valuing changes in environmental amenities when the amenity is a quality characteristic of a...
Post on 22-Dec-2015
223 views
TRANSCRIPT
Valuing Changes in Environmental
Amenities
• When the amenity is a quality characteristic of a privately consumed good
• The good’s price is not affected by quality
Exploiting “weak complementarity”
A privately purchased good (some qi) and the environmental good (b) are “weakly complementary” if individuals don’t care about b if they don’t purchase qi.
Weak complementarity..
often occurs in cases where b is a quality characteristic of the q
Example:q = recreation trips to a lake
b = water quality in that lake
And individuals only care about the water quality in the lake if they use the lake for recreation (i.e. q>0)
Why does this help us?Area between two (compensated)
demand curves for good i is:
Price
Quantity
)(~ 1bpi
)(~ 0bpi
0ip
qih(p,b0,U0)
qih(p,b1,U0)
)(~ 00
)(~ 01
0
0
1
0
),,(
),,(
bp
p ii
bp
p ii
i
i
i
i
dpp
Ubpm
dpp
Ubpm
i
i
i
i
p
p ii
p
p ii
dpp
Ubpm
dpp
Ubpm
~ 00
~ 01
0
0
),,(
),,(
=
)],,,(),,,~([
)],,,(),,,~([0000000
0100010
UbppmUbppm
UbppmUbppm
iiii
iiii
If weak complementarity between b and qi holds then 0),,,~(),,,~( 000010 UbppmUbppm iiii
Area between demands = CV =
),,,(),,,( 01000000 UbppmUbppm iiii
Can we actually find purchased goods that
relate to environmental quality in this way?
If price is bid up on units of qi with higher levels of environmental quality, then another model is appropriate (the hedonic model).
This analysis requires that price not be a function of b.
Possible example that fits:
Water quality of regulated public water suppliers.
Household Production Framework – a better fit
Suppose environmental quality is a quality dimension of a household produced good.
The maximization decision is now: max U(q,z(x),b) + (y-pq-rx)
Where z is a household produced good x is a vector of purchased inputs r is the vector of prices of x b matters only if z “produced”
In HPF framework
• If z has a constant marginal cost of production (treat as “price”) then apply previous results
• If not, then need to be creative… results exist that use areas between demands for an essential input into the production of z
Implementation Problem
How do we observe behavior in the face of varying levels of environmental quality?
Can not usually observe demand for a single site over time as quality varies.
Often, the best we can do is look at choices across sites with varying quality.
Traditional Travel Cost Model of a Single Site
Conceptually, can value existence of site:
trips
cost Individual’s lost consumer surplus from closing this site
Conceptually, can value quality change at site:
Individual’s demand function for trips
cost
trips
ci0
zi0
ci0
zi0 zi
1
demand after quality improvement
original demand function for trips
Individual’s consumer surplus from improvement in site
Traditional Travel Cost Model
• Difficult to adequately account for substitutes.
• Difficult to capture changes in behavior in the face of changing environmental quality
Correlation of Substitute Prices
Costs of access are often correlated across sites
imprecise estimates of cost coefficient.
Consumer surplus estimate is
dependent on the cost parameter. e.g. linear demand: CS = -zi
2/2c
semilog demand: CS = -zi/c
A Single Site Demand Function with Substitute Costs and Quality
AiCABAAA
CiABiAAiAiAAi
bbb
cccz
654
3210
zji = trips to site j by individual icji = costs to site j by individual ibj = quality at site j
The Remaining Two Equations in this System of Demands
CiCCBCAC
CiCBiCAiCiCCi
BiCBBBAB
CiBBiBAiBiBBi
bbb
cccz
bbb
cccz
654
3210
654
3210
Problem with the System of Demand Equations:
If the bA, bB, and bC are objective measures of site quality, then the will not vary over individuals in the sample.*
*(sometimes researchers try to use perceptions for the b’s.)
With no variation in these variables, coefficients can not be estimated.
Another problem
The costs of access are often correlated across sites, making precise estimation of the “price” coefficients difficult as well.
The Random Utility Model
has become the most popular model for modeling the choice among a finite set of alternatives with varying prices and qualities.
Many, many types of applications. We will look at:
recreational demand application
commercial fisheries application
What’s the Logic Behind the RUM?
On a given occasion in which the individual makes a choice among the finite set of substitutes available to him,
he does so by choosing alternative j among the M alternatives, such that :
U(j) =
max U(m) for all m=1,…,M
This is a simple expression for the decision maker.
But, the researcher does not observe utility nor does he observe all factors that affect utility.
So he frames the problem as a stochastic one:
Pr (individual i chooses alternative j) =
imii
ii
mVmUwhere
MmmUjU
)()(
],...,1)()(Pr[
What does this systematic portion of the utility function contain?
Vi(m) is the “utility on the choice occasion, conditional on choosing alternative m.
