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### The Hedonic Pricing Method

Professor A. Markandya

Department of Economics and International Development University of Bath

tel. +44 1225 386954

Environmental Economics 2

March 2006

Hedonic pricing method

- Revealed preferences
- No questionnaire!
- We gather data that come from the market
- No need to build a hypothetical market
- Where do we decide to live?
- Why do we choose a specific location?
- Which factors push companies to choose one location rather than another?
- Which characteristics of an area affect housing prices?
- Which are the important elements of a house that determine its price?

The choice of localization

- The choice of housing is a composite good
- Distance from work, availability of public services, distance from schools, availability of green areas, availability of sport facilities, characteristics of housing (# of bedrooms, # of bathrooms, flat, detached, etc.) etc.
- We assume that buyers choose houses that maximize their utility
- The constraints in the maximization problem are given by income, the price of the houses and the level of taxes
- => therefore, the housing market give us some information on buyers preferences for housing and for their localization

Composite goods

- In the conjoint analysis and in the multi-site travel cost model we saw that goods (or sites) can be described by a set of attributes or characteristics.
- The hedonic pricing method uses the same idea that goods are composed by a set of characteristics.
- Consider the characteristics of a house:
- Number of floors, presence of a garden, GCH, number of bedrooms, number of bathrooms, square footage of the house, type of house, age, materials, etc.
- And also:
- Distance from public transport, distance from the city centre, distance from main roads, distance from shops, distance from sport facilities, crime rate, average income of inhabitants, presence of a university, etc.
- The composite good has a price, but there is no explicit price for each characteristic that compose the good.

The hedonic pricing method

- Problem of estimating hedonic equations
- Hedonic prices are identified through a comparison of similar goods that differ for the quality of one characteristic
- The basic idea is to use the systematic variation in the price of a good that can be explained by an environmental characteristic of the good. This is the starting point to assess the WTP for the environmental characteristic
- We look at market data!
- Real transactions!

Example

- Let’s consider 2 residential properties identical in all characteristics and localization.
- The only difference is that house A has 2 bedrooms, while house B has 3 bedrooms.
- In a competitive market, the price difference between the two houses reflects the value of the additional room of house B.
- If the price difference between the two houses is less than buyers’ WTP for the additional room, then buyers will try to buy house B, driving up its price until the equilibrium is reached.
- In the same way, if house A costs much less than house B, buyers will increase the demand for house A, driving up its price.

The hedonic pricing method applies this simple concept to the environmental characteristics of residential properties

- The price difference between houses that have different levels of environmental quality, keeping constant all other characteristics, reflects the WTP for the different level of environmental quality
- => we can assess the value of an environmental quality, according to market prices of residential properties
- => variation in environmental quality affects the price of housing

A bit of history

- 1926 Waugh studies the variation of prices of vegetables
- 1938 Court looks at the car market in Detroit
- 1967 first application to the housing market: Ridker and Henning => effects of air pollution on prices of housing
- 1974 Rosen describe the first formal model of the hedonic pricing method
- Other applications:
- Agricultural goods
- Cars
- Wine
- Job market

Rosen’s model

- Consumers (buyers) have a utility function:
- U(s,n,c)
- s = house characteristics
- n = characteristics of the area where the house is located
- c = other consumption goods
- Budget constraint:
- m = c + p(s,n)
- m = income
- p(s,n) expenditure for a house
- p(s,n) is assumed to change in a non linear relationship with the characteristics of houses. That is, the cost of houses change in an unknown relationship with number of rooms, etc.
- c is the expenditure for all other goods

The maximization of the utility function subject to the budget constraint, gives the usual first order conditions.

That is, the marginal rate of substitution between each characteristic n and the consumption of other goods is equal to the ‘price’ (coefficient) of n and the price of c.

