estimation of willingness to pay in preference space …/file/...estimation of willingness to pay in...

Estimation of willingness to pay in preferencespace vs. WTP space

Arne Risa HoleUniversity of She¢ eld

Danish Choice Modelling Day - Odense 8th December 2011

Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space

Background

Increased use of mixed logit to estimate willingness to pay inapplied economics

Standard approach: assume a distribution for the coe¢ cientsand derive WTP for an attribute as the ratio of the attributecoe¢ cient to an estimate of the marginal utility of money

Can lead to WTP distributions which are heavily skewed andthat may not even have de�ned moments

Train and Weeks (2005) suggested re-formulating the modelsuch that assumptions are made regarding the distribution ofWTP

Motivating example: health workers choice betweenhypothetical jobs


Utility in preference space

The utility person n derives from choosing job j in choicesituation t is speci�ed as

eUnjt = αnwnjt + βnxnjt + εnjt/σn (1)

εnjt is assumed to be extreme value distributed with scale σn

Muliplying through by σn yields

Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt (2)

It is not possible to separately identify σn and αn/βn !standard practice to normalise σn to 1. Assumes εnjt ishomoscedastic (more on that later).


Utility in willingness to pay space

Train and Weeks (2005) suggest rewriting equation (2) as

Unjt = αn [wnjt + γnxnjt ] + εnjt (3)

Uses the fact that the WTP for the attributes is given by

γn = βn/αn

The models are behaviourally equivalent but standardassumptions regarding the distributions of αn and βn in (2)can lead to unusual distributions for WTP

The WTP space approach avoids this problem by specifyingthe distributions for WTP directly


The Generalized Multinomial Logit Model

Let�s go back to the utility function in preference space

Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt

Fiebig et al. (2010) propose that instead of imposing theσn = 1 normalisation σn speci�ed as

exp(σ+ θzn + τε0n)

where ε0n � N(0, 1) and zn is a vector of characteristics ofperson n

Relaxes the assumption of homoscedastic errors

Since only relative scale di¤erences can be identi�ed σ is setto �τ2/2 which implies that E (σn) = 1 when θ = 0

This speci�cation is called GMNL-II by Fiebig et al.


Hensher and Greene (2010) show that the GMNL model neststhe preference space and WTP space models

It is easy to see that the GMNL model reduces to thepreference space model when τ = θ = 0

We can also see that

Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt

= σn [αnwnjt + βnxnjt ] + εnjt

becomes the WTP space model when αn = 1

σn = exp(σ+ θzn + τε0n) with zn = 1 can then beinterpreted as the (log-normally distributed) wage parameter

βn can be interpreted as the WTP estimates


GMNL Stata command

Developed in collaboration with Yuanyuan Gu and StephanieKnox at the University of Technology, Sydney

Can be used to estimate models in WTP space and allvariants of the GMNL model described in Fiebig et al.

Postestimation commands for generating predictedprobabilities, individual-level parameter estimates etc.

The module, including an example dataset and a workingpaper is available at

http://www.shef.ac.uk/economics/people/hole/stata

Comments and suggestions are very welcome


Example: Households�choice of electricity supplier

Subset of the data from Huber and Train (2000)

Residential electricity customers presented with a series ofexperiments with four alternative electricity suppliers

Price is either �xed or a variable rate that depends on thetime of day or the season

The following attributes are included in the experiment:

Price in cents per kWh if �xed price, 0 if TOD or seasonal ratesContract length in yearsWhether a local company (0-1 dummy)Whether a well-known company (0-1 dummy)TOD rates (0-1 dummy)Seasonal rates (0-1 dummy)


First 16 records in dataset

. use http://fmwww.bc.edu/repec/bocode/t/traindata.dta, clear

. list in 1/12, sepby(gid)

++| y price contract local wknown tod seasonal gid pid |||

1. | 0 7 5 0 1 0 0 1 1 |2. | 0 9 1 1 0 0 0 1 1 |3. | 0 0 0 0 0 0 1 1 1 |4. | 1 0 5 0 1 1 0 1 1 |

||5. | 0 7 0 0 1 0 0 2 1 |6. | 0 9 5 0 1 0 0 2 1 |7. | 1 0 1 1 0 1 0 2 1 |8. | 0 0 5 0 0 0 1 2 1 |

||9. | 0 9 5 0 0 0 0 3 1 |

10. | 0 7 1 0 1 0 0 3 1 |11. | 0 0 0 0 1 1 0 3 1 |12. | 1 0 0 1 0 0 1 3 1 |

++


Model with �xed coe¢ cients (MNL)

