HIAS-E-123

Price Discrimination in the Transport
Industry and the Gains from Trade
Han Zheng (a)(a) Hitotsubashi Institute for Advanced Study, Hitotsubashi University

October 2022

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

All rights reserved.

Price Discrimination in the Transport Industry and
the Gains from Trade
Han Zheng*
October 12, 2022

Abstract
Shipping companies often charges nonlinear and discriminatory pricing for transportation. This paper shows that this nonlinear and discriminatory pricing inthe shipping industry could hamper the welfare gains from trade due to withinindustry allocation across heterogeneous firms. I extend a standard heterogeneous
firm trade model with variable markups by incorporating monopolistically competitive shipping companies that charge nonlinear and discriminatory pricing against
manufacturers. In a standard setting, shipping companies optimally charge a higher
transport price to the more productive firms, weakening within-industry reallocation toward productive firms. Elimination of this discriminatory practice could
potentially increase the gains from trade.
Keywords: Price discrimination, Shipping industry, Heterogeneous firms, The
gains from trade
JEL Codes: F12, L91, R13, R41* Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo

186-8601, Japan. Email address: hanzheng211@gmail.com.

1

Introduction

Very often trade literature has regarded freight rates and tariff rates as independent andholistic components of the iceberg trade cost. Though, because of the prevalence of
non-discriminative ad valorem tariff, it is generally appropriate to envision tariff rates
as a multiplicative price shifter, as it is indeed a percentage charge non-discriminatively
on well-defined tariff lines, the freight rate as an uniform multiplicative price shifter
actually and crucially depends on the assumption that the transport industry is perfectlycompetitive, which is at odds with many preceding studies showing that considerable
market power resides in the transport industry, (see, e.g. Borenstein and Rose (1994),
and Gerardi and Shapiro (2009)).
In fact, very often nonlinear pricing schedule is observed daily in life. Using pricequote data from Sagawa Express Co.,Ltd, Figure 1 shows the per size price variation forthe maximum 50kg category.1 Conditional on the weight, the per size price is clearly
increasing in the total size of the shipment regardless how far the destinations are, meaning the more bigger shipped, the higher shipping rate is. A more rigorous regression isconducted to reveal how shipments’ characteristics correlate with the variation in price,which is summarized in Table 1 in the appendix. In a nutshell, it shows that shipping
price increases disproportionately faster as the size of the shipment increases, hintingthat shipping companies tend to charge higher per unit price the bigger the shipment is,which will be the empirical regularity this theoretical framework strives to align.
Despite this discussion about market power in the shipping industry in industrial organizations literature and evidence in real life, it is only very recently that trade economists
started to document and analyze its implications in global trade flow (see, e.g. Hummels
1
Data comes from the official website of Sagawa Express Co.,Ltd: https://www.sagawa-exp.co.
jp/english/price/regular04.html#ft01, which lists rates for any 13 bilateral region blocks in Japan.
Sagawa Express Co.,Ltd is one of the largest logistic companies in Japan. Its businesses encompass
both corporate shipping and individual consumer shipping. And this price list used in this paper isfor both corporate and individual customers. There might be other ways for big corporate customers
using shipping service provided by Sagawa, such as signing long-term contracts. Nevertheless, given that85.62% of enterprises in Japan have employees less than 10 according to 2016 Economic Census for Business Activity(https://www.stat.go.jp/english/data/e-census/2012/index.html#e2016), chances
are high that the price quote this paper is using is the most standard and popular way of shipping inside
Japan, especially among small and medium enterprises.

1Figure 1: Convexity of Price Schedule across Different Distances in Japan

Note: The price displayed here is for shipments starting from Kanto area to other areas. The priceper size is calculated only for the maximum 50kg category, as typically the price varies by both size
and weight. Size refers to the summation of the length of its three dimensions of an object and it is incentimeter(cm) unit.

