Theoretical Economics Letters, 2017, 7, 63-78
http://www.scirp.org/journal/tel
ISSN Online: 2162-2086
ISSN Print: 2162-2078

Welfare Analysis of a Market Model with
External Increasing Returns andDifferentiated Commodities
Takashi Suzuki
Meiji-Gakuin University, Tokyo, Japan

How to cite this paper: Suzuki, T. (2017)
Welfare Analysis of a Market Model with
External Increasing Returns and Differentiated Commodities. Theoretical Economics Letters, 7, 63-78.
http://dx.doi.org/10.4236/tel.2017.71007
Received: November 29, 2016
Accepted: January 13, 2017
Published: January 17, 2017
Copyright © 2017 by author andScientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

Abstract
Markets with increasing returns resulting from the positive external effects of production (hereafter external increasing returns) have been discussed from both positive and normative points of view. We present a complete general equilibrium modelin which external increasing returns prevail within industries to produce differentiated commodities which are represented by real valued functions. Firms in an industry produce the commodity characteristics which are values of each commodity
function. This formulation of differentiated commodities causes the equilibrium tobe characterized as the solution of a variational problem. In order to answer thismathematically formidable question, we utilize the delta function of Dirac for the
calculus of variations to be done in an elementary way. To our knowledge, this is thefirst time the delta function appears in economic analysis.

Keywords
External Increasing Returns, Division of Labor,
Differentiated Commodities, Delta Function

1. Introduction
In this paper, we analyze a market in which there are increasing returns resulting froma positive external effect of production. We present this idea formally, in a mathematically rigorous manner, in the following section. By increasing returns, we mean theincrease in each firm’s productivity resulting from the investments of all firms in themarket. The theory of increasing returns in this sense was initiated by Alfred Marshall[1] with the following rather mysterious descriptions.
We may divide the economies arising from an increase in the scale of productionof any kind of goods, into two classes—firstly, those dependent on the general
DOI: 10.4236/tel.2017.71007

January 17, 2017

T. Suzuki

development of the industry; and secondly, those dependent on the resources of
the individual houses of business engaged in it, on their organization and theefficiency of their management. We may call the former external economies, andthe latter internal economies (op.cit., Ch. IX, p. 221).
This ambiguous “definition” immediately invited furious debate about whether theseincreasing returns were compatible with the competitive equilibrium [2]-[7]. These
discussions further degenerated into confusion. Indeed Chipman [8] reported asfollows1.
[The compatibility of increasing returns with perfectly competitive equilibrium]
was once a lively subject of debate. The debate appears to have petered out in the
1930’s, with nobody the apparent winner. That this was the outcome seems
evident from later writings of some of the participants. Thus, Sir Dennis Robertson[9] presented in 1957 an account that was substantially unaltered from his
contribution to the 1930 Symposium on Increasing Returns, supporting thecompatibility of increasing returns with perfect competition. On the other hand,
Sir Roy Harrod [10] in 1967 was able to state flatly, without any qualification as towhether economies were internal or external, that “Increasing returns can, ofcourse only occur if competition is less than perfect”. In the contemporaryinternational trade literature, some authors maintain that perfect competition canprevail under conditions of increasing returns, provided the economies of scale areexternal to individual firms; whereas others deny the compatibility of economies
of scale with perfect competition under any circumstances, and with equal
confidence ... (op.cit., pp. 347-9).
After the concept of “externalities” in the modern sense was established by Edgeworth [11], the debates were finally terminated by Chipman (op.cit.) who provided aclear definition of external increasing returns, and showed, using the example of asingle-consumer economy with Cobb-Douglas utility, that external increasing returns
were indeed compatible with the competitive equilibria, and even with Pareto optimality (the welfare formula).
Young (op.cit.) pointed out that this idea could date back to the division of labor in
Adam Smith [12]. Young described this as follows:
The division of labor resulting in the increasing returns was limited by the extent
of the market, and conversely the extent of the market is in turn enlarged by thedivision of labor.
Consequently, he wrote:the division of labor depends in large part upon the division of labor,
and added that
this is more than mere tautology.
Young claimed that increasing returns were the source of economic growth. ThisFor details of these debates and their consequences, see also Chipman (op.cit.).

164

T. Suzuki

observation was supported by Romer [13], who interpreted them as a “knowledge
spillover”. For example, when one firm succeeds in inventing a new technology or
innovation, it will not be able to keep it secret or defend it by patent for a long period.
Eventually, the discovery will be revealed to other firms, and consequently the
productivity of the entire industry (and eventually the whole economy) will increase.
Romer also insisted that the original idea dates back at least to Adam Smith.
The idea that increasing returns are central to the explanation of long-run growthis at least as old as Adam Smith's story of the pin factory (op.cit., p. 1004).
Why is it that both Young and Romer referred to the division of labor in Smith's pin
factory? How does this relate to external increasing returns? The division of labor also
increases the productivity of the firm (factory) by dividing tasks and specializing workin the production processes. However, each separate task makes no sense in and of
itself. Although each part of the process is conducted by a single worker (or a group of
workers), all other parts are interdependent. The division of labor then presupposescooperative relationships between the workers in the factory. External increasing
returns are then simply a generalization of reciprocity within the factory to the entire
industry and/or the whole economy. As a result, external increasing returns “expand”
the entire economy: they increase productivity and the range of commodities available
to consumers, in addition to ensuring the efficiency of allocations. This is the source ofeconomic growth.
In [14], the single-consumer example in Chipman (op.cit.) was generalized to a
multi-consumer model which provides a consistent and reasonable model of a market
with external increasing returns. The basic strategy for proving the existence ofcompetitive equilibrium and obtaining the welfare formula for a model with several
consumers is to characterize the equilibrium as the solution of a constrained social
optimization problem by applying the ingenious idea of Negishi [15]. As external
increasing returns arise from the externality, the Pareto optimality of the resulting
equilibrium is not obvious. According to thewelfare formula (see Theorem 3.3), it isindeed possible that equilibrium is optimal.
The purpose of the present paper is to incorporate differentiated commodities into
the model of Suzuki (op.cit.); i.e., product diversity, in the sense of Hotelling [16],Lancaster [17], and Rosen [18], etc. The economic motivation for this mathematicalgeneralization arises directly from the concept of external increasing returns. As alreadystated, (external) increasing returns are a generalization of the division of labor. The
purpose of the latter is then not only to increase productivity but also to produce morevarieties of commodities in society as a whole2. Therefore, it is appropriate to incorporate differentiated commodities into a market model to represent a (type of) divisionof labor, rather than to work with models with fixed numbers of finitely many commodities.
The optimal solution in Suzuki (op.cit.) was that of an optimization problem of afinite dimensional space. Incorporating differentiated commodities into the model,
however, causes the equilibrium to be characterized as the solution of a variational
Adam Smith (op.cit.) emphasized this.

265

T. Suzuki

problem. In order to answer this mathematical question in an infinite dimensional
vector space, we utilize the delta function of Dirac [19] for the calculus of variations to
be done in an elementary way. To our knowledge, this is the first time the delta
function appears in economic analysis.
The remainder of the paper is organized as follows. In Section 2, we present themodel and define external increasing returns precisely. In Section 3, we present thetheorems and proofs. Theorem 3.1 (the equilibrium formula) asserts that there exists acompetitive equilibrium incorporating external increasing returns by constructing theequilibrium explicitly. This shows that the market is consistent with increasing returns.
Similarly, we compute the Pareto optimal allocations (the welfare formula, Theorem3.2). This formula enables us to compare the equilibrium and optimal allocations(Theorem 3.3). Thanks to the delta function, we do not require any mathematics
beyond the level of elementary calculus to obtain the formula. In Section 4, we provide
some remarks.

2. Economy with External Increasing Returns
2.1. A Market Model
There exist two categories of the commodities, the homogeneous input commodity(labor) indexed by 0, and the differentiated consumption commodities which are
determined by theircharacteristics indexed by t ∈ T . The index set T is simply asubset of the (open) real half-line  ++ ={t ∈  t > 0} , which will be defined below.

The amount of the commodity 0 is denoted by x 0 , z 0 , etc., and that of the commodity
characteristic t > 0 by x ( t ) , y ( t ) , and so on. Mathematically, a differentiatedcommodity is a function defined on (a subset of) T . Let us call the function a commodity bundle. Then it has a continuum of coordinates, or the commodity bundle is an
infinite-dimensional vector. Let  (T ) be the set of commodity bundles.
The economic interpretation of the differentiated commodities is as follows. Whentwo values t ∈ T and s ∈ T are “close” (in the usual mathematical sense), so are thecommodity characteristics t and s (in the economic sense). In other words, the characteristics t and s are not exactly the same, but very similar. When each characteristic contained in two commodity bundles with the same set of the characteristics areclose, they are similar, or the commodities are “differentiated”. Notice that this interpretation is impossible for the usual finite-dimensional commodity bundles for which
the coordinates are necessarily discrete.
Each commodity characteristic t ∈ T is assumed to be produced from the input
commodity 0 by a firm that is also indexed by t for convenience (hence, there exist a
continuum of firms in the economy) using the production function that embodies theincreasing returns,()

y ( t ) = ft zt0 , ζ t .

ζ t is a parameter that is determined by the market equilibrium. Hence, for firm t ,the value of ζ t is given by the market and not determined by the firm. The parameterζ t is assumed to be the total input, ζ t = ∫zs0 ds ; therefore, it depends on the amountof input used by other firms in the economy (including t ). That is to say, these66

T. Suzuki

increasing returns come from (positive) externalities.
We have to specify the range of the externality to extend. For simplicity, consider afinite number of (measurable3) subsets {T1 Tn } of  such that

{

{

}

sup t j − s j t j , s j ∈ T j < inf t j − s j t j ∈ T j , sk ∈ Tk

} for all j, k = 1 n with j ≠ k .

We set T = ∪nj =1 T j and call T j the industry j . The above condition implies that any
characteristics in the same industry are closer than those of different industries.
It is important to distinguish the commodity vectors (bundles) from their characteristics. The industry j produces and supplies its own commodity bundles (hence

there exist n distinct commodities in the economy) which are functions x ( ⋅) ormembers of an appropriate function space  (T j ) on T j , and the characteristicx ( t ) is their value at t ∈ T j which is not traded among the consumers. They are“intermediate goods” which constitute the commodity j supplied by the industry j .

( )
( )

Of course for each t * ∈ T , we can consider the commodity bundle x t ; t * whichcontains only the characteristic t * defined by x t ; t * = 1 for t = t * and x t ; t * = 0( )otherwise. But this commodity would not have any economic significance, since its
market values are always 0, as shown later on4. Functions on T j can be naturally
extended to T setting as 0 outside T j . Then we can write  (T=)  (T1 ) ⊕  ⊕  (Tn )(direct sum).

Chipman, Romer and Suzuki assumes the function ft to be homogeneous of degree
1 in z 0 ,(Hom) ft λ zt0 , ζ t = λ ft zt0 , ζ t for all λ ≥ 0 ,

(

)

(

)

and it is monotonically increasing in ζ t ,(Mon) ft zt0 , ζ t ≤ ft zt0 , ζ t′ whenever ζ t ≤ ζ t′ .

(

)

)

(

Then, by these two conditions, the final output y ( t ) is produced under “increasing
returns to scale”; namely,

(

)

(

)

(

ft λ zt0 , λζ t =λ ft zt0 , λζ t ≥ λ ft zt0 , ζ t) for all λ ≥ 1.

