# Estimating Endogenous Coalitional Mergers: Merger Costs and Assortativeness
of Size and Specialization††thanks: I thank my advisor Jeremy Fox for his
valuable advice. I also thank Sen Lu for sharing an early version of code for
computing matching equilibria with externality and Fumio Nagasawa for allowing
me to refer to the firm-level data he has collected. I benefited from the
extensive comments of Robin Sickles, Isabelle Perrigne, Kei Kawai, Takeshi
Murooka, Naoki Wakamori, Takuma Matsuda, and Yuri Matsumura as well as seminar
participants at Japan Empirical Industrial Organization Workshop, 16th Applied
Econometrics Conference, Japan Society of Logistics and Shipping Economics
Kanto region, and 24th Moriguchi prize conference. I also thank Koichiro
Hayashi and Nobufuku Takahashi for sharing industry knowledge and expertise at
Syosen Mitsui and Nippon Yusen groups.

Suguru Otani Department of Economics, Rice University. Email<EMAIL_ADDRESS>
###### Abstract

I present a structural empirical model of a one-sided one-to-many matching
with complementarities to quantify the effect of subsidy design on endogenous
merger matching. I investigate shipping mergers and consolidations in Japan in
1964. At the time, 95 firms formed six large groups. I find that the existence
of unmatched firms enables us to recover merger costs, and the importance of
technological diversification varies across carrier and firm types. The
counterfactual simulations show that 20 % of government subsidy expenditures
could have been cut. The government could have possibly changed the
equilibrium number of groups to between one and six.

Keywords: Antitrust, Merger, Subsidy, Specialization, One-sided one-to-many
matching model, Shipping
JEL Codes: C78, L11, L25, L41, L50, L91

##  1 Introduction

The relationship between the matching mechanism and merger policy is a complex
and evolving area of research, and the precise nature of their interaction is
still not fully understood. One case that illustrates the success of special
merger encouragement as a mechanism is the shipping mergers and consolidations
that occurred in Japan in 1964. Japan relied on domestic shipping firms to
transport nearly 90 percent of its trade.111I am referring to page 10 of
book4. Simultaneously, Japan was beset by a chronically inefficient industrial
structure caused by postwar entry into the shipping industry. To resolve this
inefficiency problem, the Japanese government implemented “the Two
Reconstruction Laws" in 1964, which entitled merged firms to a subsidy if,
after mergers and groupings, their total tonnage exceeded a certain
threshold.222After the Peace Treaty came into effect in 1952, Japanese
shipping and shipbuilding regained complete freedom in international markets,
but this involvement also meant vulnerability to the vagaries of world
politics. The Japanese shipping industry faced the closure of the Suez Canal
in 1956 and a depression after the Korean War from 1950 to 1953. During the
Korean War, Japanese shipping tonnage had expanded at a particularly rapid
rate, and then had a large degree of overcapacity when the shipbuilding demand
came to an abrupt end. At that time, the industry needed to overcome a
shortfall in earnings within the industry. The shortfall can be attributed to
two principal factors. First, there was a sharp fall in freight rates as the
wartime boom turned to a postwar slump in demand. Second, Japanese interest
rates were higher than those in other countries, which imposed a heavy burden
on firms that had acquired tonnage with the aid of borrowed capital. For
reference, see the detailed history of the Japanese shipping industry from
1952 to 1962 in chida1990japanese, Section 4. The objective was to decrease
the number of firms. After the shipping mergers and consolidations, 27 major
firms and 68 small firms formed six mutually exclusive groups through mergers
and hierarchical relationship contracts, and 33 firms chose to remain
unmatched. These six groups satisfied the subsidy requirements of total group
tonnage. Thus, the government successfully implemented a simple subsidy
mechanism to effectively encourage firms to endogenously initiate and
implement horizontal coalitional mergers and decreased the number of firms.
After these mergers and consolidations, Japan succeeded in achieving the top
share of total tonnage in the world’s shipping industry until the 1973 oil
shock.333See global trends of tonnage in the 1960s in Figure 4 in Appendix A.

Given the implementation of laws as an ideal experiment, I estimate the
primitive parameters in the one-sided one-to-many coalitional matching model
with complementarities by fox2018qe using novel detailed data about Japanese
shipping mergers and consolidations and the processes from 1960 to 1964. The
data contain the observed characteristics of almost all firms and ships in the
industry before and after the mergers and consolidations.

Constructing a coalitional matching model poses some challenges. The first
challenge is a tractable computation. My paper contributes to the literature
on coalition formation by applying a tractable matching model. To the best of
my knowledge, only fox2019externality constructed a one-sided one-to-many
matching model to explain mergers in the United States mobile carrier
industry. The empirical matching model allows us to quantitatively explain why
one firm is more likely to merge with multiple other firms endogenously, based
on the observed matching patterns and characteristics. This cannot be
investigated by alternative non-cooperative game-theoretic models, such as
quantity competition models.

Another challenge is estimating the merger costs for each main firm. The one-
to-many matching model with unmatched firms can recover the costs that the
main firm implicitly incurs by inviting an additional firm into the group. The
balance of the subsidy amount and merger cost is crucial for evaluating merger
outcomes. To the best of my knowledge, exploiting unmatched firms to estimate
matching costs in a single, large market setting is new to the
literature.444fox2018jpe investigated the identification with unmatched agents
across many matching markets and found that the existence of unmatched agents
helped the identification of unobservable complementarities.
bajari2008evaluating also investigated the estimation of fixed cost parameters
as fixed effects in a many-markets setting.

My estimation results indicate that assortativeness of both size and
specialization contributes to merger incentives. However, the assortativeness
of specialization has a significantly different role across carrier and firm
types. This implies that the importance of technological diversification and
specialization for merger incentives varies. I also find the existence of high
merger costs for adding a target firm to a group in a single, one-sided
market.

Finally, I exercise two counterfactual simulations given the estimated
parameters. First, I investigate the effect of changes in the subsidy design
on the equilibrium number of groups and unmatched firms. The counterfactual
simulations reveal the efficiency of the government subsidy in terms of
expenditure minimization. Second, I investigate how the subsidy threshold and
amount could have changed merger configurations. The simulations open a black
box regarding how market participants are determined via coalitional mergers.
555The numerical implementation of a general matching model with
complementarities based on azevedo2018existence is new to the literature. In
Appendix C, I present identification and estimation results in a rich variety
of real-world market situations. In Appendix D, I demonstrate the numerical
results of comparative statics based on simulated data.

My simulation results illustrate that, given a subsidy amount, by shifting the
subsidy threshold, the government could have changed the number of equilibrium
matching groups to between one and six. Conversely, given a subsidy threshold,
increasing the subsidy amount would not have stimulated additional merger
incentives. The simulations of several policy mix scenarios indicate that the
actual subsidy provision in 1964 was too generous, and 20 % of the subsidy
expenditure could have been cut. The cost-savings amount would have been
approximately equivalent to the matching production of a hypothetical merger
of the two largest groups.

Second, in the counterfactual merger configurations, I find that coalitional
mergers would have been driven by large firms that could have overcome the
high merger costs given the small subsidy amount. As the subsidy amount had
increased, additional mergers of unmatched firms would have occurred, and some
groups would have split into multiple subgroups to enjoy the larger subsidy
amount shared in the subgroups. For example, for individual firms, the subsidy
benefit from a group consisting of four firms would be less than the benefit
from its subgroup consisting of two firms because the subsidy amount was
shared with all members in the group.

###  1.1 Related literature

This paper examines the effect of a subsidy mechanism on endogenous merger
matching. This paper is based on three strands of the literature, namely, one-
sided one-to-many matching with complementarities, endogenous merger analysis,
and optimal subsidy to decrease the number of firms. See chiappori2016jel for
a general reference on several empirical matching models.

First, this paper contributes to the literature on empirical one-sided one-to-
many transferable utility (TU) matching. The most related econometric model is
fox2018qe, whose model has been applied to other empirical topics such as
banking merger (akkus2015ms, chen2013ijio), faculty room allocation
(baccara2012aer), executive and firm matching (pan2017determinants), and buyer
and seller relationships in the broadcast television industry (stahl2016aer).
These papers have applied the matching maximum score estimator proposed by
fox2010qe to two-sided many-to-many and one-to-one matching in a TU matching
environment. However, in general, the one-sided matching model does not
guarantee the existence of stable matching.666Recently, chiappori2019human
demonstrated the mild conditions for the existence of a stable equilibrium in
the one-sided one-to-one matching model. The present paper takes a different
approach to convert the general results of azevedo2018existence into our
computationally feasible setup. To overcome this obstacle, the present paper
extends the approach of azevedo2018existence to the one-sided one-to-many
matching model in a single large market with unmatched agents.777As an
alternative approach to estimating the one-to-many matching model,
agarwal2015aer and diamond2017qe showed the theoretical results on
identification and estimation and demonstrated an empirical application of
medical matches. Their results crucially rely on the properties of one-to-many
matching data, whereas my model does not. Instead, my model does not
incorporate a rich structure of preference heterogeneity and the
differentiation of agents. The proposed tractable method can also be applied
to other settings. The computational applicability is demonstrated in Appendix
C.

A few papers investigated the empirical one-sided matching model and the
coalitional matching model with a government subsidy mechanism using various
approaches. weese2015qe developed a method to estimate the bounds of primitive
parameters in the coalition formation model by using data regarding the
Japanese municipal governments. gordon2009jpub investigated the effect of
merger subsidies on a geographically adjacent coalition of school districts
for economies of scale. These two papers exploited a non-transferable utility
(NTU) matching framework. I do not take their coalitional game approach
because it requires superadditivity for the payoffs assigned to each
coalition, which ignores the additional merger costs, my main subject of
interest.

Second, this paper contributes to the literature on endogenous merger
analysis. In particular, this paper focuses on endogenous mergers in a single,
static, large matching market because of the institutional background
explained in Section 2.888Endogenous merger analyses in the industrial
organization literature are divided into dynamic and static matching models.
In terms of dynamic matching models, they follow gowrisankaran1999dynamic.
stahl2011dynamic was the first to estimate a merger activity model using a
dynamic, strategic framework. jeziorski2014effects estimated the sequential
merger process to analyze ownership consolidation in the United States radio
industry after the enactment of the Telecommunications Act of 1996.
igami2019mergers applied a stochastic sequential bargaining model to the
merger processes of the hard disk industry. As the most recent paper,
hollenbeck2020horizontal enriched the Gowrisankaran-type dynamic endogenous
merger model. With different dynamic approaches, nishida2015better compared
post-merger and pre-merger beliefs and equilibrium behaviors in a Markov
perfect equilibrium in the Japanese retail chain industry. perez2015building
incorporated mergers as bidding games by incumbents and investigated the
effect of the Reagan-Bush administration’s merger policy on the reallocation
of assets in the United States cement industry. Conversely, using a static
matching model, uetake2019entry developed an empirical two-sided NTU matching
model with externalities using moment inequalities and investigated the effect
of entry deregulation on the “with whom”-decisions of bank mergers by the
Riegle-Neal Act. akkus2015ms tackled the same Act with a different approach.
They added transfer data and constructed a one-to-one matching model with
transfer utility and found that merger value increased from cost efficiencies
in overlapping markets, relaxing regulations, and the network effects
exhibited by acquirer-target matching. This paper constructs a coalitional
matching model by converting the abstract model of azevedo2018existence in
which the equilibrium solution is guaranteed to exist uniquely into a
numerically tractable model as a classical assignment problem
(shapley1971assignment). In addition, the present paper quantifies the
relative importance of post-merger size effects and technological
specialization, which are the main economic forces driving firms to pursue
mergers to gain cost efficiency in the shipping industry
(notteboom2004container). The cost-side incentives such as decreasing costs
are indirectly incorporated in the merger incentive.

Third, this paper empirically evaluates the effect of using a subsidy
mechanism to resolve the excess entry problem. mankiw1986rand proposed the so-
called “Excess Entry Theorem". This theorem states that the business-stealing
effects and fixed costs in oligopoly markets result in an excessive number of
entrants at a free-entry Cournot and Bertrand equilibrium, and they also
induce social inefficiency. In the same spirit as mankiw1986rand,
suzumura1987res proved the inefficiency of excessive entrants under a free-
entry quasi-Cournot equilibrium and stated that the paper was motivated by the
regulation of excessive competition throughout Japan’s postwar period.
Although the institutional background and history are shared with this
article, I emphasize that my cooperative game-theoretic model does not address
the excess entry problem based on non-cooperative game-theoretic models, but
rather focuses on merger patterns, which are of interest to the Japanese
government.

In many industries, excess entry is reported to be problematic, so many
researchers have analyzed it using several structural models and quantitative
methods. waldfogel1999free studied the radio industry to quantify the
inefficiency of excess entry and exercised counterfactual simulations to make
comparisons of the number of entrants under free-entry, monopoly, and socially
optimal. They proposed two alternatives to resolve excess entry – the direct
regulation of entries and levying an entry tax. By using real-world data about
special permissions and government interventions, I investigate the third
alternative – that a regulator encourages the formation of mergers and
coalitions.999nishiwaki2016horizontal and okazaki2022 answered similar policy
questions in a related industry in Japan. nishiwaki2016horizontal models the
divestment dynamics of firms with chronic excess capital in the Japanese
cement industry from 1998 to 2009 and examines whether merger-induced
divestment improves total welfare. He focuses on endogenous divestment as a
key device to resolve the excess entry problem by assuming mergers as
exogenous events. Conversely, I utilize the two laws as a natural experiment
inducing matchings and focus on endogenous mergers as a device to resolve the
excess entry problem. okazaki2022 investigate the aspect of the capacity
coordination policies. I also evaluate the efficiency of subsidy design in the
sense of minimizing government expenditures given targeted matching outcomes.
Recently, miller2019optimal empirically evaluated the optimality of subsidies
in health insurance markets by incorporating market competition and demand
changes into the channels of subsidy effects. clark2020incentivized evaluate
the impact of government-provided incentives by transfer tax reductions on
consolidation patterns of local electricity distribution companies in Ontario.
Aside from industrial and institutional differences, the focus of my paper –
the channels of a subsidy design’s effects on matching outcomes – complements
their insights.

The remainder of this article is organized as follows. I first describe the
data, institutional details, and conduct preliminary analysis in Section 2.
Next, I detail the one-to-many matching merger model in Section 3. Section 4
introduces the estimation procedures. Section 5 discusses the estimation
results. Section 6 reports the results of the counterfactual experiments. I
conclude in Section 7. The institutional and estimation details, Monte Carlo
simulations, comparative statics, supplemental estimations, and robustness
checks are shown in Appendices A, B, C, D, E, and F, respectively.

