Theoretical Economics Letters, 2018, 8, 1869-1888
http://www.scirp.org/journal/tel
ISSN Online: 2162-2086
ISSN Print: 2162-2078

A Simple Model of the Difference Principle
Takashi Suzuki
Department of Economics, Meiji-Gakuin University, Tokyo, Japan

Abstract

How to cite this paper: Suzuki, T. (2018)
A Simple Model of the Difference Principle.
Theoretical Economics Letters, 8, 1869-1888.
https://doi.org/10.4236/tel.2018.810123

In this paper, we present a socioeconomic model illustrating the difference
principle first proposed by John Rawls (1971). We demonstrate that this principle can be fulfilled by incorporating reciprocity into the basic structure ofsociety. We show its appropriate embodiment in external increasing returns
in line with traditional neoclassical economic theory, which is the exact mechanism described by Rawls (1971, 2001). Our model yields an intuition andinterpretation of the principle by showing its implementation in the market
equilibrium. Moreover, the model will show that the utilitarian principle, i.e.,the maximization of the total (average) utility leads to advantaged people
monopolizing all wealth as a just state of society. We also discuss the sociopolitical conditions necessary for the difference principle model to be stable andsustained.

Received: March 24, 2018
Accepted: June 18, 2018
Published: June 21, 2018
Copyright © 2018 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

Keywords
Justice as Fairness, Difference Principle, Reciprocity, External Increasing
Returns

1. Introduction
John Rawls [1] proposed and established the difference principle as an alternative to the utilitarian principle in social justice. The difference principle states
that:
Social and economic inequalities are to be arranged so that they are attached to the greatest benefit of the least advantaged ([1], p. 83).
Rawls [2] listed up to five kinds of social regimes: a) laissez-faire capitalism1;b) welfare-state capitalism; c) state socialism with a command economy; d)property-owing democracy; and e) liberal (democratic) socialism. Rawls asked
himself:
A regime supported by Hayek [3] and Nozick [4].

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When a regime works in accordance with its ideal institutional description,
which of the five regimes satisfy the two principles of justice? ([2], p. 137).
This question is significant, especially as Rawls observed:[W]hile a regime may include institutions explicitly designed to realize certain
values, it still may fail to do so. Its basic structure may generate social intereststhat make it work very differently than its ideal description (ibid., p. 137).
Rawls eliminated the first three regimes (a) to (c) because he felt that they violated the principles of justice from the outset and concluded:This leaves (d) and (e) above, property-owing democracy and liberal socialism: their ideal description include arrangements designed to satisfy the two
principles of justice (ibid., p. 138).
However, we note that Rawls did not provide any positive or convincing proof
to demonstrate that the second principle was indeed satisfied within (at least)
these two regimes. Thus, the purpose of this paper is to provide a proof that
supports his conclusion.
The absence of proof is no accident. Indeed, Rawls would not be able to prove
his conclusion even if he had attempted to do so because he did not have any
theoretical understanding of the underlying mechanism2, including how the difference principle is realized by the economic and political institutions in property-owing democracy (d) or liberal socialism (e).
Consider Figure 1, which corresponds to Figure 6 in Rawls ([1], p. 76). As far
as we are aware, this is the only exposition by Rawls that elaborates upon the
difference principle theoretically.
Rawls explained this as follows:
Suppose that x1 is the most favored representative man in the basic structure. As his expectations are increased so are the prospects of x2, the least
advantaged man. In Figure 6 [our Figure 1] let the curve OP represent thecontribution to x2’s expectation made by the greater expectations of x1. The
point O, the origin, represents the hypothetical state in which all social
primary goods are distributed equally. Now the OP curve is always below
the 45˚ line, since x1 is always better off. Thus the only relevant parts of theindifference curves are those below this line, and for this reason the upper
left-hand part of Figure 6 is not drawn in. Clearly the difference principle isperfectly satisfied only when the OP curve is just tangent to the highest indifference curve it touches. In Figure 6 this is at the point a ([1], p. 76).
However, the question is where does the contribution curve OP come from?
In [1], the mechanism that generates this curve is a black box, which is precisely
what we wish to explore in the remainder of this paper.
Thus, the clue is the concept of reciprocity as described by Rawls:
However, Rawls did have quite a strong intuition. This is evident from the quotation from [2] in thefollowing.
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Figure 1. Figure 6 in Theory, p. 76.

[T]he difference principle expresses a conception of reciprocity. It is a principle of mutual benefit (ibid., p. 102).
In the next section, we show that the economic concept of external increasing
returns as developed in neoclassical theory expresses reciprocity (mutual benefit), such that a society equipped with external increasing returns can generate
the type of contribution curve shown in Figure 1.
In sum, reciprocity is a social relationship of mutual advantage that can beachieved by individually rational behavior3. Reciprocity is said to exist in society
if a person acts from his/her own interest and this action gives something beneficial to others. In this sense, market trade is an example of reciprocity. However,external increasing returns do more; they increase productivity and expand theentire economy, not just those aimed at allocation efficiency.
The paper is organized as follows. After explaining how the idea of reciprocity
or mutual advantage embodied in the difference principle is appropriately expressed by external increasing returns in Section 2, the concept of production
functions with external increasing returns is formally defined in Section 3. In
Section 4, this concept will be essentially applied in our social model implementing the difference principle. The heart of this paper is Section 5, where wewill examine the economic conditions to be satisfied for the difference principle.
The political conditions required for the difference principle will be discussed in
Section 6. Section 7 concludes.

