A Simplified Test for Preference Rationality of Two-Commodity Choice

A Simplified Test for Preference Rationality of Two-Commodity Choice Samiran Banerjee and James H. Murphy December 9, 2004 Abstract We provide a simplified test to determine if choice data from a two-commodity consumption set satisfies the Generalized Axiom of Revealed Preference (GARP), and thus the preference or utility maximization hypothesis. We construct an algorithm for this test and illustrate its application on experimental choice data. JEL Categories: C91, D11, D12 Keywords: Revealed preference, GARP, laboratory experiments, rationality Banerjee (corresponding author): Department of Economics, Emory University, Atlanta, GA 30322-2240. Phone: 404-712-8168. Fax: 404-727-4639. Email: sbaner3@emory.edu. Murphy: College of Business, Western Carolina University, Cullowhee, NC 28723. Phone: 828-227-3731. Fax: 828-227-7414. E-mail: jmurphy@email.wcu.edu. We are grateful to the editor and especially an anonymous referee for detailed comments that have improved the paper. Additionally, we thank James Andreoni and Jim Miller for access to their data.

1 Introduction Much theoretical and applied work in microeconomics presumes that a consumer s observed choices result from the maximization of some utility function or preference relation subject to a binding budget. The centrality of this presumption has motivated attempts to construct and implement falsifiable tests of this utility hypothesis. However, applying the appropriate theoretical tests for finite data sets such as the Strong Axiom of Revealed Preference (SARP) or the Generalized Axiom of Revealed Preference (GARP) to field data presents some difficulties. Varian (1988) showed that axiomatic tests of the consistency of field data requires the analyst to observe all (and not just a subset of) the prices and quantities associated with the commodity space or impose strong behavioral assumptions such as separability. One way around such problems, especially in the context of controlled experimental methods, is to restrict the dimension of the commodity space. Several studies which have utilized various axioms of revealed preference to test whether experimental choice data satisfies the preference or utility maximization hypothesis do precisely this. In an early study, Battalio et al. (1973) analyzed the choices of institutionalized subjects in a token economy, work later refined and extended by Cox (1997) to incorporate labor supply and portfolio choice. Sippel (1997) elicited university students choices from a commodity space of eight common consumption goods. Harbaugh, Krause and Berry s (2001) grade school and undergraduate subjects selected from a two-commodity consumption space (boxes of juice and bags of chips); Andreoni and Miller s (2002) subjects chose from bundles consisting of payments to self and to an anonymous partner. While the SARP is the appropriate test to determine if a finite set of observed demand choices is consistent with the maximization of a strictly concave utility function (Matzkin and Richter, 1991), the GARP allows for demand behavior that is somewhat more general and is consistent with the maximization of a piecewise-linear concave utility function (Varian, 1982). Checking whether a set of m demand observations satisfy the SARP or GARP requires the identification of both direct and indirect chains of revealed preference between any k of the chosen bundles for k = 2,..., m. But when choice is from a two-commodity space, Rose (1958) proved that satisfying 2

the Weak Axiom of Revealed Preference (WARP), which requires only the pairwise comparison of demanded bundles, is equivalent to satisfying the SARP. In this paper, we provide an analogous shortcut to check for GARP-consistency in terms of a Weak Generalized Axiom of Revealed Preference (WGARP) when the commodity space consists of two goods. Thus a violation of the GARP can be demonstrated in the simple accessible form of a pairwise violation. Section 2 contains definitions and our theoretical result, that choice from a two-commodity consumption set that is WGARP-consistent is also GARP-consistent. We present an algorithm for checking the WGARP in Section 3 and apply this algorithm to the data of Andreoni and Miller (2002). Section 4 concludes. 2 Theoretical Result Let IR + n be a consumption space, where n 2 indexes the number of commodities. A subject facing a price vector p i IR ++ n chooses a bundle x i IR +; n the pair (p i, x i ) constitutes an observation. Let S be a finite set of m observations on a subject, i.e., S = {(p i, x i )} m i=1. Definition 1. The set S satisfies the Weak Generalized Axiom of Revealed Preference (WGARP) if for any two choices x i and x j, [p i x i p i x j ] [p j x j p j x i ]. Following Varian (1982), if x i is chosen when x j is also affordable, we write x i R 0 x j, where R 0 is the binary relation directly revealed preferred to. If x i is chosen when x j is less costly (i.e., p i x i > p i x j ), we write x i P 0 x j, where P 0 is the binary relation strictly directly revealed preferred to. Then an alternative statement of the WGARP is If x i R 0 x j, then it is not the case that x j P 0 x i. Definition 2. The set S satisfies the Generalized Axiom of Revealed Preference (GARP) if for any observation (p i, x i ) and any sequence {(p 1, x 1 ), (p 2, x 2 ),... (p j, x j )} of length j m drawn from S, it is the case that [p i x i p i x 1 & p 1 x 1 p 1 x 2... & p j 1 x j 1 p j 1 x j ] [p j x j p j x i ]. 3

