CSIS Discussion Paper No. 92

Is the Transport Sector Too Large?
Welfare Analysis of the Trade Model with a Transport Sector
Takaaki Takahashi∗
October, 2008

Abstract
Because the demand for the transport services stems from that for the final goods, the sizes of thetransport sector and the final goods sector in an economy depend on each other and are determined
simultaneously. If there is a distortion in the transport sector, notably a distortion arising fromthe imperfect competition, however, its size realized by the market mechanism may diﬀer from thesocially desirable one. In this paper, we examine how the consequence of the market mechanism
diﬀers from the social optimum, considering the first best policy, the price regulation and the entry
regulation.

Keywords: Cournot oligopoly; derived demand; entry and exit; number of transport firms; price oftransport services; regulation; transport industry

JEL Classification Numbers: L13 (Oligopoly and Other Imperfect Markets); L51 (Economics ofRegulation); R13 (General Equilibrium and Welfare Economic Analysis of Regional Economies); R48(Transportation Systems: Government Pricing; Regulatory Policies)∗ Center for Spatial Information Science, University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa, Chiba, 277-8568, Japan.

E-mail: takaaki-t@csis.u-tokyo.ac.jp. I thank seminar participants at various institutions for valuable comments andsuggestions. I gratefully acknowledges financial support by Ministry of Education, Science and Culture in Japan (Grant in
Aid for Research No. 18600004).

1

1

Introduction

For a long period of time, economists have been discussing government interventions in transport industries. One of the main justifications for them is that the industries are usually characterized by increasing
returns to scale or decreasing average cost, which allows only a small number of firms or more typicallythe only one firm to survive in each of the industries. In order to cope with the distortion caused by
the exercise of monopoly power, governments are expected to carry out economic policies like regulationsof the price of transport services and of the number of firms in an industry. Such regulation policies
on decreasing returns industries have been one of the most frequented topics in the field of industrialorganization, so that there is thick literature on it (see Spulber (1989), Train (1991) and the literaturecited there).
Since the drastic changes in the regulation policies toward an airline industry in the US in 1978,
however, many developed economies have been undertaking various forms of deregulation in the transportindustries. Railroad industry is a typical example. The UK and Japan had already embarked on a reform.
In the other European countries, although the speeds of deregulation diﬀer from country to country with
Sweden and Germany, for instance, being quite ahead while France much behind, a general policy of theEuropean Union is heading toward the deregulation.1 In such circumstances, reassessing the eﬀects ofregulation/deregulation in transport industries is not a matter of minor importance.
Having said that, merely applying the framework in the industrial organization literature to transportindustries can be not only insuﬃcient but also misleading, for there is a key factor which is usuallydisregarded in the literature but comes to play a critical role when transport industries are concerned. It
is one of the properties of the demand for transport services, the property that it is a derived demand,as such researchers as Rietveld and Nijkamp (2003) emphasize: the services themselves do not directly
contribute to the consumers’ utility, but they are inevitably accompanied by the consumption of final
goods.2
To recognize this property is important because it immediately implies that the level of the othereconomic activities, in particular, the production of final goods depends on the size of the transport
sector, and vice versa. On the one hand, the size of the transport sector aﬀects the price of transport
services, which alters the delivered prices of final goods and therefore their demand. An expansion and ashrinkage of the final good sector is, on the other hand, associated with the changes in trade flows between
regions/countries and thus in the demand for transport services. Such viewpoints are indispensable when
one examines the government interventions. The regulation of the price of transport services, for example,
immediately aﬀects the demand for final goods. The entry regulation in the transport sector, for anotherexample, not only harms the demand for final goods through a rise in the price of transport services
1 For the movements toward deregulation in transport industries and their consequences in industrialized countries, see

Winston (1993) and Quinet and Vickerman (2004).
2 Notwithstanding, in standard textbook applications, the determination of the amount of demand for transport services
is, in most cases, put into a black box: the explanation usually begins with a given inverse demand curve that comes fromnowhere (see Balchin et al. (2000) and O’Sullivan (2006), for instance).

2

due to less competitive behaviors of transport firms, but also encourages the production of final goods
because the transport sector is now relieved of some of the resources previously used there for a fixed
input. In this way, it is not innocuous for us to discuss, with isolating the transport sector from theothers, whether the transport sector tends to become too large or too small and how government should
regulate it. The linkages between sectors are all the more important because the transport sector has aconsiderable weight in most of the industrialized economies.3
The purpose of this paper is to evaluate the size of the transport sector realized by a market mechanism
and to examine the government policies against the ineﬃciencies due to a monopoly power prevalent in
the sector, paying special attention to the fact that the demand for transport services is a derived one. For
that purpose, we construct a simple Ricardian model of trade with two regions. In order to carry goods
between regions/countries, they need to use transport services, which are provided by transport firms.
Thus, the demand for the transport services stems from that for the final goods. Furthermore, because
of increasing returns to scale, the transport sector is characterized by the oligopolistic competition à la
Cournot. We examine the levels of transport cost and the numbers of transport firms realized at a market
equilibrium and at the equilibria under three types of government interventions: the regulation of boththe price of the transport services (i.e., transport cost) and the number of transport firms, that is, thefirst best policy; the second best policy with only the price regulation; and the second best policy with
only the entry regulation.
One of the most interesting findings in this paper is that the number of transport firms at the market
equilibrium is too large or too small depending on the level of fixed cost in the production of transport
services and the size of the economy. To understand this point, it is helpful to consider what happens if
a firm enters into the transport sector. On the one hand, it intensifies the competition among transport
firms, which causes a decline in the price of transport services (transport cost) and, consequently, theprice of imports. As a result, consumers, who now find themselves to be able to consume more final goods,
become better oﬀ. This is a competition eﬀect of the entry. On the other hand, the economy needs toallocate a certain amount of its limited resources to the fixed input of the entrant at the sacrifice of final
goods production, which aggravates the welfare of consumers. This is referred to as a fixed input eﬀectof the entry. If the competition eﬀect more than oﬀsets the fixed input eﬀect at the market equilibrium,social welfare will be improved by the entry. In this case, the number of the firms is too small. If thecompetition eﬀect is dominated by the fixed input eﬀect, to the contrary, the number is too large.
To study an optimal size of transport sector and a desirable form of its regulation with taking into
account its interdependence with the other sectors has not constituted a main concern of the researchers
in related fields of economics.
First of all, it has been rare that the transport sector is explicitly examined in the fields that study
trade of goods and services between regions and countries. In the international economics, on the one
3 Quinet and Vickerman (2004) estimate that expenditure on transport in EU countries amounted to between 9 per centto 17 per cent of total household expenditure in 2000. Furthermore, McCarthy (2001) reports that the transportation billoccupies 16.1 % of the GDP and that the share of the employment in the transport sector accounts to 8%, both in the US
in 1995.

