ b= 1;



In[18]:= phiStarEquilCond@b_, k_, Σ_, ∆_, fc_, fe_ D :=EvaluateB

PiecewiseB::phiStarEquil, fc H-1 + ΣLfe ∆ H1 + k - ΣL

>= 1>, 8Null, TrueFF

Average profit of course is equal to the average profit from the ZCP condition, under the condition that an equilibrium exists.

In[19]:= HprofitEquil =Piecewise@88FullSimplify@FullSimplify@profitavgFE@xD, x ³ bD .

x ® phiStarEquil, allAssump && phiStarAssumD, phiStarAssum

Out[23]=

Σ

k

b

∆

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

Ex ante vs. equilibrium PDF

b=1 Φ*=1.3

In[24]:= phiCDFEquil@Φ_D :=Evaluate@Piecewise@88FullSimplify@

Integrate@Simplify@phiPDFEquil@xD, phiStarAssumD,8x, x0, Φ 0D .

x0 ® phiStarEquil, allAssump && phiStarAssumD, phiStarAssum

Out[26]=

Σ

k

b

∆

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

Ex ante vs. equilibrium CDF

b=1 Φ*=1.3

2.4 Equilibrium aggregate variables

In the following section, the expressions for the aggregate variables M , Φ, P, Q and Û in equilibrium are calculated. To simplify

notation, the following section assumes that the condition for the existence of an equilibrium, fc HΣ-1L

fe ∆ Ik-Σ+1M ³ 1, holds.

In[27]:= equilCondAssum = allAssump && phiStarAssum

Out[27]= k > -1 + Σ && Σ > 1 && k > 2 && b > 0 &&

0 < ∆ < 1 && fe > 0 && fc > 0 &&fc H-1 + ΣL

fe ∆ H1 + k - ΣL³ 1

In[28]:= hasEquilibrium@k1_, Σ1_, ∆1_, fc1_, fe1_D :=TrueQ@phiStarAssum . 8 k ® k1, Σ ® Σ1, ∆ ® ∆1, fc ® fc1, fe ® fe1

Out[31]=

k

1 2 3 4 5 6 7 8Σ0

10

20

30

40

M

Mass of firms in equilibrium

It is instructive to look at the mass of varieties as a function of the elasticity of substitution Σ. The number of varieties decreaseswith Σ, as a high Σ implies that the products are close varieties (with the limiting case Σ = ¥, when they are perfect substitutes).With perfectly substitutable products consumers do not gain any additional utility from consuming even more varieties, so thenumber of varieties decreases.

2.4.2 Average / aggregate productivity

The average/aggregate productivity in equilibrium can be calculated using either Eq. (7) or (10) from the paper. The results are,of course, identical.

In[32]:= phiAvgEquil =

FullSimplifyBIIntegrateAΦΣ-1 Simplify@phiPDFEquil@ΦD, phiStarAssumD,8Φ, phiStarEquil, ¥ allAssump && Φ Î RealsEM

1

Σ-1, allAssumpF TraditionalForm

Out[32]//TraditionalForm=

bk

k - Σ + 1

1

Σ-1 fc HΣ - 1Lfe ∆ Hk - Σ + 1L

1

k

In[33]:= FullSimplify@phiavg@phiStarEquilD, allAssump && phiStarAssumD TraditionalForm


bk

k - Σ + 1

1


1

k

In[34]:= phiAvgEquilCond@b_, k_, Σ_, ∆_, fc_, fe_D :=Evaluate@Piecewise@88phiAvgEquil, phiStarAssum

Out[35]=

k

0 2 4 6 8 10Σ

5

10

15

20

25

30

Φ

Σmax

2.4.3 Price index

In[36]:= HΡ = x . FlattenSolve@Σ 1 H1 - xL, xDL TraditionalFormOut[36]//TraditionalForm=

