General revealed preference theory

Chambers, Christopher P.; Echenique, Federico; Shmaya, Eran Article General revealed preference theory Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Chambers, Christopher P.; Echenique, Federico; Shmaya, Eran (2017) : General revealed preference theory, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 12, Iss. 2, pp. 493-511, https://doi.org/10.3982/TE1924 This Version is available at: https://hdl.handle.net/10419/197195 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 12 (2017), 493–511 1555-7561/20170493 General revealed preference theory Christopher P. C hambers Department of Economics, University of California, San Diego Federico Echenique Division of the Humanities and Social Sciences, California Institute of Technology Eran Shmaya Kellogg School of Management, Northwestern University We generalize the standard revealed preference exercise in economics, and prove a sufficient condition under which the revealed preference formulation of an economic theory has universal implications and when these implications can be recursively enumerated. We apply our theorem to two theories of group behavior: the theory of group preference and the theory of Nash equilibrium. Keywords. Revealed-preference theory. JEL classification.D0. 1. Introduction Economic theories have observable and unobservable components, and one can say that an observable data set is consistent with the theory if there exists some specification of the unobservables that is consistent with the theory. The statement that observables are consistent if there exists unobservables that are consistent with the theory is the “as if” or “revealed preference” formulation of the theory. However, the revealed preference formulation of a theory may not be useful as an empirical test of the theory. For example, consider the theory of utility maximization. One can observe the choices made by an agent, and ask if there exists some utility function (unobservable) that is consistent with the choices and with the theory of utility maximization. To actually use this formulation as a test, one would have to check all possible utility functions and see if they can explain the data. This is a problem because there are infinitely many utility functions. After one has checked any finite set of utilities and verified that none of them can explain the data, one cannot conclude that the data are inconsistent with the Christopher P. Chambers: [email protected] Federico Echenique: [email protected] Eran Shmaya: [email protected] Chambers and Echenique acknowledge support from the NSF through Grant SES-0751980. We are grateful to seminar audiences at Arizona State University and Brown University. An anonymous referee provided extremely helpful feedback that greatly improved the paper. We are especially grateful to Kim Border, Faruk Gul, and Philippe Mongin for many detailed comments and suggestions. Copyright ©2017 The Authors. Theoretical Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE1924 494 Chambers, Echenique, and Shmaya Theoretical Economics 12 (2017) theory. Another example is profit maximization in a model of industrial organization: one may observe production and pricing decisions, and ask if there is some specification of firm technologies such that the observations are consistent with Nash equilibrium. Again, this revealed preference formulation does not give a practical (or effective) test of the theory because there are infinitely many possible technologies. The contribution of our paper is to give a sufficient condition on a theory, under which the revealed preference formulation of the theory enables a practical test. The existence of a test will translate into a “universal” formulation of the theory. “Practical” will translate into “effective.” We shall discuss these two terms. The first is universality. We said that observable data are consistent with the theory if there exists some specification of the unobservables that makes the data consistent with the theory. This formulation of the revealed preference question is existential because it starts with “there exists.” Because it is existential, no finite number of utility functions that fail to explain the data constitute evidence that the data are inconsistent with the theory. A testable formulation will instead start with “for all,” and therefore be a universal formulation. The following examples from Popper (1959) illustrate the basic ideas. Suppose that theory E claims, “There is a black swan,” while theory U says, “All swans are white.” Theory E, an existential theory, is not falsifiable because no matter how many finite data sets of non-black swans we find, it is still possible that there is a black swan somewhere. Theory U, a universal theory, is falsifiable because