##  2 Data and Institutional Background

This section provides data descriptions of my dataset and provides a brief
overview of the industry background before and after the mergers and
consolidations. Next, I present the descriptive statistics. Institutional
evidence and preliminary regression results to explore merger patterns are
shown in Appendix A.101010Before introducing specific data details, I make
some remarks about the broad historical background. This paper focuses on the
period between 1960 and April 1964, which is before global containerization
(1966) and just after the start of the foreign-flag convenience ship boom in
Japan (1962). This paper also focuses on the shipping industry, not the
shipbuilding industry, although the main firms overlapped both industries
indirectly through their stockholders during this period.

###  2.1 Data

My dataset consists of two parts. The first part records information about the
firms and ships owned by each firm. The dataset is based mainly on “The list
of shipping tonnage of ocean-going ship industry "(“Gaikousen-Unkougyoubetsu-
Senpuku-Kouseihyou," listgaikou, in Japanese), published biannually by the
Japanese Shipowners’ Association (JSA). The data source provides ship-level
characteristics, which consist of contract type and carrier types as follows.

First, a ship’s contract type can fall into three different categories – (i)
time charters, (ii) bareboat charters, and (iii) owned ships. A time charter
is a contractual agreement between a shipowner and a charterer, where the
charterer hires the vessel for a specified period of time and pays a fixed
rate for each day the vessel is under charter. A bareboat charter, also known
as a demise charter, is a contract in which the charterer takes full control
and responsibility for the vessel, including crewing and maintenance. These
categorizations are aggregated because contract types are not important from
the viewpoint of subsidy requirements.

Second, according to the classifications of the syuuyaku, the carrier type of
a ship is categorized into four types.111111According to listgaikou, the
carrier type is categorized into nine detailed types as follows: (1) cargo,
(2) ore carrier, (3) oil tanker, (4) bauxite carrier, (5) nickel carrier, (6)
lumber carrier, (7) LPG carrier, (8) passenger cargo, and (9) refrigerated
transportation. For example, Ship A is classified as a pair (time charter, oil
tanker) and Ship B is classified as a pair (bareboat charter, lumber carrier).
However, because there are many missing values in the classification
information in listgaikou, I follow the categories in syuuyaku. The first
carrier type is a cargo liner, which is operated regularly to carry cargo
according to a schedule and route set in advance. The second type is a special
vessel that specializes in transporting a particular type of cargo such as
timber or iron ore. This special vessel is in service with an optimal
structure and equipment for this purpose and works exclusively on certain
routes. The third is a tramper, which is irregularly operated without a
specific route, depending on the availability of cargo. The fourth is a
tanker, which is equipped with a tank in its belly to carry oil or other
liquids.

Carrier types, such as the special vessel and tanker, determine exclusively
what types of goods the owner firms can carry. For example, only tankers can
carry oil, whereas only some special vessels can carry bauxite. Therefore, the
carrier type represents the degree of shipping ability specialization. For
each carrier type, I collect data regarding the firm’s key characteristics
measured in dead-weight (D/W) tonnage tons.121212Although the data about
shipyard id, ship class, power (in horsepower), speed (in knots), age, and
main shipping route are recorded, I focus on the key independent variables
because the present model considers firm-level decisions based on the firm’s
tonnage. Technically, the ship class records the inspector company’s name,
which is similar to car inspection but does not imply a vertical quality
measure. The main shipping route records specific destinations such as country
or ocean, but is not consistent within the data, and some ships are defined as
irregular tramper ships, so I omit this information. I filled in the missing
values for some small firms by referring to nostalgic.131313I confirmed that
the information about the main ships in nostalgic is precise and consistent
with my raw data, although he collects a consistent dataset privately. The
owner of the website kindly allowed me to refer to the dataset. The present
paper focuses on ocean-going ships and related subsidies; therefore,
information about non-international, coastal trading vessels operated by small
domestic coastal trading firms is omitted.

The second data source is “Documents related to the consolidation of shipping
companies" (“Kaiun Kigyou Syuyaku Kankei Shiryo," syuuyaku, in Japanese),
published by the syuuyaku, which includes information about firm type. Firms
are of three firm types: (i) a main firm that has the right of capital control
over the group, (ii) an affiliate firm that receives capital controlled led
through the merged company, and (iii) a wholly controlled firm that has a
close relationship with the merged company or its affiliates and is on a long-
term charter or that delegates the operation of a vessel to the merged company
or its affiliates. The firm types are important for satisfying subsidy
requirements, which is explained in Section 3.141414For preliminary analysis
in Appendix A.4, we use “The report of shipowner’s management” (“Keiei no
jissou”, jissou, in Japanese), published by the JSA, which includes basic
financial data. For the analysis of coalitional mergers, the information about
the main banks of each firm is potentially important to explain the incentives
for coalitional matchings. Finally, to create Figure 4, I use world shipping
data published by the Ministry of Transport Shipping Bureau, Lloyd statistics,
and nostalgic’s complementary auxiliary dataset, which is partially common
with recent papers (kalouptsidi2014aer, kalouptsidi2017res, jeon2017learning,
barwick2019china). I refer to book3 and book4 for institutional background
knowledge and detailed histories of consolidations of my interest.

This paper focuses on firm-level mergers and group-level subsidy criteria.
Therefore, I use firm-level total tonnage for each carrier type in the main
analysis after aggregating ship-level total tonnage. I manually construct a
dataset combining these data sources from the Japan Maritime Center library.
Note that this paper focuses on the static matching model in a single large
market. Thus, I mainly use the data in 1963 and 1964. I use data from other
years only to check the accuracy and consistency of the main pre- and post-
merger data.

###  2.2 Reconstruction Maintenance Law and Interest Subsidy Law

In this subsection, I explain the institutional background of shipping mergers
and consolidations and the details of Japan’s policy according to
chida1990japanese, Sections 4 and 5. See Table 7 in Appendix A for reference
of major events related to my sample.

The story of the modern Japanese shipping industry dates back to the postwar
period. After World War II, the Japanese government implemented a planned
shipbuilding policy (“Keikaku Zousen" in Japanese) to compensate for major
losses of domestic ships during the war. Many entrants chose to enter the
shipping and shipbuilding market after Japan participated in the global ocean-
going shipping market. Japanese shipping firms enjoyed the fruit of high
demand for shipping and shipbuilding during the Korean War from 1950–1953 and
the Suez Crisis in 1956, but they depended heavily on planned shipbuilding and
government loans with a low interest rate. As a result, firms did not make an
effort to catch up to technological innovation and reduce their operational
costs, and their financial condition worsened in the 1950s.151515The detailed
history is as follows. The shipping industry recovered to its prewar level as
early as 1957 in terms of ship tonnage by implementing planned shipbuilding,
but most funding for construction still depended on loans. Although the
government had launched policies such as the interest subsidy system and
improved the financing conditions of the fiscal fund for planned shipbuilding,
these policies alone were insufficient to achieve satisfactory results.
Moreover, since this interest subsidy was suspended due to the temporarily
high profits of the Suez boom period, the difference in international
competitiveness quickly led to a deterioration in performance. After 1956,
when the economy recovered after the war, the shipping capacity required by
the Japanese economy continued to increase rapidly even during the shipping
recession after the summer of 1957, due to the rapid growth of the heavy
chemical industry. In December 1960, the Cabinet approved a plan to double the
shipping industry’s national income, which called for the construction of
13.35 million gross tons of ocean-going shipping capacity by the target year
of 1970. This tonnage was determined to be the minimum amount necessary to
bring Japan’s international balance of payments into the black by $200 million
in the target year.

To change the chronically inefficient industrial structure, the Japanese
government planned to implement the so-called “Two Laws for the Reconstruction
of the Shipping Industry" (“Kaiun Saiken Niho" in Japanese) in 1960 as one of
the industrial policy interventions in the Income-Doubling Plan announced
nationally that year.161616suzumura2012excess explained the institutional
details of the relationship between industrial policy interventions and the
excess entry theorem of suzumura1987res in major industries in postwar Japan.
The laws, which were proclaimed and enforced on July 1, 1963, are divided into
two parts.171717For reference, chida1990japanese, Section 5, stated: “the main
points of the new legislation can be summed up under three headings:  1. There
should be mergers between the major operators to form new companies, each
owning at least half a million d.w.t. The new firms, called “nucleus
companies” (Chukaku-tai), would then be expected to act as a kind of “parent”
company within the new groupings. 2. Each nucleus company would include other
operators and owners within its group so that its total capacity was over one
million tons d.w.t. 3. Government assistance of varying types: • The provision
of capital on favorable terms for building vessels under the Programmed
Shipbuilding Scheme. • Favorable terms under the Interest Subsidy Scheme for
new construction with capital raised from both public and city banks. • The
right to postpone over five years interest payments on outstanding past loans
received from the Development Bank of Japan. The city banks were expected to
adopt similar policies. The cash equivalent of these sums was to be used to
reduce their capital debt with respective financial institutions.”(p.141) Note
that the above tonnage is calculated based on ocean-going ships.

The first part was the Reconstruction Maintenance Law, which addressed
shipping industry’s reconstruction (“Saikenseibi" in Japanese) and was to take
effect for 5 years starting in April 1964. The Reconstruction Maintenance Law
set a goal and method for the rationalization of an industry. The objective of
the law was to resolve the underdepreciation of domestic firms in 5 years. The
government announced that firms were granted a 5-year moratorium period for
interest expenses related to shipbuilding if the firms satisfied the following
two requirements. The first requirement was that a merged group must own more
than 0.5 million tons of shipping tonnage of an ocean-going ship after
mergers. The second requirement was that the group whose central firm was the
merger firm must own more than 1 million tons of shipping tonnage of an ocean-
going ship, including that of affiliate and wholly controlled firms.

The second part was the Interest Subsidy Law, which provided the interest
subsidies. This was a revision of an earlier law that offered terms that were
more favorable. The Interest Subsidy Law extended the term of interest subsidy
for a development bank loan from 5 to 10 years and subsidized 1.0 % out of 5.0
% of its interest expenses. Furthermore, the law extended the term of interest
subsidies for a city bank loan to 7 years and subsidized 1.5 % out of 7.5 % of
its interest expenses. Hereafter, I refer to these two financial supports
simply as “subsidies" for the sake of simplicity.

To receive subsidies under the law, qualified merger firms submitted a merger
plan or extension petition by December 31, 1963. As a result, by May 31, 1964,
44 firms had submitted and completed merger plans, and 53 firms participated
in grouping without submitting a plan. This paper treats these firms as
merging and merged firms because the operation of the latter 53 firms’ ships
was shared within the group, so that merged firms and group-joining firms
become indistinguishable.181818I noticed that the number of firms
participating in mergers is slightly different across papers and data sources
because some firms retained their names after mergers. For example,
chida1990japanese showed that 88 ocean-going shipping firms formed coalitional
groups while keeping the name of the buyer firm. Strictly speaking, the market
consisted of 94 matched firms (six main buyer firms, six main seller firms, 27
affiliated firms, and 55 wholly controlled firms) and 31 unmatched firms after
dropping seven missing unmatched firms. This is consistent with my data, as
shown in Table 1.

In this study, I interpret the amount of two types of subsidies with
calculated real interest payments for the total cost of shipbuilding within 10
years for each firm as a single constant dummy variable indicating whether the
group qualified for the subsidy.191919Alternative specifications of the
subsidy are shown in Appendix C. Note that the firm’s decision about building
ships is dynamic (kalouptsidi2014aer, jeon2017learning, barwick2019china) and
merger decisions must be affected by the cost-side incentives of the
expectation of future cost savings and subsidies. However, based on the
institutional background of the law, it is reasonable to construct a static
matching model with complete information.

###  2.3 Assortativeness of size and technological specialization as proxies
for economies of scale and scope.

The anecdotal main drivers of mergers in the shipping industry are economies
of scale and scope (notteboom2004container). Theoretically, economies of scale
and scope are defined as cost efficiencies. Economies of scale exist when the
average production cost of a single product decreases with the number of units
produced, whereas economies of scope are cost-saving externalities between
product lines. Analogously, a merger between two firms will generate economies
of scale if, post-merger, the merged firm can provide the same quality at a
lower cost or higher quality at the same cost. This implies that even if a
merger decision is modeled as a single agent problem, the decision is based on
the expectation of post-merger cost efficiencies in all possible future paths.
In a matching environment like my data, the dynamic merger consideration is
more complicated. In addition, measuring post-merger cost efficiencies in
itself is challenging.202020As far as I know, jeziorski2014estimation is a
notable and exceptional study quantifying post-merger cost efficiencies with
assumptions on a firm’s sequential dynamic acquisition and repositioning
decisions. His model assumes that merger execution costs are zero, aside from
the merger transfer price based on the institutional background, so that he
can estimate a cost-efficiency parameter. My model takes a different approach
to explicitly quantify merger execution costs. From the different approach
focusing on indivisibility in shipping, holmes2018indivisibilities
investigated the relationship between consolidation and scale economies in
distribution.

Instead, I focus on how much the observed proxy variables of post-merger size
and technological specialization affect merger incentives as actual pre-merger
firms would expect before the enactment of the subsidy laws. Specifically,
quantifying the assortativeness of the proxy variables in a single matching
market is the main interest of this paper.

I specify the proxy variables as follows. The proxy variable for economies of
scale is measured by firm-level ship tonnage for each carrier type. Since
shipping firms gain profits from transportation and the shipping industry is
modeled as a homogeneous good or a competitive market, tonnage size reflects a
potential scale effect that buyer firms expect at the time of their merger
decision. The proxy variable for economies of scope is measured by the firm-
level ship tonnage share of each carrier type. This proxy captures firm-level
pre-merger technological specialization. If a firm with a low level of
specialization in liners merges with another firm with a low level of
specialization in liners, the merged group may gain cost efficiency through
technological diversification across carrier types because of technology
spillover. Similarly, if a firm with a high level of specialization in tankers
merges another firm with a high level of specialization in tankers, the merged
group may also achieve cost efficiency through technological specialization.
Thus, the technological share information is useful for pre-merger firms to
predict potential (dis-)economies of scope. In Section 4, I evaluate which
carrier types contribute to merger incentives.

###  2.4 Descriptive statistics

To understand the industry-level background, I begin by discussing group-level
information. Table 1 and Figure 1 summarize the final configuration of mergers
and consolidations.212121I found that Table 1, which was calculated from
syuuyaku and nostalgic and the corresponding final merger configuration shown
in chida1990japanese are somewhat different in terms of unknown aggregation,
different recorded dates, and some missing firms. For consistency, I continue
to use individual firm data collected from the syuuyaku and listgaikou to
compute aggregate measures. These indicate that all six groups satisfy the
subsidy criteria and that unmatched firms have a larger amount of total
tonnage than the Showa Line group, which would imply the existence of fixed
merger costs. Figure 1 also shows post-merger evidence of assortativeness of
size and specialization for some groups. For example, Nippon Yusen and
Mitsui-O.S.K Lines have balanced portfolios, whereas Showa Line specializes in
special shipping. However, it is hard to disentangle the relative importance
of post-merger size effects and technological specialization from group-level
information.