Σ - 1

Σ

In[37]:= price@Φ_D := EvaluateB1

Ρ ΦF

In[38]:= IpriceIdx = FullSimplifyAmassFirmsEquil1H1-ΣL price@phiAvgEquilD,allAssumpEM TraditionalForm


Σ J lsizefc ΣN

1

1-Σ J fc HΣ-1Lfe ∆ Hk-Σ+1L N

-1k

b HΣ - 1LIn[39]:= priceIdxCond@b_, k_, Σ_, ∆_, fc_, fe_, lsize_D :=

Evaluate@Piecewise@88priceIdx, equilCondAssum

Out[40]=

k

L

2 4 6 8 10Σ0.0

0.2

0.4

0.6

0.8

1.0

PPrice index

Σmax

k

Σ

0 20 40 60 80 100L

0.05

0.10

0.15

0.20

PPrice index

The price index is decreasing in the country size, which is due to the larger amount of varieties M available in larger countries.Equivalently, the real income (or utility, which is the same in this model), is higher in larger countries.

Also, the price index is initially increasing in Σ: varieties become closer substitutes when the elasticity of substitution increases,which leads to less utility from additional varieties, hence fewer varieties produced, thus resulting in a higher weighted priceindex. However, there is also an offsetting effect as individual product prices fall for a given productivity (as products becomemore substitutable, firms lose their monopolistic pricing power).

The relationship between P and M is negative, as in the standard Dixit−Stiglitz model. More varieties result in greater utility(which in these models is equivalent to real income), hence the price index must fall for a given nominal income. This is shown

in the following graph which shows different equilibrium combinations of M and P for a given set of exogenous parameters.



Out[41]=

k

L

0 10 20 30 40 50M0.0

0.1

0.2

0.3

0.4

0.5

P

Comparative statics: M vs. P

2.4.4 Aggregate profits

In[42]:= HaggrProfitEquil = FullSimplify@ massFirmsEquil * profitEquil,phiStarAssumDL TraditionalForm


lsize HΣ - 1Lk Σ

2.4.5 Aggregate quantities

Aggregate quantities are determined from the formula Q = R P = L P, as R = L.In[43]:= HaggrQuantEquil = FullSimplify@lsize priceIdxDL TraditionalForm


b fc HΣ - 1L lsizefc Σ

Σ


1

k

3 Open economy modelComment on notation: following Melitz, symbols referring to variables from the closed−economy case will from now on have asubscript a (for autarky). All other variables refer to the open−economy scenario.

3.1 Assumptions1. Initial sunk costs fex > 0 have to be paid by each exporting firm for every country it exports to after learning its productivity

level.These one−time sunk costs can alternatively be modeled as per−period fixed costs fx incurred by every exporting firm, with

fex = fxI1 + H1 - ∆L + H1 - ∆L2 + ¼ Mfex = fx

1

1 - H1 - ∆L =fx

∆



which arises from the fact that per−period costs have to be paid with ever−decreasing probability in future periods.

2. Iceberg trade costs Τ

3. The world (or trade block) consist of n ³ 2 identical countries (hence factor prices and aggregate variables are identical)

4. Fixed costs f are incurred regardless of the export status

As fixed export costs fex and variable trade costs Τ are identical for each country, a firm will either export to all countries or not

export at all.

3.2 Cutoff conditions and average productivity

The probability of a successful market entry is pin = 1 - GΦHΦ*L, as before (however, Φ* ¹ Φa* , as shown below!). Additionally, thereis a second cutoff productivity level Φx

