HIAS-E-112

Nonlinear Pricing in the Transport
Industry and the Gains from Trade
Han Zheng (a), Daisuke Fujii (b)(a) Hitotsubashi Institute for Advanced Study, Hitotsubashi University(b) The University of Tokyo

September, 2021

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
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Nonlinear Pricing in the Transport Industry and theGains from Trade
Han Zheng*

Daisuke Fujii

September 18, 2021

Abstract
Recent empirical research documented that there exists a nonlinear pricing phenomenon in the shipping industry. This paper strives to show how this empiricalregularity would alter conventional results in trade literature. This paper also showsthat when nonlinear pricing in the shipping industry is considered, while the average productivity is higher conducive to the higher welfare level, the gains fromtrade are generally lower than the situation without. In addition, the model built
in this paper offers micro foundations for the additive trade cost and features anendogenous response of shipping charges to the iceberg trade cost, an empiricalfinding emphasized in Hummels et al. (2009). In a much broader sense, this paperargues that the heterogeneous firm model offers a lens through which traditional
results on some interesting objects, for example, gains from trade, could be altered.
Keywords: Nonlinear pricing, Shipping industry, Heterogeneous firms, The gains
from trade
JEL Codes: F12, F14, L25, L91, R41* Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo

186-8601, Japan. Email address: hanzheng211@g.ecc.u-tokyo.ac.jp.
 Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033. Email address: dfujii@e.u-tokyo.ac.jp.

11Introduction

Though trade economists have long acknowledged that the shipping charge is part of the
trade cost, many view the components of trade cost (e.g. tariff rate, shipping charge) as
independent, which is clearly at odds with the empirical findings emphasized by Hummels
et al. (2009) (i.e., the shipping charges endogenously respond to tariff rate1 ). Based on
the heterogeneous firm international trade model, this paper offers a framework that
accommodates an endogenous response of shipping charges to the iceberg trade costat the same time aligning well with the nonlinear pricing phenomenon in the shipping
industry, recently documented by Ignatenko (2020). Moreover, this framework showsthat the nonlinear pricing in the shipping industry could potentially dampen the gains
from trade, though the average productivity is higher after opening to trade.
Among the four components of trade cost, transport attrition, time cost, transaction
cost, and tariff, it is sensible to claim that the technological advances in logistics would
shrink the proportion of transport attrition and time cost out of the total trade cost,
meanwhile constant trade negotiations would limit the size of tariff, leaving the transaction cost as the major component of the barrier to trade flows. This time trend echos
Hummels (2007)’s documentation that in the US the transportation expenditure, initiallyonly half of the tariff duties in 1958, grew to be three times higher than the concurrent
tariff duties paid by 2004. Additionally, using custom data Hummels (2007) shows thatthe transportation cost is at least as important as the tariff in terms of impeding tradeflows. Therefore, naturally more delicate treatment on the shipping industry is requiredon how the industry policy on transportation sector and trade policy would stem theflows of world trade.
However, the majority of prevalent trade literature follows the simplification initiatedby Samuelson (1954) and does not explicitly model the transport industry, where this paper attempts to fill the gap and examine how this would reshape the conventional results.
This paper shows that the nonlinear pricing practice prevailing in shipping industry, an
1

The anatomy of the trade cost is beautifully synthesized by Spulber (2007) as “the four Ts”—
transport attrition, time cost, transaction cost, and tariff.

1

empirical regularity emphasized by Ignatenko (2020), would provide productive firms withmore favorable deals, enabling them to expand more than they would otherwise. Thisempowerment reinforces the productivity advantages of the productive firms and cripples
the survivability of the unproductive firms, thereby raising the entry cutoff of firms, a
crucial statistics to the measurement of the gains from trade. Also, more favorable deals
would translate into a distorted firm size distribution which many researches attempt tomatch. In contrast to the preceding work by Ignatenko (2020) and Ardelean and Lugovskyy (2020), this paper yields closed-form general equilibrium effects which clearly shows
its welfare implication, after showing the same theoretical results using the Melitz-Chaneymodel which also hold in the Melitz and Ottaviano (2008) (thereafter MO) framework.
The importance of the general equilibrium effects manifests in this paper’s results: thepositive welfare effect from a “winner-favoring” essence of nonlinear pricing conducive toa higher average productivity is generally offset by the market power distortion from theshipping industry, which is invisible without general equilibrium settings.
This paper connects with many strands of literature. It contributes to the strandof trade literature striving to assess the gains from trade. Starting from the seminal
and influential paper by Arkolakis et al. (2012) (thereafter ACR), many other papers
discover channels or mechanisms which could potentially modify the gains from trade.
Melitz and Redding (2015) argue that the ACR’s results are sensitive to firm productivity
distribution in a sense that even an truncated version of the Pareto distribution would
introduce additional elements into the ACR formula and the routine applying of an
intrinsically changing partial trade elasticity with respect to the variable trade cost to
the ACR formula would undervalue the gains from trade, especially under the range ofvalues when the variable trade cost is high. Similar results are also uncovered by both
Head et al. (2014) and Bas et al. (2017)—a departure of firm heterogeneity from thePareto distribution fitting the upper tail of firm sales to log-normal distribution whichfits the complete distribution of firm sales, naturally gives rise to variable trade elasticity
with respect to trade cost. Under the translog preference, Novy (2013) also finds the tradeelasticity is a variable. Therefore, without the varying trade elasticity, the canonical trade2

literature might underestimate the gains from trade. Additionally, Caliendo and Parro(2015) derives analytical expressions for the gains from trade in a world with mutisectors
and vertical linkages, showing that the gains from trade would be underestimated withoutthem. In line with the aforementioned literature, this paper discovers a new channel, thenonlinear pricing practice in the transport industry, modifying the assessment of the gains
from trade.
This paper also adds to the line of literature attempting to explain the vast existenceof an additive trade cost documented by Irarrazabal et al. (2015) and investigates how
the additive trade cost would steer the trade flows differently from the conventional multiplicative trade cost. Irarrazabal et al. (2015) show that the elasticity of the quantity
elasticity of price with respect to additive trade cost changes along the additive trade cost,
the property helps in identifying the additive trade cost. They show quantitatively thatthe additive trade cost do change the welfare gains assessment. Coşar and Demir (2018)
and Fan et al. (2020) consider the additive trade cost when firms choose both quantityand quality of their products. Coşar and Demir (2018) finds that containerization explains a significant amount of the global trade increase. Fan et al. (2020) show that theadditive trade cost would affect firms’ quality choice differently from the multiplicativetrade cost, thereby generating two different trade elasticities in the traditional structural
gravity equation. However, all aforementioned papers incorporate the additive trade costwithout micro foundations. This paper together with Ignatenko (2020) lay out the micro
foundations for the additive trade cost and assign economic meanings to its components
in both the Melitz-Ottaviano and Melitz-Chaney frameworks.
This paper also sheds some light on the recent discussion about superstar firms andtheir implications on the labor share. Freund and Pierola (2015) document that the
top several firms account for extraordinarily huge portion of one countries’ export andrevealed a comparative advantage in a sector can be created by a single firm. Autor et al.
(2020) argue that the creative destruction in a heterogeneous firm model contributes tothe sales concentration of top firms, which have high markups and a low labor share ofvalue added. The framework in this paper speaks to the very cause of “export superstars”

