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An Analytical Theoryof Just Market Exchange

Ricardo Andrés GuzmánCentro de Investigación en Complejidad Social

and Facultad de GobiernoUniversidad de Desarrollo

[email protected]

Michael C. MungerDepartments of Economics

and Political ScienceDuke University

[email protected]

Prepared for presentation at the “September School of Philosophy,Economics, and Politics,” September 5-9, 2012, Moeclu, Brasov,sponsored by C.A.D.I., Bucharest, Romania. Draft: Please do not quote

without permission.



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An analytical theory of just market exchange

Ricardo Andrés Guzmán Michael C. Munger†

September 1, 2012

Otto “Toby” Davis’s believed that market exchange, and the institutions that foster marketexchange, were crucial to human development and flourishing. In two famous papers (Wu and Davis,1999a and 1999b), Toby and his coauthor Wenbo Wu argued that economic freedom is

fundamental, and can precede political freedom on a growth path to development. But critics protested, plausibly, that this aggregate result assumes too much about the justice of market exchanges between

particular individuals. If the disparity between market participants is too vast, if the access to thebenefits to markets are too restricted to those with market power, then the aggregate growth maybenefit the nation, but will exploit and damage those too weak to negotiate on fair terms. In this

paper, we take on the issue directly, investigating the moral obligations of individual participants inmarket exchange. We are able to show that it is possible for market exchange to be just, even if itmay not be fair. Therefore, on both material consequentialist and moral justice grounds, market

freedoms can support the claims made by Davis and other advocates.

1 Introduction

Any plausible theory of just market exchange must balance two conflicting moral considerations:euvoluntariness (true voluntariness) and Pareto efficiency . Voluntariness requires that neither party iscoerced into exchange by threat of violence or other form of direct harm. Euvoluntarinessimposes the additional requirement that neither party is coerced by the lack of a decent alternativeto a negotiated agreement. Pareto efficiency, on the other hand, requires that voluntary, mutuallybeneficial exchanges should always be allowed, even if they are not euvoluntary.

The conflict between euvoluntariness and efficiency is illustrated by Michael Sandel, whodescribes “coercion by scarcity of alternatives.”

The... objection [to the claim that an exchange is truly voluntary] is an argumentfrom coercion. It points to the injustice that can arise when people buy and sell thingsunder conditions of severe inequality or dire economic necessity. According to this

Centro de Investigación en Complejidad Social and Facultad de Gobierno, Universidad de Desarrollo.† Departments of Economics and Political Science, Duke University.



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objection, market exchanges are not necessarily as voluntary as market enthusiastssuggest. A peasant may agree to sell his kidney or cornea in order to feed his starvingfamily, but his agreement is not truly voluntary. He is coerced, in effect, by the

necessities of his situation (Sandel 1998).Sandel notes that there are two conditions under which one party to an exchange might be

coerced by scarcity of alternatives: “severe inequality or dire necessity.” Severe inequality meansthat the disparity in alternatives to the exchange is too large to be equitable. Dire necessity meansthat for one party the alternative to exchange is disastrous. If at least one of these two conditionsis met the exchange is not euvoluntary.

However, it is not obvious that non-euvoluntary exchanges are morally objectionable, and evenless obvious that they should be prohibited. In Sandel ’s example, the peasant would likely beg for achance to exchange, arguing that the loss of one kidney is better than having his family starve.

When viewed this way, denying him the right to sell his kidney appears to be a form of injustice.

The question naturally arises: Are non-euvoluntary market exchanges ever morally permissible? Ifso, does justice impose any limits on the terms of exchange?1 A sketch of an answer can be traced back to John Locke’s essay “Venditio” , where he poses

the following moral dilemma:

A ship at sea that has an anchor to spare meets another which has lost all heranchors. What here shall be the just price that she shall sell her anchor to thedistressed ship? To this I answer the same price that she would sell the same anchor to a ship thatwas not in that distress. For that still is the market rate for which one would part with anything toanybody who was not in distress and absolute want of it. And in this case the master of the

vessel must make his estimate by the length of his voyage, the season and seas he sailsin, and so what risk he shall run himself by parting with his [extra] anchor, which allput together he would not part with it at any rate, but if he would, he must then takeno more for it from a ship in distress than he would from any other. (Locke,1661/2005, Venditio, pp. 445 – 6; emphasis added).

In a sense, Locke maintains that the safe ship must devise a “fictitious negotiation” in which thedistressed ship is not in distress, and behave in the actual negotiation as he would behave in thefictitious one.

Inspired by all these ideas, we propose an analytical theory of just market exchange that guidesreasoned moral action in balancing euvoluntariness and efficiency. Our method is to postulate aset of plausible moral axioms, and then use deductive logic to infer a series of moral imperatives

1 For our purposes, the validity of Sandel’s example depends on whether a kidney qualifies as transferable property or not(Sandel himself would raise this objection). As this matter is the subject of another debate, we will refrain from taking a position.In what follows, we will restrict the analysis to the exchange of conventional market goods and services.



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with respect to two issues: the circumstances under which exchange is morally optional orobligatory, and the just allocation of the gains from exchange.

The theory is operationalized through a “fictitious negotiation model.” Briefly, the fictitious

negotiation model is as follows:1. Neither party is morally obliged to suffer harm by an act of market exchange.

2. The negotiation will be fair and exchange will be euvoluntary if and only if the disparitybetween the parties’ outside options 2 does not exceed a certain threshold. The magnitude ofthe disparity threshold depends on the abjectness of the weaker party. The direr the weakerparty’s outside option, the lower the disparity threshold.

3. If the negotiation is fair, all non-supererogatory outcomes are just. A non-supererogatoryoutcome is either a mutually beneficial agreement or the disagreement outcome, in whichthe parties get their respective outside options.

4. If the negotiation is unfair, the stronger party must devise a fictitious negotiation, in whichthe weaker party has an improved outside option. The outside option must be improveduntil one of two things happens:

(a) The disparity in outside options is reduced until it is no longer unfair. This does notrequire that all the disparity disappears; only that it equals the disparity threshold.