Table 1: Summary of the total tonnage size for each group.

| Firm type | Total tonnage | Num of firms
---|---|---|---
Nippon Yusen |  |  |
| (1) Main | 1.51 | 2
| (2) Affiliate | 0.84 | 7
| (3) Wholly controlled | 0.23 | 7
| Total | 2.58 | 16
Mitsui OSK Line |  |  |
| (1) Main | 1.92 | 2
| (2) Affiliate | 0.4 | 5
| (3) Wholly controlled | 0.55 | 20
| Total | 2.87 | 27
Japan Line |  |  |
| (1) Main | 1.12 | 2
| (2) Affiliate | 0.13 | 1
| (3) Wholly controlled | 0.08 | 4
| Total | 1.33 | 7
Kawasaki Kisen Kaisha |  |  |
| (1) Main | 1.66 | 2
| (2) Affiliate | 0.39 | 9
| (3) Wholly controlled | 0.29 | 7
| Total | 2.34 | 18
Yamashita Shinnihon Kisen |  |  |
| (1) Main | 0.9 | 2
| (2) Affiliate | 0.6 | 5
| (3) Wholly controlled | 0.17 | 10
| Total | 1.67 | 17
Showa Line |  |  |
| (1) Main | 0.55 | 2
| (2) Affiliate | 0.46 | 3
| (3) Wholly controlled | 0.03 | 4
| Total | 1.04 | 9
Unmatched |  |  |
| Unmatched | 1.13 | 24
Total |  |  |
|  | 12.96 | 118

  * a

Note: Source : listgaikou and nostalgic. After dropping missing ten firms, the
data contain matched 94 firms (12 main firms, 30 affiliates, and 52 wholly
controlled firms) and unmatched 24 firms chose to stay unmatched. Total
tonnage is measured by million (D/W) tons.

Figure 1: Configuration of tonnage for each carrier type.

  * a

Note: Source : listgaikou and nostalgic. Observation units: Total group-level
tonnage for each carrier type after mergers. The dotted horizontal line
indicates the subsidy threshold (1 million (D/W) tons).

Table 2: Summary statistics for independent variables.

| N | mean | sd | min | q25 | q50 | q75 | max
---|---|---|---|---|---|---|---|---
Size variables (million (D/W) tons) |  |  |  |  |  |  |  |
total tonnage size | $118$ | 0.110 | 0.207 | 0.000 | 0.011 | 0.026 | 0.091 | 1.023
total tonnage size of liner | $118$ | 0.031 | 0.112 | 0.000 | 0.000 | 0.000 | 0.000 | 0.721
total tonnage size of special | $118$ | 0.015 | 0.033 | 0.000 | 0.000 | 0.000 | 0.008 | 0.177
total tonnage size of tanker | $118$ | 0.030 | 0.071 | 0.000 | 0.000 | 0.000 | 0.024 | 0.417
total tonnage size of tramper | $118$ | 0.035 | 0.055 | 0.000 | 0.002 | 0.013 | 0.034 | 0.246
Specialization variables (% share) |  |  |  |  |  |  |  |
share of liner | $118$ | 0.104 | 0.235 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
share of special | $118$ | 0.117 | 0.244 | 0.000 | 0.000 | 0.000 | 0.088 | 1.000
share of tanker | $118$ | 0.192 | 0.351 | 0.000 | 0.000 | 0.000 | 0.208 | 1.000
share of tramper | $118$ | 0.587 | 0.417 | 0.000 | 0.135 | 0.705 | 1.000 | 1.000
HHI based on carrier types | $118$ | 0.815 | 0.241 | 0.258 | 0.584 | 1.000 | 1.000 | 1.000

  * a

Note: Source : listgaikou and nostalgic. Note that the fact that minimum of
some size variable equal to zero implies that the tonnage size is less than
1,000 (D/W) tons. Total tonnage is measured by (D/W) tons.

To investigate firm-level incentives, I focus on firm-level information. Table
2 shows the summary statistics for the independent variables from all 118
samples. Note that I observe the names of 128 firms but cannot collect
variables of 10 firms, so I drop these samples. First, the tonnage of almost
all the firms is less than 1 million (D/W) tons, so almost all pre-merger
firms have an incentive to be matched with other firms to satisfy the subsidy
criteria. Second, the Herfindahl-Hirschman Index (HHI) based on the relative
composition of carrier types for each firm is 0.815 on average.222222The
Herfindahl-Hirschman Index (HHI) for firm $i$ is calculated by
$HHI_{i}=\sum_{k\in K}(s_{ik})^{2}.$ where
$s_{ik}=\frac{\text{tonnage}_{ik}}{\sum_{k\in K}\text{tonnage}_{ik}}$ and
$K=\\{Liner,Special,Tramper,Tanker\\}$. The index captures the concentration
level as a single index measure. This is defined formally in Section A.3.
This implies that each firm specialized in one or two shipping categories. For
example, the medians of the firm-level tonnage of liners, special vessels, and
tankers are zero, which means that more than half of all firms did not have
these types of ships. This evidence supports the idea that merger incentives
might depend on some measure of post-merger specialization.

##  3 Model

###  3.1 Setup

I construct a one-sided one-to-many matching model with complementarities by
extending azevedo2018existence and fox2019externality. I consider a single
large market before and after a subsidy. For convenience, I denote the former
pre-merger market by $t=1$ and the latter, post-merger market by $t=2$. Firms
$i=1,\cdots,N$ exist at $t=1$. For notational convenience, I use type $i$ firm
in a continuum market and individual firm $i$ in a finite market
interchangeably.232323The formal discussions are postponed in Section 3.2
after introducing the definitions and assumptions. Let $\mathcal{N}$ be the
set of firms at $t=1$. Let $\mathcal{J}$ be a set of all possible coalitions,
that is, $\mathcal{J}=\mathcal{P}(\mathcal{N})$ which is the power set of
$(\mathcal{N})$. Let $\widehat{J}\in\mathcal{J}$ be a set of realized
coalitions. Let $x_{i}$ be a vector of continuous or discrete observable
characteristics for each firm $i$ at $t=1$. Suppose that firm $i$ acquires
coalition $J\in\mathcal{J}$. Let a finite set of rival’s coalitions be denoted
as $\widehat{J}_{-J}$ where $J\cup\widehat{J}_{-J}=\widehat{J}$. I specify a
vector $X$ of observables in post-merger valuation in the market at $t=2$ as a
function of pre-merger observables $x=(x_{1},\cdots,x_{N})$,

$\displaystyle X((x_{j})_{j\in J},\widehat{J}_{-J}),$

and $X_{i}((x_{j})_{j\in J},\widehat{J}_{-J})$ with subscript $i$ indicates
observable variables related to firm $i$. This implies that the post-merger
valuation is determined by the own coalition and all rival’s coalitions.

For simplicity, I assume that the “main" firm $i$ in coalition $J$ is an
acquirer. The “main" firm, introduced in Section 2, has the right of capital
control over the group. I specify the post-merger valuation for coalition $J$
with acquirer $i$ conditional on realized coalitions $\widehat{J}$ as a linear
form as follows:

$\displaystyle F_{i,J}((x_{j})_{j\in
J},\widehat{J}_{-J})+\epsilon_{i,J}=X_{i}((x_{j})_{j\in
J},\widehat{J}_{-J})^{\prime}\theta+\epsilon_{i,J},$

where $F_{i,J}((x_{j})_{j\in J},\widehat{J}_{-J})$ is the production function
from coalition matching $(i,J)$, and $\epsilon_{i,J}$ is an unobservable
valuation shock to acquirer $i$ for coalition $J$. Note that $(x_{j})_{j\in
J}$ includes observed characteristics of firm $i$ belonging to coalition $J$.
The post-merger valuation is interpreted as a payoff that is shared with firm
$i$ and the members of coalition $J$.

Post-merger profit for coalition $J$ with acquirer $i$ is defined as follows:

$\displaystyle F_{i,J}((x_{j})_{j\in
J},\widehat{J}_{-J})+\epsilon_{i,J}-\sum_{j\in J\backslash\\{i\\}}p_{j},$

where $p_{j}$ is the price for target firm $j\in J\backslash\\{i\\}$. For
target firm $j$, the equilibrium acquisition price $p_{j}=p_{ij}$ equals the
shadow price on the constraint that target firm $j$ may be acquired by at most
one acquirer.242424The appendix of akkus2015ms considers a similar approach
for merger equilibrium price $p_{j}$. The equilibrium price can be modeled as
a bargaining outcome of buyer firm $i$ and target firms within coalition $J$.
Since this modeling requires merger price information for both realized and
non-realized mergers, this article does not take this approach. Acquirers
demand coalition $J$ to maximize the post-merger profits. To satisfy the
sufficient conditions of fox2018qe for (set-)identification and rank order
property, we assume the following theoretical assumption.

###### Assumption 1

(1) The observations $(i,J)$ are independent and identically distributed
(i.i.d.). (2) The term $\epsilon_{i,J}$ has full support in
$\mathbb{R}^{\text{dim}(\epsilon_{i,J})}$. (3) The term $\epsilon_{i,J}$ has
an exchangeable distribution for each type $i$.252525Denote
$\epsilon=\\{\epsilon_{i,J}\\}_{i\in\mathcal{N},J\in\mathcal{J}}$. Let $\rho$
be a permutation of the elements of $\epsilon.$ An exchangeable distribution
satisfies $F\left(\epsilon\mid i,J\right)=F\left(\rho\left(\epsilon\right)\mid
i,J\right)$ for all such permutations $\rho.$ This is equivalent to Assumption
2 (iii) in fox2018qe when the dimension of unobservable type, $k=1$. (4) The
parameter space of $\theta$ is compact. (5) The cumulative distribution
function of unobservable valuation shock conditional of acquirer $i$ for
coalition $J$, $F\left(\epsilon_{i,J}|i,J\right)$ has bounded, continuous
derivatives.

The equilibrium concept of fox2019externality is a competitive equilibrium
where a price vector with no excess demand for merger target firms exists with
externalities. Their full model incorporates strategic interactions across
coalitions and allows multiplicity of equilibria. Instead, I impose the
following assumption to exclude the possibility of strategic decisions.

###### Assumption 2

The post-merger valuation for coalition $J\in\mathcal{J}$ depends on observed
characteristics $(x_{j})_{j\in J}$ and unobserved characteristics
$\epsilon_{i,J}$, but not on characteristics of rival’s coalition
$\widehat{J}_{-J}$.

Assumption 2 excludes the possibility of strategic decisions of all firms to
participate in coalitional groups, like a static entry game. Although
fox2019externality and uetake2019entry incorporated the strategic decisions
into the TU and NTU matching models in different ways, the externality issues
are beyond the state of the methodological literature.262626As discussed in
agarwal2021market, the condition of the existence of the equilibrium with
externality is an active area of theoretical research. The conditions of the
uniqueness of the equilibrium are not investigated. Also, a global market for
each carrier type is characterized differently so that incorporating post-
merger competition in the model is difficult in a unified way. For example,
tramper bulk carriers operate like taxi drivers, whereas container carriers
operate like bus drivers employed by large firms and groups. Assumption 2
allows me to drop $\widehat{J}_{-J}$ and reformulate the post-merger profit
into the following:

$\displaystyle F_{i,J}((x_{j})_{j\in J})+\epsilon_{i,J}-\sum_{j\in
J\backslash\\{i\\}}p_{j}=X_{i}((x_{j})_{j\in
J})^{\prime}\theta+\epsilon_{i,J}-\sum_{j\in J\backslash\\{i\\}}p_{j},$

where production function $F_{i,J}$ depends only on the pre-merger
characteristics of firm $i$ and coalition $J$. Additionally, Assumption 2
allows me to implement tractable counterfactual simulations in Section 6.

###  3.2 Equilibrium Concept

For convenience, I introduce the formal definition of competitive equilibrium
with a continuum of agents according to Section 6 of azevedo2018existence. An
allocation is a measurable map
$A:\mathcal{N}\rightarrow\Delta(\mathcal{P}(\Omega))$ specifying for each type
$i\in\mathcal{N}$ a distribution $A_{i}$ over the power set
$\mathcal{P}(\cdot)$ of trade bundles $\Omega$. It allows type $i$ agent to be
a single firm. Let $\eta_{i}$ be a measure of type $i$ firm. The element of
$\Omega$ is called a merger bundle in my setting. Let $\mathcal{A}$ be the
space of allocations. A price vector $p\in\mathbb{R}^{\Omega}$ assigns a price
$p_{\omega}$ for each merger bundle $\omega\in\Omega$. Given a vector of
prices $p$, define expenditure as the vector
$e_{p}\in\mathbb{R}^{\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)}$ such that
for $\Phi\subseteq\Omega$ and $\Psi\subseteq\Omega,$

$e_{p}(\Phi,\Psi)=\sum_{\varphi\in\Phi}p_{\varphi}-\sum_{\psi\in\Psi}p_{\psi},$

where $\Phi$ (resp. $\Psi$) is the set of merger bundles in which agents
participate as a buyer (resp., a seller). That is, $e_{p}(\Phi,\Psi)$ is the
net transfer paid by an agent buying $\Phi$ and selling $\Psi.$ The utility of
type $i$ agent who buys the set of merger bundles $\Phi\subseteq\Omega$ and
sells the set of merger bundles $\Psi\subseteq\Omega$ at prices $p$ is given
as follows:

$u^{i}(\Phi,\Psi)-e_{p}(\Phi,\Psi),$

where $u^{i}(\Phi,\Psi)$ is the gain from buying the set of merger bundles
$\Phi$ and selling the set of merger bundles $\Psi$.

Given allocation $A\in\mathcal{A}$, define the excess demand for each
$i\in\mathcal{N}$ and for each merger bundle $\omega\in\Omega$ as follows:

$\displaystyle
Z_{\omega}^{i}(A)\equiv\sum_{\\{\omega\\}\subseteq\Phi\subseteq\Omega}A^{i}(\Phi)-\sum_{\\{\omega\\}\subseteq\Psi\subseteq\Omega}A^{i}(\Phi),$

Define the excess demand for each merger bundle $\omega\in\Omega$ for the
entire economy as

$\displaystyle Z_{\omega}(A)\equiv\int Z_{\omega}^{i}(A)\mathrm{d}\eta,$

where $\eta$ is the distribution of the population of full type agents. I
introduce competitive equilibrium as a solution concept.

###### Definition 1

An arrangement $[A;p]$ is a competitive equilibrium if it satisfies two
conditions:

  1. 1.