the observation of a single non-white swan contradicts the theory. The second term is effectiveness. For universal theories, effectiveness means that there has to be an algorithm that detects, after a finite number of steps, whether a data set is inconsistent with the theory. An effective test comes with an algorithm that one can run on a data set and that will stop after a finite number of steps when the data are incompatible with the theory. Later (see Example 12) we provide an example of a universal theory that is not effective in this sense. For such a theory, even if a data set is inconsistent with the theory, there may not be a way to demonstrate the inconsistency. Our paper provides a very general result on revealed preference theory. Our result says that whenever an economic model has a certain kind of axiomatization, its revealed preference formulation can be translated into a universal and effective test for the theory. Essentially our result says the following. Consider the theory, with its observable and unobservable components. An axiomatization of the theory can talk about observable and unobservable components. Whenever the theory has a universal and effective axiomatization, then there is a “projection” of the axiomatization onto the theory’s observable components. The projection gives a universal and effective test for the theory. Moreover, as we illustrate in this paper, this condition on a theory is widely applicable in economics. In fact, many papers in revealed preference theory set out to accomplish universal and effective tests for particular economic models. One of the best known examples is Brown and Matzkin (1996), who show that there exists a test for general equilibrium theory. There are many other papers with a similar agenda, but no general result that Theoretical Economics 12 (2017) General revealed preference theory 495 encompasses all of them. Our main theorem gives a general result that is applicable to all economic environments studied in the revealed preference literature. We proceed to illustrate our framework and our result by considering the example of utility maximization in more detail. The idea is that there is a theoretical object (a preference or a utility) that is not observable, but that places restrictions on observable data. The theory was originally developed for consumer choice (Samuelson 1938,Houthakker 1950): the restrictions placed on data by the theory are captured by the strong axiom of revealed preference (SARP). SARP is the universal and effective test we are talking about. In principle, one needs to check all utility functions before one decides that a data set is inconsistent with the theory of utility maximization, but the result of Samuelson and Houthakker means that one can check SARP instead of checking all possible utility functions. The purpose of our paper is to give a general result, in the same spirit as Samuelson and Houthakker’s, establishing the existence of universal and effective tests for the revealed preference formulation of an economic theory. We use a simple and abstract formulation of the problem of consumer choice (Richter 1966). Assume that one can observe the binary comparisons (weak and strict) between objects made by an agent. Let us refer to the observed comparisons as Rand P: Ris a binary relation, often called a revealed weak preference, and Pis a revealed strict preference. The theory of preference maximization says that the agent has a weak order (complete and transitive relation) governing these comparisons (what economists call a rational preference). Each weak order ≤is associated with its strict part, <.Thus,weposit that there exists a pair of binary relations ≤and <, which are theoretical (and hence not directly observable) for which the following axioms are satisfied: Axiom 1 (Completeness). ∀x∀y(x ≤y∨y≤x). Axiom 2 (Transitivity). ∀x∀y∀z(x ≤y∧y≤z→x≤z). Axiom 3(<strict part of ≤). ∀x∀y(x<y)↔(x ≤y)∧¬(y ≤x). Axiom 4(TheR-rationalization). ∀x∀y(xRy →x≤y). Axiom 5(TheP-rationalization). ∀x∀y(xP y →x<y). We wish to emphasize that the theory hypothesizes the existence of unobservable ≤ and <for which the axioms in the list are satisfied, and that all of Axioms 1–5are universal. The first point means that the theory does not directly provide a test of when observed revealed preference relations Rand Pare inconsistent with preference maximization. There are infinitely many weak orders ≤.Soforany≤that cannot explain Rand P, there may exist a different weak order that can explain it. In other words, this formulation of the theory does not provide a proof that an inconsistent pair Rand Pis indeed inconsistent with preference maximization. The second point to emphasize is that all of Axioms 1–5are universal: This is clear as they begin with the universal