(b) The surplus of the fictitious negotiation is reduced to zero. This can happen because thesurplus of any negotiation decreases as the parties’ outside options improve.

We term condition (b) the “non-worseness”3 clause of the fictitious negotiation.

5. The just agreements of the actual negotiation correspond to the non-supererogatoryoutcomes of the fictitious negotiation that are implementable in practice.

The fictitious negotiation model is parametric . The free parameter is the observer’s revulsiontoward the imbalance in bargaining power, which is captured by the disparity threshold. Ratherthan being a constant, the disparity threshold is a decreasing function of the direness of the weakerparty’s outside option, but the particular shape of the threshold function varies from observer toobserver. Clusters of similar disparity threshold functions constitute a culture. A thresholdfunction that captures the essential features of a culture constitutes an ideology.

2 In the jargon of game theory, the parties’ outside options are their best alternatives to a negotiated agreement. 3 “Non - worseness” is a potential constraint on moral action, on consequentialist grounds. Suppose that, in order for the

stronger party to act morally, the weaker party must actually be harmed in some material sense. Non-worseness is described byZwolinski (2008, 2009 ) interpreting Wertheimer (1996), this way: “In cases where A has a right not to transact with B, a nd wheretransacting with B is not worse for B than not transacting with B at all, then it cannot be seriously wrong for A to engage in thistransaction, even if its terms are judged to be unfair by some external standard.” (p. 357).



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Several moral imperatives follow logically from the fictitious negotiation model. The mostimportant are:

1. If the negotiation is unfair but has a positive surplus, the stronger party is morally obligedto exchange with the weaker party. He cannot refuse to exchange by invoking his propertyrights, or by claiming that the exchange would not be euvoluntary.

2. Moreover, the stronger party must negotiate as if the weaker party had a better outsideoption.

(a) This fictitious outside option should eliminate all the unfair disparity, provided this doesnot rule out the possibility of a mutually beneficial exchange.

(b) Otherwise, the stronger party must give the entire surplus to the weaker party, andcontent himself with a payoff equal to his outside option.

These logical implications of the model are imperatives because they will necessarily trigger when the weaker party’s outside option is “ sufficiently dire.” The particular disparity threshold justspecifies the meaning of sufficiently dire. For some observers, even small disparities are sufficientto trigger the obligation to give away the entire surplus. For other observers, who find propertyrights and ownership paramount, the obligation may only trigger at very large disparities. But theobligation to give away the surplus is always triggered, at the limit of infinite direness, even on theeye of the least disparity-averse beholder.

The theory of just exchange is a coherent alternative to radical egalitarianism and radicallibertarianism. When they are carried to their logical conclusions, both moral philosophies produceuntenable moral judgments. This indicates that these philosophies are founded on flawed sets ofpremises. In contrast to the radical moral philosophies, our theory of just exchange producesperfectly intuitive moral judgments. Importantly, these judgments are logical consequences of thepremises of the theory. There is no need to patch the theory with a long list of exceptions. Below

we provide two brief examples (the detailed workings of the fictitious negotiations will bepresented in the following sections).

Radical egalitarians consider that voluntary exchange is unjust if the resulting allocation isunequal, despite being beneficial to both parties. For example, radical egalitarians firmly opposesweatshops, even if the workers themselves prefer the low-paid factory jobs to starvation in thefields.4 In contrast, our theory entails that it is unjust to let the workers starve. Moreover, thefactory owner is obliged to hire them and pay substantially more than they would earn in the field,but he is not obliged to incur an economic loss (i.e., he must recover at least his opportunity cost

of capital).

4 See, for example, Sample 2003, and SASL 2001.



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Radical libertarians condone exchanges that violate even the most basic notions of humandignity and mutual obligation. For example, a man lost in the desert is on the verge of dying ofthirst. Out of nowhere appears another man carrying plenty of water. From the point of view of a

radical, logically consistent libertarian, the passerby has the right to sell his water at any price hechooses, even if that price is exorbitant and patently abusive. And what is more absurd, thepasserby is entitled to refuse the sale and let the lost man die of thirst. Our theory entails it isunjust to deny water to a thirsty man. The passerby is obliged to sell his water, but he is notobliged to give it away. He must offer the lowest price that covers the full cost of the water,including the purchase price, transport costs, and opportunity costs of possibly needing the waterhimself.

The paper is organized in six sections. In Section 2 we define the notion of “euvoluntaryexchange.” In Section 3 we present an analytical negotiation model and formalize the notions offairness and euvoluntariness. In Section 4 we develop the fictitious negotiation model and ouranalytical theory of just market exchange, which makes a distinction between equitable and just

negotiated agreements. In Section 5 we apply the fictitious negotiation model to four hypotheticalexamples, each an abstract version of a real life moral dilemma. The examples are: Locke’s anchorproblem, competitive market exchange, price-gouging, and sharecropping contracts. In Section 6

we offer some conclusions. Importantly, we argue that all reasonable moral theories must beparametrical, rather than strictly deductive (à la Kant) or sentimentalist (à la Hume). Our analytictheory of just market exchange is an example of a parametric theory, because its axioms aremodulated by a free parameter: the disparity threshold function.

Before we proceed, a clarification must be made. In this paper, we do not advance acomprehensive theory of justice, or even a theory of just exchange. Our theory is only concerned

with marketexchange. Any other form of exchange is outside the theory’s domain of application.Market exchange takes place within a particular structure of social norms and legal constrains.

Practices of negotiation and bargaining that are perfectly acceptable in market exchange might beconsidered distasteful, offensive, or plainly immoral in other contexts, such as romance, the family,military service, politics, the school, sports, or religious worship. Furthermore, the law or socialcustom may prohibit the commerce of certain goods (e.g., recreational drugs). But even if thecommerce of a good is socially accepted and legal, the good may be degraded or corrupted bymarket exchange; sexual intercourse being a favorite example of this phenomenon (Sandel 1998).

