Structural Model and Additional Materials

Appendix A
Structural economic and econometric model

The Demand for Internet Access
The conventional labor-leisure choice model is extended to include the benefits from
Internet access. The consumer is assumed to maximize a utility function of consumption
and leisure, subject to a monetary budget constraint that includes the household
production input Internet bandwidth, and subject to a time budget constraint that includes
the household production input time online. Both inputs are used to produce reductions
in essential time, defined as the non-remunerated time lost when participating in the labor
market, plus time doing fundamental living activities such as banking, bill-paying,
maintaining health, shopping, etc.
Essential time is represented by the household production function X Ð2ß ,ß >à +Ñ,
where 2 is the number of hours worked, , is Internet bandwidth, > is time spent online,
and + is an efficiency parameter that reflects the technical ability of the individual. The
function X is convex in , and >, and , and > are assumed to be complements in production
so that increasing , will raise the marginal productivity of >. Similarly, + augments the
productivity of , and >, decreasing essential time for a given input level. As such, X , , X > ,
X + , X ,> , X ,+ , X >+  ! and X ,, ß X >>  !, where subscripts indicate partial derivatives.
Some of the time costs of work may be fixed. Others, including commuting time, costs
associated with the stress of work, the preparation and recovery period, and training and
child care costs, may be linear or concave functions of the number of hours worked
(Heim and Meyer, 2004). Essential time is concave in 2 so that X 2  ! and X 22  !.
The consumer's maximization problem is:
7+B
2ß ,ß >

Y Ð-ß PÑ

A1

=Þ>Þ - œ C  A2  :, ,  :> >

P œ X  2  >  X Ð2ß ,ß >; +Ñ
where Y is utility, - is consumption, P is leisure, C is non-wage income, A is the wage
rate, :, is the per-unit price of bandwidth, :> is the per-unit price of time online, and X is
total time available.

Structural Econometric Models and Likelihoods
The individual's utility of an Internet service is assumed to be a function of the attributes
of the service and a random error (known to the individual but not the researcher). This
is the Random Utility Model (RUM) as it is applied in environmental economics,
transportation research, health economics, and marketing.

It is assumed that respondents maximize their household's conditional utility of the
service option (conditional on all other consumption and time allocation decisions):
5

5

5

Y3434 œ " w B3434  %3434 ß 3 œ "ß á ß 8; 4 œ "ß á N ß 534 œ "ß #

A2

5

where Y3434 is the utility of alternative 534 chosen by individual 3 during occasion 4Þ1 The
5

vector B34 contains the observed attributes of the alternatives. It is assumed that the %3434
are independent, and identically distributed mean zero normal random variables,
uncorrelated with B34 , with constant unknown variance 5%# .2 The probability of choosing
alternative ", for example, is:
T341 œ T ÐY "  Y # Ñ
œ T Ð"

w

B134



1
%34

A3
"

w

B#34



#
Ñ
%34

#
1
1
œ T Š%34
 %34
  " w ÐB#34  B34
Ñ‹
1
œ F’  " w ÐB#34  B34
ÑÎÈ#5% “

#
1
and similarly for alternative 2, where È#5% is the standard deviation of %34
 %34
and
F( † ) is the univariate standard normal cumulative distribution function. Note that
equation A2 comprises the usual probit model for dichotomous choice under the
assumption the individual knows the random component and maximizes utility. The
parameter vector " is identified only up to the scale factor È#5% , and 5% is not identified,
since only the sign and not the scale of the dependent variable (the utility difference) is
observed. If the N observations for each respondent are simply “stacked” to produce a
data set with N 8 observations, the unit of observation is an 3ß 4 pair and the likelihood is
the product of the N 8 probabilities like equation A2:
5
œ $$ T3434 .
8

PÐ534 ß 3 œ "ß á ß 8ß 4 œ

#
"ß á ß N lB"34 ß B34
à " ß 5% Ñ

N

A4

3œ" 4œ"

