Afriat and arbitrage

This paper presents a proof of Afriat’s (Int Econ Rev 8:67–77) theorem on revealed preference by using the idea that a rational consumer should not be vulnerable to arbitrage. The main mathematical tool is the separating hyperplane theorem.


Introduction
There are a number of proofs of Afriat (1967)'s famous theorem on revealed preference. This paper proposes another one. It is based on the idea that a consumer should not be vulnerable to arbitrage or money pumps if her behavior is internally consistent. The only non-trivial mathematical tool it uses is the separating hyperplane theorem. The current proof has the advantage that it is based on familiar economic ideas and so is perhaps more accessible than some others. The paper also links the proof to arbitrage arguments in other areas of economics. Echenique et al. (2011) propose a measure of the seriousness of violations of the Generalized Axiom of Revealed Preference (GARP) by using a money pump argument. This idea can be used to motivate the proof in the current paper. Suppose that the consumer chooses x i at prices p i and x j at prices p j but x j is affordable at prices p i and x i is also affordable at prices p j . The Weak Axiom of Revealed Preference, and a fortiori GARP, is violated as x i is revealed preferred to x j but x j is also revealed preferred to x i . Echenique et al. (2011) argue that the consumer is vulnerable to a money pump. An arbitrageur could buy x i and sell x j at prices p j and sell x i and buy x j at prices p i . Each transaction is profitable and the arbitrageur would make an overall profit of mp , or equivalently the consumer makes an overall loss of −mp where: − mp = p i · (x j − x i ) + p j · (x i − x j ) < 0 ( 1 ) Echenique et al. (2011) use mp as measure of the seriousness of violations of GARP. Instead of invoking an arbitrageur one could simply regard this as a measure of how much money the consumer could save by consuming x j at prices at p i instead of x i and x i instead of x j at prices p j , which given her cyclic preferences she would be willing to. Now a consumer does not care about a money pump directly but rather utility and it is possible that (1) holds even if GARP is not violated. GARP only rules it out if preferences are quasi-linear (see Brown and Calsamiglia 2007). One can however show that if GARP holds there exist numbers λ i and λ j such that for any pair of observations λ i and λ j can be thought of as the value of money, or the marginal utilities of money, in states i and j so that the consumer is not vulnerable to a money pump in utility terms.
One can give an interpretation in terms of asset pricing. In the example above, when GARP is violated the arbitrageur gains at each step of the trade (buying x j and selling x i to the consumer at prices p i and then buying x i and selling x j at prices p j ). This is analogous to an asset which is profitable whatever the state (here the states are the observations i and j) and as such is a 'free lunch'. According to the fundamental theorem of asset pricing [see for example Dybvig andRoss 2008 or Ross (2005, Chapter 1)], if arbitrage is not possible, that is there are no free lunches, then there exist a set of state prices or risk neutral probabilities such that any portfolio with zero cost has non-positive expected return. The λ i are analogous to these state prices or risk neutral probabilities.
These arguments on arbitrage can also be connected to subjective probability. Nau and McCardle (1991) show that if there does not exist a bet which is always favorable to the buyer, then there must exist subjective probabilities such that every 'acceptable' gamble has non-negative expected value. There is also some analogy with the work Cournot (1927) Chapter III on exchange rates-see Ellerman (1984) for discussion of Cournot's work.
There are a number of proofs of Afriat's theorem. Afriat (1967)'s original proof was combinatorial in nature and quite complicated. It contained a gap which was filled by Diewert (1973) and Varian (1982). Teo and Vohra (2003) give a simpler combinatorial proof and bring out clearly the connection with finding negative length cycles in graphs, a theme initially explored by Varian (1982). Shiozawa (2016) further extends the link made by Teo and Vohra (2003) to shortest path problems and the condition that there should exist weights so that that are no negative length cycles. His proofs are combinatorial in nature. The existence of weights so that there are no negative length cycles can be thought of as implying the existence of marginal utilities so that arbitrage cycles cannot exist, so (2) holds. Shiozawa (2016) however concentrates on connections to shortest path problems rather than links to conditions for absence of arbitrage.