This means that some goods are inherently non-marketable (e.g., friendship). The fictitious negotiation model applies to the exchange of marketable goods, provided the

exchange is socially accepted and legal. In many cases, there will be little dissent in this regard. Forexample, most people would agree that trade in food, clothing, housing, and labor are legitimatebusinesses, and that these particular goods are not degraded or corrupted by market exchange. Atother times, dissent will be profound, as in the cases of prostitution and organ trade. For thisreason, we remain agnostic, and leave to readers the task of agreeing on the boundaries of markets.



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2 Euvoluntary exchange

For a market exchange to be considered euvoluntary (truly voluntary) six conditions must be met

(Munger, 2011):1. The parties own the objects being exchanged, according to the conventional interpretation

of ownership.

2. The parties have the capacity to transfer the ownership of these objects.

3. There is no fraud in the exchange, and no psychological compulsions such as addiction orneuropathy.

4. The exchange does not produce large-scale uncompensated non-pecuniary externalities, anddoes not impose costs on third parties without their express euvoluntary consent.

5. Neither party is coerced in the sense of being forced to exchange by threat of violence orother form of direct harm.

6. Neither party is coerced in the alternative sense of being harmed by the dire consequencesof failing to exchange.

Conditions 1 – 6 are standard requirements for a valid contract in the common law. Conditions 5and 6 could be summarized as “no duress.” An exchange will be voluntary if and only if conditions1 – 5 are satisfied, but the exchange will not be euvoluntary unless condition 6 is satisfied as well.

As stated by condition 5, an exchange is not voluntary if a party exercises its power to force anagreement that benefits him. Here, “power” is understood in its colloquial sense, as the ability to

impose one’s will on others through the threat of violence or other form of direct harm.In negotiations, power has a different meaning. “Bargaining power” is the ability to cause harm

indirectly by refusing to exchange. Condition 6 states that euvoluntariness requires that no partyhas excessive bargaining power over the other. Note that condition 6 says nothing about theexercise of bargaining power during the negotiation, only that the parties are close in power. Thebalance in bargaining power makes a negotiation fair, in the similar sense that weight classes makeboxing matches fair. The boxers need not weigh the same, but fairness requires that the disparityin weight must be within a reasonable range.

For example, most of us have a sense that monopoly is not only economically inefficient, butfundamentally unfair. It is true that the monopolist does not hold a gun to the consumer’s head and shouts “buy!” But if the product is desperately needed, a consumer’s decision to buy cannot

be said to be truly voluntary. Put another way, the sale is essentially an armed robbery.David Hume gives another example of coercion by scarcity of alternatives:



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A man, dangerously wounded, who promises a competent sum to a surgeon tocure him, wou’d certainly be bound to performance; tho’ the case be not so muchdifferent from that of one, who promises a sum to a robber. (Hume, 2000, Book III,

Pt. II, Page 11) The surgeon has a tremendous bargaining power because the injured man is about to die and doesnot have the time to call another doctor. Hume’s view on this case coincides with that of mostpeople: the exercise of bargaining power by the surgeon is a form of extortion.

3 A two-parties negotiation model

3.1 Definitions

Define a two-party negotiation as a three-tuple 21 ,, d d v P , where 0v is the total value to bedivided between the parties, and

1d and

2d are the parties’ outside options. The (potential)

surplus of the negotiation is thus

.Σ21 d d v (1)

We will assume that the surplus is nonnegative, so 21 d d v . Otherwise, any exchange wouldnecessarily harm one of the parties. A negotiation with a positive surplus is profitable . Anegotiation with zero surplus is unprofitable .

We will also assume that the negotiation satisfies conditions 1 – 5 from the previous section. These were, briefly, that ownership is exclusive and transferrable, there is no fraud orpsychological compulsion, there are no uncompensated non-pecuniary externalities, and no partycan force the other to accept a particular allocation.

Let 1x and 2x be the payoffs to party 1 and party 2, respectively. A negotiation outcome isan allocation X x x ),( 21 , where

.: ),( ),,(

,

2121

21

v x x x x A

d d d

d A X

(2)

Allocation d is the disagreement outcome , in which both parties obtain their outside options. The allocations in set A are called (potential) agreements . The set of agreements can also be written as follows,

,Σ: ),(

212121 d d x x x x A

(3)since .Σ

21 d d v



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As an example, consider the following negotiation. Ada has carved a sculpture which she values at $1,000. The rest of her wealth amounts to $1,000,000. Bob, a collector, is interested inthe sculpture, which he values at $1,200. Before the negotiation, his wealth amounts to $1,500,000.

The surplus equals the difference in valuations; that is, $200. Formally,

.200$000,1$200,1$Σ

,000,500,1$

000,001,1$000,000,1$000,1$

Bob

Ada

d

,d

(4)

If Ada and Bob reach an agreement, then

, , ,$ px

, , ,$ px

0005001200,1$0000001

Bob

Ada (5)

where p is the price agreed by the parties. Thus, the disagreement outcome, the set of

agreements, and the set of all possible outcomes are

. )000,500,1$,000,001,1($200$: ),(

,200,501,2$: ),(

),000,500,1$,000,001,1($ ),(

Bob AdaBob Ada

Bob AdaBob Ada

Bob Ada

x x x x X

x x x x A

d d

(6)

An agreement is mutually beneficial (i.e., Pareto-improving or neutral) if and only if 11 d x and 22 d x . This condition is “strong” if both inequalities are strict, and “weak” if at least one ofthe constraints is met with equality (i.e., that party is not strictly benefitted, but is indifferentbetween exchanging and not exchanging). The set of mutually beneficial agreements is thus

}.,,Σ

: ),{( 2211212121 d x d x d d x x x x M

(7)

In what follows, we will assume that all mutually beneficial agreements are achievable throughnegotiation, although this may not always be the case in reality. It may be that a potential surplusexists, but that it cannot be divided between the parties; for instance, if it happens that one of theparties lacks sufficient liquidity and cannot borrow.