Incentive Compatibility: Each agent obtains an optimal bundle given prices $p$
for all $i\in\mathcal{N},A^{i}(\Phi,\Psi)>0$ only if

$\displaystyle(\Phi,\Psi)\in\underset{(\tilde{\Phi},\tilde{\Psi})\in\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)}{\arg\max}u^{i}(\tilde{\Phi},\tilde{\Psi})-e_{p}(\tilde{\Phi},\tilde{\Psi}).$

  2. 2.

Feasibility: $Z_{\omega}(A)=0$ for all $\omega\in\Omega$.

azevedo2018existence prove the existence of unique competitive equilibrium in
a regular economy with a continuum of agents with a finite set of
trades.272727An economy is regular if (1) The integral of absolute values of
utility is finite, as long as the agents are not given bundles for which they
have a sufficiently negative utility. That is,
$\int_{I}\max_{\Phi,\Psi\subseteq\Omega,u^{i}(\Phi,\Psi)\neq-\infty}\left|u^{i}(\Phi,\Psi)\right|\mathrm{d}\eta<\infty$
and (2) Agents can supply any sufficiently small net demand for trades. That
is, $\left\\{\mathrm{Y}\in\mathbb{R}^{\Omega}:\exists A\in\mathcal{A}\right.$
such that $\mathrm{Y}=\mathrm{Z}(A)$ and
$\left.(u^{i}(\Phi,\Psi)=-\infty\text{, then ,}A^{i}(\Phi,\Psi)=0)\right\\}$
contains the neighborhood of 0. The competitive equilibrium is unique if the
distribution of full types $\eta$ ensures the uniqueness of the maximizer of
the social welfare function. Also, every competitive equilibrium is efficient.
A regular economy where measure $\eta$ has full support has a unique vector of
competitive equilibrium prices. fox2017specifying discusses the difference in
the data between a truly finite market and a continuum market. Following his
reasoning, I assume that the observed single large matching market in the data
is a finite subset of a continuum of firms and target firm
bundles.282828fox2017specifying clearly states that working with a continuum
of agents is a way of stating an approximation while ensuring the existence of
the equilibrium. The assumption on a continuum market is imposed such that the
aggregate matching outcome is a function of the distribution of
$\epsilon_{i,J}$ for all $i$ and $J$ but not specific realizations of
$\epsilon_{i,J}$. Thus, the market outcome is econometrically deterministic in
the aggregate. This is a conceptual assumption for interpreting the matching
market(s) in the data.

These theoretical results allow us to find competitive equilibrium
conceptually by discrete assignment linear programming formulation introduced
by shapley1971assignment. However, the competitive equilibrium above allows
each agent to be a seller and buyer over any possible trades in a many-to-many
matching environment so that the dimension of $|\Omega|\times|\Omega|$ goes to
the size of universe for even small $N$. To overcome the numerical
impossibility while applying the fundamental theorems of azevedo2018existence
to my one-sided one-to-many coalitional matching model, I assume Assumption 3.

###### Assumption 3

$\mathcal{J}=\mathcal{P}(\mathcal{N})=\Phi=\Psi$, i.e., the both sets of
trades are summarized as the power set of firms.

Assumption 3 means that the sets of trades as a seller and buyer are
indistinguishable and summarized as the set of possible coalitions
$\mathcal{J}$. Also, it explicitly specifies the set of trades $\Omega$ as a
set of all possible merger combinations, not arbitrary trades. Assumption 3 is
innocuous in practical merger situations. The assumption allows us to
represent the set of merger bundle $\Omega$ as a list of $N$-dimensional
binary digits in the power set $\mathcal{P}(\mathcal{N})$. Also, the power set
of $\Omega$ denoted by $\mathcal{P}(\Omega)$ is reduced to the set with a
tractable size of dimensions. To understand how the model works, I introduce a
simple toy model.

Example 1: Suppose $N=3$, i.e., three firms exist in a market. Each firm can
be a buyer and a seller. Then, each firm faces the set of merger bundles
described as

$\displaystyle\mathcal{P}(\Omega)=\\{(000),(100),(010),(001),(110),(101),(011),(111)\\},$

where the element $(ijk)$ captures whether firm $i$,$j$, and $k$ are included
in the merger bundle. Concretely, merger bundle $(000)$ means a null merger
bundle where no firm participates in the merger (i.e., all firms stay
unmatched), merger bundle $(100)$ means the merger bundle where only firm 1
participates in the merger, and merger bundle $(110)$ means the merger bundle
where firm 1 and 2 participate in the merger. Then, the dimension of
$\mathcal{P}(\Omega)$ for each firm is $|\mathcal{P}(\mathcal{N})|=2^{N}=8$.

Under the assumptions, I define a competitive equilibrium in the coalitional
merger matching game as a simple version of Definition 1.

###### Definition 2

Under Assumptions 1, 2, and 3, an allocation is a measurable map
$A:\mathcal{N}\rightarrow\Delta(\mathcal{J})$ specifying for each type
$i\in\mathcal{N}$ a distribution $A_{i}$ over the power set of merger bundles
$\mathcal{J}$, denoted by $\\{A_{i,J}\\}_{J\in\mathcal{J}}$. Suppose that the
observed matching market comes from a continuum of matching market. I describe
the full distribution as $A=\\{A_{i,J}\\}_{i\in\mathcal{N},J\in\mathcal{J}}$.
A merger configuration $[A;p]$ is a competitive equilibrium if it satisfies
two conditions:

  1. 1.

Incentive Compatibility: Each firm obtains an optimal merger bundle $J$ given
prices $p$ for all $i\in\mathcal{N},A_{i,J}>0$ only if, for all
$i\in\mathcal{N}$,

$\displaystyle J\in\underset{J^{\prime}\in\mathcal{J}}{\arg\max}\quad
F_{i,J^{\prime}}((x_{j})_{j\in J^{\prime}})+\epsilon_{i,J^{\prime}}-\sum_{j\in
J^{\prime}\backslash\\{i\\}}p_{j}.$

  2. 2.

Feasibility: $Z_{J}(A)=0$ for all $J\in\mathcal{J}$.

The competitive equilibrium allocation can be computed via the following
constrained linear programming problem.

$\displaystyle\max_{\\{A_{i,J}\\}_{i\in N,J\in\mathcal{J}}}$
$\displaystyle\sum_{i\in\mathcal{N}}\sum_{J\in\mathcal{J}}A_{i,J}U_{i,J}$ (1)
$\displaystyle\text{s.t. }U_{i,J}$ $\displaystyle=F_{i,J}((x_{j})_{j\in
J})+\epsilon_{i,J},$ $\displaystyle A_{i,J}$
$\displaystyle\in\mathbb{R}\quad\forall i\in\mathcal{N},J\in\mathcal{J},$
$\displaystyle 0\leq A_{i,J}$ $\displaystyle\leq\eta_{i}\quad\forall
i\in\mathcal{N},J\in\mathcal{J},$ $\displaystyle\sum_{J\in\mathcal{J}}A_{i,J}$
$\displaystyle=\eta_{i}\quad\forall i\in\mathcal{N},$ $\displaystyle Z_{J}(A)$
$\displaystyle=0\quad\forall J\in\mathcal{J},$

where the objective function is called a social welfare function, $\eta_{i}$
is the total population measure over type $i$ firm, and allocation $A_{i,J}$
is allowed to be real-valued. In my empirical application, I assume
$\eta_{i}=1$ for all $i\in\mathcal{N}$ because I interpret the type as firm’s
identity.292929azevedo2018existence use a finite number of trades and a
continuum of full agent types. In my empirical application, the number of
observable types of firm $i\in\mathcal{N}$ is also finite. I interpret that in
mapping the continuum model to the finite data, there is a continuum of firms
of Nippon Yusen’s observable type, but I observe only one such firm in the
data. Note that the term $\sum_{j\in J\backslash\\{i\\}}p_{j}$ is cancelled
out to calculate joint production function $U_{i,J}$ for firm $i$ and
coalition $J$. Except the constraints $Z_{J}(A)=0$, the problem is the natural
extension of the classical assignment problem of shapley1971assignment.

###  3.3 Specification of feasibility constraint $Z_{J}(A)=0$

To compute competitive equilibrium, I need to explicitly specify the
feasibility constraints $Z_{J}(A)=0$ for all $J\in\mathcal{J}$. Recall that
the merger bundle $J\in\mathcal{J}$ identifies participants in the merger
bundle $J$ by Assumption 3, so I can assign the roles such as buyers and
sellers in a merger market to all participants. For assignment $A_{i,J}$ for
all $i\in\mathcal{N},J\in\mathcal{J}$, type $i$ firm is a seller if merger
bundle $J\in\mathcal{J}$ includes only $i$, i.e., only $i$-th element of $J$
is one and is zero otherwise. I denote the set of merger bundles by
$J_{i}^{s}$ as type $i$ seller firm’s bundle. On the other hand, type $i$ firm
is a buyer if merger bundle $J\in\mathcal{J}$ includes at least $j$-th ($j\neq
i$) element of $J$, where the $j$-th element is one. I denote the set of
merger bundles by $J_{i}^{b}$ as type $i$ buyer firm’s bundle. For assignment
$A_{i,J}$ where $J$ consists of only zeros, type $i$ firm chooses to stay
autarky. These specifications of assignment $A$ capture all possible merger
configurations that are mutually exclusive.

Under the specifications, the feasibility constraints $Z_{J}(A)=0$ for all
$J\in\mathcal{J}$ in Definition 2 are specified as

$\displaystyle A_{i,J_{i}^{s}}$ $\displaystyle=\sum_{i^{\prime}\neq
i,i^{\prime}\in\mathcal{N}}\sum_{J^{\prime}\in
J_{i^{\prime}}^{b}}A_{i^{\prime},J^{\prime}}\cdot\mathbbm{1}(i\text{-th
element of }J^{\prime}\text{ is }1),\forall i\in\mathcal{N},$

where $\mathbbm{1}(\cdot)$ is the indicator function which equals to 1 if
$(\cdot)$ is satisfied and 0 otherwise, the left-hand side can be interpreted
as a supply side for merger bundle $J$, and the right-hand side can be
interpreted as a demand side for merger bundle $J$. Example 2 illustrates the
simple version of $Z_{J}(A)=0$.

Example 2: As Example 1, suppose $N=3$, i.e., three types of firms exist in a
market. Each type of firm can be a buyer and seller. Then, each type of firm
faces the set of merger bundles described as

$\displaystyle\mathcal{J}=\\{(000),(100),(010),(001),(110),(101),(011),(111)\\},$

where the element $(ijk)$ captures whether type $i$, $j$, and $k$ firms are
included in the merger bundle. Then, assignment
$\\{A_{i,J}\\}_{i\in\mathcal{N},J\in\mathcal{J}}$ is expressed as $3\times
2^{3}$-dimensional matrix. For example, $A_{1,(000)}$ means that type 1 firm
joins null merger bundle $(000)$, i.e., the firm stays autarky. $A_{1,(100)}$
means that type 1 firm joins merger bundle $(100)$, i.e., type 1 firm sells
type 1 firm himself. $A_{1,(011)}$ means that type 1 firm joins merger bundle
$(011)$, i.e., type 1 firm buys type 2 and 3 firms.303030Some merger bundles
seem unreal for some firms. For example, suppose type 1 firm joins merger
bundle $(101)$ which means that the type 1 firm sells the type 1 firm and
acquires a type 3 firm. Such an unreal case can be eliminated innocuously by
assigning sufficiently large negative values to these assignments. Here, I
allow the possibility for completeness of the model description. Note that the
type of firm can be interpreted as an individual firm, but the expression
allows for non-integer and probabilistic allocation on $A$. Then, all
feasibility constraints $Z_{J}(A)=0$ for all $J\in\mathcal{J}$ can be
described as

$\displaystyle A_{1,(100)}$
$\displaystyle=A_{2,(100)}+A_{2,(110)}+A_{2,(101)}+A_{2,(111)}+A_{3,(100)}+A_{3,(110)}+A_{3,(101)}+A_{3,(111)}$
$\displaystyle A_{2,(010)}$
$\displaystyle=A_{1,(010)}+A_{1,(110)}+A_{1,(011)}+A_{1,(111)}+A_{3,(010)}+A_{3,(110)}+A_{3,(011)}+A_{3,(111)}$
$\displaystyle A_{3,(001)}$
$\displaystyle=A_{1,(001)}+A_{1,(101)}+A_{1,(011)}+A_{1,(111)}+A_{2,(001)}+A_{2,(101)}+A_{2,(011)}+A_{2,(111)},$

where left-hand sides of the equalities capture the supply side for each firm
and right-hand sides capture the corresponding demand side.

Under the specification of $Z_{J}(A)=0$, I can calculate competitive
equilibrium numerically by solving (1) via linear programming under
equilibrium constraints.

###  3.4 Specification of production function $F_{i,J}$

The specification issues about production function $F_{i,J}((x_{j})_{j\in
J})$, merger cost $C_{i,J}$, and subsidy $s_{i,J}$ remain. These
specifications are highly dependent on the data and the researcher’s interest,
although the specification of $Z_{J}(A)=0$ is a common simple trick on any
one-sided one-to-many matching market. In Sections 3.4, 3.5, and 3.6, I
specify $F_{i,J}((x_{j})_{j\in J})$, $C_{i,J}$, and $s_{i,J}$ to reflect the
matching environment in my data.

To make it consistent with the estimation steps, I specify
$F_{i,J}((x_{j})_{j\in J})$ as

$\displaystyle F_{i,J}((x_{j})_{j\in J})$ $\displaystyle=X_{i}((x_{j})_{j\in
J})^{\prime}\theta$ $\displaystyle=\beta(x_{i}\circ x_{J})+\delta
s_{i,J}-C_{i,J}(\gamma),$ (2)

where $X_{i}((x_{j})_{j\in J})^{\prime}\theta$ is a linear single index of
interactions of observed characteristics of firm $i$ and all coalition members
$j\in J$ with composite parameter $\theta=[\beta,\delta,\gamma]$ which is a
vector of parameters for $X_{i}((x_{j})_{j\in J})$ and subsidy and cost terms,
$s_{i,J}$ is the subsidy term, and $C_{i,J}(\gamma)$ is the merger cost term
explicitly dependent on parameter $\gamma$ in a flexible way. Following the
literature, I incorporate the match-specific interaction term $(x_{i}\circ
x_{J})$ via element-wise product of observed characteristics of firm $i$ and
coalition $J$ and the dimension of $\beta$ corresponds to that of the
interaction term. The vector of observed characteristics $x_{i}$ includes firm
$i$’s size and share of each carrier type. The vector of $x_{J}$ includes the
corresponding observed characteristics of coalition $J$ which summarizes
$x_{i}$ for all $i\in J$.313131In this specification, I can interpret that the
production function captures some reduced-form features about ex-post
competition under the transportation demand of downstream firms. The post-
merger size effects and technological specialization affect quantity
competition dynamically in not only the domestic but also the global shipping
market, as in jeon2017learning. These points are beyond the scope of this
article. If firm $i$ stays unmatched, I assume that
$F_{i,\emptyset}((x_{j})_{j\in\emptyset})=\beta(x_{i}\circ x_{i})$ which means
that the payoff of being unmatched equals to that of matching with a
hypothetical firm which has the same composition of carriers.323232In the
model, an opt-out action corresponds to the action of selling firms to some
buyer and exiting the market, not an unmatched action that incurs operating
costs. So, normalizing an unmatched payoff to zero is inappropriate. If firm
$i$ sells itself to some buyer firm, I assume that
$F_{i,\\{i\\}}((x_{j})_{j\in\\{i\\}})=0$ as the payoff of an outside option.

