Slide 13.1 Random Utility Models MathematicalMarketing Chapter 13 Random Utility Models This chapter covers choice models applicable where the consumer must pick one brand out of J brands. The sequence we will go through includes  Terminology  Aggregate Data and Weighted Least Squares  Disaggregate Data and Maximum Likelihood  Three or More Brands  A Model for Transportation Mode Choice  Other Choice Models  Patterns of Competition

Slide 13.2 Random Utility Models MathematicalMarketing Key Terminology  Dichotomous dependent variable  Polytomous dependent variable  Income type independent variable and the polytomous logit model.  Price type independent variable and the conditional logit model.  Aggregate data  Disaggregate dat

Slide 13.3 Random Utility Models MathematicalMarketing A Dichotomous Dependent Variable According to the regression model y i =  0 + x i  1 + e i We define

Slide 13.4 Random Utility Models MathematicalMarketing How Do Choice Probabilities Fit In? From the definition of Expectation of a Discrete Variable 

Slide 13.5 Random Utility Models MathematicalMarketing Two Requirements for a Probability Logical Consistency Sum Constraint

Slide 13.6 Random Utility Models MathematicalMarketing A Requirement for Regression V(e i ) = E[e i – E(e i )] 2 V(e) =  2 I Gauss-Markov Assumption Two possibilities exist Since E(e i ) = 0 by the Definition of E(  )

Slide 13.7 Random Utility Models MathematicalMarketing Heteroskedasticity Rears It’s Head Note that the subscript i appears on the right hand side!

Slide 13.8 Random Utility Models MathematicalMarketing Two Fixes Linear Probability Model Probit Model

Slide 13.9 Random Utility Models MathematicalMarketing The Logit Model Is A Third Option

Slide Random Utility Models MathematicalMarketing The Expression for Not Buying where u i =  0 + x i  1

Slide Random Utility Models MathematicalMarketing The Logit Is a Special Case of Bell, Keeney and Little’s (1975) Market Share Theorem a i1 = and a i2 = 1 For J = 2 and the logit model,

Slide Random Utility Models MathematicalMarketing My Share of the Market Is My Share of the Attraction where a 1 is a function of Marketing Variables brought to bear on behalf of brand 1

Slide Random Utility Models MathematicalMarketing The Story of the Blue Bus and the Red Bus Imagine a market with two players: the Yellow Cab Company and the Blue Bus Company. These two companies split the market 50:50. Now a third competitor shows up: The Red Bus Company. What will the shares be of the three companies now?

Slide Random Utility Models MathematicalMarketing The Model Can Be Linearized for Least Squares

Slide Random Utility Models MathematicalMarketing Aggregate Data and Weighted Least Squares Response PopulationYes (y i = 1)No (y i = 0)x 1 f 11 f 12 x1x1 2 f 21 f 22 x2x2 … …… … i f i1 f i2 xixi … …… … N f N1 f N2 xNxN

Slide Random Utility Models MathematicalMarketing Some Definitions n i = f i1 + f i2 p i1 = f i1 / n i

Slide Random Utility Models MathematicalMarketing Assumptions About Error

Slide Random Utility Models MathematicalMarketing Weighted Least Squares: Scalar Presentation

Slide Random Utility Models MathematicalMarketing Weighted Least Squares: Matrix Presentation

Slide Random Utility Models MathematicalMarketing The Model Expressed in Matrix Terms

Slide Random Utility Models MathematicalMarketing Putting the Weights in Weighted Least Squares

Slide Random Utility Models MathematicalMarketing Minimizing f leads to the WLS Estimator

Slide Random Utility Models MathematicalMarketing The Variance of the WLS Estimator So this allows us to test hypotheses of the form H 0 : a  - c = 0

Slide Random Utility Models MathematicalMarketing Multiple DF Tests Under WLS H 0 : A  - c = 0

Slide Random Utility Models MathematicalMarketing ML Estimation of the Logit Model Two equivalent ways of writing the likelihood We will use the left one, but isn't the right one clever?

Slide Random Utility Models MathematicalMarketing Likelihood Derivations These first order conditions must be met:

Slide Random Utility Models MathematicalMarketing Second Order ML Conditions When arranged in a matrix, the second order derivatives are called the Hessian. Minus the expectation of the Hessian is called the Information Matrix.

Slide Random Utility Models MathematicalMarketing Three Choice Options p i1 + p i2 + p i3 = 1

Slide Random Utility Models MathematicalMarketing Multinomial Logit Model the above model is a special case of the Fundamental Theorem of Marketing Share

Slide Random Utility Models MathematicalMarketing The Likelihood for the MNL Model

Slide Random Utility Models MathematicalMarketing I i Income of household i Cost (price) of alternative j for household i CAV i Cars per driver for household i BTR i Bus transfers required for member of household i to get to work via the bus Classic Example

Slide Random Utility Models MathematicalMarketing MNL Example Model

Slide Random Utility Models MathematicalMarketing GLS Estimation of the Transportation Example

Slide Random Utility Models MathematicalMarketing Other Choice Models Simple Effects Differential Effects Fully Extended MNL MCI

Slide Random Utility Models MathematicalMarketing Share Elasticity or

Slide Random Utility Models MathematicalMarketing The Derivative Looks Like de a /da = e a Here we have used the following two rules:

Slide Random Utility Models MathematicalMarketing The Elasticity for the Simple Effects MNL Model Putting the derivative back into the expression for the elasticity yields: