Signature
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
| Inputs | Definition | Unit |
|---|---|---|
a | Value of one unit to its user before any loss from use of the resource | pounds per unit |
c | Cost of one unit borne by its user before the fee, below a | pounds per unit |
N | Number of identical users who choose their use independently, a whole number of at least 1 | users |
k | Fall in the value of every unit for each unit used by any user, above zero | pounds per unit per unit used |
t_s | Charge on each unit used that makes the users' equilibrium total equal to the joint optimum, zero for a single user | pounds per unit |
|---|---|---|
q_t | Units each user chooses at the symmetric equilibrium when every unit carries the fee t_s, the q_E of HE-FM-TOTC-001 with c plus t_s in place of c | units per user |
Q_t | Use of all N users under the fee, N times q_t, equal to the joint optimum (a less c) / (2k) | units |
R_t | Fee per unit times total use under the fee, a transfer paid by the users to the authority that collects it | pounds |
Function
Equilibrium use, rent dissipation and corrective fee function for a shared depletable resource
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.
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Implementations
Excel
Corrective fee, use under the fee and revenue in four Excel cells
With inputs in cells named ValuePerUnitFull, CostPerUnit, LossPerUnitUsed and NumberOfUsers, the four formulas fill cells named CorrectiveFeePerUnit, UsePerUserWithFee, TotalUseWithFee and FeeRevenue, in that order. A blank NumberOfUsers cell is read as 0 and the fee returns #DIV/0!, and a blank LossPerUnitUsed cell gives #DIV/0! in the use cell.
=(ValuePerUnitFull-CostPerUnit)*(NumberOfUsers-1)/(2*NumberOfUsers); =(ValuePerUnitFull-CostPerUnit-CorrectiveFeePerUnit)/(LossPerUnitUsed*(NumberOfUsers+1)); =NumberOfUsers*UsePerUserWithFee; =CorrectiveFeePerUnit*TotalUseWithFee
Assumptions
Uniform per-unit fee passed in full to every user's cost
The same fee is charged on every unit and each user treats it as part of the cost of a unit, so the first-order condition becomes a less c less t_s less k Q less k q_i equal to zero. Delaney and Jacobson pay their subsidy on each token not used, which changes a user's payoff by a lump sum less the subsidy times own use, so it acts on choices as a fee per unit of use (derived).
Revenue not returned in proportion to each user's own use
The fee changes choices only if its revenue does not come back to each user in proportion to that user's use. Revenue returned as a fixed sum, or spent on conservation as Giubilini proposes for an antibiotic tax, leaves the incentive in place, while a rebate per unit used cancels it (derived).
Users, values and loss of value as in the equilibrium formula
The users are identical, choose independently and care only about their own net value, a exceeds c, k is above zero and N is a whole number of at least 1, as in HE-AS-TOTC-001 and HE-AS-TOTC-002. The fee assumes the authority knows a, c, k and N; with a misjudged k or N the fee misses the optimum.
Worked examples
Charge of 30 pounds per course for four prescribers in the article's example
With the article's illustrative values the fee is 30 pounds per course, the loss one more course imposes on the other three prescribers' 3,000 courses at the optimum. Each prescriber then chooses 1,000 courses, 4,000 in all, the joint optimum, and the 120,000 pounds raised is a transfer, as the article states.
a = 100; c = 20; k = 0.01; N = 4; t_s = 30; q_t = 1000; Q_t = 4000; R_t = 120000
Pigouvian subsidy in the three-person laboratory game of Delaney and Jacobson
Delaney and Jacobson set a starting return of 0.36 dollars per token, an outside return of zero, a fall of 0.01 dollars per token for each token in the resource and groups of three. The formula gives their subsidy of 0.12 dollars, 6 tokens each and a group total of 18, the social optimum, as the paper reports. R_t is the fee equivalent. The paper pays its subsidy on tokens kept out of the resource, 12 of the group's 30, so it pays out 1.44 dollars, a lump-sum difference that leaves choices unchanged (derived).