* ³ Φ* such that firms with Φ* £ Φ < Φx* produce only for the domestic market, while firms

with Φ ³ Φx* produce for the domestic market and additionally export to all other countries. Let

px = PIΦ ³ Φx* Φ ³ Φ*M = PIΦ ³ Φx* M PHΦ ³ Φ*L be the probability of a firm being an exporter conditional on successful marketentry. Then px = I1 - GΦIΦx* MM I1 - GΦHΦ*LM.From this it follows that the exporting firms’ productivity PDF is given by gΦHxL = I1 - GΦHΦ*LM I1 - GΦIΦx* MM ÙΦx*¥gΦHxL â x . Theaverage productivity of exporting firms Φ

x = Φ IΦx* M is defined analogously to Φ as

(2)Φ

x =1 - GΦ HΦ*L1 - GΦ IΦx*M àΦx*

¥

xΣ-1 gΦHxL â x1

Σ-1

Furthermore, from Eq. (19) in the paper, Φx* is defined as a function of the cutoff level Φ*: Φx

* = Φ* ΤK fxfO

1

Σ-1.

In[44]:= phistarX@phistar_D := phistar * Α1

Σ-1

Here we use the substitution Α = ΤΣ-1 fx f , as this makes the Mathematica expressions less complex. Additionally, the partitioninto exporting and non−exporting active firms only occurs if the fixed export costs are sufficiently high. Otherwise, the productiv−

ity of any active firm, Φ ³ Φ*, would be sufficient to cover export costs and yield a non−negative profit. The necessary condition

for the partition to exist is Α > 1.

In[45]:= partitionAssump = Α > 1;

In[46]:= allAssumpOpen = allAssump && partitionAssump && Τ > 1 &&

fx > 0 && n ³ 2 && n Î Integers

Out[46]= k > -1 + Σ && Σ > 1 && k > 2 && b > 0 && 0 < ∆ < 1 && fe > 0 &&

fc > 0 && Α > 1 && Τ > 1 && fx > 0 && n ³ 2 && n Î Integers

For later use we also define the following expression:

In[47]:= a =fx

fcΤΣ-1;

In[48]:= px@phistar_D :=EvaluateBPiecewiseB::FullSimplifyB

SimplifyB 1 - phicdf@xD1 - phicdf@phistarD

, x ³ b && phistar ³ bF .

x ® phistarX@phistarD, allAssumpOpenF, phistar ³ b>>FF



In[49]:= TraditionalForm@px@Φ*DDOut[49]//TraditionalForm=

Αk

1-Σ Φ* ³ b

0 True

From the above expression it can be seen that the probability of an active firm being an exporter is constant (i.e. independent

from the cutoff productivity level) for given exogenous parameters k, Σ, Τ, f and fx . This immediately follows from the fact that

the export cutoff productivity level is a linear function of Φ* and the Pareto CDF used here.

The average export firm productivity conditional on firm market entry can be calculated as follows. Just like the average

productivity in a closed economy, it is a linear function of the cutoff productivity level, but Φ

x > Φ

for every Φ*. Thus export firms

are more productive.

In[50]:= TraditionalForm 8phiavg@Φ*D, phiavg@phistarX@Φ*DD<

Out[50]= :Φ*k

k - Σ + 1

1

Σ-1

, Φ*k

k - Σ + 1

1

Σ-1

Α

1

Σ-1>

From this it can be seen that Φ

x = Φ

Α1HΣ-1L = Φ ΤK fxfO

1

Σ-1, where the right−most term in brackets is strictly greater than 1 from

the condition for the existence of firm partitioning.

Out[51]=

k

Σ

fx

1 2 3 4 5Φ*

2

4

6

8

10

Φ, Φ

x

Avg. prod. of exporters and nonexporters

3.3 Average profit / zero cutoff profit condition

In[52]:= kfun@phistar_D :=phiavg@phistarD

phistar

Σ-1

- 1



In[53]:= profitAvgZCPOpen@phistar_D := Evaluate@Piecewise@88FullSimplify@fc kfun@phistarD +

px@phistarD * n * fx * kfun@phistarX@phistarDD,allAssumpOpen && phistar > bD, partitionAssump

In[58]:= FullSimplify@phistar . Flatten@Solve@eqn, phistar, InverseFunctions ® TrueDD,allAssumpOpenD . Α ® a