3

hand, even transport costs (or more broadly trade costs) have seldom been incorporated to models, notto mention the transport sector.4 In the regional economics, on the other hand, it is true that transport
costs have been playing a key role in the explanation of locations of economic activities, in particular,
in a subfield called new economic geography (Krugman (1991), Fujita, Krugman and Venables (1999),
and Fujita and Thisse (2002), among others). For all that, most of studies in that literature treat thelevel of transport cost as given, which is one of many consequences of abstracting away the transport
sector. Even when they discuss endogenous determination of transport costs, it is not, in most cases,
dealt as a result of individual behaviors of transport firms.5 There are two exceptions. First, Behrens etal. (2006) examine the impacts of the regulation of a transport sector upon welfare paying attention to
the incentives of transport firms, though in a very simple way. Second, Takahashi (2007) discusses the
behaviors of individual transport firms to obtain welfare implications in a general equilibrium setting.
Second, in a field of urban economics, a number of researchers have accumulated studies on thegovernment interventions to a transport sector aimed at the congestion externalities (see Small (1992),
and Small and Gómez-Ibáñez (1999) for the review). Notwithstanding, it seems that there is a big gap
between their concerns and ours. First, they consider the commuting cost from residences to workplaces
rather than the transport cost to ship goods. This is because their main interest lies in a spatial structureof cities but not in regional trade. Second, as its consequence, they do not attempt to fully embody theidea of the derived demand property for transport services: it is usually assumed that each worker makes agiven amount of trip, often normalized to one trip, between their homes and oﬃces. True the commuting
distance for each worker and its aggregate are variables to be determined, but with the number of trips
given, the assertion that the demand for transport services is the derived one holds only partially.
Lastly, it is worth adding another literature in relation to the conclusion that the number of transport
firms can be too small for some values of parameters, as has been mentioned. It makes a sharp contrast
to the result that oligopoly involves excessive entry because of the existence of a business stealing eﬀect(Mankiew and Whinston (1986), and Suzumura and Kiyono (1987), among others). The reason for thisdisparity is that the literature focuses on a partial equilibrium whereas this paper on a general equilibrium.
That is, we consider the situation in which the market of transport services is interrelated to that of final
goods: an entry of a firm into the transport sector aﬀects not only the demand for transport services but
also that for final goods. More specifically, an entry stimulates the consumption of final goods througha decline in the prices of transport services (transport cost) and final goods, as we have explained as thecompetition eﬀect. This mitigates the business stealing eﬀect and consequently, the result is not always
the excess entry.
The paper is organized as follows. In the subsequent section, basic framework is presented. In section
4 The attempts to take into account transport costs include a classical article by Samuelson (1952), Herberg (1970),

Falvey (1976), Cassing (1978), Casas (1983), and Casas and Kwan Choi (1990). See Steininger (2001) for the overview.
5 Mori and Nishikimi (2002) have discussed the situation in which transport cost decreases with the volume of trade. In
Takahashi (2006), I have tackled the question of how economic geography aﬀects transport cost. Finally, Kanemoto andMera (1985), Bougheas, Demetriades and Morgenroth (1999), and Mun and Nakagawa (2005) have analyzed the impacts
of infrastructure investment upon transport cost and the resulting change in the amount of goods transported.

4

3, we formulate the behaviors of the oligopolistic transport firms that make a decision on the quantity oftransport services to produce and on the entry and exit. In section 4, government policies are examined.
Three types of policies, namely, price and entry regulation (the first best policy), price (transport cost)
regulation and entry regulation are examined. We show that the price regulation is better than theentry regulation. Section 5 deals with a market equilibrium. We ask whether the market equilibrium isassociated with too many or too few transport firms. Section 6 concludes.

2

Basic framework

We consider a simple Ricardian economy with two regions, East and West (which may be denoted asregion E and region W when it is more convenient), and two goods, E and W . The total population of
the economy is fixed at L. The share of the population in East, which is endogenous in the long-run, isdenoted by θ.
Both goods are produced through a constant returns to scale technology. Let aij be the unit labor
requirement to produce good i in region j. East has an absolute advantage in the production of good
E while West in the production of good W : aW E = βE aEE for some βE > 1 and aEW = βW aW W forsome βW > 1. Obviously, East (West, resp.) has not only the absolute advantage but also a comparativeadvantage in the production of good E (good W , resp.). For simplicity, furthermore, we focus on thesymmetric case where the degrees of the absolute advantage are equal: βE = βW ≡ β. Finally, one canchoose units so that aEE = aW W = 1. Then, aW E = aEW = β.
Furthermore, workers have Cobb-Douglas preference with the share of spending on each good being
equal to 1/2. This rather restrictive assumption of symmetry is made for the sake of simplicity, because
without it the model would become too complicated for us to use as a basis for the subsequent generalequilibrium analysis with the explicit consideration of the behaviors of transport firms. Besides, thesymmetric case is a useful benchmark and yields the results that are qualitatively similar to those obtainedin asymmetric cases. Under this preference, the demand functions for the two goods become
Xij =

Yj
,
2pij

(i, j = E, W )(1)where pij and Xij are price and aggregate consumption of good i in region j, respectively, and Yj isaggregate income of region j.
In addition, because good E is produced in East while good W is in West no matter whether the
economy engages in trade or not, we always havepii = aii wi = wi ,(i = E, W )(2)where wi is wage rate in region i. Let us consider good E in East a numéraire, that is, wE = 1 andwW = p, where p is a relative price of good W in West. On the other hand, the price of good E in Westand that of good W in East hinge on whether trade occurs or not. If it does not occur, they are given
by pij = aij wj = βwj for i, j = E, W with i 6= j. If the trade occurs, instead, pij ’s depend on transport
costs.
5

Thus, let us introduce a transport sector, which consists of n transport firms. They provide transport
services necessary to ship goods between regions, and charge an ad valorem price. Let tE be the priceof the transport services that ship 1 dollar of good E from East to West: the transport cost to have 1
unit of that good shipped in that direction is equal to pEE tE . tW is similarly defined. Then, when tradeoccurs, the price of the good i produced in region i and consumed in region j becomes equal topij = (1 + ti )pii = (1 + ti )wi ,(i, j = E, W ; i 6= j)(3)where (2) is used. In what follows, we focus on the case where trade occurs and each region specializes
in the production of one good, namely, East produces only good E while West produces only good W .
This occurs if the size of the comparative advantage is suﬃciently large, that is, if
1 + tE < βand

1 + tW < β,which is assumed.
Transport firms are owned by workers in the economy. For the sake of simplicity, we assume thateach of them has an equal share. Then, the profit raised in the transport sector is distributed to the tworegions in proportion to their populations. That is, out of total profit, Π, portion θ goes to East and therest, portion 1 − θ, goes to West. Since there is only one factor of production, labor, regional income
consists of labor income and profit:
YE= θ(LwE + Π),

and

YW

= (1 − θ)(LwW + Π).