Incorporating the Status Quo Question
After choosing 534 , individuals answer a question stating whether alternative 534 would be
chosen over the status quo. Let the status quo be indicated by 0. There are now four
kinds of observationsÞ Let the binary variable ^34" indicate the choice of alternative 1 or 2
1This

notation, especially the use of 534 to indicate either a 1 or a 2, is a bit cumbersome at first, but will
make precise many of the concepts below.
2We allow for correlation of errors for an individual when it comes to choices involving the status quo–see
section 3.2. For the hypothetical choices, there is no question of correlation since the effective errors that
enter the likelihood are the difference in the two errors for any choice occasion, and the attribute sets are
randomly assigned to choice “A” or choice “B”. That is, the relevant distribution theory for forming the
"
#
 %3"
likelihood is based on %3"
, for example (person 3, first choice occasion–see equation A7). In addition,
any additive systematic component of the error is then eliminated. This is similar to the arguments of
Heckman and Robb (1985) in their evaluation of social interventions.

for individual 3 on occasion 4, and let the binary variable ^34# indicate the chosen
alternative or the status quo. These are defined by:
^34" œ œ

!
1

^34# œ œ

choose 1
choose 2

! choose 1 or 2 over status quo
1 choose status quo over 1 or 2

A5

Note that there is an information asymmetry here: when the status quo is chosen over 1 or
2 (^34# œ "), a complete ranking of the three alternatives has been determined; when 1 or
2 is chosen over the status quo (^34# œ !), all that is known is that 1 or 2 is the most
preferred alternative.
Utility for the status quo, Y3! under the model assumption (equation A1) is given
by:
Y3! œ " w B!3  %3! ß

A6

where %3! are disturbances and B! are the attributes of the individual's current Internet
access. The attributes of the status quo vary over individuals, but not over choice
occasions, and the utility of the status quo is evaluated only once by each individual (Y3!
and %3! are subscripted with 3 only). The %3! are assumed to be independent, identically
distributed normal random variables with zero expectation and variance 5!# , uncorrelated
5
with %3434 .
The probability of choosing alternative 534 Ð"ß #Ñ over alternative $  534 Ð#ß "Ñ
and then choosing alternative 534 over the status quo (^34# œ !) is the bivariate
probability:
5

$534

T ÐY3434  Y34
œ T Š%34

$534

5

ß Y3434  Y3! Ñ

5

$534

 %3434   " w ÐB34

œ F# ’  " w ÐB34

$534

 B3434 Ñß %3!  %3434   " w ÐB!  B3434 Ñ‹
5

5

A7

5

 B3434 ÑÎÈ#5% ß  " w ÐB!  B3434 ÑÎÉ50#  5%# à 3“
5

5

$534

where 3 is the correlation between %34
3œ

5

5

 %3434 and %3!  %3434 ,

5%#
5%
œ
,
#
#
#
È#5% Ð50  5% Ñ
È#Ð50#  5%# Ñ

A8

and F# is the standard bivariate normal cumulative distribution function. Similarly, the
probability of choosing alternative 534 over alternative $  534 and then choosing the
status quo over alternative 534 (^34# œ ") is:
5

$534

T ÐY3434  Y34
œ T Š%34

$534

5

ß Y3434  Y3! Ñ

5

$534

 %3434   " w ÐB34

œ F# ’  " w ÐB34

$534

 B3434 Ñß %3!  %3434   " w ÐB!  B3434 Ñ‹
5

5

 B3434 ÑÎÈ#5% ß " w ÐB!  B3434 ÑÎÉ50#  5%# à  3“
5

5

where the symmetry of the normal distribution has been utilized.

5

A9

One normalization is required: let 5% œ "ÎÈ#. Define -# œ 5!# /5%# œ #5!# Þ Then
equation A8 can be written as:
5

$534

T ÐY3434  Y34
F# ’  "