Out[58]= bfe ∆ H1 + k - ΣL

H-1 + ΣL fc + fx n J fx Τ-1+ΣfcN

k

1-Σ

-1k

This can be transformed into

(3)Φ* = bf HΣ - 1L

fe ∆ Hk + 1 - ΣL

1

k fx

fn ΑkH1-ΣL + 1

1

k

= Φa*

fx

fn ΑkH1-ΣL + 1

1

k

where the first term on the right−hand side is Φa* from the closed economy solution. The second term is strictly greater than one

given the initial assumptions and parameter restrictions. Therefore the cutoff productivity level in the open economy is always

higher than in the closed economy scenario. For sufficiently high export costs fx , the term fx f × nΑkH1-ΣL becomes very smalland thus the value of Φ* tends to Φa

* as each country effectively becomes a closed economy.

In[59]:= LimitBfx

fcn ΑkH1-ΣL . Α ® a, fx ® ¥,

Assumptions ® HallAssumpOpen . Α ® aLF

Out[59]= 0

The cutoff productivity level for firms to be exporters is given as:

In[60]:= HphiStarXEquil = FullSimplify@phistarX@phiStarOpenEquilD,allAssumpOpenDL TraditionalForm


b Α1

Σ-1

fe ∆ Hk - Σ + 1LHΣ - 1L fc + fx n Α k1-Σ

-1k

Alternatively, Φx* can be written as a function of the autarky cutoff productivity Φa

* : Φx* = Φa

* fx

fcn + Α

k

Σ-1

1

k

:

In[61]:= phiStarXEquil1 = phiStarEquil *fx

fcn + Α

k

Σ-1

1

k

Out[61]= bfx n

fc+ Α

k

-1+Σ

1

k

-fc H-1 + ΣL

fe ∆ H-1 - k + ΣL

1

k

In[62]:= FullSimplify@phiStarXEquil phiStarXEquil1, allAssumpOpenDOut[62]= True



In[63]:= phiStarXEquilCond@b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_D :=Evaluate@Piecewise@88phiStarXEquil . Α ® a,

phiStarAssum && HpartitionAssump . Α ® aL

Out[68]=

Σ

k

fx

Τ

n

0 1 2 3 4 5Φ

0.2

0.4

0.6

0.8

1.0

Ex ante PDF vs. equilibrium PDF of Φ

b=1 Φ*=1.7

Φx*=1.9

In[69]:= phiCDFOpenEquil@Φ_D :=Evaluate@Piecewise@88FullSimplify@

Integrate@Simplify@phiPDFOpenEquil@xD, phiStarAssumD,8x, x0, Φ 0D .

x0 ® phiStarOpenEquil, allAssumpOpen && phiStarAssumD,phiStarAssum

In[71]:= HmassFirmsOpenEquil =Together

FullSimplify@lsize HΣ HSimplify@profitAvgOpenEquil, phiStarAssumD + fc +

HSimplify@px@xD, x ³ bD . x ® phiStarOpenEquilL * n * fxLL,allAssumpOpenD L TraditionalForm


lsize Hk - Σ + 1Lk Σ fc + fx n Α

k

1-Σ

In[72]:= HmassFirmsXEquil =Together

FullSimplify@HSimplify@px@xD, x ³ bD . x ® phiStarOpenEquilL *massFirmsOpenEquil, allAssumpOpenDL TraditionalForm


lsize Hk - Σ + 1L Α k1-Σk Σ fc + fx n Α

k

1-Σ

In[73]:= HmassFirmsTEquil =FullSimplify@massFirmsOpenEquil + n * massFirmsXEquil,phiStarAssumDL TraditionalForm


lsize Hk - Σ + 1L n Α k1-Σ + 1

k Σ fc + fx n Αk

1-Σ

(The following functions define the mass of firms conditional on the existence of an equilibrium and will be used in calculationsfurther below.)

