Rakesh V. Vohra

Teo Chung Piaw

English

Research Papers in Economics

Afriat’s Theorem and Negative Cycles
Teo Chung Piaw∗and Rakesh V. Vohra†
July 3, 2003
Abstract
This note provides a short proof of Afriat’s theorem and shows that the problem is
equivalent to the problem of identifying a negative length cycle in a related graph.
1 Introduction
Afriat’s (1967) theorem is an answer to the question of when a sequence of purchase decisions
is consistent with the purchaser maximizing a concave utility function u(·). Suppose a se-
quence of purchase decisions (pi,x i), i =1 ,...,n, where pi ∈  n
+ and xi ∈  n
+ are price vectors
and purchased quantity vectors respectively. Asume the purchaser makes purchase decision
based on utility maximization. If pi·(xj −xi) ≤ 0, the utility function u must satisfy u(xj) ≤
u(xi), otherwise, with purchase prices of pi, bundle xj costs less but provides higher utility to
the purchaser. If we have a sequence of decisions (pi,x i),(pj,x j),(pk,x k),...,(pr,x r), with
pi · (xj − xi) ≤ 0,p j · (xk − xj) ≤ 0,..., p r · (xi − xr) ≤ 0,
then u(xi)=u(xj)=...= u(xr), and
pi · (xj − xi)=0 ,p j · (xk − xj)=0 ,..., p r · (xi − xr)=0 .
The above necessary condition can be described in graph theoretic terms as follows: let
A be a n × n matrix of real numbers with all zero’s on the diagonals. We associate with the
matrix A a directed graph D(A) as follows: introduce a vertex for each index and for each
ordered pair (i,j) an edge with length aij = pi·(xj −xi). The matrix A will be said to satisfy
the Afriat condition (AC) if every negative length cycle in D(A) contains at least one edge
of positive weight.
Associated with A is an inequality system:
yj ≤ yi + siaij ∀i  = j, 1 ≤ i,j ≤ n
∗Department of Decision Science, National University of Singapore
†Department of Managerial Economics and Decision Sciences, Kellogg Graduate School of Management,
Northwestern University, Evanston IL60208. Research supported in part by NSF grant ITR IIS-0121678.
1si > 0 ∀1 ≤ i ≤ n
which we label L(A). We now state Afriats theorem:
Theorem 1 L(A) is feasible iﬀ. D(A) satisﬁes AC.
Whenever D(A) satisﬁes AC, we use the solution to L(A) to construct a utility function u(·)
consistent with the sequence of purchase decisions (pi,x i) by setting:
u(x) = min{y1 + s1p1(x − x1),y 2 + s2p2(x − x2),...,s npn(x − xn)}.
A number of proofs of the theorem exist.1 This note provides a new proof that makes
explicit the network structure inherent in L(A). This structure appears to have been over-
looked (but used implicitly) in previous theorems. In fact, the most recent paper on the
subject by Fostel, Scarf and Todd (2003) describes the system L(A) as being unusual.
To each s ∈  n
+ and matrix A with zeros on the diagonals, we associate a directed graph
D(A,s) as follows: introduce a vertex for each index and for each ordered pair (i,j) an edge
with length siaij. Notice that D(A)=D(A,e) where e is the n-vector of all 1’s.
Now ﬁx s ∈  n
+. Then feasibility of L(A) reduces to identifying y ∈  n such that
yj − yi ≤ siaij for all i  = j. Readers familiar with network ﬂows will recognize this system
as the constraint set to the dual of a shortest path problem. A standard result is that this
system is feasible iﬀ. D(A,s) contains no negative cycles. Assuming feasibility, we can choose
the y’s as follows: set y1 = 0 and yj to be the length of the shortest path from 1 to i in
D(A,s). Afriat’s theorem can be rephrased as;
Theorem 2 There is an s ∈  n
+ such that D(A,s) contains no negative cycles iﬀ. D(A,e)
satisﬁes AC.
Proof: If there is an s ∈  n
+ such that D(A,s) contains no negative cycles, the system
of inequalities L(A) (with s ﬁxed) is feasible. So we can construct a utility function u(·)
consistent with the sequence of purchase decisions. Therefore, D(A,e) must satisfy AC. We
next prove the non-trivial direction. Suppose D(A,e) satisﬁes AC. We prove there exists
s ∈  n
+ such that D(A,s) has no negative cycles.
Let S = {(i,j):ai,j < 0}, E = {(i,j):ai,j =0 }, and T = {(i,j):ai,j > 0}. Consider the
weighted digraph G with edges in S ∪ E, where arcs (i,j) ∈ S are given weight wij = −1,
and arcs (i,j) ∈ E are given weight wij = 0. Since D(A,e) satisﬁes AC, G does not contain
a negative length cycle. Hence there exists a set of potentials {φj} on the nodes such that
φj ≤ φi + wij, ∀ (i,j) ∈ E(G).
Without loss of generality, we relabel the vertices so that φn ≤ φn−1 ≤ ...≤ φ1. Choose {si}
non-decreasing so that
si × min
(i,j)∈T
aij ≥ (n − 1) × si−1 max
(i,j)∈S
(−aij)i fφi <φ i−1,
1Afriat’s orginal proof assumed that aij  =0∀i  = j. This was relaxed by Diewert (1973) and Varian (1982).
2and
si = si−1 if φi = φi−1
for all i>2, with s1 =1 .
For any cycle C in the digraph D(A,s), let (v,u) be an edge in C such that (i) v has the
smallest potential among all vertices in C, and (ii) φu >φ v. Such an edge exists, otherwise φi
is identical for all vertices i in C. In this case, all edges in C have non-negative edge weight
in D(A,s).
By selection, φu >φ v.I f( v,u) ∈ S∪E, then we have φu ≤ φv+wvu ≤ φv, a contradiction.
Hence (v,w) ∈ T. Now, note that all vertices q in C with the same potential as v must be
incident to an edge (q,t)i nC such that φt ≥ φq. Hence the edge (q,t) must have non-negative
length. i.e., aq,t ≥ 0. Let p denote a vertex in C with the second smallest potential. Now, C
has length
svavu +
 
