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.