For example, Andrew is broke, homeless, and starving. Beatrice is a prosperous baker. Heenters Beatrice’s bakery and begs her for bread , which would save him from starvation. There is a

very large surplus at stake, since the value of a human life is much higher than the cost ofproducing a loaf of bread. However, there are only two possible negotiation outcomes, neither of

which is mutually beneficial: Beatrice ignores Andrew and he starves, or she gives him a loaf of

bread for free (losing the opportunity cost) and he survives. Since a mutually beneficial agreementis not achievable, this negotiation is not a true case of standard market exchange. Thus, our modelis silent about its moral content.



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In principle, any non-mutually beneficial agreement could be achieved with the voluntaryconsent of both parties. In our model, an agreement in which one party sacrifices itself to benefitthe other is considered supererogatory. There are two sets of supererogatory agreements ,

depending on which party makes the sacrifice: .2,1for,,: ),( 2121 i d x v x x x x S i i i (8)

Note that 21 S M S X , where 1S , M , and 2S are disjoint sets. The set of non-supererogatory negotiation outcomes contains all mutually beneficial

agreements and also the disagreement outcome:

. ),( 2121 d d M S S X NS (9)

The sets M and ),( 21 d d need not be disjoint. If M d d ),( 21 , then the negotiation has zerosurplus, and vice versa .

3.2 Euvoluntary exchange and the disparity threshold

Without loss of generality, let us assume that 21 d d . The fact that the outside option for party 1is worse means that party 1 is the weaker party to the negotiation and that party 2 is the stronger

party . Euvoluntariness does not require that there be zero disparity in outside options, but onlythat the disparity not be “too large.” The specific meaning of “too large” depends on theobserver’s revulsion toward the imbalance in bargaining power.

An observed disparity in outside options is fair if and only if it does not exceed the disparitythreshold function θ ; that is, if and only if θ d d 12 , where 0θ . The smaller is θ , the lesslikely that any given exchange will be euvoluntary. To quantify the magnitude of the unfairness, wedefine the (degree of) unfair disparity :

0,maxΔ12 θ d d (10)

We distinguish between a negotiation , which is the process of reaching an agreement or adisagreement, and an exchange , which is the act of consummating an agreement. If there is nounfair disparity (i.e., if 0Δ ), the negotiation is fair . Any exchange between the parties will beconsidered euvoluntary . If there is an unfair disparity, the negotiation is unfair . Any exchangebetween the parties will be considered non-euvoluntary .

We will assume that the disparity threshold is a non-decreasing function of the weaker party’soutside option:

,: )( 1

D d θ (11)



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where 0D . Intuitively, the disparity threshold function should increase with the weakerparty’s outside option (in absolute terms, not in relation to the other party’s outside option). Thereason is that an imbalance of bargaining power strikes most people as more unfair when the

weaker party is desperate.Figure 1 shows what may be considered a reasonable disparity threshold function. As 1d

grows smaller, the value of )( 1d θ asymptotically approaches zero. Consequently, all negotiations will violate the unfair disparity threshold, because even the smallest differences between 1d and 2d will be larger than )( 1d θ . But for larger values of 1d , the value of )( 1d θ might rise very quickly,implying that even exchanges with significant disparity in outside options will be euvoluntary.

Radical egalitarianism and radical libertarianism can be modeled as particular cases of the

threshold function. On the side of radical egalitarians, Olsaretti (2003)5 would argue that )( 1d

θ should be close to zero for all values of 1d . This implies that an exchange can only be euvoluntary

if the difference in outside options is negligible. On the side of radical libertarians, Robert Nozick(1974) (worried about “preventing capitalist acts between consenting adults”) would advocate foran “infinite dispari ty threshold,” regardless how low the value of 1d . This is to say that voluntaryand euvoluntary exchanges are two names for the same thing. In Nozick’s worldview, there is nosuch thing as coercion by scarcity of alternatives.

To summarize, in this section we posited a specific condition under which negotiations are fairand exchanges are euvoluntary:

.00 ),( )( maxdisparity unfair 112 d θ d d (12)

5 See also Colburn 2008, and Gaus 2007.

Figure 1: A reasonable disparity threshold function

θ ( d 1 )

d 1



12

This condition would be true if the parties have similar bargaining power (i.e., 1d and 2d are closein magnitude), but the condition could also be true if the disparity threshold is so large that we donot care that the weaker party lacks a decent alternative to a negotiated agreement. We allow the

disparity threshold to increase with the value of the weaker party’s outside option, which capturesthe intuition that disparity in outside options is more unfair when the weaker party is desperate.

4 The fictitious negotiation model of just market exchange

Formally, a fictitious negotiation is a three-tuple 21 ,, d d q P . Its only difference with theactual negotiation P is that P assigns an improved fictitious outside option 1d to the weakerparty. The fictitious outside option is defined as follows:

,Σ,Δmin11 d d (13)

whereΔ

is the unfair disparity of the actual negotiation, andΣ

is its surplus.Properties of the fictitious negotiation

1. If the actual negotiation is fair, then the actual and the fictitious negotiations are identical.

.0Δ P P (14)

2. If the actual negotiation is unfair and has zero surplus, then the actual and the fictitiousnegotiations are identical. Both have a unique mutually beneficial agreement, whichcoincides with the actual disagreement outcome.

.0Σ0Δ d M M P P (15)

3. If the actual negotiation is unfair but potentially profitable, and the unfair disparity does notexceed the surplus, then the fictitious negotiation completely eliminates the unfair disparityand is potentially profitable. The fictitious negotiation has infinitely many mutuallybeneficial agreements, which constitute a proper subset of the actual mutually beneficialagreements. The actual disagreement outcome is not a feasible outcome of the fictitiousnegotiation.

.c0Σ0ΔΣΔ0 X d M M M (16)

4. If the actual negotiation is unfair but potentially profitable, and the unfair disparity exceeds

the surplus, then the fictitious negotiation reduces the unfair disparity but does noteliminate it. The fictitious negotiation is unfair, but less unfair than the actual negotiation.