3through emphasizing the “winner-favoring” effect of nonlinear pricing in the shipping
industry, which amplifies the productivity advantage of big firms abroad and aggravatesthe lower labor share.
Admittedly, this paper is not the first one to endogenize the transport industry.2
Ishikawa and Tarui (2018) consider an monopolistic international transportation industry
arguing that the freight rate will endogenously respond to tariff changes, thereby, modifing
the desired effects that follow. Brancaccio et al. (2020) build a spatial model in global
shipping and reveal three mechanisms that would be otherwise absent in the exogenous
trade cost models. The work by Kleinert and Spies (2011) and Asturias (2020) point out
that the profit maximizing behavior of the shipping companies enable them to switch
shipping technology between trade routes of different sizes. Notably, Asturias (2020)
assembles an unique data set and its calibration shows that shipping companies tend touse a higher fixed cost but a lower marginal cost technology at routes with increasing
trade flows. Despite all their efforts, none of them pay attention to the firmwise shipping
charges variations and consider the nonlinear pricing as a main force in modeling thetransportation industry’s pricing strategy.
The next section lays out the primitives and environment needed to set up the model.
Section 3 describes the shipping firm’s profit maximizing strategy. Section 4 investigates
the welfare implication of nonlinear pricing in the shipping industry in general equilibrium. Section 5 discusses its implication in firm size distribution and gravity. The lastsection concludes.

2

Primitives and Environment

The specification for consumer utility is the one used in Melitz and Ottaviano (2008) andthe individual representative preference is given as:Z1
q c (ω) dω − γ
Ujc = α
2
ω∈Ωj

Z

1
(q c (ω))2 dω − η
2
ω∈Ωj

2

Z

!2
q c (ω) dω

+ q0.

(1)

ω∈Ωj

A vast strand of literature investigates this phenomenon in urban economics (see, e.g. Behrens et al.
(2009)).

4

where ω is the index of goods in the set Ωj —the set of goods available in country j.
The production technology of the outside good q 0 is linear whose per unit production
requires one unit of labor and it’s being freely traded. The market for the outside goods
is assumed to be perfect competition and we assume consumers have positive demand of
the outside goods. From here on, we treat this outside goods as numeraire.
Then utility maximization of an individual preferences yields the following demand
for a particular good ω:Ljq(ω) =γZ
α − p(ω) − η!
q c (ω) dω ,

(2)

ω∈Ωj

where Lj is the population of the destination country. We define the aggregate variables
R
R
Qcj := ω∈Ωj q c (ω) dω and Pj := ω∈Ωj p(ω) dω. Then the demand becomes:q(ω) =Lj
α − p(ω) − ηQcj .
γ(3)

Summing across goods available in country j, whose mass is Mj , yields the following:Qcj =αMj − Pj
γ + ηMj(4)

which connects the two aggregate variables Qcj and Pj . Furthermore, the quantity produced by the cutoff manufacturer q(φ∗ ) = 0, which will be rationalized by the shipping
companies’ profit maximization problem, implies there exist p∗j such that:p∗j = α − ηQcj .

(5)

The shipping industry bears a resemblance to the production sector specification in
Krugman (1980), where the shipping firms engage in monopolistic competition which
rebates all the profits back to the individuals through their wages.3 It incurs a fixed costfs for the shipping companies to serve a route. The number of shipping firms operating
3

The monopolistic competition assumption in the shipping industry is to some extent innocuous as
Hummels et al. (2009) discover evidence of considerable market power in the shipping industry.

5

a route from country i to country j is denoted as Nij , which will be determined endogenously by the free-entry conditions for the shipping companies. Because the shipping
industry per se is not at the very center of interest in this paper, we simply assume they
are homogeneous and having the same productivity. With the above assumptions, it isreasonable to assume the heterogeneous manufacturers who use shipping services pairwith shipping firms randomly. Therefore, given a shipping firm, the productivity distribution of its potential customers G(φ) (with g(φ) being is p.d.f) is the same across all the
shipping companies which is common information for all the players. The equilibrium isreached in sequence: in the first stage, free entry from both the shipping industry andthe manufacturing industry occur foreseeing their actions in the second stage; then in thesecond stage, both the manufacturers and the shipping firms optimize their quantity andservice given the macro variables.
As in Melitz and Ottaviano (2008) the manufacturers do not incur a fixed cost toexport its products to each market. The realization of productivity draw is private
information to each manufacturer which can not be observed by the shipping companies.
After observing the shipping pricing menu, the manufacturers choose the quantity toexport and pay the corresponding shipping fees.

3

Shipping Firm Screening

A typical shipping firm would use a type contingent contract to maximize their profits.
Thanks to the revelation principle, the optimal contract for the shipping companies would
be the one specifying the outcome pairs (q(φ), T (φ)) and inducing the manufacturers toreveal their true types. The problem can be formulated as follows:Z
maxq(φ),T (φ),Ω

subject to:(T (φ) − wvq(φ)) dG(φ)φ∈Ωwp(φ)q(φ) − τ q(φ) − T (φ) ≥ 0 ∀φ ∈ Ω (IR)
φ
{z
}
|
π(φ)

p(φ)q(φ) − τww
q(φ) − T (φ) ≥ p(φ̂)q(φ̂) − τ q(φ̂) − T (φ̂) ∀φ, φ̂ (IC)
φ
φ
6

(6)

where p(φ) is the inverse demand for the manufacturer goods from the MO preferences,τ is the standard iceberg trade cost and v is the unit shipping cost in terms of labor.
G(φ) (with g(φ) being its p.d.f) is the distribution of manufacturers.
After checking the single-crossing condition and some simple manipulation (the details
are in the appendix), the objective function of shipping companies can be reduced to thefollowing:
Z ∞

Ji
max∗
q(φ),φφ∗ Nijw
p(φ)q(φ) − τ q(φ) −
φ

Z φ


τ wq(x)dx − wvq(φ) dG(φ),
x2
φ∗

(7)

where Ji is the mass of potential manufacturers in country i and Nij is the mass ofshipping firms operating the route from country i to country j, both of which are macro
variables and taken as given by the shipping firms.
The transformed version after integration by parts is given by:
Z ∞

Ji
max∗
q(φ),φφ∗ Nijτ wq(φ)w[1 − G(φ)] dφ.
(p(φ) − τ − wv)q(φ) g(φ) −
φ
φ2

(8)

The Euler Lagrange condition requires a maximization of q(φ) over the term underintegral in equation (8), which yields:p′ q + p| {z }

marginal revenue=w
τ + wvφ| {z }+

1 − G(φ) τ w.
g(φ) φ2(9)

marginal social cost

As is standard in information economics literature, the incomplete information plugs in awedge between the marginal revenue and the marginal social cost, thereby distorting thefirst best allocation. Here in this paper, when the “consumers” become the manufacturers,
one additional implication appears—given the same demand, every manufacturer will
charge a higher mark-up than the one under perfect competition. The higher mark-up
stemming from the market power of the shipping firms has a negative impact on the gains
from trade. This higher mark-up, as Autor et al. (2020) claim, is conducive to higher
profits, thereby aggravating the already low labor share from the standard heterogeneous
firm model.
7

If the productivity distribution is assumed to be the Pareto distribution with shapeparameter θ: θb,G(φ) = 1 −φ(10)then the equation (9) is simplified as:

α−

τw τw
2γ
q − ηQcj =
+
+ wv.
Lj
θφ
φ

(11)

The optimization over φ∗ yields the following:wτ wq(φ∗ )∗∗(p(φ ) − τ ∗ − wv)q(φ ) g(φ∗ ) −[1 − G(φ∗ )] = 0.
φ(φ∗ )2(12)

The above equation means the shipping companies with the monopoly power will configure their customer basis to the extent that profits earned from the marginal cutoff
manufacturers with productivity φ∗ , adjusted by its mass, will be zero. This equationshows the trade-off to include additional manufacturers, which is the marginal gains byadding marginal manufacturers into the customer base and the resulting inframarginalloss by uniformly lowering the transport charges on the inframarginal manufacturers,
which resembles the optimal exclusion condition—equation 7 in Luo et al. (2018).4
The manufacturers with cutoff productivity φ∗ will naturally satisfy both equations(11) and (12), which, together with the Pareto distribution assumption for productivity,
lead to the following:τ w φ∗w∗q(φ∗ ) = 0
p(φ ) − τ ∗ − wv − ∗ 2φ(φ ) θ
γ(q(φ∗ ))2 = 0,L(13)which shows necessarily the manufacturers with cutoff productivity φ∗ will choose to∗)
In fact, the term τ wq(φin equation (12) represents the change of shipping charges to each in(φ∗ )2framarginal manufacturers, in order to include the marginal manufacturers with productivity φ∗ into
the customer base. When it is multiplied by 1 − F (φ∗ ), together they capture the inframarginal loss.
Appendix B offers more intuitive explanations.
48

produce a zero quantity. This naturally yields the following:

p∗ =

τw
τw
+
+ wv = α − ηQcj .
θφ∗
φ∗

(14)

Substitute equation (14) back into equation (11) yields the following:
1 ∗ τw τw
p +++ wv ,
p=
2
θφ
φ


Lj
τw τw
∗
q=
p −
−
− wv .
2γ
θφ
φ

(15)

The equation (11) defines the producing quantity q(φ) in terms of the productivity
φ and it is increasing in φ, therefore, it satisfies the monotonicity constraint given byequation (52).
Also notice that the first line of equation (15) specifies the pricing scheme for eachmanufacturers of different levels, which is the average of its own effective marginal costτw
+ τφw +wv and cost cut off p∗ . Compared with the counterpart in Melitz and Ottaviano
θφ(2008), the effective marginal cost of the manufacturers is lifted up with an additional
two terms τθφw and wv, which together can be understood as the last unit price charged byshipping companies to the manufacturers. It’s worthwhile to take a note that equation(15) echos one of the main advantages of the additive trade cost models, highlighted
in Irarrazabal et al. (2015), that the additive trade cost models can naturally yield zerobilateral trade flows with just parsimonious assumptions. They show that a zero bilateral
trade flow could happen under additive trade cost with the CES demand if the additive
trade cost is so high that even the infinitely productive firm might find it unprofitable
to export to some regions. Although here I show the same property holds for the linear
demand with this special model specification, in fact it holds for more general demand
functions.
PROPOSITION 1. The incomplete information structure between shipping companies
and manufacturers yields a higher mark-up than the perfect competition, given the demand
function and naturally yields additive trade cost components which is the unit cost of
shipping regardless of the demand specification and the productivity distributions.
9The shipping companies would offer a menu of pair (q(φ), T (φ)) to induce self-selection(φ)
would be priced at τθφw + wv, which can be shown using
and the last unit of quantity dTdq(φ)

equation (53):dT (φ)
τw
=
+ wv.
dq(φ)
θφ

(16)

If the information is complete and the shipping industry is competitive, the last unit
price would be constant and equal to wv. However, in this environment, the well-known
phenomenon “Nonlinear Pricing” in Industrial Organization is captured by the τθφw + v
term. The most competitive manufacturers get closet to competitive price wv and thelast unit price is decreasing in φ meaning the more quantity shipped the less the marginalunit price would be, which in turn reduces the average unit price. This decreasing unit
price observed in reality reinforces the advantage of the productive firms, empowering
them further to capture a larger market shares.
PROPOSITION 2. If the common knowledge productivity distribution is Pareto distribution and the realized productivity is private information, the optimal self-selecting(φ)
contract for the principal is a set of pairs (q(φ), T (φ)), which has the property that dTdq(φ)

is decreasing in φ.
The formula for the transfer from the manufacturers to the shipping companies can bederived by simultaneously manipulating equations (15) and (16). Equation (15) defines amonotonic relationship of equilibrium q and φ denoted as φ = ϕ(q). Then equation (16)can be transformed into:

dT =


τw
+ wv dq.
θϕ(q)

(17)

Taking indefinite integration over both sides of the above equation yields the following:
q p∗ − Lγ q + θwv(p + θwv)q
T (q) =
+C =
θ+1
θ+1

(18)

where C is a constant and it will be equal to zero because the manufacturers with cutoff
productivity will produce zero quantity in this model.5 After the manufacturers obtain
5

This approach is essentially the same as in Maskin and Riley (1984). To see this, following the

10the productivity draws, the ones whose draws are above the cutoff level φ∗ will enter themarkets. Just as the IR’ condition implies, the shipping companies will accommodate all
the entering manufacturers by making the marginal player indifferent between quit andstay. Also the marginal player will have a demand exactly equals to zero, indicating that
there is no surplus which the shipping companies can extract in the means of a lump-sumtransfer.
Inspection of equation (18) reveals that the total amount charged by the shipping
firms in this model is no longer proportional to the quantity shipped in a conventional
assumption in traditional trade literature, but a weighted average of the sales of the
manufacturer and the variable cost incurred in shipping. This shipping payment potentially affects the profitability of the manufacturers and in turn impacts the mass ofmanufacturers in the general equilibrium.
Also notice that the average unit price is given as:(p(q) + θwv)
T (q)
=
q
θ+1

(19)

which is indeed decreasing in q comfirming the “Nonlinear Pricing” result and it differsfrom the otherwise competitive average unit price v. It is also increasing in the price ofgoods which is in line with the empirical finding in Hummels et al. (2009). Conveniently,
substituting equation (15) into equation (19) yields a formula showing the relationshipbetween the unit freight rate and the iceberg trade cost τ showing that freight rate
endogenously responds to the change in the iceberg trade cost.
In conventional trade literature, however, where both the tariff and the freight rate
are conceived to be two multiplicative components of τ , the changes in the tariff would
not affect the freight rate, because they are independent from each other, a property
which is at odds with the empirical findings documented by Hummels et al. (2009) that(p+θwv)q∗
method in Maskin and Riley (1984), taking definite integral over dT,
dq yields T (q) − T (q ) =θ+1∗
where T (q ) is the transport charges paid by the least productive firm and its value is given by W (φ∗ ) =r(φ∗ ) − τφw∗ q(φ∗ ) − T (q(φ∗ )). Equation (13) shows that the cutoff manufacturers will produce zero
quantity, which immediately implies both r(φ∗ ) and q(φ∗ ) are 0. After setting W (φ∗) = 0 as in Maskin
and Riley (1984), T (q(φ∗ )) = 0 ensues, which gives the same profit maximizing pricing scheme as in themain content.

11

shipping companies tend to set a higher freight rate on goods facing a higher tariff. The
inability of capturing the positive endogenous change of the freight rate with respect totariff in the traditional trade model does not merely stem from the lack of modeling thetransport industry. Even imposing a monopolistic structure on the transport industry inthe traditional trade model, the holistic multiplicative nature of both the tariff and thefreight rate would imply that a higher tariff leads to a lower freight rate, which is notsupported by the empirical findings in Hummels et al. (2009).
Equations (11) and (18) together characterize the surplus-splitting mechanism between shipping companies and manufacturers.