. clogit y price contract local wknown tod seasonal, group(gid)

Iteration 0: log likelihood = 1379.3159(output omitted)Iteration 4: log likelihood = 1356.3867

Conditional (fixedeffects) logistic regression Number of obs = 4780LR chi2(6) = 600.47Prob > chi2 = 0.0000

Log likelihood = 1356.3867 Pseudo R2 = 0.1812

y | Coef. Std. Err. z P>|z| [95% Conf. Interval]

+price | .6354853 .0439523 14.46 0.000 .7216302 .5493403

contract | .13964 .0161887 8.63 0.000 .1713693 .1079107local | 1.430578 .0963826 14.84 0.000 1.241672 1.619485wknown | 1.054535 .086482 12.19 0.000 .8850338 1.224037

tod | 5.698954 .3494016 16.31 0.000 6.383769 5.01414seasonal | 5.899944 .35485 16.63 0.000 6.595437 5.204451


Willingness to pay estimates

Based on the MNL results we can calculate the WTPestimates using the Stata command wtp:

. wtp price contract local wknown tod seasonal

contract local wknown tod seasonalwtp .21973759 2.2511589 1.6594175 8.9678781 9.2841551ll .27319536 1.8855365 1.3653397 9.2764201 9.6128415ul .16627982 2.6167813 1.9534953 8.6593361 8.9554687


Model with �xed coe¢ cients in WTP space

. gen mprice = price

. gen const = 1

. constraint 1 [Mean]mprice = 1

. constraint 2 [tau]_cons = 0

. matrix start = 1,0,0,0,0,0,0,0

. gmnl y mprice contract local wknown tod seasonal, group(gid) id(pid) het(const)constraint(1 2) from(start, copy) nrep(1)


Iteration 0: log likelihood = 6814.2393 (not concave)(output omitted)Iteration 14: log likelihood = 1356.3867

Generalized multinomial logit model Number of obs = 4780Wald chi2(5) = 5133.89

Log likelihood = 1356.3867 Prob > chi2 = 0.0000

( 1) [Mean]mprice = 1( 2) [tau]_cons = 0

(Std. Err. adjusted for clustering on pid)

y | Coef. Std. Err. z P>|z| [95% Conf. Interval]+Mean |

mprice | 1 . . . . .contract | .2197376 .0272749 8.06 0.000 .2731953 .1662798

local | 2.251159 .1865454 12.07 0.000 1.885537 2.616781wknown | 1.659417 .1500424 11.06 0.000 1.36534 1.953495

tod | 8.967877 .1574222 56.97 0.000 9.276419 8.659336seasonal | 9.284155 .1677001 55.36 0.000 9.612841 8.955469

+Het |

const | .4533659 .0691633 6.56 0.000 .5889235 .3178082+

/tau | (omitted)The sign of the estimated standard deviations is irrelevant: interpret them asbeing positive


The price coe¢ cient is given by

. nlcom (price: exp([Het]const))

price: exp([Het]const)


+price | .6354856 .0439523 14.46 0.000 .7216305 .5493407


Model with random coe¢ cients in WTP space

. matrix start = e(b),0.1,0.1,0.1,0.1,0.1

. gmnl y mprice, group(gid) id(pid) rand(contract local wknown tod seasonal)het(const) constraint(1) from(start, copy) nrep(100) gamma(0)


Iteration 0: log likelihood = 1352.9782 (not concave)(output omitted)Iteration 11: log likelihood = 1122.3142

Generalized multinomial logit model Number of obs = 4780Wald chi2(5) = 2303.48

Log likelihood = 1122.3142 Prob > chi2 = 0.0000

( 1) [Mean]mprice = 1(Std. Err. adjusted for clustering on pid)


+Mean |

mprice | 1 . . . . .contract | .2535117 .0562202 4.51 0.000 .3637012 .1433221

local | 2.09998 .2201434 9.54 0.000 1.668507 2.531454wknown | 1.535489 .1582956 9.70 0.000 1.225235 1.845743

tod | 9.455161 .2898365 32.62 0.000 10.02323 8.887092seasonal | 9.379777 .2524339 37.16 0.000 9.874539 8.885016

+Het |

const | .0332425 .0856309 0.39 0.698 .201076 .1345909+SD |

contract | .4615008 .0593945 7.77 0.000 .5779119 .3450897local | 1.706872 .2254088 7.57 0.000 1.265079 2.148665wknown | 1.126835 .1862598 6.05 0.000 .7617721 1.491897

tod | 2.398136 .3486618 6.88 0.000 3.081501 1.714771seasonal | 2.205005 .329771 6.69 0.000 2.851344 1.558666