The surplus of the fictitious negotiation is reduced to zero. The fictitious negotiation has a



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unique mutually beneficial agreement, which corresponds to the fictitious disagreementoutcome. This outcome assigns the entire surplus to the weaker party. The actualdisagreement outcome is not a feasible outcome of the fictitious negotiation.

. ),Σ( 0ΣΔΔ0ΔΣ0 21 X d d d d d M (17)

Proofs to these properties are provided in Appendix A.1

A negotiation outcome can be either equitable or inequitable . Three requirements must bemet for an outcome to be equitable. First, it must be feasible (i.e., it bust belong to X ) Second,the outcome must be a non-supererogatory outcome of the fictitious negotiation. Third, thefictitious negotiation must completely eliminate the unfair disparity.

In formal terms, the set of equitable outcomes is

.0Δ

if

,0Δif S N X E (18)

The inequitable outcomes , on the other hand, correspond to the non-supererogatory outcomesthat are not equitable:

E NS IE (19)

The sets of equitable and inequitable outcomes are decomposed in Table 1.

Properties of the equitable outcomes

1. If the actual negotiation is fair, then all non-supererogatory outcomes are equitable(including the disagreement outcome).

.0Δ d M NS E (20)

Table 1: Decomposition of the sets of equitable, inequitable, just, and unjust outcomes

Case Equitableoutcomes

Inequitableoutcomes Just outcomes Unjust outcomes



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2. If the actual negotiation is unfair and has zero surplus, then no outcome is equitable.

.0Σ0Δ E (21)

3. If the actual negotiation is unfair but potentially profitable, and the unfair disparity does notexceed the surplus, then the negotiation has infinitely many equitable outcomes. Thesecorrespond to the fictitious mutually beneficial agreements. These agreements constitute aproper subset of actual mutually beneficial agreements. The actual disagreement outcome isinequitable.

.cΣΔ0 UEd M M E E (22)

4. If the actual negotiation is unfair but potentially profitable, and the unfair disparity exceedsthe surplus, then no outcome is equitable.

.ΔΣ0 E (23)

Proofs to these properties are provided in Appendix A.2.

It may seem natural that a negotiation outcome must be equitable to be considered just. Butthis requirement is too stringent to be reasonable, as it would prohibit inequitable mutuallybeneficial agreements when no equitable agreement is available (property 4). Worse yet, theprohibition will take effect precisely when the weaker party more desperately needs to exchange.

This is because the disparity in outside options is more unfair when the weaker party’s outsideoption is more disastrous.

We propose a less stringent definition of justice that avoids this problem. We affirm that thejust outcomes correspond to the implementable non-supererogatory outcomes of the fictitiousnegotiation. These are the outcomes that two self-interested and rational parties could reacheuvoluntarily if the fictitious negotiation actually took place.

In formal terms, the set of just outcomes is

S N X J (24)

The unjust outcomes , on the other hand, correspond to the non-supererogatory outcomes thatare not just:

. J NS UJ (25)

The sets of just and unjust outcomes are decomposed in Table 1.



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Properties of the just outcomes

1. If the actual negotiation is fair, then all non-supererogatory outcomes are both equitableand just (including the disagreement outcome). Therefore, exchange is morally optional.

.0Δ d M NS E J (26)

2. If the actual negotiation is unfair and has zero surplus, the disagreement outcome is just,although it is not equitable. This outcome coincides with the unique mutually beneficialagreement of the actual negotiation, which is also just. Therefore, exchange is morallyoptional.

.0Σ0Δ UF d M d J (27)

3. If the actual negotiation is unfair but potentially profitable, and the unfair disparity does notexceed the surplus, then the negotiation has infinitely many just outcomes. Just andequitable outcomes are the same, and correspond to the fictitious mutually beneficialagreements, which constitute a proper subset of actual mutually beneficial agreements. Theactual disagreement outcome is unjust. Therefore, the stronger party is morally obliged toexchange with the weaker party.

.cΣΔ0 UJ d M M E J J (28)

4. If the actual negotiation is unfair but potentially profitable, and the unfair disparity exceedsthe surplus, then the fictitious disagreement outcome is the only just outcome, although it isinequitable. The just outcome assigns the entire surplus of the actual negotiation to the

weaker party, so the stronger party is left indifferent between exchanging and notexchanging. The actual disagreement outcome is unjust. Therefore, the stronger party ismorally obliged to exchange with the weaker party.

. ),Σ( ΣΔ0 21 UJ d d d d UEd d M J (29)

Proofs to these properties are provided in Appendix A.3.

As a corollary to these properties, we have:

1. All negotiations have at least one just outcome.

2. All equitable outcomes are just.

3. If a negotiation has at least one equitable outcome, then the just and equitable outcomescoincide.



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4. If a negotiation does not have an equitable outcome, then it has a unique just outcome, which assigns the entire surplus to the weaker party.

5. If the negotiation is unfair but potentially profitable, then a disagreement is unjust.

5 Examples

5.1 A ship without an anchor

A ship at sea that has an anchor to spare meets another which has lost all heranchors. What here shall be the just price that she shall sell her anchor to thedistressed ship? (Locke, 1661/2005, Venditio, p.p. 445 – 6)

Denote by 01 u the use value of a first anchor, and by 02 u the use value of a spare. Sincethe lack of an anchor is dangerous, the use value of a main anchor is quite high, approaching the

value of the ship plus the value of its cargo and a “value” for the lives of the passengers and crews. A spare anchor is inessential, but provides valuable insurance against the accidental loss of the firstanchor. Hence, 21 u u . At port, anchors are bought and sold at 0mkt p .

