-
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
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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
Printed by Mathematica for Students
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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;
Richard Foltyn melitz_pareto.nb 4
Printed by Mathematica for Students
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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
Out[33]//TraditionalForm=
bk
k - Σ + 1
1
Σ-1 fc HΣ - 1Lfe ∆ Hk - Σ + 1L
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
Out[38]//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.
Richard Foltyn melitz_pareto.nb 10
Printed by Mathematica for Students
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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
Out[42]//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
Out[43]//TraditionalForm=
b fc HΣ - 1L lsizefc Σ
Σ
Σ-1 fc HΣ - 1Lfe ∆ Hk - Σ + 1L
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
∆
Richard Foltyn melitz_pareto.nb 11
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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
Richard Foltyn melitz_pareto.nb 12
Printed by Mathematica for Students
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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
Richard Foltyn melitz_pareto.nb 13
Printed by Mathematica for Students
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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
Out[60]//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
Richard Foltyn melitz_pareto.nb 15
Printed by Mathematica for Students
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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
Out[71]//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
Out[72]//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
Out[73]//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
Out[77]//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
Out[78]//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
Out[79]//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
Richard Foltyn melitz_pareto.nb 21
Printed by Mathematica for Students
-
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
Richard Foltyn melitz_pareto.nb 22
Printed by Mathematica for Students
-
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
Richard Foltyn melitz_pareto.nb 23
Printed by Mathematica for Students
-
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)).
Richard Foltyn melitz_pareto.nb 24
Printed by Mathematica for Students
-
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
Richard Foltyn melitz_pareto.nb 25
Printed by Mathematica for Students
-
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.
Richard Foltyn melitz_pareto.nb 26
Printed by Mathematica for Students
-
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.
Richard Foltyn melitz_pareto.nb 27
Printed by Mathematica for Students