2. External Increasing Returns and Reciprocity
When we refer to external increasing returns, we intuitively mean the increase in
In Rawls [5], the concept of reciprocity is extended into the idea of fair terms of cooperation rather
than just mutual advantage:
3[A]ll who engaged in cooperation and who do their part as the rules and procedure require,
are to benefit in an appropriate way as assessed by a suitable benchmark of comparison ([5], p.
16).
However, this finer concept of reciprocity was basically developed from the idea of mutual advantage, even though Rawls remarked that both ideas are not identical ([5], p. 17). The revised conceptof reciprocity should then be based on mutually reasonable behaviors rather than just rational ones.
We will come back to this point later.

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each firm’s productivity resulting from the investments of all firms in the market. This can be achieved not by market trade between firms, but by an external
effect from outside the market (hence external increasing returns). We formallypresent this idea in a more mathematically rigorous manner in the next section.
The idea dates back at least to Adam Smith [6] and the division of labor in a pin
factory4. Indeed, reciprocity can be observed for external increasing returns evenin this prototype.
The division of labor increases the productivity of the firm (factory) by dividing tasks and specializing work in the production processes. However, each separate task makes no sense in and of itself. Although each part of the process isconducted by a single worker (or group of workers), all other parts are interdependent. The division of labor then presupposes cooperative relationships between the workers in the factory. We also remark that these cooperative actions
are generally not intentionally altruistic or benevolent; rather, they generally result from each person’s rational behaviors. Thus, there is a form of mutual advantage or reciprocity within the factory. External increasing returns are thensimply the generalization of the reciprocity within the factory to the entire industry and/or the whole economy. We must consider that the problem is not the“efficiency of markets,’’ but of the reciprocity incorporated into liberal societies.
In other words, this is not a problem within markets but outside markets, i.e., itis the influence of external factors. The crucial aspect is the basic structure of theliberal societies in which the reciprocity is incorporated. This is exactly what enters our consideration of external increasing returns. Generally, the cooperative
relationship in the division of labor is recognized retrospectively. For instance,
there are generally no “central planners’’ in any factory to form relationships
between workers. The division of labor is self-organizing; therefore, so are external increasing returns, which do not come from any particular human mind.
Thus, the reciprocity embodied in external increasing returns is the work of theinstitutions in a liberal society, which develop naturally to increase the welfare of
the members of the society5. In this sense, external increasing returns exist everywhere in a liberal society.
The tendency to form such an institution is because people in any society
cannot do everything they want or receive everything they need. Thus, people dowhat they can do and rely on other people for the rest; i.e., they have naturalcomplementarities with each other. This is exactly what Rawls called a social

union. This idea later deeply impressed Kenneth Arrow, who also pointed out its
similarity to Adam Smith’s pin factory:
Indeed, one of the most brilliant passages in Rawls’s book is that on what hecalls “social union’’ ([1], pp. 520-530). He argues that no human life isenough to encounter more than a small fraction of the experiences needed
for completeness, so that individuals have a natural complementarity with
Refer to Romer [7] and Young [8] for this point.
Of course, this is also the case for the formation of markets.

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each other (a more mundane version of this idea is Adam Smith’s stress on
the importance of the division of labor) ([9], p. 262).
Our fundamental argument is that this kind of mechanism could be responsible for the contribution curve OP in Figure 1. Indeed, Rawls himself stated thatthe increasing portion of the contribution curve arises from the social relationships of mutual advantage (reciprocity), and not just the locus of the redistribution of goods. Instead, Rawls suggested that it results from the cooperative actions among people made possible by the basic structures or institutions of society:
Note that the contribution curve, the curve OP, supposes that social cooperation defined by the basic structure is mutually advantageous. It is nolonger a matter of shuffling about a fixed stock of goods ([1], p. 77).
It would be impossible to consider the difference principle using any standard
market model lacking this sort of mechanism because the difference principle isreduced to the (average) utilitarian principle in such models. We will expand on
this point at a later stage. However, we must stress that a mechanism caused byexternal increasing returns is an example of reciprocity. We do not assert that itis the mechanism by which the second principle is fulfilled. What we hope toaccomplish in this paper is to show how mutual benefits are naturally built intosocial increasing returns and that they will then give us an intuition how thesecond principle can be realized in societies characterized by property-owing
democracy or liberal socialism.

3. Production Functions with External Increasing Returns
In this section, we explain the idea of external increasing returns following
Chipman [10], Romer [7], and Suzuki [11] [12]. Suppose that there are two typesof commodities: a consumption commodity x and an input commodity z. The
consumption commodity is produced by firms with a production function in theform y = f ( z; ζ ) , where ζ is a parameter with an economic meaning explained later. The function f ( z; ζ ) is assumed to be homogeneous of degree 1
in z, (Hom) f ( λ z; ζ ) = λ f ( z, ζ ) for all λ ≥ 0 , and it is monotonically increasing in ζ , (Mon) f ( z; ζ ) ≤ f ( z; ζ ′ ) whenever ζ ≤ ζ ′ .
Given these two conditions, the final outputy is produced under “increasing
returns to scale,” i.e.:f ( λ z; λζ ) =λ f ( z; λζ ) ≥ λ f ( z; ζ ) for all λ ≥ 1.