The ships’ outside options correspond to the status quo:

,2

,0

mkt212

1

pu u d

d (30)

where ship 1 is the distressed ship, and ship 2 is the rescuer. The outside option of the rescuer shipincludes the market price of her anchors, because at the end of her voyage the ship will sell themat the port. Or, what amounts to the same thing, the ship could “sell” the anchors to herself at the

market price to use again on a future voyage. The distressed ship values the anchor at mkt1 pu , while the rescuer ship values it in

mkt2 pu . The surplus of the negotiation is equal to the difference in valuations:

.0Σ21 u u (31)

Following Locke, we will assume that the prospect of a shipwreck is so dire that the unfairdisparity is greater than zero:

.0 )0( 2Δmkt21 θ pu u (32)

Locke argues in his Venditio that the only just price for the spare anchor is its market price plus

its use value for the rescuer ship. At that price, the rescuer is indifferent between selling and notselling. Locke claims that the rescuer is not obliged to perform a supererogatory act and simply



17

give away the anchor. But at the same time he claims that the rescuer ship may not profit from thedistress of the other ship, recognizing that “profit” in this case means taking a price above theprice required to make the rescuer ship indifferent between selling and not selling.

Under reasonable parameterizations, a fictitious negotiation would lead to the same conclusionreached by Locke. Recall that the non-worseness clause is activated if the unfair disparity is largerthan the surplus. In the case at hand, this condition reduces to

.0 )0( 2ΣΔmkt2 θ pu (33)

Most people would agree that the value of human life is so high that the disparity threshold in thiscase must be extremely low, probably close to zero. This implies that the previous condition ismet. As Locke intuited, the rescuer ship has a duty to hand over the entire surplus to the distressedship, but not a penny more. Moreover, and contrary to Locke’s intuition 6, the rescuer boat ismorally obliged to exchange with the distressed ship. The rescuer cannot argue that property rightsallow her to refuse the sale.

5.2 Competitive market exchange

Is it just to charge high prices for an essential commodity? We argue that the answer is yes,provided the market for the commodity is competitive. All competitive market exchanges are just,because they do not generate a surplus. We show this below.

A producer and a consumer are bargaining over the price of a certain commodity, which is alsotraded in a competitive market. Let 0c be the cost of producing the commodity, and let u be itsuse value for the consumer. The parties’ alternative to the exchange is trading with someone else inthe market. Denote by mkt p the market price of the good, and assume that u pc mkt . Let

prodw be the wealth of the producer, and let consw be the wealth of the consumer. It follows that

the outside options are

.

,

consmktcons

prodmktprod

w pu d

w c pd (34)

Also, if an agreement is reached, then

,

,

conscons

prodprod

w pu x

w c px (35)

6 As Locke put it: “ And in this case the master of the vessel must make his estimate by the length of his voyage, the seasonand seas he sails in, and so what risk he shall run himself by parting with his [extra] anchor, which all put together he would not part withit at any rate , but if he would,he must then take no more for it from a ship in distress than he would from any other. ” In other

words, there is no obligation to sell the extra anchor, though the value of the first anchor to the distress ship is clearly higher thanthe second anchor to the other ship.



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where p is the price agreed between the parties. Therefore, the surplus created by the exchange isgiven by

,0

Σ

consmktprodmktconsprod

consprodconsprod

w pu w c pw pu w c p

d d x x

(36)

since all terms are cancelled in the subtraction. Because the surplus of the negotiation is zero, theunique just agreement is equivalent to the disagreement outcome. Also, the disagreement outcomeand the unique mutually beneficial agreement coincide. Hence, exchange is morally optional.

The above reasoning is valid even in scenarios of extreme shortage of basic necessities. Lockeprovides the following example.

[L]et us suppose a merchant of Danzig sends two ships laden with corn, whereofthe one puts into Dunkirk, where there is almost a famine for want of corn, and therehe sells his wheat for 20s a bushel, whilst the other ship sells his at Ostend just by for5s. Here it will be demanded whether it be not oppression and injustice to make suchan advantage of their necessity at Dunkirk as to sell them the same commodity at 20sper bushel which he sells for a quarter the price but twenty miles off? (Locke,1661/2005, Venditio, p. 444)

In this example, the market price is 20s a bushel, while the “production cost” is 5s , because thecost of “producing” a bushel of corn is the alternative cost of not selling it for 5s at Ostend.Despite the great sales margin, the corn market is competitive in Dunkirk. This is evidenced by thefact that the merchant is a mere price taker. It follows that corn sales yield zero surplus, which inturn implies that the market price is just.

Locke reaches the same conclusion, although his makes a slightly different argument. Hisargument is based on the concepts of price discrimination and arbitrage:

[The merchant commits no injustice], because he sells at the market rate at theplace where he is [Dunkirk], but he sells there no dearer to Thomas as he would toRichard. And if there he should sell for less than his corn would yield, he would onlythrows his profit into other men’s hands, who buying of him under the market rate

would sell it again to others at the full rate it would yield. (Locke, op. cit., p. 444.)

5.3 Price gouging in Dunkirk

Consider the alternative scenario in which our merchant is the first to arrive at Dunkirk. This givesthe merchant an enormous bargaining power: If the people do not accept his prices, the merchantcan take his business to Ostend, while they go hungry waiting for the arrival of more supplies. At



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the same time, the intense hunger of the people of Dunkirk, combined with the low cost of corn,results in a huge negotiation surplus. Given these circumstances, the analytical theory of justmarket exchange stipulates that the merchant must sell his corn at a low price, perhaps even at

cost.Suppose that Thomas desperately needs corn for his own consumption, but he also detects a

business opportunity: Richard, his neighbor, is unaware of the arrival of the merchant. So thecunning Thomas goes to the just merchant’s shop and asks for two bushels of corn, with thesecret intention of keeping one bushel for himself and selling the other to Richard at a higherprice.

The merchant feels obliged to sell the bushels to Thomas at a low price. But the merchant, who is not a fool, anticipates Thomas’ intentions. The merchant knows that Thomas will resell onebushel at an abusively high price, to someone who also has a desperate need of corn. What shallthe merchant do? This contradiction between two imperatives, make a just exchange and preventan abusive exchange, appears to invalidate our theory.

But the contradiction is only apparent. It arises from the violation of a simplifying assumptionof the fictitious negotiation model, not from an incoherence of its moral premises. The fictitiousnegotiation model assumes the existence of only two agents, and rules out externalities. Thisassumption ceases to be valid in scenarios in which the exchange may unjustly harm third parties.In Locke’s example, that third party is Richard.