(2007), Hummels et al. (2009), Kleinert and Spies (2011), Ishikawa and Tarui (2018),
Brancaccio et al. (2020), Ignatenko (2020), Asturias (2020), and Ardelean and Lugovskyy (2020)). However, none of the above discusses the market power phenomenon in
the transport industry within the heterogeneous firm framework which accumulated empirical evidence has proven to be true and has become a starting point in international
trade. This paper strives to fill the gap and shows that this nonlinear and discriminatory pricing in the shipping industry could hamper the welfare gains from trade due towithin-industry allocation across heterogeneous firms.
I extend a standard heterogeneous firm trade model with variable markups by incorporating monopolistically competitive shipping companies that charge nonlinear anddiscriminatory pricing against manufactures. In a standard setting, shipping companies optimally charge a higher transport price to the more productive firms, weakening
within-industry reallocation toward productive firms.2 Intuitively, the more productive
2This result echos what Yoshida (2000) finds in a vertically-linked industry structure, where the2manufacturing firms’ innate productivity advantage is sabotaged by the higher transport
prices charged to them. As a result, those productive firms would not be capable of expanding as well as they otherwise would do under uniform transport fees, leaving enoughspace for the less productive firms to survive. Therefore, the within-industry reallocation
toward productive firms conducive to higher gains from trade is dampened. Eliminationof this discriminatory practice could potentially increase the gains from trade.
This paper relates to multiple strands of literature. First, it connects with studies
discussing the welfare implications of trade. For a large class of mostly used trade models, Arkolakis et al. (2012) (thereafter ACR) provide an unified framework claiming that
despite a wide range of settings in the trade literature, two empirical statistics are sufficient for welfare implications: (i) the share of expenditure on domestic goods; and (ii)the trade elasticity, implying that in terms of the total size of gains from trade, new trademodels yield the same result as the old one. However, Melitz and Redding (2015) argue
that the ACR’s results are sensitive to firm productivity distribution in the sense thateven an truncated version of Pareto distribution would introduce additional elements into
the ACR formula and the routine application of an intrinsically changing partial tradeelasticity, with respect to the variable trade cost to the ACR formula, would undervalue
the gains from trade, especially under the range of values where the variable trade cost is
high. Similar results are also uncovered by Head and Mayer (2014)—a departure of firm
heterogeneity from Pareto distribution fitting the upper tail of firm sales to log-normal
distribution which fits the complete distribution of firm sales, naturally gives rise to variable trade elasticity with respect to trade cost. Under a translog preference, Novy (2013)
also finds that the trade elasticity is a variable. Therefore, Melitz and Redding (2015),
together with the others, discover the varying trade elasticity without which the canonicaltrade literature might underestimate the gains from trade. Considering the multi-productfirms setting, Bernard et al. (2011) suggest that the within firm reallocation could havecomparable impact on the gains from trade as the across firm reallocation does. Additionally, Caliendo and Parro (2015) derive analytical expressions for gains from trade in
upstream firms will charge a higher unit price to the more productive downstream firms.

3

a world with mutisectors and vertical linkages, showing that the gains from trade would
be underestimated without. Based on the roundabout model developed by Caliendo andParro (2015), Antràs and Gortari (2020) show that the evaluation of the gains from tradediffers even doubles for certain countries, when the property of sequential production
stages in the value chain is incorporated. In line with the aforementioned literature, thispaper strives to discover another channel hampering the gains from trade or its quantification, which the trade literature might overlook. This paper offers a possible channel,
hampering the endogenous selection of firms into both domestic and export markets,
which is touted by Melitz and Redding (2015) as the extra margin of adjustment leading
to higher gains from trade, apart from the ones in homogeneous firms model by Krugman(1980).
This paper also contributes to literature documenting and analyzing market powers
in the transport industry. Both Borenstein and Rose (1994) and Gerardi and Shapiro(2009) use a reduced-form approach to show there are considerable market powers in theairline industry. Several other studies, such as Busse and Keohane (2007), Hughes (2011),
MacDonald (2013), and Hughes and Lange (2020) particularly document the market
power phenomenon in domestic railroad shipping and estimate its impact on the power
generating industry. Despite its vastness, none of the aforementioned studies investigates
the impact of market power in the transport industry on international trade. One notable
exception is Hummels et al. (2009) who estimate how the freight rate of maritime shipping
responds to the goods price, tariff rate, and demand elasticity. However, the homogeneousfirms setting in Hummels et al. (2009) precludes some interesting results which will be
uncovered in this paper with a heterogeneous firm setting.
The next section specifies the primitives and environment needed to set up the model,
followed by a list of the results inherited from Melitz and Ottaviano (2008) as the closed
economy benchmark. Section 3 discusses the properties of an open economy when adiscriminatory pricing scheme in the freight rate is implemented. Section 4 presents theresults when an uniform pricing scheme is applied, reflecting the conventional assumption that heterogeneous manufacturers face the uniform freight rate in cross-border trade.

4Section 5 compares the results under two different pricing regimes and yield the implication of the market power in the transport industry on the gains from trade. The final
section concludes.

