Industrial Organization

Transcription

1 In the Name of God Sharif University of Technology Graduate School of Management and Economics Industrial Organization ( st term) Dr. S. Farshad Fatemi Product Differentiation Part 3

2 Discrete Choice Models In this section, we examine models with more than 2 possible choices Examples How to get to work (bus, car, subway, walk) How you treat a particular condition (bypass, heart cath., drugs, nothing) Which car to buy (Samand, Peugeot, L90) Industrial Organization Dr. F. Fatemi Page 93

3 In these examples, the choices reflect tradeoffs the consumer must face Transportation: More flexibility usually requires more cost Health: more invasive procedures may be more effective In contrast to ordered probit, no natural ordering of choices Industrial Organization Dr. F. Fatemi Page 94

4 Modeling choices Model is designed to estimate what cofactors predict choice of 1 from the other J-1 alternatives Motivated from the same decision/theoretic perspective used in logit/probit modes (Just have expanded the choice set) j indexes choices (J of them) i indexes people (N of them) Yij=1 if person i selects option j, =0 otherwise Uij is the utility or net benefit of person i if they select option j Industrial Organization Dr. F. Fatemi Page 95

5 Suppose they select option 1 Then there are a set of (J-1) inequalities that must be true Ui1>Ui2 Ui1>Ui3.. Choice 1 dominates the other Ui1>UiJ We will use the (J-1) inequality to help build the model Industrial Organization Dr. F. Fatemi Page 96

6 Two different approaches Multinomial logit Utility varies only by i characteristics People of different incomes more likely to pick one mode of transportation Conditional logit Utility varies only by the attributes of the option Each mode of transportation has different costs/time Mixed logit combined the two Industrial Organization Dr. F. Fatemi Page 97

7 Multinomial Logit Utility is determined by two parts: observed and unobserved characteristics However, measured components only vary at the individual level Therefore, the model measures what characteristics predict choice Are people of different income levels more/less likely to take one mode of transportation to work Industrial Organization Dr. F. Fatemi Page 98

8 Uij = Xiβj + εij εij is assumed to be a type 1 extreme value distribution f(εij) = exp(- εij)exp(-exp(-εij)) F(a) = exp(-exp(-a)) Choice of 1 implies utility from 1 exceeds that of options 2 (and 3 and 4.) Focus on choice of option 1 first Ui1>Ui2 implies that Xiβ1 + εi1 > Xiβ2 + εi2 OR εi2 < Xiβ1 - Xiβ2 + εi1 Industrial Organization Dr. F. Fatemi Page 99

9 There are J-1 of these inequalities εi2 < Xiβ1 - Xiβ2 + εi1 εi3 < Xiβ1 Xiβ3 + εi1 εij < Xiβ1 - Xiβj + εi1 Probability we observe option 1 selected is therefore [Prob(εi2 < Xiβ1 - Xiβ2 + εi1 εi3 < Xiβ1 Xiβ3 + εi1. εij < Xiβ1 - Xiβj + εi1)] Recall: if A, B and C are independent Pr(A B C) = Pr(A)Pr(B)Pr(C) And since ε1 ε2 ε3 εk are independent,he term in brackets equals Pr(Xiβ1 - Xiβ2 + εi1) Pr(Xiβ1 Xiβ3 + εi1) Industrial Organization Dr. F. Fatemi Page 100

10 But since ε1 is a random variable, must integrate this value out k j 2 F( X X ) f ( ) d 1i i 1 i j 1i 1i k exp( X ) j 1 i 1 exp( X ) i j The probability you choose option j is Prob(Yij=1 Xi) = exp(xiβj)/σk[exp(xikβk)] Each option j has a different vector βj Industrial Organization Dr. F. Fatemi Page 101

11 To identify the model, must pick one option (m) as the base or reference option and set βm=0 Therefore, the coefficients for βj represent the impact of a personal characteristic on the option they will select j relative to m. If J=2, model collapses to logit Log likelihood function Yij=1 of person I chose option j 0 otherwise Prob(Yij=1) is the estimated probability option j will be picked L = Σi Σj Yij ln[prob(yij)] Industrial Organization Dr. F. Fatemi Page 102

12 Notice there is a separate constant for each alternative Represents that, given X s, some options are more popular than others Constants measure in reference to the base alternative Parameters by themselves are not that informative We want to know how the probabilities of picking one option will change if we change X Two types of X s Continuous Discontinuous Industrial Organization Dr. F. Fatemi Page 103

13 Independent of Irrelevant alternatives IIA Suppose two options to get to work Car (option c) Blue bus (option b) What are the odds of choosing option c over b? Since numerator is the same in all probabilities Pr(Yic=1 Xi)/Pr(Yib=1 Xi) = exp(xiβc)/exp(xiβb) Note two thing: Odds are independent of the number of alternatives independent of characteristics of alt. not appealing Industrial Organization Dr. F. Fatemi Page 104

14 Example Pr(Car) + Pr(Blue Bus) = 1 (by definition) lets assume Pr(Car) = 0.75 & Pr(Blue Bus) = 0.25, So odds of picking the car is 3/1. Suppose that the local govt. introduces a new bus. Identical in every way to old bus but it is now red (option r) Choice set has expanded but not improved Commuters should not be any more likely to ride a bus because it is red Should not decrease the chance you take the car Industrial Organization Dr. F. Fatemi Page 105

15 In reality, red bus should just cut into the blue bus business Pr(Car) = 0.75 Pr(Red Bus) = = Pr(Blue Bus) Odds of taking car/blue bus = 6 What does model suggest? Since red/blue bus are identical βb =βr ; Therefore, Pr(Yib=1 Xi)/Pr(Yir=1 Xi) = exp(xiβb)/exp(xiβr) = 1 But, because the odds are independent of other alternatives Pr(Yic=1 Xi)/Pr(Yib=1 Xi) = exp(xiβc)/exp(xiβb) = 3 still Industrial Organization Dr. F. Fatemi Page 106

16 With these new odds, then Pr(Car) = 0.6 Pr(Blue) = 0.2 Pr(Red) = 0.2 Note the model predicts a large decline in car traffic even though the person has not been made better off by the introduction of the new option Poorly labeled really independence of relevant alternatives Implication? When you use these models to simulate what will happen if a new alternative is added, will predict much larger changes than will happen Industrial Organization Dr. F. Fatemi Page 107

17 How to solve IIA problem? Conditional probit models. Allow for correlation in errors Very complicated. Not pre-programmed into any statistical package Nested logit Group choices into similar categories IIA within category and between category Industrial Organization Dr. F. Fatemi Page 108

18 Mixed Models Most frequent type of multiple unordered choice Z s that vary by option X s that vary by person Uij = Xiβj + Zijγ + εij Prob(Yij=1 Xi Zij) = exp(xiβj + Zijγ)/Σk[exp(Xiβk + Zik γ)] Industrial Organization Dr. F. Fatemi Page 109