4

Welfare Implications

After knowing the screening strategy of the shipping firms and the optimal response fromthe manufacturers, it is ready to analyze its effect on the gains from trade. Because
nonlinear pricing in the shipping industry introduces two additional terms, namely τθφw
and vw, a variant of Melitz and Ottaviano (2008), capable of incorporating the unit costof shipping, is established as a comparison benchmark. To streamline the analysis, fromnow on, the variables will have a subscript, when necessary, with the form ij meaning
the manufacturers from the home country i to the destination country j.

4.1

BenchmarkThe benchmark model is a variant of the original Melitz and Ottaviano (2008) model withan additional requirement that goods need to incur a shipping cost to be distributed, or
alternatively it can be interpreted as the shipping industry is perfectly competitive. The
profits maximization of the manufacturers yields the following:p′ q + p =

12

w
+ wv
φ

(20)

The right hand side of the equation is the total social cost consisting of the unit cost of
production and the unit cost of shipping, which can be transformed into the following:


w
1
c
α − ηQj + + wv ,p(φ) =
2
φ


w
Lj
c
q(φ) =α − ηQj − − wv .
2γ
φ

(21)

Comparing to the counterparts in the original Melitz and Ottaviano (2008) model, theadditional additive term wv, surprisingly does not break the tractability of the generalequilibrium objects. Their expressions can be found in the appendix. Typically, theexpected profits under a closed economy have the following expression:

πjD =
=

Z ∞

Lj
φ∗jj 4γw
α − ηQcj − − wv

2
θ

φ

bθj
dφ
φθ+1

bθj Lj w2,2γ(θ + 1)(θ + 2)(φ∗jj )θ+2

(22)

which is the same as the one in Melitz and Ottaviano (2008). The inclusion of an unit costof shipping does not diminish the manufacturers’ profitability and it is perfectly absorbed
into the price cutoff. As a result, the benchmark here shares the same closed-economy
productivity cutoff:"bθj Lj w
φDC=
jj
2γ(θ + 1)(θ + 2)fe

1
# θ+2

.

(23)

The indirect utility function is given as follows:

U

D

1
=1+
2η



1
α−v− ∗
φii


α−v−

θ+1
(θ + 2)φ∗ii


.

(24)

The inclusion of the unit cost of shipping does not structurally change the welfare expression and the expected profits remain the same as in the original MO model. In thesame vein, if the unit shipping cost is assumed to be the same, both domestically andabroad, the domestic productivity cutoff under an open economy will be the same as in13the original MO model, which can be expressed in this paper’s notation as:

φDO
jj =

θ λLj (τjiθ τijθ − 1)
θ + 1 τijθ (τjiθ − 1)

1
! θ+2

,

(25)

θ

b
where λ := (4γθ+2γθ
2 )f . The indirect utility function under an open economy is the sameeas the one under a closed economy and the gains from trade manifests as a higher domestic
productivity cutoff.

4.2

Nonlinear Pricing on Foreign Shipping Routes – The Intermediate Model

This setting can be viewed as an intermediate step to understanding the inner workingsof nonlinear pricing in the shipping industry. It can be interpreted as the manufacturers
distributing locally themselves, whereas, shipping services are used to ship goods abroad,which typically captures the real situation of small countries or the situation where the entry cost of shipping domestically is close to zero, therefore, the domestic shipping market
is almost perfectly competitive. Under the closed economy where the manufacturers bear
the cost of shipping themselves, the equilibrium remains the same as in the benchmark.
After the countries in the two-country model are open to trade, under the assumptionthat the unit shipping cost is the same both domestically and abroad, the productivity
cutoff is connected as follows:θφ∗ji = (θ + 1)τji φ∗ii

∀ i, j.

(26)

The above equation reflects the distortion from the market power in the foreign shipping
market. It shows that under nonlinear pricing in international shipping, ceteris paribus,
only more productive manufacturers are able to overcome the additional wedge from
nonlinear pricing to export, thereby reducing the expected profits from exporting. A
manufacturer in country j with productivity φji = τji φ∗ii —who is able to export when
there is no market power in the shipping industry—cannot afford to pay the information14

wedge to export. This market power distortion dampens the gains from trade.
To examine whether a single sufficient statistics exists as in the benchmark model, we
denote the term τθφw + τφw + wv as the effective marginal cost c:c(φ) =

τw τw
θ + 1 τw+
+ wv =
+ wv.
θφ
φ
θ φ

(27)

Then the marginal cost distribution of manufacturers selling domestically is given as
follows:w
w
P (cii ≤ C) = P ( + wv ≤ C) = P (φ ≥)φC − wvθ ∗w
φii (C − wv)
with support: cii ∈ (wv, ∗ + wv].
=
w
φii

(28)

The marginal cost distribution of manufacturers selling abroad is given as follows:
τji w τji w
τij w(θ + 1)+
+ wv ≤ C) = P (φ ≥)θφφ
θ(C − wv) ∗
φji θ(C − wv) θτji w(θ + 1)=
with support: cji ∈ (wv,+ wv]
τji w(θ + 1)
θφ∗ji
 ∗
θ
φii (C − wv)w=with support: cii ∈ (wv, ∗ + wv].
w
φii

P (cji ≤ C) = P ((29)

The price distribution of manufacturers, regardless of origin, is the same, so the macro
variables including the indirect utility function in a closed economy remains the same inan open economy, where nonlinear pricing is exercised on foreign shipping. As a result,
like in the benchmark model, the only sufficient statistic for a welfare evaluation is thecountries’ domestic productivity cutoff.
The free entry conditions that determine the productivity cutoffs in the open economy
are given as follows:bθ Li τji2 w2bθ L j w 2+= wfe
2γ(θ + 1)(θ + 2)(φ∗jj )θ+2 2γθ(θ + 2)(φ∗ji )θ+2bθ Lj τij2 w2bθ L i w 2+= wfe
2γ(θ + 1)(θ + 2)(φ∗ii )θ+2 2γθ(θ + 2)(φ∗ij )θ+2

15

(30)

which gives the following domestic productivity cutoff:(φ∗ii )θ+2 =τijθ τjiθ −θ λLi
θ + 1 τ θ (τ θ −
ji

ij

2θ+2
θ
θ+1
θ+1
θ
)
θ+1


∀i, j.

(31)

The relative magnitude of the above formula and the one in the benchmark model depends
on the relative magnitude of τij , τji and θ. At first sight, this result seems at odds with thenegative effect of the market power in the shipping industry, because the market power
distortion and the higher mark-up thereof implies a lower productivity cutoff. This oddity
reveals a countervailing effect from nonlinear pricing positively affects the welfare, whichis specific to the heterogeneous firms model. As emphasized in the section of shipping
firms’ pricing strategy, the nonlinear pricing in the shipping industry offers rents to moreproductive manufacturers empowering their already advantageous productivity strengthand crowding out the more unproductive manufacturers than without, thereby lifting
up the average productivity of the manufacturing sector. Comparing the productivity
cutoff in equation (31) and its counterpart in the benchmark model, we can arrive at thefollowing proposition:
PROPOSITION 3. In a model where only international shipping uses shipping services,∗D
1. τji ≥ τij ⇒ φ∗ii < φDOii ⇒ U < U∀i, j ;∗
D
2. τji < τij and θ is reasonably valued ⇒ φ∗ii > φDOii ⇒ U > U∀i, j.