a = 0.36; c = 0; k = 0.01; N = 3; t_s = 0.12; q_t = 6; Q_t = 18; R_t = 2.16
No corrective fee needed for a single user of the shared resource
With one prescriber the fee is zero, use is the joint optimum of 4,000 courses and no revenue is raised, since the only user already bears the whole loss. The values are illustrative and the case checks an implementation.
a = 100; c = 20; k = 0.01; N = 1; t_s = 0; q_t = 4000; Q_t = 4000; R_t = 0
Corrective fee for ninety-nine prescribers close to the whole external cost
With 99 prescribers the fee is 39.596 pounds per course, close to the 40 pounds of the whole marginal external cost at the optimum, the t_star of HE-FM-PIGT-001 with b and g equal to 0.01 and m equal to zero. Each prescriber chooses 40.404 courses, 4,000 in all, and 158,383.84 pounds is raised. The values are illustrative.
a = 100; c = 20; k = 0.01; N = 99; t_s = 39.596; q_t = 40.404; Q_t = 4000; R_t = 158383.84
Common errors
Charging the whole marginal external cost when users are few
Setting the fee at k Q_s, the t_star of HE-FM-PIGT-001 with b and g equal to k and m equal to zero, ignores the share of the loss each user already bears. In the article's example a fee of 40 pounds for four prescribers cuts use to 800 courses each, 3,200 in all, below the optimum, and the rent to 153,600 pounds instead of 160,000 (derived). The whole external cost is the right fee only as N grows.
Setting the fee at the external cost observed before the fee
Valuing the loss on the other users at the untaxed equilibrium, k (N less 1) q_E, gives 48 pounds per course in the article's example, the external loss of its step 3. Each prescriber then chooses 640 courses, 2,560 in all, and the rent is 139,264 pounds, below the 160,000 pounds of the optimum (derived). The fee is the external cost at the optimum, as HE-FN-PIGT-001 sets it and as HE-FM-PIGT-003 shows for a tax set at the pre-tax external cost.
Counting the fee revenue as a loss of value
The 120,000 pounds raised in the article's example is paid by the prescribers to the authority, and the group's net value at the optimum is still 160,000 pounds. Subtracting the revenue would show 40,000 pounds and make the fee look worse than leaving the four prescribers alone at 102,400 pounds. Delaney and Jacobson likewise treat their subsidy as a transfer with no efficiency cost in itself, apart from the cost of raising the funds.
Sources
Delaney and Jacobson on the optimal Pigouvian subsidy in a common pool resource game
Delaney J, Jacobson S. Payments or persuasion: common pool resource management with price and non-price measures. Working paper, March 2015 (read in full); published as Environmental and Resource Economics. 2016;65(4):747-772. doi:10.1007/s10640-015-9923-z (published version not seen). Section 2.1 states the optimal (Pigouvian) subsidy delta = (beta less alpha)(n less 1) / (2n), and section 2.2 adds delta per token to the return of the outside option and cites Pigou (1920) for the optimal subsidy as the difference between marginal private and marginal social cost at the optimal quantity. The Reviewers' Appendix gives the equilibrium with the subsidy, (beta less alpha less delta) / ((n + 1) gamma), and finds delta by setting it equal to the social optimum. Section 3 sets beta of 0.36, alpha of zero, gamma of 0.01 and n of 3, and section 3.1 sets delta at 0.12 dollars, with a symmetric Nash equilibrium of 6 tokens that yields the social optimum, and footnote 8 treats the subsidy as a transfer.
Falk, Fehr and Fischbacher on the social optimum of the common-pool resource game
Falk A, Fehr E, Fischbacher U. Appropriating the Commons: A Theoretical Explanation. Chapter 5 in National Research Council. The Drama of the Commons. Washington, DC, The National Academies Press; 2002. doi:10.17226/10287 (chapter read in the publisher's online edition). The Appendix to Chapter 5 (page 186) maximises the sum of the players' payoffs and gives the social optimum of total appropriation a / (2b), the target the fee reaches, and the section The Standard Common-Pool Resource Game (pages 164 and 165) gives 36 for the Walker and colleagues parameters and states that selfish subjects ignore the negative externality they impose on the other players.
Canonical Identity
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