Bài 6: Mô hình Random Utility

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THEORY OF INDIVIDUAL CHOICE

AND APPLICATION:

THE RANDOM UTILITY MODEL

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Development of RUM

 Lancaster’s attribute based utility theory

 The Law of Comparative Judgment (Thurstone 1927)

 Individuals react to stimuli

 When making choices among alternatives, individuals choose

the one with highest level of stimulus

 Stimulus comprises an objective level and a random error

 Economists interpret stimulus as utility (Marschak

1960, Manski1977)

 systematic component

 the random component

 Individuals choose the alternative with the highest level

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Development of RUM (cont)

 The specification of random and systematic

utility

 makes the model probabilistic
 estimate the utility function

 The model was made popular by McFadden

(1974)

 multinomial logit (MNL) model
 nested logit model

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Application of RUM

 Transportation demand: choice among

transportation modes

 Environmental valuation:

 choice data generated from real market (actual

choice, or Revealed Preference data)

 choice data from hypothetical market (Stated

Preference data)

 Experimental data that involves choice among

options

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Utility function

Alternative specific constants
Error terms

The probabilistic choice

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Structure of RUM

Alternatives

Alternative 1 Alternative 2

… Alternative J

1 1 1

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The utility function

 Utility from alternative j include
 systematic component

 the random component

V
j



j j j

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The utility function

 Utility from alternative j is assumed to be a

function of attributes of alternative j

 utility of from alternative j

 level of attribute k of alternative j

 marginal utility of attribute k (to be estimated)
 Note: is constant specific to alternative j

j 0 j 1 j1 1 j2 ... K jK

V 







x 



x  



x

j
V

jk

x

k



0 j

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The alternative specific constant

(ASC)

 The constant term of each alternative is ASC
 The model is unidentifiable if all the ASCs are

estimated. We have to fix one of them at 0

1 1

V  X



2 2 2

V  ASC  X



J J

V  ASC  X



...

ASC reflects 
the preference 
on each

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The error term

 Gumbel distributed random

variable

 location parameter

 scale parameter



,



G



:

 

 

e

F





e

   





0

 





 

e

f







e

  

e

   


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The error term

 Properties of the Gumbel distribution
 1. Mode ; Mean where

(Euler’s constant); and variance

 2. If then

 3. If and then

is logistically distributed













0.577

2
2

6 





,



G



:

 

V





:

G







V

,









G

,



:

 



2

:

G



 

2

,



*
1 2



 

 

 





2 1
*

1 F

e

   



 



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The error term

 Properties of the Gumbel distribution

 4. If ; ; …;

are independent, then





1 G 1,

 :   2 : G



 2,





,



J G J

 :  



1 2



1 max

,

,...,

ln

j

,

J
J

j

G

e



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The probability of choosing

alternative

 The utility function

 The probability of choosing j


j j j

U  V 







Pr j is chosen among C

j

p 





Pr U , , j,l

j j l

p  U  l j C

 





Pr U max Pr U

j j l j

l C j

p U U

 

 

     

 



* *





* *



Pr V + Pr V

j j j j j

p   V      V

 * 

1 j

j V V

p

e 

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The probability of choosing

alternative

 Properties of the Gumbel distribution result in

j

l

V

j J

V

l

e

p

e





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Estimation of the RUM

 Log-likelihood



 the choice of individual i on alternative j (1 =

chosen)

 Coefficients of the utility functions are estimated by

maximizing the log-likelihood function

1
j
l
V
j J
V
l
e
p
e






 

1 1
log ln
N J
ij ij
i j

L Y p

 





ij

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Method of estimation

Data: Revealed preference and Stated preference
Data organization

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Estimation of the RUM model

 Assume location and scale

 Likelihood function

 Log-likelihood function

 The model is estimated

by maximizing the log-likelihood function

   1

1
j
l
V
j J
V
l
e
p
e





 

1 1
ij

N J Y

ij
i j
L p

 




 

1 1
log ln
N J
ij ij
i j

L Y p

 





is actual observed choice,
= 1 if j is chosen,

= 0 otherwise

ij

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Data for RUM – Stated preference

 Choices are obtained in a hypothetical situation
 respondents are presented with a set of alternatives
 each alternative are characterized by a set of

attributes’ levels

 respondents are asked to choose among the

presented alternatives

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Stated preference data – Example 1

 Harper (2012) estimates WTP for the

conservation of endangered species (caribou)

 Respondents were asked to choose between 2

alt

 status quo: current management strategy
 the proposed management strategy

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Organization of RUM data

Resp Choice

set Alt Herd Cost Choice

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Stated preference data – Example 2

 Pham and Tran (2005) used choice modelling to

analyze the demand for water service improvement

 Each respondent were asked to make several

choices between:

 the current situation (status quo)
 the improved service plan

 Each alternative is characterized by 3 attributes:

 water quality (with 3 levels: low, medium, high)
 water pressure (low; medium; high)

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Organization of RUM data

Resp Choice

set Alt qualityWater pressureWater Cost Choice

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Coding of qualitative variables

 2 levels: water quality (2 discrete levels Low and

High)

 create a dummy variable WQ

 WQ = 1 if high quality; 0 otherwise

 How to interpret the estimated coefficient of WQ?

 3 levels (or more): water pressure (Low, Medium

and High

 2 dummy variables PM and PH
 PM = 1 if medium; 0 otherwise
 PH = 1 if high; 0 otherwise

 Similar for the case of more than 3 levels

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What RUM can do?

 Probability of choosing
 Demand analysis

 Predict the changes in
 probability

 quantity demanded

when an attribute changes.

1
j

l

V
j J

V

l

e
p

e



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What RUM can do?

 Estimate welfare changes resulted from a change

in attributes

 is the attribute under consideration, p is the price
 increases by 1 unit 

 p increases by 1 unit ($) 

 1 unit increase in is equivalent to ($)

increase in price

V





x





p

x

x  U 

U 

 

x 



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Collect data

Input and organize data

Estimate the RUM using Stata

Calculate the probability of choosing a product
Illustration of how to calculate log-likelihood
value

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Example: demand for chocolate bar

 A producer considers introducing a new product (chocolate

bar) to the market

 The producer found that the following attributes are important

 weight (gr): 50, 100, 200
 type: milk or dark

 ingredient: with or without nuts
 price (1000 VND): 15, 30, 45

 Target market

 Questions:

 How to decide the levels of attributes?

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Sample questionnaire

Design #: 1

You are requested to consider a chocolate bar with the characteristics presented below
Chocolate bar

Weight (gram) 50

Type (milk chocolate or dark chocolate) Dark 
Ingredients (With nuts or not) No nuts

Price (thousand VND) 15

Would you buy the chocolate bar? □ Yes □ No
Please let us know:

Your gender □ Male □ Female

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Bài 6: Mô hình Random Utility

THEORY OF INDIVIDUAL CHOICE

AND APPLICATION:

THE RANDOM UTILITY MODEL

Development of RUM

Development of RUM (cont)

Application of RUM

Structure of RUM

The utility function

The utility function









The alternative specific constant

(ASC)







The error term



,



<i>G</i>



:

 

 

<i>F</i>





<i>e</i>





0

 





<i>f</i>







<i>e</i>

<i>e</i>

The error term











<sub></sub>

<sub></sub>

<sub>0.577</sub>

6







,



<i>G</i>



:

 

<i>V</i>





:

<i>G</i>







<i>V</i>

,









<i>G</i>

,