The proof of the above proposition is offered in the appendix. Inspection of equations(11) and (26) reveals that the iceberg trade cost performs like a multiplier of the market
power distortion. Higher τji means a higher mark-up distortion from manufacturers incountry j. Consumers are worse off in country i, whereas the additional profitability isrebated to country j, thereby lowering the overall gains from trade for country i. For the
positive effect to reverse the negative effect from nonlinear pricing, a sufficient condition,
on top of τji < τij , is to have θ reasonably valued, so that the negative effect is relatively
small, as highlighted in equation (16). Intuitively, the above proposition shows that there
is a possibility the positive effect could outweigh the negative effects, though it requires
16

far more strict condition for welfare being higher than being lower. This stringencyof conditions is natural, given that incomplete information adds a cost wedge to every
manufacturer, except the most productive one compared to the benchmark.

4.3

Nonlinear Pricing on both Domestic Shipping and Foreign
Shipping Routes – The Full Model

This situation corresponds to the case where both domestic shipping and foreign shipping
use shipping services, which is more appropriate to capture the reality in large countries.
It provides some additional effects which are active but not visible in the previous analysis.
To ensure consistency with the previous analysis and highlight the impact of nonlinear
pricing, the unit cost of shipping domestically and internationally is assumed to be thesame.6 Additional analysis about an unequal unit cost of shipping is offered in the
appendix. Whether this assumption is too strong or approximately captures the reality
is a valuable empirical question which is not the focus of this paper.
To start with, consider a closed economy where the shipping service is used to distribute goods. All the macro variables can be derived following the steps in the benchmark model, which are given in the appendix. Typically, the expected profits are given
as following:
πi =bθi Lj τii2 w2(4γθ + 2γθ2 )(φ∗ii )θ+2

(32)

which is higher than its counterpart in the benchmark model, provided φ∗ii is fixed.
Though the nonlinear pricing in the shipping service enables shipping firms to split more
surplus, as shown in equation (18), the profitability of manufacturers is actually higher
because nonlinear pricing reinforces the productivity advantage of the productive manufacturers by lower marginal shipping charges. These decreasing marginal shipping charges
renders more advantages over the unproductive firms, enabling the productive ones toearn more profits relatively. This relative profitability from the nonlinear pricing em6

The empirical analysis in Asturias (2020) shows that the marginal cost of shipping is only weakly
decreasing along the trade flows which is inversely related with distance. Therefore, it is relatively safe
to assume that the unit cost of shipping is constant, regardless of the distance of shipping.

17

powers the productive ones to crowd out more unproductive firms. As a result, a higher
productivity cutoff ensues:φ∗ii =

Li bθi2θ(θ + 2)γfe

1/(θ+2)

> φDC
ii .

(33)

Higher profitability and higher productivity cutoff confirm that the nonlinear pricing inthe shipping industry though stemming from the market power does lead to a higher
productivity cutoff which is conducive to a higher welfare level. However this higher
average productivity has a welfare cost which is active but veiled by multiple forces in
the intermediate model. Higher productivity cutoff naturally implies fewer varieties ofavailable goods, which is confirmed as following:Mi =2γ(αθφ∗ii − θ − 1 − vθφ∗ii )< MiD .
η(34)

Up to this stage, all the components impacting the gains from trade, as argued by Feenstra (2018), have been shown as being affected by the nonlinear pricing in the shipping
industry—higher mark-up, higher average productivity, and fewer goods varieties. Their
combined effects will be discussed later.
The profits earned by the shipping firms is given as:Z ∞Π(Nii ) = max∗q(φ),φ

=

φ∗ii



Ji (p + θwv)q− wvq dG(φ)
Niiθ+1

Li wi (θ + 1)(αθφ∗ii − wi (vθφ∗ii + θ + 1)).
Nii ηθ2 (θ + 2)(φ∗ii )2(35)

The free entry condition for the shipping industry Π(Nii ) = wi fs yields the expression
for the number of active shipping firms:

Nii =Li (θ + 1)(αθφ∗ii − wi (vθφ∗ii + θ + 1)).
fs ηθ2 (θ + 2)(φ∗ii )2(36)

This is because all the profits accrued in the shipping industry are rebated to consumers
as wages and wage is the only source of income for each individual. Thereafter, the18

indirect utility function under a closed economy could be expressed as the parametervalues and entry cutoff φ∗ii :
1
U =1+
2η




θ+1
(θ + 1)2
α−v−
α−v−
< UD
θφ∗ii
θ(θ + 2)φ∗ii(37)

The welfare level is lower than the counterpart in the benchmark model, meaning thenegative effects from nonlinear pricing outstrips the positive one.
Like in the intermediate model, the open economy version shares the same macro
variables as the closed economy, including the indirect utility function. The productivity
cutoffs are given as follows:1! θ+2

φ∗ii =λLi (τijθ τjiθ − 1)τjiθ (τijθ − 1)φ∗ji =λLi τji2 (τijθ τjiθ − 1)τijθ − 1

,

(38)

1
! θ+2

.

(39)

Comparing the entry cutoff under both the closed economy and the open economy reveals
that, as in the standard trade model, opening to trade increases the entry cutoff selecting
more productive firms into the market. The counterpart in the benchmark model is asfollows:
φDOii =θ λLi (τijθ τjiθ − 1)
θ + 1 τjiθ (τijθ − 1)

1
! θ+2

< φ∗ii .

(40)

The conclusion reached in the closed economy holds in the open economy, as the negative
effects from nonlinear pricing offset the positive one. Nonlinear pricing will dampen thegains from trade.

5

Firm Size Distribution and Gravity

The nonlinear pricing phenomenon has a famous property in that it offers an increasing
rent along the type of manufacturers to induce self-revealing, which should translate intoa sales advantage for the higher type of firms because they are given a favorable deal by19the shipping company. The firm sales is given as follows:
"

2 #
Lj
τ
w
τ
w
ij i
ij i
rij (φ) = pij (φ)qij (φ) =(p∗j )2 −++ wi v
.
4γ
θφ
φ

(41)

The partial derivative of the firm destination-specific sales with respect to productivityis:
Lj wi2 τij2 (θ + 1)2 Lj wi2 (θ + 1)τij v
Lj wi2 τij2
∂rij (φ)
=
+
>
∂φ
2γφ3
θ2
2γθφ2
2γφ3
where

2
Lj wi2 τij
2γφ3

(42)

is the counterpart in Melitz and Ottaviano (2008). As anticipated in the

beginning of this section, the same level of productivity difference would translate intoa higher sales advantage in the presence of nonlinear pricing. Even when v is set to be0, the other functioning term

τij wiθφ

introduced by nonlinear pricing elevates the effect of2

the marginal productivity growth on sales to a constant rate (θ+1)
over its counterpart
θ2
in MO, regardless of the initial productivity level.
To highlight the crucial role of the nonlinear pricing practice from the shipping industry in shaping the firm size distribution, we consider two manufacturers whose productivity are connected as φ1 = kφ2 (k > 1). To yield a stark contrast to conventional
results (or to obviate the unnecessary complexity), manufacturers’ selling to the local
market follow the same structure as in the original MO model. Similar qualitative results
can be attained using the setting in the benchmark and the marginal insights do not
warrant the additional complexity incurred. The domestic sales difference between these
two manufacturers is given as:
2 #
wi
−
φ1
"
  2 #
Li
1
wi
=
1− 2
.
4γ
k
φ2