+/tau | .3172959 .1406606 2.26 0.024 .0416063 .5929856

The sign of the estimated standard deviations is irrelevant: interpret them asbeing positive


Price is assumed to be log-normally distributed. The mean ofthe underlying normal distribution is

. nlcom (price_mean: [Het]const[tau]_cons^2/2)

price_mean: [Het]const[tau]_cons^2/2


+price_mean | .0835809 .0715274 1.17 0.243 .223772 .0566102

The "-[tau]_cons^2/2" term is due to the normalisation ofsigma in the GMNL model


Example: Health workers choice of job

Joint work with Julie Riise Kolstad at the University of Bergen

Discrete choice experiment on the choice of health servicejobs among Tanzanian �nal-year students training to beClinical O¢ cers (COs)

The aim of the experiment was to elicit the students�preferences for di¤erent features of health service jobs

320 �nalists (around 60% of all CO �nalists in Tanzania in2007) from 10 randomly selected schools participated in theDCE

After excluding incomplete responses we were left with anestimation sample of 296 respondents


The attributes in the choice experiment were chosen followingextensive literature searches and early in-depth interviews

The attributes included the wage of the job, educationprospects and other characteristics related to the location ofthe job and the facilities of the workplace

We used a D-optimal design to construct the hypotheticalchoice situations

The result was a set of 32 choice situations that wererandomly divided into two blocks

Each respondent was presented with 16 choice situationswhere each of these represented the choice between twohypothetical jobs


An example choice set

Job AAvailability ofequipment &drugs:

Housing: Education opportunities/possibility of upgradingqualifications:

Workload: Infrastructure: Salary andallowances:

Location:

Sufficient No house isprovided.

Education offered after 6 yearsof service.

Normal: Nearly enough time tocomplete duties. One hour ofextra work per day.

The place has mobilecoverage, electricity andwater.

350,000TSH permonth

Regionalheadquarters

Job BAvailability ofequipment &drugs:

Housing: Education opportunities/possibility of upgradingqualifications:

Workload: Infrastructure: Salary andallowances:

Location:

Insufficient A decenthouse isprovided.

Education offered after 2 yearsof service.

Heavy; barely enough time tocomplete duties. Three hours ofextra work per day.

The place has unreliablemobile coverage, noelectricity or water.

500,000TSH permonth

A 3hour or morebus ride from thedistrictheadquarters

Considering your current situation, which of the two jobs would you choose?

Job A: Job B:


Model speci�cation

We estimated six di¤erent choice models:

1: Logit2: ML with �xed wage coef.3: ML pref. space, all coefs. random4: ML pref. space, all coefs. random and some correlated5: ML WTP space, all coefs. random6: ML WTP space, all coefs. random and some correlated

Coe¢ cients for wage, education, infrastructure and equipmentare given log-normal distributions, all other coefs. arenormally distributed

We use 1000 Halton draws to estimate the ML models withindependent coefs. and 2500 draws for the models withcorrelated coefs.


Key �ndings

Evidence of substantial amount of heterogeneity in preferencesfor wage, further education, equipment, infrastructure andworkload

Suggests that model 2 (�xed wage coef.) is too restrictive

Mean WTP estimates derived from pref. space models arevery high:

1 2 3 4 5 6

Education 2yrs 357 416 849 561 390 369Infrastructure 222 237 466 353 205 262


WTP for further education after 2 years

0.0

005

.001

.001

5.0

02.0

025

Den

sity

0 500 1000 1500WTP (1000 TSH per month)

Model 3 Model 4Model 5 Model 6


WTP for infrastructure

0.0

01.0

02.0

03.0

04.0

05D

ensit

y

0 200 400 600 800 1000WTP (1000 TSH per month)

Model 3 Model 4Model 5 Model 6


Goodness of �t

1 2 3 4 5 6

LL -2424 -2335 -2267 -2226 �2278 -2228AIC 4872 4715 4580 4499 4601 4501BIC 4950 4857 4728 4647 4750 4650

Preference space models �t data better

But: bigger di¤erence between models allowing/not allowingfor preference heterogeneity in wages and correlated vs.uncorrelated coe¢ cients


Conclusions

Although it is common in practice to specify that thecoe¢ cient for the monetary attribute is �xed, this may beunrealistic

Allowing for preference heterogeneity is not straightforward asit can lead to implausible WTP estimates

In line with the literature from other �elds our evidencesuggest that models estimated in WTP space produce morerealistic WTP estimates

Preference space models �t data better but best �ttingmodels in the two regimes have similar GOF

Our results suggest that sensitivity testing is important


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