(4)

When the profit of the transport sector is negative, the loss is borne by workers, probably in the form oftax. In this case, (4) implies that the tax burden to finance the loss is equally shared by workers.
Transport technology exhibits increasing returns to scale. For one thing, transport firms use laborfor a variable input. In particular, they use the labor in the region of the origin of shipment. In order toship 1 unit of good from East to West (from West to East, resp.), each firm needs to employ c units oflabor in East (in West, resp.).
In addition, they use labor for a fixed input as well. Each of them uses F units. In what follows,
however, we rather use a per capita variable, f ≡ F/L, instead of F for the sake of convenience. Here,
two assumptions are imposed for the fixed input. First, notice that the total amount of labor used as afixed input in the economy is equal to F n = Lf n. In order to focus on the non-trivial cases where at least
some final goods is produced, we assume that the fixed cost is so small that Lf n < L, or equivalently,
f<

1
,
n

(5)

for any admissible value of n. The second assumption is that the regional distribution of the labor used
for a fixed input coincides with that of workers. In other words, θLf n units of labor are employed in
East while (1 − θ)Lf n units are employed in West.6 There are three reasons for the assumption. First,
6 As another plausible hypothesis, we could assume that firms are located in the region where the wage rate is no higher

than the counterpart in the other region. However, in what follows, our analysis turns out to be confined to the cases inwhich the wage rates become equal between the two regions. For such cases, the hypothesis is of no use and a stronger one
is necessary.

6it is often observed that the fixed input is used mainly for the construction of the facilities necessary
to handle shipment in each region, such as terminal buildings in airports and railway stations. In thiscase, it is natural to suppose that the amount of fixed input used in each region depends on the sizeof that region. At its extreme, it may be permissible to consider the amount to be proportional to theregional size. Second, it may be the case that the firms owned by the workers in a particular region are
somehow destined to employ for a fixed input the labor in that region. One possibility is that the firms
are located in the region where their shareholders are located, with no access to the labor in the otherregion, probably due to the existence of commuting costs or by other technological, institutional and/or
social reasons. In such a case, the regional distribution of the labor used for a fixed input coincides
with that of shareholders and, therefore, that of population. Third and finally, the assumption greatly
simplifies the analysis.
Profit of the sector is, therefore, given byΠ = (pEE tE − wE c)XEW + (pW W tW − wW c)XW E − Lf n [θwE + (1 − θ)wW ] .

(6)Finally, because we are focusing on the case of complete specialization, the markets clear ifθL = XEE + (1 + c)XEW + θLf n,

and

(7)

(1 − θ)L = XW W + (1 + c)XW E + (1 − θ)Lf n.
This completes the description of a short run, in which the regional distribution of workers, θ, isgiven. In this paper, we concentrate on the equilibrium at which the prices of transport services areequal with respect to the two directions, that is, tE = tW = t. Although there remains a possibility thatasymmetric prices are supported by an equilibrium, the analysis becomes much simpler by focusing on
the cases with symmetric prices. Later, we will come back to this point to show that this symmetry is
indeed consistent with the behaviors of transport firms. Solving the system of (2), (3), (4), (6) and (7)along with tE = tW = t, we obtain
p=

AE θ
,
AW (1 − θ)

(8)

where
h
i (1 − θ)(t − c)2
AE ≡ f n(1 + t) 1 + t − θ(t − c) −− (1 + c)(1 + t)and2hi θ(t − c)2AW ≡f n(1 + t) 1 + t − (1 − θ)(t − c) −− (1 + c)(1 + t).
2
In a long run, regional distribution of workers is determined. Here, we follow the convention in theliterature that workers’ decisions on their regions of residence are based on the relative levels of indirect
utility achieved in the two regions. It is straightforward to compute the indirect utility of a worker ineach region:
vE

=

vW

=

µ
∂
1
Π
p
1+
,
L
2 p(1 + t)
≥
1
Π¥
p
.
p+
L
2 p(1 + t)
7

and

The long-run equilibrium must satisfy vE = vW , which implies that p = 1: At the long-run equilibrium,
the prices of the two goods become equal to each other. Then, (8) is reduced to
AE θ = AW (1 − θ).

(9)

<
>
It is readily verified that AE θ = AW (1 − θ) if θ = 1/2, provided that (5) holds. Therefore, (9) has a><unique solution, θ = 1/2. This establishes the following lemma:
Lemma 1. When the prices of transport services are symmetric with respect to the directions of shipment,
workers are distributed equally in the two regions at the long-run equilibrium.
It immediately follows that, at the long-run equilibrium, the wage rates in the two regions become
equal. Furthermore, the aggregate consumptions are reduced to:
XEE = XW W=

XW E = XEW=

L(1 − f n)(1 + t),
2(2 + c + t)
L(1 − f n).
2(2 + c + t)

andThe condition (5) guarantees that all of these amounts are positive. The equilibrium level of the profit
per transport firm, Π/n, is given by£§
L t − c − 2f n(1 + t)π(t, n) =,n(2 + c + t)

(10)which decreases with n for given t.
Moreover, the total amount of labor used in the transport sector is equal to c(XEW + XW E ) + Lf n.
Therefore, the share of labor in that sector is given by
λ(t, n) ≡c + f n(2 + t).
2+c+t

Furthermore, the indirect utility of each consumer in each region becomes equal tovE = vW = v(t, n) ≡1 − fn √1 + t.
2+c+t(11)

In order to understand the structure of the model, it would be helpful to examine the eﬀects of changes
in cost parameters, c and f , upon the level of indirect utility. We can decompose the total eﬀects into
three parts:
dv(t, n)dx==∂v(t, n)∂t ∂v(t, n) ∂n ∂v(t, n)
+
+
∂x
∂x ∂t
∂x ∂n
µ
∂
∂v(t, n)∂t ∂v(t, n) ∂n ∂t ∂v(t, n) ∂v(t, n)
+
+
+
∂x
∂x ∂t
∂x ∂n ∂t∂n

(12)x ∈ {c, f }.