5

ß Y3434  Y3! Ñ œ

A8w

" w ÐB!  B3434 Ñ
"
à
“
ÈÐ"  -# ÑÎ#
È#Ð"  -# Ñ
5

w

$5
ÐB34 34



5
B3434 Ñß

and similarly for equation A6. The additional parameter to be estimated is -. When
- œ ", 5%# œ 5!# and the A versus B question and the question comparing A or B to the
status quo have equal weight in the likelihood. When -  " the question relating to the
status quo contains more information, as there is more variability in the errors for the A
:
<
vs. B question (5%#  5!# Ñ, and conversely. Let B<:
34 œ ÐB34  B34 Ñ for <ß : œ !ß ". Then
the probabilities of the four data types are:
w !"
T Ð^34" œ !ß ^34# œ !Ñ œ F# ’  " w B#"
34 ß  " B34 Î-à

"
“
#"
w !"
T Ð^34" œ !ß ^34# œ "Ñ œ F# ’  " w B#"
“
34 ß " B34 Î-à 
#"
w !#
T Ð^34" œ "ß ^34# œ !Ñ œ F# ’" w B#"
“
34 ß  " B34 Î-à
#"
w !#
T Ð^34" œ "ß ^34# œ "Ñ œ F# ’" w B#"
“
34 ß " B34 Î-à 
#-

A10

The likelihood is the product of these N 8 probabilities:
œ $$T Ð^34" ß ^34# Ñ A11
8

PÐ^34" ß ^34# ß

3 œ "ß á ß 8ß 4 œ

"
#
"ß á ß N |B34
ß B34
ß B! à " ß -Ñ

N

3œ" 4œ"

which, upon substitution of equations 9 can be written
"
#
PÐ^34" ß ^34# ß 3 œ "ß á ß 8ß 4 œ "ß á ß N |B34
ß B34
ß B! à " ß -Ñ œ

A12

!"
!#
"^34
$$F# œÐ  "Ñ"^34 " w B#"
 ^34" " w B34
’Ð"  ^34" Ñ" w B34
“Î-à Ð  "Ñ^34
34 ß Ð  "Ñ
8

N

"

#

#

3œ" 4œ"

"
-

References
Heckman J. J., and R. Robb (1985). Alternative methods for evaluating the impact of
interventions. In Longitudinal Analysis of Labor Market Data (eds. J. J. Heckman and B.
Singer), 156-245. Cambridge: Cambridge University Press.
Heim, B. and Meyer, B. (2004), “Work Costs and Nonconvex Preferences in the
Estimation of Labor Supply Models,” Journal of Public Economics, 88, 2323-2338.

Appendix B
Estimating the standard error of WTP measures
from discrete choice experiments

Ignoring interactions, the utility model for Internet access choice is
‡

Y34 œ ": :34  \34w "+  "= ,34  %34 ß 3 œ "ß á ß 8à 4 œ "ß á ß ).

B1

where :34 is price, ,34 is bandwidth, and "+ is a O ‚ " vector of attributes of the service
s + /s
" : and
other than price and bandwidth. The estimates of WTP for these attributes are "
s = Îs
the estimated WTP for bandwidth is A
": .
s, œ "
Since the estimates of willingness-to-pay are nonlinear function of parameter
estimates, their exact standard errors are unknown. While it would be possible to
bootstrap the distribution of these estimators, since the normally distributed estimator of
": is the denominator, the simulation would not converge to anything useful (see Kling
and Sexton, 1990; Morey and Waldman, 1994). Instead, we use a linear approximation
to the variance (sometimes known as the “delta method”). This approximation for
elasticities has been examined in Krinsky and Robb (1986).
Define the ÐO  "Ñ ‚ " vector
s +ã s
A
" = Ñ /s
": .
sœÐ"

B2

s w+ ã s
s be the
Define the ÐO  #Ñ ‚ " vector of parameter estimates s
) œ Šs
": ã "
" = ‹ . Let D
estimated variance-covariance matrix of s
). The linear approximation to the variance of
A
s is
w

`A w s `A
s ÐA
Z
“ D’
“
sÑ ¸ ’
`)
`)

B3

where the derivatives are evaluated at the parameter estimates. The square root of the
s ÐA
diagonal elements of Z
sÑ are the estimated standard errors of the estimates of WTP.
These derivatives are
Î  +#"
s
":
Ð
Ð
s
Ð  "+##
Ð
s
":
`A
s Ð
Ð
œÐ ã
`s
)
Ð
s
Ð  "+#O
Ð
s
":
Ð
s
"=
Ï  s#
s
"

":

"
s
":

!