The proofs closest to this one are that of Fostel et al. (2004) and that of Geanakoplos (2013). Fostel et al. (2004) give an elegant proof using the duality theorem of linear programming. Geanakoplos (2013) appeals to the minimax theorem in zero-sum games.
This paper instead uses the separating hyperplane theorem. The duality theorem of linear programming, the minimax theorem and the separating hyperplane theorem are of course closely connected-see for example Trustrum (1971) for a concise account. The separating theorem is, however, familiar to economists and so the proof is perhaps more readily accessible than Fostel et al. (2004)'s and Geanakoplos (2013)'s.
The second fundamental theorem of welfare economics can of course be proved using the separating hyperplane theorem. Krasa and Yannelis (1994) show that a separation argument can be used to prove the Gale-Nikaido-Debreu lemma and thus the existence of competitive general equilibrium (see also Yannelis 1985). This paper shows that a separation argument can be used to prove another classical result, Afriat's theorem.
The paper proceeds as follows. Section 2 outlines the main result and gives an application to quasi-linear utility. Section 3 shows that the same method used in proving in Afriat's theorem also yields the extension of Afriat's theorem to general budget sets given by Forges and Minelli (2009). It also gives an application to homothetic utility functions.

Results
Consider a sequence of observations ( p i , x i ) on a consumer, where i = 1, . . . , n and p i ∈ R K ++ and x i ∈ R K + are vectors of prices and of purchased quantities respectively. Afriat (1967) asks when this sequence is consistent with rational behavior, that is whether there is a locally non-satiated utility function u such that for each i, x i maximizes u over the budget set then x i is said to be revealed preferred to x j : x j was affordable at prices p i and budget p i · x i but was not chosen, so x i must be weakly preferred. The data satisfies the Generalized Axiom of Revealed Preference if the induced relation is acyclic.
For a cycle c if i is an element of the cycle denote the next element in the cycle by then they all hold with equality.
Theorem 1 (Afriat 1967) The following are equivalent: The difficult part of Afriat's theorem is the implication 2) ⇒ 3). Here this is proven by an arbitrage argument.
For any cycle c = {i, j, k, . . . , r } its elements a i j etc. can be thought of as (minus) the returns to an arbitrageur who engages in cyclic trade of selling x i and buying x j at prices p i etc. Each cycle can therefore be thought of as a potential trade or asset. The idea outlined in the introduction is that if revealed preference is satisfied, there must exist prices for money, or marginal utilities, so that the overall gain for a consumer from any cyclic trade is non-negative (and for the arbitrageur non-positive) in utility terms. Given this, 3) follows easily.
Recall the following version of the separating hyperplane theorem [see for example Ok (2007, Chapter G.3

, Proposition 5)]:
Lemma 1 Let C and D be disjoint non-empty convex sets in R L with C closed and D compact. There exists q ∈ R L such that The assumption that D is compact guarantees that the inequality in (5) can be taken to be strict.
The following well-known result is a simple consequence of Lemma 1 Lemma 2 Let A be a convex, closed subset of R L with A∩R L − = {0}. If A is polyhedral or a cone, there exists q >> 0 such that q · a ≥ 0 for all a ∈ A.
Proof The result follows from Theorem B.3.5 in Appendix B of Karlin (1959). A simple proof can be given following Appendix B of Duffie (2010). Note that if A is polyhedral and generated by a 1 , . . . , a r then the cone generated by a 1 , . . . , a r satisfies the assumptions of the lemma, so it is enough to prove the result when A is cone. Let M be the convex hull of the vectors f l , l = 1, . . . , L, where f l has lth entry −1 and all other entries 0.
M ⊂ R L − . A and M are disjoint and M is compact so by Lemma 1 there exists q strictly separating A and M: q.a ≥ α > β ≥ q · m for all a ∈ A and m ∈ M, for some α and β. Since A is a cone with 0 ∈ A, α can be taken to be 0. Considering the vectors f l in M shows that q >> 0.
Recall that a set is polyhedral if it is the convex hull of a finite number of vectors. The restriction that A be a cone or polyhedral is used to guarantee that q can be taken to have all components strictly positive.