(1)

The condition (Mon) means that the external effect is positive, which plays acrucial role in this analysis. Through this positive externality, the cooperative
arrangements leading to reciprocity in this society will be manifested as shown
below. As ζ is a parameter for each firm, the condition (Hom) implies that thefirm subjectively operates under constant returns; therefore, no profits will exist
at equilibrium. Thus, the condition (Hom) also plays an important role, which
we discuss in the following section.
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Chipman [10], Romer [7], and Suzuki [11] [12] interpreted ζ as total input∑ z , where the summation covers the entire economy. Therefore, this parame-

ter depends on the amount of input used by other firms in the economy; i.e.,these increasing returns come from (positive) externalities. For each firm, thevalue of ζ is given by the market, and the firm cannot determine it itself.
The most significant difference between this and an ordinary production
function is that the function f ( z; ζ ) not only describes the production technology of the firm, but also reflects the background institutions of society. As
stated in the previous section, background institutions determine externalities
that increase the productivity of each firm, but their interpretation is not specificand is completely open. For instance, Paul Romer interpreted these as a “knowledge spillover effect’’:
The creation of new knowledge by one firm is assumed to have a positive
external effect on the production possibility of the other firms because
knowledge cannot be perfectly patented or kept secret. More important,production of the consumption goods as a function of the stock of knowledge and other inputs exhibits increasing returns; more precisely, knowledge may have an increasing marginal product ([7], p. 1003).
The network of information transmissions by which technologies will prevail
in society is indeed a typical example of background institutions. It is obviousthat the more a society expands, the larger the societal network. The knowledge
spillover effect asserts the converse: the formation and growth of the information network will bring mutual advantage to the members of the society, andconsequently, will contribute to the prosperity of society. This is a typical mechanism governing how reciprocity works. Undoubtedly, there are many more
examples of institutions generating this sort of reciprocity. Of course, we cannot
list them all, nor specifically describe each in our theoretical model. Nevertheless, as we show in the next section, it is exactly this type of reciprocity that produces the contribution curve OP in Figure 1.

4. A Toy Model of Liberal Societies
In this section, we provide a simple toy model of a society with free markets toprovide an interpretation of the difference principle. We stress that this is amodel of a liberal society, rather than just an (ordinary) economic market.
As a market model, we show that the difference principle is economically andtheoretically possible. However, we do not believe it will be sufficient. We ask if
the difference principle will prevail stably in the society. To conclude that thiswill indeed be the case, some political and legal conditions beyond the (usual)
economic conditions (such as consumer rationality) are required. In particular,
the conditions that the difference principle is recognized and honored in the society, and that citizens are reasonable rather than merely rational seem crucial.
Rawls [5] already emphasized these conditions for well-ordered societies. We
will explain these conditions in more detail in the following.
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In this sense, our model serves as a device of representation6. We should consider that these are distinct from the description of actual societies and we only
construct these to obtain some intuition to help us to explain the reciprocity
mechanism underpinning the difference principle. To this end, we present themodel as simply possible. Therefore, the “reality” of the model should not bequestioned too severely.
In society, two kinds of goods are traded in markets. Consumption goods are
consumed by all members of society and its amount is denoted by x. Consumption goods determine the current level of quality of life for citizens and lead to
expectations for their future lives. The other type of good is input goods, which
are used as an input (resource) to produce the consumption good and its
amount is denoted by z. The input good is not consumed, which means that theconsumers’ utilities do not depend on it. Therefore, the value of the consumption good can be identified as the utility level if we restrict ourselves within theclass of monotonous and continuous utility functions. This allows us to avoid
the problem of the interpersonal comparability of utilities. We can also interpretthe input good as a primary good in Rawls’s sense [1] because it is required todevelop each person’s life plan; i.e., advantaged citizens will use it as “money”
and increase their “productivity” (see below). For simplicity, we assume that
there is initially no consumption good in the society (market) and so it must be
produced by the social cooperative production activity described as follows.
There are two groups of citizens (consumers). Citizen 1 represents the groupof more-advantaged (or talented) persons and citizen 2 represents the group of
less-advantaged persons. Here we assume that the populations of the bothgroups are the same to facilitate a comparison of the results with those of Rawls.
However, our results are far more general in this restricted setting because they
do not depend on the population ratio of the two groups (see the Appendix).
Citizens 1 and 2’s consumption levels are denoted by x1 and x2, respectively(see the variables x1 and x2 in Figure 1). Citizens own nothing but possess someamount (possibly 0) of the primary good as an initial endowment. Let Ω > 0 be
the total amount of the primary good that initially exists in society. We denotethe share of citizen 1 by ω; therefore, the share of 2 is Ω − ω . We identify ω as
the index of the policy parameter through which the government arranges justice in the society. When ω = Ω , then the advantaged person owns all, andwhen ω = 0 , the least advantaged person owns all. We then search for the valueof ω that achieves the social state in which the difference principle is satisfied.
The reason why citizen 1 is the more “talented’’ or better-endowed person isthat he/she has a “production function’’ specified by a function with an externality that satisfies the conditions (Hom) and (Mon)f ( z; ω ) = αω κ z,(2)