It is usually specified linearly as:
.......)~()( 21 mimii bcymV
mb
msic
siy
m
im
i
of sticcharacteriquality
accessingofcost'
occasionchoice
thewithassociatedincome'~
Conditional Logitvs Multinomial Logit
The Random Utility Model is McFadden’s Conditional Logit.
The Multinomial Logit is a related model in which the explanatory variables are individual, rather than alternative, characteristics.
In the MNL, different coefficients are estimated for different alternatives, normalizing on one alternative.
An Example of the Difference in Models
Suppose you wished to model individuals’ occupation choice:
• MNL: Models the choice as a function of the individual’s characteristics (e.g. age, education, parents’ education…)
• RUM: Models the choice as a function of the characteristics of available jobs (e.g. wage rate, education required, vacation days…)
The Form of the Conditional Logit (or RUM)
If im is distributed according to a Type I Extreme Value Distribution
(usual assumption),
Then the probability that individual i will choose alternative j is given by:
(Note: this is the individual’s contribution to the likelihood function.)
M
mmimi
jijii
bcy
bcyj
121
21
...])(exp[
...])(exp[)(Pr
An Important Specification Issue in the
RUM
Mathematically, the expression is equivalent to:
If a variable does not vary over alternatives, it falls out of the specification.
M
mjmijiimi
i
bbcycyj
121 ...]}{)}(){(exp[
1)(Pr
• In the current form, income falls out of the specification.(You can introduce income term
non-linearly, but welfare difficult to calculate).
• Individual characteristics will drop out as well. (You can include them “crossed” with site specific or other site varying variables.)
• There is no constant term
in the model.
(You can include site specific constants
in a similar manner to including dummy
variables in regression.)
The Role of Income
We saw that income falls out of the model in the linear form.
Question: do you expect income to matter in a choice among these substitute sites?
Often, it does not.
Question: do you know how to measure income associated with a “choice occasion”?
It’s a difficult concept.
You can easily include
non-linearly in model, but CV measure is very difficult to calculate.
imi cy ~
Welfare Measurement Using the Simple RUM
If valuing a quality change at one or more sites:
1
1
021
1
1
121
...]}exp[ln{
...]}exp[ln{
M
mmim
M
mmim
bc
bcCV
Where b0 is the original level of quality and b1 is the subsequent level of quality.
For example, suppose site 1 is eliminated, then the CV measure is simply:
1
121
1
221
...]}exp[ln{
...]}exp[ln{
M
mmim
M
mmim
bc
bcCV
We can also use the RUM to value the loss
of a site.
An Increasingly Complex Example for
Illustration
Starting Simply….
Choice variable:
Choice of beach in St Lucia, by residents of St. Lucia.
Site attributes include:
monetary travel cost to each site
time cost to each site
beach size
Choice Set: the alternatives that an
individual views as feasible
Consequences of mis-specifying the choice set:
• The welfare estimates will be incorrectly calculated
e.g. suppose too many substitutes are
included so reduced quality at a site
or loss of a site is undervalued
• The parameter estimates may
be inconsistent
e.g. suppose there is an alternative of
high quality, but the individual
doesn’t know about it.
How Does One Determine What the Relevant Choice Set Should Be?
Options in the literature:
Distance based choice sets(Researcher defines)
Parsons and Hauber
Familiarity based choice sets(Respondents define) Hicks and Strand
Peters, Adamowicz and Boxall
Endogenous choice sets(Data define)
Manski Swait and Ben-Akiva
What if You Have Large Number of Alternatives?
Sampling Alternatives
Estimates less efficient but consistent
Welfare calculations require all alternatives
Grouping Alternatives
Deal with a smaller number of geographically aggregated sites
If large variation in number of “elemental sites” across groups, then adopt Ben-Akiva and Lerman approach.
Coeff. Std.Err. t-ratio P-value
TIMECOST -1.861 .534 -3.486 .00049
MONEYCOST .203 .069 2.907 .00365
BEACHSIZE .0004 .0002 1.845 .06503
Estimating a Simple Model Using theSt. Lucia Data
5 alternative sites; 70 individuals
Estimation performed using LIMDEP,software package of William Greene, Columbia University.
Interpretation of Coefficients in RUMs
ttimejk
ttimej
ttime
ttime
j
jk
jksite
kjforkj
kjforkj
ksite
cos
cos
cos
cos
]timecost-timecost[
)]Pr(/)ln[Pr(
)Pr(timecost
)choosingPr(ln
)Pr(*))Pr(1(
)Pr(*)Pr(
timecost
)choosingPr(
Not especially intuitive…..
Evaluating some marginal effects…
Example: what’s the effect of an increase in the time costs of accessing site 1 by one hour?
Choice=Site1 -26.153 -4.67
Choice=Site2 6.168 .572
Choice=Site3 5.506 .572
Choice=Site4 6.068 .572
Choice=Site5 8.411 .572
Change inProbability
Elasticity
LIMDEP will calculate these for you.