The first term, which is negative for both x = c and x = f , represents an adverse eﬀect due to the shiftof labor from the final goods sector to the transport sector (direct cost eﬀect): if the price of transport
services and the number of transport firms remain unchanged, the rise in the production cost in thetransport sector reduces the amount of labor available in the final goods sector. Next, the second term8measures an eﬀect through the change in the price of transport services, provided that the number of firms
remains unchanged (transport price eﬀect). Because a higher price is obviously associated with a lower
level of the indirect utility (∂v(t, n)/∂t < 0), the eﬀect has an opposite sign of ∂t/∂x. Finally, the lastterm represents an eﬀect through the change in the number of transport firms (number-of-firms eﬀect).
It is further decomposed into two parts. First, the number of firms aﬀects the indirect utility througha change in the price of transport services. This is represented by the first term in the parentheses inthe second line. When transport firms are free to choose their quantity of production, a shrinkage of thetransport sector will mitigate the competition among them, which renders them a stronger monopoly
power and results in a higher price. To the contrary, an expansion of the sector will provoke fiercer
competition, which brings about a lower price. Because ∂v(t, n)/∂t < 0, this competition eﬀect has an
opposite sign of ∂n/∂x · ∂t/∂n. Second, the change in the size of the transport sector directly aﬀects theindirect utility. This eﬀect is represented by the last term in the parentheses in the second line. As thetransport sector shrinks, a certain amount of labor previously used for a fixed input is released from thesector and becomes available for the final good production, which raises the level of the indirect utility:∂v(t, n)/∂n < 0. This fixed input eﬀect has an opposite sign of ∂n/∂x.

3

Oligopolistic transport sector

In this section, we give more concrete structure to the transport sector to obtain some key results on theprices of transport services and the number of transport firms when the sector is characterized by theoligopolistic competition à la Cournot.
Suppose that n transport firms produce transport services, which are distinguished only by the direction of shipments, that is to say, they are not diﬀerentiated as long as the direction is the same. Let zE (k)be the quantity of the service produced by the k-th firm to carry good E from East to West. zW (k) issimilarly defined. For simplicity, we choose the unit so that 1 unit of transport services can ship exactly
1 unit of final goods. Then, the market for the transport services clear whenXzE (k) = XEWkXzW (k)= XW E .

kGiven the amounts of services produced by each firm, their aggregates determine the prices of the services,tE and tW , through inverse demand functions: Xij = αi Yj /pii (1 + ti ), where Yj and pii depend on tE
and tW . (As has been mentioned, tE and tW become equal to each other ex post.)
In the rest of this section, we first analyze the decision making of each transport firm on the scale ofproduction given the number of its competitors, and then examine the number of firms determined as aresult of free entry and exit.

9

3.1

Scale of production and prices of transport services

First, let us discuss the decision making of each transport firm on the scale of production when the numberof its competitors is given. Firm k chooses zE (k) and zW (k) that maximize its profit, given by:
π(k) =

X

i=E,W[ti zi (k)pii − czi (k)wi ] − Lf [θwE + (1 − θ)wW ] .

Two assumptions are made. First, we assume that each firm makes Cournot conjecture, or, to put it
another way, that a coeﬃcient of conjectural variation for each firm is equal to 1. It means that eachfirm sticks to its belief that the other firms will not change their behaviors (see Seade (1980)).7 Then,
we have ∂ti /∂zi (k) = ∂ti /∂Xij . Second, we assume that transport firms do not take into account all theeﬀects of their decisions but only a part of them. Theoretically, a choice of a firm aﬀects its profit throughmany channels: it aﬀects the prices of transport services, which, in turn, alters all the other variables
including the price of final goods, wage rates and regional incomes. However, it seems farfetched andunnecessary to suppose that transport firms well consider such far-flung eﬀects. In this paper, therefore,
we assume that they consider only two types of eﬀects, namely, a direct eﬀect, ∂π(k)/∂zi (k), and theeﬀect transmitted through the change in the prices of transport services, ∂π(k)/∂ti · ∂ti /∂zi (k). These
two assumptions imply that ∂ti /∂zi (k) = ∂ti /∂Xij = −(1 + ti )/Xij , because the inverse demand is givenby ti = Yj /(2Xij pii ) − 1 (see (1) and (3)). Assuming the symmetry with respect to transport firms,
zi (k) = zi = Xij /n, we obtain the condition for the profit maximization: ti = (pii + cn)wi /[pii (n − 1)].
Then, (2) and w1 = w2 = p = 1 give the following prices of transport services:tE = tW = t∗ (n) ≡1 + cn.
n−1(13)

Two observations follow. First, the prices of transport services indeed become equal to each other, as
has been assumed in the previous section. Therefore, the symmetry of the prices of final goods, p = 1, isconsistent with that of the prices of transport services, tE = tW : p = 1 and tE = tW actually consistute anequilibrium. Second, the price of transport services decreases with the number of firms: dt∗ (n)/dn < 0.
This is because competition among firms becomes severer when there are a greater number of firms.
For the price given by (13), the indirect utility of workers, given by (11), is reduced tor°∗¢ 1 − f n n(n − 1)∗
v (n) ≡ v t (n), n =.
2n − 1
1+c

Furthermore, the share of the labor employed in the transport sector becomes equal to°¢ c(n − 1) + f n(2n − 1 + cn)
λ∗ (n) ≡ λ t∗ (n), n =(1 + c)(2n − 1)

In the earlier discussion on the number-of-firms eﬀect, we have paid an attention to the impact ofa change in the number of transport firms upon the indirect utility. It is worth studying the impact
7 Suzumura and Kiyono (1987) discuss the entry and exit behaviors of firms for a general level of the coeﬃcient ofconjectural variations. As a matter of fact, however, we could not treat the coeﬃcient as given, because it usually depends
on the number of firms in the transport sector. Yet explicitly incorporating this eﬀect into the model brings about too
much complication.

10

again for the specific case with the price being given by (13), because it plays a key role in the following
sections. It is straightforward to see that°¢dv t∗ (n), n
dv ∗ (n)
=
=
dn
dn

°
¢
°
¢
∂v t∗ (n), n
dt∗ (n) ∂v t∗ (n), n
+
dn h
∂t
∂n

i
= Kv (n) (1 − f n) − 2f n(n − 1)(2n − 1)