quantification ∀. Following Popper, then, if one could 496 Chambers, Echenique, and Shmaya Theoretical Economics 12 (2017) observe ≤and <,sothatallofR,P,≤,and<were observable entities, then the theory described by Axioms 1–5(the axioms of rationalization with weak order) would be universal and, therefore, falsifiable. Now, what does revealed preference theory say about Axioms 1–5?ItsaysthatRand Pare consistent with Axioms 1–5for some weak order ≤, if and only if Rand Pjointly satisfy the following countably infinite list of axioms: The Strong Axiom of Revealed Preference (SARP). We have that ∀x1∀xk¬ k  i=1 (xiSix(i+1)mod k) for every kand every S1Sk,whereS1=Pand Si∈{RP}for all i∈{2k}. Here we wish to again emphasize several points. First of all, the strong axiom (formally, a countably infinite collection of axioms) is also universal, but unlike Axioms 1–5, it does not refer to the unobservable objects ≤or <. It is a statement only about the observable Rand P. Second, there is an algorithm that decides whether an observable data set satisfies the strong axiom of revealed preference. So the strong axiom constitutes a universal and effective axiomatization of the theory of utility maximization.1 The purpose of our paper is to generalize these results beyond utility maximization. We prove that if a theory hypothesizes the existence of a collection of unobservable relations, but it does so in such a way that the theory would have a universal and effective axiomatization were these relations observable, then the theory has an equivalent universal and effective axiomatization purely in terms of observables. Put differently, if the theory has a universal and effective axiomatization when unobservable relations are assumed to be observable, then there is a universal and effective axiomatization that only refers to observables. One such equivalent universal axiomatization consists of all the logical consequences of the original theory that are universal and refer only to observables. As in the case of preference maximization and SARP, it is straightforward to establish that the universal consequences referring only to observables must be satisfied by the theory. The converse, that if all universal consequences are satisfied, then there exists unobservable relations such that the original universal axiomatization is satisfied, relies on the axiom of choice. This result is not trivial; it is possible to write down theories involving unobservables whose projection onto observables has no axiomatization whatsoever. For example, consider a theory claiming that every man in the universe can be matched to exactly one woman. We can describe the revealed preference formulation of the theory as involving statements about who is a man, who is a woman, and who is matched to whom. 1We could repeat the exercise assuming only that revealed weak preference Rwere observable, and seek rationalization by a linear order (complete, transitive, and antisymmetric). This would involve introducing only one new symbol, ≤, which would be required to satisfy antisymmetry. Further, the requirement of Prationalization would be dropped. The resulting version of the strong axiom would be similar, except that all instances of Siwould be P. Theoretical Economics 12 (2017) General revealed preference theory 497 Suppose now that the matching itself is not observable. Given that the universe of men and women is infinite, there is no axiom we could write down that would preclude the set of men and women from each being infinite, but of different cardinalities. And this could never be observed with finite data. The reason our result would fail in this case is because if the matching function were observable, then the theory would hypothesize existential statements; that for each man, there exists auniquewomantowhomheis matched. We discuss this in more detail in Remark 5. After laying out the formal structure of the model, and presenting the main result (Theorem 1), we demonstrate applications for two economic approaches to collective decision making. We first discuss an application to preference aggregation in which group preferences are assumed to be some function of individual preferences. Then we turn to a framework in which group choice is modeled as the outcome of strategic interaction between the agents, and discuss the testable implications of Nash equilibrium. Section 6 describes the meaning of our results for testing and falsifying data sets. 2. Main results 2.1 Preliminary definitions Our results are about axiomatizations of possible data. We use model theory to study these ideas. The framework used here is developed in more detail in Chambers et al. (2014), where it is used for a different purpose. At the end of this paper, we discuss the relation to Chambers et