###  3.5 Specification of merger cost $C_{i,J}(\gamma)$

Both $s_{i,J}$ and $C_{i,J}(\gamma)$ are important terms for eliminating the
possibility of a grand coalition (i.e., in a grand coalition, all firms form a
single coalitional group). First, as explained in Section 2, no merger
occurred before the laws. This implies that if $s_{i,J}=0$ for all $i$ and
$J$, no merger occurred. Since I observe a single large matching market before
and after mergers, the subsidy term is critical to tilt the playing field
before and after the enactment of subsidy laws.

Second, the estimation of fixed costs such as merger costs is critical for
policy implications. However, researchers do not observe the proxy variables
directly.333333The literature on a static and dynamic entry game provides the
point-estimate or bounds of fixed costs regarding a firm’s entry decision in
the market. The main difference between merger costs in a matching merger
market setting and fixed costs in an entry game setting is that merger costs
must be determined by a comparison of the payoffs of being unmatched, selling
the firm, and buying a single or multiple firms through feasibility condition
$Z_{J}(A)=0$. In particular, few papers estimate endogenous merger costs.
igami2019mergers treat the merger cost as a single common constant parameter
independent of any observed characteristics, which is incurred only when a
firm chooses a merger decision. In the shipping industry allowing a one-to-
many matching, this specification is not adequate because the cost of
acquiring a single firm is expected to be smaller than that of acquiring many
firms.

To reflect the above features, I specify $C_{i,J}(\gamma)$, a merger cost that
depends on the number of target firms as follows:

$\displaystyle
C_{i,J}(\gamma)=\begin{cases}0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{
if }J\in\\{\emptyset,i\\},\\\ \gamma\cdot[\text{the number of firms in
coalition }J\backslash i]\quad\text{ otherwise },\end{cases}$

where $\gamma$ is a cost parameter that captures merger costs for adding one
firm to the coalition $J$, and firm $i$ can remain as an autarky with no
merger costs. The case of $J\in\\{\emptyset,i\\}$ means that firm $i$ is an
autarky or selling itself. This specification imposes the condition that
$C_{i,J}(\gamma)$ is linearly increasing in the total number of acquired firms
in the coalition. The linearity of $C_{i,J}(\gamma)$ might be inadequate
because the merger costs of acquiring two firms might not be approximately
twice the cost of acquiring a single firm. However, this is unknown to
researchers. Thus, I follow and extend the linearly additive separable merger
cost of acquiring a single firm in gowrisankaran1999dynamic.

###  3.6 Specification of subsidy $s_{i,J}$

The institutional background helps us to specify the subsidy term $s_{i,J}$
explicitly.343434The model has the same spirit of kawai2011wp and weese2015qe.
kawai2011wp considered the policy implications of designing procurement
auctions, that may yield social gains by inducing investment by firms. He
provided empirical evidence on how different subsidy mechanisms affected the
incentives of agents to invest using data from plastic recycling auctions in
Japan. The main difference between kawai2011wp and the present paper is the
subsidy requirement. His paper specifies the incentive of firms to invest
individually. Conversely, my paper specifies the incentive of acquirers to
match targets considering total tonnage. The subsidy plan depends on the total
shipping tonnage of an ocean-going ship owned by coalition $J$ so that
$s_{i,J}$ is specified as

$\displaystyle s_{i,J}=\begin{cases}M_{i,J}\quad\quad\quad\mbox{ if
}[\sum_{i\in J}\text{tonnage}_{i}>1.0\text{(million tons)}]\\\
0\quad\quad\quad\mbox{ otherwise }\end{cases},$

where $M_{i,J}$ denotes the amount of subsidy and the subsidy threshold level
(1 million tons) is determined by the government. The key difference from the
standard one-to-one matching model is that positive amounts in the joint
production function are shared by all members of coalition $J$ including
$i$.353535If I reflect the real subsidy criteria, $s_{i,J}$ is written as
$\displaystyle s_{i,J}=\begin{cases}M_{i,J}\mbox{ if }[\sum_{i\in
J}\text{tonnage}_{i}>1.000.000]\wedge[\sum_{i\in J,i\text{ is the main
firm}}\text{tonnage}_{i}>500.000]\\\ 0\mbox{ otherwise }\end{cases}.$ However,
under Assumption 5, only 12 main firms can become buyers and Table 1 implies
that the second subsidy condition is satisfied for almost all merger scenarios
if the first subsidy condition is satisfied. Thus, I take the simplification
of $s_{i,J}$ for tractability. Appendix C shows the alternative subsidy
specifications in simulated data.

Importantly, the subsidy amount $M_{i,J}$ was not fixed for all firms because
$M_{i,J}$ was dependent on future loans for each group through interest
subsidy.363636The details of the subsidy amounts are as follows. The period of
the interest subsidy is extended from 5 to 10 years for commercial bank loans
and from 5 to 7 years for municipal loans, and the interest rate of the
interest subsidy is decreased from 5 years for commercial bank loans to 4
years for shipowners. Under rational expectation assumption, all firms
anticipated all future paths of all possible matchings and investment games,
which is infeasible to incorporate in the model. Also, under the data
limitation about $M_{i,J}$ for all possible matching pairs $(i,J)$, I assume
Assumption 4.

###### Assumption 4

Subsidy amount $M_{i,J}$ is fixed, observed by all firms ex-ante, and
specified as

$M_{i,J}=M,$

where $M$ is a positive constant.

Note that, in general, $M_{i,J}$ could be flexibly dependent on the observed
characteristics such as the capital size of all possible matching pairs
$(i,J)$ if the calculation formula and data were available. However, the
subsidy in my data is the interest subsidy involving the future investment
cost and benefits. Imposing the ex-ante dependency of the subsidy amount on
the observed characteristics will not be innocuous, and the current model
could not estimate potential correlation of the unknown subsidy amount with
the size of the group. Instead, I focus on the qualification of the subsidy
under Assumption 4 which is restrictive but the most innocuous. Practically, I
assume that $M=1$ so that the subsidy effect term in specification (2) is
measured by $\delta$.373737Another specification of the subsidy term is shown
in Appendix C.  Under Assumption 4, we interpret $\delta$, the coefficient of
$M_{i,J}$, as the sensitivity to subsidy qualification. By shifting the level
of $M$ from one given a fixed $\delta$, I can simulate the counterfactual
scenarios.

Under Assumptions 1, 2, 3, and 4, I can characterize the competitive
equilibrium fully and compute it numerically.

##  4 Estimation

###  4.1 Matching maximum rank estimator

The estimation procedure is based on fox2018qe and fox2019externality. I have
the data in a single, large market before and after a subsidy, so I need to
use a matching maximum rank estimator instead of a matching maximum score
estimator. In this approach, pairwise stability matchings construct
inequalities of incentive compatibility conditions for obtaining consistent
estimators, even though researchers do not have transfer data. Recall that
incentive compatibility for a type $i$ firm given prices $p$ is described as
follows:

$\displaystyle J\in\underset{J^{\prime}\in\mathcal{J}}{\arg\max}\quad
F_{i,J^{\prime}}((x_{j})_{j\in J^{\prime}})+\epsilon_{i,J^{\prime}}-\sum_{j\in
J^{\prime}\backslash\\{i\\}}p_{j}.$

I can construct the following five types of inequalities corresponding to
incentive compatibility as follows:

  1. 1.

inequalities from two observed coalitions;

  2. 2.

inequalities from one observed coalition;

  3. 3.

inequalities from one unmatched target;

  4. 4.

inequalities from for the individual rationality (IR) conditions of unmatched
agents; and

  5. 5.

Inequalities from for the IR conditions with and without subsidies.

The last two inequalities are derived from the special features of my data. I
explain how the inequalities are constructed in Appendix B.1. 383838To hold
these inequalities true, I introduce Assumption 6 known as the rank order
conditions (fox2010qe).

I estimate and simulate using all or a sub-sample of the data containing at
most 118 firms. If all firms can be either buyers, sellers, or unmatched
firms, I need to construct inequalities for all possible combinations of
$\mathcal{N}=\\{1,\cdots,118\\}$ and $|\mathcal{J}|=2^{118}$, which is
infeasible to compute an equilibrium even once. To make the model tractable in
the estimation step, I additionally impose Assumption 5 for eliminating
unreasonable merger scenarios.

###### Assumption 5

Only the main firms defined in Section 2 are eligible to be buyer firms. In
other words, affiliated or wholly controlled firms that arise after the merger
cannot act as buyer firms.

Note that I require Assumption 5 only for estimation with the full sample.
Although Assumption 5 seems natural since small and medium-sized firms could
not acquire much larger and many firms in general due to financial
constraints, Assumption 5 eliminates the possibility that many small and
medium-sized firms could form a single coalitional group with tonnage that
would be comparable to that of the six groups. Assumption 5 also ignores the
possibility of the profitable deviation of affiliated and wholly controlled
firms from belonging to a coalition to joining the other coalitions or staying
unmatched.

Based on these assumptions, I construct the objective function as follows:

$\displaystyle Q(\theta)=\sum_{g\in G_{N}}1\left[Z_{g}^{\prime}\theta\geq
0\right],$ (3)

where $G_{N}$ is the set of observed inequalities and $Z_{g}^{\prime}\theta$
is the corresponding inequality of matching pairs $g$. The set of maximizers
of $Q(\theta)$ is a vector of point-estimated parameters.

###  4.2 Identification, consistency, and inference

The present paper follows fox2018qe and fox2013aej who both applied the
subsampling approach to the matching maximum score and rank estimator, based
on the literature developed by manski1975maximum, han1987non, and
kim1990cube.393939The alternative approach is a set identification method. The
subsampling method developed by romano2008inference is used in a single-agent
maximum score and rank estimator (bajari2008evaluating) and a many-to-many
matching maximum score estimator (fox2018qe).

I repeat the notion of pairwise stability in matches only from fox2010qe and
fox2013aej. I assume that the econometrician observes some finite number of
recorded firms from a continuum matching market, as in azevedo2018existence.
This means that I introduce the fiction that the real-life matching market
with $N$ firms is a subset of some very large matching market. I formally
assume a rank order property without price data as in fox2018qe:

###### Assumption 6

Suppose that type $i$ firm acquires firms $J_{i}$ in the data, including firm
$h\in J_{i}$, whereas type $l$ firm acquires firms $J_{l}$ in the data,
including firm $k\in J_{l}$. Define swapped coalitions as
$\overline{J_{i}}=J_{i}\cup\\{k\\}-\\{h\\},\overline{J_{l}}=J_{l}\cup\\{h\\}-\\{k\\},$.
Then,

$\displaystyle\text{Pr}[\text{observed matching}|X]$
$\displaystyle\geq\text{Pr}[\text{swapped counterfactual matching}|X],$

if and only if

$\displaystyle X_{i}\left(\left(x_{j}\right)_{j\in
J_{i}}\right)^{\prime}\theta+X_{l}\left(\left(x_{j}\right)_{j\in
J_{l}}\right)^{\prime}\theta$ $\displaystyle\geq
X_{i}\left(\left(x_{j}\right)_{j\in\overline{J_{i}}}\right)^{\prime}\theta+X_{l}\left(\left(x_{j}\right)_{j\in\bar{J}_{l}}\right)^{\prime}\theta.$

Assumption 6 says that if the exchange of a subset of merger members produces
a lower sum of deterministic valuations, then the frequency of observed
matching pairs with the same characteristics as the exchange of a subset of
merger members must be lower than that of observed matching pairs with
characteristics that yield higher valuations.404040As shown in Appendix B.1,
Assumption 6 corresponds to inequality (5), and the corresponding rank order
conditions with inequalities (6), (7), (8), and (9) hold analogously.

Given additional assumptions on the support of $\beta$ and $x,$ fox2010qe
shows that the set of all maximizers of $Q(\theta)$ are identified as true
parameters and the matching maximum score estimator is consistent. Continuous
support also ensures that the maximum is unique. sherman1993limiting
demonstrated that the maximum rank correlation estimator proposed by
han1987non is $\sqrt{N}$ -consistent and asymptotically normal if the
parameter is point identified. The asymptotic variance matrix in
sherman1993limiting is complex to implement in that it requires additional
nonparametric estimates of the components that appear in the variance matrix.
To avoid this complexity, fox2013aej proposed a subsampling approach
(politis1994large), which is consistent under weak conditions.

For the point estimate, I use all possible pairwise inequalities to construct
the objective function $Q(\theta)$ and numerically find a maximizer
$\hat{\theta}$. For inference, I compute the confidence intervals via
bootstrap. An advantage of bootstrap over subsampling is that researchers do
not need to tune the subsampled size. subbotin2007asymptotic showed that a
nonparametric bootstrap is valid for inference under point-
identification.414141I confirm that both bootstrap and subsampling approaches
with one-third of the full sample generated similar confidence intervals with
the same signs using my data, although this is not a theoretical finding. To
construct a meaningful 95 % confidence interval, I calibrate subsidy
sensitivity $\delta$ because Appendix C finds that a dummy variable like the
subsidy term can fail its identification because the identification set may be
unbounded on one side.424242As shown in Appendix B.1, since parameter $\delta$
mainly evaluates inequality (9) before and after the subsidy period, the value
of calibrated $\delta$ can take a positive infinity to achieve inequality (9).
So, as the most insensitive scenario to the subsidy, I calibrate $\delta$ to
the lower bound, achieving the best fitting, i.e., the most correct prediction
of pairwise inequalities. For the government, this scenario is the most
expensive and worrisome case. Thus, for the main estimations, I estimate nine
parameters: coefficient parameter $\beta$ for eight-dimensional observed
characteristics and merger cost parameter $\gamma$. The estimation details are
provided in the footnote of the estimation result tables.

##  5 Results

###  5.1 Estimation results on two observed characteristics

Table 3 reports the estimation results with two observed characteristics and
merger cost.434343In Appendix E.1, I investigate an alternative simple
specification of aggregate-level technological specialization (i.e., carrier-
type HHIs of buyer firms and coalitions in Table 2). The results are
consistent with the results in Table 3. The numbers in parentheses are 95 %
confidence intervals via bootstrap. Estimation details are summarized in the
footnote of Table 3. I normalized $\beta_{0}$ to $+1$ for identification. I
estimate the other parameters separately under the normalization and pick the
vector with the highest number of satisfied inequalities. The number of
inequalities for the full sample is $17,864$. As found in Section C.2, the
identification of the upper bound of $\delta$ is difficult. To deal with this
problem, I find in the preliminary point estimation that $\delta=20$ achieves
the maximum score as the integer lower bound of $\delta$ so I calibrate
$\delta=20$ for all specifications. Thus, I assume the worst case regarding
$\delta$ from the perspective of a government.

