Supposing a car driver’s demand for car usage, measured in car- km driven, for 12 months (X12) and 24 months (X24) can be described by the following Cobb-Douglas Marshallian demand curves:
X12 = aP−d
a, d > 0
X 24 = bP−λd
b, λ > 0
where P are variable or km dependent car usage costs per km. The –d and ( − λd ) parameters are the elasticities of car driving with respect to car usage costs for intervals of 12 months and 24 months, respectively. The long-run elasticity is higher, in absolute value, than ( − λd ) because we ignore any adjustments made after two years. Based on previous research regarding the magnitudes of elasticities for car usage, we assume that that λ > 1.
Assume that P = P0 is the car usage cost per km for the driver when he holds his driving license. If car usage is in equilibrium with respect to these prices in all time horizons, it follows that X24 = 2X12 . Using equations (1) and (2) , we thus have:
2 ) 1 ( 0 − = λ d a P b
From (3) follows that b ≥ 2a when λ > 1 and P0 ≥ 1. In the special case where the demand response is independent of the time period (λ = 1), then b = 2a.
Using the demand functions above, the driver’s willingness to pay for not losing his driving license for 12 months (WP12) and 24 months (WP24) can, thus be described by the following formulas: 1
1 In the following we assume, in line with most empirical analyses that the area under the Marshallian demand function is a good approximation for the true value of WP, see for example Hausman et al. (1993).