where α > 0 and κ ≥ 0 are constants, and z is the value of the input to proThis term was borrowed from Rawls [1] [2] [5]. It is well known that Rawls repeatedly emphasizedthat the original position of Justice as Fairness was as a device of representation.
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duce the consumption good. This function was originally adopted by Chipman[10]. We also use this function here because it is easy to handle without any lossof essential points of discussion. However, our results do not depend upon thisspecific functional form (see Appendix). A natural interpretation for (2) is a potential ability such as natural talents. However, the production function (2) reflects the background institutions explained before; therefore, its interpretation
should not be restricted to natural talents. Instead, it will include one’s own environmental elements such as family and education. As to the latter, while opportunities in education are certainly open to all citizens in liberal societies, theymust choose (and be chosen by) an appropriate school (consider schools of law,
medicine, engineering, and music). The production function in this case resultsfrom a combination of one’s natural talent and school education. Indeed, educational systems are a good example of institutions for information transmission
or knowledge spillover of a Romer type.
When citizen 1 acts as a “producer”, he/she takes the value of ω as a parameter, which means that it is given to him/her; therefore, he/she will subjectively
produce under constant returns to scale. Here the role of “primary good’’ isthreefold; it not only serves as an input commodity in the usual sense, but also
develops the ability of citizen 1 which works through the background institutions and enters the model as a positive external effect, and finally it is used as“money’’ to purchase the consumption good. As stated, positive externalities, aswell as market trades, convey an idea of reciprocity. The value of κ is the degreeof reciprocity. As citizen 1 is a representative agent of the class of talented
people, the share value of ω included in the production function is also the valueof the other members of the class represented by citizen 1. The functional form(2) restricts the range of the externality (mutual advantage) to the group of talented persons.
We set up a political and cooperative scheme for this society as follows. Suppose that the entire amount Ω of the primary good is initially owned by citizen 2. Other initial distributions of the primary good are discussed in the following section. The “government” levies a tax ω on this and redistributes it tocitizen 1. As there is initially no consumption good, the less-endowed citizen 2
must give some amount of the primary good to the better-endowed citizen 1 tohelp him/her produce the consumption good. Using the remainder of the primary commodity (disposable income), citizen 2 purchases the consumption
good in the market. In contrast, citizen 1 can undertake production activity only
when some amount of the primary commodity ω > 0 is supplied by society(government), otherwise they can produce nothing given Equation (2). The difference principle requires ω to be set, which maximizes the consumption (welfare level) of citizen 2 at the equilibrium. Obviously, this sort of “fiscal policy” isrealizable in property-owing democracies and liberal socialism regimes.

5. Economic Conditions for the Difference Principle
Let p be the price of the consumption good and we normalize the price q of the
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primary good as q = 1 . Given the primary good is not consumed, we can set

z = Ω at the equilibrium. The competitive equilibrium of this economy is theneasily obtained from the budget equations=
px1 pαω κ z − z + ω ,(3)px2 = Ω − ω ,(4)where x1 is the consumption level of citizen 1 and x2 is that of citizen 2 and themarket condition

x1 + x2 = αω κ Ω.

(5)

Given the production function f ( z; ω ) is (subjectively) constant returns toscale, it follows from the profit condition that pαω κ = 1 . Therefore, the equilibrium price is

p = 1 αω κ ,(6)and the equilibrium consumption levels are=( x1 , x2 )(αω ,αω ( Ω − ω ) ) .
κ +1

κ

(7)

Normally, we would expect that x1 ≥ x2 , but not always (see below). Eliminating ωfrom equation (3), we obtain
κ +1

 x1 + x2  αΩ κx = 1 ,
α (8)which is the locus of equilibria depicted for κ = 2 in Figure 2. We refer to thecurve (8) as the equilibrium curve.
The fundamental properties of the equilibrium curve are stated in the following theorem.
Theorem 1 For κ > 0 , the equilibrium curve starts from the origin when

ω = 0 , intersects with the 45˚ line when ω = Ω 2 and ends at the x1-axis,
x1 = αΩ k +1 when ω = Ω . Moreover, it is concave to the x1-axis and has a peakk +1κΩ (κ + 1) .
at x1 = α (κΩ (κ + 1) )when ω =Proof. The first three properties are verified by substituting 0, Ω/2, and Ωforω, respectively. We then obtaindx1
dx2= α (κ + 1) ω κ , =αω κ −1 (κΩ − (κ + 1) ω ) .
dω
dω

(9)

It follows from (9) thatdx2 κΩ − (κ + 1) ω=,dx1
(κ + 1) ω

(10)

−1

d 2 x2  dx1  d  dx2 
−κΩ
=
< 0.


 =

2
dx1  dω  dω  dx1  α (κ + 1)2 ω κ +2

(11)

Theorem 1 follows from (10) and (11).
We note that Theorem 1 (and Theorems 2 and 3 below) can be extended togeneral cases using conditions (Hom) and (Mon). In particular, the intersection
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Figure 2. Equilibrium curve for κ > 1 .

property with the curve and the 45˚ line at ω = Ω 2 generally follows from thecondition (Hom) rather than the specific functional form in the current model.
See the proof of Theorem 4 in the Appendix.
Suppose for a moment that the degree of reciprocity κ is greater than one (seeFigure 2). The relation between the contribution curve OP in Rawls and theequilibrium curve (8) is explained as follows. As we mentioned earlier, Rawls
sets the origin as the reference point where each individual is assigned the sameamount of the primary good.
The point O, the origin, represents the hypothetical state in which all social
primary goods are distributed equally ([1], p. 76).
Thus, the point at which the equilibrium curve intersects the 45˚ line corresponds to the origin O in Figure 1, since by Theorem 1, we have ω = Ω 2 atthat point. The difference principle is applied to the equilibrium (7), thereby instructing society (or the government) how to set the value of ω to obtain thelargest value of x2.