Now, I will discuss the estimated signs in Table 3. First, all specifications
report the robust result that the estimated coefficient of merger cost
$\gamma$ is positive, which means that a merger incurs significant costs.
Second, in Columns 1 through 4, the point-estimated coefficients of
assortativeness of scale for $\beta_{1}$ to $\beta_{4}$ are positive. This
implies that assortativeness of scale exists. In Columns 5 through 8, the
estimated coefficients of assortativeness of specialization are negative for
$\beta_{5}$ to $\beta_{8}$. This means that technological diversification on
carrier types contributes to merger incentives. The 95 % confidence intervals
of these parameters indicate that the signs are robust.

Table 3: Results of matching maximum rank estimator with two observed
variables

|  | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
---|---|---|---|---|---|---|---|---|---
|  | Point Est | Point | Point | Point | Point | Point | Point | Point
|  | [95% CI] | [95% CI] | [95% CI] | [95% CI] | [95% CI] | [95% CI] | [95% CI] | [95% CI]
Scale variables |  |  |  |  |  |  |  |  |
totalb $\times$ totalt | $\beta_{0}$ | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1
|  | (S) | (S) | (S) | (S) | (S) | (S) | (S) | (S)
linerb $\times$ linert | $\beta_{1}$ | 93.61 |  |  |  |  |  |  |
|  | [41.7, 294.7] |  |  |  |  |  |  |
tramperb $\times$ trampert | $\beta_{2}$ |  | 34.71 |  |  |  |  |  |
|  |  | [13.5, 295.0] |  |  |  |  |  |
specialb $\times$ specialt | $\beta_{3}$ |  |  | 227.29 |  |  |  |  |
|  |  |  | [39.9, 297.2] |  |  |  |  |
tankerb $\times$ tankert | $\beta_{4}$ |  |  |  | 34.71 |  |  |  |
|  |  |  |  | [10.0, 289.6] |  |  |  |
Share variables |  |  |  |  |  |  |  |  |
linerb $\times$ linert | $\beta_{5}$ |  |  |  |  | -90.68 |  |  |
|  |  |  |  |  | [-280.6, -41.1] |  |  |
tramperb $\times$ trampert | $\beta_{6}$ |  |  |  |  |  | -285.49 |  |
|  |  |  |  |  |  | [-296.0, -91.5] |  |
specialb $\times$ specialt | $\beta_{7}$ |  |  |  |  |  |  | -261.56 |
|  |  |  |  |  |  |  | [-296.7, -86.6] |
tankerb $\times$ tankert | $\beta_{8}$ |  |  |  |  |  |  |  | -240.9
|  |  |  |  |  |  |  |  | [-294.5, -125.8]
merger cost | -$\gamma$ | -1.24 | -0.94 | -2.05 | -0.94 | -1.35 | -0.84 | -1.14 | -0.89
|  | [-2.5, -0.9] | [-2.2, -0.7] | [-3.0, -0.9] | [-3.0, -0.8] | [-2.7, -0.6] | [-12.0, -0.5] | [-9.0, -0.3] | [-5.7, -0.2]
subsidy sensitivity | $\delta$ | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20
$\sharp$ Inequalities (Point) |  | 17864 | 17864 | 17864 | 17864 | 17864 | 17864 | 17864 | 17864
% Inequalities |  | 0.8226 | 0.798 | 0.8465 | 0.8032 | 0.9126 | 0.9102 | 0.9267 | 0.9079

  * a

Note: The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-20,10]$ for $\gamma$ and $[-300,300]$ for other
parameters, and the latter to 100. For point estimation, 100 runs were
performed for all specifications. The reported point estimates are the best-
found maxima. The parentheses are 95 % confidence intervals which are computed
by taking the 2.5th percentile and the 97.5th percentile of the empirical
sampling distribution of estimated parameters. Bootstrap uses 200 replications
with 200 population seeds and 118 firms (all 12 main firms and 106 non-main
firms sampled with replacement out of 106 non-main firms) per replication.
Confidence intervals are not necessarily symmetric around the point estimate.
Parameters that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, then they are superconsistent (denoted by (S)).
The unit of measure of assortativeness of scale is a million (D/W) tonnage
tons. The sub-indices $b$ and $t$ mean the buyer’s and target coalition’s
covariates.

Table 4: Comparable results of matching maximum rank estimator in Table 3

$m$ | $x_{0}$ | $\beta_{m}$ | $\bar{x}_{m}$ | $1SD(x_{m})$ | $-\gamma$ | (1) $V(\bar{x}_{m})$ | (2) $V(\bar{x}_{m}+1SD(x_{m}))$ | (2) - (1)
---|---|---|---|---|---|---|---|---
1 | 0.11 | 93.61 | 0.031 | 0.112 | -1.24 | -1.138 | 0.686 | 1.824
2 | 0.11 | 34.71 | 0.015 | 0.033 | -0.94 | -0.92 | -0.848 | 0.072
3 | 0.11 | 227.29 | 0.03 | 0.071 | -2.05 | -1.833 | 0.281 | 2.114
4 | 0.11 | 34.71 | 0.035 | 0.055 | -0.94 | -0.885 | -0.647 | 0.238
5 | 0.11 | -90.68 | 0.104 | 0.235 | -1.35 | -2.319 | -11.759 | -9.44
6 | 0.11 | -285.49 | 0.117 | 0.244 | -0.84 | -4.736 | -38.033 | -33.297
7 | 0.11 | -261.56 | 0.192 | 0.351 | -1.14 | -10.77 | -78.249 | -67.479
8 | 0.11 | -240.9 | 0.587 | 0.417 | -0.89 | -83.885 | -243.709 | -159.824

  * a

Note: $V(\bar{x}_{m})=x_{0}x_{0}+\beta_{m}\bar{x}_{m}\bar{x}_{m}-\gamma$. The
parameters are shown in Table 3. Models $m=1,\cdots,8$ corresond with columns
in Table 3. I denote the mean and one standard deviation of observed
characterisitics $x_{m}$ as $\bar{x}_{m}$ and $1SD(x_{m})$ shown in Table 2.

To obtain the intuition of the relative importance of each variable, Table 4
reports the value of the joint production function for each specification with
the corresponding units of observed characteristics. To pin down the
situation, suppose that there is a single firm that has a mean observed
characteristic. Then, the value of acquiring a single firm with the same
characteristic is shown in Column 6 as a benchmark. Column 7 shows the
hypothetical value when their observed characteristics change by one standard
deviation.444444The matching outcomes under different specifications are not
comparable because the competitive equilibrium in the market is determined by
these joint production functions for all possible matching configurations
under equilibrium conditions, as shown in equation (1). Recall that the value
of the joint production function of staying unmatched that is dependent on the
specification varies, whereas the value of selling itself is invariant and
normalized to zero. Thus, direct comparison of the effect of the parameters on
matching outcomes is inadequate in my estimation. See Appendix E.2 for details
of the further comparison.

As comparable results, Column 8 summarizes the changes in the values. This
illustrates that the size of a liner is 30 times more important than the size
of a tramper. In addition, the share of a tramper is twice as important as the
share of a liner. Although it would not be adequate to compare the results
based on size and share because of crucial differences in the units of the
measures, I would interpret that the size of a liner is 5.5 times more
important than the share of a liner. In Appendix E.2, I investigate this
comparison further.

###  5.2 Estimation results on two observed characteristics for 12 main firms

Finally, I report the estimation results on two observed characteristics by
focusing only on 12 main firms, each of which can be either a buyer, a seller,
or stay unmatched. This is the most flexible way to model matching behaviors
instead of using a small subsample. The 12 main firms hold about 60 % of the
total tonnage in the industry, and are the main merger decision makers.
Ideally, one would solve a matching equilibrium for the whole sample with
$N=118$. Computational constraints, however, preclude this approach. Instead,
I show that the estimation procedure is not only applicable to market(s) in
which large firms compete even in a single market but also implementable for
counterfactual analysis in a tractable size market as shown in Section 6. I
find the lower bound of $\delta$ in the preliminary point estimation and fix
$\delta$ to the value as a calibrated parameter ($\delta=400$).454545The value
is much greater than the previous value in Table 3. This is because the
subsample includes only matched groups that consist of much fewer firms, and
the interaction terms of the explanatory variables are much less than the full
sample. Thus, to explain the difference between pre-subsidy and post-subsidy
matching outcomes, the subsidy term will play a more important role.
Robustness checks are provided in Appendix F. The goodness of fit in Table 5
is similar to the results presented in Table 3.

Table 5 reports the maximum rank estimation results for the subsample. Since I
use 12 main firms in this section, I could not construct a 95 % confidence
interval with enough subsampled data. Instead, the numbers in brackets
indicate the lower and upper bounds of a set of maximizers of the objective
function. The estimated signs of parameters such as $\beta_{1}$ and
$\beta_{2}$ are consistent with Table 3. On the other hand, as different
points, other parameters such as $\beta_{3}$ and $\beta_{4}$ show negative
signs, which implies that regarding special and tanker ships, the main firms
consider the size of their tonnage relative to the total tonnage a
disincentive to merge. Columns 5 to 8 show that all parameters regarding the
assortativeness of specialization have positive signs, which implies that main
firms aim for technological specialization. These results indicate that a few
main firms specializing in special and tanker shipping want to match with each
other, whereas many non-specialized main firms considered special and tanker
tonnage as a disincentive to merger.464646The results look consistent with
some institutional facts. For example, the main firms in the Showa Line group
specialized in special shipping. Furthermore, the main firms in the Japan
Lines group and the Kawasaki Kisen Kaisha group specialized in tankers.

Combining the results in Table 5 with those in Table 3, I find that the main
firms improve the scale effects and specialization by merging main large firms
while aiming for total diversification by merging small and middle-sized non-
main firms. This captures well the actual post-merger configuration, in
particular, regarding the special and tanker tonnage of small firms for each
group.474747See Figure 6 in Appendix A. The decomposition of merger incentives
illustrates how and why mergers occur at each firm size level.

Finally, some parameters such as $\beta_{1}$, $\beta_{4}$, and $\beta_{8}$ are
point-identified in the small subsample. This is a desirable feature for
solving matching outcomes. In Section 6, I simulate counterfactual analyses of
12 main firms based on the estimated parameters in Table 5 to answer several
policy questions regarding the effects of subsidies on merger configurations
and matching outcomes.

Table 5: Results of matching maximum rank estimator with two observed
variables for 12 main firms.

|  | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
---|---|---|---|---|---|---|---|---|---
|  | [LB, UB] | [LB, UB] | [LB, UB] | [LB, UB] | [LB, UB] | [LB, UB] | [LB, UB] | [LB, UB]
Scale variables |  |  |  |  |  |  |  |  |
totalb $\times$ totalt | $\beta_{0}$ | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1
|  | (S) | (S) | (S) | (S) | (S) | (S) | (S) | (S)
linerb $\times$ linert | $\beta_{1}$ | [124.0, 124.0] |  |  |  |  |  |  |
tramperb $\times$ trampert | $\beta_{2}$ |  | [54.1, 293.4] |  |  |  |  |  |
specialb $\times$ specialt | $\beta_{3}$ |  |  | [-151.8, -42.6] |  |  |  |  |
tankerb $\times$ tankert | $\beta_{4}$ |  |  |  | [-220.8, -220.8] |  |  |  |
Share variables |  |  |  |  |  |  |  |  |
linerb $\times$ linert | $\beta_{5}$ |  |  |  |  | [33.9, 37.4] |  |  |
tramperb $\times$ trampert | $\beta_{6}$ |  |  |  |  |  | [241.9, 279.9] |  |
specialb $\times$ specialt | $\beta_{7}$ |  |  |  |  |  |  | [278.9, 292.6] |
tankerb $\times$ tankert | $\beta_{8}$ |  |  |  |  |  |  |  | [286.4, 286.4]
merger cost | -$\gamma$ | [-15.4, -15.4] | [-10.3, -5.2] | [-9.1, -1.8] | [-8.4, -8.4] | [-13.6, -3.5] | [-9.1, -3.8] | [-7.8, -5.2] | [-8.9, -8.9]
subsidy sensitivity | $\delta$ | 400 | 400 | 400 | 400 | 400 | 400 | 400 | 400
$\sharp$ Inequalities (Point) |  | 330 | 330 | 330 | 330 | 330 | 330 | 330 | 330
% Inequalities |  | 0.9636 | 0.9271 | 0.9394 | 0.9152 | 0.9756 | 0.9514 | 0.9635 | 0.9515

  * a

Note: The numbers in parentheses are lower and upper bounds of a set of
maximizers of the maximum rank estimator. The implementation details are the
same as the footnote of Table 3.

##  6 Counterfactual simulations

Could the government have stimulated endogenous merger incentives more
efficiently? How would a merger configuration have changed in such possible
scenarios? Given the estimated parameters in Section 5, I exercise two
counterfactual simulations to answer these questions. First, I investigate the
effect of a change in the subsidy threshold on matching formation outcomes.
Second, I investigate the effect of shifting the subsidy amount on matching
outcomes. These simulations evaluate the efficiency of an actual subsidy
provision. Finally, I investigate how counterfactual merger configurations
would have been formed under different subsidy thresholds and amounts by
focusing on the most frequently observed simulated matching outcomes. The
numerical comparative statics experiments with the simulated data are shown in
Appendix D.