Li
rii (φ1 ) − rii (φ2 ) =
4γ

"

20

wi
φ2

2



(43)
(44)

The sales difference of those two firms in country j is given as follows:
"
2 
2 #
Lj
τij wi τij wi
τij wi τij wi
rij (φ1 ) − rij (φ2 ) =++ wi v −++ wi v
4γ
θφ2
φ2
θφ1
φ1
"
2 

2



#
θ+1
1
τij wiθ+1
1
τij wiLj
1− 2+ 2wi v
1−
.
=
4γ
θ
k
φ2
θ
k
φ2

(45)
(46)

Taking the ratio of these two differences yields the following:rij (φ1 ) − rij (φ2 )Lj=rii (φ1 ) − rii (φ2 )
Li

"

θ+1
θ

2

τij2 + 2v



θ+1
θ

#

−1
1
1+
τij φ2 .
k(47)

To investigate the effect from two novel terms introduced by Nonlinear Pricing one by one,h
2 2 i2Lτij which is θ+1
times
first we set v = 0. The sales difference ratio becomes Lji θ+1θθLlarger than Lji τij2 the counterpart in the Melitz and Ottaviano (2008) model. This result
remains even when rij (φ2 ) and rii (φ2 ) are replaced by average sales in the corresponding
markets. This implies that nonlinear pricing indeed magnifies the productivity advantageof the more productive firms and translate it into higher sales than it otherwise would be.
This effect is reinforced when v is not zero by adding a new term which is increasing in v.
This magnification effect from the nonlinear pricing practice in the shipping industry helps
to explain the empirical fact that the top exporters are so big that they are dominant
in shaping a country’s comparative advantage, as documented by Freund and Pierola(2015). They are big not because they are way more productive but because they receive
more favorable deals than their underdogs when procuring intermediate services, which
reinforces their advantage.
Additionally, equation (47) implies the sales-magnifying effect is responsive to thevalue of θ—the productivity dispersion parameter. As θ increases, the manufacturers are
increasingly homogeneous, the sales-magnifying effect from the nonlinear pricing dwindles. As a side note, the other effects affecting the welfare evaluation, as in last section,
are not active here, because those effects affects manufacturers in the destination market
from the same country equally, and the sales difference eliminates them.

21

The bilateral trade flow from country i to country j is given by:
Z ∞Ji pij (φ)qij (φ) dGi (φ)

Xij =
φ∗ij



bθi Ji Lj wi2(1 + θ)2v=+
2γθ2 (2 + θ)(φ∗jj )θ+2 2γθ(φ∗jj )θ+1
τijθ

(48)

This bilateral trade flow resembles the gravity equation in that it decreases with respect
to the iceberg trade cost and increases with the destination size when holding destination
entry cutoff φ∗jj as fixed. Whereas, if φ∗jj is not held to be fixed, though it is still robust
the bilateral trade flow decreases with the iceberg trade cost τij , it will only increase withthe destination size in a relatively small order. Generally, an increase in v will reduce thebilateral trade flow Xij because an increase of v will dampen both the manufacturers’
sales and the number of entrants, which directly affects the volume of bilateral trade flow,
provided that v does not affect cutoff levels, as shown in equation (39). However, there
exist certain cutoff levels which lift the bilateral trade flow by increasing the numberof entrants, which is particularly so when the selection into the export market effect issmall. Intuitively, the number of entrants in the home country is jointly determined by
the number of active firms in the home and in the foreign, the latter of which can bebranded as import competition effect from the foreign markets. The hike of v will reduce
the active firms in both the home and foreign markets. The ensued abatement of theimport competition will stimulate the entering of domestic entrants, which potentiallycould outweigh the negative effect stemming from the hike of v.
Equation (48) can be rewritten as follows, so that it could be used in empirical analysis:bθ Ji wi2 −θ
Xij
τ .
= θi
Xjj
bj Jj wj2 ij(49)

The above equation shows that the trade elasticity of multiplicative trade cost θ can be
estimated using the standard fixed effect gravity approach.

22

6Conclusion

This paper provides a closed-form general equilibrium framework building on the workhorse
Melitz-Ottaviano model to show that the nonlinear pricing practice in the shipping industry, an empirical regularity just documented recently, does reshape the conventional
results. More favorable deals toward the more productive manufacturers would enable
them to expand their sales more than they would be able to do otherwise. This helps
to explain why the top firms are so extraordinarily large that they are crucial in shaping one country’s comparative advantage and why there is a shift from labor returns to
profits as globalization deepens, as documented by Autor et al. (2020). The harsher competition crowds out the more unproductive firms lifting up the average productivity inthe manufacturing industry but the lesser variety of available goods ensues and a higher
mark-up from the market power in the shipping sector offsets the productivity gain. This
result would not be possible without a closed-form general equilibrium model and it suggests a policy intervention into the transport industry to amplify the gains from tradeliberalization.
By introducing a monopolistic competitive shipping industry, this novel model yields
several properties which are more in line with some empirical findings. The endogenous
additive pricing structure naturally gives rise to the possibility of zero-trade volumes
between country pairs with only parsimonious assumptions. The optimal unit pricing
formula shows that the shipping firms would set a higher unit freight rate for goods
facing a higher tariff, a property traditional trade model unable to generate even having
a shipping industry with market powers.
Fitting this paper into a broader picture, contrary to what ACR claims, it showsthat a heterogeneous firm model does introduce new implications and it opens a valley
through which several channels could potentially yield alternative results, which are notpossible under the homogeneous firm Ricardian setting.
This paper has established a framework aligned well with various empirical studies
and has shown that nonlinear pricing has a complicated but significant impact on thegains from trade. Therefore, a possible extension to this paper is to use real-time data to23

quantify the model and assess how countries with different characteristics will be affected
heterogenously.

24

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AppendicesA

Solving the Optimization Problem of Shipping Firms

If all the IC constraints of the initial problem hold, then the whole set of IR constraints
can be reduced to the following constraint:π(φ∗ ) ≥ 0,

(IR’)

(50)

> 0, the downward IC constraints
where φ∗ is the entry cutoff productivity. Because ∂π(ω)∂φ
automatically mean the IR constraint hold for any other types higher than the lowestone.
The single-crossing condition is satisfied, as shown below:




∂
∂π/∂q
∂
w
′
−
=
p(φ) − τ + p q > 0.
∂φ
∂π/∂T
∂φ
φ

(51)

Then all the IC constraints are reduced to following two sets of constraints: the Monotonicity
dq(φ)
≥ 0,
dφ

27

(52)

Local incentive compatibility



w
dq(φ)
′
p(φ) − τ + p q= T ′ (φ) ∀φ ∈ [φ∗ , ∞).
φ
dφ

(53)

Define the following function as W (φ):

W (φ) := p(φ)q(φ) − τww
q(φ) − T (φ) = max p(ω̂)q(ω̂) − τ q(ω̂) − T (ω̂).
ω̂
φ
φ

(54)

By the envelop theorem,dW (φ)
τ wq(φ)
=
.
dφ
φ2

(55)

Integrate the expression (55) over [φ∗ , φ] yields the following:
Z φτ wq(x)dx + W (φ∗ ) =W (φ) =x2φ∗

Z φ

τ wq(x)
dx.
x2
φ∗

(56)

The second equal sign is valid because the IR’ constraint must hold with equality, otherwise the shipping companies can be better off by increasing T (ω). Substitute theexpression of W (φ) back into the original objective function of the shipping firms andperform an integration by parts will yield expression (8).

