(Behavioral Model and Duality Theory)

Transcription

1 2018 (Behavioral Model and Duality Theory)

2 2 / / (Rational Inattention)

3 6 max u(x 1,x 2 ) x 1,x 2 s.t. p 1 x 1 + p 2 x 2 = I u = V (p 1,p 2,I) I = M(p 1,p 2, ū) x H j (p 1,p 2, ū) V / V p j I M p j u = V (p 1,p 2,I) I = M(p 1,p 2, ū) I u

4 7 e.g.

5 8 Miyagi (1986) (1986) Logit Model Entropy Model S(V, θ) = max P j J s.t. j J { P j V j 1 } θ P j (ln P j 1) max P j =1 exp(θv j ) P j = j J exp(θv j ), j J S(V, θ) = 1 θ ln j J exp(θv j )

6 9 Anderson, de Palma, & Thisse (1988) Verboven (1996) Entropy (Nested) Logit [Logit Representative Consumer Behavior] U = max X s.t. J j=1 Y = X j = N X 0 + J a j X j 1 θ j=1 J p j X j + p 0 X 0 j=1 J j=1 X j ln X j N Shannon Entropy Entropy term Variety-seeking behaviour exp[θ(a j p j )] X j = N J j =1 exp[θ(a j p j )], U = Y + Nθ ln p 0 j J J ( ) aj p j /p 0 exp θ j=1

7 11 Logit, Entropy & Random Utility Miyagi, ADT, Verboven Shannon Entropy Logit logsum [ " ] (McFadeen 1974)

8 12 Logit Example Additive random utility model (ARUM) with Gumbel error term u j = δ j + ϵ j, J McFadden (1974) Surplus (Indirect utility) Logsum G(δ) E ( ) max u j j J Williams (1977) Daly & Zachary (1978) Ben-Akiva (1973) McFadden (1978) Small & Rosen (1981) =ln j J e δ j Choice Probability (Demand) Logit model q : ( ) d : 0, 1,..., J: (0: )

9 13 General Additive Random Utility Model (ARUM) (ARUM) (Surplus) G(δ) E u j = δ j + ϵ j, J ( max j J u j 1 [Williams (1977) Daly & Zachary (1978) Theorem ] q j (δ) = G(δ) δ j WDZ Theorem ARUM )

10 14 ARUM-Logit Model Entropy Model ARUM-Gumbel u j = δ j + ϵ j, J Surplus as Logsum ( ) G(δ) E max u j j J =ln j J e δ j G (q) = max δ { q δ G(δ) } Shannon Entropy Logsum (Convex conjugates) Choice probability as Multinomial Logit q j (δ) = G(δ) δ j = e δ j j J eδ j Entropy Demands Shannon Entropy G (q) = j J q j ln q j Representative Consumer max q δ j q j + G (q) j J where = {q R J+1 + : J j=0 q j =1}

11 GEM ARUM GEM ARUM GEM max q δ j q j + G (q) j J 16 Generalized Entropy (GE) Generator S G (q) q j ln S (j) (q) j J Logit Generator S(q) =q H = S 1 (Shannon Entropy) H (j) (e δ (δ) )= eg δ j j P j (δ) = H (j) (e δ ) J j =0 H(j ) (e δ ) ( ) G(δ) =ln H (j) (e δ ) j J

12 17 GEM (1) 1. ARUM GEM 2. GE ( Generator) / ( ) ARUM GEM (e.g. GEN) 1. Nested Logit S (j) (q) =q µ j j g j q j 1 µ 2. Generalized { Nested Entropy (GEN) S (j) q0, (j = 0) (q) = q µ 0 C j c=1 qµ c σ c (j), (j > 0) 3. Berry, Levinsohn & Pakes (1995) Method ( ) ln ( qjt q 0t ) = β 0 + X jt β αp jt + C µ c ln c=1 ( qjt q σc (j),t ) + ξ jt

13 18 GEM (2) 4. (e.g. Rust 1987) Chiong et al. (2016) δ(x) a. G (q(x)) Choice-Specific Monge-Kantorovich b. δ(x) δ(x) ū(x)+βe [V (x )] ū(x) Foegerau et al. (2013) Hotz & Miller (1993) 5. Entropy= (Rational Inattention: RI)

14 19 (Rational Inattention) Sims (2003) (RI) : Entropy Matějka & McKay (2015): Mutual Shannon Entropy Logit Fosgerau, Melo, de Palma & Shum (2018): Generalized Entropy IIA Fosgerau & Jiang (2017): ( )

15 20 Rational Inattention Model RI Matějka & McKay (2015) max p( ) ( ) {E(V A) λ information cost} ( ) Information cost Mutual Shannon Entropy { { = max p(v) [v + λ log p 0 ] λp(v) log p(v) } } µ(v) p( ) v V N s.t. p i (v) 0 i, p i (v) =1 i=1 (MNL) p i (v) = p 0 i ev i/λ N j=1 p0 j ev j/λ = e (v 0 i+log pi )/λ = N j=1 e(v j+log p 0 j )/λ eṽi/λ N j=1 eṽj/λ

16 23 ARUM (Unobserved Taste Heterogeneity) (n) (Train & Weeks 2005) U njt = α n p njt + β n x njt + e njt where Var(e njt )=k 2 n(π 2 /6) k n *WTP: Willingness-to-pay A) Model in Preference-Space B) Model in WTP-Space U njt =(α n /k n )p njt +(β n /k n )x njt + ε njt U njt = λ n p njt +(λ n w n )x njt + ε njt = λ n p njt + c n x njt + ε njt where Var(ε njt )=π 2 /6 k n w n c n /λ n WTP* ( ) WTP w n ( ) l n Model A Model B (Scarpa et al. 2008)

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