Slide 13.Slide 13.11Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
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.Slide 13.22Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
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.Slide 13.33Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
A Dichotomous Dependent Variable
Noif0
Yesif1yi
According to the regression model
yi = 0 + xi1 + ei
1i0ii x)y(Ey
We define
Slide 13.Slide 13.44Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
How Do Choice Probabilities Fit In?
]YesPr[]1yPr[p i1i
]NoPr[]0yPr[p i2i
1i
2i1ii
p
p)0(p)1()y(E
1i01i xp
From the definition of Expectation of a Discrete Variable
Slide 13.Slide 13.55Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
Two Requirements for a Probability
1p01i
1pp2i1i
Logical Consistency
Sum Constraint
Slide 13.Slide 13.66Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
A Requirement for Regression
0yfor)x(0
1yfor)x(1e
i1i0
i1i0i
V(ei) = E[ei – E(ei)]2
21i02i
21i01i
2i )x(p)x1(p)e(E
V(e) = 2I Gauss-Markov Assumption
Two possibilities exist
Since E(ei) = 0
by the Definition of E()
Slide 13.Slide 13.77Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
Heteroskedasticity Rears It’s Head
)x1)(x(
)p1(p
)p)(p1()p1(p
1i01i0
1i1i
21i1i
21i1i
21i02i
21i01i
2i )x(p)x1(p)e(E
Note that the subscript i appears on the right hand side!
Slide 13.Slide 13.88Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
Two Fixes
1xif1
1x0ifx
0xif0
p
1i0
1i01i0
1i0
1i
1i0 x 2
1i01i0p1i
dz2
zexp
2
1
)x()x(Fp
Linear Probability Model
Probit Model
Slide 13.Slide 13.99Random Random
Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing
The Logit Model Is A Third Option
1i0
1i0
1i0L1i xe1
xe
)x(Fp
0.0
0.2
0.4
0.6
0.8
1.0
ix
Slide 13.Slide 13.1010
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Expression for Not Buying
i
i
1i ue1
uep
i
i
i
i
i
1i2i
ue1
ue1
ueue1
ue1p1p
1
where ui = 0 + xi1
Slide 13.Slide 13.1111
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Logit Is a Special Case of Bell, Keeney and Little’s (1975) Market Share Theorem
J
mim
ij
ija
ap
ai1 = and ai2 = 1iue
For J = 2 and the logit model,
Slide 13.Slide 13.1212
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
My Share of the Market Is My Share of the Attraction
21
1
aaa
)1Pr(
where a1 is a function of Marketing Variables brought to bear on behalf of brand 1
Slide 13.Slide 13.1313
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
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 13.Slide 13.1414
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Model Can Be Linearized for Least Squares
i
i
1i ue1
uep
i
i
i
i
2i1i
ue
ueue1
uepp
1
1
1i0i2i1i xu)ppln(
Slide 13.Slide 13.1515
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Aggregate Data and Weighted Least Squares
Response
Population Yes (yi = 1) No (yi = 0) x
1 f11 f12
x1
2 f21 f22
x2
…… …
…
i fi1 fi2
xi
…… …
…
N fN1 fN2
xN
Slide 13.Slide 13.1616
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Some Definitions
ni = fi1 + fi2
pi1 = fi1 / ni
)ppln(ˆ2i1i12,i l
)ppln( 2i1i12,i l
Slide 13.Slide 13.1717
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Assumptions About Error
)ˆ(x 12,i12,i1i012,i lll
12,i12,iˆ)(E ll
2i1ii
12,i12,i12,i
ppn
1
)ˆ(V)(V
lll
2i1ii12,i12,i ppn1,0N~ll
Slide 13.Slide 13.1818
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Weighted Least Squares: Scalar Presentation
2
i1i0iiError )xy(wSS
2i1iii ppnw
Slide 13.Slide 13.1919
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Weighted Least Squares: Matrix Presentation
N
i
2
1
2N1N
2i1i
2221
1211
xx1
xx1
xx1
xx1
x
x
x
x
X
2
1
0
β
12,N12,212,1ˆˆˆ lll l
12,N12,212,1 lll l
Slide 13.Slide 13.2020
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Model Expressed in Matrix Terms
βx
βx
i
i
22i11i0
22i11i0
1i
e
e
e
e
1
xx1
xxp
Xβl
Slide 13.Slide 13.2121
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Putting the Weights in Weighted Least Squares
2N1NN
22212
12111
ppn100
0ppn10
00ppn1
V
N
2
1
1
w00
0w0
00w
V
Slide 13.Slide 13.2222
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Minimizing f leads to the WLS Estimator
)()(
)ˆ()ˆ(f
1
1
XβVXβ
V
ll
llll
l111 ][ˆ VXXVXβ
Slide 13.Slide 13.2323
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Variance of the WLS Estimator
][
][][)ˆ(V
1
111111
XVX
VXXVXVVXXVXβ
So this allows us to test hypotheses of the form H0: a - c = 0
aXVXa
a11 )(
cˆˆ
β
t
Slide 13.Slide 13.2424
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Multiple DF Tests Under WLS
H0: A - c = 0
)ˆ(])([)ˆ(SS 111
HcβAAXVXAcβA
)ˆ()ˆ(SS 1
ErrorβXVβX ll
kn,q
Error
H F~kn/SS
q/SS
Slide 13.Slide 13.2525
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
ML Estimation of the Logit Model
βx
βx
i
i
11i0
11i0
1i
e
e
e
e
1
x1
xp
1
11 1210 ˆˆ
N
i
N
Niii ppl
N
i
ii
ii
yp
yp
1210
1)ˆ()ˆ(l
Two equivalent ways of writing the likelihood
We will use the left one, but isn't the right one clever?