(2.1)

As ζ t is a parameter for the firm t , the condition (Hom) implies that firm t
operates subjectively under constant returns; hence, there exist no problems involved
with the “internal increasing returns (convex production functions)5.” Furthermore,since ζ t = ∫ zs0 ds for t ∈ T j , the condition (Mon) means that the more inputs in theTjindustry j , the more productive the technology of each firm becomes. This is thesource of the “increasing returns for a whole industry”, and we can interpret it to be akind of reciprocity that is naturally incorporated into the markets. “Naturally
incorporated into the market” means that the value of the parameter ζ t is determined
endogenously, not given or postulated from outside of the market.

There exists m consumers indexed by i ( = 1 m ) . Following Chipman (op. cit.),A measurable set is a set for which its (Lebesgue) measure is well defined. Hence the integral3∫ x ( t ) dt isTjalso well defined for any integrable function x ( t ) .
In order to facilitate this point, we have to treat the commodity bundles as the equivalent classes of functions

4

modulo null sets on T . Then x ( t ; t * ) is equivalent with the commodity 0, the constant 0 function.

For the theoretical problems arising from convex (not concave!) production functions, see Scarf [20] andOddou [21]. Arrow [22] discussed the possibility that a kind of increasing returns exists in a more specificmodel.
567

T. Suzuki

consumer i ’s utility function takes the log-linear (Cobb-Douglas) form,=ui ( xi ( t ) )β ( s ) log xi ( s ) ds, i 1 m,
∫=
T i

(2.2)

where for all t ∈ T , βi ( t ) ≥ 0 for each i and ∑ i =1βi ( t ) > 0 , and ∫ βi ( s ) ds = 1 ,Ti = 1 m . Precisely speaking, this utility function is not a function but a (non-linear)mfunctional that is a map assigning a real value to the function x ( ⋅) .

It is well known and can be verified easily that for log-linear utility functions, all of
the standard assumptions such as continuity (in an appropriate sense6), (strong)
concavity and monotonicity (the more consumption, the higher utility) and so on aresatisfied. Note that ui does not include commodity 0 consumption. Hence, theconsumers do not demand commodity 0; rather, they supply it inelastically for as longas they own it. Chipman (op.cit.) observed that this assumption makes formulae very
clean without losing essence of discussion. We will then follow him.
We assume that consumer i is endowed with ωi ( > 0 ) units of commodity 0 as an
initial endowment, but does not have an initial endowment for commodity t . SetΩ =∑ i =1ωi .
m

The market price of commodity 0 is denoted by w , and the price of commodity t
by p ( t ) . Then we can define the competitive equilibrium of this market in thestandard manner.

(

Definition 2.1. An m + 1 -tuple of the input and the consumption bundleszˆt0 , xˆi ( t ), i = 1 m , and the prices ( wˆ , pˆ ( t ) ) are called the competitive equili-)t∈Tbrium if and only if the following conditions are satisfied.
(E-1) xˆi ( t ) maximizes ui ( ⋅) subject to∫T pˆ ( s ) xi ( s ) ds ≤ wˆ ωi , i = 1 m , ˆ ˆ ˆ0(E-2) pˆ ( t ) ft  z , ∫ zˆs0 ds  − wz
≤ pˆ ( t ) ft  zˆt0 , ∫T zˆs0 ds  − wz
0 for all z ≥ 0 and for
t =
Tj
j




all t ∈ T j , j = 1 n ,
(E-3)





∑ i =1xˆi ( t ) = ft  zˆt0 , ∫T j zˆs0 ds  for all t ∈ T j , j = 1 n , and ∫T zˆs0 ds = Ω .
m

The economic meanings of these conditions are clear and do not require detailed
explanations, apart from the production externality. Condition (E-1) is the utility
maximization condition (under the budget constraint) and condition (E-2) is the profit
maximization condition. As the firms perceive themselves that they operate underconstant returns to scale, they earn zero profit in the equilibrium. The first condition of(E-3) states that the total amount of the characteristic t which is consumed is equal towhat is produced. Recall that the second variable (parameter) of the production
function is the total input used within each industry T j . These inputs sum to the total
input of the whole economy∑ j =1∫T j zs0 ds = ∫T zs0 ds , and the latter is equal to the totalnendowment of the economy (the resource is not wasted). This is consistent with thefirst condition of (E-3), which states that supply equals demand in the input
commodity market.
An important remark on the concept of prices in this economy should be in order.
We have to specify a topology on the space of functions x ( t ) to make the meaning of ``continuity'' ma-6thematically rigorous. We set aside of this problem here, because we do not discuss topologies in the present
paper. For a precise discussion, see [23].

68T. Suzuki

Strictly speaking, given the price function p ( t ) as in Definition 2.1, the market priceis a (non-negative) linear functional p on  (T ) defined by px ( t ) = ∫ p ( s ) x ( s ) ds
Tfor every x ( t ) ∈  (T ) . Since the characteristic t is not traded by consumers, eachvalue p ( t ) of the functional is not directly observable for them, or p ( t ) is thehedonic price in the sense of Rosen (op.cit). To see this, suppose that the price functional is strictly positive, p ( t ) > 0 for all t ∈ T . If p ( t * ) is observable in the market
for some t * ∈ T , it must be the market value of the commodity x ( t ; t * ) defined above.

