Testable Implications of Models of Intertemporal Choice Exponential - PowerPoint PPT Presentation

Testable Implications of Models of Intertemporal Choice Exponential Discounting and Its Generalizations Federico Echenique Taisuke Imai Kota Saito Cowles F. conference, June 9 2015

Utility and behavior Model: max U ( x ) x ∈ R T + s.t p · x ≤ I Echenique-Imai-Saito Exp. Discounting

Utility and behavior (Market) behavior: Echenique-Imai-Saito Exp. Discounting

x 1 x 2 Echenique-Imai-Saito Exp. Discounting

Utility and behavior ◮ Q: When is observable behavior consistent with utility max.? ◮ A: When SARP is satisfied. Echenique-Imai-Saito Exp. Discounting

This paper: max x ∈ R T + U ( x ) s.t p · x ≤ I ◮ Exponential discounting : � δ t u ( x t ) U ( x 0 , . . . , x T ) = t ∈T Importantly: u assumed to be st. increasing & concave. Echenique-Imai-Saito Exp. Discounting

This paper: max x ∈ R T + U ( x ) s.t p · x ≤ I ◮ Exponential discounting : � δ t u ( x t ) U ( x 0 , . . . , x T ) = t ∈T ◮ Quasi-hyperbolic discounting : T � δ t u ( x t ) U ( x 0 , . . . , x T ) = u ( x 0 ) + β t =1 Importantly: u assumed to be st. increasing & concave. Echenique-Imai-Saito Exp. Discounting

This paper: max x ∈ R T + U ( x ) s.t p · x ≤ I ◮ Exponential discounting : � δ t u ( x t ) U ( x 0 , . . . , x T ) = t ∈T ◮ Quasi-hyperbolic discounting : T � δ t u ( x t ) U ( x 0 , . . . , x T ) = u ( x 0 ) + β t =1 ◮ Time-separable utility : � U ( x 0 , . . . , x T ) = u t ( x t ) t ∈T Importantly: u assumed to be st. increasing & concave. Echenique-Imai-Saito Exp. Discounting

This paper. ◮ Q: When is observable behavior consistent with model M.? ◮ A: When SA-M is satisfied. M ∈ { TSU, QHD, EDU } Application to experimental data. Echenique-Imai-Saito Exp. Discounting

This paper. Application to experimental data from Andreoni and Sprenger “Estimating Time Preferences from Convex Budgets” (AER 2012). Fits our framework: ◮ “Economic” budget sets; ◮ Planned (not realized) consumption. later 25 20 15 10 5 sooner 5 10 15 20 Echenique-Imai-Saito Exp. Discounting

Warmup Echenique-Imai-Saito Exp. Discounting

Warmup The 2 × 2 case. ◮ 2 dates ◮ 2 observations ◮ Exp. discounting. Echenique-Imai-Saito Exp. Discounting

What is the meaning of this: max u ( x 0 ) + δ u ( x 1 ) p 0 x 0 + p 1 x 1 ≤ I model for market behavior ? Unobservables: ◮ Utility u : R + → R st. inc. Observable: & conc. ◮ choices at different budgets ◮ δ ∈ (0 , 1] Echenique-Imai-Saito Exp. Discounting

x 1 x 2 Echenique-Imai-Saito Exp. Discounting

x 1 x 2

MRS = u ′ ( x 0 ) δ u ′ ( x 1 ) x 1 x 2 Echenique-Imai-Saito Exp. Discounting

x 1 x 2

MRS = u ′ ( x 0 ) δ u ′ ( x 1 ) x 1 x 2

MRS = u ′ ( x 0 ) δ u ′ ( x 1 ) x 1 x 2 Echenique-Imai-Saito Exp. Discounting

Axiom 1 Axiom 2 Not: Not: x 1 x 1 x 2 x 2 Echenique-Imai-Saito Exp. Discounting

END of Warmup Echenique-Imai-Saito Exp. Discounting

Main theorem(s): A dataset is M-rationalizable iff it satisfies the “Strong Axiom of M” (SA-M). M = ◮ TSU ◮ Q-Hyperbolic discounting (QHD) ◮ Exp. discounting (EDU) Echenique-Imai-Saito Exp. Discounting

