Car Ownership? Evidence from the Copenhagen Metropolitan Area

In this equation the levels of , and , that occur in the numerator refer to the
situation before the extension of the metro network. 25
corresponds with the change ∆ln ,, in log income can be determined as:
The change in income itself that
∆ ,,
1 ∆
,,
(A5)
If households would not change their location, housing type or car ownership position, this
would suffice to compute the welfare impact of the change in the metro network. However, the
changes in the utilities of the choice alternatives that occur as a consequence of the extension of
the metro network will induce some households to change their residential location, car
ownership position or perhaps even their housing type.
To take this into account as well, we use the results of De Palma and Kilani (2003). We
start by defining ∆
as the largest of the conditional compensating variations:
∆ max ,, ∆ ,, (A6)
DePalma and Kilani (2003) show that the unconditional compensating variation is always
between the minimum and the maximum of the conditional compensating variations.
To be able to compute it exactly we define functions ,, as follows:
,,
,, , , , , , , ; , (A7)
where the values of , and , refer to the situation with the extended metro network.
Clearly, if ∆ ,, , ,, is equal to the utility household I experienced before the
∗
extension of the metro network. Now define: ,,
max ,, , ,, ∆ ,, and
let ∗ denote the logit choice probabilities defined on the basis of these utilities: 26
25 The income level that occurs in the denominator refers (from the derivation of (A.3)) to the situation after the
extension of the metro network, but we assume throughout that household income does not change because of the
extension of the metro network.
26 This equation is analogous to (2) for the generator function of the multinomial logit model.
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