Equilibrium use, rent dissipation and corrective fee function for a shared depletable resource
f(a, c, k, N) = (q_E, Q_E, V_E, d_E, t_s)
Maps the value of one unit of a shared resource at full effectiveness, the private cost of a unit, the loss of value that each unit used imposes on every unit and the number of identical users to the use each user chooses when all act alone, the net value left to the group, the share of the attainable value lost and the per-unit fee that brings use back to the joint optimum. Each user keeps the whole value of a unit but bears only their own share of the loss, so the symmetric Nash equilibrium (HE-FM-TOTC-001) exceeds the joint optimum whenever there is more than one user, and the fee of HE-FM-TOTC-002 charges the part of the loss that falls on the others. The joint optimum is the efficient quantity Q_star of HE-FM-PIGT-001 with the slope of marginal private benefit b and of marginal external cost g both set to k and m set to zero, and the open-access limit (a less c) / k as the number of users grows is its Q_0 with b set to k, where the deadweight loss of HE-FM-PIGT-002 equals the whole attainable value. For a finite number of users the equilibrium total, the fee and the lost rent take the form of Q_0 and t_star of HE-FM-PIGT-001 and the deadweight loss of HE-FM-PIGT-002 with b set to k (N + 1) / N, g to k (N less 1) / N and m to zero (derived, and checked here on 448 parameter sets). Those records compute them only for b the slope of a demand curve in an otherwise undistorted market (HE-VR-PIGT-005, HE-AS-PIGT-001); these slopes exist only along the symmetric path of the users' game and depend on N, which no live record holds. The marginal external cost of one more unit is the MEC of HE-FM-EXT-001. Notation follows the Tragedy of the Commons article.
Symmetric Nash equilibrium use of a shared resource by N users and the share of its rent dissipated
q_E = (a - c) / (k * (N + 1)); Q_E = N * q_E; v_E = a - c - k * Q_E; V_E = Q_E * v_E; Q_s = (a - c) / (2 * k); V_s = Q_s * (a - c - k * Q_s); d_E = 1 - V_E / V_s
Each of N identical users chooses q_i to maximise q_i (a less c less k Q), where Q is the use of all users together. Setting the derivative to zero gives a less c less k Q less k q_i equal to zero, and with identical users the unique equilibrium gives each user (a less c) / (k (N + 1)), the result Falk, Fehr and Fischbacher state as Proposition 1. The chain then gives total use, the net value of each unit at that use and the rent V_E, the net value to the group. Q_s is the joint optimum, the efficient quantity of HE-FM-PIGT-001 with b and g equal to k and m equal to zero (also the equilibrium at N = 1), V_s the rent it yields, (a less c) squared / (4k), which equals the open-access deadweight loss of HE-FM-PIGT-002 with the same b, g and m (derived), and d_E the share of V_s lost at the equilibrium, which equals ((N less 1) / (N + 1)) squared (derived, checked on a grid). Use rises and the rent falls as users are added, and the rent tends to zero as N grows, the open-access outcome.
Per-unit fee that brings N users of a shared resource to the joint optimum, with use and revenue under the fee
t_s = (a - c) * (N - 1) / (2 * N); q_t = (a - c - t_s) / (k * (N + 1)); Q_t = N * q_t; R_t = t_s * Q_t
A fee t_s on each unit raises every user's cost to c plus t_s, so use under the fee is the equilibrium of HE-FM-TOTC-001 with that cost. Setting the resulting total equal to the joint optimum (a less c) / (2k) gives t_s, the optimal Pigouvian amount Delaney and Jacobson state for their common pool resource game. It equals the loss one more unit imposes on the other N less 1 users at the optimum, k (N less 1) Q_s / N, the marginal external cost of HE-FM-EXT-001 valued at the efficient quantity as in HE-FN-PIGT-001 (derived). The user already bears their own share, so the fee is below the whole marginal external cost k Q_s, the t_star of HE-FM-PIGT-001 with b and g equal to k and m equal to zero, and tends to it, (a less c) / 2, as N grows (derived). Revenue R_t is paid by the users to the authority as a transfer and does not reduce the group's net value. A quota of Q_s / N units per user reaches the same use.