( )

(

)

( )

*
However, we have px t ; t * =0 ≠ p t * ( > 0 ) . This is exactly what
∫T p ( s ) x s; t ds =we stated earlier. Here the industry j is acting as a fictitious producer “selling” theconsumption bundle x ( ⋅) ∈  (T j ) as a final output at the price p and “buying”

infinitely many characteristics x ( t ) ∈  as inputs at the prices p ( t ) .
Our formulation of the differentiated commodities forces us to present the set offirms producing commodity characteristics as a continuum T . On the contrary, the setof consumers is a finite set {1 m} . Therefore one might say that the latter should also
be a continuum. We answer this conceptual question by noticing that our model can
include a continuum of consumers, following the well known idea (e.g., [24]). Let

I = [ 0,1] be the set of consumers which is the unit interval. Define the sets
i −1
i
m −1



≤ a < =≤ a ≤ 1 .
, i 1 m − 1 , I m = a ∈ I
I i ( i = 1 m ) by I i = a ∈ I
m
m
m




We define the consumption sector of the economy as a “simple map”(a  ui ( xi ( t ) , ωi )) for a ∈ I ( i = 1 m ), here we have defined the allocationsixa ( t ) for a ∈ I is also as a simple map; xa ( t ) = xi ( t ) for a ∈ I i . Then this model

presenting a continuum economy with “ m types” of consumers formally coincides
with the finite economy of the present paper.
The next example, although economically simple, illustrates the mathematicalstructure of the problem and a technical devise called the delta function for elucidating
it will be introduced.

2.2. A Simple Example
There exists one consumer m = 1 with the utility functionu ( x ( t ) ) = ∫ β ( s ) log x ( s ) ds
T

(2.3)

such that β ( t ) ≥ 0 for all t ∈ T and ∫ β ( s ) ds = 1 . The consumer’s initial endowT
ment ω ( > 0 ) is the total endowment (resources) of the economy. Suppose for simplicity that n = 1 so that T1 = T (there exists only one industry in the economy).
In order to elucidate the functional calculus in an elementary way, the celebrated

Dirac’s delta function is used. For any t ∈ T , the “function” δ ( t ) on  is definedas 0 for t ≠ 0
δ (t ) = 
+∞ for t = 0
and assumed to satisfy(2.4)∫δ ( s ) ds = 1 . From this and the definition, we obtain thatg ( t ) for any function g ( s ) (Dirac (op.cit., p.59)).
∫T g ( s ) δ ( t − s ) ds =Let f ( x ) be a differentiable function. We define69

T. Suzuki

df ( x ( s ) )≡ lim

dx ( t )f ( x ( s ) + hδ ( t − s ) ) − f ( x ( s ) )
h

h →0

=
f ′ ( x ( s ))δ (t − s ).

(2.5)

This is a fundamental mathematical formula that will be used throughout the paper.

∫T f ( x ( s ) ) ds with respect to x ( t ) ,df ( x ( s ) )dds ∫ds ∫ f ′ ( x ( s ) ) δ (=
=
=
f
x
s
t − s ) ds
(
)
(
)
∫
T
T
dx ( t ) T
dx ( t )Then we can differentiatef ′ ( x (t )).

(2.6)

The consumer maximizes the utility function (functional) (2.3) subject to the budget
constraint∫T p ( s ) x ( s ) ds = wω . Mathematically speaking, this requires us to solve a

constrained variational problem. To this end, we differentiate the constrained Lagrangian with the multiplier µ=

∫T β ( s ) log x ( s ) ds + µ ( wω − ∫T p ( s ) x ( s ) ds )(2.7)

in x ( t ) and obtain the first-order condition (FOC)β (t )d=− µ p ( t ) =0.
dx ( t ) x ( t )(2.8)

It follows from the FOC that ∫ β ( s ) ds − µ ∫ p ( s ) x ( s ) ds =0 ; hence, µ = 1 wω .
T
T
Therefore, the demand function for x ( t ) is given by x ( t ) = ( β ( t ) ω ) ( w p ( t ) ) . The
equilibrium relative price wˆ pˆ ( t ) can be obtained from the firm’s profit conditionpˆ ( t ) ft (1, ω ) = wˆ .

The above calculation of equilibrium for the one-consumer economy is straightforward. Our economic problem is to compute the competitive equilibrium for themulticonsumer economy. This will be achieved in the next section, using the method of
Negishi (op.cit.).

3. Existence and Efficiency of the Equilibrium
3.1. The Competitive Equilibrium
Given
=
ζ

(ζ 1 ζ n ) ∈ [0, Ω] , consider the constrained variational problemnm

P (ζ ) :Maximize ∑α i ui ( xi ( t ) ) subject toi =10
1 n, and ∫ zs0 ds ≤ Ω,
∑xi ( t ) ≤ f=t ( zt , ζ j ) , t ∈ T j , j
T
m

i =1

where α1 α m are welfare weights of consumers satisfying α i ≥ 0 and we normalizemmγ 1} be the unit simplex. Then α ∈ ∆ . Notice∆ {( γ i ) ∈  m+ ∑ i ==∑ i =1α i = 1 . Let =1 ithat the normalization of the welfare weights is arbitrary, but each normalization
determines a price normalization, as we will see later on. The solution of this problem isa saddle point of the constrained Lagrangianα ,ζ( ( x ( t ) ) , z , p ( t ) , w) = ∑α ∫ β ( s ) log x ( s ) ds
i

0
t

m

i =1

i T

i

i

n
m


+ ∑ ∫ p ( s )  f s zs0 , ζ j − ∑xi ( s )  ds
Tj
=j 1 =
i 1



(

(

)

+ w Ω − ∫ zs0 ds ,
70

T

)

(3.1)

T. Suzuki

where p ( t ) and w are the multipliers (a fortiori they will be the equilibrium prices).
The saddle point is unique by the strict concavity of the utility functional. Let thesaddle point be ( xˆi ( t ) ) , zˆt0 , pˆ ( t ) , wˆ which satisfies()( ( x ( t ) ) , z , pˆ ( t ) , wˆ ) ≤  ( ( xˆ ( t ) ) , zˆ , pˆ ( t ) , wˆ ) ≤  ( ( xˆ ( t ) ) , zˆ , p ( t ) , w) (3.2)
for every ( ( x ( t ) ) , z , p ( t ) , w ) . For each α ∈ ∆ , define a map
α ,ζ

0
t

i

α ,ζ

0
t

i

α ,ζ

0
t

i

0
t

i
n

Φ : ∆ × [ 0, Ω ] → ∆ × [ 0, Ω ] ,
n




0
Φ (α i ) , ( ζ j ) = (αi ) ,  ∫T j zˆs ds  




)

(

where α = (αi ) is defined by

{

(3.3)

}

max 0, α i + wˆ ωi − ∫T pˆ ( s ) xˆi ( s ) ds=
, i 1 m.
m
ˆ
ˆ
ˆ
α
+
w
ω
−
p
s
x
s
s
max
0,
d
∑
i
i
∫T ( ) i ( )

=
αi

{

i =1

}

(3.4)

∑ i =1 max {0, α i + wˆ ωi − ∫T pˆ ( s ) xˆi ( s ) ds} ≠ 0 , since otherwise we would
have α i + wˆ ωi − ∫ pˆ ( s ) ωi ( s ) ds ≤ 0 for all i = 1 m . Then it follows from (3.2) that
T
m
m
m
ˆ
ˆ
1
=
α
≤
p
s
α (αi ) ∈ ∆∑i 1 =ii ( s ) ds − w∑ i 1ωi ≤ 0 , a contradiction. Hence =∫T ( ) ∑ i 1xˆ==m

Note thatis well defined.

Suppose that there exists a fixed point7(((αˆ ) , (ζˆ )) of the map Φ ,
i

j




( )) = (αˆ ) ,  ∫ zˆ ds   .
Φ (αˆi ) , ζˆ j

0
Tj s

i

( ( xˆ ( t ) ) , zˆ , pˆ ( t ) , wˆ ) associated with the fixed

We will show that the saddle point

0ti

point is a competitive equilibrium. We first note that the equilibrium price vector( pˆ ( t ) , wˆ ) is strictly positive. Suppose pˆ ( t0 ) = 0 for some t0 . Then defining a newallocation ( yi ( t ) ) by yi ( t ) = xˆi ( t ) for t ≠ t0 , y=xˆi ( t0 ) +  for some  > 0 ,i ( t0 )
0
0
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
α ,ζ ( yi ( t ) ) , zt , p ( t ) , w > α ,ζ ( xi ( t ) ) , zt , p ( t ) , w , contradicting the first inequalityof (3.2). wˆ > 0 can be proved similarly. From the saddle point property (3.2) together

(

(

)

)

with the strict positivity of prices, we conclude that the constraints of the problem
P (ζ ) hold with the exact equalities,



m



ft  zˆt0 , ∫ zˆs0 ds  , t ∈ T j , j =1 n, and ∫ zˆs0 ds =Ω,∑xˆi ( t ) =
Tj
T


i =1

or the equilibrium condition (E-3) of Definition 2.1 is met.
Next we claim αˆi > 0 for all i = 1 m . If not, αˆl = 0 for some l . It follows from
(3.4) that 0 < wˆ ωi ≤ ∫ pˆ ( s ) xˆi ( s ) ds , hence xˆi (t ) > 0 . Since α̂ ∈ ∆ , αˆ h > 0 for someTh . Define a new allocation ( yi ( t ) ) by yl ( t ) = 0 ,y=xˆh ( t ) + xˆl ( t ) and y k ( t ) = xˆk ( t ) for k ≠ l , h . Thenh (t )()

f s zˆs0 , ∫ zˆs0 ds = ∑ i =1yi ( s ) andTm() ∑αˆ u ( xˆ ( t ) ) < ∑αˆ u ( y ( t ) )=  ( ( y (( t ) ) , zˆ , pˆ ( t ) , wˆ ) ) ,αˆ ,ζˆ ( xˆi ( t ) )=, zˆt0 , pˆ ( t ) , wˆ

m

m

i i
i
=i 1 =i 1αˆ ,ζˆ

i

i i

i

0
t

We don’t have to prove the existence, since we will compute it explicitly in the following; see Theorem 3.1.

771

T. Suzuki

contradicting the first inequality of (3.2).

Setting (α i ) = (αˆi ) and (ζ j ) =  ∫ zˆt0 dt  in the FOC's for α ,ζ , we obtain Tj∂αˆ ,ζˆ=αˆi βi ( t ) xˆi ( t ) − pˆ ( t ) =0, i =1 m, t ∈ T ,

(3.5)


 ˆ ˆ01 n.
=pˆ ( t ) ft  zˆt0 , ∫T zˆs0 ds  − wz
t =0, t ∈ T j , j =
j



(3.6)

∂xi ( t )∂αˆ ,ζˆ∂zt0

Since α̂ is a fixed point, it follows from the definition (3.4) that

=
αˆi

(
(

αˆi + wˆ ωi − ∫T pˆ ( s ) xˆi ( s ) ds))=
, i 1 m,
m
1 + ∑ wˆ ωi − ∫T pˆ ( s ) xˆi ( s ) ds
i =1

(3.7)

and from this we obtain thatm()

wˆ ωi − ∫ pˆ ( s ) xˆi ( s ) ds =αˆi ∑ wˆ ωi − ∫ pˆ ( s ) xˆi ( s ) ds , i =
1 m.
T

i =1

T

(3.8)

The conditions αˆ i > 0 , (3.6) and (E-3) imply that the right-hand side of equation(3.8) is equal to 0; hence, the budget constraint ∫ pˆ ( s ) xˆi ( s ) ds = wˆ ωi follows. Let
xi ( t ) be such thatT

∫T pˆ ( s ) xi ( s ) ds ≤ wˆ ωi . Setting xh ( t ) = xˆh ( t ) for h ≠ i , we obtainfrom the first inequality of (3.2)αˆi ui ( xi ( s ) ) − ∫T pˆ ( s ) xˆi ( s ) ds ≤ αˆi ui ( xˆi ( s ) ) − ∫T pˆ ( s ) xˆi ( s ) ds
thenαˆi ui ( xi ( s ) ) ≤ αˆi ui ( xˆi ( s ) ) + ∫T pˆ ( s ) ( xi ( s ) − xˆi ( s ) ) ds ≤ αˆi ∫T βi ( s ) log xˆi ( s ) ds,
or ui ( xi ( s ) ) ≤ ui ( xˆi ( s ) ) , hence the equilibrium condition (E-1) is met. Similarly theequilibrium condition (E-2) follows from (3.2) and (3.6).
Integrating (3.5) and using the budget constraint, we have=αˆi wˆ=ωi , i 1 m.

(3.9)

This is the celebrated Negishi condition (op.cit., p. 97); the welfare weight of aconsumer is the inverse of his/her marginal utility of income, which is the Lagrangian
multiplier of the consumer’s maximization problem (see the example of the lastmsection). As ∑ i =1αˆi = 1 , the equilibrium price has been normalized aswˆ = Ω −1 .

(3.10)

Summing (3.5) over i with help of (3.6), (3.9) and (E-3), we obtainmm ˆ ˆ0ˆ ∑βi ( t ) ωi pˆ=w=zˆt0 , ∫T zˆs0 ds  wz
( t ) ∑xˆi ( t ) pˆ ( t ) ft =
t ,
j


=i 1 =i 1

hence zˆt0=m∑βi ( t ) ωi , t ∈ T .
i =1(3.11)

For convenience, we set φ j zˆt0 = ft  zˆt0 , ∫ zˆs0 ds  , t ∈ T j . Then we have
Tj



( )

( )

( )

pˆ ( t ) =zˆt0 wˆ φ j zˆt0 =zˆt0 Ωφ j zˆt0 , t ∈ T j , j =1 n,
72(3.12)

T. Suzuki β (t )ω i
 φ j zˆt0 , t ∈ T j , j= 1 n, i= 1 m.
xˆi ( t )=  m i
 β t ω ∑ i( ) i  i =1( )Therefore, we have proved
Theorem 3.1 (Equilibrium Formula) The competitive equilibrium

uniquely exists and is given by (3.10), (3.11), (3.12) and (3.13).

(3.13)( zˆ , xˆ ( t ) , wˆ , pˆ ( t ) )0ti

3.2. The Welfare Formula
In this section, we consider the social optimization problemm

P :Maximize ∑α i ∫T βi ( s ) log xi ( s ) ds subject toi =1) ≤ φ j ( zt0 ) , t ∈ T j , j 1 n,∑xi (t=m∫T zs ds ≤ Ω.
0

and

i =1

As before, α1 α m are the welfare weights of consumers satisfying α i ≥ 0 andm∑ i =1α i = 1 . The solution8 zt0 , ( xi ( t ) ) of this problem is a Pareto optimal allocation,()

and it is a saddle point of the constrained Lagrangian

(

)

m

α ( xi ( t ) ) , zt0 = ∑α i ∫T βi ( s ) log xi ( s ) ds
i =1

(

)

n
m


+ ∑ ∫T µ ( s )  φ j zs0 − ∑xi ( s )  ds + ν Ω − ∫T zs0 ds ,j1
i
=j 1 =( )where µ ( t ) and ν are the multipliers. The FOC's for α are∂α1 m, t ∈ T ,=α i βi ( t ) xi ( t ) − µ ( t ) =0, i =∂xi ( t )(3.14)∂α= µ ( t ) φ ′j zt0 −ν = 0, t ∈ T j , j = 1 n,∂zt0(3.15)

( )





=
φ ′j zt0ft 1, ∫T zs0 ds  + ∂ζ ft  zt0 , ∫T zs0 ds  . The second term represents the
where
j
j




externality effect of production.

( )The problem is: For which value of α i should we compare the competitive allocazˆt0 , ( xˆi ( t ) ) and the efficient allocation zt0 , ( xi ( t ) ) ? The answer is obviously

tion

(

)

(

)

given by the Negishi condition (3.9). From the normative point of view, this seems tobe justifiable only when the initial endowments of the consumers are at least nearly
identical. We will discuss this point further in the next section.
Setting α i = wˆ ω in (3.14) and summing over i , we havem

( )(3.16)( )  =ν , t ∈ T , j =1 n.
( ) 

(3.17)mwˆ ∑βi=xi ( t ) µ ( t ) φ j zt0 , t ∈=T j , j 1 n.
( t ) ωi µ ( t ) ∑==i 1 =i 1

Substituting this into (3.15), it follows that φ ′j zt0mwˆ ∑βi ( t ) ωi 
 φ j zt0
i =1


j

Remember that the solution is unique.

873

T. Suzuki

Multiplying zt0 with (3.17), integrating over T and summing over j , we obtain
m

( )
( )

 φ ′j zs0 zs0 
 ds.
 φ j zs0 nν = wˆ Ω −1 ∑∑ ∫T βi ( s ) ωi j=i 1 =j 1(3.18)( ) of firm t in industry j as
 φ′ ( z ) z  t== (z ) ∈ T , j 1 n.
 φ (z ) We define the elasticity  j zt0
j

0
t

j

0
t

j

0
t

(3.19)

j0t

Then substituting (3.18) and (3.19) into (3.17), we obtain



0

 m


z
j
t
 ∑βi ( t ) ωi , t ∈ T j , j 1 n.
=zt0 
= −1 m n i =10 Ω ∑∑ ∫T j βi ( s ) ωi  j zs ds 
=
i
1
=
j
1



( )

( )

(3.20)

From (3.16) and (3.14), we finally obtain β (t )ω i
==
φ j zs0 , t ∈ T j , j 1 n.
xi ( t )  m i
 β t ω ∑ i( ) i  i =1( )We have thus proved the following.
Theorem 3.2 (Welfare Formula) The optimal allocation

Formulaes (3.20) and (3.21).