Nevertheless, the logic of the theory of just market exchange extends naturally to the multi-agent case, as illustrated by the following argument. Given his privileged position, the merchant ismorally obliged to sell corn to Thomas at a discount price, but he has the same obligation towardevery famished person in Dunkirk. Therefore, if Thomas resold one of his bushels at a high price,the merchant wound fail in his duty to an anonymous buyer, who is as desperate as Thomas. Tofulfill his duty, all that the merchant needs to do is sell a single bushel to each costumer. In times

of famine, price caps and rationing are the moral market strategy.

5.4 Sharecropping

The harvest has just ended and the hacendado and his inquilino renegotiate their sharecroppingagreement. What agreement would be just?

Suppose that the parties expect that the value of the next harvest will be 0h . If an agreementis reached between the parties, the inquilino will have to incur a cost of 0c (this includes food,effort, etc.), while the hacendado will have to invest to invest a total of 0i (these includes seeds,fertilizers, etc.). The value of the crop is enough to cover all the costs of production: i c h .

Thus, the set of potential agreements is given by

,: ),( HIHI i c h x x x x A (37)



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where subscript i identifies the inquilino and subscript H identifies the hacendado.If the hacendado and his inquilino fail to reach an agreement, the inquilino may move out from

the hacienda and seek work elsewhere as a day laborer. The hacendado, on the other hand, is free to

evict the inquilino and hire day laborers at market wage w . However, the harvest would suffer dueto the day laborers’ lack of sk ills and lower effort. It is expected that the harvest will fall from h toh . The outside options are thus,

.

,

H

I

w i h d

c w d (38)

Both outside options are positive. The surplus of the negotiation is given by

.ΣHI h h d d v (39)

This surplus is small but not negligible, because day laborers are imperfect substitutes for theinquilino.

As is usual in the countryside, the hacendado is the stronger party to the negotiation, and thenegotiation is unfair. That is,

.0 )( 2 )( ΔIIIH d θ w i c h d θ d d (40)

The unfair disparity is positive but not dramatically high, because the inquilino has a poor butrelatively decent outside option.

What is the minimum payoff to the inquilino that would satisfy the requirements of justice? According to our theory, the answer is his fictitious outside option.

.Σ,ΔminΣ,ΔminII c w d d (41)

This fictitious outside option may or may not eliminate the unfair disparity, depending on whetheror not ΣΔ . Both scenarios are plausible, since Δ is positive but not dramatically high, and Σ is small but not negligible.

Let Iw be the implicit wage that the inquilino would earn if he stayed at the hacienda. Thepayoff to the inquilino would be his wage minus the costs he incurs. That is, c w x II . We alsoknow that, the minimum just payoff to the inquilino is his outside option Id . Hence, a justimplicit wage satisfies:

.Σ,ΔminII w c d w (42)



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This means that the hacendado is not only morally forbidden to evict the inquilino from hishacienda. He is also obliged to pay the inquilino a higher salary than he would earn working as aday laborer. In the extreme case that ΣΔ , the inquilino must earn exactly

,ΣI h h w w w (43)

That is, the wage of a day laborer plus the harvest that would be lost if the inquilino moved outfrom the hacienda. The payoff to the hacendado would thus be

.H w i h w i h x H (44)

In words, the hacendado must content with the same profit that he would obtain if he hired daylaborers instead of keeping the inquilino.

6 Concluding remarks

The main modern theories of justice commit themselves to a strictly deductive approach (à laKant). The theorist first dictates a finite set of universal moral axioms, which are self-evident andtherefore must be accepted by every morally-sane person. The axiomatic theory is supposed to

work as an automated reasoning machine that receives as an input a moral dilemma and producesas an output a moral judgment. Many times, perhaps most of the time, the moral judgments appearto be reasonable, or at least defensible, as general propositions. But inevitably, the machine willface a confounding moral dilemma, of the sort in which philosophers delight. At this point, insteadof delivering an obvious or intuitively sensible result, the axiom machine will deliver a judgment sorepugnant that the axioms are rendered untenable. And so the machine is scrapped, thrown into apile with all the other broken axiomatic systems.

Confronted with this predicament, the theorist takes one of two ways out. The first is to stickto the logical consequence of his theory, no matter how absurd. This was Kant ’s gallant reaction

when posed with the following dilemma: “would you tell the truth to a murderer that knocks onyour door, and asks you if his victim is hiding inside your house?” Kant (1797) maintained that he

would not lie, because not to lie is a categorical imperative. Instead, he would give an “ambiguous”answer to the murderer: “I do not know if he is. ” Needless to say, few people consider this to be arightful answer in such a situation.

The second way out for the theorist is to patch his theory with a list of exceptions. Considerthe libertarian traditional justification of private property rights . In Locke’s words,

Though the earth, and all inferior creatures, be common to all men, yet every man

has a property in his own person: this no body has any right to but himself. Thelabour of his body, and the work of his hands, we may say, are properly his.



22

Whatsoever then he removes out of the state that nature hath provided, and left it in,he hath mixed his labour with, and joined to it something that is his own, and therebymakes it his property. It being by him removed from the common state nature hath

placed it in, it hath by this labour something annexed to it, that excludes the commonright of other men: for this labour being the unquestionable property of the labourer,no man but he can have a right to what that is once joined to, at least where there isenough, and as good, left in common for others . [Locke, 1661/2005, Second Treatise ofGovernment, Chapter V, Sec. 27; emphasis added).]