al. (2014) in more detail. The following presentation of results is terse, but (we hope) fairly self-contained. In Section 6, we take stock and interpret our findings. We collect the symbols that we need into a language. The language is a primitive, and specifies the properties and relations (both observable and unobservable) that one can make statements about. A relational first-order language Lis given by a set Rof relation symbols and a positive integer nR,thearity of R, for every R∈R.Forexample,if we wish to talk about preference, we may use a language with a single binary symbol R. We can then write axioms and make sense of when a set Xand a specific binary relation on Xsatisfy these axioms. The example in the Introduction used the language R≤, in which the arity of Rand ≤is 2(both are binary relation symbols). The statements Axioms 1–5in the Introduction are axioms in language R≤. A structure is a universe of possible objects (called a domain) and an interpretation of the relation symbols of the language in that universe. An L-structure Mis given by a nonempty set Mcalled the domain of M,andforeveryn-ary relation symbol R∈R, an n-ary relation,2the interpretation RMover Mof R. When the language Lis understood, we refer to an L-structure simply as a structure. Structures provide the appropriate framework for interpreting the symbols in the language. For example, when L= has a single binary relation, then one possible structure is (R≥); the structure of the real numbers with the usual greater-than binary relation. Another example is (2X⊇), the power set of Xwith set containment. 2An n-ary relation RMover Mis a subset RM⊆Mn. 498 Chambers, Echenique, and Shmaya Theoretical Economics 12 (2017) Suppose that Mand Nare L-structures with universes Mand N, respectively. Then Mand Nare isomorphic if there exists a bijective map η:M→Nthat preserves the interpretations of all relation symbols, i.e., such that (m1mn)∈RM↔η(m1)  η(mn)∈RN for every n-ary relation symbol Rof Land m1mn∈M. Given a language L, we can write sentences using the relation symbols in L.Inaddition to the relation symbols specified by L, we shall use certain logical symbols. These symbols are fixed, and we are allowed to use them regardless of the language under consideration. The logical symbols are the quantifiers “exists” (∃)and“forall”(∀), “not” (¬), the logical connectives “and” (∧)and“or”(∨), a countable set of variable symbols xyzuvw, parentheses (and ), and the equality symbol =. Certain strings of symbols can be put together to form sentences, or axioms. Rules for forming sentences are given in, for example, Marker (2002). Such rules are intuitive and immediately recognizable: The string ∀x∃yxRyis a legitimate sentence; the string ∀y∃Rx is not. We refer to rules of forming legitimate sentences as rules of syntax. In a given structure, sentences can be either true or false. Again we skip the formal definition of what it means for a sentence to be true in a structure since it is intuitively clear: The sentence ∀x∃yxRyis true in the structure (R≥): For every x∈Rthere exists y∈Rsuch that x≥y. The same sentence is false in the structure with domain X= {123}when the relation symbol Ris interpreted as >: it is not true that for every x∈X there exists y∈Xsuch that x>y. 2.2 Main result The objects we study in this paper are classes of structures (over some language) that are closed under isomorphism. Such a class of structures captures our idea of a theory. For example, the theory of preference maximization encompasses all structures M=(M RM)for which the observed RMcan be extended to a linear order over all elements of M. We caution that the term “theory” is somewhat misleading because it means something else in model theory. For this reason, we do not use the term “theory” in our formal definitions. WesaythataclassofstructuresTover some language is (formally) axiomatized by a collection of sentences if Tconsists exactly of the structures for which each sentence in is valid. Given two classes of structures, Tand T,whereT⊆T, we say that Tis (formally) axiomatized by a collection of sentences with respect to Tif Tconsists of exactly those structures in Tfor which each sentence in is valid. Auniversal sentence is a sentence that includes only universal quantifiers.3The axioms of reflexive, complete, transitive, and antisymmetric relations in the Introduction are all universal sentences in the language with two binary relation symbols ≤R. A universal axiomatization is an axiomatization that consists entirely of universal sentences. 