Plug “Savage in the market” Echenique - Saito (2014) Subjective Expected Utility � U ( x ) = µ s u ( x s ) s ∈ S Echenique-Imai-Saito Exp. Discounting

Data ◮ Time: T = { 0 , 1 , . . . , T } ; so T + 1 periods. ◮ Consumption path x ∈ R T + . ◮ Prices (interest rates): p ∈ R T ++ . k =1 , where x k ∈ R T A dataset is a collection { ( x k , p k ) } K + is a consumption path and and p k ∈ R T ++ a price vector. Echenique-Imai-Saito Exp. Discounting

Rationalizable data ◮ Let M be a set of functions U : R T + → R . Echenique-Imai-Saito Exp. Discounting

Rationalizable data ◮ Let M be a set of functions U : R T + → R . ◮ B ( p , I ) = { y ∈ R T + | p · y ≤ I } Echenique-Imai-Saito Exp. Discounting

Rationalizable data ◮ Let M be a set of functions U : R T + → R . ◮ B ( p , I ) = { y ∈ R T + | p · y ≤ I } Definition Dataset { ( x k , p k ) } K k =1 is M-rational if ∃ U in the class M s.t. y ∈ B ( p k , p k · x k ) ⇒ U ( y ) ≤ U ( x k ) , Echenique-Imai-Saito Exp. Discounting

Notation Let ◮ C = { u : R + → R | u is st. increasing and concave } Echenique-Imai-Saito Exp. Discounting

Models: M ∈ { EDU,QHD,TSU } 1. EDU: set of U s.t � δ t u ( x t ) , U ( x 0 , . . . , x T ) = t ∈T for some u ∈ C , and δ ∈ (0 , 1]. 2. QHD: set of U s.t T � δ t u ( x t ) , U ( x 0 , . . . , x T ) = u ( x 0 ) + β t =1 for some u ∈ C , β > 0 and δ ∈ (0 , 1] 3. TSU: set of U s.t � U ( x 0 , . . . , x T ) = u t ( x t ) , t ∈T for some u t ∈ C , t ∈ T . Echenique-Imai-Saito Exp. Discounting

Main result: EDU Derivation of SA-EDU. Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. Derivation of SA-EDU ◮ K = 1 ◮ δ = 1 (fixed and known) ◮ u differentiable. Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. � max x ∈ R T t ∈T u ( x t ) + � t ∈T p t x t ≤ I FOC: u ′ ( x t ) = λ p t Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. u ′ ( x t ) = λ p t So, u ′ ( x t ) u ′ ( x t ′ ) = p t p t ′ Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. u ′ ( x t ) = λ p t So, u ′ ( x t ) u ′ ( x t ′ ) = p t p t ′ Axiom (Downward sloping demand): x t > x t ′ ⇒ p t p t ′ ≤ 1 Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. Consider two pairs of observations: x t 1 > x t 2 ⇒ p t 1 ≤ 1 p t 2 and x t 3 > x t 4 ⇒ p t 3 ≤ 1 p t 4 Or, when x t 1 > x t 2 and x t 3 > x t 4 then p t 1 p t 3 ≤ 1 . p t 2 p t 4 Echenique-Imai-Saito Exp. Discounting

K = 1 and δ = 1. k ′ A sequence ( x k i i ) n t i , x i i =1 has the downward sloping demand t ′ property if n p k i k ′ x k i � t i t i > x i , i = 1 , . . . , n ⇒ i ≤ 1 . t ′ k ′ p i i =1 t ′ i Echenique-Imai-Saito Exp. Discounting

Derive SA-EDU; K > 1 and δ = 1. Now: K > 1. u ′ ( x k t ) = λ k p k t So, u ′ ( x k t ′ ) = λ k p k t ) t u ′ ( x k ′ λ k ′ p k ′ t ′ Echenique-Imai-Saito Exp. Discounting

