Stanford Econ 266: International Trade
— Lecture 10: Firm-Level Trade (Theory I) —
Stanford Econ 266 (Dave Donaldson)
Winter 2016 (lecture 10)
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
1 / 49
Today’s Plan
1
Introduction
2
A brief introduction to firm-level trade facts (more in Lectures 12-13)
3
Melitz (2003)
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
2 / 49
Introduction
Hallak and Levinsohn (2005): “Countries don’t trade. Firms trade.”
Since around 1990, trade economists have increasingly used data from
individual firms in order to better understand:
Why countries trade.
The nature of trade costs.
The mechanisms of adjustment to trade liberalization: mark-ups, entry,
exit, productivity changes, factor price changes.
How important trade liberalization is for economic welfare.
Who are the winners and losers of trade liberalization (across firms,
across workers)?
This has been an extremely influential development for the field.
These are all new and interesting questions that a firm-level approach
has enabled access to.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
3 / 49
Today’s Plan
1
Introduction
2
A brief introduction to firm-level trade facts (more in Lectures
12-13)
3
Melitz (2003)
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
4 / 49
Stylized Facts about Trade at the Firm-Level
Exporting is extremely rare.
Exporters are different:
They
They
They
They
are larger.
are more productive.
use factors differently.
pay higher wages.
We will go through some of these findings first.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
5 / 49
Exporting is Rare
Two papers provide a clear characterization of just how rare exporting
activity is among firms:
1
2
Bernard, Jensen, Redding and Schott (2007, JEP) on US
manufacturing.
Eaton, Kortum and Kramarz (2011, Ecma) on French manufacturing.
(We will have more to say about this paper in the next empirical
lecture, when we discuss how exporting varies across firms and partner
countries.)
It has been hard to match firm-level datasets (which typically contain
data on total output/sales, but not sales by destination) to
shipment-level trade datasets, but fortunately this has now been
achieved (by researchers such as the above authors).
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
6 / 49
BJRS (2007)
Firms in InternationalTrade 109
Table2
Exporting By U.S. Manufacturing Firms, 2002
NAICS industry
311
312
313
314
315
316
321
322
323
324
325
326
327
331
332
333
334
335
336
337
339
Food Manufacturing
Beverage and Tobacco Product
Textile Mills
Textile Product Mills
Apparel Manufacturing
Leather and Allied Product
Wood Product Manufacturing
Paper Manufacturing
Printing and Related Support
Petroleum and Coal Products
Chemical Manufacturing
Plastics and Rubber Products
Nonmetallic Mineral Product
Primary Metal Manufacturing
Fabricated Metal Product
Machinery Manufacturing
Computer and Electronic Product
Electrical Equipment, Appliance
Transportation Equipment
Furniture and Related Product
Miscellaneous Manufacturing
Aggregate manufacturing
Percentof
firms
Percentof
firms that
export
Mean exportsas a
percentof total
6.8
0.7
1.0
1.9
3.2
0.4
5.5
1.4
11.9
0.4
3.1
4.4
4.0
1.5
19.9
9.0
4.5
1.7
3.4
6.4
9.1
12
23
25
12
8
24
8
24
5
18
36
28
9
30
14
33
38
38
28
7
2
15
7
13
12
14
13
19
9
14
12
14
10
12
10
12
16
21
13
13
10
15
100
18
14
shipments
Sources:Data are from the 2002 U.S. Census of Manufactures.