In[74]:= massFirmsOpenEquilCond@b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_, lsize_D :=EvaluatePiecewise@88massFirmsOpenEquil . Α ® a, phiStarAssum

In[77]:= HphiAvgOpenEquil =FullSimplify@phiavg@xD . x ® phiStarOpenEquil, allAssumpOpenDL

TraditionalForm


bk

k - Σ + 1

1

Σ-1 fe ∆ Hk - Σ + 1LHΣ - 1L fc + fx n Α k1-Σ

-1k

In[78]:= HphiAvgOpenXEquil = FullSimplify@phiavg@xD . x ® phiStarXEquil,allAssumpOpenDL TraditionalForm


bk Α

k - Σ + 1

1

Σ-1 fe ∆ Hk - Σ + 1LHΣ - 1L fc + fx n Α k1-Σ

-1k

However, the first definition does not take into account the greater (world−wide) market share of more productive exporters,while the average export firm productivity ignores losses due to iceberg trading costs Τ. Therefore, Melitz defines a third average

productivity, Φ

t , taking into account both effects.

In[79]:= phiAvgOpenTEquil =

FullSimplifyB1

massFirmsTEquilJmassFirmsOpenEquil * phiAvgOpenEquilΣ-1 +

n * massFirmsXEquil IΤ-1 phiAvgOpenXEquilMΣ-1N1

Σ-1

,

phiStarAssum && allAssumpOpenF TraditionalForm


Hk - Σ + 1L k HΣ+1L+HΣ-1L2

k-k Σ Hk ΑL Σ+1Σ-1 J k2 ΑHk-Σ+1L2 N1

1-ΣbΣ Α

Σ

1-Σ + n Τ J bΤNΣ Α k+11-Σ fe ∆

HΣ-1L fc+fx n Αk

1-Σ

1-Σ

k

b n Αk

1-Σ + b

1

Σ-1

(Again, the following functions only compute the average productivities if an equilibrium exists for the given parameters.)

In[80]:= phiAvgOpenEquilCond@b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_D :=EvaluatePiecewise@88

phiAvgOpenEquil . Α ® a, phiStarAssum

In[81]:= phiAvgOpenXEquilCond@b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_D :=EvaluatePiecewise@88phiAvgOpenXEquil . Α ® a, phiStarAssum

Out[86]=

k

Σ

fx

Τ

n

Φa* Φ* Φx

*Φ*

Π

4.1 Trade effects on the mass of varieties

Using the results obtained above, the mass quantities for equilibrium firms/varieties can be written as:

(5)

Ma =L Hk + 1 - ΣL

f k Σ

M =L Hk + 1 - ΣL

f Σ k J fxf

n ΑkH1-ΣL + 1N=

1

fx

fn ΑkH1-ΣL + 1

Ma < Ma

Mx = px M = ΑkH1-ΣL L Hk + 1 - ΣL

f Σ k J fxf

n ΑkH1-ΣL + 1N< M

Mt = H1 + n pxLM = I1 + n ΑkH1-ΣLM L Hk + 1 - ΣLf Σ k J fx

fn ΑkH1-ΣL + 1N

=I1 + n ΑkH1-ΣLMJ fx

fn ΑkH1-ΣL + 1N

Ma

where Ma is the mass in autarky. From this we see that in the open economy, the number of varieties produced by domestic

firms is always smaller than the number produced in autarky: Ma > M . The last line shows the relationship between the mass of

overall product varieties Mt and the mass of product varieties in autarky, Ma. It is evident that whether Mt > Ma and thus

whether trade results in more varieties for consumers only depends on the relative value of fixed costs and fixed export costs, f

and fx :

fx = f Mt = Ma

fx > f Mt < Ma

fx < f Mt > Ma



Thus, if there is to be any partition into exporting and non−exporting firms and therefore ΤΣ-1 fx > f holds, and trade is to

provide more choice to consumers, fx must be in the interval f Τ1-Σ < fx < f .