ST1 ST2 ST3 ST4 ST5
ST1 5 2 2 2 4
ST2 5 3 3 4 5
ST3 1 1 3 3 1
ST4 1 1 2 2 2
ST5 2 2 2 3 10
Actual vs Predicted Choices
ACTUAL
P R E D IC T E D
23 of 70 choice predicted correctly
Choice-Based Sampling
Random utility models depend on a
random sample of the population.
Parameter estimates are based on
the proportion of individuals who
choose different sites (conditioned
on their explanatory variables).
If you sample on-site, you tamper with this relationship and bias the estimated parameters.
Choice-Based Sampling is Sometimes the Only
Feasible Survey Approach
Solution to Problem:
Reweighting to correct for choice-based sampling is possible if you know the following:
sample proportion of interviews at site j
population proportion of trips to site j
j
js j sitetotripsofnumber
siteatinterviewsofnumber
s
jw j sitealltotripsofnumber
sitetotripsofnumber
The Contribution to the Likelihood Function is Now…
Prob of intercepting individual i at site j=
k
ikk
ijj
kws
jws
)(Pr)(
)(Pr)(
kh
ih
ikkk
hih
ijjj
xxws
xx
ws
)}'exp()'exp()/({
)'exp()'exp(
)(
kkik
jij
xx
)'exp()'exp(
where k = ln(sk/wk)
The good news is….
If each site is randomly sampled and
you know the proportions of total trips
taken to each site (i.e. visitation rates),
then LIMDEP* will do the work for you.
Visitation rates can often be gotten independently either through site authorities or through a random telephone survey of the population.
How Much of A Difference Do Choice-
Based Sampling Corrections Make?
The St. Lucia study was really a choice-based sample.
Visitation rates - separate random telephone survey of the population.
Coeff. P-value
TIMECOST -3.35 .0000056
MONYCOST 0.318 .0000173
BEACHSZ 0.0012 .000000003
Coeff. P-value
-1.861 .00049
.203 .00365
.00036 .06503
Corrected Uncorrected
ST1 ST2 ST3 ST4 ST5
ST1 5 1 1 2 6
ST2 5 2 2 3 9
ST3 1 0 2 4 2
ST4 1 0 1 1 3
ST5 2 1 1 2 14
No Real Improvement in Percent Predicted Correctly
P R E D I C T E D
ACTUAL
24 of 70 predicted correctly
Assessment:
• Choice-based sampling correction changed results
• Problems remain:– Wrong sign on money costs– No real improvement in
proportion predicted correctly
• Something is still wrong
Suppose We Wanted to Do Welfare Measurement Using
this Simple RUM?
The general form of the WTP measure is:
WTPi = {E[max(Ui(final))]-E[max(Ui(initial))]}/Muy
where: E[max(Ui(initial))] =
M
mmimim btc
1
03
02
01 ]}exp[ln{
And, 1 is the marginal utility of income
If valuing a quality change at one or more
sites:
1
1
0321
1
1
1321
]}exp[ln{
]}exp[ln{
M
mmimim
M
mmimim
btc
btcWTP
Where b0 is the original level of quality and b1 is the subsequent level of quality and 1 is the –1*coefficient on money cost.
For example, suppose site 1 is eliminated, then the WTP measure is :
1
121
1
221
...]}exp[ln{
...]}exp[ln{
M
mmim
M
mmim
bc
bcWTP
We can also use the RUM to value the loss
of a site.
Again, 1 is coefficient on money cost.
Welfare Measurement with RUMs
• Unlike the linear or semi-log regression model, the WTP measure is a function of all parameters in the model.
• BUT, similar to the linear and semi-log regression models, WTP is very dependent on the money cost parameter.
• Our estimate of the money cost parameter has the wrong sign!
What’s the Cause of Our Difficulties in the St Lucia Model?
One Possibility:
We are not taking into account differing modes of transportation by different people – car, bus, walking, so costs are not really comparable.
Suppose we model the choice of
mode and the choice of site
simultaneously?
Coeff. P-value
TIMECOST -.476026 .00027
MONYCOST .019520 .58543
BEACHSZ .000338 .06473
Uncorrected for Choice-Based Sampling
Results from Including 15 Alternatives:5 sites*3 modes
Coeff. P-value
TIMECOST -.549212 1.00298e-012
MONYCOST -.087839 .00892492
BEACHSZ .001034 2.88658e-015
Corrected for Choice-Based Sampling
A Hypothetical Policy Evaluation
In an effort to raise funds to protect the beach, the authorities add a car parking fee of $5 (5.5 EURO or 2 DKK) for entrance to site 1.
Using our formula for WTP the ave. loss per individual per choice occasion is only $.25 (.27 EURO or 41 DKK) in this model.
There’s lots of opportunity for substitution in this model.
Why so small?
New Hypothetical Policy
Add a $5 parking fee at all sites.
Average welfare loss per individual per choice occasion is now $2.25
(about 2.5 EURO or 18.50 DKK)
Now, substitution is possible only through changing modes of transportation.