Notes:The first column of numbers summarizes the distribution of manufacturing firms across threedigit NAICS manufacturing industries. The second reports the share of firms in each industry that
export. The final column reports mean exports as a percent of total shipments across all firms that
export in the noted industry.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
7 / 49
BJRS (2007)
124 Journal of EconomicPerspectives
Table 7
Exporting and Importing by U.S. Manufacturing Firms, 1997
NAICS industry
311
312
313
314
315
316
321
322
323
324
325
326
327
331
332
333
334
335
336
337
339
Food Manufacturing
Beverage and Tobacco Product
Textile Mills
Textile Product Mills
Apparel Manufacturing
Leather and Allied Product
Wood Product Manufacturing
Paper Manufacturing
Printing and Related Support
Petroleum and Coal Products
Chemical Manufacturing
Plastics and Rubber Products
Nonmetallic Mineral Product
Primary Metal Manufacturing
Fabricated Metal Product
Machinery Manufacturing
Computer and Electronic Product
Electrical Equipment, Appliance
Transportation Equipment
Furniture and Related Product
Miscellaneous Manufacturing
Aggregate manufacturing
Percentof all
firms
Percentoffirms Percentoffirms
that import
that export
Percentoffirms
that import&
export
7
1
1
2
6
0
5
1
13
0
3
5
4
1
20
9
4
2
3
6
7
17
28
47
19
16
43
15
42
10
32
56
42
16
51
21
47
65
58
40
13
31
10
19
31
13
15
43
5
18
3
17
30
20
11
23
8
22
40
35
22
8
19
7
13
24
9
9
30
3
15
2
14
26
16
7
21
6
19
37
30
18
5
15
100
27
14
11
Sources:Data are for 1997 and are for firms that appear in both the U.S. Census of Manufactures and the
Linked-Longitudinal Firm Trade Transaction Database (LFTTD).
Notes:The first column of numbers summarizes the distribution of manufacturing firms across threedigit NAICS industries. Remaining columns report the percent of firms in each industry that export,
import, and do both.
Stanford Econ 266 (Dave firm
Donaldson) is now available.10
Firm-Level
Trade
(Theory I)
The data
on firm
Winter
2016 (lecture 10)
of the same
8 / 49
EKK (2011)
Out of 229,9000 French manufacturing firms, only 34,035 sell abroad
Figure 1: Entry and Sales by Market Size
Panel A: Entry of Firms
FRA
# firms selling in market
100000
10000
1000
100
CEN
SIE
BELSWI
GER
ITA UNK
NETSPA
CAN
AUT
SWE
CAM
COTMOR
POR
ALGDEN
TUN
GRE
NOR
SEN
FIN
SAU
ISRHOK
AUL
IRE
SIN EGY
SOU
TOG
KUW
BEN
TUR YUG
IND
NIG MAL
MADBUK
TAI
BRA
KOR
ZAI JOR
MAS
NZE
ARG
HUN
VEN
CHI
THA
MEX
MAU
MAY
SYR
PAK
NIA
CHN
IRQ
CZE
INO
COL
OMA
CHA
ANG
BUL
URU KEN
PER
PAN PHI
ROM
IRN
GEE
RWA
ECU LIY
BUR
SUD
SRI
CUB
PAR
ETH
DOM
ZIM
COS
BAN
MOZ GHALIB
TRI
GUA
TAN
ZAM
HON
ELS
VIE
NIC
BOLJAM
PAP
SOM
ALB
MAWUGA
AFG
NEP
USA
JAP
USR
10
.1
1
10
100
market size ($ billions)
1000
10000
rket share
Panel B: Normalized Entry
5000000
1000000
Stanford Econ
266 (Dave Donaldson)
JAP
USA
Firm-Level Trade (Theory I)
CAN
USR
Winter 2016
(lecture 10)
9 / 49
Exporters are Different
The most influential findings about exporting and intra-industry
heterogeneity have been related to:
Exporters being larger.
Exporters being more productive.
But there are other “exporter premia” too.
Clearly there is an issue of selection versus treatment here that is of
fundamental importance (for policy and for testing theory).
This difficult issue has been best tackled with respect to the correlation
between exporting and productivity, covered in Lectures 12-13.
For now, we focus on the stylized fact that concerns the association
between exporting and some phenomenon (like size).
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
10 / 49
ExporterAndrew
Premia
inJ.the
United
States
B. Bernard,
Bradford
Jensen, StephenJ.
Redding, and PeterK. Schott
111
BJRS (JEP, 2007)
Table3
Exporter Premia in U.S. Manufacturing, 2002
Exporterpremia
(1)
(2)
(3)
Log employment
Log shipments
Log value-added per worker
Log TFP
Log wage
Log capital per worker
Log skill per worker
1.19
1.48
0.26
0.02
0.17
0.32
0.19
0.97
1.08
0.11
0.03
0.06
0.12
0.11
0.08
0.10
0.05
0.06
0.04
0.19
Additional covariates
None
Industry fixed
effects
Industry fixed
effects, log
employment
Sources:Data are for 2002 and are from the U.S. Census of Manufactures.