If κ > 1 , this state can be achieved at ω =Ω
κ (κ + 1)−1

by Theorem 1 and theequilibrium allocation permitted as just for this case is given by:



κ +1

κΩ 

  κ +1

( x1 , x2 ) =  α 

κ +1

,

α  κΩ κ  κ + 1 

The value of x1 is indicated by a = α (κΩ (κ + 1) )

κ +1


.



(12)

in Figure 2.

What about the case where κ ≤ 1 ? In this case, point a in Figure 3 is placed to
the left of the 45˚ line. If point a is chosen, this means that talented person 1 is“sacrificed’’ for society. Obviously, this is unjust. In this case, therefore, the difference principle would order the point on the 45˚ line. The idea behind this observation is as follows. We always start from the reference point of ω = Ω 2 .
Any deviation from this point is admitted only when both of the more- andless-advantaged persons gain by this deviation, which will be possible under thecondition κ > 1 . In other words, the difference principle will order the states thatare distinct from that of perfect equality only for those societies where reciprocity
works sufficiently or the degree of reciprocity is greater than one. Rawls stated:
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Figure 3. Equilibrium curve for κ < 1 .

[T]he difference principle is a strongly egalitarian condition in the sensethat unless there is a distribution that makes both persons better off (limiting ourselves to the two person case for simplicity), an equal distribution is
to be preferred. The [social] indifference curves [...] are actually made up ofvertical and straight [horizontal] lines that intersect at right angles at the45˚ line (again supposing an intertemporal and cardinal interpretations of
the axes). No matter how much either person’s situation is improved, there
is no gain from the standpoint of the indifference principle unless the othergains too ([1], p. 76).
We summarize these results as a theorem.
Theorem 2 The difference principle orders the allocation associated withω=κΩ (κ + 1) on equilibrium curve when κ > 1 , and the allocation on the 45˚line ( ω = Ω 2 ) when 0 < κ ≤ 1 .
For the following analysis, we need to distinguish two concepts of optimality.
The first concept is the standard notion of Pareto optimality:
Definition 1 A feasible allocation ( x1 , x2 ) , which satisfies the Equation (5) is

said to be globally Pareto optimal (or Pareto optimal for short), if and only
another feasible allocation exists ( y1 , y2 ) such that ( x1 , x2 ) < ( y1 , y2 ) 7.
Notice that globally Pareto optimal allocations are exactly the ones which satisfy x1 + x2 =αΩκ +1 . Our second concept of optimality is a local one, defined
only for the equilibrium allocations:Definition 2 An equilibrium allocation ( x1 , x2 ) , which satisfies the equation(8) is said to be locally Pareto optimal (or locally optimal for short), if and only if
the derivative dx2 dx1 is less than or equal to 0 at ( x1 , x2 ) .
Obviously the allocations on the equilibrium curve over the domaina ≤ x1 ≤ αΩκ +1 are locally optimal. Hence the allocation (11) stipulated by thedifference principle for the case κ > 1 is locally optimal. Rawls was concernedwith local optimality when he discussed the efficiency of allocations [1]. We
trust that the local optimality is more appropriate than the global one for liberalsocieties, given the latter will generally need to compare very distant allocations.

( x , x ) ≤ ( y , y ) is defined by x ≤ y and x ≤ y . We then define ( x , x ) < ( y , y ) by
( x , x ) ≤ ( y , y ) and ( x , x ) ≠ ( y , y ) .

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To attain an improving allocation, the society must gather all produced commodities at once and redistribute them among citizens. Although such drastic
reallocations might be possible in societies with centrally planned economies,
they would be unrealistic for liberal societies with market economies.
Next, we would like to identify allocations determined by the utilitarian principle that select the equilibrium allocations maximizing total (average) utility

x1 + x2 . For this, the relevant statements in Rawls are:If there are but two persons, then assuming an interpersonal cardinal interpretation of the axes, the utilitarian’s indifference lines for distributions are
straight lines perpendicular to the 45˚ line. [...] Drawing the same contribution curve OP as before, we see that the best distribution from a utilitarianpoint of view is reached at the point which is beyond the point b where theOP curve reaches its maximum (ibid., p. 77).
Thus, we obtain a more precise result.
Theorem 3 For any value of κ > 0 , the utilitarian principle orders the alloca-

tion (αΩκ +1 , 0 ) .
Proof. Computing the derivative, we haveκ −1
dx2 dx2 dω αω (κΩ − (κ + 1) ω ) κΩ − (κ + 1) ω.
= ==dx1 dx1 dω
α (κ + 1) ω κ(κ + 1) ω(13)Then at ω = Ω , one has dx2 dx1 = −1 (κ + 1) > −1 whenever κ > 0 . Given

the equilibrium curve is concave to the x1-axis, the whole curve is below the line

x1 + x2 =αΩκ +1 and αΩκ +1 is the maximum value of total utility; see Figure 5.
The result follows.
Theorem 3 might look surprising given it states that the utilitarian principle
stipulates as just an allocation in which talented people monopolize all of society’s wealth! It is interesting that the set of equilibrium allocations that are globally Pareto optimal is just(αΩ , 0 ) because every other point on the equiliκ +1

brium curve can be moved to an optimal point on x1 + x2 =αΩκ +1 , which is located at its north-east direction (see Figure 5). However, we cannot help but
hold quite strong skepticism about the utilitarian principle as a principle of justice for societies with any degree of reciprocity, even if Pareto optimal allocations are realized by it. Indeed, Theorem 3 might be considered as yet anotherreason for the justification of the difference principle over the utilitarian principle from the original discussion.
Rawls remarked on the slope of the utilitarian indifference line:
However, since x1 and x2 are representative men, their gains must be
weighted by the number of persons that they each represent. Since presumably x2 represents rather more persons than x1, the indifference line
become more horizontal, as seen in Figure 8 [Figure 4] (ibid., p. 77).
In the Appendix, we show that Theorem 3 is true even if this remark is considered (Theorem 6).
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Figure 4. Figure 8 in Theory, p.77.