Derive SA-EDU; K > 1 and δ = 1. Now: K > 1. u ′ ( x k t ) = λ k p k t So, � u ′ ( x k λ k p k t ′ ) = � t ) t ✚ u ′ ( x k ′ p k ′ λ k ′ ✚ t ′ Axiom (Downward sloping demand): t ′ and k = k ′ ⇒ p k t > x k ′ x k t ≤ 1 p k ′ t ′ Echenique-Imai-Saito Exp. Discounting

u ′ ( x k ′ λ k ′ p k ′ u ′ ( x k p k t 1 ) t 3 ) t 4 ) = λ k t 1 t 3 u ′ ( x k ′ u ′ ( x k λ k ′ λ k p k ′ p k ′ t 2 ) t 2 t 4 Echenique-Imai-Saito Exp. Discounting

✚ u ′ ( x k ′ � p k ′ u ′ ( x k λ k ′ p k λ k t 1 ) t 3 ) ✚ t 4 ) = � t 1 t 3 ✚ u ′ ( x k ′ p k ′ p k ′ u ′ ( x k λ k ′ � t 2 ) ✚ λ k � t 2 t 4 Hence p k ′ t 4 ⇒ p k t 1 > x k ′ t 2 and x k ′ x k t 3 > x k t 1 t 3 ≤ 1 p k ′ p k ′ t 2 t 4 Echenique-Imai-Saito Exp. Discounting

k ′ A sequence ( x k i i ) n t i , x i i =1 is balanced if each k appears as k i (on the t ′ left of the pair) the same number of times it appears as k ′ i (on the right). Axiom (for δ = 1 and K ≥ 1): Any balanced sequence has the downward sloping demand property. Echenique-Imai-Saito Exp. Discounting

Derive SA-EDU - general K and δ Agent solves � δ t u ( x t ) max x ∈ R T + t ∈T � s.t. p t x t ≤ I , t ∈T Echenique-Imai-Saito Exp. Discounting

FOC δ t u ′ ( x t ) = λ p t , So we need to find δ , u and λ k s.t δ t u ′ ( x k t ) = λ k p k t , for all k Echenique-Imai-Saito Exp. Discounting

u ′ ( x k ′ λ k ′ p k ′ t ) = δ t t ′ ) t ′ . δ t ′ u ′ ( x k λ k p k t t > x k ′ Suppose that x k t ′ . Then: λ k ′ p k ′ δ t t ′ ≤ 1 , δ t ′ λ k p k t But δ , λ k ′ and λ k are unobservable. Echenique-Imai-Saito Exp. Discounting

x k 1 t 1 > x k 2 t 2 and x k 2 t 3 > x k 1 t 4 . such that t 1 + t 3 ≥ t 2 + t 4 . u ′ ( x k 1 · u ′ ( x k 2 � λ k 1 p k 1 � � λ k 2 p k 2 � δ t 2 δ t 4 t 1 ) t 3 ) t 1 t 3 = · u ′ ( x k 2 u ′ ( x k 1 δ t 1 λ k 2 p k 2 δ t 3 λ k 1 p k 1 t 2 ) t 4 ) t 2 t 4 = δ ( t 2 + t 4 ) − ( t 1 + t 3 ) p k 1 p k 2 t 1 t 3 p k 2 p k 1 t 2 t 4 Echenique-Imai-Saito Exp. Discounting

x k 1 t 1 > x k 2 t 2 and x k 2 t 3 > x k 1 t 4 . such that t 1 + t 3 ≥ t 2 + t 4 . ✚ u ′ ( x k 1 · u ′ ( x k 2 � λ k 1 p k 1 � � λ k 2 p k 2 � δ t 2 ✚ δ t 4 t 1 ) t 3 ) t 1 t 3 = · ✚ u ′ ( x k 2 u ′ ( x k 1 δ t 1 λ k 2 p k 2 δ t 3 λ k 1 p k 1 t 2 ) t 4 ) ✚ t 2 t 4 = δ ( t 2 + t 4 ) − ( t 1 + t 3 ) p k 1 p k 2 t 1 t 3 p k 2 p k 1 t 2 t 4 Echenique-Imai-Saito Exp. Discounting