(3.21)( z , x ( t ) ) obeys the0tiFrom Theorems 3.1 and 3.2, we can immediately deduce the next theorem which was,for economies with finitely many homogeneous commodities, first stated by Chipman(op.cit., p.365) for the case m = 1 with homogeneous commodities.
Theorem 3.3. Suppose that every consumer has a Cobb-Douglas utility functional

and the same amount of labor as an endowment. Then, the optimal output (with thewelfare weights proportional to the endowments) of the t -th product is greater than,equal to, or less than the competitive level of output according as the elasticity (at the
optimum)  j ( zt0 ) of the firm t in the industry j is greater than, equal to, or lessthan the weighted average of the elasticities of all industries,
m
n
Ω −1=
βi ( s ) ωi  j ( zs0 ) ds . In particular, if all firms’ elasticities are equal to the∑ i 1∑=j 1∫Tjweighted average, the competitive equilibrium is Pareto optimal.

4. Discussion
4.1. Formalism of Differentiated Commodities
Ideas of differentiated commodities similar to that of the present paper date back at
least to Hotelling (op.cit.). He considered a duopoly market model in which consumersof a commodity are distributed uniformly on a segment of the real line, and the two
firms supplying this commodity are located at fixed places (their shops) on the segment.
Each consumer demands one unit of the commodity and must go to a shop, thereby
incurring a fixed cost of c for each unit of distance moved. Therefore, the consumers
will go to the nearest shop when the two firms charge the same price. Firms aim to set
74

T. Suzuki

their prices to attract more customers and consequently to earn more profits. In thismodel, it is possible to interpret the commodity to be differentiated according to thecoordinate of the real line (the location of the consumers), t ∈  .
Our formulation of commodity differentiation, however, follows Lancaster (op.cit.)and Rosen (op.cit.). According to them, consumers are assumed to obtain their utility
not directly from the commodity itself, but from the commodity’s characteristics
exactly as in this paper. They assumed that there exist two (indexes of) characteristics
and that the space of the commodity characteristics is a compact subset of T of  2 .
In particular, in the Rosen model, every characteristic t ∈ T has its own “price” p ( t ) ,although it is not revealed in the market; only the commodities—rather than theircharacteristics—are traded in the market. Rosen called p ( t ) the hedonic price (thetitle of his paper) of characteristic t .
Mas-Colell [25] and Jones [26] generalized these ideas of commodity differentiation
into an abstract concept that treats the differentiated commodity as a measure (distribution) m on a compact metric space T . The measure m is a countably additiveand real valued set function on T , meaning that for each set (precisely each “measur-able” set) B , m ( B ) is a real value interpreted as an amount of the commodity m ,that contains a portion B ⊂ T of the characteristics. Countable additivity means that∞m ∪i∞=1 Bi =∑ i =1m ( Bi ) for all countably many pairwise disjoint Bi s.

()

In this definition, the commodities are set functions, not simply functions, as in ourpresent paper. Mas-Colell justified this concept by saying “the choice problem [of thedifferentiated commodities model] is not typically how much of each (perhaps individually insignificant) commodity [characteristic] to buy, but which commodity to buy.”
We agree that the measures are conceptually more appropriate than the functions.
However, there are mathematically formidable complications in the commodity spaceof the measures, and the paper of Mas-Colell is very difficult to digest. Jones (op.cit.)
simplified Mas-Colell’s proof considerably. Khan-Suzuki [27] further elaborated theseworks.

4.2. Relation with Monopolistic Competition
The competitive equilibrium presented in this paper can be interpreted as amonopolistically competitive equilibrium in the long run. For simplicity, consider the
one consumer economy of the example in Section 2.2.

If the firm t behaves monopolistically, it does not take the price p ( t ) as given,
but uses the true (inverse) demand function p ( t ) = ( β ( t ) ω w ) x ( t ) to compute profit.
Substituting this into the profit function=
π ( t ) p ( t ) ft ( zt , ω ) − wzt , with=π ( t ) w ( β ( t ) ω − zt ) . The maximum profit is equal to 0,x ( t ) = ft ( zt , ω ) , yields

rather than wβ ( t ) ω . This will be explained as follows. To maximize the profit, firmt must set zt = 0 , or leave the market, in which case profit is π ( t ) = 0 . Consequently,=
x ( t ) f=
( 0, ω ) 0 . However, this situation is unstable because p ( t ) = +∞ forx ( t ) = 0 , which implies that a firm supplying something earns an infinite profit. Then,the firm enters the market, and sets zt = β ( t ) ω (any other values are inconsistent
with profit maximization). The resulting profit is equal to 0, and this is a monopolistically competitive equilibrium that is stable in the long run. Because
75

T. Suzuki=x ( t ) f=
( β ( t ) ω , ω ) f (1, ω ) β ( t ) ω , the monopolistically competitive equilibriumcoincides with the competitive equilibrium. This is exactly what we wanted to show.

5. Conclusions
In this paper, we have considered a theoretical model in which the external increasing
returns increase the productivity of firms that produce products which are the characteristics of differentiated commodities. We have demonstrated that such a market
model is compatible with competitive equilibrium, and even with Pareto optimality.
The external increasing returns are essentially a type of positive externality thatworks outside markets through the background institutions of societies. Romer’s“knowledge spillover effect” is an example of a positive externality. We can also include
educational systems, as workers’ skills gained through education contribute significantly to a firm’s productivity. These institutions organize participants’ activities, integrating them such that they can contribute to the whole economy (and society). Whenthe institutions work well, society behaves reciprocally and is well ordered. Adam Smith
recognized reciprocity occurring within the division of labor in pin factories and placedthese observations at the start of his book, suggesting its importance; his true motive
was not to promote unrestricted laissez-faire.
Neoclassical economic theory has emphasized the efficiency of markets, realized
through the price mechanism. It has been inclined to shun externalities as a cause of
market failures. The presence of external increasing returns, however, supports an
opposing view that highlights the significance of background institutions. We mustbear in mind that economies do not simply work through market mechanisms; theymust be supported by various institutions, both social and political. Unlimited laissezfaire is not appropriate, so it is essential to establish effective institutions that facilitate
reciprocity in societies.
Finally, we point out some open questions that remain unanswered in regard to ourmodel. The first issue is our restriction of utility functions in order to compute exact
formulae. Obviously, proving the existence of equilibria in a general setting is the firsttask for pure theorists. Moreover, we may ask if we can obtain any general results
concerning optimality. The welfare formulae seem to suggest that the optimal states are“on a knife edge” and are attained “by accident.” If so, can this be stated more precisely?
In other words, can the “size” of the optimal states in an appropriate parameter space
be estimated? The second issue is the incompleteness of our formulation for the
production process of differentiated commodity bundles as final outputs. As statedabove, the theoretical status of industry j was “fictitious,” and thus a precise andcomplete description of it is strongly desired in future work.