Nozick (1974) calls Locke’s final qualification the “Lockean proviso”. This can be summarized asfollows: homesteading private property by mixing labor with natural resources is legitimate only

where there is enough for everyone.In his own libertarian manifesto, Nozick applies the Lockean proviso to the following

examples:

Once it is known that someone ’s ownership runs afoul of the Lockean proviso,there are stringent limits on what he may do with (what it is difficult any longerunreservedly to call) “his property.” Thus a person may not appropriate the only waterhole in a desert and charge what he will. Nor may he charge what he will if hepossesses one, and unfortunately it happens that all the water holes in the desert dryup, except for his. This unfortunate circumstance, admittedly no fault of his, bringsinto operation the Lockean proviso and limits his property rights. Similarly, anowner’s property right in the only island in an area does not allow him to order acastaway from a shipwreck off his island as a trespasser, for this would violate theLockean proviso. (Locke, 1974, p. 180)

In other words, Nozick backs himself into a corner, and then invokes to his defense a deux exmachina . Property rights are absolute, unless they are not. No one is obliged to share a glass of

water, except that one may be obliged under some circumstances to do just that. The fundamental flaw with Nozick’s line of argument is that the specific meaning of

“sufficient” is subjective, casting down the claim to universality of the libertarian axioms. It caneven be argued, as economists do, that resources are always scarce (Robbins 1939). In this limitingcase, the libertarian justification of property rights collapses into one big exception that ironicallyprecludes all private property.

The natural alternative to strict moral deductivism is sentimentalism, a moral philosophy thatposits that moral judgments are based on feelings rather than universal moral principles. One ofthe foremost sentimentalists is David Hume 7 (2000, p. 325), who wrote:

7 Hume’s sentimentalist beliefs were shared by his friend Adam Smith (Smith, 1759), and both friends were influenced by theearlier writings of Anthony Ashley Cooper (Cooper, 1737), and Francis Hutcheson (Hutcheson, 1728).



23

Morals excite passions, and produce or prevent actions. Reason itself is utterlyimpotent in this particular. The rules of morality, therefore, are not conclusions of ourreason.

Sentimentalism is the ultimate version of moral relativism: anything is right, as long as you feelit is right. For its refusal to deliver judgments, sentimentalism is useless as a moral philosophy. Butthis is a minor flaw compared to its absolute empirical failure. Systematic observation of humanbehavior falsifies the sentimentalist claim that in matters of moral values, anything goes (Rachels1995). Certainly, moral values vary broadly across people and cultures, but they do not varyhaphazardly. There is a structure to values (Rachels 1995, Brown 1991).

We argue that moral universals exist, but not in the form of a minimal list of principles sharedby all cultures, as suggested by Rachels (1999) and others. Moral universals are not principles butlogical structures that constrain the set of plausible moral theories. The scope of disagreement is

tightly limited to the specific values of parameters that modulate a set of moral axioms. Therefore,this approach to moral philosophy is neither strictly deductive nor sentimentalist, but parametric. To be more precise, universals are captured by parametric meta-theories of morality. Dependingon the values taken by its parameters, a meta-theory produces a particular, concrete moral theory.

The analytical theory of just market exchange is an example of a parametric meta-theory. Thefree parameter of the theory is the disparity threshold function, which quantifies the observer’srevulsion toward unfairness. The fictitious negotiation model provides the logical scaffolding.

The analogy between the analytical theory of just market exchange and the utility theoryclarifies the parametric approach. Preferences can vary broadly among people, but utility theoryrequires four conditions for rationality: completeness, reflexivity, transitivity, and continuity. Manynonsensical preference rankings are ruled out by these axioms.

For example: Carlos cannot prefer simultaneously an apple to an orange, an orange to abanana, and a banana to an apple. The utility theory then dictates that a rational person mustmaximize utility subject to his budget constraint. Every rational person must obey this dictation,regardless of his particular utility function. Otherwise, he would not be rational. In a similarfashion, the analytical theory of just market exchange admits that aversion to unfairness (e.g., todisparity in outside options) can vary broadly among people. But the theory also establishes that alldisagreements in aversion to unfairness are captured by differences in the non-decreasing disparitythreshold functions θ.

The theory then dictates that a just person is obliged to use the fictitious negotiation model tojudge the morality of a market exchange. Every just person must obey this imperative, regardless ofhis particular disparity threshold function.



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The fictitious negotiation model produces logical consequences that are morally intuitive. Thisis appealing, if the theory unifies conflicting commonsense principles under a single framework.8 Of course, consistency is not necessary for acceptance; existing moral intuitions do not constitute

an “empirical test” of an axiomatic theory. In fact, axiomatic theories can re-educate intuition bymeans of persuasion. They can also guide action when several moral principles come into conflictand none seems to dominate over the others. In fact, this is precisely why we need moralphilosophy on the first place: the need to assist intuition in the “hard cases” afforded to us bymoral dilemmas.9

For its parametric character, our theory can be considered part of the market democracyresearch program initiated by Tomasi:

I propose we treat justice as fairness…as a living research program. The generaldefinition of justice as fairness… leaves room for a significant (but bounded) range ofdisagreement about the requirements of liberal justice. These ambiguities and tensions

are fundamental features of justice as fairness. They spring from the complex natureof the bedrock liberal idea that citizens are free and equal moral beings. (Tomasi 2012;p. 179).

In this spirit, we recognize that there are substantial discrepancies about what makes a fairnegotiation and a just market exchange. But these differences are bounded by the seeminglyuniversal moral sentiment that all persons posses an essential dignity. In market interactions, theprohibition of voluntary exchanges tramples this dignity, as does the arbitrary imposition of termsof exchange by an enlightened third party. In the market, human dignity is honored by respectingthe liberty of others, and by refraining from abusing power and from harming others, either by actor omission.

These are the axioms that constitute our theory.

8 As Kamm (1993, p. 26) explains: “[One] philosophical method (there may others) begins with responses tocases… This procedure is attracti ve because it permits recognition of new factors that may be morally relevant incertain cases, factors emphasized by no theory yet developed. Philosophers using this method try to unearth thereasons for particular responses to a case and to construct more general principles from these data. They thenevaluate these principles in three ways: Do they fit the intuitive responses? Are their basic concepts coherent anddistinct from one another? Are the principles or basic concepts in them morally plausible and significant, or evenrationally demanded? The attempt to determine whether the concepts and the principles are morally significant andeven required by reason is necessary in order to understand why the principles derived from cases should be

endorsed.” 9 Insofar as they are truly hard. Kant’s murderer dilemma, for example, is not really a dilemma, but aprovocation.



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