Signature
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
| Inputs | Definition | Unit |
|---|---|---|
a | Value of one unit to its user before any loss from use of the resource, such as the value of an antibiotic course when no resistance has built up | pounds per unit |
c | Cost of one unit borne by the user who takes it, including its adverse effects, the same for every unit and below a | pounds per unit |
k | Fall in the value of every unit for each unit used by any user, above zero, such as the loss of effectiveness of every course from one more course prescribed | pounds per unit per unit used |
N | Number of users who choose their use independently, a whole number of at least 1 | users |
q_E | Units each of the N identical users chooses at the symmetric Nash equilibrium, above zero when a exceeds c | units per user |
|---|---|---|
Q_E | Use of all N users together at the symmetric Nash equilibrium, N times q_E | units |
v_E | Value of a unit less its private cost and less the loss from total equilibrium use, a less c less k times Q_E | pounds per unit |
V_E | Total net value of all units used at the equilibrium, the sum of the users' net values, Q_E times v_E | pounds |
Q_s | Total use that maximises the group's net value, (a less c) / (2k), the efficient quantity of HE-FM-PIGT-001 with b and g equal to k and m equal to zero | units |
V_s | Total net value at the jointly optimal use Q_s, (a less c)^2 / (4k) | pounds |
d_E | Proportion of V_s lost when users act alone, 1 less V_E / V_s, zero at N = 1 and rising towards 1 as N grows | proportion |
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
Equilibrium use, rent and dissipated share of a shared resource in six Excel cells
With inputs in cells named ValuePerUnitFull, CostPerUnit, LossPerUnitUsed and NumberOfUsers, the six formulas fill cells named EquilibriumUsePerUser, EquilibriumTotalUse, EquilibriumRent, OptimalTotalUse, OptimalRent and RentShareDissipated, in that order. A blank LossPerUnitUsed cell is read as 0 and returns #DIV/0!. A blank NumberOfUsers cell is read as 0 and returns zero total use and a dissipated share of 1, so the cell must hold a whole number of at least 1.
=(ValuePerUnitFull-CostPerUnit)/(LossPerUnitUsed*(NumberOfUsers+1)); =NumberOfUsers*EquilibriumUsePerUser; =EquilibriumTotalUse*(ValuePerUnitFull-CostPerUnit-LossPerUnitUsed*EquilibriumTotalUse); =(ValuePerUnitFull-CostPerUnit)/(2*LossPerUnitUsed); =OptimalTotalUse*(ValuePerUnitFull-CostPerUnit-LossPerUnitUsed*OptimalTotalUse); =1-EquilibriumRent/OptimalRent
Assumptions
Identical users whose use lowers the value of every unit by the same amount
Every user faces the same a, c and k, and the value of each unit falls in a straight line with total use, so user i's net value is q_i (a less c less k Q). This is the payoff of Falk, Fehr and Fischbacher, e plus a x_i less b x_i times total appropriation, with their a equal to a less c here and their b equal to k, and of Delaney and Jacobson with beta equal to a, alpha to c and gamma to k. Users with different values or costs need the asymmetric first-order conditions and do not share one q_E.
Independent one-shot choices by selfish users with an interior equilibrium
Users choose independently and at the same time and each cares only about their own net value, the setting of the standard game and Proposition 1 of Falk, Fehr and Fischbacher. No communication or binding agreement between users is allowed for. Use is continuous and a exceeds c, so q_E is above zero; with a at or below c no unit is worth taking and the formula does not apply. In the experiments Ostrom reviews, allowing users to communicate reduced overharvesting, so the equilibrium describes unmanaged open access and not a governed commons.
One period with no benefit of use to non-users
All values are in one currency for one season, with no carry-over of resistance to later seasons and no benefit of treatment to people other than the patient. The article notes that transmission benefits, future seasons and differences in need can change the answer, and Porco and colleagues found that more antibiotic use benefits society in some circumstances.
Worked examples
Four prescribers sharing one antibiotic in the article's example
With the article's illustrative values, a course worth 100 pounds at full effectiveness, a cost of 20 pounds and a loss of 0.01 pounds per course for each course prescribed, four prescribers each choose 1,600 courses, 6,400 in all. Each course is then worth 16 pounds net and the rent is 102,400 pounds, against 160,000 pounds at the joint optimum of 4,000 courses, so 36% of the attainable value is lost, as the article reports.
a = 100; c = 20; k = 0.01; N = 4; q_E = 1600; Q_E = 6400; v_E = 16; V_E = 102400; Q_s = 4000; V_s = 160000; d_E = 0.36
Single steward of the antibiotic as the limiting case with no dissipation
With one prescriber, or one steward choosing for the region, the equilibrium is the joint optimum of 4,000 courses, each worth 40 pounds net, and none of the 160,000 pounds is lost, the first row of the article's table. The values are illustrative and the case checks an implementation.