≡ Kv (n)ψ(n),

(14)

where
Kv (n) ≡

p

1

> 0,
(1 + c)n(n − 1)
ψ(n) ≡ 1 − f n − 2f n(n − 1)(2n − 1).
2(2n − 1)2

and

°
¢
The term dt∗ (n)/dn · ∂v t∗ (n), n /∂t = Kv (n)(1 − f n) in (14) measures the competition eﬀect, whichis positive because of (5). As the number of firms increases, the competition in the transport sector

becomes fiercer, which induces them to produce more and results in a lower price. This has a favorable°¢
eﬀect on the welfare of workers. The term ∂v t∗ (n), n /∂n = −K(n)2f n(n − 1)(2n − 1), on the other

hand, measures the fixed input eﬀect, which is negative as has been explained. Since the directionsof the two eﬀects are opposite, sign of the total eﬀect is ambiguous. When the competition eﬀect
dominates the fixed input eﬀect, the total eﬀect is positive. When the opposite holds, it is negative. Asn increases, moreover, the competition eﬀect shrinks whereas the fixed input eﬀect grows, and, therefore,dψ(n)/dn < 0. Consequently, when the number of firms is greater, it is more likely that the total eﬀect
becomes negative, and that a rise in the number of firms gives an adverse impact on the welfare ofworkers.
Similarly, we can investigate the impact of a change in the number of transport firms upon the shareof labor in the transport sector, λ, when the price is given by (13). It can be shown that the impact isunambiguously positive: as the number increases, the share of labor always rises. To see this, note that
°
¢
°
¢
°
¢
dλ t∗ (n), n∂λ t∗ (n), ndλ∗ (n)dt∗ (n) ∂λ t∗ (n), n
=
=
+
dn
dn
dn h
∂t
∂n
i
= Kλ (n) c(1 − f n) + f (2n − 1 + cn)(2n − 1) > 0,

where
Kλ (n) ≡1> 0.
(1 + c)(2n − 1)2

The total eﬀect is decomposed into two parts. First, as the number of transport firms increases, theprice of transport services declines. This stimulates the trade between regions and therefore increases
the demand for the labor used for a variable input in the transport sector. However, it follows that theamount of final goods production declines because only a smaller amount of labor becomes available for it,on the other hand. This reduces the demand for transport services and thus the labor used for a variable
input in the transport sector. Although part of the increase in the labor demand in the sector is, in thisway, oﬀset by the secondary eﬀect, the primary eﬀect dominates it. Consequently, the labor used for avariable input in the transport sector increases. This eﬀect is captured by the first term. Second, as the11number of transport firms increases, the transport sector needs to hire more labor for a fixed input. To
the extent that the amount of the labor available for the final good production decreases, the final good
production shrinks. We have again a counteractive eﬀect. However, this counteractive eﬀect is not toolarge, and consequently the share of labor in the transport sector rises. This direct eﬀect is represented
by the second term.

3.2

Number of transport firms

Next, we discuss the number of transport firms, n0 , determined as a result of free entry and exit. Whatis important is that it is an integer. Thus, it can diﬀer from a 0-profit number of firms, n, which is asolution to π(t, n) = 0. In this profit function, the price of transport services is either a constant or afunction of the number of firms, depending on the situation we analyze.
Two observations follow with respect to n0 . First, free entry/exit implies that the profit of eachfirm becomes no lower than 0: π(t, n0 ) ≥ 0. In what follows, it will be always the case that the profit

decreases with the number of firms, that is, dπ(t, n)/dn < 0. Then, π(t, n0 ) ≥ 0 is equivalent to n0 ≤ n.
Second, the profit would be negative when one additional firm entered, that is, π(t, n0 + 1) < 0. This isequivalent to n0 > n − 1 as long as dπ(t, n)/dn < 0. Putting these two observations together, we haven − 1 < n0 ≤ n. Now, let us introduce function I(x) which gives the maximum integer that does notexceed x, that is,x − 1 < I(x) ≤ x with I(x) ∈ Z,

(15)

where Z is a set of integers. Then, the above finding implies thatn0 = I(n),(16)

because n0 is an integer. In addition, it immediately follows that a necessary and suﬃcient condition for
n0 to be no smaller than 1 is that n ≥ 1.
In the following analysis, we will use some properties of the function I(·), which are summarized asfollows:
Lemma 2.
i) dI(x)/dx ≥ 0 for any x;ii) I(x + z) = I(x) + z for any x and any z ∈ Z; andiii) x < z if and only if I(x) < z for any x and any z ∈ Z.
The proof is relegated to the appendix.

4

Social optima

In order to judge whether the transport sector is too large or not, we need a yardstick. In this section,
three types of social optima are introduced as such yardsticks. They are the first best optimum when12

a social planner can choose both the price of transport services and the number of transport firms, thesecond best optimum when the planner sets the price while the number of firms is determined throughindividual decision makings by transport firms, and finally the second best optimum when the plannersets the number of firms and firms determine the price.
First best optimum. When a social planner can set both the price of transport services and thenumber of transport firms, it chooses marginal cost pricing having one transport firm operate. This is
obvious since distortion arises only from the transport sector, and, furthermore, the welfare increases by
the relocation of the labor used for a fixed input in that sector to the final good sector as long as themarginal cost pricing prevails.8 Indeed, we can solve the problem to maximize the indirect utility givenby (11) subject to constraint n ≥ 1 by choosing t and n, to obtain t = c and n = 1. It is worth addingthat at the first best optimum, the profit of the (unique) transport firm is negative and the loss is equal
to the fixed cost. Therefore, to achieve the first best, the tax whose total amount is equal to the fixed
cost must be collected.
Second best optimum when the social planner can set only the price of transport services.
The second case to consider is the second best optimum where a social planner can set only the price oftransport services. This is described by a two stage game in which, at the first stage, the social plannersets the price, and, at the second stage, transport firms make entry and exit until incumbent firms earnnonnegative profit but a potential entrant cannot do so.
Notice that the number of transport firms determined at the second stage is prescribed by (16). Here,£§the 0-profit number of firms, n, can be regarded as a function of t and written as nP (t) ≡ (t−c)/ 2f (1+t)

where superscript P stands for the second best with the price regulation. Therefore, the equilibrium°¢
number of firms, I nP (t) , also becomes a function of the regulated price. Thus, we denote it as nP (t).

The problem of the planner is, then, to choose the price that maximizes the indirect utility, v(t, n), given°¢by (11), subject to the constraint on the number of transport firms, nP (t) = I nP (t) . It turns out thatthe solution is the price that entices exactly one firm to operate in the transport sector. We can prove
the following result whose proof is relegated to the appendix:
Lemma 3. When the social planner can control only the price of transport services, it chooses the pricetP ≡c + 2f,
1 − 2fwhich induces one firm to operate.
It is not diﬃcult to explain this result intuitively. Suppose that the planner lowers the price oftransport services. It aﬀects the indirect utility through two channels. First, it enables workers to spend
8 In this paper, we do not explicitly deal with the outside option that no transport firm operates in the economy.

Inother words, we assume that social welfare is always higher when at least one firm operates than when no firm operates.
The latter situation might be explicitly formulated by introducing a ‘traditional transport technology’ that enables eachconsumer to carry goods by themselves, which, however, makes the analysis much more complicated without yielding any
additional insight.