3These quantifiers can only come at the very beginning of the sentence. Theoretical Economics 12 (2017) General revealed preference theory 499 Finally, a set of sentences is called recursively enumerable (r.e.) if there exists a Turing machine that enumerates over the elements of . If a recursively enumerable axiomatizes a class of structures T, we say that Tadmits an effective axiomatization. A Turing machine is a formalization of the intuitive idea of algorithm or effective procedure, without any requirements about computational resources. So when we say that there is a Turing machine that enumerates over the elements of , we mean that there is a procedure that outputs an exhaustive list ϕ1ϕ2 of all the elements of . This also means that if ϕ∈, then there is a way to demonstrate this membership. In Section 6, we argue that existence of effective axiomatization captures our idea of falsifiability of a theory. For example, the collection of axioms in SARP is a r.e. set of sentences. See Sipser (2012) for formal definitions. Let F=R1RNand L=R1RNQ1QKbe languages, where all the Rnand Qkare relation symbols. Note that F⊆L. The languages Fand Lcapture the difference between observable and unobservable objects. The relations Rnare assumed to be observable in the data, while the relations Qkare unobservable. In our applications below, we choose Fand Lwith this interpretation in mind. Let TbeaclassofL-structures, closed under isomorphism. Define F(T) to be the class of F-structures (X∗R∗ 1R∗ N)for which there exist relations Q∗ 1Q∗ Ksuch that (X∗R∗ 1R∗ NQ∗ 1Q∗ K)∈T.Thatis,F(T) is the projection of Tonto the language F.4 We are now in a position to state our theorem. Theorem 1. Let Tbe a class of structures that is closed under isomorphism. If Tadmits a formal universal axiomatization, then F(T)admits a formal universal axiomatization. Moreover, if Tadmits a universal effective axiomatization, then F(T) admits a universal effective axiomatization. As an example of Theorem 1, recall the revealed preference example in the Introduction. In that example, F=RP, which consists of the observed relation and L=RP≤<, and also includes the unobserved preference relation of the agent as well as its strict part. The class of structure T, which represents the theory of preference maximization, is axiomatized by Axioms 1–5in the Introduction.TheclassF(T) is axiomatized by the strong axiom of revealed preference, which is in fact a r.e. sequence of axioms. The following corollary, which follows immediately from Theorem 1,extends the theorem to the case of axiomatization of a class of structures with respect to a larger class. Corollary 2. Let Tbe a class of L-structures and let Tbe a class of F-structures that are closed under isomorphism. If Tadmits a formal universal axiomatization, then F(T)∩T 4If Thas a finite first-order axiomatization, then F(T)is an example of an existential second-order theory for language Fin that it allows existential quantification over relations. That is, if σis a first-order L-axiom axiomatizing a class of structures T,thenF(T)is axiomatized by ∃Q1∃QKσ 500 Chambers, Echenique, and Shmaya Theoretical Economics 12 (2017) admits a formal universal axiomatization with respect to T.Moreover,ifTadmits a universal effective axiomatization, then F(T)∩Tadmits a universal effective axiomatization with respect to T. 3. Proof of Theorem 1 3.1 Preliminaries We recall some terminology from model theory used in the proof. An atomic formula over a language Lis a string of the form P(x1xn),wherePis an n-ary relation symbol in Land x1xnare variable symbols. As usual, when n=2, we sometimes write xPyfor P(xy).Aquantifier-free formula is a string of symbols that is composed of atomic formulas and the connective symbols ¬∨∧→under the rule of syntax. For example the string ¬(x y) →(y z) is a quantifier-free formula with variables xyz in the language with a binary predicate . Every universal sentence can be written in the form ∀x1∀xnϕ(x1xn),whereϕ(x1xn)is a quantifier-free formula with variables x1xn. If ϕ(x1xn)is a quantifier-free formula in a language L,andMis an L-structure with domain M, then for every m1mn∈M, there is a well defined sense in which the expression ϕ(m1mn), obtained by substituting the elements mifor the variables xi,is true in M. Again we provide an example instead of formal definition: if ϕ(xyz) =(x y)∧(y z),thenϕ(431)is true in the structure (R≥)(since 4≥3and 3≥1)butϕ(175)is false. In particular, for atomic formulas, P(m1mn)is true in Mif and only if (m1mn)∈PM. With this notation, a universal sentence ∀x1∀xn ϕ(x1xn),whereϕ(x1xn)is a quantifier-free formula with variables x1xn, is true in Mif and only if ϕ(m1mn)is true in Mfor every m1mn∈M. Let Fbe a relational language and let Mbe an L-structure with domain M.Asubstructure Mof Mis a structure Mwhose domain satisfies M⊆Mand such that RM=(m1mn)∈Mn|(m1mn)∈RM for every n-ary relation symbol R. We use the following theorem of Tarski (1954). Theorem 3. Let Fbe a relational language and let Tbe a class of structures of F.Then