Covariates and "left right effect" in WTP space

[cybey]

Hello Stephane,

I got another question regarding the interpretation of the covariates of the price coefficient when estimating a model in wtp space. Let's say my utility function looks like this:

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V[['alt1']] = b_Price_value * ( wtp_Att1Lvl2 * Att1Lvl2 + wtp_Att2Lvl2 * Att2Lvl2 + Price )

And my price coefficient looks like this:

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b_Price_value = b_Price *
      ( 1 + (b_TariffSwitched_Price * TariffSwitched) ) *
      ( Income / Income_Mean ) ^ elast_Price_Income

... and my estimation results are:

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b_TariffSwitched_Price = 0.10
elast_Price_Income = -0.15

Now I wonder if my interpretations are correct:

• For a 10% increase in income, we would observe a 1.5% increase in wtp (on average across all wtp coefficients).

• If a respondent has changed his/her tariff in the past, this lowers his/her wtp by 10% (on average across all wtp coefficients).

The reason I ask is that when estimating the model in utility space, the covariates only affect the beta coefficient of price (b_Price_value). Next, I divide the beta coefficients of the attributes by the price coefficient, while the latter changes as a function of the covariates, i.e. the betas of the attributes won't be affected by the covariates. In wtp space, however, the covariates of the price coefficient affect the whole utility function. On the one hand, the wtp coefficients are multiplied by the price coefficient (including covariates), but so is the price attribute.

What I mean is:

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b_Price * covariates * ( wtp_Att1Lvl2 * Att1Lvl2 + wtp_Att2Lvl2 * Att2Lvl2 + Price )

is not the same as

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b_Price * ( wtp_Att1Lvl2 * Att1Lvl2 * covariates + wtp_Att2Lvl2 * Att2Lvl2 * covariates + Price )

Best
Nico

[stephanehess]

Hi Nico

the issue is that you're interaction the covariate with the scale of the model as opposed to only with the cost sensitivity. You can understand it most easily by looking at the partial derivatives.

Here's a case with two attributes, namely price (p) and an attribute x, and we have some covariate (cov) and want the wtp for this.

Preference space specification

Vpref = bp * cov * p + bx * x

dVpref/dx = bx
dVpref/dp = bp * cov
wtp = bx/(bp * cov)

Incorrect WTP space specification (as in your example)

Vwtp = bp * cov * ( p + vx * x )
dVwtp /dx = bp * cov * vx
dVwtp /dp = bp * cov
wtp = vx

==> this gives us a different wtp

Correct WTP space specification

Vwtp = bp * ( p * cov + vx * x )
dVwtp /dx = bp * vx
dVwtp /dp = bp * cov
wtp = vx/cov

we then have that this is the same as in preference space, as vx=bx/bp

Hope this helps

Stephane

[cybey]

Hi Stephane,

as always thank you very much! I must be a moron for not having noticed that myself.

Best
Nico