A characterization of combinatorial demand

Transcription

1 A characterization of combinatorial demand C. Chambers F. Echenique UC San Diego Caltech Montreal Nov 19, 2016

2 This paper Literature on matching (e.g Kelso-Crawford) and combinatorial auctions (e.g Milgrom): D(p) = argmax{v(a) a A p a : A X } (*) When is * true? What is the behavioral content of the combined assumptions of rationality and quasilinearity?

3 Notation Let X be a finite set (of items). Let S be the set of all nonempty subsets of 2 X (so the empty set is not in S, but { } is). Identify A X with 1 A R X. If p R X then p, A = x A p x.

4 Demand A demand function is D : R X ++ S s.t. p R X ++ with D(p) = { } for all p p. ( p a choke price)

5 Demand D is quasilinear rationalizable if v : 2 X R s.t D(p) = argmax A X v(a) p, A

6 Suppose D is QL-rationalizable Let A D(p) and B D(q). Thus: p q, A B 0. v(a) p, A v(b) p, B v(b) q, B v(a) q, A.

7 Suppose D is QL-rationalizable Let A D(p) and B D(q). v(a) p, A v(b) p, B v(b) q, B v(a) q, A. Thus: p q, A B 0. The law of demand!

8 Demand A demand function D satisfies the law of demand if for all p, q R X ++, and all A D(p) and B D(q), p q, A B 0; is upper hemicontinuous if, p R X ++, nbd V of p s.t. D(q) D(p) when q V.

9 Main result Theorem A demand function is quasilinear rationalizable iff it is upper hemicontinuous and satisfies the law of demand.

10 Identification Theorem For any quasilinear rationalizable D, there is a unique monotone v : 2 X R for which v( ) = 0 which rationalizes D. Utility is identified up to an additive constant.

11 Monotone rationalization D is monotone, concave, quasilinear rationalizable (MCQ-rationalizable) if a monotone, concave g : R X + R s.t v(a) = g(1 A ), and D(p) = argmax{v(a) p, A : A X }. Corollary If a demand function is quasilinear rationalizable, then it is MCQ-rationalizable.

12 Proof ideas D(p) = argmax A X v(a) p, A If A D(p) then we want p to be the gradient of v at A. Can recover v by integrating over p.

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18 Cyclic monotonicity D satisfies cyclic monotonicity if, for all n (using summation mod n), n p i, A i A i+1 0, i=1 where A i D(p i ), for all sequences {p i } n i=1.

19 Cyclic monotonicity Define: v(a) = inf p 1, A A p, A k, inf is taken over all finite seq. (p i, A i ) k i=1 with A i D(p i ).

20 Cyclic monotonicity Define: v(a) = inf p 1, A A p, A k, inf is taken over all finite seq. (p i, A i ) k i=1 with A i D(p i ). Observe, by CM, { p 1, A A p, A k } + p, A 0. So v(a) is well defined (and 0).

21 Let A D(p) and B X (B D(R X ++) need a different arg. otherwise). By defn. of v, v(b) p, B A + v(a). Thus v(a) p, A v(b) p, B.

22 Let A D(p) and B X (B D(R X ++) need a different arg. otherwise). By defn. of v, v(b) p, B A + v(a). Thus v(a) p, A v(b) p, B. Proof that if A D(p) and B / D(p) then v(a) p, A > v(b) p, B requires more.

23 D satisfies condition if p and B / D(p) A D(p) and p s.t A D(p ) and p, A B > p, A B. Lemma If D is upper hemicontinuous, then it satisfies condition.

24 Cyclic monotonicity Lemma If D satisfies cyclic monotonicity, and condition, then it is quasilinear rationalizable. Based on ideas in Rochet/Rockafellar (but plays a technical role).

25 Lemma A demand function satisfies cyclic monotonicity if it satisfies the law of demand. Follows from recent results in mech. design (Lavi, Mu alem, and Nisan; Saks and Yu; and Ashlagi, Braverman, Hassidim, and Monderer).

26 Related literature Rochet/Rockafeller Brown and Calsamiglia Sher and Kim Lavi, Mu alem, and Nisan; Saks and Yu; Ashlagi, Braverman, Hassidim, and Monderer

27 Conclusions Quasilinear rational demand is a ubiquitous assumption. Our result is the first characterization in terms of observable behavior. Identification enables welfare analysis. New use for recent results in mech. design.