Tadmits a universal axiomatization if and only if the following conditions are satisfied. (i) The class Tis closed under isomorphism. (ii) The class Tis closed under substructure. (iii) For every F-structure M,ifM∈Tfor all finite substructures Mof M,then M∈T. The proof uses basic ideas from sentential logic.5A sentential logic is given by a set Sof sentence symbols.A(well founded) formula of Sis a string built from sentence 5For more on this, see, e.g., Chapter 1 of Enderton (2001). Theoretical Economics 12 (2017) General revealed preference theory 507 Consider the class of structures Taxiomatized by the following sentences: For each γ∈and k∈γ, ∀x1∀xn∀y1∀yn∀z1∀zn  i∈γ ∈(ziyi)∧ i∈N ∈(xiyi)∧¬ i∈γ (xi=zi)∧R(y1ynx1xn)∧≥ k(zγx−γ)x → i∈γ\{k} ≥ix(zγx−γ) and the universal axioms that express that ≥kis a linear order (complete, transitive, reflexive, and antisymmetric). As Thas a universal axiomatization, so does F(T). Since the axiomatization of Tis finite, F(T) has a recursively enumerable universal axiom by Theorem 1,andT=F(T)∩TG.SobyCorollary 2,Thas a r.e. universal axiomatization within TG. 6. Discussion The notions of universal and r.e. universal axiomatization capture the idea of falsifiability of a theory in the following way: Suppose that a scientist postulates a theory T by providing a universal axiomatization for T. Suppose that we observe the elements a1anof some structure M, and the relationships between them. We call these observations a data set. If there exists some universal axiom ∀x1∀xnϕ(x1xn)∈ such that ϕ(a1an)is not true, then Thas been falsified. Thus, a violation of an axiom of the theory can be demonstrated by presenting a data set. The fact that the theory is given by a universal axiomatization means that any violation of the theory can be demonstrated. In the terminology introduced in Chambers et al. (2014),suchatheory is identical to its empirical content.12 If, in addition, is recursively enumerable, then the scientist can describe by providing the algorithm (or Turing machine) that generates . In this case, if a data set falsifies the theory, then this falsification can be demonstrated by pointing to the index of the axiom that is violated in the recursive enumeration of the axioms. The following example illustrates this issue. Example 12. Consider a language with a single binary relation symbol L,wherexLy is supposed to represent the relationship xloves y.Alove cycle of size kis a sequence of people x1xksuch that, for every 1≤i j ≤k, it holds that xiloves xjif and only if j=(i +1)mod k.LetC⊆N. Suppose that a scientist postulates the theory that there arenolovecyclesofsizekfor any k∈C. Such a theory has a universal axiomatization. If Cis recursively enumerable, then the scientist can describe the theory by providing a computer program that enumerates over C. If the theory is incorrect, i.e., if there exists alovecycleofsizekfor some k∈C, then an antagonist can demonstrate this violation and falsify the theory by pointing to a data set that violates the theory (i.e., to a love 12Under a caveat that in the current paper we assume that absence of relationship can be observed. See Section 4.3 in Chambers et al. (2014). 508 Chambers, Echenique, and Shmaya Theoretical Economics 12 (2017) cycle of size kfor some k∈C) and, by pointing to the index of kin the enumeration of Cto show that k∈C.If,however,Cis not recursively enumerable,13 then, while the scientist’s theory has a universal axiomatization, he has no way to formally describe the theory. An antagonist who has an access to a love cycle of size kfor some k∈Cmay not be able to demonstrate that k∈C.♦ Example 12 shows that a theory may have a universal axiomatization, but the ability of a theory to be falsified in practice depends on the existence of an effective or recursively enumerable axiomatization. However, if a data set is consistent with the theory, then this consistency cannot necessarily be effectively demonstrated, since such a demonstration requires checking that the countably infinite list of axioms is satisfied. Consider Example 12 again. If Cis recursively enumerable but not recursive, then the fact that a given data set is consistent with the theory cannot necessarily be demonstrated since if Cis not recursive, the researcher has no way to effectively demonstrate for a given love cycle of size kthat k/∈C. Note that, by Remark 6, the theory also admits a recursive axiomatization, but this additional feature does not seem to be related to falsifiability. In particular, as the example shows, it does not mean that the researcher can demonstrate that a given data set is consistent with the theory. Thus, recursively enumerable universal theories have the property that any violation of the theory can be demonstrated. In logical terminology, if a sentence ϕis an