(k,l)∈C,(k,l) =(v,u)
skak,l ≥ svav,u + sp(n − 1) max
(i,j)∈S
{aij}≥0,
i.e., C has non-negative length.
Since D(A,s) is a digraph without any negative length cycles, L(A) is feasible.
Given the matrix A, we can verify if condition AC is violated by checking if the graph G has
a negative length cycle. This can be done in O(n3) time (see Ahuja, Magnanti and Orlin
(1993)) using standard algorithms.2 Assuming AC holds, the node potentials can be com-
puted by a standard shortest path algorithm in O(n2) (again see Ahuja, Magnanti and Orlin
(1993)). This gives the s’s in O(n2) steps and another O(n2) steps to determine the value of
the y variables.
References:
1. Afriat, S. N. ‘The Construction of a Utility Function from Expenditure Data’, Inter-
national Economic Review, 8, 1967, 67-77.
2. Ahuja, R., T. Magnanti and J. Orlin, ‘Network Flows: Theory, Algorithms, and Appli-
cations’, Prentice-Hall, NJ, 1993.
3. Diewert, E. ‘Afriat and Revealed Preference Theory’, Review of Economic Studies,
40, 1973, 419-426.
4. Fostel, A., H. Scarf and M. J. Todd. ‘Two New Proofs of Afriat’s Theorem’, Cowles
foundation discussion paper, 1415, 2003.
5. Varian, H. ‘The Non-parametric Approach to Demand Analysis’, Econometrica, 50,
1982, 945-974.
2An improvement to O(n
2.5) using a more spohisticated algorithm due to Gabow is possible.
3

Afrait's Theorem and Negative Cycles

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