a = 100; c = 20; k = 0.01; N = 1; q_E = 4000; Q_E = 4000; v_E = 40; V_E = 160000; Q_s = 4000; V_s = 160000; d_E = 0
Ninety-nine prescribers sharing the antibiotic near open access
With 99 prescribers each chooses 80 courses, 7,920 in all, close to the open-access limit of 8,000. Each course is worth 0.80 pounds net, the rent is 6,336 pounds and 96.04% of the attainable value is lost, the last row of the article's table. The values are illustrative.
a = 100; c = 20; k = 0.01; N = 99; q_E = 80; Q_E = 7920; v_E = 0.8; V_E = 6336; Q_s = 4000; V_s = 160000; d_E = 0.9604
Eight appropriators in the laboratory resource of Walker and colleagues
Falk, Fehr and Fischbacher restate the laboratory design of Walker and colleagues with a revenue of 23 per token less 0.25 times total tokens, a cost of 5 per token and eight players. The prediction is 8 tokens each and 64 in all against a social optimum of 36, as the chapter states. The net value per token is 2, the group rent 128 against 324 at the optimum, so 60.49% of the attainable rent is lost (derived from the printed values).
a = 23; c = 5; k = 0.25; N = 8; q_E = 8; Q_E = 64; v_E = 2; V_E = 128; Q_s = 36; V_s = 324; d_E = 0.6049
Common errors
Dropping each user's own share of the loss from the first-order condition
A user who takes the value of a course as fixed sets a less c less k Q to zero, the average-return behaviour of the rent dissipation literature, and the total becomes (a less c) / k for every N, the open-access limit Q_0 of HE-FM-PIGT-001 with b equal to k. In the article's example it gives 8,000 courses and a rent of zero for four prescribers instead of 6,400 courses and 102,400 pounds (derived). The full open-access loss holds only as N grows.
Measuring dissipation on gross yield rather than net value
Delaney and Jacobson define gross efficiency on the yield of the resource, which leaves out the opportunity cost of use, and note that it equals net efficiency in their experiment because the outside return is set to zero. With the article's values the gross yield is 240,000 pounds at the optimum and 230,400 pounds at the equilibrium, a loss of 4%, while the net rent falls by 36% (derived). The share d_E uses net values, as the article does.
Reading every rise in antibiotic use as a loss to society
The formula counts only the loss of effectiveness from use and no benefit to anyone but the patient. Where treatment stops transmission, use protects others, and Porco and colleagues found that more antibiotic use benefits society in some circumstances, so the predicted overuse and dissipated share hold only when the external effect of use is a cost, as assumed in HE-AS-TOTC-003.
Sources
Falk, Fehr and Fischbacher on the selfish Nash equilibrium 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 section The Standard Common-Pool Resource Game (pages 164 and 165) states Proposition 1 (Selfish Nash Equilibrium), that with purely selfish preferences the unique Nash equilibrium is symmetric with individual appropriation a / (b (n + 1)), for payoffs written e plus a x_i less b x_i times total appropriation after the unit cost is netted out, and gives 8 per player, a total of 64 and a social optimum of 36 for eight players with the Walker and colleagues revenue 23 times total tokens less 0.25 times its square and a cost of 5. The Appendix to Chapter 5 (pages 185 and 186) derives the best reply, the total n a / (b (n + 1)) and the social optimum a / (2b). The records write their a as a less c.
Delaney and Jacobson on the Nash equilibrium and social optimum of 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, A common pool resource appropriation game, sets each agent's payoff as alpha times unused tokens plus (beta less gamma times total extraction) times own extraction, states that it is mathematically identical to the payoff of Ostrom and colleagues (1994), and gives the symmetric Nash equilibrium (beta less alpha) / ((n + 1) gamma) and social optimum (beta less alpha) / (2 n gamma) per agent, equal only for n = 1 or beta at or below alpha. The Reviewers' Appendix derives both from the first-order conditions.
Walker, Ostrom and Gardner on rent dissipation in a laboratory common-pool resource
Walker JM, Ostrom E, Gardner R. Rent-dissipation in common-pool resource environments: experimental evidence. Workshop in Political Theory and Policy Analysis working paper W88-7, Indiana University, September 1988 (read in full in the Digital Library of the Commons). Its introduction traces the rent dissipation hypothesis to Gordon (1954). Section III, Theoretical Predictions, contrasts group optimality (maximum rents, where marginal revenue product equals marginal cost), the Nash equilibrium and complete rent dissipation (where average revenue product equals marginal cost), and prints the payoff of eight players with a revenue of 25 times total tokens less 0.3 times its square and an outside return of 5. Its pure-strategy Nash equilibria total 59 tokens and earn approximately 41% of the rents possible, against 33 tokens for maximum rents and 67 for zero rents. The continuous equilibrium of HE-FM-TOTC-001 gives 59.3 tokens and keeps 39.5% (derived).
Canonical Identity
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