element of , then there exists a formal deduction (a proof) for this fact. Gradwohl and Shmaya (2015) go one step further and require in addition that this proof be short. Consider now the case in which a theory Tadmits a finite universal axiomatization. This is the case of the consumer choice example discussed in the Introduction,aswell as the applications to group choice and Nash equilibrium developed above, and to all the other natural economic applications of which we are aware. In this case, there is a way to check whether a given data set is consistent with the theory (and, a fortiori, to demonstrate consistency of the data set with the theory): go over all the axioms and check that they are all satisfied. Moreover, in the framework of Section 2.2,ifTis an Ltheory that admits a finite and universal axiomatization, then there is a way to check whether a given finite data set is consistent with F(T):onehasto go over all possibilities for the unobserved relations Q1QKand check whether it is possible to define them in a way that is consistent with all of the axioms. In logical terminology, if Thas a finite universal axiomatization, then the set of semantic implications of F(T) is recursive. Summing up, for the case in which Thas a finite universal axiomatization (which it does in all our examples), there is a way to check whether a finite data set is consistent with the theory F(T),andTheorem 1 implies that if the theory is incorrect, then one can point to a finite data set that is not consistent with it. 7. Relation to previous literature One relevant antecedent to our paper is Brown and Matzkin (1996). These authors exploit a famous model-theoretic result, the Tarski–Seidenberg theorem (Tarski 1951), to 13For example, if Cis the set of codes of computer programs that do not halt. Theoretical Economics 12 (2017) General revealed preference theory 509 study the empirical restrictions placed by the competitive equilibrium hypothesis on observable data. Roughly, they use Afriat’s theorem to show that testing the consistency of data with the theory of Walrasian equilibrium boils down to verifying whether there exists a solution to a finite collection of polynomial inequalities. The Tarski–Seidenberg theorem establishes that the existence of a solution to this set of inequalities is equivalent to the satisfaction of another set of polynomial inequalities (which can be algorithmically determined) in which no theoretical variables appear. The second system of inequalities only depends on data. What we accomplish is similar in spirit to what Brown and Matzkin do in their paper, but our techniques are different (our result follows from Tarski’s theorem on universal axiomatization, not on the theory of quantifier elimination in systems of real polynomials). Many papers in revealed preference theory are interested in the specific form of the axiomatization of F(T), but there are also many studies that are primarily interested in the existence of a test. We have already mentioned Brown and Matzkin (1996),butthere are many other papers based on developing so-called Afriat inequalities. Two recent developments are Quah (2012) and Polisson and Quah (2013), which seek to show that certain economic theories are falsifiable by establishing what in our papers would be universal axiomatizations of F(T). Our paper is also related to a model-theoretic literature that studies when a theory can be given an axiomatization using additional relations. See, for example, Craig and Vaught (1958), who provide conditions under which a class of structures is of the form F(T) for some finitely axiomatizable T. The work also contains a finite model-theoretic result related to Theorem 1. The type of issues we discuss here have previously been studied by philosophers of science. Without going into full detail, Ramsey (1931) was one of the first to discuss the elimination of “theoretical” terms from scientific theories. Various authors give different interpretation to the notion of “Ramsey elimination.” Herbert Simon wrote a sequence of papers on falsifiability and empirical content. For example, Simon (1985) discusses some of the issues we discuss here: Simon argues that the theory of rational choice is falsifiable, even though its usual formulation existentially quantifies over unobservables (what he calls theoretical). As Simon (1985) states, “although existential quantification of an observable is fatal to the falsifiability of a theory, the same is not true when the existentially quantified term is a theoretical one.” While this may seem obvious, it has led to a large degree of confusion among economists. 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Manuscript received 2 August, 2014; final version accepted 7 June, 2016; available online 21 June, 2016.