- The price of c is our numeraire and we put it equal to 1.
- The price of n describes the price of a marginal change in n.
- The first order conditions are:
- (Un is the partial derivative of U with respect to n)
- First order conditions simply say that the consumer (buyer) is willing to pay pn for a marginal change of n

Utility maximization and budget constraint

- This looks like a normal example from your microeconomic class. We only add a non linear constraint for a given value of s, s*:

c

U(s*,n,c)

m=c+p(s*,n)

n

The hedonic price function

- The function that describes how housing price changes when housing characteristics change:

p(s,n)

is the hedonic price function

- The derivative of the function with respect to one of the characteristics n is the ‘implicit price’ of n.
- If we knew the hedonic price function and the implicit price of n, we could estimate buyers’ WTP for n, given that this is equal to the marginal rate of substitution between n and the other goods (numeraire)

Indifference curves

- The budget constraint says that what we don’t spend for other goods is spent for housing:

p(s,n): c = m – p(s,n)

- The utility function can be written in this way:
- U(s,n,c)=U(s,n,m – p(s,n))
- Therefore we can describe the utility function of consumers (buyers) with indifference curves (for given values of m and s):
- Each indifference curve gives for a constant level of utility the expenditure on housing and n for a given level of income and s.

p(s*.n)

U

n

Heterogeneous consumers

- People with different incomes have different indifference curves, even if they have the same preferences (U has the same functional form for all respondents)
- People with different preferences have different indifference curves
- In a world of heterogeneous consumers (buyers) that have different levels of income, we have a continuum of indifference curves:

p(s*.n)

n

Hedonic equilibrium

- Suppose that consumers (buyers) consider exogenous the hedonic price function
- Consumers (buyers) maximize utility subject to the budget constraint and to the hedonic price function:

Hedonic price function

p(s*.n)

n

Hedonic equilibrium considering the supply

- The hedonic price function comes from the equilibrium of demand and supply of housing. Both are considered exogenous.
- Sellers have isoprofit curves (π)

πb

p(s*.n)

Sellers

πa

Uk

Buyers

Ui

n

Marginal Willingness To Pay

- The main characteristic of the model is that buyers and sellers are efficiently matched along the hedonic price function
- At any point along the hedonic price function, buyers marginal willingness to pay (and sellers willingness to accept) for a change in n is given by the derivative of the hedonic price function with respect to n.
- This implicit price changes with n if the hedonic price function is non linear.
- The model can be generalized to the case where we consider several characteristics of residential properties and of the area where houses are located:

p(x1,x2,…xk)

Model estimate

- Now we need to specify a functional form for p.
- A common functional form is the semi-log:
- The coefficients of the regression function give the implicit price, in natural logarithm terms, of the characteristics of the house
- The implicit price can be estimated for specific value of the characteristics of houses (for example, the average value)
- For the semi-log function, the implicit price of x1 is given by:
- β1 gives the percentage change in the price of housing given a percentage change in x1
- We usually estimate the implicit price at the average value of housing

Some limitations and assumptions

- Perfect information:
- Buyers observe the characteristics of houses and are able to perfectly describe the hedonic price function
- Buyers can purchase whatever combination of characteristics they desire.
- They can always find the combination of bedrooms, bathrooms, location of the house that they want
- Implicit prices allow us only to assess marginal variations in the characteristics of houses (but if we consider that all buyers are identical then we can consider non marginal changes as well – too strong assumption!)
- Example: if the average house has 3 bedrooms and costs X, I cannot say that buyers are willing to pay Y for a house that has 7 bedrooms. We can’t say that an increase of 4 bedrooms is a marginal change
- The estimate of non-marginal variations requires the estimate of individual demand parameters, which is very difficult

Econometric problems

- Multicollinearity
- if a house has several bedrooms, it will likely have several bathrooms, etc.
- distances: don’t use too many distances in your function
- Heteroskedasticity
- Spatial autocorrelation
- The value of one house will be influenced by the value of surrounding houses
- Market extension: homogeneous markets => bias
- If I only use the data of sold properties and do not consider the characteristics of unsold properties, my coefficient can be biased (sample selection bias)
- Solution: 2 steps estimate 1) Probit model for the probability of a sale with both sold and unsold properties 2) regression model with only sold properties + Inverse Mills Ratio calculated in 1. Check if the coefficient of the inverse mills ratio is significantly different from zero. If it is not, then delete it from the regression

Sample selection problem

- Formally, we observe the sale of a property i only if the profit from the sale is greater than the discounted sum of profits from keeping the property. Therefore, the net profits from selling a property can be described by the following:
- where zi is a vector of site characteristics and location attributes, and ηi is the error term assumed to be i.i.d. standard normal. However, we only observe net profits when , that is only when a sale occurs. If the error terms in the above equation and in the hedonic equation are correlated, the coefficient estimates of the hedonic equation may be biased. The solution to this problem is to explicitly model the selection problem by conditioning the logarithm of price on the fact that a sale occurs. Indicating a sale by yi , we have that yi=1 if a sale occurs, and 0 otherwise. Formally, we have:

where φ() is the standard normal pdf and () is the standard normal cdf. The last part of the above equation, , is the inverse Mills ratio. The dependence on ρηε, the correlation between the error terms of the hedonic price function and the above equation, is apparent through this formulation:

- When there is none, estimate the hedonic price function as usual.
- However, if there is correlation, include the inverse Mills ratio as an independent variable in the hedonic price function.
- This requires estimating δ, which can be estimated with a probit model that predicts the probability of observing a sale:
- Introducing the inverse Mills ratio in the hedonic price function produces consistent estimation of the parameters in the hedonic pricing equation and also provides a test for the presence of selection bias: If the coefficient on the inverse Mills ratio is not statistically distinguishable from zero, then the hypothesis that ρηε is not zero can be rejected and analysis can proceed as usual.

Mortality how do we estimate the value of a life?

- Key concept: the Value of a Statistical Life (VSL).
- Suppose each member of a group of 10,000 is willing to pay $30 for a 1 in 10,000 (0.0001) reduction in the risk of death (=1 life saved in this group). Then, the VSL is $30/0.0001=$300,000.
- VSL is used to estimate the life-saving benefits of proposed environmental policies and other safety regulations.

How do we estimate VSL?

- Traditional approach: discounted stream of income from future years of life. Disadvantage: places very low values on the life of homemakers and retirees.
- The appropriate measure is Willingness to Pay for the risk reduction.
- WTP for the risk reduction can be estimated using
- Compensating wage studies (labor market)
- Other hedonic approaches (housing market, price of a car)
- Revealed preference approaches
- Contingent valuation

Compensating wage studies

- Based on the idea that firms must pay workers more for them to accept higher workplace risks
- Use wage data and information about fatal risks in one’s job to estimate the regression equation:

Where w=wage rage, p=fatal risk in the workplace faced by worker i, q=risk of non-fatal injury, WC = worker comp., and the x’s are other worker and job characteristics (age, experience, etc.)

Compensating wage studies

- Many studies available in the literature.
- VSL ranges from a few hundred thousand Dollars to several million dollars, best range is 1-9 million.
- But…risk premium might in reality be inter-industry wage differential.
- But…approach relies on workers knowing exactly the correct risks they face.

Problems with compensating wage studies

- But…this is risk for an accident in the next year, while most environmental Risks are incurred later in life.
- But…this is risk for males in their prime age, whereas environmental Risk affects mostly older and sick people.
- But…workplace risks are voluntary, while environmental Risks are not.
- And finally, compensating wage studies do not obtain WTP from the demand function of individuals, but only points at the tangency between demand for safety and supply of safety by firms.

Example 1: Air quality

- Air pollution is one of the first application of the hedonic pricing model
- Ridker, Henning (1967) “The determinants of property values with special reference to air pollution” Review of Economics and Statistics.
- No residential properties sale prices, but census tract data from St. Luis, 1960.
- Dependent variable: median value of property prices
- Independent variables: median characteristics of houses in a census tract, quality of schooling, access to highway, neighbourhood characteristics, tax levels, public services
- Air quality (SO2, SO3, H2S, H2SO4) measured as direct effects on houses and on human health.

Results (some variables)

Linear model

House price falls by 245US$ if pollution is present

Ridker and Henning estimate the environmental damage of air pollution in St. Louis to be 82 million dollars => need to compare this estimate with the cost of a public program to clean pollution

Problems of Ridker e Henning example

- Multicollinearity
- Omitted variables
- Positive sign for the coefficient of the distance from the city centre
- They do not consider the price of single houses, but the median value of the houses sold in a census tract
- …

Example 2: Water quality

- Leggett and Bockstael (2000) Journal of Environmental Economics and Management
- 741 observations
- Effects of Chesapeake Bay water quality on prices of houses located along the bay
- Rather than using the characteristics of houses (rooms, bathrooms, etc.), Leggett and Bockstael use the appraised value of houses.
- Water quality is measured using information on the level of pollution of the bay publicly given by the Department of Health of Maryland

‘Welfare’ change

- The presence of fecal coliform is equal to -0.052 dollars per 1,000 dollars of the value of the house
- Suppose fecal coliform increase from 109 (average value) to 159:
- The welfare change is equal to:
- (159-109)*(-0.052) = -2.6
- This means that a person that is buying a house is willing to pay $2,600 more to avoid the increase in the concentration of fecal coliform.