To make the counterfactual situation clear, I specify the objective function
of the government as follows.484848To characterize the data generating process
for all economic environments in which all firms and the government behave
rationally, I ideally need to solve the government problem and obtain the
optimal solution $M^{*}_{J}$ for all $J$ and $\kappa^{*}$. I do not take this
direction because $M^{*}_{J}$ in my data was the form of an interest subsidy
not measured as a lump sum amount. In addition, subsidy threshold $\kappa^{*}$
in my data was equal to 1 million tons, which seemed to be roughly determined
by the government in the 1960s without an exact calculation, as some
institutional evidence such as book4 has described. Given data about all
observed firm’s characteristics $X$, parameters $\theta$, and thresholds
$\underline{\mathcal{J}},\overline{\mathcal{J}}\in\mathbb{N}$ such that
$\underline{\mathcal{J}}\leq\overline{\mathcal{J}}$, the government solves the
expenditure minimization problem as follows:

$\displaystyle\min_{\\{M_{J}\\}_{J\in\mathcal{J}}\in\mathbb{R}_{+}^{|\mathcal{J}|},\kappa\in\mathbb{R}_{+}}$
$\displaystyle\sum_{J\in\mathcal{J}(M,\kappa,\epsilon)}M_{J},$ (4) s.t.
$\displaystyle\underline{\mathcal{J}}\leq|\mathcal{J}(M,\kappa,\epsilon)|\leq\overline{\mathcal{J}},$
$\displaystyle\mathcal{J}(M,\kappa,\epsilon)=\Gamma(M,\kappa,\epsilon|\theta,X),$
$\displaystyle\epsilon=\\{\epsilon_{i,J}\\}_{i\in\mathcal{N},J\in\mathcal{J}},\quad\epsilon_{i,J}\sim_{iid}F_{\epsilon},$

where $M_{J}$ is the subsidy amount paid to group $J$ from the government,
$\kappa$ is the predetermined subsidy threshold, $\overline{\mathcal{J}}$ and
$\underline{\mathcal{J}}$ are the predetermined maximum and minimum numbers of
groups that the government must satisfy with its subsidy design,
$\Gamma(M,\kappa,\epsilon|\theta,X)$ is the operator that determines the
matching allocation $\mathcal{J}(M,\kappa,\epsilon)$ conditional on $M$ and
$\kappa$ by equation (1) given parameters $\theta$, observed characteristics
$X$, and random term $\epsilon$ drawn from the distribution $F_{\epsilon}$.
The computation of a competitive equilibrium as a central planner’s problem is
nested in $\mathcal{J}(M,\kappa,\epsilon)$. The analytical or approximate
solution to equation (4) is impossible to obtain because $\Gamma$ involves a
random term $\epsilon$ in the equilibrium computation and the domain of the
control variables, $\mathbb{R}_{+}^{|\mathcal{J}|}$ varies via the computation
of $\Gamma$. Instead, I focus on how the equilibrium number of groups, i.e.,
$|\mathcal{J}(M,\kappa,\epsilon)|$ would change if the government changed
$M=M_{J}$ for all $J\in\mathcal{J}$ and $\kappa$ given the $\epsilon$ drawn.

For subsequent counterfactual simulations, I focus on 12 main firms (i.e.,
$N=12$) in six groups for computational feasibility, as in Section 5.2. I
recompute equation (1) 20 times based on the observed characteristics in the
data under the estimated parameters in Table 5.494949I confirmed from 20
simulated equilibria that increasing the number of simulations would not
affect the simulated results because the error term is relatively small in the
specification. In addition, unlike dynamic game papers computing the
equilibrium paths such as a conditional choice probability under 1000
simulated draws for each scenario, my model needs to compute the matching
allocations for 60 scenario points (i.e., 6 thresholds $\times$ 10 amounts)
under one simulated draw. Thus, the cost of a single equilibrium computation
of my model is very low, but the cost of computing a counterfactual “path” is
high, depending on the number of counterfactual scenarios. This is a common
feature of a matching model. For each simulation, I draw
$\varepsilon_{i,J}\sim_{iid}N(0,5)$.

One aspect of our matching model requires fine-tuning. This aspect is the
credibility of the estimated parameters in Section 4.505050Another potential
aspect is the level of matched payoffs and unmatched payoffs. The estimated
parameters in Table 5 are based on 12 main matched firms, which implies that
these estimated parameters do not reflect any information about unmatched
firms. To correct this for counterfactual equilibrium computation, researchers
may need to add a positive common constant to each matched payoff that fits
the market environment in the data but does not affect the evaluation of
pairwise inequalities. This fine-tuning will enable them to generate an actual
matching outcome before a merger, which is consistent with the fact that no
mergers occurred without subsidies. In this paper, the estimated model
generates consistent outcomes before and after mergers without the correction.
Appendix C suggests that in a small matching market, the bounds of parameters
such as merger costs are correctly identified, but those of others may not be
identified. Fortunately, I can use point-identified estimates in Column 1 of
Table 5, which achieves the second-highest score of the objective function and
the scale variable is easy to interpret. Concretely, I use
($(\beta_{1},-\gamma)=(124.0,-15.4)$) as a middle scenario.515151As another
alternative for the counterfactual simulations, it is possible to simulate
equilibrium outcomes based on the lower and upper bounds of the estimates for
the cheapest and most expensive cases, like in ciliberto2009market.

###  6.1 Changes in subsidy threshold ($\kappa$) and subsidy level ($M$)

First, I consider the effect of the magnitude of the subsidy threshold
$\kappa$ on the matching outcome, especially the number of groups.
igami2019mergers showed that the optimal policy should stop mergers when five
or fewer firms exist, highlighting a dynamic welfare trade-off between ex-post
pro-competitive effects and ex-ante value-destroying side effects.525252The
applications of dynamic games to a merger incorporate different merger
incentives. stahl2011dynamic modeled a firm’s station acquisition as a
portfolio choice to maximize the discounted expected future profit.
igami2019mergers modeled six mergers in their data as the proposer firm’s
discrete choice to maximize expected future profit. These dynamic merger
models are too complex to explain a firm’s merger incentive since the papers
involve constructing value functions built on state and action variables with
industry-specific features. Instead of dropping the firm’s dynamic
consideration, matching models such as in fox2019externality and akkus2015ms
capture the merger incentive explicitly and straightforwardly unlike the
dynamic merger model. In addition, book_kiseki recorded a detailed discussion
about how the number of groups after consolidation would be favorable. For
example, some council members insisted that three or four groups were the best
number of groups. In fact, until 1999, six groups had been consolidated into
three groups. Given these suggestions, I recomputed the matching outcome under
different subsidy thresholds conditional on fixed subsidy amounts. This
simulation explores the optimal subsidy issue and disentangles the incentives
of endogenous matching.

Figure 2: Simulation results of the number of groups under different subsidy
thresholds and amounts.

  * a

Note: I set grid points of the subsidy amount in the X-axis to
$[0,0.1,0.25,0.5,0.6,0.7,0.8,0.9,1.0,2.0]$. The threshold level of 7.5 million
tons can be satisfied if and only if all firms form a single grand coalition.
I use the estimated parameters in Column 1 of Table 5, for each scenario
point. If the solution is a non-integer, I add small perturbations drawn from
a standard normal distribution to the allocation matrix and then pick up the
maximum one as a probabilistic matching. I simulate 20 times and take the
median of the number of groups and unmatched firms.

Figure 2 reports in the top panels the number of groups and unmatched firms
under different subsidy thresholds. First, I confirm that if the subsidy
amount had been zero, all firms would have remained unmatched, which is
consistent with institutional fact. Second, the top-left panel shows that
shifting the subsidy threshold would have significantly changed the number of
groups given the benchmark subsidy amount level (the black dotted vertical
line). For example, if the subsidy threshold had been set to 2 million tons,
the number of groups would have been four. This implies that a higher subsidy
threshold would have encouraged firms to form larger groups. Thus, the
government would have been able to control the number of groups by setting the
subsidy threshold under a fixed budget. The right panel also indicates that if
the subsidy threshold had been high, mergers would have been relatively less
sensitive to an increase in the subsidy amount because it might be more
difficult to compensate for merger costs by forming large groups under small
subsidy amounts.

To find an alternative way to control the merger outcome, I investigate the
effect of subsidy amount level $M$ on the matching outcome. Figure 2 reports
in the top left panel that the level of subsidy provision would have changed
the number of groups. First, if matched firms have satisfied the subsidy
requirement at the benchmark subsidy amount (black dotted vertical line) and
the subsidy threshold (1 million), increasing the subsidy amount would not
have affected the matching outcomes because each group had no incentive for
additional mergers, which induces splitting the fixed amount of the subsidy
with more group firms. Second, the number of groups would have increased if
the amount of subsidies had increased. The case of a subsidy threshold of two
million tons (purple line) illustrates that increasing the subsidy amount from
0.5 million to 0.8 million would have split two large groups into four mid-
sized groups because the split groups would have obtained a larger amount of
the subsidy shared with fewer group firms and the mid-sized groups could have
compensated for merger costs. Third, the upper right panel shows that
increasing the subsidy amount would have reduced the number of unmatched
firms.

###  6.2 Efficiency of the actual merger-induced subsidy

As a by-product, the above counterfactual simulations reveal the efficiency of
the government subsidy. The efficient subsidy generates a targeted matching
outcome with minimum expenditure. First, I evaluate the efficiency of the
actual subsidy provided in 1964. Suppose that the government set the target
equilibrium number of firms to six, as in the data. The case of a subsidy
threshold of 1 million, as shown by the green line in the top left panel of
Figure 2 shows that the government could have cut 20 % of the expenditures to
achieve the same matching outcome. The cost-savings amount would have been
equivalent to the joint matching production that a hypothetical firm with 2
million tons in a liner would have generated by acquiring a firm having 2
million tons in a liner.535353The calculation of the total production function
in (1) is as follows. In the matching production function, $20$ % cost savings
is equal to 80 when $\delta=400$ (i.e., $400\times 0.2=80$). The data observed
six groups, so the total cost savings is 480 (i.e., $80\times 6=480$). Since
$\beta_{1}=124.0$ and $-\gamma=-15.4$, adding a firm with 2 million tons in a
liner to a group that has 2 million tons in a liner yields about 480 (i.e.,
$124.0\times(2.0\times 2.0)-15.4\times 1\approx 480$). Thus, the cost-savings
amount is equivalent to the matching production generated by the hypothetical
matching of a firm having 2 million tons in a liner with another firm with 2
million tons in a liner. Note that this cost-savings amount cannot be
converted into a monetary amount such as JPY. Concretely, the impact is
approximately equivalent to that of the two biggest mergers (i.e., the Mitsui-
OSK Line merger and the Kawasaki Kisen Kaisha merger) shown in Table 1. The
cost-savings amount is negligible for the government.

In contrast, suppose that the government set the target equilibrium number of
firms to four, which was achieved 25 years later in the industry. The case of
a subsidy threshold of 1 million, as shown by the green line in the top left
panel of Figure 2 shows that the hypothetical target equilibrium number of
firms could have been achieved in 1964, given a half of the total
expenditure.545454The actual expenditure is $6=1\times 6(\text{groups})$ and
the hypothetical expenditure is $2.0=0.5\times 4(\text{groups})$.

Second, I explore the feasible target equilibrium number of firms given the
same government budget constraint. The bottom left panel in Figure 2 shows the
total expenditure incurred by a policy mix combining shifting the subsidy
threshold with the subsidy amount. This shows that under the same government’s
budget level (i.e., the dotted horizontal line), the government could have
changed the equilibrium number of firms to between one and six using a policy
mix. For example, per the top left panel, fixing the subsidy threshold at two
and the subsidy amount at 0.9 would have generated four groups, while fixing
the subsidy threshold at three and a subsidy amount at 0.5 would have
generated two groups. The bottom-right panel summarizes the number of post-
merger firms in the market for each scenario. Therefore, subsidy design is
crucial to inducing endogenous mergers for any targeted number of groups.

###  6.3 Investigation of a counterfactual merger configuration

Figure 3: Merger simulation across different subsidy thresholds and amounts.

  * a

Note: Given my data, the estimated parameters in Column 1 of Table 5, and the
match-specific errors drawn, the upper panel picks up the most frequently
observed counterfactual mergers under different subsidy thresholds, and the
lower panel picks up those most frequently observed under different subsidy
amounts. To show the Sankey diagrams, if the simulated matching outcome
includes non-integer allocations, I assign firms in the allocations to the
existing groups equally, as in a probabilistic matching. The names shown in
the figures indicate the firm’s identity and hypothetical independent group’s
ID. I denote an unmatched firm by $U$.

Finally, I investigate how a counterfactual merger configuration would have
been constructed under different subsidy thresholds and amounts by focusing on
the most frequently observed simulated matching outcomes. Figure 3 illustrates
counterfactual merger configuration under different subsidy thresholds and
amounts.

First, it would be desirable for the merger configurations, such as merger
pairs with firms’ identities, to be perfectly predicted by the estimated
parameters in the full sample. However, given the data and estimated
parameters, the merger configuration in the subsample may not reflect the
entire configuration because the merger configuration uses a tiny fraction of
all inequalities. In addition, the estimation procedure utilizes pairwise
stability based on observed characteristics without firms’ identities so that
simulated stable matching does not need to fit the actual merger configuration
with the firm’s identities. Second, the merger configuration varies only if
the match-specific error $\epsilon_{i,J}$ varies and turns over pairwise
inequalities. Given the estimated parameters in Column 1 of Table 5, the error
does not affect the inequalities significantly because the terms of the
observed characteristics are much larger than $\epsilon_{i,J}$ in the matching
production function.

Figure 3 shows the most frequently observed simulated matching configurations.
First, the upper panel of Figure 3 demonstrates that if the subsidy threshold
had increased from 1 million tons to 2 million tons, the six existing groups
would have formed a few large groups, and ultimately, an oligopoly market with
three large groups and one unmatched firm would have resulted. As the subsidy
threshold had increased, the market would have become close to a monopoly
market.

Second, the lower panel of Figure 3 shows that even if the subsidy amount had
increased from the baseline level, the merger configurations would not have
varied because the six existing groups had already satisfied the subsidy
threshold. On the other hand, if the subsidy amount had decreased from the
baseline level, the merger configuration would have consisted of some
unmatched firms and four groups because only four firms could have overcome
the high merger costs while satisfying the subsidy threshold. Although these
hypothetical simulated merger scenarios would have been blocked by competition
policy authority, the simulations provide a cut-off point regarding whether
endogenous mergers could have occurred innocuously or harmfully.

##  7 Conclusions

This paper proposes a structural model to investigate the effect of subsidy
thresholds and amounts on endogenous coalition mergers by constructing a one-
sided one-to-many matching model with complementarities, an extension of
azevedo2018existence. I apply the model to shipping mergers and consolidations
and the related subsidy laws enacted between 1960 and 1964 in Japan. I find
that the assortativeness of both size and technological specialization
contributes to merger incentives. The assortativeness of technological
specialization varies across carrier types, which implies that the importance
of technological diversification and specialization varies across
technological types. The counterfactual simulations under the estimated
parameters indicate that the real subsidy provision was far from optimal to
achieve the same matching outcome in terms of cost minimization. It also shows
that a policy mix of shifting the subsidy threshold and amount under a fixed
budget constraint could have controlled matching outcomes.

Finally, I should mention possible extensions as well as some shortcomings of
this study. First, this paper focuses on disentangling the endogenous merger
incentives under the unique and specific subsidy provision policy while
ignoring future competition in the market due to data limitations. Thus, a
welfare evaluation of the post-merger market was not investigated. Combining a
firm’s strategic interactions with estimations of demand and supply sides with
the endogenous matching merger model remains a challenging and open research
question in the field of industrial organization (agarwal2021market). Second,
this paper does not consider the potential effect of global competition in a
foreign transportation market. My current data could not address this
potentially interesting question. Third, my matching model does not
incorporate unobserved heterogeneity which is identified nonparametrically
(fox2018jpe). Pursuing this direction will require a different econometric
approach such as the simulated method of moments in fox2018jpe and multiple
market data. Fourth, because the subsidy law did not specify subsidy amount
flexibly, I could not incorporate potentially heterogeneous subsidy effect
although the theoretical model in itself allows it. Further empirical research
in this direction will be fruitful.