Lemma 2 underlies the result that the absence of arbitrage opportunities implies the existence of state prices and the existence of an equivalent risk neutral measure in finance: see for example the proof of Theorem 2.7 in Harrison and Pliska (1981) or of Theorem 1.A of Duffie (2010). A general discussion of the so-called fundamental theorem of asset pricing can be found in Dybvig and Ross (2008).
In the context of asset pricing the set A is the set of all possible portfolios. If there are S states, which all occur with positive probability, then a portfolio can be represented as a = (−b 1 , . . . , −b S , p a ), where the final component is the price of the portfolio, so that L = S + 1, and the other components represent minus the payoff in each state. The condition that A ∩ R L − = {0} is the no arbitrage condition: any portfolio with guaranteed positive returns, that is non-negative returns in all states and strictly positive in some, has a strictly positive price (and any portfolio with zero returns in all states has a non-negative price).
Provided short sales are allowed, A is in fact a linear space. One therefore deduces that q · a = 0 for all a ∈ A. Normalizing q S+1 = 1, it follows that for a portfolio a, p a = s q s b s , that is the price of any portfolio equals its expected payoff in terms of risk-neutral probabilities or state prices. If short sales are not possible, so that portfolios can only be traded in non-negative amounts, A is simply a cone not a linear space and from q · a ≥ 0 one deduces p a ≥ s q s b s , that is the price of any portfolio is at least equal to its expected payoff in terms of risk-neutral probabilities or state prices. In particular if a portfolio has zero price it must have non-positive expected payoff. See for example Leroy and Werner (2014, Chapter 7) for further discussion of arbitrage with sales restrictions.
A similar result is used in Theorem 1 of Nau and McCardle (1991) showing the existence of subjective probabilities if arbitrage is not possible. There A is the set of all acceptable gambles and is polyhedral.
Lemma 2 implies that if there is no arbitrage opportunity, that is a gamble with strictly negative payoffs, then there exist subjective probabilities q such that every acceptable gamble has non-negative expected payoffs.
In the present article A is the convex hull of the set of the returns in each state generated by all possible cyclic trades and is polyhedral. Each observation is thought of as a state and a cycle c is thought of as generating return of (minus) a ic(i) = p i · (x c(i) − x i ) in state i to a hypothetical arbitrageur conducting cyclic trade, where c(i) is the next element of the cycle, if i belongs to the cycle (and return 0 if it does not). Cyclic trades have zero price in this interpretation and cannot be traded in negative amounts so, as in the case of asset-pricing with short sales restrictions, there must exist marginal utilities or state prices so that every cycle has non-positive expected return to the arbitrageur.
The convex hull may be thought of as mixtures of trades if it is possible for the hypothetical arbitrageur to make fractional sales and purchases of bundles. If randomization over trades is possible then it may represent expected returns from randomizing over trades. It may also simply be treated as an artificial construction.
There is of course no actual securities market on which the consumer can trade or indeed an actual arbitrageur who carries out cyclic trades. In particular the consumption bundles observed may not be those which would emerge if the consumer could transfer income between observations.
The interpretation of observations as states and marginal utilities as state prices in a hypothetical market simply provides a useful if imperfect analogy which is helpful in understanding Afriat's theorem.
Proof For any cycle c let a c be the element of R n with i-th element a ic(i) if i belongs to the cycle, 0 if i does not belong to the cycle. Denote the set of all cycles by C. Let A be the convex hull of the a c , that is a ∈ A if there exist μ c ≥ 0, c∈C μ c = 1, such that a = c∈C μ c a c . The null cycle, involving no swaps, has all entries zero, so 0 ∈ A. The result follows from Lemma 2 provided it is shown that A ∩ R n − = {0}. So suppose a ∈ A and a ∈ R n − but a = 0. Now a = c∈C μ c a c , for some μ c , so if for some i, a i < 0 there exists a c with μ c > 0 a c i < 0. Follow the cycle c from i. If there is no element j of the cycle with a c j > 0 then GARP is violated, so let j be the first element in the cycle after i with a c j > 0. Now a j ≤ 0, so there must exist another cycle c with a c j < 0. Now switch to following the cycle c from j onwards. If for some some subsequent k, a c k > 0 then switch to following a new cycle c with a c k < 0 and so on. Since the number of observations is finite, at some point indices must repeat and so if l is the first index at which there is a repetition, if we consider the subset of the path we have constructed which joins l to l we have constructed a cycle c * from l to l with a c * i ≤ 0 for all i and strict inequality for some i. This contradicts GARP.