2

Primitives

The configuration follows the pathway of Melitz and Ottaviano (2008) (thereafter MO)
and inherits their notations whenever possible.3 In order to isolate the effect of thetransport industry’s price on the firms’ decisions in the manufacture industry, the pricecharged by the transport industry is abstracted to be the only component of the trade
cost τ (ω), where the ω indexed trade cost hints that the transport industry is capable ofcharging differentiated prices. The game played by both the transport industry and theMelitz industry is delineated as follows:
In stage (i), the productivity of firms is realized and observed by the transport industry.4
In stage (ii), the transport industry charges a price deemed as an iceberg type τ (ω) toeach firm willing to export.
In stage (iii), taking the price charged as the trade cost, correspondingly firms produce
and export.
The subsection 2.1 to subsection 2.3 enumerate and remind the pivotal structure
and results in MO model upon which the later analysis will be built. Meanwhile thosesubsections introduce the notation convention used throughout this paper. The last
subsection describes the structure imposed on the transport industry. To streamline thelogic flows and at the same time keep the analysis as concise as possible, only important
expressions and equations will be kept in the main text, the rest will be delegated to thetheory appendix.
3

The CES type demand function will preclude the possibility of a heterogeneous transport cost becauseof its property of constant mark-up pricing. Despite its popularity in the quantification analysis, the
CES demand system falls short to capture the variable mark-up aspect of the reality which has been
documented by numerous industrial organization literatures.
4
Though in reality shipping firms typically cannot observe the exact productivity of their customers,
they could classify their customers into several productivity categories based on past history, because of
their local monopolistic power. This model provided here approximates this situation.

5

2.1

Preferences

The representative consumer in each country has the utility type developed by Ottaviano
et al. (2002):

U = q0c + α

Z
i∈Ω

1
qic di − γ2Z1(qic )2 di − η

i∈Ω

2

Z

2

qic di.

i∈ΩThen the inverse demand for each variety i is given as:pi = α − γqic − ηQc ,(1)

indicating the maximum price for any variety to have a non-negative demand is:pD := α − ηQc .

2.2

Production

Following the terminology in MO, cD is used to denote the cost cut-off which is themarginal cost of the manufacturer with zero demand. The profit maximizing behaviorof manufacturers implies the manufacturers in a country with population size L will
necessarily satisfy:q(cc ) =L(p(cc ) − cc ),γ(2)meaning the least productive manufacturers active in the market with zero demand will
effectively set its price equal to its marginal cost: pD = cD .
It is worthwhile noting that cc represents the effective marginal cost when consumers
receive the goods. Under the scenario of a closed economy and costless trade where aninternational shipping service is not used, cc = c the cost facing the consumer equals its
cost of production. In the case of costly trading discussed in the paper, cc = τ (c)c to
account for the iceberg type transport cost.
The specification of production technology also follows MO and the distribution of c6

is repeated here as:G(c) =

2.3

c
cM

k
,

c ∈ [0, cM ].

(3)

Closed Economy

The closed economy as the benchmark in this paper is the same as the one in the original
MO model. Some preponderant results in the MO model will be listed below to facilitate
a comparison with analysis later on. Given the parametrization of cost distribution, thefree-entry condition yields the cutoff cost level under the closed economy as MO:2(k + 1)(k + 2)ckM γfE
cD =
L


1/(2+k),(4)

where fE is the cost of entry.
The indirect utility function, regardless of the parametrization of cost distribution, isgiven as follows:
γ −1
N
1
η+
(α − p̄)2 + σp2 ,
U =I+
2
N
2γ

(5)

where N is the number of active manufacturers in a particular economy. p̄ is defined asRp̄ := 1/N Ω p(ω) dω, namely the average price in that particular economy. σp2 is definedR
as σp2 := 1/N Ω (p(ω) − p̄)2 dω, which can be intuitively interpreted as the variance of
the available price in that particular economy. All the general equilibrium objects N , p̄,
σp2 can be solved as functions of cD . The indirect utility function can then be transformedas:(α − cD ) α − k+1
c
D
k+2
U =1+
.
2η

(6)

Notice that the utility level is decreasing at cD , confirming the intuition that the lower
cD , meaning higher market-wise average productivity is conducive to a higher welfare
level. MO show that when countries are open to trade, the only sufficient statistics cD
becomes lower, implying a higher level of average productivity, which is also touted as
the new margin of gains from trade by Melitz (2003).