Appendix (For Online Publication)

This Appendix is prepared for online publication and contains supplementary
materials for Sections 2 through 6. Table 6 lists the contents.

Appendix | Contents | Corresponding main text
---|---|---
A | Institutional background |  1 and 2
A.1 | Supplemental figures |
A.2 | Institutional analysis: what mergers occurred and why these mergers occurred. |
A.3 | Preliminary analysis: Merger patterns |
A.4 | Preliminary regression analysis: Who matches whom? |
B | Estimation algorithm | 4
B.1 | Construction of inequalities based on pairwise stability |
B.2 | Mathematical representation of the score function |
B.3 | Procedures for the score function evaluation |
B.4 | Implementations for inference |
C | Monte Carlo simulation |  4 and 6
C.1 | Data generating process |
C.2 | Identification results for a single market |
C.3 | Estimation results for a single market |
C.4 | Identification power with a no-merger market |
C.5 |  Identification power with a small $N$ market |
C.6 | Identification and Estimation results for a single market with only large firms |
D | Comparative statics |  4 and 6
D.1 | Equilibrium properties in a benchmark market |
D.2 | Equilibrium properties in an oligopoly market with only large firms |
E | Supplemental estimation results | 5
E.1 | Estimation results based on assortativeness of HHI levels |
E.2 | Estimation results on multivariate observed characteristics |
F | Robustness Checks | 5
F.1 |  Different calibrated $\delta$ |
Table 6: Contents of the Appendix

##  Appendix A Institutional background

###  A.1 Supplemental figures

Figure 4 depicts the country-level trends of total tonnage, which is referred
in the footnote in Section 1. Table 7 documents the major events related to my
sample.

Figure 4: The trend of the world’s shipping gross tonnage (1000 gross weight
tons).

  * a

Note: Source: syuuyaku which borrows data from the Statistical Tables in
Lloyd’s Statistics. The data contain ships with tonnage sizes of at least 100
tons and include fishing vessels. The dotted vertical line indicates the
period before and after mergers. I omit Liberia. This is because Liberia had
the most shipping tonnage through flag of convenience ships, but this feature
did not reflect actual economic activity in the shipping industry.

Timing / Period | Event | Japanese shipping industry
---|---|---
1945 (Aug) | The end of WW2 | lost 80% of tonnage
1950 – 1953 | Korean War | shipbuilding boom
1956 – 1957 | Suez Crisis | shipbuilding boom
1960 – 1971 | Income-Doubling Plan | targeted as a subsidized industry
1961 (Dec) | The 1st bill of the subsidy laws was submitted | shared information with all firms
1962 (Dec) | The 2nd bill of the subsidy laws was submitted | shared information with all firms
1963 (July) | The subsidy laws were enforced | prepared for merger plans
1964 (April) | The laws were implemented | mergers and coalition formations
Table 7: The industry historical background

###  A.2 Institutional analysis: What mergers occurred and why these mergers
occurred.

Before quantitatively investigating merger patterns, I provide institutional
facts based on book_kiseki who recorded implicit communications between some
firms and the government’s detailed plans for merger processes. First,
book_kiseki reported that although newspaper companies predicted matching
pairs for the main firms based on shareholder information before the laws were
enacted, the actual matching pairs differed somewhat from their
predictions.555555For example, the birth of Mitsui O.S.K Lines formed by Osaka
Merchant Ship and Mitsui Lines was not predicted based on shareholder
information. book_kiseki mentioned that Osaka Merchant Ship parted from ex-
ante Japan Lines and found unmatched Mitsui Lines sequentially. book_kiseki
also recorded talks by the JSA and some corporate executives that the main
firms considered post-merger size effects and technological specialization
important on different levels. Therefore, the matching outcome is not easy to
predict via simple reasoning.

Second, the actual matching processes were determined sequentially and
certainly involved sequential private bargaining and communication, which were
essentially not reported. However, book_kiseki recorded that the newspapers
reported many specific private and public meetings of corporate executives of
many firms between 1960 and July 1, 1964. These reports about sharing
information among firms for finite but sufficiently long periods helped me to
approximate the sequential matching process to the static matching process in
a single large market with complete information. While these institutional
reports enable us to infer the qualitative reasons why some specific firms
were matched to another specific firm, it highlights the significance, from
the government’s perspective, of quantitative reasoning of matching outcomes
and evaluation of the subsidy design.

Shipping firms have a substantial incentive for post-merger size effects and
technological specialization, which was a general feature of the shipping
industry that was not specific to the Japanese at that time.565656See
branch2013maritime for industry-specific background information as a general
reference.  notteboom2004container maintained that the economic rationality
for mergers and acquisitions is rooted in the objectives of economies of scale
and power: to gain instant access to markets and distribution networks in the
liner shipping industry. The industry-specific features support the above
institutional narratives.

###  A.3 Preliminary analysis: Merger patterns

Based on the institutional provided in Appendix A.2, I illustrate some
quantitative features of the ex-post matching results in the following
figures. Due to shipping mergers, 27 main firms and 68 affiliate and wholly
controlled firms formed six mutually exclusive groups through mergers and
acquisitions, while 33 firms chose to stay unmatched. I was unable to collect
variables for ten firms, so the final sample size is 118. Explicitly, the six
groups are Nippon Yusen, Mitsui-O.S.K Lines, Japan Lines, Kawasaki Kisen
Kaisha, Yamashita-Shinnihon Kisen, and Showa Line. Preliminary regression
results are shown in Appendix A.4.

Figure 5: Distribution of tonnage for each firm type.

  * a

Note: Observation units: Firm-level tonnage size for each firm type of each
group after mergers. I omit the observations that have less than 10,000 (D/W)
tons in total tonnage.

Figure 5 reports the post-merger distribution of the firm-level tonnage for
each firm type. This distribution is evidence that all the groups have two
main large firms that played the role the main merger decision-makers. Some
groups also have many wholly controlled firms, whereas others do not. Finally,
unmatched firms are not significantly small. These facts support the idea that
the main firms in each group and the unmatched firms might have different
capacities for marginal merger costs.

Figure 6: Distribution of tonnage for each carrier type.

  * a

Note: Source : listgaikou and nostalgic. Observation units: Firm-type-level
tonnage size of each group after mergers. I omit observations which have less
than 10,000 (D/W) tons in the total tonnage.

Figure 6 reports the post-merger distribution of firm-level tonnage for each
carrier type. In particular, this illustrates group-specific features about
the configuration of carrier types. The remarkable finding is that many large
unmatched firms own special vessels or tankers. In addition, Showa Line has
many small special vessels and tankers. This implies that the preferences for
the assortativeness of specialization are heterogeneous at the group level.
Figure 7 illustrates the post-merger tonnage share of segregated shipping
markets. It indicates that more than half of the liner ships were owned by two
groups, whereas the other carrier types seem to be more competitive. The
findings imply that with these carrier types, concentration seems to be
important for potential economies of scale.

Figure 7: Shares of each carrier type and each group.

  * a

Note: Source : listgaikou and nostalgic. Observation units: The total tonnage
for each carrier type of each group after mergers.

###  A.4 Preliminary regression analysis: Who matches with whom

To obtain some intuition about the matches that occurred quantitatively, I run
two preliminary regression models for predicting pairwise match probability:
(1) the logit model, and (2) the linear probability model. Here, I simplify
the coalitional relationships into pairwise matchings. If two firms belonged
to the same group, I interpret that these two firms were matched. I observe
$N=118$ firms, so I can construct $13,806$ combinations as possible pairwise
matchings. For notational convenience, I assign the index $b$ (buyer) and $t$
(target) to each agent in all possible matching combinations. As a dependent
variable, define the indicator variables $Y_{bt}$, that is, $Y_{bt}=1$ if
buyer $b$ and target $t$ are matched in the data and $Y_{bt}=0$ otherwise.

For each match, I identify several observed characteristics $X_{bt}$. First, I
use the interaction terms of buyer $b$ and target $t$ ’s log of the tonnage of
each carrier type, which captures the economies of scale at the pairwise
level.

Second, to capture the assortativeness of specialization, I include the
interaction terms of buyer $b$ and target $t$ ’s specialization index for each
carrier type which is defined as follows:

$\text{share}_{ik}=\frac{\text{ton}_{ik}}{\sum_{k\in K}\text{ton}_{ik}},\quad
K=\\{Liner,Special,Tramper,Tanker\\},$

where $\text{ton}_{ik}$ is the tonnage of carrier type $k$ owned by firm $i$.

To measure total diversification, I also include the interaction terms of
buyer $b$ and target $t$ ’s Herfindahl-Hirschman-Index (HHI), which is defined
as follows:

$HHI_{i}=\sum_{k\in K}(\text{share}_{ik})^{2}.$

Third, I can observe banking relationships that provide information on capital
efficiency, encouraging mergers.575757I can observe the banking relationships
for only 55 samples which includes almost all the main and wholly controlled
firms. I assume that the bank coverage similarity ratio is zero for missing
samples. 585858I confirmed that the information about major shareholders and
the main banks overlapped because at that time, large banks were operated by
“Zaibatsu” (a financial conglomerate). In this sense, the bank coverage
similarity ratio implicitly captures the effects of shareholders. Based on the
variables, I calculate the bank coverage similarity ratio between buyer $b$
and target $t$ as follows:

$\sum_{z\in Z}(\frac{\mathbbm{1}[b\text{ and }t\text{ have the relationship
with bank }z]}{|Z|})$

where $Z$ is the set of banks, $\mathbbm{1}(\cdot)$ is an indicator function,
and $|\cdot|$ is a cardinality number. I observe 18 banks in the data, so
$|Z|=18$. The ratio also captures the geographic proximity of the head offices
because the contracted local banks were located in regions such as Tokyo,
Osaka, and Kobe, which were close to the head offices. I also include the
dummy variable that indicates whether buyer $b$ and target $t$ are from the
same firm type.

Table 8: Preliminary regression results for predicting matchings.

| Dependent variable:
---|---
| 1(match)
| (1) | (2) | (3) | (4) | (5)
log(linerb *linert+1) |  $-$0.002 |  |  $-$0.013 |  $-$0.029 |  $-$0.003
| (0.006) |  | (0.009) | (0.010) | (0.001)
log(tramperb *trampert+1) | 0.005 |  | 0.004 | 0.018 | 0.002
| (0.002) |  | (0.005) | (0.006) | (0.001)
log(specialb *specialt+1) |  $-$0.009 |  |  $-$0.002 |  $-$0.017 |  $-$0.002
| (0.004) |  | (0.006) | (0.006) | (0.001)
log(tankerb *tankert+1) |  $-$0.003 |  |  $-$0.017 |  $-$0.026 |  $-$0.003
| (0.004) |  | (0.007) | (0.007) | (0.001)
log(totalb *totalt+1) |  $-$0.021 |  |  $-$0.010 | 0.049 | 0.006
| (0.013) |  | (0.017) | (0.018) | (0.002)
bank coverage similarity ratio |  | 1.598 | 2.052 | 0.649 | 0.088
|  | (0.525) | (0.575) | (0.617) | (0.076)
log(HHIb *HHIt+1) |  | 0.525 | 0.372 |  $-$0.123 |  $-$0.019
|  | (0.148) | (0.221) | (0.231) | (0.028)
log(share of linerb *share of linert+1) |  | 0.334 | 1.159 | 2.140 | 0.253
|  | (0.473) | (0.739) | (0.788) | (0.096)
log(share of specialb *share of specialt+1) |  |  $-$0.996 |  $-$0.990 |  $-$0.529 |  $-$0.041
|  | (0.519) | (0.667) | (0.694) | (0.072)
log(share of tramperb *share of trampert+1) |  | 0.308 | 0.165 |  $-$0.558 |  $-$0.058
|  | (0.091) | (0.188) | (0.200) | (0.024)
log(share of tankerb *share of tankert+1) |  | 0.257 | 0.992 | 1.311 | 0.158
|  | (0.210) | (0.335) | (0.354) | (0.043)
same type |  |  |  | 1.600 | 0.229
|  |  |  | (0.052) | (0.007)
Intercept |  $-$1.271 |  $-$2.041 |  $-$1.748 |  $-$3.312 |  $-$0.033
| (0.260) | (0.083) | (0.395) | (0.425) | (0.051)
Model | Logit | Logit | Logit | Logit | OLS
Observations | 13,806 | 13,806 | 13,806 | 13,806 | 13,806
Akaike Inf. Crit. | 12,056.510 | 12,037.120 | 12,034.740 | 11,053.080 | 10,230.180

  * a

Note: Observation unit: a one-to-one matching pair. The sample size is
determined by all possible matching pairs from 118 firms in my data set.
Parentheses show standard errors.

Table 8 reports the preliminary regression results. I find robust results for
predicting matches. First, if the firms have the same main banks and are of
the same type, then they are likely to belong to the same group. Second, the
coefficients of assortativeness of size in trampers (resp. in liners and
tankers) are positive (resp. negative). The coefficients of economies of scope
also have similar positive signs. These findings imply the heterogeneity of
matching incentives across carrier types.

Although the preceding reduced-form analysis is useful as a first cut of the
data, the estimated signs and significance levels are ambiguous and not
robust. As fox2010qe mentioned, the matching outcomes are determined by
considering the possible matching patterns and error distributions, which
essentially involves a high-dimensional numerical integration, so the simple
logit regression is not adequate in a structural sense. Specifically, the
reduced-form analysis ignores important features such as one-to-many matches
with complementarities, the one-sidedness of matches, and trade-offs between
merger costs and subsidies.

To understand the trade-offs and merger patterns, a structural model is
necessary to quantify matching production and quantitatively investigate the
balance of subsidies and costs. In addition, this enables us to simulate
counterfactual policies. This is the purpose of the structural model described
in Section 3.

##  Appendix B Estimation algorithm for evaluating the score function

###  B.1 Construction of inequalities based on pairwise stability

Suppose that type $i$ firm acquires firms $J_{i}$ in the data, including $h\in
J_{i}$, whereas firm $l$ acquires firms $J_{l}$ in the data, including $k\in
J_{l}$. For notational convenience, I define swapped coalitions as follows:

$\displaystyle\begin{array}[]{l}\overline{J_{i}}=(J_{i}\cup\\{k\\})\backslash\\{h\\},\\\
\overline{J_{l}}=(J_{l}\cup\\{h\\})\backslash\\{k\\}.\end{array}$