Denoting the transitive closure of the binary relation R 0 by R, 1 an alternative statement of the GARP is If x i Rx j, then it is not the case that x j P 0 x i. Proposition. When the consumption space contains two commodities (i.e., n = 2), the WGARP is equivalent to the GARP. Proof. Sufficiency follows from the definition of GARP. We show necessity by induction on the length of the sequence following Rose (1958). The theorem is trivially true for any sequence of length one. Suppose it is true for any sequence of length j 2. Suppose an observation (p i, x i ) and a sequence {(p 1, x 1 ), (p 2, x 2 ),..., (p j+1, x j+1 )} is such that each pair of observations (p i, x i ) and (p 1, x 1 ), (p 1, x 1 ) and (p 2, x 2 ),..., (p j, x j ) and (p j+1, x j+1 ) satisfies the WGARP, and it is the case that [p i x i p i x 1 & p 1 x 1 p 1 x 2... & p j x j p j x j+1 ]. We need to show that [p j+1 x j+1 p j+1 x i ]. For each price vector, normalize the price of the second commodity to unity. Now if p j+1 is the smallest or the largest in the sequence, then either (a) p i lies between p j and p j+1, or (b) p j lies between p i and p j+1. If p j+1 is neither the smallest nor the largest in the sequence, then (c) it lies between two consecutive prices p k and p k+1, k < j. Thus there are three cases to consider: (a) p i = λp j + (1 λ)p j+1 for some 0 λ 1, or (b) p j = λp i + (1 λ)p j+1 for some 0 λ 1, or (c) p j+1 = λp k + (1 λ)p k+1 for some 0 λ 1 and some k < j. In all cases we need to show that p j+1 (x j+1 x i ) 0. To show this in case (a), note that p i (x i x j+1 ) = p i (x i x 1 ) + p i (x 1 x j+1 ) = p i (x i x 1 ) + [λp j + (1 λ)p j+1 ](x 1 x j+1 ) = p i (x i x 1 ) + λp j (x 1 x j+1 ) + (1 λ)p j+1 (x 1 x j+1 ) = p i (x i x 1 ) + λp j (x 1 x j ) + λp j (x j x j+1 ) + (1 λ)p j+1 (x 1 x j+1 ). 1 For any preference relation R 0 on a commodity space X, let R 0 denote the set of all possible transitive extensions of R 0, i.e., for all Ri 0 R 0, (1) Ri 0 X X, (2) Ri 0 is transitive and (3) R 0 Ri 0. The transitive closure of R0 is then the smallest transitive extension of R 0 defined as R = R 0 i R 0R0 i. 4

