I agree with the argument you have used here for multiple agents – however the substitute road only influences the reservation value of those agents, so can be simplified out. On Thursday I was suggesting that the alternate road could be important if we had scope for “multiple equilibrium”.

If we add search costs, and some type of bounded rational rule following, there is definite scope for a multiple pareto ranked equilibrium given the existence of difference types of roads. In the absence of rule following I’m not sure if government action (outside of education) can improve outcomes – but if we can define a rule that becomes suboptimal we may be able to justify it.

However, this involves assumptions about the behaviour of agents I might not be comfortable with – and I’m wondering if there is a different way.

I’m not sure this is the multiple equilibrium argument – but it is an argument for congestion charging.

Fundamentally you are saying that there are two types of agents those that place a high value on time and those that place a low value on time. In this case there are drivers that place a low value on time that are being held up for x minutes that are also holding up a bunch of agents with a high value of time for a cumulative x minutes – as a result even though the average externality cancels out, at the margin there are people who are placing a higher cost on society from driving than they face.

In this case, a toll gets these people off the congested road leading to a net benefit (given our assumption of values).