13more money upon the final goods. This corresponds to what we have seen as the transport price eﬀect.
Second, it reduces the number of firms in the transport sector; and therefore, some of the labor used fora fixed input in that sector is released and becomes available for the final goods production, namely, thefixed input eﬀect. Both eﬀects are favorable and, consequently, the planner attempts to lower the priceas much as possible. Provided that at least one firm operates in the transport sector, this implies that itchooses such a low price that only one firm enters into the sector.
Three comments are in order. First, it is important to note that with the price in Lemma 3, the(unique) transport firm earns 0 profit. Therefore, it is actually the average cost pricing or Ramsey-Boiteux
pricing (Ramsey (1927), Boiteux (1956)).9 To put it diﬀerently, tP solves the problem to maximize thewelfare subject to the constraint that the profit of transport firms does not become negative. Second,tP increases with c and f : a rise in the marginal and/or fixed cost makes the price higher. Finally, theindirect utility under the price regulation is given by1v ≡ v(t , 1) =
2
P

P

r

1 − 2f.
1+cSecond best optimum when the social planner can set only the number of transport firms.
Next, I examine the situation in which the social planner regulates the number of transport firms. The
following game depicts this situation. At the first stage, the planner issues permits for firms to operate
in the transport sector. For the sake of simplicity, I assume that they are allocated to firms randomly
and free of charge. At the second stage, the firms that have acquired them produce transport services.
Here, the planner controls the number of firms by changing the number of permits to issue.
Let us solve the planner’s optimization problem. Because firms’ decision on the amount of transport
services to produce is made competitively à la Cournot, their price is given by t∗ (n) for any n set by theplanner. The planner’s problem is thus to choose an integer n ≥ 1 that maximizes v ∗ (n). Now, let us
suppose for a while that the constraint that the number of firms be no smaller than 1 (n ≥ 1) is satisfied.

The first order condition without considering the integer constraint is that dv ∗ (n)/dn = Kv (n)ψ(n) = 0(see (14)). Recall that ψ(·) is a decreasing function. Since ψ(1) = 1 − f > 0 (see (5)) and ψ(1/f ) =−2(1 − f )(2 − f )/f 2 < 0, there is unique n ∈ (1, 1/f ) that solves ψ(n) = 0 or dv ∗ (n)/dn = 0. We denotesuch n by nb. Because dv ∗ (n)/dn > 0 for n < nb and dv ∗ (n)/dn < 0 for n > nb, nb gives a unique maximumof v ∗ (n). By construction, the optimal number of firms, denoted by nE , is nb when it happens to be an

integer, or either of the two integers that surround nb otherwise. In other words,

°
¢
°
¢
E


= I(b
n)
if
v ∗ I(bn) > v ∗ I(bn) + 1
 n

n
o
°
¢
°
¢
nE ∈
I(b
n), I(bn) + 1ifv ∗ I(bn) = v ∗ I(bn) + 1


°
¢
°
¢

 nE = I(bn) + 1ifv ∗ I(bn) < v ∗ I(bn) + 1 .

In addition, we need to take into account the constraint that the number of firms be no smaller than
1. It is straightforward to see that nE indeed satisfies it: because nb > 1 implies that I(bn) ≥ 1, it must
9 Indeed, one can confirm that the total cost, c (zE wE + zW wW ) + Lf [θwE + (1 − θ)wW ], divided by the amount of

production, zE + zW , becomes equal to tP , provided that n = 1, wE = wW = 1, θ = 1/2 and zE = XEW = XW E = zW .

14

be the case that nE ≥ 1. Now, we denote the indirect utility realized under the entry regulation withn = nE by v E . By construction, we have v E = v ∗ (nE ).

The mechanism working behind this result is quite simple. Suppose that the social planner reduces theregulated number of firms. This works against workers through the competition eﬀect: the competition
among firms in the transport sector becomes less severe, which results in a higher price of transport
services. On the other hand, it works in favor of workers through the fixed input eﬀect: the transport
sector needs a smaller amount of labor for a fixed input, and therefore, more labor becomes available in
the final goods sector. In this way, there are two forces working in the opposite directions. The solutionnE is a point where these two are balanced with each other (as long as we ignore the integer constraint).
Lemma 4. When the social planner can controls only the number of transport firms, it sets the number
at nE .
Comparison of the two second best optima.

One of the important policy implications of this paper

is related to a comparison of the two second best optima, that with the price regulation and that with theentry regulation. We sometimes encounter the situations in which the first best cannot be implemented
due to some political, institutional and/or historical reasons. In such situations, however, a government
might still have an option to attain one of the second best optima. In this part, we answer the questionof which is more desirable.
One may notice that there is an asymmetry between the two second best optima in the following
sense. The ideal for the social planner is that the price of transport services is as low as possible while
the number of firms is as small as possible so that the economy can save the fixed cost. In the case of
the price regulation, it can suppress the number of firms by imposing a lower price. Thus, there is notradeoﬀ between lowering the price and reducing the number of firms. In the case of the entry regulation,
however, this is no longer true. As the planner diminishes the number of transport firms, the price oftransport services determined in the market rises. Here, it faces a tradeoﬀ between a higher price andfewer firms. This asymmetry brings us a conjecture that the price regulation can achieve higher welfare
than the entry regulation.
To verify this conjecture, let us define function ν(n) as ν(n) ≡ v P − v ∗ (n), where v P is independentof n. By construction, ν(nE ) gives the diﬀerence between the level of indirect utilities at the priceregulation and that at the entry regulation: ν(nE ) = v P − v E . From the preceding discussion about theentry regulation, we know that v ∗ (n) has a unique maximum at nb ≥ 1. It immediately follows that ν(n)reaches a unique minimum at that point. Formally,<dν(n)/dn = 0>
which leads to the following result.

if

<
n = n
b
>

for any n ≥ 1,

Proposition 1. The price regulation always yields a higher level of indirect utility than the entry regulation, that is, v P > v E .
15

The proof is tedious and relegated to the appendix.