7

2.4

Trade Cost and the Transport Industry

To align the model structure as close to reality as possible, a Krugman type monopolisticcompetition industry structure is employed in the transport industry where the transport
firms’ profits are rebated to consumers in terms of wages. Additionally, to highlight theconsequence of the market power in the transport industry, the matching process between transport firms and manufacturers is simplified as random matching, thereby the
distribution of manufacturers that each transport company is facing is the same. The
market power enables transport firms to charge discriminatory prices to any firms in the
manufacture industry. Those firms do not have any bargaining power when negotiating with the monopoly and simply take those prices as given, which, to some extent,
can be justified by the commonly accepted infinitesimal firm setting in the monopolisticcompetition model.

3

Discriminatory Pricing

The analysis now moves from enumerating the results in MO as the closed economy
benchmark in this paper to an open economy where exporters have to use shipping
services offered by the monopolistic competitive shipping industries.

3.1

Discriminatory Pricing Schedule

Under discriminatory pricing, to be consistent with the iceberg trade cost formulation,
the profit maximization problem facing a representative shipping company is formulated
as:5
Z cX

c

(τ (·)cq(c ) − cq(c )) Ns d

max Π(τ (·)) =
τ (·)

c

0



G(c)
G(cX )

,(7)

where Ns is the number of customers of this shipping company and cX is the unit cost
export cutoff. From the perspective of manufacturers, τ (·)cq(cc ) is the total variable cost
incurred from exporting and cq(cc ) is the total variable production cost from exporting.
5

To obviate from unnecessary complications and yield stark contrast to the MO model, the unit costof shipping is assumed to be zero. Because of the monopolistic competitive nature of shipping industry,each individual shipping firm takes cX and Ns as macro variables, which it takes as given.

8

The difference between these two cost terms yields the payment as well as the profits
received by the shipping company, given the assumption that unit cost of shipping iszero. Summing across the profits received from different manufacturers indexed by theirunit cost of production c with its corresponding measures Ns d (G(c)/G(cX )) yields thetotal profits received by a typical shipping company. Using the technique in calculus ofvariation, the profit maximizing function τ must satisfy the following condition, which,in light of (7), is given by:
ck kNs L (c + cD − 2cτ (c))
= 0,
2γckX
which implies the following pricing rule:τ (c) =

cD 1
+ .
2c 2

(8)

Notice that cD ≥ c for all exporters guarantees that τ (c) ≥ 1 and τ (c) is a decreasing
function in c, implying that the more productive manufacturers will be charged withhigher transportation fees. τ (c) reaches its lower bound value, when c = cX .
Lemma 1 The discriminatory pricing schedule exercised by shipping firms is given as:τ (c) =

cD 1
+ ,
2c 2

implying that the more productive manufacturers will be charged higher freight rates andthis pricing schedule is independent of k.
The above lemma is somewhat surprising in the way that the discriminatory pricing
schedule is independent from the dispersion of the distribution of manufacturers. It seems
that knowing the distribution of its customers does not procure additional gains to theshipping firms. In fact, this essentially comes from the independence of decision making
facing each heterogeneous manufacturers—the freight rate charged on manufacturers withproductivity c1 will not affect the profitability of manufacturers with productivity c2 , aproperty intrinsic to an infinitesimal manufacturers assumption. Though the calculus9of variation allows finding an optimal path with the interdependence across individuals,
under the current setting, each manufacturer is independent from each other. To atypical shipping firm, the one optimal path finding problem is equivalent to the infinite
static optimization problems facing each manufacturer with different productivity levels
which then are added up together. In fact, one can check that the solution to the profits
maximization problem is even invariant to any specification of the unit cost distributionG(c).
A second noteworthy point regarding the above lemma is that the lower unit cost
manufacturers are coupled with a higher unit freight rate, effectively rendering the cost
cc facing consumers an average of unit cost of production c and unit cost cutoff cD . This
theoretical result mirrors the real life situation discussed in the introduction—the moreproductive manufacturers tend to send more and larger shipments, therefore they are
more likely facing higher unit freight rate imposed by shipping firms with market power.
The higher freight rate offered to the more productive manufacturers effectively alters
the distribution of cc , perceived by the consumers and sabotages the competitive edge ofthe more productive manufacturers, hinting at an alternative welfare evaluation.