Notes:All results are from bivariate ordinary least squares regressions of the firm characteristic in the first
column on a dummy variable indicating firm's export status. Regressions in column 2 include industry
fixed effects. Regressions in column 3 include industry fixed effects and log firm employment as
controls. Total factor productivity (TFP) is computed as in Caves, Christensen, and Diewert (1982).
"Capital per worker" refers to capital stock per worker. "Skillper worker" is nonproduction workers per
total employment. All results are significant at the 1 percent level.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
11 / 49
Exporter Premia in the United States
Firms in InternationalTrade 125
BJRS (JEP, 2007)
Table8
Trading Premia in U.S. Manufacturing, 1997
Log
Log
Log
Log
Log
Log
Log
employment
shipments
value-added per worker
TFP
wage
capital per worker
skill per worker
(1) Exporterpremia
(2) Importerpremia
(3) Exporter& importerprem
1.50
0.29
0.23
0.07
0.29
0.17
0.04
1.40
0.26
0.23
0.12
0.23
0.13
0.06
1.75
0.31
0.25
0.07
0.33
0.20
0.03
Sources:Data are for 1997 and are for firms that appear in both the U.S. Census of Manufacturers and
the Linked-Longitudinal Firm Trade Transaction Database (LFTTD).
Notes:All results are from bivariate ordinary least squares regressions of the firm characteristic listed on
the left on a dummy variable noted at the top of each column as well as industry fixed effects and firm
employment as additional controls. Employment regressions omit firm employment as a covariate. Total
factor productivity (TFP) is computed as in Caves, Christensen, and Diewert (1982).
In Table 8, we compare the characteristics of exporting and importing firms.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
12 / 49
The Exporter Premium: Productivity
Bernard, 1272
Eaton, Jensen and Kortum
(AER, 2003)
forREVIEW
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ratio of labor productivity
* Exporters
I
J l Nonexporters
FIGURE 2A.
Stanford Econ 266 (Dave Donaldson)
RATIO OF PLANT LABOR PRODUCTIVITY TO OVERALL MEAN
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
13 / 49
The Exporter Premium: Productivity
Bernard, VOL.
Eaton,
and Kortum
(AER,
2003) for
US data TRADE
93 NO.Jensen
4
BERNARD
ET AL.: PLANTS,
PRODUCTIVITY
IN INTERNATIONAL
1273
16
14
12
10
|
8
4
<0.25 0.25- 0.30- 0.35- 0.42- 0.50- 0.59- 0.71- 0.84- 1.00- 1.19- 1.14- 1.68- 2.00- 2.38- 2.83- 3.36- >4.00
0.30 0.35 0.42 0.50 0.59 0.71 0.84 1.00 1.19 1.41 1.68 2.00 2.38 2.83 3.36 4.00
ratio of labor productivity
l NonexportersU Exporters
FIGURE2B. RATIOOF PLANTLABORPRODUCTIVITY
TO 4-DIGIT INDUSTRYMEAN
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
14 / 49
The Exporter Premium: Productivity
EKK (2011) on France (we’ll talk about the model in Lectures 12-13)
Figure 6: Productivity and Markets Penetrated
Model Versus Data
2.25
average productivity
2
1.75
data
model
1.5
1.25
1
1
2
Stanford Econ 266 (Dave Donaldson)
4
8
16
32
64
minimum number of markets penetrated
Firm-Level Trade (Theory I)
128
Winter 2016 (lecture 10)
15 / 49
The Exporter Premium: Domestic Sales
EKK (2011) on France
Figure 3: Sales in France and Market Entry
Panel A: Sales and Markets Penetrated
Panel B: Sales and # Penetrating Multiple Markets
10000
average sale in France ($ millions)
average sale in France ($ millions)
10000
1000
100
10
110
109