5

Market equilibrium

In this section, we derive the market equilibrium realized when transport firms freely make decisions
on the amounts to produce and on the entry and exit, and then evaluate the associated outcome, inparticular, the number of firms in the transport sector, in terms of the social optima discussed in thepreceding section.
Consider the two stage game in which transport firms make entry and exit at the first stage anddecide the price of transport services they produce at the second stage. Earlier discussion suggests thatthe solution to the second stage subgame is given by t∗ (n). This implies that the 0-profit number of firms
is equal to n = nM ≡ (2f )−1/2 , where superscript M stands for the market equilibrium (see (10)). Since
the solution to the entire game is given by (16), we haveµ∂1nM = I(nM ) = I √
,
and
2f

tM = t∗ (nM ),(17)

where tM and nM are the price and the number of firms at the market equilibrium, respectively.
Now, let us conduct some comparative statics analyses. In the first place, we examine a change inc. Suppose that it rises. First, the equilibrium number of transport firms is independent of c. Second, because the equilibrium number remains unchanged and ∂tM /∂c > 0, the equilibrium price of thetransport services rises. Lastly, let us examine the eﬀect upon the indirect utility, which consists ofthree components as has been explained (see (12)). First, we have a negative direct cost eﬀect. Second,∂tM /∂c > 0 implies that the transport price eﬀect is negative. Third and finally, the number-of-firms
eﬀect disappears since nM does not depend on c. Therefore, the total eﬀect, which becomes equal to thesum of the negative direct cost eﬀect and the negative transport price eﬀect, is negative: the rise in themarginal cost necessarily aggravates the welfare of workers.
Next, we turn to a change in the per capita fixed cost, f . Suppose that it rises. First, the rise in
the fixed cost deters entry of transport firms and/or prompts their exit and results in a decline in thenumber of firms: ∂nM /∂f ≤ 0 (see Lemma 2 i) and (17)). Second, the shrinkage of the transport sector
brings about a higher price because the competition among transport firms becomes less fierce. (Noticethat tM does not directly depend on f .) Third and finally, the direction of the eﬀect upon the indirect
utility is ambiguous. For one thing, the direct eﬀect is negative because more labor needs to be allocatedto the transport sector as a fixed input. Moreover, the transport price eﬀect disappears since tM does
not directly depend on f . In addition, the direction of the number-of-firms eﬀect is ambiguous. As has
been explained earlier, the shrinkage of transport sector gives an adverse eﬀect on the indirect utility byrelaxing the competition among transport firms (negative competition eﬀect), on the one hand, and afavorable eﬀect by saving the fixed cost (positive fixed cost eﬀect), on the other hand. When the former
dominates the latter, the indirect utility goes down as a result of the decline in the number of firms, thatis, the number-of-firms eﬀect is negative. When the opposite holds, it is positive. Because the direct16

cost eﬀect is negative, consequently, the entire eﬀect is negative when the fixed input eﬀect is too weak,whereas it is positive when the latter eﬀect is suﬃciently strong.
In addition, since ∂nM /∂f ≤ 0, the necessary and suﬃcient condition for nM ≥ 1 is that f ≤ 1/2. Inorder to focus on non-trivial cases, we assume that it is satisfied.
Now, we are ready to ask the question of whether the size of the transport sector realized at themarket equilibrium is too large or too small.
For one thing, we can say that in light of the first best optimum, too many firms enter into thetransport sector at the market equilibrium, because the first best solution involves only one firm. The
same statement holds true when we evaluate the market equilibrium in light of the second best optimum
when the social planner can control only the price of transport services.
However, if we use the second best optimum when the planner can control only the number of transport
firms as a benchmark, the result changes: depending on the values of parameters, the number of firms
in the transport sector at the market equilibrium can be too small. By the comparison of nM and nE ,we can obtain the following results: When the fixed cost in the transport sector is too high and/or the
economy is too small, too few firms enter the transport sector at the market equilibrium. Instead, when
the fixed cost is too low and/or the economy is too big, there is an excess entry.
Proposition 2.
i) A suﬃcient condition for nM ≤ nE , is that f ≥ 2/9. A necessary condition for nM < nE is, on theother hand, that f > g, where g is a real number.10ii) A suﬃcient condition for nM ≥ nE is that f ≤ g. A necessary condition for nM > nE is, on the other
hand, that f < 2/9.
The proof is relegated to the appendix.
To understand these results intuitively, note that as f rises, both nM and nb decline:£§nb 1 + 2(bn − 1)(2bn − 1)∂nM1∂bn=− √ <0
=−
< 0.
and
∂f
∂f
3f (2bn − 1)2
2f 2f
Ø
Ø
Ø
Ø
n/∂f Ø, however, nM declines faster than n
Since Ø∂nM /∂f Ø > Ø∂bb.11 Because nM is a maximum integerthat is no greater than nM and nE is one of the two integers surrounding nb, nM tends to decline fasterthan nE . Therefore, as f rises, it becomes more likely that nM becomes lower than nE , and consequently

that there are too few firms at the market equilibrium.
Two comments are worth adding.

First, when too few firms enter the transport sector at the market equilibrium, we have dv ∗ (n)/dn > 0
at n = nM . As has been discussed in Section 3.1, this implies that the competition eﬀect dominates the
fixed input eﬀect. That is, an additional firm would intensify the competition so much that the benefit
from the decline in price more than oﬀsets the loss caused by the additional expense for the fixed cost.
.
10 Numerically, we can solve for g to obtain g =0.0484.
11 Solving ψ(n̂) = 0 for f yields f = [n̂ {1 + 2(n̂ − 1)(2n̂ − 1)}]−1 . Substituting this value of f into ∂nM /∂f and ∂ n̂/∂f ,and rearranging, we can conclude that |∂nM /∂f | > |∂ n̂/∂f | if 9(2n̂ − 1)4 > 8n̂ [1 + 2(n̂ − 1)(2n̂ − 1)] > 0, which is indeed
satisfied for any n̂ ≥ 1.

17

On the other hand, when there are too many firms, the fixed input eﬀect dominates the competitioneﬀect.
Second, we have shown that the number of firms at the market equilibrium can be too small. This
makes a sharp contrast to the ‘excess entry’ result by Mankiw and Whinston (1986), and Suzumura
and Kiyono (1987), among others. They insist that the oligopolistic competition involves an excessive
number of firms if evaluated in light of the second best outcome with the entry regulation. The reason is a
so-called business stealing eﬀect: when an additional firm enters, the marginal revenue of the incumbents
decreases. This externality eﬀect, according to them, causes a gap between the firm’s benefit and thebenefit for the economy as a whole from the entry of an additional firm. Their conclusion is, however,based on a partial equilibrium setting. In the model presented here, the market of the transport services
is not independent of the market of the final goods. Indeed, entry of a firm into the transport sector
stimulates the consumption of the final goods through the competition eﬀect, that is, through the decline
in the price of transport services. This mitigates the business stealing eﬀect and consequently, there is
not always the excess entry.
Now, recall that the share of the labor employed in the transport sector, λ, increases with the numberof firms when the price is given by t∗ (n): dλ∗ (n)/dn > 0. Therefore, one can conclude that, when thenumber of transport firms at the market equilibrium is smaller (larger, resp.) than that at the entry
regulation, the share of labor in the transport sector is lower (higher, resp.) than the counterpart at theentry regulation. Hence, the next corollary immediately follows from Proposition 2.
Corollary 1. When f ≥ 2/9, the share of labor in the transport sector at the market equilibrium is nohigher than that at the entry regulation. When f ≤ g, instead, it is no lower than the counterpart at theentry regulation.