Then the first and third terms on the righthand side are non-negative by assumption and the second and fourth terms are non-negative because of WGARP. Thus p i (x i x j+1 ) 0 and hence p j+1 (x j+1 x i ) 0 by WGARP. For case (b), note that p j (x i x j+1 ) = p j (x i x j ) + p j (x j x j+1 ) is non-negative because the first term on the righthand side is non-negative by WGARP and the second by assumption. Therefore, substituting for p j, we get p j (x i x j+1 ) = λp i (x i x j+1 ) + (1 λ)p j+1 (x i x j+1 ) 0. If p i (x i x j+1 ) 0, it follows from the inequality above that p j+1 (x i x j+1 ) 0 and we are done. On the other hand, if p i (x i x j+1 ) 0, then by WGARP p j+1 (x j+1 x i ) 0. Finally for case (c), note that p j+1 (x i x j+1 ) = p j+1 (x i x k+1 ) + p j+1 (x k+1 x j+1 ) = λp k (x i x k+1 ) + (1 λ)p k+1 (x i x k+1 ) + p j+1 (x k+1 x j+1 ) = λp k (x i x k ) + λp k (x k x k+1 ) + (1 λ)p k+1 (x i x k+1 ) + p j+1 (x k+1 x j+1 ). Then the first, third and fourth terms on the righthand side are non-negative by WGARP, while the second term is non-negative by assumption. Thus p j+1 (x i x j+1 ) 0, i.e., p j+1 (x j+1 x i ) 0 as desired. Why is it that in the two-commodity case there must be a pairwise violation of GARP if there is any violation of GARP at all? At the heart of this result lies a crucial difference between a choice scenario with two versus more than two commodities: in the case of two goods and a k-wise violation of GARP, the k budget lines can be ranked from steepest to flattest by normalizing the price of the second good, something not possible when there are three or more commodities. Then the inequalities in the k-wise GARP violation can be shown to impose restrictions implying at least one pairwise GARP violation. To illustrate this, consider the simplest instance when there are three observations S = {(p 1, x 1 ), (p 2, x 2 ), (p 3, x 3 )} and where all prices and chosen bundles are distinct. Suppose that 3-wise GARP is violated: x 1 R 0 x 2 and x 2 R 0 x 3 hold, while x 3 P 0 x 1. The inequalities expressed by these revealed preferred relations are [p 1 x 1 p 1 x 2 ] & [p 2 x 2 p 2 x 3 ] & [p 3 x 3 > p 3 x 1 ]. 5

Since we wish to demonstrate that this implies at least one WGARP violation, assume that WGARP is not violated for any pair of observations from S, i.e., for any pair of observations i and j from S (i, j = 1, 2, 3, i j), [p i x i p i x j ] & [p j x j p j x i ]. Ranking the budget lines from steepest to flattest, there are six possibilities: (i) p 1 > p 2 > p 3, (ii) p 1 > p 3 > p 2, (iii) p 2 > p 1 > p 3, (iv) p 2 > p 3 > p 1, (v) p 3 > p 1 > p 2, and (vi) p 3 > p 2 > p 1. We analyze the first of these possibilities below. 2 In Figure 1, OAD is the budget set from which x 1 is chosen, BE the budget line p 2 x 1 and CF the budget line p 3 x 1. From x 3 P 0 x 1 (i.e., p 3 x 3 > p 3 x 1 ), x 3 lies strictly to the right of CF. However, x 3 cannot lie in Ax 1 C because that would result in a WGARP violation between x 1 and x 3. Thus x 3 must lie strictly to the right of Ax 1 F. Since x 1 R 0 x 2 (i.e., p 1 x 1 p 1 x 2 ), x 2 must lie in OAD; however, x 2 cannot lie in Ax 1 B because that would result in a WGARP violation between x 1 and x 2. Thus x 2 must lie in OBx 1 D (excluding the segment Bx 1 ). For any x 2 in OBx 1 D (excluding segment Bx 1 ) and x 3 in the area to the right of Ax 1 F, one can verify that x 2 R 0 x 3 (i.e., p 2 x 2 p 2 x 3 ) is not possible, a contradiction. 3 Algorithm and Application Given a data set of m price vectors and m corresponding consumption bundles, we provide an algorithm to calculate the number of WGARP violations in a given set of observations. Let U be an m m matrix where a typical element u ij is defined as u ij = 2 if p i x i > p i x j 1 if p i x i = p i x j 0 otherwise. The elements of U then represent the R 0 and P 0 relations: u ij = 1 iff x i R 0 x j but not x i P 0 x j, and u ij = 2 iff x i P 0 x j. 2 A complete analysis of all six cases is available from the authors upon request. 6