B

The Determination of Cutoff Manufacturers

Equation (18) shows that the shipping companies have the option to maximize profits
over the fixed component C of transport charges. The adjustment of C changes the
productivity level of cutoff manufacturers φ∗ . Therefore, this section of the appendix is
to show that the shipping firms’ optimization over C is the same as opimization over φ∗ .
One can easily show that equation (18) holds regardless of whether q(φ∗ ) = 0 is
exogenously imposed beforehand. In case the cutoff productivity of the manufacturers is
endogenously determined, equation (18) has the following form:q α − ηQcj − Lγ q + θwv(p + θwv)q
+C =
+ C.
T (q; C) =
θ+1
θ+1
28

(57)

Substituting equation (57) back into equation (7) yields the following:
Z ∞
max
C

Ji
(T (q, C) − wvq(φ)) dG(φ),φ∗ (C) Nij(58)

where φ∗ is implicitly defined as a function of C through the following equation:p(φ∗ )q(φ∗ ) −τij wiq(φ∗ ) − T (q, C) = 0.
φ∗

(59)

Using the General Leibniz rule, the first order condition of above objective function w.r.t
C gives the following:dφ∗− [T (q(φ ), C) − wvq(φ )] g(φ )
+
dC}
|
{z
∗

∗

∗

marginal loss

Z ∞
φ∗

|

dG(φ) = 0,
{z
}

(60)

inframarginal gain

where the first underbrace shows how much profits lose from marginal unit increase
in C, while the second underbrace shows how much profits gain from the inframarginal
manufacturers by one unit increase of C. Essentially, this first order condition shows thatthe shipping firms face a trade-off when setting how much C they should charge. After∗dC
and substituting for dφ
dividing both sides of equation (60) by dφ∗ with the expressiondC

generated through the total differentiation equation (59), the above first order condition
arrives at equation (12).
Figure 1 is a numeric example showing how the shipping firms’ profits vary withrespect to the changes in the fixed component of the shipping charges C. The parametervalues are given with proper discretion.7
The profits of the shipping firms reach their highest when the fixed component of theshipping charges vanishes to zero, which justifies the result in equation (13).
7

The parameter values are set as follows: L = 1, τ = 1.2, w = 1, v = 2, γ = 4, θ = 5, α = 1000, η =3, Qcj = 5, b = 0.001. The range of C is [0, 100]. Here the value of Qcj is exogenously given, though it
should be determined within the model. When the shipping firms are making decision, however, theytreat Qcj as exogenously given. The assignment of exogenous values to Qcj here does not change theshipping firms’ decision on their customer base.

29

Figure 1: Shipping Firms’ Profits Variation with respect to CC

Solving the Benchmark Model

Supposing manufacturers do not use the shipping services to distribute domestically, thenthe marginal cost of production is wφ + wv. The profits maximization yields the following
conditions:
w
+ wv
φ

w
− α − p − ηQcj + p = + wv
φ


w
1
c
p(φ) =α − ηQj + + wv
2
φ


Lj
w
c
q(φ) =α − ηQj − − wv .
2γ
φp′ q + p =If the mass of potential manufacturers ex ante in country i is denoted by Ji , then themass of entrants Mij can be expressed as follows:
Mij = Ji (1 − Gi (φ∗ij )),

30

(61)

where Gi (φ) = 1 −

 θ
bi
φ

is assumed to be the Pareto distribution where the potential

manufacturers draw productivity and we assume bi is low enough to ensure a cutoff
threshold for every destination country j is active, namely φ∗ij > bi . It is straight forward
to show Fi (φ) and Gi (φ) have the following relation:

Fi (φ) =Gi (φ) − Gi (φ∗ ).
1 − Gi (φ∗ )The aggregate variable Pj is expressed as:Zp(ω) dωPj =ω∈Ω
Z ∞ ijJj p(φ) dGj (φ)

=
φ∗jj


Z ∞ 
bθj
1
w
c
= Jj
α − ηQj + + wv θ θ+1 dφ
φφ
φ∗jj 2bθj w(1 + 2θ + 2(θ + 1)vφ∗jj ).
= Jj
2(θ + 1)(φ∗jj )θ+1
The profits formula is given as follows:w
πjj (φ) = pq − q − wvq
φ


1
w
c
=
α − ηQj − − wv q
2
φ

2
Lj
w
c
=
α − ηQj − − wv
4γ
φ
The expected profits are given as follows:
Z ∞
πj =

πjj (φ) dGj (φ)
φ∗jj

Z ∞

Lj
=
φ∗jj 4γ


2
bθj
w
c
α − ηQj − − wv θ θ+1 dφφφbθj Lj w2=.
2γ(θ + 1)(θ + 2)(φ∗jj )θ+2

31

(62)

The free-entry condition implies the following:
πj = wfe"bθj Lj w
φD=
jj
2γ(θ + 1)(θ + 2)fe

1
# θ+2

(63)

which is the same productivity cutoff as in Melitz and Ottaviano (2008). Similarly,
the productivity cutoff in the open economy in this paper is the same as in Melitz andOttaviano (2008). The active mass of firms is given as follows:MjD =2γ(θ + 1)(αφ∗jj − vwφ∗jj − w)
.
wη

(64)

Correspondingly, σp2 is given as:
σp2 :=

Z ∞(p(φ) − p̄i )2 dFi (φ)
φ∗ii


2
Z ∞ 
1w(1 + 2θ + 2(θ + 1)vφ∗ii )wc=
α − ηQj + + wv −dFi (φ)φ2(θ + 1)φ∗ii
φ∗ii 22
Z ∞ 
w(1 + 2θ + 2(θ + 1)vφ∗ii )w(φ∗ii )θ1cdφ
α − ηQj + + wv −θ=φ2(θ + 1)φ∗ii
φθ+1
φ∗ii 2=w2 θ.
4(θ + 1)2 (θ + 2)(φ∗ii )2

The welfare expression is then given as follows:

U

D

D

1
=1+
2η



1
α−v− ∗
φii


α−v−

θ+1
(θ + 2)φ∗ii


.

(65)

θ λLi (τijθ τjiθ − 1)
θ + 1 τjiθ (τijθ − 1)

(66)

Proof of Proposition 3

To compare the following two cutoffs:(φ∗ii )θ+2 =

θ
θ+1


λLi τijθ τjiθ −

2θ+2
θ
θ+1

τjiθ (τijθ −

θ+1
θ
)
θ+1



32

θ+2
, (φDO=ii )it is equivalent to assessing whether the following function is increasing or decreasing in
the range of [θ/(θ + 1), 1]:
f (x) =

τijθ τjiθ − x2.
τijθ − xIts derivative is:f ′ (x) =x2 − 2xτijθ + τijθ τjiθ.
(τijθ − x)2

A sufficient condition for the numerator of the above to be positive is to have τji ≥ τij
and for it to be negative is to have τji ≪ τij , at the same time θ is large enough to ensure:qθθ> τij − (τijθ )2 − τijθ τjiθ
θ+1

E

and 1 − 2τijθ + τijθ τjiθ < 0.

Solving the Full Model

The model can be solved following the steps in the benchmark model. The aggregate
variable Pj is expressed as:

Pj =

X

=

X

i

i


Z ∞ 
bθi
1 ∗ τij w τij w
pj +++ wv θ θ+1
dφ
Ji
θφ
φ
φ
φ∗ij 2Jibθi w(τij + 2θτij + 2θvφ∗ij ).
2θ(φ∗ij )θ+1

(67)

Given the quantity and payment menu (q(ω), T (ω)) specified in section 3, the profits ofthe manufacturers in country i earned in country j is given by:

πij (φ) =θ Lj ∗ τ w τ w(p −−− wv)2
θ + 1 4γ j
θφ
φ

(68)

Then, the expected profits for the manufacturers in country i is given as:
X Z ∞ θ Lj
τw τw
bθi
πi =
(p∗ij −−− wv)2 θ θ+1
dφ
θφ
φ
φ
φ∗ij θ + 1 4γ
j
=

X
j

bθi Lj τij2 w2.
(4γθ + 2γθ2 )(φ∗ij )θ+2

33

(69)

The free-entry condition implies that the productivity cutoff under a closed economy:

φ∗ii =

Li bθi2θ(θ + 2)γfe

1/(θ+2).

To generate the formula of the masses of active firms, we utilize equation (14) to express
Qci as a function of entry cutoff:Qci =αθφ∗ii − vwθφ∗ii − wτ − wθτ.
ηθφ∗ii(70)

The macro variable Pi under the closed economy is given by equation (67):Mi w(τ + 2θτ + 2vθφ∗ii ).
2θφ∗iiPi =Substituting the expressions of Qci and Pi into the equation (4), we can arrive at theexpression for the mass of active firms:

Mi =2γ(αθφ∗ii − θ − 1 − vθφ∗ii )
.
η

(71)

p̄i is given as:p̄i :=

τ + 2θτ
Pi
=
+ v.
Mi
2θφ∗ii

σp2 is given as:
σp2 :==

Z ∞(p(φ) − p̄i )2 dFi (φ)φ∗iiw2 τ 2.
4θ(θ + 2)(φ∗ii )2

Because all the profits accrued in the shipping industry are rebated to consumers as
wages, wage is the only source of income for each individual. Therefore, the indirect
utility function under a closed economy could be expressed as the parameter values and34the entry cutoff φ∗ii :

−1
1
γ
1 Mi 2U =1+
η+(α − p̄i )2 +
σ
2
Mi
2 γ p



1
θ+1
(θ + 1)2
=1+
α−v−
α−v−
.
2η
θφ∗ii
θ(θ + 2)φ∗ii
Consider a two-country open economy where the unit shipping costs are the same both
domestically and internationally. Equation (14) yields the connection between φ∗ij andφ∗jj :
φ∗ij = τij φ∗jj

∀i, j.

(72)

The free entry condition of both countries implies:
Lj τij2
Li1+ ∗ θ+2 = ,∗ θ+2(φij )(φii )λ(73)

Li τji2Lj
1+ ∗ θ+2 = .
∗ θ+2(φji )(φjj )λTogether with equation (72), the cutoffs are given as:1! θ+2

φ∗ii =λLi (τijθ τjiθ − 1)τjiθ (τijθ − 1)φ∗ji =λLi τji2 (τijθ τjiθ − 1)τijθ − 1

,

(74)

1
! θ+2

.

(75)

As in the benchmark model, it is straightforward to verify that in any destination country
i the price distribution facing the consumers is the same for goods from any source
countries, which indicates that the expression for macro variable Pi is the same as in aclosed economy. Equation (14) shows that the formula for macro variable Qci should be
the same as in the closed economy in equation (70). Thereafter, every macro variablein the open economy inherits the same formula as in the closed economy, including thewelfare expression.
The mass of firms in country i consists of firms coming from domestic and foreign

35

markets which gives the following:

Mji + Mii = Mi ,(1 − G(φ∗ji ))Jj + (1 − G(φ∗ii ))Ji = Mi .

(76)

The equations (76) and (72) can solve Ji as functions of cutoffs:

Ji =
=

1
bθ (τijθ τjiθ − 1)

(Mi (φ∗ii )θ τijθ τjiθ − Mj (φ∗jj )θ τijθ )2γηbθ (τijθ τjiθ − 1)[(αθφ∗ii − θ − 1 − vθφ∗ii )(φ∗ii )θ τijθ τjiθ− (αθφ∗jj − θ − 1 − vθφ∗jj )(φ∗jj )θ τijθ ].

The value of Mij and Mii can be easily derived, after knowing Ji , and correspondingly,
the equilibrium number of the shipping firms on each route can be pinned down by thefree-entry condition as in the closed economy.
Just as in traditional heterogeneous firms trade literature, opening to a costly tradefeatures selection of the more productive firms into export market, Ji > 0 implies:(αθφ∗ii − θ − 1 − vθφ∗ii )(φ∗ji )θ > (αθφ∗jj − θ − 1 − vθφ∗jj )(φ∗jj )θ
φ∗φ∗(αθ τjiji − θ − 1 − vθ τjiji )(φ∗ji )θ
(αθφ∗jj − θ − 1 − vθφ∗jj )(φ∗jj )θ

>1

φ∗ji > φ∗jj .

F

Unequal Unit Cost of Shipping

When the unit cost of domestic shipping does not equal to international shipping, equation(14) implies a variant of equation (72):φ∗ij = τij φ∗jj + Aij
θ(vij − vjj )φ∗ij φ∗jj > 0.
where Aij = θ+1

36∀i, j.

(77)

Total differentiate equations (73) and (77) and consider a perturbation around vij =vjj ∀i, j yielding the following:
Lj τij2 [−(θ + 2)](φ∗ij )−(θ+3) (τij dφjj + dAij ) + Li [−(θ + 2)](φ∗ii )−(θ+3) dφii = 0
|
{z
}
|
{z
}

(78)

Li τji2 [−(θ + 2)](φ∗ji )−(θ+3) (τji dφii + dAji ) + Lj [−(θ + 2)](φ∗jj )−(θ+3) dφjj = 0
{z
}
|
{z
}
|

(79)

B12

B11

B21

B22

Using Cramer’s rule, we arrive at the following:dφjj−B11 B21 τji
=
>0
dAij
B11 τij B21 τji − B22 B12
dφii−B11 B21 τij
=
>0
dAji
B11 τij B21 τji − B22 B12dφjj
B12 B21=<0dAji
B11 τij B21 τji − B22 B12dφii
B11 B22=<0dAij
B11 τij B21 τji − B22 B12
The first two equations have some interesting implications—the increasing trade barrier
will increase the destination’s productivity cutoff, contradicting the intuition that theincreasing trade barrier will shield domestic firms from foreign competition. The intuition
behind this is indeed somewhat sophisticated. When vij goes up, a smaller fraction of
manufacturers export (dφij /dAij > 0), thereby lowering the expect profits, which leads toa smaller number of potential entrants. This allows for more unproductive manufacturers
to survive (dφii /dAij > 0), at the same time attracting more manufacturers from countryj (dφji /dAij > 0). The resulting expected high profits encourage more manufacturers toenter, thereby crippling the survivability of the existing ones (dφjj /dAij > 0).
How the cutoff level φ∗ii will change depends on the level of change dAij and dAji , aswell as Lj , Li , τij and τji . What can be asserted here is that the inclusion of a unit costof shipping from nonlinear pricing does change the welfare evaluation.

37