Out[87]=

k

Σ

L

fxfΤ

n

Ma M Mx Mt

10

20

30

40

50

Product varieties in autarky and with trade

From the interactive bar chart it is easy to see that once fx < f Þ Mt > Ma, the variety−increasing effect is further magnified

with low variable trade costs Τ, many countries/large n, large k and low values of Σ.

4.2 Trade affects on average productivity

Again using Α = ΤΣ-1fx

f> 1, the average productivities in autarky, and for all firms and exporters in the open economy, Φ

a, Φ

and

Φ

x , respectively, can be written as

(6)

Φ

a = bf HΣ - 1L

fe ∆ Hk + 1 - ΣL

1

k k

k + 1 - Σ

1HΣ-1L

Φ

= bf HΣ - 1L

fe ∆ Hk + 1 - ΣL

1

k fx

fn ΑkH1-ΣL + 1

1

k k

k + 1 - Σ

1HΣ-1L= Φ

a

fx

fn ΑkH1-ΣL + 1

1

k

> Φ

a

Φ

x = bf HΣ - 1L

fe ∆ Hk + 1 - ΣL

1

k

ΑkHΣ-1L +fx

fn

1

k k

k + 1 - Σ

1HΣ-1L=

= bf HΣ - 1L

fe ∆ Hk + 1 - ΣL

1

k

1 +fx

fn ΑkH1-ΣL

1

k

Α1HΣ-1Lk

k + 1 - Σ

1HΣ-1L= Α1HΣ-1L Φ



As Α > 1 ì Σ > 1 Þ Α1HΣ-1L > 1, we get the productivity ordering Φ a < Φ < Φ x , which always holds in an equilibrium withexporting and non−exporting firms.

Out[88]=

k

Σ

L

fxfΤ

n

Φ

a Φ Φ

xΦ

t

0.5

1.0

1.5

2.0

Average productivity in autarky and with trade

4.3 Welfare effects of trade

Welfare in autarky and the open economy is defined as the real wage, i.e. with wages standardized at 1, as the inverse priceindex:

Wa = Pa-1 = M1HΣ-1L Ρ Φ

W = P-1 = Mt1HΣ-1L

Ρ Φ

t

The exact equation for the price index with a Pareto distribution can be derived as follows: Using the expressions for Mx and Φ

x

from above, we get

P =1

ΡM Φ

Σ-1+ n ΑkH1-ΣL M

Α1HΣ-1L Φ

Τ

Σ-11

1-Σ

=

=1

Ρ Φ M

1H1-ΣL 1 + n Τ1-Σ Αk+1-Σ

1-Σ

1

1-Σ

For the Pareto distributions and the values for M and Φ

given above, this results in



P =1

Ρ b

L

f Σ

1

1-Σ f HΣ - 1Lfe ∆ Hk + 1 - ΣL

-1

k fx

fn ΑkH1-ΣL + 1

k-Σ+1

k HΣ-1L1 + n Τ1-Σ Α

k+1-Σ

1-Σ

1

1-Σ

=

=1

Ρ b

L

f Σ

1

1-Σ f HΣ - 1Lfe ∆ Hk + 1 - ΣL

-1

k fx

fn ΑkH1-ΣL + 1

-1

k

Recalling the expressing for Pa from above, this can also be written as

P = Pa

fx

fn ΑkH1-ΣL + 1

-1

k

Melitz claims that regardless of the effect of trade on the number of total varieties Mt , the welfare effect is always positive. For

this to be true, the term in parentheses must be smaller than one:

fx

fn ΑkH1-ΣL + 1

-1

k

< 1

fx

fn ΑkH1-ΣL + 1 > 1

Given our assumptions, this always holds . Therefore, Pa > P Þ W > Wa.

