This model is probably the simplest possible model that can analyze both HOV lanes and HOT lanes. In some cases, converting HOV lanes to HOT lanes reduces each commuter's utility for all positive tolls. However, in some cases, the introduction of HOV lanes may weaken the social welfare by increasing the social costs.

When K is very large, it is not surprising that the introduction of HOV lanes can worsen social welfare. Now, let's return to the equilibrium with HOV lanes and consider converting HOV lanes to HOT lanes. Thus, at some tolls lower than tHOV, converting HOV lanes to HOT lanes improves social welfare under (5).

The above condition shows that when K is high and m is low, HOT lanes tend to dominate HOV lanes. It seems to be widely accepted that converting underused HOV lanes to HOT lanes is a good idea to reduce traffic in the conventional lane. An example (Example 2) in the appendix shows that the last negative e¤ect can strongly dominate the first positive e¤ect (in the Pareto sense), and that HOT lanes perform worse than HOV lanes.

Proposition 4 The allocation under the optimal uniform pricing can be achieved by using HOV lanes with capacity.

## Aggregate Social Costs under HOV and HOT lanes

Note that under uniform congestion pricing, the congestion levels for HOV and regular lanes are the same: ie. Therefore, we can conclude that under the optimal uniform congestion pricing, the level of shared driving is too high compared to the efficient level. The above proposition states that the introduction of HOV lanes improves social welfare as congestion costs exceed a marginal value, which is a highly nonlinear function of K and m.

Although it is difficult to deal with the inequality directly, we can say that HOV lane policy is more likely to improve the social welfare as capacity of HOV lanes is smaller, or the number of people sharing the car is larger9. As for the effect of m, the condition that HOV lane policy reduces the social cost is expressed asm >me (c; K. Proposition 6 The conversion of HOV lanes to HOT lanes (with some toll rate) improves the social welfare if and only if the following condition applies.

Proposition 6 implies that converting HOV lanes to HOT lanes improves social welfare if (i) the unit cost of congestion, c, is small, (ii) m is small, and (iii) K is large. Condition (ii) means that an SOV (single passenger vehicle) does not impose much more congestion than an HOV (say, if m = 2).10 Condition (iii) means that if the lane has large capacity , the social cost of unused lanes is high. Combining the results so far, we classify possible models regarding the effects of introducing HOV lanes and converting HOV lanes to HOT lanes, as depicted in Figure 1A-C.11 The range of parameters of each model is shown in Figure 2 in which the letters A, B, C attached to the areas correspond to the patterns in Figure 1 A, B, C, respectively.

According to Figure 2, an HOV lane policy is wasteful (the case of Figure 1A emerges) when K is relatively large. This condition is consistent with the statement based on casual observations: in this case, HOV lanes are likely to be underutilized. It is also true that converting HOV to HOT lanes is not always effective in reducing congestion: it only improves social welfare if the capacity of the HOT lanes is greater and the level of congestion is not too heavy (Figure 1A and 1B).

On the other hand, HOT lanes can be effective even when the introduction of HOV lanes worsens the situation (Figure 1A). In the cases in Figures 1A and B, there exists an optimal HOT tax that minimizes the total social costs. 11 Although the curve in Figure 1C is concave, it can be convex in some cases, as discussed in the proof of Proposition 6.

## Comparing Alternative Policies

Since @K@t = c(mm1)22+m > 0, the indirect effect acts in the opposite direction to the direct effect, but the latter is greater than the former. This distortion can be reduced by converting HOV lanes to HOT lanes: it allows solo drivers in HOV lanes. Consequently, contrary to popular belief, converting HOV lanes to HOT lanes can reduce social welfare under conditions that are not too unrealistic.

Tolling regular lanes encourages carpooling, increases the number of HOV lane users, and decreases the number of regular (now toll) lane users. First, we assumed that there is no additional cost in introducing HOV lanes and converting HOV lanes to HOT lanes. Here, we provide two examples that show that the results in the special case (F(t) =t and C(q) = cq) may not be consistent.

The first example shows that even if K is small enough (K = 1=4), the introduction of HOV lanes can increase the cost of each passenger. 13If the capacity of the HOV lanes can be freely chosen, then the second policy always dominates the first, uniform congestion pricing. Thus, all commuters pay more than 1, which can be achieved without HOV lanes (unit population with road capacity 1 makes q = 1).

Thus, this example shows that if there are no commuters with low carpool organizational costs, and if there is a large population of middle-high types, then introducing HOV lanes tends to reduce social welfare. The next example shows that converting HOV lanes to HOT lanes can increase any commuter's cost for any nontrivial toll level. Introducing HOV lanes is thus socially welfare-reducing (but not in the Pareto sense in this example: some lowest commuters become better o¤ by introducing HOV lanes).

It is easy to construct an example where all commuters are worse off when HOV lanes are introduced. CHOT( )is a convex function that is monotonically decreasing in the relevant interval.) All commuters witht are thus worsened o¤ by the conversion. This proves that in our numerical example, converting HOV lanes to HOT lanes increases all commuters' commuting costs.

Thus, if the condition in the proposition is satisfied by a strict inequality (with an equality), then type t strictly prefers HOV lanes (is independent between HOV and regular lanes). Traffic congestion in both HOV and regular lanes is therefore eased, and commuters are better o¤ in a Pareto sense. Since the congestion ratio in the regular lanes is always higher than in the HOT lanes, we have F(.

Thus, SCHOT is monotonically decreasing in , and SCHOT( ) > SCHOT(tHOV) for all