The remainder of the proof is standard. Lemma 3 constructs the marginal utilities of money, λ i . Utilities consistent with these can be constructed by ensuring that if a consumer starts with x i and then is given enough money to buy x j at these prices and current marginal utilities then her utility must be at least the utility of consuming x j , u j . The marginal utility of money may not be constant but this will still be an over-estimate of u j if utility is concave.
The proof is based on this idea but also considers indirect ways of giving the consumer enough to buy x j , for example first giving them enough money to buy x k at price p i then enough money to buy x j at prices p k , to ensure the estimates are mutually consistent.
Lemma 4 If the conclusion of Lemma 3 holds then there exist u i , i = 1, . . . , n, such that u j ≤ u i + λ i a i j for all i, j, i = j.
Proof The λ i are those constructed in Lemma 3. Set u 1 = 0. A path π from 1 to j is a sequence of observations π k , k = 1, . . . , π * , with first element 1 and final element j. For j = 1 let That is u j is the least-cost path from 1 to j, where the cost of moving from i to l directly is λ i a il . By Lemma 3, any loop has non-negative cost, so this is well-defined. These u j have the desired property as any path to from 1 to i can be extended to a path to j by moving directly from i to j.
Lemma 3 and Lemma 4 in combination prove the implication 2) ⇒ 3) in Afriat's theorem. The proof of the implication 3) ⇒ 1) is standard: Lemma 5 There exists a continuous, concave, increasing utility function u(x) which rationalizes the data.
As the infimum of a set of increasing, linear functions u is continuous, concave and increasing. Lemma 4 implies that for any j, u(x j ) = u j . On the other hand if p j · x ≤ p j · x j then u(x) ≤ u j + λ j p j · (x − x j ) ≤ u(x j ), so u rationalizes the data. Brown and Calsamiglia (2007) provide conditions under which observations can be rationalized by a quasi-linear utility function. That is there is a utility function u(x) such that for each i, x i maximizes u(x) − p i x.
Definition 3 The observations are said to satisfy the Quasi-Linear Generalized Axiom of Revealed Preference (QGARP) if for any cycle c, i∈c a c ic(i) ≥ 0.
In other words, a money pump in the sense of Echenique et al. (2011) is not possible. This implies immediately that Lemma 3 holds with λ i = 1, that is with constant marginal utilities.

Corollary 1
The following are equivalent: Proof The proof is is almost identical to that of Lemma 1 but simpler as Definition 3 immediately implies that Lemma 3 holds with λ i = 1 and hence so does Lemma 4. The only part that requires proof is the quasi-linearity of utility but note that since the utility function constructed in Lemma 5 is u(x) = min i u i + p i (x − x i ), for any i u(x) − p i x ≤ u(x i ) − p i x i as was to be shown.
This proof presented in this section is standard except that it uses Lemma 2 to prove Lemma 3. Its merit is that it uses a standard tool widely familiar to economists, the separating hyperplane theorem, and brings out connections with arbitrage arguments.

Extensions
This section shows that the same method as in the previous section can be used to prove Forges and Minelli (2009)'s generalization of Afriat's theorem. It applies this result to prove Varian (1983)'s result on rationalization by homothetic utility functions.
Forges and Minelli (2009) prove a generalization of Afriat's theorem. They consider a set of observations (x k , B k ) k=1,...,n , with the consumer able to choose from the budget set B k , where B k = {x ∈ R K + |g k (x) ≤ 0} with g k a continuous, increasing function such that g k (x k ) = 0. They say that the the utility function v rationalizes the observations if for all k v(x k ) ≥ v(x) for all x ∈ B k . Afriat's original theorem corresponds to the case g k (x) = p k · (x − x k ).
For any i and j let a i j = g i (x j ). The following is the natural generalization of GARP to this context: Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.