6

Concluding remarks

In this paper, we have evaluated the size of the transport sector realized by the market mechanism andhave studied the government policies against the ineﬃciencies caused by the monopoly power prevalent
in the sector. In doing so, special attention is paid to the fact that the demand for the transport services
is derived from that for the final goods. Constructing a simple Ricardian model of trade with two regions,
we have examined the price of transport services and the number of firms in the transport sector realized
at the market equilibrium and at the equilibria under three types of government interventions, namely,
the first best optimum, the price (transport cost) regulation and the entry regulation. It is shown thatthe number of firms in the transport sector at the market equilibrium is too large or too small dependingon the amount of fixed cost in the production of transport services and the size of the economy. It is also
revealed that the price regulation always yields higher welfare than the entry regulation.

18

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20

Appendix
Proof of Lemma 2.
i) Suppose that dI(x)/dx < 0 for some x. Then, there exist x0 and x00 such that x0 < x00 and I(x0 ) > I(x00 ).
The last inequality implies that I(x0 ) ≥ I(x00 ) + 1 because, by definition, I(x0 ) and I(x00 ) are integers.
Furthermore, by definition, I(x0 ) ≤ x0 and that I(x00 ) > x00 − 1 (see (15)). Combining the last threeinequalities yields x0 ≥ I(x0 ) ≥ I(x00 ) + 1 > x00 . Consequently, x0 > x00 , which is a contradiction. Hence,dI(x)/dx ≥ 0 for any x.
ii) From the definition of I(·), it immediately follows that x + z − 1 < I(x + z) ≤ x + z and thatx + z − 1 < I(x) + z ≤ x + z (see (15)). However, both I(x + z) and I(x) + z are integers. Therefore, thetwo inequalities imply that I(x + z) and I(x) + z coincide with each other.
iii) Look at the ‘if’ par, first. Since z is an integer, I(x) < z implies that I(x) ≤ z − 1. Using the factthat x − 1 < I(x) ≤ z − 1 (see (15)), we have x < z. The ‘only if’ part, on the other hand, immediately
follows from (15).
Proof of Lemma 3.
First, note that dnP (t)/dt ≥ 0 for any t, because nP (t) is an increasing function of t and I(·) is a
non-decreasing function (see Lemma 2 i)). Second, it is readily verified that°¢√
dv t, nP (t)1+t hdnP (t) (t − c)(1 − f nP (t)) i.
=−+
2f (2 + c + t)2
dt2(2 + c + t)
dt
1+t
It must be the case that nP (t) ≤ nP (t) by (16). In order that nP (t) = (t − c)/ [2f (1 + t)] ≥ 1, therefore,
we need nP (t) ≥ 1 > 0. Consequently, t > c. Hence, dnP (t)/dt ≥ 0 implies dv T /dt < 0. The social
planner chooses t that is as low as possible. However, since we must have 1 ≤ nP (t) ≤ nP (t), nP (t)cannot be lower than 1 and, therefore, t cannot be lower than tP . Hence, the optimal t is given by tP
and the corresponding number of firms is equal to 1.
Proof of Proposition 1.
It is convenient to distinguish three cases. First, suppose that nE = 1. Then, it is straightforward to see° √¢√that ν(nE ) = 1 − 2f / 2 1 + c > 0. Second, suppose that nE = 2. In this case, we have
h p
i
√
1
ν(nE ) = √3 1 − 2f − 2 2 (1 − 2f ) > 0.
6 1+c¢2 £ √§2° √Here, the inequality follows from the fact that 3 1 − 2f − 2 2(1 − 2f ) = (1 − 2f )(1 + 16f ) > 0,
because 1 − 2f > 0 due to (5). Finally, suppose that nE ≥ 3. Recall that nE is equal to either of the two
integers that surround nb, that is, either I(bn) or I(bn) + 1. If nE = I(bn), on the one hand, nE ≥ 3 implies
Ethat nb ≥ 3, because nb ≥ I(bn) (see (15)). If n = I(bn) + 1, on the other hand, nE ≥ 3 (or equivalently
I(bn) ≥ 2) implies that nb ≥ 2. Therefore, in both cases, nb ≥ 2. Now, we know that n
b is a solution toh ©
™i−1ψ(n) = 0. Solving ψ(bn) = 0 for f yields f = n
b 1 + 2(bn − 1)(2bn − 1). Substituting this value off into ν(bn), we can verify that ν(bn) > 0 if ξ(bn) > 0 where ξ(n) ≡ 12n3 − 28n2 + 21n − 6. Note that√ ¢°.
dξ(n)/dn > 0 for n > 14 + 7 /18 = 0.63. Therefore, ξ(n) is increasing in n for any n ≥ 1. This, alongwith the fact that ξ(2) = 20 > 0, implies that ξ(bn) > 0 for any nb ≥ 2. Consequently, ν(bn) > 0 for any
Enb ≥ 2. Because nb gives a minimum of ν(·), however, ν(n ) ≥ ν(bn) > 0 for any nb ≥ 2.

Proof of Proposition 2.
First, we prove the suﬃcient condition for nM ≤ nE . Suppose that f ≥ 2/9. It follows that ψ(nM ) =£ √§ √4 2f − 3f − 2 / 2f ≥ 0. This implies that nM ≤ nb because ψ(bn) = 0 and ψ(·) is a decreasing
function. Since I(·) is a non-decreasing function, however, we have I(nM ) ≤ I(bn) and nM ≤ nE by
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definition. Hence, f ≥ 2/9 is a suﬃcient condition for nM ≤ nE . Next, let us obtain the necessary
condition for nM < nE . Suppose that nM < nE . Since nE ≤ I(bn) + 1 by definition, nM < nE
implies that I(nM ) < I(bn) + 1. However, I(bn) + 1 = I(bn + 1) as has been discussed in Lemma 2 ii).
Because I(·) is a non-decreasing function, we establish nM < nb + 1. The definition of nb implies thatnM < nb + 1 or, equivalently, n̄M − 1 < nb, if and only if ψ(nM − 1) > 0. The last inequality is reduced to
338f 3 − 209f 2 + 92f − 4 > 0, which turns out to be equivalent to f > g for some real number g. Hence,f > g is a necessary condition for nM < nE . One can similarly prove that the suﬃcient condition for
nM ≥ nE and the necessary condition for nM > nE are ψ(nM − 1) ≤ 0 and ψ(nM ) < 0, respectively,
which establishes the proposition.

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