Slide 13.Slide 13.2626
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Likelihood Derivations
N
1Ni 11i0
N
1i 11i0
11i0
01
1
x1
1x
1
x
ee
el
.x
1lnx
)ln(L
1N
1i
N
1i
1i0
1i0
00
e
l
0LL
1
0
0
0
These first order conditions must be met:
Slide 13.Slide 13.2727
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Second Order ML Conditions
01
0
2
10
0
2
1
0
0
LLL
00
0
00
0
2
0
0
0
11
LLLh
1)](E[)ˆ(V Hβ
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 13.Slide 13.2828
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Three Choice Options
pi1 + pi2 + pi3 = 1
3i
2i
3i
1i
2i
1i
p
pln
p
pln
p
pln
Slide 13.Slide 13.2929
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Multinomial Logit Model
J
m
mim
jij
ij
e
ep
βx
βx
J
mim
ij
ij
a
ap
)exp(ajijij
βx
the above model is a special case of the Fundamental Theorem of Marketing Share
Slide 13.Slide 13.3030
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Likelihood for the MNL Model
N
1NiiJ
N
1Ni2i
N
1i1i0
1J
2
1
1
ppp l
Slide 13.Slide 13.3131
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Ii Income of household i
Cost (price) of alternative j for household i
CAVi Cars per driver for household i
BTRi Bus transfers required for member of
household i to get to work via the bus
Classic Example
j
iC
Slide 13.Slide 13.3232
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
MNL Example Model
4i3
3
i
1
i1i1
3i
1i CAV)CC(Ip
pln
5i3
3
i
2
i2i2
3i
2i BTR)CC(Ip
pln
J
m
jim
jij
ij
e
ep
βx
βx
Slide 13.Slide 13.3333
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
GLS Estimation of the Transportation Example
5
4
3
2
1
2
1
32
232
222
131
211
31
232
122
131
111
23,
23,2
23,1
13,
13,2
13,1
0010
0010
0010
0001
0001
0001
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
NNNN
NNNN
N
N
BTRCCI
BTRCCI
BTRCCI
CAVCCI
CAVCCI
CAVCCI
l
l
l
l
l
l
Xβl
Slide 13.Slide 13.3434
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Other Choice Models
J
mim
ij
ij
a
ap
)xexp(ak
kijkij
k
kijkij
xa
)xexp(ak
jkijkij
k
jk
ijkij
xa
)xexp(aJ
m kmjkimkij
k
mjk
imkm
ijxa
Simple Effects Differential Effects Fully Extended
MNL
MCI
Slide 13.Slide 13.3535
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
Share Elasticity
ij
ij
ij
ij
ij p
x
x
pe
)xx(expaiJijjij
.)xexp(aijjij
1
mimij
ijij
ij aaxx
p
or
Slide 13.Slide 13.3636
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Derivative Looks Like
dea/da = ea )x(f)x(f
edx
)x(fde
)p1(p
pp
aaaaadx
pd
ijij
ij
2
ij
1
mimijij
2
mimij
ij
ij
Here we have used the following two rules:
Slide 13.Slide 13.3737
Random Random Utility ModelsUtility Models
MathematicalMathematicalMarketingMarketing
The Elasticity for the Simple Effects MNL Model
)p1(xeijijij
Putting the derivative back into the expression for the elasticity yields:
Top Related