3.2

Cutoff Condition and Free Entry Condition

To maintain a parsimonious difference from the original MO model, and illustrate the idea
in a simplified way, from here onward, a symmetric and two-country setting is assumed.6
Given the pricing schedule as in equation (8), for a particular country i, the domestic
firm with productivity ciD will earn zero profits as the foreign firm with productivity
cjX , meaning the domestic productivity cutoff ciD and the foreign export cutoff cjX areconnected in the following way similar to the one in original MO model:

ciD = τ (cjX )cjX .
6(9)

In fact, this symmetry assumption is not a restrictive assumption at all. In the MO model, theprimary asymmetry comes from the exogenous asymmetric iceberg trade cost. However, under thispaper’s setting where iceberg trade cost is abstracted to contain only freight rate, the freight rate is
endogenously determined to be symmetric across countries. And it is irresponsive to other asymmetry
assumed in MO model such as the population size.

10

The symmetric country assumption and the freight rate pricing schedule in equation (8)transforms the above equation into the following relationship:cD = cX .

(10)Equation (10) indicates that all the manufacturers who are active domestically will export,
meaning the freight rate charged to the highest unit cost manufacturers is effectively
zero. Therefore, according to equation (8) all manufacturers active in domestic market
will export. Under the current setting where the freight rate is the only component of
the iceberg trade cost and zero unit cost of shipping, there is no endogenous selection
into the export market.7 A natural result follows in equation (10)—the mass of active
manufacturers in any country contains the operating manufacturers of all the origins:

NT = 2N,(11)

where NT is the total mass of manufacturers active in a particular country and N is themass of active manufacturers from one particular country.
The free entry condition requires that the expected profit is zero:
L
4γ

Z cD
Z
L cX
2
(cD − c) dG(c) +(cX − τ (c)c)2 dG(c) = fE
4γ
0
0

(12)

which together with equation (10) determine the cost cut-off cD given as:
8(k + 1)(k + 2)γ(cM )k fE
cD =
5L


1/(k+2).

(13)

It is interesting to compare the cutoff level above with the one in equation (4) undera closed economy, because under this particular scenario the endogenous selection into
the export market is shut. A simple comparison reveals that the domestic unit cost cutoff level is lower, indicating potential gains from higher average productivity. Though7

Relaxing any of the assumptions would not change the direction of pricing schedule, which complicates the comparison without providing additional insights, because both discriminatory pricing anduniform pricing are equally affected by those settings.

11this result might look counter intuitive at first sight, it is very natural because endogenous selection into the domestic market is still operative—the relatively productive firms,
though handicapped by the increasing freight rate, gain from accessing the foreign market
thereby crowd out the least productive firms in the domestic market.

3.3

Welfare under Discriminatory Pricing

As in the original MO model, the domestic unit cost cutoff is the only sufficient statistic
for the welfare, meaning all the other macro variables including NT , p̄, and σp2 can be
expressed as functions cD whose derivations are relegated to the appendix.
Because the welfare formula in equation (5) is general to both a closed and an open
economy, to whatever the unit cost distribution specification is, the welfare expression
under the discriminatory pricing is given as follows:

Ud = 1 +α − cdD(6k + 12) α − (6k + 7) cdD,
12 (k + 2) η

(14)

where cdD denotes the domestic unit cost cutoff given as in equation (13). Similar to theclosed economy benchmark, the domestic unit cost cutoff is the single sufficient statistic
to the welfare level. The property that welfare is decreasing in cdD remains, though theformula itself is very different from the one in the closed economy benchmark.

4

Uniform Pricing

The basic setup is the same as the one under discriminatory pricing, except that the
only difference is the shipping firms in the transport industry are confined to imposing
an uniform price on the manufacturers, reflecting the conventional uniform freight rate
assumption in trade literature.

12

4.1

Uniform Pricing Schedule

The profit maximization problem of a typical shipping firm, under this scheme, is formulated as
Z cX(τ − 1) cq(c )Ns d

max Π(τ ) =
τ

c

0



G(c)
G(cX )

,(15)

where τ is no longer a function of c. The first order condition of the above yields thefollowing:
Z cX kc kN L (c + cD − 2cτ )dc = 02γckX
012k + 3=1+.
τ=
2k + 22k + 2(16)

The optimal freight rate is a constant greater than one and it is decreasing in the valueof structural paramter k.
Lemma 2 Under the uniform pricing scheme, the optimal freight rate pricing scheduleis given as:
τ =1+

1
,
2k + 2

which is decreasing in the unit cost dispersion parameter k.
Unlike the pricing schedule in discriminatory pricing, the freight rate is a decreasingin the unit cost dispersion parameter k, because now the freight rate is applied to every
exporters, rendering the freight rate decision making interdependent across individuals.
The intuition behind this result is fairly intricate, as it connects to the demand elasticity
those heterogeneous manufacturers are facing. As k decreases, more productive manufacturers take more mass weight out of the total manufacturers population. Meanwhile,
those more productive manufacturers are facing a relatively less elastic manufacturing
goods demand, meaning they have higher potential rent ready to be extracted than thoseless productive manufacturers. Therefore, the overall freight rate tends to be higher whenmore productive manufacturers are taking up higher weight out of the total population.

13

4.2

General Equilibrium under Uniform Pricing

1
is a special case in the open economy discussed in the MO model, theAs τ = 1 + 2k+2

cutoff condition and the free entry condition are basically the same as in the MO model.
The domestic unit cost cutoff is connected with the export unit cost cutoff in the followingway:
cD = τ cX .
The free entry condition is given as:
L
4γ

Z cD
0

L
(cD − c) dG(c) +
4γ
2

Z cX

(τ cX − τ c)2 dG(c) = fE .

0

Together with the symmetric country assumption, the domestic unit cost cutoff can besolved as:2(k + 2)(k + 1)γ(cM )k fE τ k
cD =L(1 + τ k )1/(k+2).

(17)

Because of the exact same reason mentioned in the MO model, the observed price distribution of manufacturers is the same regardless of the origins of those manufacturers,
the general equilibrium objects, including NT , p̄, σp2 , have the same functional form as in
the closed economy benchmark. As a result, the welfare expression is the same as in theclosed economy:


1
k+1 u
u
U =I+
(α − cD ) α −
c
.
2η
k+2 D
u

(18)

where cuD is the domestic unit cost cutoff under uniform pricing, whose value is given inequation (17).

5

Uniform Pricing versus Discriminatory Pricing

Given the portraits of general equilibrium under the two different scenarios, it is straightforward to compare their domestic unit cost cutoff level, which are single sufficient statis-

14

tics of welfare in both situations: d k+2
cD
4(τ k + 1)
=
> 1,
cuD
5τ k

∀k ∈ [1, 20].

(19)

Though k is the dispersion parameter of the unit cost of production, given the quasilinear utility function and the wage rate are taken as the numeraire, this dispersion
parameter is effectively the dispersion parameter of productivity—the number of goods
produced per unit of labor. The range of k from 1 to 20 covers not only the values fromthe trade elasticity estimate literature, (e.g., Eaton and Kortum (2002), Waugh (2010)
and Simonovska and Waugh (2014)) but also the value from the firm size distribution
estimate, (e.g., Gabaix (2016)).
Proposition 1 The domestic unit cost cutoff under discriminatory pricing is higher thanthe one under uniform pricing, for commonly accepted value of dispersion parameter k.
The intuition behind the above proposition is that, as mentioned in previous section,
the nonlinear freight rate under discriminatory pricing sabotages the innate productivity
advantage of the more productive manufacturers, making them unable to expand asmuch as they would do under uniform pricing, leaving enough market shares for the lessproductive manufacturers to survive. As an illustration, at the least productive extreme,
the pricing schedule specified in equation (8) permits 0 freight rate, whereas, at the otherextreme, the freight rate goes to infinity when the innate productivity of manufacturers
goes to infinity.
As mentioned in previous section, the comparison of the domestic unit cost cutoff
under a discriminatory pricing with the counterpart under a closed economy hints that
there is gains from trade from the increase in average productivity. Moreover, one canshow that the mass of available goods varieties, which is assumed to be the mass ofoperating manufacturers, is higher when open to trade under a discriminatory pricing,whose proof is given in the theory appendix.
Corollary 1 The mass of operating manufacturers in a particular country under discriminatory pricing is higher than the mass of operating manufacturers under the closed
15

economy benchmark.
However, what is surprising is that the gains from productivity and the gains fromthe available variety does not guarantee the overall gains from trade under discriminatory
pricing is positive, meaning the welfare level under discriminatory pricing can be lower
than the closed economy benchmark. The welfare comparison of those two situations
above is very technical and depends on the relative magnitude of parameter k, α, ηand γ. The countervailing force reducing welfare actually comes from the market power
distortion of the shipping industry. The discriminatory pricing distorts the innate unitcost distribution and reduces the sales of manufacturers, thereby, creates dead-weightlosses.
The comparison of the open economy under uniform pricing with the closed economy
benchmark is very straightforward, as the uniform pricing situation is just a special caseof the open economy discussed in the MO model. Therefore, all the gains from trade
forces in the MO model apply here. The distortion created by the market power in thetransport industry manifests as an iceberg trade cost type cost wedge—markup pricing
over the shipping firms’ unit cost, but it is not strong enough to offset the gains fromtrade.
To compare the welfare level under two different pricing regimes, a two step approachis taken. First, assume there exist a hypothetical state where the welfare formula takes
the form of uniform pricing but the domestic productivity cutoff takes the value under
discriminatory pricing. We denote this hypothetical welfare level as U dc . Then it isstraight forward to show that:

U dc − U d =(α − cdD )cdD
> 0.
12(k + 2)η

(20)

Then notice that the welfare formula under the uniform pricing is a decreasing function indomestic unit cost cutoff and together with the inequality of unit-cost cutoff in equation(19), it implies
U u > U dc .