108
107
1000
106
105
104
103
102
101
100
99
98
97
96
9590
92
91
94
93
89
88
87
86
8482
83
85
100
81
80
79
78
77
76
75
74
73
70
72
71
69
68
67
66
65
64
63
62
61
60
59
58
57
56
54
55
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
98
7
65
4
10
3
2
1
1
1
2
4
8
16
32
minimum number of markets penetrated
64
128
Panel C: Sales and # Selling to a Market
average sales in France ($ millions)
1000
MAW
UGA
BOL
SIE
JAM
SOM
ELS
ALB
ZAM
GUA
AFG NIC
VIE
COS
HON
GHA
LIB
BAN
DOM
PAPTAN
ETH
ZIM
SRI
SUD
MOZ
LIY
GEE
ECU
PAR
PER
TRI
BUR
KEN
CUB
ROM
IRN
NEP
PHI
COL
INO
CHN
RWA
BUL
URU
OMA
ANG
PAK
USR
IRQ
MAY
PAN
NIA
JOR
CZE
ARG
MEX
THA
SYR
CHI
VEN
HUN
MAU
TAI
NZE
MAS
YUG
KOR
TUR
ZAI
MAD
MAL
CHA
BRA
EGY
IND
BUK
CEN
NIG
KUW
BEN
SOU
SIN
TOG
IRE
AUL
FIN
SAU
HOK
SEN
GRE
ISR
NOR
TUN
POR
COT
DEN
JAP
SWE
MOR
AUT
ALG
CAM
CANSPA
NET
USA
UNK
ITA
GER
SWI
BEL
100
10
FRA
1
20
100
1000
10000
# firms selling in the market
Stanford Econ 266 (Dave Donaldson)
100000
500000
1
percentiles (25, 50, 75, 95) in France ($ millions)
1
10
100
1000
10000
# firms selling to k or more markets
100000 500000
Panel D: Distribution of Sales and Market Entry
10000
MAW
UGABOL
ELS
SIE
ALB
SOM
JAM
ZAM
PAP
HON
GUA
COS
NIC
VIELIB
ETH
SUD
BAN
LIY
DOM
PAR
SRI
AFG GHA
MOZ
ECU
ZIM
GEE
PER
KEN
BUR
IRN
CUB
PHI
COL
CHN
ROM
INO
TAN
BUL
ANG
PAK
MAY
VEN
URU
USR
JOR
OMA
IRQ
SYR
CHI
TRI RWA
NIA
THA
MEX
HUN
ARG
NEP
PAN
CZE
CHA
TUR
MAU
NZE
MAS
BRA
TAI
ZAI
YUG
IND
MAL
MAD
KOR
EGY
CEN
BUK
NIG
SOU
KUW
BEN
SIN
TOG
IRE
AUL
SAU
FIN
GRE
HOK
SEN
ISR
NOR
TUN
POR
DEN
COT
JAP
MOR
SWE
ALG
CAM
AUT
CANSPA
NET
ITAGER
USA
UNK
SWI
BEL
1000
100
10
FRA
1
.1
20
Firm-Level Trade (Theory I)
100
1000
10000
# firms selling in the market
100000
500000
Winter 2016 (lecture 10)
16 / 49
Other Exporter Premia
Examples of other exporter premia seen in the data (and there are
many more):
Produce more products: BJRS (2007) and Bernard, Redding and
Schott (QJE, 2011)
Higher Wages: Frias, Kaplan and Verhoogen (2009 wp) using
employer-employee linked data from Mexico (i.e., when a given worker
moves from a purely domestic firm to an exporting firm, his/her wage
rises).
More expensive (i.e. probably indicating higher quality) material
inputs: Kugler and Verhoogen (REStud, 2012) using very detailed data
on inputs used by Colombian firms.
Innovate more: Aw, Roberts and Xu (AER, 2011).
Pollute less: Holladay (2015)
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
17 / 49
Firm-Level Heterogeneity and Trade Facts: Summary
What’s wrong with previous theories?
Problem: previous theories in this course are at odds with (or cannot
account for) many micro-level facts (some described above, others
that we’ll see in Lectures 12-13):
1
Within a given industry, there is firm-level heterogeneity in export
behavior.
2
Fixed costs of exporting appear to matter in export-related decisions.
3
Within a given industry, more productive firms are more likely to
export.
4
Trade liberalization leads to intra-industry reallocation across firms.
5
These reallocations are correlated with productivity and export status.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
18 / 49
Today’s Plan
1
Introduction
2
A brief introduction to firm-level trade facts (more in Lectures 12-13)
3
Melitz (2003)
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
19 / 49
Firm-Level Heterogeneity and Trade
What does Melitz (2003) do about it?