Figure 5. Utilitarian principle.

Finally, we consider the case κ = 0 (no externalities). This economy might
appear in a standard microeconomics textbook. Then equation (8) simply reduces to the market condition, which is the (usual) utility possibility frontier:x1 + x2 =αΩ.

(14)That is, we have only the pure trade-off between the consumption of bothconsumers. In this case, contribution curves like OP do not appear. For the
curve to have the upward portion, some sort of mechanism is required, such asexternalities, which are considered to represent social cooperation rendering
mutual advantage. Rawls’s understanding of this point was exact. Recall his
statement:
Note that the contribution curve, the curve OP, supposes that social cooperation defined by the basic structure is mutually advantageous. It is nolonger a matter of shuffling about a fixed stock of goods (ibid., p. 77).
The lack of social cooperation mechanisms is exactly why any insights for determining just allocations cannot be obtained from standard market models
without externalities. In such models, the difference principle will mechanicallyorder the perfect equality x1 = x2 = αΩ 2 as the intersection of the curve (14)with the 45˚ line. Thus, the difference principle is simply reduced to the perfectequality principle. However, the utilitarian principle is indifferent between every
allocation on the utility possibility frontier (14). It does not select any specificallocations as just when the commodity is produced under constant returns to
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scale.

6. Political Conditions for the Difference Principle
In previous sections, we have described a model of liberal societies, which wasabsent from Rawls’s original discussions. In the model, each realization of a social state is determined by the policy parameter ω, which is citizen 1’s share of
the primary good. Given these institutional and political mechanisms for thedifference principle provided by our model, we can now proceed to discuss thepolitical conditions for the difference principle to be satisfied in socially stableways.
Up to this point, we have assumed that citizen 2 initially owned the complete
resource, on which the citizen was taxed by the government. The citizen would
do so whenever he or she is rational because the tax will be used to make thembetter off. In contrast, consider the situation where the whole of the resource isinitially owned by citizen 1. In this case, the government must also collect taxes
to achieve the allocation ordered by the difference principle. How is this possible?
It seems difficult to assume that the citizen will agree to pay the tax voluntarily
if he or she is only rational, as is usually assumed in ordinary economic market
theory. The government will be able to collect money only through some lawthat coerces the citizens to pay taxes. For the citizens to endorse this law, thedifference principle must necessarily be already accepted in society as a principleof justice. The citizens must also be convinced that the government’s fiscal policy will execute the difference principle. Further, citizens must hold these facts as
public information. Those conditions coincide essentially with the first two conditions for a well-ordered society raised by Rawls:
To say that a society is well-ordered conveys three things; first (and impliedby the idea of a publicly recognized conception of justice), it is a society inwhich everyone accepts, and knows that everyone else accepts, the verysame principles of justice; and second (implied by the idea of the effective
regulation such a conception), its basic structure—that is, its main politicaland social institutions and how they fit together as one system of cooperation—is publicly known, or with good reason believed, to satisfy these principles ([5], p. 35).
Rawls required a third condition for society to comply with social rules:
And third, its citizens have a normally effective sense of justice and so theygenerally comply with society’s basic institutions, which they regard as just(ibid., p. 35).
However, for this condition to be satisfied, it seems insufficient that the citizens are simply rational. Instead, we must consider that they are reasonable.
That is, citizens are ready to propose principles and standards as fair terms ofcooperation and to abide by them willingly, given the assurance that others will
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likewise do so (ibid., p. 49). We recall that the “productivity’’ of citizen 1 is increased by external effects from background institutions, which means that they
owe the improvement of their abilities to society. This yields fair terms of cooperation in Rawls’s sense:[... W]e cannot view the talents and abilities of individuals as fixed natural
gifts. To be sure, even as realized there is presumably a significant genetic
component. However, these abilities and talents cannot come to fruition
apart from social conditions, and as realized they always but one of many
possible forms. Developed natural capacities are always a selection, a small
selection at that, from the possibilities that might have been attained (ibid.,
pp. 269-70).
Thus, if the citizens were reasonable, they would realize that the current situation where rich people monopolize all of the social resources and wealth is unjust8. Moreover, they would endorse Rawls’s thesis concerning natural talents as
a common asset [1]. Consequently, they would agree with his following statement:
Here it is crucial that the difference principle includes an idea of reciprocity: the better endowed (who have a more fortunate place in the distributionof native endowments they do not morally deserve) are encouraged to acquire still further benefits—they are already benefited by their fortunate
place in the distribution—on condition that they train their native endowments and use them in ways that contribute to the good of the less endowed(whose less fortunate place in the distribution they also do not morally deserve) ([2], pp. 76-77).
Rawls concluded:
In such a [well-ordered] society the publicly recognized conception of justice establishes a shared point of view from which citizens’ claims on society
can be adjudicated9 ([5], p. 35).
Following Rawls, we also conclude that under those conditions, the difference

principle will be effective as a principle of justice for regulating the economic
differences in a society characterized by property-owing democracy or liberalsocialism.

