Revealed Preference Theory for Indivisible Goods Fran coise Forges - PowerPoint PPT Presentation

Revealed Preference Theory for Indivisible Goods Fran¸ coise Forges Universit´ e Paris-Dauphine & IUF Vincent Iehl´ e Universit´ e Paris-Dauphine September 27, Institut Henri Poincar´ e Working Paper: Afriat’s Theorem for Indivisible Goods.

Outlines • Introduction: Revealed Preference Theory and Afriat’s Theorem – Motivation – GARP – Afriat’s constructive approach – Extensions • The case of indivisible goods – Intuition – (new) Axiom – Main results and sketch of the proof – Related literature 1

Introduction - Revealed Preference Theory • Samuelson (1937): “ [Revealed Preference] . . . is a topic that people will be discussing a hundred years from now ” (from Mas-Colell (1982)) • Samuelson’s general postulate: the consumer theory based on preferences (or utility functions) must be operational, i.e., refutable by observable data generated from feasible experiments. • Conversely: given observable data, underlying preferences consistent with data. • Importance for Welfare Economics: from what is observed, one wants to evaluate and predict what would happen with a change in the environment • ⇒ very basic axiom on demand data: WARP 2

Introduction - Revealed Preference Theory • General formulation: Richter (1966) based on Houthakker (1950) – Possibly abstract framework – In the competitive framework he defines SARP – SARP reinforces WARP to a chain of observations (transitive clo- sure). – SARP is necessary and sufficient for the existence of underlying pref- erences (assuming single-valued demand functions) • Another Approach: Integrability (or differentiable approach) – The whole demand is observed. – Testable restrictions are based on Slutsky relations. – See Chiappori and Ekeland (2009) 3

Introduction - Revealed Preference Theory Richter’s approach is general but not constructive. Alternative approach: Afriat (1967) (Diewert (1973) and Varian (1982) in its modern formulation) Main ingredients: • Fully constructive and operational • Competitive framework: standard consumer problem with linear prices. • Finite set of observations • axiom on demand data: GARP • Result: the data satisfy GARP iff there exists a well-behaved utility function consistent with the data 4

Model and Afriat’s Theorem An analyst observing at each date t = 1 , . . . , n the bundle x t ∈ ❘ K + purchased by a single consumer, and positive prices p t ∈ ❘ K ++ . The consumption set is X ⊆ ❘ K + . The budget set at any date t is: B t := { x ∈ ❘ K + : p t · x ≤ p t · x t } Definition 1 A utility function u : X → ❘ is called a rationalization of the observations ( x t , p t ) t =1 ,...,n if, at each date t , x t solves max u ( x ) subject to x ∈ B t ∩ X (1) 5

Model and Afriat’s Theorem The bundle x i is said to be directly revealed preferred to x j , x i Rx j , if x j ∈ B i . The transitive closure of R is denoted by H , � That is, x i Hx s if there exists an ordered subset { i, j, k, . . . , r, s } ⊂ { 1 , . . . , n } � such that x i Rx j , x j Rx k , . . . , x r Rx s . x i is said to be revealed preferred to x s if x i Hx s . Definition 2 The observations ( x t , p t ) t =1 ,...,n satisfy GARP if for any i, j = 1 , . . . , n x i Hx j ⇒ p j · x i ≥ p j · x j Theorem 1 (Afriat) Let X = ❘ K + . The observations ( x t , p t ) t =1 ,...,n satisfy GARP if, and only if, there exists a continuous, concave and strictly mono- tonic rationalization of the observations. 6

GARP x 1 Budget at date 1 − → • x 2 • Budget at date 2 − → 7

Violation of GARP Budget at date 1 − → x 2 • Budget at date 2 − → • x 1 8

Afriat’s constructive approach (Afriat’s inequalities) If the observations satisfy GARP it can be shown that the following system admit a solution with ¯ ψ 1 , . . . , ¯ ψ n and ¯ δ 1 , . . . , ¯ δ n > 0: ψ k ≤ ¯ ¯ ψ j + ¯ δ j α jk ∀ j, k = 1 , . . . , n ( ∗ ) where α jk = p j · x k − p j · x k . Can be shown by using linear programming (Fostel et al. (2004)) or graph theory (Fujishige and Yang (2013)) The following function provides a well-behaved rationalization of the obser- vations: u ( x ) = min � ¯ ψ 1 + ¯ δ 1 p 1 · ( x − x 1 ) , . . . , ¯ ψ n + ¯ δ n p n · ( x − x n ) � ¯ 9

Afriat’s Theorem - Extensions Other models: • General Equilibrium (multi-agent): Brown and Matzkin (1996) • Household consumption: Cherchye, De Rock and Vermeulen (2007) • Production: Varian (1982) and Cournot competition: Carvajal, Deb, Fenske and Quah (2013) Other specifications of the consumer model (on p , u or X ): • General budget sets: Forges and Minelli (2009) • Additive preferences: Quah (2012) • Indivisible goods (this paper) 10