Melitz (2003, Ecma) will develop a model featuring facts 1 and 2 that
can explain facts 3, 4, and 5
This is probably the most influential trade paper in the last 15 years
Two building blocks:
1
2
Krugman (1980, AER): CES, IRS technology, monopolistic competition
Hopenhayn (1992, JPE): equilibrium model of entry and exit
From a normative point of view, Melitz (2003) may also provide
“new” source of gains from trade if trade induces reallocation of labor
from least to most productive firms (more on that in later lectures).
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
20 / 49
Melitz (2003)
Demand
Like in Krugman (1980), representative agent has CES preferences:
σ
Z
σ−1
σ−1
σ
dω
U=
q (ω)
ω∈Ω
where σ > 1 is the elasticity of substitution
Consumption and expenditures for each variety are given by
p (ω) −σ
q (ω) = Q
P
p (ω) 1−σ
r (ω) = R
P
(1)
(2)
where:
Z
P≡
1−σ
p (ω)
ω∈Ω
Stanford Econ 266 (Dave Donaldson)
dω
1
1−σ
Z
,R≡
r (ω) , and Q ≡ R/P
ω∈Ω
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
21 / 49
Melitz (2003)
Production
Like in Krugman (1980), labor is the only factor of production
L ≡ total endowment, w = 1 ≡ wage
Like in Krugman (1980), there are IRS in production
l = f + q/ϕ
(3)
Like in Krugman (1980), monopolistic competition implies
p (ϕ) =
1
ρϕ
(4)
CES preferences with monopoly pricing, (2) and (4), imply
r (ϕ) = R (Pρϕ)σ−1
(5)
These two assumptions, (3) and (4), further imply
π (ϕ) ≡ r (ϕ) − l (ϕ) =
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
r (ϕ)
−f
σ
Winter 2016 (lecture 10)
22 / 49
Melitz (2003)
Production
Comments:
1
Higher productivity ϕ in the model implies higher measured
productivity (which is important, as measuring productivity is hard,
especially in differentiated product industries like this one).
1
f
r (ϕ)
=
1−
l (ϕ)
ρ
l (ϕ)
2
More productive firms produce more and earn higher revenues
q (ϕ1 )
=
q (ϕ2 )
3
ϕ1
ϕ2
σ
r (ϕ1 )
and
=
r (ϕ2 )
ϕ1
ϕ2
σ−1
ϕ can also be interpreted in terms of quality. This is isomorphic to a
change in units of account, which would affect prices, but nothing else
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
23 / 49
Aside: decentralized equilibrium is efficient
To see this, note that (following Spence (1976) and Dhingra and
Morrow (2013)) the decentralized equilibrium solves:
Z n
pi (qi ) qi di
max
qi ,n 0
Z n
qi
subject to:
nf +
di ≤ L.
0 ϕ
A social planner would solve:
Z n
σ−1
max
(qi ) σ di
qi ,n 0
Z n
qi
subject to:
nf +
di ≤ L.
0 ϕ
1− σ1
Under CES, pi (qi ) qi ∝ qi
⇒ two solutions coincide
This is unique to CES (in general, entry is distorted)
So inherits many properties (essentially, all aggregate properties) of
perfectly competitive models, conditional on L.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
24 / 49
Melitz (2003)
Aggregation
By definition, the CES price index is given by
Z
1−σ
P=
p (ω)
1
1−σ
dω
ω∈Ω
Since all firms with productivity ϕ charge the same price p (ϕ), we
can rearrange CES price index as
+∞
Z
P=
p (ϕ)
1−σ
1
1−σ
Mµ (ϕ) dϕ
0
where:
M ≡ mass of (surviving) firms in equilibrium
µ (ϕ) ≡ (conditional) pdf of firm-productivity levels in equilibrium
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
25 / 49
Melitz (2003)
Aggregation
Combining the previous expression with monopoly pricing (4), we get
1
P = M 1−σ /ρϕ
e
where
Z
ϕ
e≡
+∞
σ−1
ϕ
1
σ−1
µ (ϕ) dϕ
0
One can do the same for all aggregate variables
σ
R = Mr (ϕ)
e , Π = Mπ (ϕ)
e , Q = M σ−1 q (ϕ)
e
Comments:
1
2
These are the same aggregate variables we would get in a Krugman
(1980) model with a mass M of identical firms with productivity ϕ
e
But productivity ϕ
e now is an endogenous variable which may respond
to changes in trade cost, leading to aggregate productivity changes
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Melitz (2003)
Entry and exit
In order to determine how µ (ϕ) and ϕ
e get determined in equilibrium,
one needs to specify the entry and exit of firms.
Timing is similar to Hopenhayn (1992):
1
2
3
4
There is a large pool of identical potential entrants deciding whether to
become active or not.
Firms deciding to become active pay a fixed cost of entry fe > 0 and
get a productivity draw ϕ from a cdf G .
After observing their productivity draws, firms decide whether to
remain active or not.
Firms deciding to remain active exit with a constant probability δ.
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
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Melitz (2003)
Aside: Pareto distributions
In variations and extensions of Melitz (2003), most common
assumption on the productivity distribution G is Pareto:
θ
ϕ
for ϕ ≥ ϕ
G (ϕ) ≡ 1 −
ϕ
g (ϕ) ≡ θϕθ ϕ−θ−1 for ϕ ≥ ϕ
Pareto distributions have two advantages:
1
2
Combined with CES, it delivers closed-form solutions.
Distribution of firm sizes remains Pareto, which is not a bad
approximation empirically (at least for the upper tail).
But like CES, Pareto distributions will have very strong implications
for equilibrium properties (more on this in Lecture 14).
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
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Melitz (2003)
Productivity cutoff
In a stationary equilibrium, a firm either exits immediately or produces
and earns the same profits π (ϕ) in each period.
Expected value of a firm with productivity ϕ is then
( +∞
)
X
π (ϕ)
t
v (ϕ) = max 0,
(1 − δ) π (ϕ) = max 0,
.
δ
t=0
So there n
exists a unique productivity
level
o
π(ϕ)
∗
ϕ ≡ inf ϕ ≥ 0 : δ > 0 .
Productivity cutoff ϕ∗ can also be written as:
π (ϕ∗ ) = 0
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
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Melitz (2003)
Aggregate productivity
Once we know ϕ∗ , we can compute the pdf of firm-productivity levels
(
g (ϕ)
if ϕ ≥ ϕ∗
1−G (ϕ∗ )
µ (ϕ) =
0
if ϕ < ϕ∗
Accordingly, the measure of aggregate productivity is given by
1
ϕ
e (ϕ ) =
1 − G (ϕ∗ )
∗
Stanford Econ 266 (Dave Donaldson)
Z
+∞
ϕ
σ−1
1
σ−1
g (ϕ) dϕ
ϕ∗
Firm-Level Trade (Theory I)
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Melitz (2003)
Free entry condition
Let π ≡ Π/M denote average profits per period for surviving firms.
Free entry requires the total expected value of profits to be equal to
the fixed cost of entry:
0 × G (ϕ∗ ) +
π
× [1 − G (ϕ∗ )] = fe
δ
Free Entry Condition (FE):
π=
δfe
1 − G (ϕ∗ )
(6)
Holding constant the fixed costs of entry, if firms are less likely to
survive, they need to be compensated by higher average profits.
Stanford Econ 266 (Dave Donaldson)
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Melitz (2003)
Zero cutoff profit condition
Definition of ϕ∗ can be rearranged to obtain a second relationship
between ϕ∗ and π.
By definition of π, we know that
∗
π = Π/M = π [ϕ
e (ϕ )] ⇔ π = f
r [ϕ
e (ϕ∗ )]
−1
σf
By definition of ϕ∗ , we know that
π (ϕ∗ ) = 0 ⇔ r (ϕ∗ ) = σf
Two previous expressions imply ZCP condition:
"
#
r [ϕ
e (ϕ∗ )]
ϕ
e (ϕ∗ ) σ−1
π=f
−1 =f
−1
r (ϕ∗ )
ϕ∗
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(7)
32 / 49
Melitz (2003)
(2003)
Closed economy
economy equilibrium
equilibrium (assuming ZCP is downward-sloping; see next slide though)
Closed
Stanford Econ
266 (Week
(Dave 8)
Donaldson)
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(Theory I)
Melitz
(2003)
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Fall10)
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Melitz (2003)
Aside: the shape of the ZCP schedule
FE and ZCP, (6) and (7), determine a unique (π, ϕ∗ ), and therefore
ϕ,
e independently of country size L.