Out[89]=

k

Σ

L

fxfΤ

n

Wa W

10

20

30

40

50Wellfare effects of trade

4.4 Revenue and profit in autarky and with trade

Finally we examine revenue and profits in the closed and open economy (the figure shown here is equivalent to figure 2 inMelitz (2003)).



In[90]:= rev@Φ_, phistar_D :=Φ

phistar

Σ-1

Σ fc

In[91]:= revAutarky@Φ_, b_, k_, Σ_, ∆_, fc_, fe_D :=Evaluate

PiecewiseB:8rev@Φ, phiStarEquilD, Φ ³ phiStarEquil && phiStarAssum>, 0F

In[92]:= revTrade@Φ_, b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_D :=EvaluateBBlockB8phi = HphiStarOpenEquil . Α ® aL,

phiX = HphiStarXEquil . Α ® aL, 0FFF

In[93]:= profitAutarky@Φ_, b_, k_, Σ_, ∆_, fc_, fe_D :=Evaluate

PiecewiseB

:: rev@Φ, phiStarEquilDΣ

- fc, Φ ³ phiStarEquil && phiStarAssum>,

:Null, fc H-1 + ΣLfe ∆ H1 + k - ΣL

< 1>>, 0F

In[94]:= profitTrade@Φ_, b_, k_, Σ_, ∆_, fc_, fe_, fx_, Τ_, n_D :=EvaluateBBlockB8phi = HphiStarOpenEquil . Α ® aL,

phiX = HphiStarXEquil . Α ® aL,

: rev@Φ, phiDΣ

- fc + n Τ1-Σrev@Φ, phiD

Σ- fx ,

Φ ³ phiX && phiStarAssum>,

:Null, fc H-1 + ΣLfe ∆ H1 + k - ΣL

< 1>>, 0FFF



Out[95]=

k

Σ

fxfΤ

n

Φa* Φ* Φx

*Φ

r

Revenue in autarky and with trade

Φa* Φ* Φx

*Φ

Π

Profit in autarky and with trade

The effects of trade on firm revenue and profits are identical to those described in Melitz (2003) and depend on the firm

productivity Φ. Four different types of firms can be distinguished (again, these are comparative statics results; nothing is said

about the dynamics when moving from autarky to trade):

1.



1.

Firms with productivity Φa* £ Φ < Φ* exit the market in the open economy.

2. Firms with productivity Φ* £ Φ < Φx* produce for the domestic market only and incur both revenue and profit losses (as

fixed costs do not change).

3. Firms with productivity Φx* £ Φ < Φxx

* export and increase revenue, but incur earn lower profits due to additional fixed

export costs fx . (the value of Φxx* can be determined by setting DΠ = 0 and solving for Φ in Melitz (2003, p. 1714).

4. Firms with productivity Φ > Φxx* export and increase revenues as well as profits in the open economy scenario.

4.5 Trade liberalization

The effects of trade liberalization (increasing number of countries in a trading block, lower fixed and variable export costs) can

be easily examined by manipulating the parameters n, fx , and Τ of the graphs shown in the previous section. The effects are

identical to those described by Melitz.

ReferencesDixit, Avinash K.and Stiglitz, Joseph E.(1977) : úMonopolistic Competition and Optimum Product Diversityø.In : American

Economic Review 67 (3), 297−308.

Evans, Merran / Nicholas Hastings and Brian Peacock (1993): Statistical Distributions. Wiley.

Helpman, Elhanan/ Melitz, Marc J. and Yeaple, Stephen R. (2004): úExport versus FDI with Heterogeneous Firmsø. In: AmericanEconomic Review 94(1), 300|316.

Melitz, Marc J. (2003): úThe Impact of Trade on Intra−Industry Reallocations and Aggregate Industry Productivityø. In: Economet−rica 71(6), 1695.



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Melitz’ heterogeneous firm trade model with Pareto ... · Melitz (2003) examines the effects of trade on productivity and welfare in a heterogeneous firm framework. However, Melitz