!

á

!

"
s:
"

!

á

ã

ã

!

!

!

á

!

!

!

á

!Ñ
Ó
Ó
!Ó
Ó
Ó
ã Ó
Ó
Ó
!Ó
Ó
Ó

"
s
":

Ò

B4

Focusing on bandwidth, the estimated variance of the WTP for bandwidth from equation
B2 is
Z A
s= œ Š

% ‹ s ::
s
":

s #=
"

5

 #Š

s :=
$ ‹5
s
":
s
"=



"
#
s
":

5
s ==

The utility model for access, with interactions, is
‡

Y34 œ ": :34  \34w "  Ð"=  +3w $ Ñ,34  %34 ß 3 œ "ß á ß 8à 4 œ "ß á ß )ß

B5

where +3 is a vector of P demographic variables for individual 3 and the elements of $ are
additional parameters to be estimated. The estimate of WTP for bandwidth from this
model is
A
" =  +w3s$ Ñ/s
":
s = œ Ðs

B6

where the vector of individual-specific demographic variables is evaluated at their means.
w w
s: ã s
s œ Š"
s ‹ œ D‡ Þ The variance of A
Define 9
" = ã s$ ‹ , and define Z Š9
s= is
`A
s= s‡ `A
s=
s ÐA
Z
“D ’
“
s= Ñ ¸ ’
`9
`9
w

B7

where
`A
s=
œŒ 
`9

s
" = +w3s
$
#
s
":

"
s
":

+"
s
":

+#
s
":

á

+P
s
":

w

.

B8

Reference:
Kling, C., and R. Sexton (1990). “Bootstrapping in Applied Welfare Analysis.” American
Journal of Agricultural Economics 72: p.
Krinsky, I., and A. Robb (1986). “On Approximating the Statistical Properties of
Elasticities.” Review of Economics and Statistics 68(4): p. 715-19.
Morey, E., and D. Waldman (1994). “Functional Form and the Statistical Properties of
Welfare Measures–A Comment.” American Journal of Agricultural Economics 76(4): p.
954-57.

Appendix C
Details on the study design: within subjects

The likelihood as it is written in equation A12 does not take into consideration the
fact that the formation of that part of the likelihood involving the comparison of the
5
chosen alternative to the status quo involves the error difference %3!  %3434 , where 534 œ "
or # (depending upon the choice), and from choice occasion to choice occasion these
error differences are correlated. This correlation is induced by the common occurrence
of %3! , since respondents need evaluate their utility of the status quo only onceÞ This point
is generally missed in conjoint analysis. An econometric innovation of this study is to
treat the person, and not the person-choice occasion, as the unit of observation, so that we
may explicitly model this correlation. The likelihood is now written
"
#
PÐ^34" ß ^34# ß 3 œ "ß á ß 8ß 4 œ "ß á ß N |B34
ß B34
ß B! à " ß -Ñ œ

"
#
$T Ð^3"" ß ^3"# ß ^3#" ß ^3## ß á ß ^3N
ß ^3N
Ñ .

C1

8

3œ"

The probability in equation C1 would appear to be computationally intractable, as it
involves a 16-fold (# ‚ N œ )Ñ integration of the multivariate normal density function.
"
#
Fortunately, this is not the case, as the correlation between %3!  %34
and %3!  %34
, for
!
example, is a result of the common occurrence of %3 . This means that we can follow a
familiar conditioning argument to express the probability in equation C1 as the integral
of the product of eight bivariate probabilities, integrated against the univariate normal
density (see Waldman, 1985). But the cost of this generality is in programming and
computer time, as the likelihood must be maximized by simulation or with quadrature
methods. We used Hermite polynomial quadrature (Abramowitz and Stegun, 1964, p.
890).

References
Abramowich, M., and Stegun, J. (1964). "Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables." National Bureau of Standards, Applied
Mathematics Series - 55.
Waldman, Donald M., 1985. "Computation in Duration Models with Heterogeneity."
Journal of Econometrics, Vol. 28, 127-134.