The only variable left to compute is M, which can be done using free
entry and labor market clearing as in Krugman (1980).
However, ZCP is not necessarily downward sloping:
It depends on whether ϕ
e or ϕ∗ increases relatively faster
ZCP is downward-sloping for most common distributions
In the Pareto case, it is easy to check that ϕ/ϕ
e ∗ is constant:
So ZCP is flat and average profits are independent of ϕ∗
Stanford Econ 266 (Dave Donaldson)
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Melitz (2003)
Number of varieties and welfare
Free entry and labor market clearing imply:
L = R = rM
We can rearrange the previous expression as:
M=
L
L
=
r
σ (π + f )
Like in Krugman (1980), welfare of a representative worker is given by:
1
U = 1/P = M σ−1 ρϕ
e
Since ϕ
e and π are independent of L, growth in country size and
costless trade will also have the same impact as in Krugman (1980):
welfare % because of % in total number of varieties in each country
Stanford Econ 266 (Dave Donaldson)
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Melitz (2003)
Open economy model
In the absence of trade costs, we have seen trade integration does not
lead to any intra-industry reallocation (ϕ
e is fixed)
In order to move away from such (counterfactual) predictions, Melitz
(2003) introduces two types of trade costs:
1
2
Iceberg trade costs: in order to sell 1 unit abroad, firms need to ship
τ ≥ 1 units
Fixed exporting costs: in order to export abroad, firms must incur an
additional fixed cost fex (information, distribution, or regulation costs)
after learning their productivity ϕ
In addition, Melitz (2003) assumes that c = 1, ..., n countries are
symmetric so that wc = 1 in all countries
Stanford Econ 266 (Dave Donaldson)
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Melitz (2003)
Production
Monopoly pricing now implies
pd (ϕ) =
τ
1
, px (ϕ) =
ρϕ
ρϕ
Revenues in the domestic and export markets are
rd (ϕ) = Rd [Pd ρϕ]σ−1 , rx (ϕ) = τ 1−σ Rx [Px ρϕ]σ−1
Note that by symmetry, we must have
Pd = Px = P and Rd = Rx = R
Let fx ≡ δfex . Profits in the domestic and export markets are
πd (ϕ) =
Stanford Econ 266 (Dave Donaldson)
rd (ϕ)
rx (ϕ)
− f , πx (ϕ) =
− fx
σ
σ
Firm-Level Trade (Theory I)
Winter 2016 (lecture 10)
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Melitz (2003)
Productivity cutoffs
Expected value of a firm with productivity ϕ is
( +∞
)
X
π (ϕ)
t
v (ϕ) = max 0,
(1 − δ) π (ϕ) = max 0,
δ
t=0
But total profits of are now given by
π (ϕ) = πd (ϕ) + max {0, πx (ϕ)}
n
o
π(ϕ)
∗
Like in the closed economy, we let ϕ ≡ inf ϕ ≥ 0 : δ > 0
n
o
In addition, we let ϕ∗x ≡ inf ϕ ≥ ϕ∗ : πx δ(ϕ) > 0 be the export cutoff
In order to have both exporters and non-exporters in equilibrium,
ϕ∗x > ϕ∗ , we assume that:
τ σ−1 fx > f
Stanford Econ 266 (Dave Donaldson)
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ction into exports
Melitz (2003)
Selection into exporting
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Melitz (2003)
Are exporters more productive than non-exporters?
In the model, more productive firms (higher ϕ) select into exports
Empirically, this directly implies larger firms (higher r (ϕ))
Question: Does that also mean that firms with higher measured
productivity select into exports?