Measuring Welfare Via Hedonic Methods

- The above calculations give the change in the house price due to fecal coliform changes but do not really give the changes in welfare, in the sense of the addtional consumer surplus you get from the change.
- The following example shows how that consumer surplus can be calculated. It entails a two stage estimate procedure, the first stage is the hedonic price estimates as above and the second stage is a follow on regression based on households willingness to pay the implict price.

Case Study - Overview

- This exercise presents an application of the Hedonic Price Method for the valuation of benefits brought about by the improvement of the broadleaf coverage rate in an urban area.
- The local government decided to improve the quality of urban parks and green spaces near residential areas.
- For details see Markandya et al. 2002.

Case Study - Overview

- This study focuses on the valuation of only one of them, namely the increase of broadleaf coverage.
- To elicit the value assigned to a change in broadleaf coverage, the prices of houses in areas with different coverage rates are observed.

Methodology

- Collection of data:
- Resident households were randomly selected from the council directory of resident households.
- The broadleaf coverage rate within the ray of 300 meters for every house was calculated.
- The price of the house was determined looking at estate agency bulletins and recent transactions. Expert advice on prices for some properties was also required. (Nb sample selection bias not addressed).
- The collection of information on the socio-economic features of the household was done looking at recent census data.

Methodology

- Calculation of the value of environmental quality:
- Estimation of the House Price Function.
- Calculation of the Implicit Marginal Price of the environmental good (the responsiveness of the house price function with respect to the environmental quality, ie the first derivative: c x P/Zm) for each observation.
- Estimation of the Implicit Inverse Demand Function for the environmental good. (implicit price as a function of the environmental good and socio-economic features of individuals).
- Calculation of the Consumer Surplus

Key

- OBSERV - Number of the observation
- PRICEX - Price of house (US$, 1995)
- NUMROO - Number of rooms in the house
- INDEPE - Dummy variable: INDEPE=1 Detached house. 0 otherwise.
- DISTAN - Distance from downtown (Km)
- MURDER - Murder rate of the area (murders/year per 1000 residents)
- BROADL - Broadleaves tree coverage rate (covered area/total area)
- REDHOU - Annual income of the household (Usd, 1995)
- COMPON - Number of components in the household

Estimation of Hedonic House Price Function

- Regression of LnPriceX on other variables:
- Most of the variables included in the model are highly significant from a statistical point of view in explaining the variability of the price of houses.
- The coverage rate of broadleaves exhibits a positive relevant relationship with the price, other things equal.

Estimation of Hedonic House Price Function

- Estimated Hedonic Price function is not linear:

Calculation of Implicit Price

- This function, as described in the methodology, is the first derivative of the house price function with respect to the broadleaf tree rate. The implicit price function is:

IMPLIP = (0.16871 / BROADL) PRICEX

- This function is used for estimating, observation by observation, the implicit price of an additional unit of broadleaf coverage:

Estimation of inverse demand curve

- The estimation of the inverse demand function is a second stage estimation based on the result of the first estimation (i.e. the house price function and related first derivative).
- The estimated implicit price of the broadleaf coverage unit (in this case, the percent point) is regressed on the observed coverage rate and the socio-economic features of the owners.

Estimation of inverse demand curve

- Results of regression analysis:
- The inverse demand function is therefore estimated as:

IMPLIP = EXP( 6.34) REDDHOU 0.76 COMPON 0.11 BROADL -0.85

Inverse Demand Function

- The inverse demand function can be shown for three different levels of income: the sample first quartile, the sample median and the sample upper quartile. The variable COMPON is fixed at the sample mean level.

Calculation of Consumer Surplus

- The consumer surplus is calculated by estimating the area under the demand curve.
- This is done by integrating the inverse demand curve with respect to the implicit price and calculating the definite integral observation by observation (variable CONSUR) between the present coverage rate (E1=BROADL) and the coverage rate (UPLIM) planned by the policy maker.

Discussion

- This information can be used first to calculate the average consumer surplus per household and can be multiplied by the number of households to get a measure of the total benefits which can be compared with the cost of the intervention.
- On distributional grounds, notice that a 10% increase for people living in areas with high coverage rate does not change the consumer surplus much, compared to those living in low coverage rate areas, which could be lower income neighbourhoods.

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