Next define another m m matrix V whose typical element v ij is defined as u ij u ji if i < j v ij = 0 otherwise. Finally define W as an m m matrix with element w ij defined as 1 if v ij 2 w ij = 0 otherwise. Then the data contain m i=1 mj=1 w ij WGARP violations. 3 Unlike the GARP algorithm (Varian, 1982 p. 949 and Appendix 2), the WGARP algorithm does not require computation of the transitive closure of the binary relation R 0. We apply this WGARP algorithm to Andreoni and Miller s (2002) data from Sessions 1-4, their base-case scenario. 4 In the first two columns of Table 1, we list (by subject number and reported number of GARP violations) the 13 subjects out of 142 who violated the GARP. Each subject faced eight budgets and selected bundles consisting of payments to self and to an anonymous partner, a two-good commodity space. In the third column of Table 1 we provide the corresponding number of WGARP violations calculated using our algorithm. Since Theorem 1 establishes that WGARP implies GARP, and because GARP-consistent choice can be rationalized by a piecewise linear utility function which is continuous, concave and monotonic, we conclude that the WGARP-consistent choice of the remaining 129 subjects is similarly rationalized. In columns four and five of Table 1 we reproduce the number of WARP and SARP violations. Matzkin and Richter (1991, Theorem 2, p. 291) show that SARPconsistent choice can be rationalized by a continuous, strictly concave and strictly monotonic utility function. Rose s (1958) result implies that two-commodity WARPconsistent choice is SARP-consistent; consequently, two-commodity WARP-consistent choice can be rationalized by a continuous, strictly concave and strictly monotonic utility function. As pointed out by Matzkin and Richter (1991, p. 299), the GARP implies a weaker notion of rationality than SARP; in particular, GARP-consistent 3 For spreadsheet applications, the algorithm can be shortened by using conditional statements to replace the V and W matrices. A spreadsheet template is available upon request from the authors. 4 The data can be downloaded from http://www.ssc.wisc.edu/ andreoni/getdata.html. 7

choice can be rationalized by any constant utility function (which is trivially continuous, concave and monotonic), and by our Theorem 1, the same can be said of WGARP. However, because the number of WGARP and WARP violations is identical for the Andreoni and Miller data set, 5 the full force of SARP-consistency may be used to conclude that the 129 subjects behaved as if they were maximizing a continuous, strictly concave and strictly monotonic utility function. Table 1: Violations of Revealed Preference Axioms for Andreoni and Miller Data Number of Violations Subject Number GARP WGARP SARP WARP 3 2 1 3 1 38 7 2 7 2 40 7 3 8 3 41 1 1 1 1 47 1 1 1 1 61 3 1 4 1 72 1 1 1 1 87 1 1 1 1 90 1 1 1 1 104 1 1 2 1 126 1 1 3 1 137 1 1 1 1 139 1 1 1 1 As can be seen from Table 1, there are generally more violations of the GARP than the WGARP since the former counts both pairwise and k-wise violations (k > 2), while the latter only counts pairwise violations. To see a concrete instance of this, consider the choices of subject 3 reproduced in Figure 2. The eight budgets faced by the subject are labeled from B 1 to B 8, while the choices made under each budget are 5 In general, any WARP violation implies an automatic violation of WGARP, but not necessarily the other way around. 8

shown by the bundles numbered from 1 to 8. These observed choices are summarized in terms of the revealed preference relations R 0 and P 0 in Table 2. Table 2: Choices of Subject 3 in Andreoni and Miller Data Other Bundles Available In Budget Chosen Bundle 1 2 3 4 5 6 7 8 1 P 0 P 0 P 0 2 3 R 0 P 0 R 0 R 0 P 0 P 0 4 P 0 R 0 5 P 0 P 0 P 0 P 0 P 0 P 0 P 0 6 P 0 P 0 P 0 7 P 0 8 P 0 R 0 P 0 P 0 P 0 Table 2 shows that bundle 1 was chosen over bundles 2, 7 and 8 under budget B 1, all of which were less costly; under budget B 2, none of the other bundles chosen under the other budgets were available when bundle 2 was chosen; bundle 3 was chosen when bundles 1, 4 and 6 cost the same, while bundles 2, 7 and 8 were less costly; etc. From the fact that 8R 0 3 and 3P 0 8, we have a WGARP violation which automatically constitutes a GARP violation. From the direct choices 8R 0 3 and 3R 0 1, we obtain the indirect relationship 8R1 between bundles 8 and 1; since 1P 0 8, we have another GARP violation. Note that for this to be a WGARP violation, we would need either 8R 0 1 or 8P 0 1 neither of which is possible since bundle 1 was not available under budget B 8. 4 Conclusion The growing use of experimental methods in economics and renewed interest in the behavioral foundations of choice has led to a number of recent studies testing experimental choice data for consistency with the axioms of revealed preference. The 9