Answer: Yes. For this to be true, we need
rd (ϕ) + nrx (ϕ)
rd (ϕ)
>
,
ld (ϕ) + nlx (ϕ)
ld (ϕ)
which always holds if τ σ−1 fx > f
Comment: Like in the closed economy, this crucially relies on the
fact that fixed labor costs enter the denominator
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
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Melitz (2003)
Aggregation
In the open economy, aggregate productivity is now given by
ϕ
et =
1
i σ−1
1 h
σ−1
σ−1
Mϕ
e
+ nMx (ϕ
ex /τ )
Mt
where:
Mt ≡ M + nMx is the total number of varieties
1
h
i σ−1
R +∞
ϕ
e = 1−G1(ϕ∗ ) ϕ∗ ϕσ−1 g (ϕ) dϕ
is the average productivity
acrosshall firms
1
i σ−1
R +∞
ϕ
ex = 1−G1(ϕ∗ ) ϕ∗ ϕσ−1 g (ϕ) dϕ
is the average productivity
x
x
across all exporters
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
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Melitz (2003)
Aggregation
Once we know ϕ
et , we can still compute all aggregate variables as:
1
P = Mt1−σ /ρϕ
et ,
R = Mt r (ϕ
et ) ,
Π = Mt π (ϕ
et ) ,
σ
Q = Mtσ−1 q (ϕ
et )
Comment:
Like in the closed economy, there is a tight connection between welfare
(1/P) and average productivity (ϕ
et )
But in the open economy, this connection heavily relies on symmetry:
welfare depends on the productivity of foreign, not domestic exporters
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
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Melitz (2003)
Free entry condition
The condition for free entry is unchanged
Free Entry Condition (FE):
π=
δfe
1 − G (ϕ∗ )
(8)
The only difference is that average profits now depend on export
profits as well
π = πd (ϕ)
e + npx πx (ϕ
ex )
where:
px =
1−G (ϕ∗
x)
1−G (ϕ∗ )
Stanford Econ 266 (Dave Donaldson)
is probability of exporting conditional on successful entry
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Melitz (2003)
Zero cutoff profit condition
By definition of the cut off productivity levels, we know that
πd (ϕ∗ ) = 0 ⇔ rd (ϕ∗ ) = σf
πx (ϕ∗x ) = 0 ⇔ rx (ϕ∗x ) = σfx
This implies
rx (ϕ∗x )
fx
=
⇔ ϕ∗x = ϕ∗ τ
∗
rd (ϕ )
f
1
fx σ−1
f
By rearranging π as a function of ϕ∗ , we get new ZCP condition:
"
#
"
#
ϕ
ex (ϕ∗ ) σ−1
ϕ
e (ϕ∗ ) σ−1
− 1 + npx fx
−1
π=f
ϕ∗
ϕ∗x (ϕ∗ )
Stanford Econ 266 (Dave Donaldson)
Firm-Level Trade (Theory I)
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mpact of Trade
Melitz (2003)
The Impact of Trade
Stanford Econ 266 (Dave Donaldson)
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Melitz (2003)
The Impact of Trade
In line with empirical evidence (see Lectures 12-13), exposure to trade
forces the least productive firms to exit: ϕ∗ > ϕ∗a
Intuition:
For exporters: Profits % due to export opportunities, but & due to the
entry of foreign firms in the domestic market (P &)
For non-exporters: only the negative second effect is active
Comments:
The % in ϕ∗ is not really a “new source” (depending on your
definition of that phrase) of gains from trade. It’s because there are
gains from trade (P &) that ϕ∗ %increases
Welfare unambiguously %, even though number of domestic varieties
&
R
L
M=
=
< Ma
r
σ (π + f + px nfx )
Stanford Econ 266 (Dave Donaldson)
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Melitz(2003)
(2003)
Melitz
The Impact of Trade
The Impact of Trade
14.581 (Week 8)
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Melitz (2003)
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47 / 49
he Impact of Trade
Melitz (2003)
The Impact of Trade
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Melitz (2003)
Other comparative static exercises
Starting from autarky and moving to trade is theoretically standard,
but not empirically appealing
Melitz (2003) also considers:
1
2
3
Increase in the number of trading partners n
Decrease in iceberg trade costs τ
Decrease in fixed exporting costs fx
Same qualitative insights in all scenarios:
Exit of least efficient firms
Reallocation of market shares from less from more productive firms
Welfare gains
Melitz and Redding (2013, Handbook of International Economics)
offers an excellent overview of various extensions of Melitz (2003)
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49 / 49