Acknowledgements
I would like to thank Sonia Tu for her warm help and encouragement. Discussions with
M. Ali Khan and Nobusumi Sagara are also helpful. Comments of an anonymous referee are gratefully acknowledged. Remaining errors and shortcomings are of course myown.
76

T. Suzuki

Supported
This research is supported by a Grant-in-Aid for Scientific Research (No. 15K03362)
from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

References[1]

Marshall, A. (1890) Principles of Economics. Macmillan, London.

[2]

Graham, F.D. (1923) Some Aspects of Production Further Considered. Quarterly Journal of
Economics, 37, 199-227. https://doi.org/10.2307/1883929[3]

Graham, F.D. (1925) Some Fallacies in the Interpretation of Social Cost. A Reply. Quarterly

Journal of Economics, 39, 324-330. https://doi.org/10.2307/1884879[4]

Knight, F. (1924) Some Fallacies in the Interpretation of Social Cost. Quarterly Journal of
Economics, 38, 582-606. https://doi.org/10.2307/1884592[5]Knight, F. (1925) On Decreasing Cost and Comparative Cost. A Rejoinder. Quarterly Journal of Economics, 39, 331-333. https://doi.org/10.2307/1884880[6]

Sraffa, P. (1926) The Laws of Return under Competitive Conditions. Economic Journal, 36,
535-550. https://doi.org/10.2307/2959866[7]

Young, A. (1928) Increasing Returns and Economic Progress. Economic Journal, 38, 527542. https://doi.org/10.2307/2224097[8]

Chipman, J.S. (1970) External Economies of Scale and Competitive Equilibrium. Quarterly
Journal of Economics, 84, 347-385. https://doi.org/10.2307/1879425[9]Robertson, D.H. (1957) Lectures on Economic Principles. Staples Press, London.

[10] Harrod, R.H. (1967) Increasing Returns. In: Kuenne, R.E., Eds., Monopolistic Competition
Theory: Studies in Impact; Essays in Honor of Edward H. Chamberlin, John Wiley andSons, Inc., New York, 63-79.
[11] Edgeworth, F.Y. (1925) Papers Relating to Political Economy I-III. Macmillan and Co Ltd.,
London.
[12] Smith, A. (1776/1976) The Wealth of Nations. Reprint Edited by Cannan, E., University of
Chicago Press, Chicago.
[13] Romer, P. (1986) Increasing Returns and Economic Growth. Journal of Political Economy,
94, 1002-1037. https://doi.org/10.1086/261420[14] Suzuki, T. (2009) General Equilibrium Analysis of Production and Increasing Returns.
World Scientific, Singapore. https://doi.org/10.1142/6953[15] Negishi, T. (1960) Welfare Economics and Existence of an Equilibrium for a Competitive
Economy. Metroeconomica, 12, 92-97. https://doi.org/10.1111/j.1467-999X.1960.tb00275.x[16] Hotelling, H. (1929) Stability in Competition. The Economic Journal, 39, 41-57.
https://doi.org/10.2307/2224214[17] Lancaster, K. (1975) Socially Optimal Product Differentiation. The American Economic
Review, 65, 567-585.
[18] Rosen, S. (1974) Hedonic Prices and Implicit Markets: Product Differentiation in Pure
Competition. Journal of Political Economy, 82, 34-55. https://doi.org/10.1086/260169[19] Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. 4th Edition, Oxford University Press, London.
[20] Scarf, H.E. (1986) Notes on the Core of a Production Economy. In: Hildenbrand, W. and
Mas-Colell, A., Eds., Contributions to Mathematical Economics: In Honor of Gerard Debreu, North-Holland, Amsterdam, 401-429.
[21] Oddou, C. (1976) Theoremes d’Existence et d’Equivalence pour des Economies avec Pro77

T. Suzuki
duction. Econometrica, 44, 265-281. https://doi.org/10.2307/1912723[22] Arrow, K.J. (1962) The Economic Implications of Learning by Doing. Review of Economic
Studies, 29, 155-173. https://doi.org/10.2307/2295952[23] Suzuki, T. (2016) Market Equilibrium as a Constrained Optimal Solution. Forthcoming inInternational Journal of Applied and Experimental Mathematics.
[24] Hervés-Beloso, C. and Moreno-Gracía, E. (2008) Competitive Equilibria and the Grand
Coalition. Journal of Mathematical Economics, 44, 697-706.
https://doi.org/10.1016/j.jmateco.2006.11.002[25] Mas-Colell, A. (1975) A Model of Equilibrium with Differentiated Commodities. Journal of
Mathematical Economics, 2, 263-296. https://doi.org/10.1016/0304-4068(75)90028-2[26] Jones, L. (1984) A Competitive Model of Commodity Differentiation. Econometrica, 52,
507-530. https://doi.org/10.2307/1911501[27] Ali Khan, M. and Suzuki, T. (2016) On Differentiated and Indivisible Commodities: An
Expository Re-Framing of Mas-Colell’s 1975 Model. Advances in Mathematical Economics,
20, 103-128. https://doi.org/10.1007/978-981-10-0476-6_5

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