What we do: indivisible goods X = ◆ K Consumption bundles, not items (Ekeland Galichon (2012))! In practice, goods are often indivisible, in the field or in the laboratory. Local nonsatiation becomes meaningless so that GARP, in its usual form, is no longer a necessary condition of rationalization. We identify a natural counterpart of the standard GARP for demand data in which goods are all indivisible. We show that the new axiom (DARP, for ”discrete axiom of revealed pref- erence”) is necessary and sufficient for the rationalization of the data by a well-behaved utility function. 11

Illustration Problem: max u ( x ) subject to x ∈ B t ∩ ◆ K (2) 12

Illustration 13

Main result Definition 3 The observations ( x t , p t ) t =1 ,...,n satisfy the discrete axiom of re- vealed preference (DARP) if for any i, j = 1 , . . . , n x i Hx j ⇒ x i + 1 / ∈ B j where 1 = (1 , . . . , n ) . Proposition 1 The observations ( x t , p t ) t =1 ,...,n satisfy DARP if, and only if, there exists a discrete quasi-concave and monotonic rationalization of the observations. 14

Strict monotonicity One shortcoming of the previous result is that we do not obtain a strictly monotonic rationalization. Given the set of observations, let c ( t ) be one of the cheapest goods at date t , that is, c ( t ) ∈ argmin { p g t : g = 1 , . . . , K } . Definition 4 The observations ( x t , p t ) t =1 ,...,n satisfy DARP* if for any i, j = 1 , . . . , n x i Hx j ⇒ x i + e c ( j ) / ∈ B j where e c ( j ) = (1 , . . . , 0 , 1 , 0 , . . . , 0) , with 1 in the c ( j ) -th component. Proposition 2 The observations ( x t , p t ) t =1 ,...,n satisfy DARP* if, and only if, there exists a discrete quasi-concave and strictly monotonic rationalization of the observations. 15

About the proof An interesting (and perhaps unexpected) feature of the proof of our main results (propositions 1 and 2) is that the construction of an explicit utility function from Afriat’s inequalities goes through in our discrete framework. We deduce the existence of a solution for adequately chosen Afriat’s inequali- ties. We obtain then ψ 1 , . . . , ψ n and δ 1 , . . . , δ n > 0 such that u : ◆ K → ❘ defined as follows is a well-behaved rationalization of the observations: � � u ( x ) = min ψ 1 + δ 1 p 1 · ( x − x 1 ) ✶ A c 1 ( x ) , . . . , ψ n + δ n p n · ( x − x n ) ✶ A c n ( x ) where A t = { x ∈ ◆ K : p t · ( x t − 1 ) < p t · x ≤ p t · x t � . A potential difficulty with our construction is that the desirable properties of a utility function are not a priori granted here... 16

Remark: nonconvexities The next picture describes a nonconvex and continuous budget generated by the revenue p t · x t . By using existing result of the literature (Forges and Minelli, 2008), one cannot get particular property beyond monotonicity. 17

Slight generalization: (possibly) non binding budgets Under monotonic preferences, the rational consumer holds an unobserved revenue r t at date t such that p t · x t ≤ r t and p t · x > r t for every x ≫ x t , with x ∈ ◆ K . Contrary to the perfectly divisible case, this does not imply in our framework that p t · x t = r t . It follows that, instead of B t , the analyst may be willing to consider larger budget sets, which are compatible with such typical losses. Formally, a family of budget gap parameters is θ = ( θ t ) t =1 ,...,n , with θ t ∈ [0 , 1). Given θ , the budget at date t is: � � B θ x ∈ ❘ K t := + : p t · x ≤ p t · ( x t + θ t 1 ) Our previous results are virtually not affected by allowing for such budgets: θ -DARP iff θ -rationalization. (idem with θ -DARP*) 18

Related literature with X = ◆ K 19

Polisson and Quah (2013): unobserved continuous good Consumer Model: at each date t = 1 , . . . , n , there exist M t ≥ 0 , q t > 0 such that ( x t , M t − p t · x t ) solves q t max u ( x ) + y subject to p t · x + q t y ≤ M t (3) ( x,y ) ∈ X × ❘ + Axiom: GARP. 20

Fujishige and Yang (2012): cost efficiency Consumer Model: at each date t = 1 , . . . , n , x t solves max x ∈ X u ( x ) subject to p t · x ≤ p t · x t (4) and min x ∈ X p t · x subject to u ( x ) ≥ u ( x t ) (5) Axiom: GARP. 21

Brown and Calsamiglia (2007) and S´ akovics (2013): money Consumer Model: there exists M ≥ 0 such that at each date t = 1 , . . . , n , ( x t , M − p t · x t ) solves max u ( x ) + y subject to p t · x + y ≤ M (6) ( x,y ) ∈ X × ❘ + Axiom: ARV (axiom of revealed valuation), stronger than GARP. Considering continuous or indivisible goods is innocuous here. 22