静学的ロード・プライシング理論における超混雑と待ち行列理論
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The static theory of road pricing was theoretically proposed by Walters in 1960s, and it aimed to moderate road traffic congestion by levying a congestion toll. However, this work caused much debate in the literature, because of its theoretical defects such as its inconsistency of arguments between demand function and cost function, lack of dynamic consistency of equilibria on the backward-bending segment of the cost curve, and so on.Verhoef (1999) focuses on the difficulties arising with the backward-bending cost curve defined over traffic flows in the context of continuous congestion, and questions the relevance of the backward-bending segment of this curve by demonstrating that the equilibria on this segment of the cost curve are dynamically inconsistent and so infeasible. Although his model seems to resolve almost all theoretical defects in the static theory of road pricing, it is not consistent with the classical theory of road pricing because it depends on the bottleneck model (Vickrey, 1969, and Arnott, de Palma & Lindsey, 1998), which does not aim to regulate queue size by levying toll.Naor (1969) has examined the relationship between Pareto optimal and revenue maximizing tolls for a queuing model that permits balking. Consider an M/M/1 queue with gross accession rate λ and service rate μ. Each customer has a cost per unit of service and waiting time, C, and receives a benefit R if he is served by the facility. If an arriving customer finds q people ahead of him, he faces an expected waiting plus service time of (q+1)/μ. The toll charged by the facility, θ, determines a critical queue size, n, such that the customer balks if q≥n. Assuming that there is no balking cost, the customers decision rule is : join queue if θ+C(q+1)/μ≤R (if q<n) ; balk if θ+C(q+1)/μ>R (if q≥n).The point of Naors article is to show that the value of n which maximizes his social welfare function is greater than that which maximizes the facilitys expected revenue per unit time, i.e., the revenue maximizing toll exceeds the socially optimal toll.In this paper, we will apply the above Naors queuing model to the theory of road pricing by levying toll and regulating queue size at the entrance of the road. Although Naors model permits balking, the classical road pricing theory does not seem to permit balking because it assumes that a potential driver will use the road if his expected benefit exceeds his expected cost of using the road. So we do not permit balking and obtain an optimal value of congestion tax.JEL classification : R41, R48, D62
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