7. Conclusions
1) The main contribution of this paper is that it clarifies the mechanism of
implementing the difference principle and enhances the economic and politicalconditions required for that principle. However, this paper is just a starting
point for investigations. For instance, we have assumed that there are the only
two groups—advantaged and less advantaged—of citizens in our society. Future
However, we notice that this state is Pareto optimal.
For a more precise exposition, see Rawls [1], pp. 48-50.

8
9

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studies should explore the more complex effects working among several groups,
such as “chain connection,’’ described by Rawls [1] (see also 3 below).
2) For simplicity, we have discussed the economic conditions under the as-

sumption that the production function is of the form f ( z , ω ) = αω κ z . However, our economic discussions do not depend on any specific functional form andwill essentially hold under the conditions (Hom) and (Mon) for completely
general settings. In other words, the class of production functions representing
the difference principle is exactly characterized by functions with externalities
that satisfy these two conditions. Moreover, our theorems hold for models withany arbitrary population ratio. We illustrate this general discussion in the Appendix.
3) The difference principle concerns economic inequality. Economic markets
are the cause of this difference and a means to regulate it. Therefore, economic
theory is required. In this paper, we presented an example of such a market
model with the assistance of a purely economic concept in the form of external
increasing returns.
Conversely, the difference principle is also a principle of justice. Thus, somepolitical and philosophical considerations that surpass theoretical economic
thinking are also required. To obtain positive results, we presented a sociopolitical model (which is the same as the market model). The results, if any, must be
confirmed by the test of reflective equilibrium. That is, how well this perspective
articulates our more considered convictions of political justice at all levels of generality after due examination once all adjustments and revisions that seemcompelling have been made ([5], p. 28). We trust that our conclusion in the preceding section passed this test.
It goes without saying that the actual society is far more complex than the toymodel described in the previous section, which ignored many economic and political aspects. Therefore, a concern could be that citizens must consider how
every political and economic matter affects the prospects of the least-advantaged
people, to which Rawls responded:
A useful reply is this: we are to proceed by selecting a few instruments, as
we may call them, that can be adjusted so as to meet the difference principle, once the whole family of policies is given ([2], p. 161).
In our simplified society, we highlighted a tax-subsidy policy, but there are
obviously many other welfare policies in the actual society.
Nonetheless, the production function as a natural endowment of citizen 1 isnotably generally unobservable by others, including the government; therefore,
the production function would not be able to compute the equilibrium and the
optimal value of ω. In response, we restate that the purpose of our model is todevelop an intuition and understanding of the implementation of the difference
principle, confirm that the fulfillment of the principle is indeed possible, and also confirm that the conditions given in the preceding section are sufficient (butnot necessary) conditions for the difference principle.
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Unlike with the first principle, we should also consider that it is generally difficult in an actual society to judge whether the second principle is fulfilled. Rawls
admitted as much:
Whether the aims of the second principle are realized is far more difficult[than the first principle] to ascertain. To some degree these matters are always to open to reasonable differences of opinion; they depend on inference
and judgement in assessing complex social and economic information (ib-id., p. 162).
Therefore, Rawls recommended that the difference principle be excluded fromthe essentials for a constitution ([2], p. 49). We also agree with this opinion.
Nevertheless, Rawls also said that equality in democracies requires something
like the difference principle:
While the difference principle does not fall under the constitutional essentials, it is nevertheless important to try to identify the idea of equality mostappropriate to citizens viewed as free and equal, and as normally and fully
cooperating members of society over a complete life. I believe this idea involves reciprocity at the deepest level and thus democratic equality properly
understood requires something like the difference principle (ibid., p. 49).
We have nothing further to add to this statement.

Acknowledgements
An earlier version of this paper was presented at the 76th annual meeting of thePhilosophical Association of Japan held at Hitotsubashi University, May 20,
2017. I would like to thank the session participants. This research is supported
by Grants-in-Aid for Scientific Research (No. 15K03362) from the Ministry of
Education, Culture, Sports, Science and Technology, Japan.

References

DOI: 10.4236/tel.2018.810123[1]

Rawls, J. (1971) A Theory of Justice. Harvard University Press, Cambridge.

[2]

Rawls, J. (2001) Justice as Fairness: A Restatement. Harvard University Press, Cambridge.

[3]

Hayek, F.A. (1944) The Road to Serfdom. Routledge & Kegan Paul, London.

[4]

Nozick, R. (1974) Anarchy, State and Utopia. Basic Books, New York.

[5]

Rawls, J. (1993) Political Liberalism. Columbia University Press, New York.

[6]

Smith, A. (1776/1976) The Wealth of Nations. University of Chicago Press, Chicago.