16

Then it follows that:U u > U d.

(21)

Proposition 2 The measured gains from trade under discriminatory pricing in the shipping industry is lower than the measured gains from trade under the uniform pricing inthe shipping industry.
As before, there are multiple countervailing forces behind the above proposition. The
average productivity is higher under uniform pricing, whereas the available variety is
ambiguous depending on the relative magnitude of the parameters. The average price ishigher under the discriminatory pricing, meaning the markup tends to be higher under thediscriminatory pricing. The proof of the following corollary is omitted because its logic
is exactly the same as the proof of the above proposition—by assuming a counterfactual
state where the formula of average price takes the form of average price under uniform
pricing but with the domestic productivity cutoff under discriminatory pricing.
Corollary 2 The average price of the operating manufacturers in a particular country
under the discriminatory pricing is higher than the average price of the operating manufacturers under the uniform pricing.
To summarize, in terms of the gains from trade, the productivity and markup favor
uniform pricing, whereas the available variety is ambiguous.
Moreover the above proposition has important implication on the vast trade literature
striving to quantifies the gains from trade. It shows that imposing an uniform freightrate across heterogeneous manufacturers will bias upwards the measured gains from trade,when the shipping firms have discretion to price discriminate. Though the iceberg trade
cost might be appropriate as a hands-on approach to international trade questions, persistent disregard of the shipping service provider as an industry with the market power
might produce misleading quantification results. The shipping industry, as an atom whencompared to the entire economy, could possibly has preponderant ramification, because
every single manufacturing goods about to move across space requires shipping services.

17

6Conclusion

Motivated by a nonlinear pricing example of the transport industry in the real world,
this paper provides a heterogeneous firm model with the transport industry and produces
pricing schedule aligning well with the empirical fact. Furthermore, it analytically showsa new channel where the gains from trade can be mismeasured, which is that if thefreight rate is heterogeneous across manufacturing firms, the welfare will differ from theconventional situation where every manufacturer is assumed to face with the same freightrate. In particular, if the trade cost is higher for the more productive firms, they would
not expand as they would otherwise do under the uniform freight rate assumption, leaving
enough space for the less productive firms to survive, dampening the new margin of thegains from trade—firms selection into production.
One promising avenue to extend this paper is to round up the evidence to test the
hypotheses or results in this paper empirically. Another fruitful way of extension isto quantify this model in numbers to show whether the gains from trade would bias
upward or downwards under the alternative pricing specification and whether the pricediscrimination in the shipping industry affects countries heterogeneously.

18

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21

Appendices
A

Theory Appendix

A.1

Production, Closed Economy Benchmark and Open Economy under Uniform Pricing

The profits maximization under the demand function of (1) gives:p(cc ) =α − ηQc + cc,
2

which can be substituted back into equation (1) for α − ηQc to arrive at equation (2).
After replacing pD with cD , all the performance measures can then be written as functionsof cc and cD only, as shown in MO:
1p(cc ) = (cD + cc ),2L(cD − cc ),q(cc ) =2γ
Lπ(cc ) =(cD − cc )2 .
4γ
Given the expression for profit, the free entry condition implies ex ante the expected
profits equal to entry cost:
L
4γ

Z cD
(cD − c)2 dG(c) = fE ,

(22)

0

where cc = c under the closed economy. This leads to the expression of cD under theclosed economy as in equation (4).
From here onward, the mathematical derivation of general equilibrium objects applies
for both the closed economy benchmark and the open economy under uniform pricing inthis paper because, as mentioned in the main text the situation discussed under uniform
pricing is a special case of MO open economy model. The general equilibrium objects in22the open economy of MO model inherits the same structure of its closed economy version.
Summing equation (1) over the available varieties will leads to an expression connects
aggregate variable Qc and p̄:


η+

γ c
Q = α − p̄.
N(23)

Taking the above equation (23) back to the original utility function and replacing Qc
whenever possible yields equation (5). Similarly, plugging equation (23) back to equation(1) gives the following:pi =1(γα + ηN p̄) − γqic .
ηN + γThe above expression holds for any variety i. Therefore, for the variety with zero demand,
it follows thatcD =1(γα + ηN p̄).
ηN + γ(24)

From the definition of p̄, we know p̄ is a function of cD and N :p̄ =2k + 1cD
2k + 2

Therefore, equation (24) can be used to solve N as a function of cD :N=

2(k + 1)γ α − cD
.
η
cD

(25)

Then it follows that σp can also be expressed as a function of cD by its definition. Finallyreplacing N , p̄, and σp as functions of cD , the equation (5) will be transformed to equation
(6).