GARP has played a pivotal role in several studies. In this paper, we defined an alternative axiom, the WGARP, which requires only the binary comparison between chosen bundles. Paralleling Rose s (1958) findings concerning the equivalence between the WARP and SARP, we have shown that choice from a two-commodity consumption set that satisfies the WGARP also satisfies the GARP (and vice-versa). Because checking for GARP-consistency requires an iterative procedure such as Warshall s (1962) algorithm to compute the transitive closure of a revealed preference relation, while WGARP involves only pairwise comparisons of chosen bundles, our finding offers a conceptually simpler test of whether choices over a two-commodity space satisfy the hypothesis of preference maximization (as satisfying the GARP implies). Consequently, the WGARP is particularly valuable in contexts where transparency in identifying violations of preference rationality is useful, such as classroom demonstrations of the axioms of revealed preference. Additionally, we have provided some intuition as to why the case of two goods is special in that any violation of GARP must entail one pairwise violation of GARP. We have also provided an algorithm to test for WGARP-consistency. On a cautionary note, ours is an equivalence result for WGARP- and GARP-consistency. While data which contain at least one GARP violation (and hence fails the GARP) must necessarily fail the WGARP by exhibiting at least one WGARP violation, the total number of GARP violations need not equal the number of WGARP violations. This is because each GARP violation is based on direct (i.e., pairwise) and indirect chains of revealed preference between chosen bundle. Each WGARP violation, on the other hand, depends only on the direct pairwise comparison between chosen bundles; hence the number of GARP violations will exceed or equal the number of WGARP violations. Our algorithm is not appropriate if the number of violations rather than their mere presence or absence is of interest. References [1] Andreoni, J. and J. Miller. (2002). Giving According to GARP: An Experimental Test of the Consistency of Preferences for Altruism. Econometrica. 70, 737-753. 10

[2] Battalio, R. C. and J. H. Kagel, R. C. Winkler, E. B. Fisher, Jr., R. L. Basmann and L. Krasner. (1973). A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases. Western Economic Journal. 11, 411-428. [3] Cox, J. C. (1997). On Testing the Utility Hypothesis. The Economic Journal. 107, 1054-1078. [4] Harbaugh, W. T., K. Krause and T. R. Berry. (2001). GARP for Kids: On the Development of Rational Choice Behavior. American Economic Review. 91, 1539-1545. [5] Matzkin, R. and M. Richter. (1991). Testing Strictly Concave Rationality. Journal of Economic Theory. 53, 287-303. [6] Rose, H. (1958). Consistency of Preference: The Two-Commodity Case. Review of Economic Studies. 25, 124-125. [7] Sippel, R. (1997). An Experiment on the Pure Theory of Consumer s Behaviour. The Economic Journal. 107, 1431-1444. [8] Varian, H. R. (1982). The Nonparametric Approach to Demand Analysis. Econometrica. 50, 945-973. [9] Varian, H. R. (1988). Revealed Preference with a Subset of Goods. Journal of Economic Theory. 46, 179-185. [10] Warshall, S. (1962). A Theorem on Boolean Matrices. Journal of the American Association of Computing Machinery. 9, 11-12. 11

good 2 A B AD = p 1 x 1 BE = p 2 x 1 CF = p 3 x 1 C x 1 O D E F good 1 Figure 1: The Case of Three Observations With p 1 > p 2 > p 3

150 B6 B4 100 Payment to other B8 50 B7 4 B1 6 3 B3 5 B2 2 7 8 1 B5 50 Payment to self 100 150 Figure 2: Choices of Subject 3