[7]

Romer, P. (1986) Increasing Returns and Economic Growth. Journal of Political
Economy, 94, 1002-1037. https://doi.org/10.1086/261420[8]Young, A. (1928) Increasing Returns and Economic Progress. Economic Journal,
38, 527-542. https://doi.org/10.2307/2224097[9]

Arrow, K.J. (1973) Some Ordinalist-Utilitarian Notes on Rawls’s Theory of Justice.
The Journal of Philosophy, 70, 245-263. https://doi.org/10.2307/2025006
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T. Suzuki[10] Chipman, J.S. (1970) External Economies of Scale and Competitive Equilibrium.
Quarterly Journal of Economics, 84, 347-385. https://doi.org/10.2307/1879425[11] Suzuki, T. (2009) General Equilibrium Analysis of Production and Increasing Returns. World Scientific, Singapore.
[12] Suzuki, T. (2016) Welfare Analysis of a Market Model with External Increasing Returns and Differentiated Commodities. Theoretical Economics Letters, 7, 63-78.
https://doi.org/10.4236/tel.2017.71007

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Appendix
For notational convenience, we set f (1, ω ) = φ (ω ) and assume φ (ω ) to be
twice-continuously differentiable. We also assume that φ ( 0 ) = 0 , or that no
outputs can be produced without the primary good. Note that the condition(Mon) implies that φ ′ (ω ) > 0 . Suppose that there are n1 citizens of group 1
and n2 citizens of group 2, and assume that each member of group 1 own thesame amount of initial endowment ω1 , and similarly citizens are endowed with

ω2 in group 2; hence we have n1ω1 + n2ω2 =Ω . Let θ = n1 n2 be the population ratio, and we denote Ω1 =Ω n1 and Ω 2 =Ω n2 ; then it follows thatθω1 + ω2 =
Ω2 .

(15)

The market equilibrium is now determined by the following equations:=px1 pφ (ω1 ) z − z + ω1 ,(16)

px2 = ω2 ,(17)

θ x1 +=x2 θφ (ω1 ) Ω1.

(18)

and
As before, the equilibrium price is determined as p = 1 φ (ω1 ) . Hence, thederivatives are calculated as

dx1
dx2
φ ′ (ω1 ) ω1 + φ (ω1 ) , =φ ′ (ω1 )( Ω 2 − θω1 ) − θφ (ω1 ) .
=
dω1
dω1

(19)

0 is obtained by eliminating p, ω1 , andThe equilibrium curve Φ ( x1 , x2 ) =ω2 from (15) to (18). Theorem 1 is now generalized to:
Theorem 4 The equilibrium curve starts from the origin when ω1 = 0 , intersects with the 45˚ line when ω1 = ω2 and ends at the x1-axis, x1 = φ ( Ω1 ) Ω1
when ω1 = Ω1 . Moreover, it is concave to the x1-axis if

φ ′′ (ω1 ) φ (ω1 ) < 2φ ′2 (ω1 )(20)

and has the peak at x1 = φ (ωˆ ) ωˆ where ω̂ satisfies

φ ′ (ωˆ )( Ω 2 − θωˆ ) − θφ (ωˆ ) =0.

(21)

Proof. The first three properties are verified by substituting 0, Ω/2, and Ω forω in (15), (16), and (17), respectively. We notice that the intersection of thecurve with the 45˚ line at ω = Ω 2 comes from the condition (Hom). Given
the “profit” of citizen 1 is 0, her income to purchase the consumption good issolely determined by her initial endowment; therefore, citizens 1 and 2 will purchase the same amount when ω1 = ω2 . From (19), we can compute the following equations:dx2 φ ′ (ω1 )( Ω 2 − θω1 ) − θφ (ω1 ),=dx1
φ ′ (ω1 ) ω1 + φ (ω1 )()φ ′′ (ω1 ) φ (ω1 ) − 2φ ′ (ω1 ) Ω 2
d 2 x2.
=2
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0 and the concavity
The value of x1 which attains the peak of Φ ( x1 , x2 ) =of the curve follow from (22) and (23), respectively.
Note that the concavity condition (20) is mild and does not depend on θ.
In general, no counterpart to the degree of reciprocity κ exists. Instead, we
obtain a measure of reciprocity from the derivative of the equilibrium curve on
the 45˚ degree line.
Definition 3 We say that reciprocity works effectively or reciprocity is effec-

tive if and only if dx2 dx1 > 0 when ω1 = ω2 .
By (22), reciprocity is effective if and only if

(

)

(

)

φ ′ ( n1 + n2 ) Ω ( n1 + n2 ) Ω − θφ ( n1 + n2 ) Ω > 0 .
−1

−1

−1

It is easy to see in particular that for f ( z , ω ) = αω κ z , reciprocity is effective if
and only if κ > θ . It is interesting that the smaller the proportion of the advantaged (talented) persons, the more likely the reciprocity is to be effective.
The next theorem is immediately drawn from Theorem 4.
Theorem 5 Suppose that reciprocity works effectively. Then the difference
principle orders the allocation associated with ω = ωˆ on equilibrium curvewith x1 > x2 , where ω̂ is a solution of the equation (21).
Note that we do not assume the concavity condition (20) in Theorem 5. Finally, we state a generalization of Theorem 3.
Theorem 6 Suppose that the concavity condition (20) holds. Then the averageutilitarian principle orders the allocation (φ ( Ω1 ) Ω1 , 0 ) for any θ > 0 .
Proof. We first notice that the average utilitarian principle orders to maximize
θ x1 + x2 . Set ω1 = Ω1 in (22). Then we have θω1 = Ω 2 ; therefore, it follows that−θφ ( Ω1 )dx2=> −θ .
dx1 ω =Ω φ ′ ( Ω1 ) Ω1 + φ ( Ω1 )
1

(24)

1

Given the equilibrium curve is concave to the horizontal axis, the result follows as in Theorem 3.

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