A.2

Equilibrium Objects under Discriminatory Pricing

Similar to the closed economy, p(cD ) = cD is also the zero demand price threshold:cD =1ηNT + γ(γα + ηNT p̄),

23

(26)

where following the definition of p̄, it can be expressed as:
1
p̄ =
NT

Z cD


Z cX


c
N p(c) dG(c) /G(cD ) +N p(c ) dG(c) /G(cX ) .

0

0

Together with equation (11) and the unit cost distribution specification, the expressionof p̄ can be reduced to the following:8p̄ =(8k + 5)cD.
8(k + 1)

(27)

Substituting equation (27) back into equation (26) yields:

NT =8(k + 1)(α − cD )γ.
3cD η

Because σp2 is defined similarly to p̄, in the same vein, σp2 can be expressed as:

Z cX

1
1
2
2
c
(p(c) − p̄) dG(c) /G(cD ) +(p(c ) − p̄) dG(c) /G(cX )2200c2D (2 + 11k)=.
64(1 + k)2 (2 + k)

σp2 =

A.3

Z cD

Proof of Corollary 1

Assuming there exist a hypothetical state where the formula of active mass of firm takes
the form of the closed economy but the domestic productivity cutoff takes the value under
discriminatory pricing, which is denoted as NTdc and given as:
NTdc =

2(k + 1)γ α − cdD
.
η
cdD

8

The logic of deriving p̄ in the open economy in MO model no longer applies here, because, as
mentioned in subsection 3.2, the price distribution perceived by consumers is altered by the nonlinear
discriminatory pricing schedule and the price distributions of manufacturers of different origin are now
different.

24

It is straightforward to show that the mass of active firms under discriminatory pricing
is higher than its counterpart in the hypothetical state:

NTd − NTdc =2(k + 1)(α − cdD )γ> 0.
3cdD η

Notice that NTdc is a decreasing function in its argument cdD and the domestic productivity
cutoff under discriminatory pricing is lower than its counterpart in the closed economy,
namely cdD < cD . Therefore, NTdc > N and then we arrive at NTd > N .

B

Empirics Appendix

B.1

Regression of Shipping Prices on Shipments’ Characteristics

The Table 1 shows how shipping prices vary with the shipments’ characteristics.
In line with the intuition, other than the two factors (size and weight) specified in thepricing menu, the distance between origin and destination is a factor positively correlateswith the price. On the contrary, GRP (Gross Regional Product) in the destination region
does not weigh on the price determination, suggesting no evidence of economy of scale atregional level.9 Though weight exhibits slightly negative correlation with price variation,
the significant and greater than one coefficient of size variable indicates that price tends
to be higher, the more and larger the shipments are.

9

The inclusion of destination’s GRP is inspired by Clark et al. (2004) who use country GDP to
instrument bilateral trade flow to identify the possible economy of scale at country level. Confirming theirprediction, the instrumented bilateral trade flow is indeed significant across their different specifications.

25

Table 1: Regression of Price on Its Factors
Dependent variable:
OLS
Distance
Size
WeightDestination GRP
Constant
Observations
R2
Adjusted R2
Note:

∗

Price
Fixed Effect

(1)

(2)

0.128∗∗∗(0.009)2.385∗∗∗(0.046)−0.401∗∗∗(0.021)
0.012(0.010)−3.681∗∗∗(0.256)0.150∗∗(0.060)2.385∗∗∗(0.046)−0.401∗∗∗(0.020)

495
0.953
0.953

495
0.959
0.955

−3.649∗∗∗(0.334)p<0.1; ∗∗ p<0.05; ∗∗∗ p<0.01

1

Robust standard error is reported. All variables are in log.
Price is in Japanese Yen. Distance is in kilometer(km) unit.
Size refers to the summation of the length of its three dimensions of an object and it is in centimeter(cm) unit. Weight isin kilogram(kg) unit. GRP is in million Yen current price.
226