Why Open Access Can Dissipate Resource Rents
A common resource can be difficult to exclude users from, while one user’s activity reduces what remains available to others. In an open-access setting, potential users can enter when returns appear attractive. Their entry may increase total production, but it also spreads the resource’s returns across more effort and can raise the cost of obtaining each unit. Entry can therefore continue even when the resulting total surplus is below what the resource could generate under a socially efficient level of use.
The key distinction is between the average return that guides entry and the marginal return that guides the efficient choice. Open access stops attracting effort when average revenue just covers the cost of effort. The efficient level, by contrast, equates the marginal revenue from one more unit of effort to its marginal cost. With diminishing returns, the marginal return falls below the average return, so the efficient level of effort is generally lower.
Model Assumptions and the Two Conditions
The model treats effort as a continuous, nonnegative choice. Assume \(R(0)=0\), revenue is differentiable, and additional effort has diminishing marginal returns over the relevant range: \(R'(E)>0\) and \(R''(E)<0\). Each unit of effort costs the same amount \(c\). Users can enter or leave, take the aggregate resource conditions as given when making an individual decision, and do not account for the effect their own effort has on the returns available to others. There are no other benefits, costs, or external effects in this basic model.
For \(E>0\), total revenue per unit of effort is \(R(E)/E\), and average net profit per unit is \(R(E)/E-c\). If that amount is positive, entry is attractive; if negative, effort has an incentive to exit. With free entry and exit, the positive open-access equilibrium \(E_{\mathrm{OA}}\) therefore satisfies
The social planner instead chooses effort to maximize total surplus, \(R(E)-cE\). At an interior maximum, the first-order condition is \(R'(E^*)=c\). Diminishing marginal returns ensure that this is a maximum rather than a minimum; the boundary \(E=0\) must also be considered if the assumptions or parameters make positive effort unprofitable.
Why the Efficient Condition Uses Marginal Revenue
The social objective is total revenue less total effort cost, not revenue per user or revenue per unit of effort. To see the efficiency condition, consider a small increase \(\Delta E\) in effort. Its change in total surplus is approximately \([R'(E)-c]\Delta E\). If \(R'(E)>c\), this increase adds more revenue than cost and raises surplus. If \(R'(E)<c\), it costs more than it adds and reduces surplus. At an interior maximum, neither a small increase nor a small decrease should improve surplus, which requires \(R'(E)=c\).
The open-access condition has a different logic. An individual entrant compares the revenue associated with a unit of effort with that unit’s cost, while treating the overall return per effort as given. Under the model’s free-entry assumption, entry stops when this average return equals cost. The difference between average and marginal revenue is central: with diminishing returns, an additional unit of effort produces less revenue than the average unit. Entry can be privately worthwhile according to the average return while the aggregate effort already exceeds the surplus-maximizing level.
Worked Example: Comparing Open Access and Efficient Effort
Suppose a hypothetical harvesting activity has total revenue \(R(E)=120E-E^2\), for \(0\leq E\leq120\), and each unit of effort costs \(c=20\) monetary units. Find open-access effort, efficient effort, the total surplus at each level, and the surplus lost under open access.
Method and assumptions. Use the free-entry average-revenue condition for open access and maximize total surplus for efficiency. Revenue is concave because \(R''(E)=-2<0\); effort cost is linear.
Open access. Average revenue is
Set average revenue equal to the effort cost: \(120-E_{\mathrm{OA}}=20\), so \(E_{\mathrm{OA}}=100\). At that effort, revenue is \(R(100)=120(100)-100^2=12{,}000-10{,}000=2{,}000\). Total effort cost is \(20(100)=2{,}000\), and total surplus is \(2{,}000-2{,}000=0\).
Efficient effort. Marginal revenue is \(R'(E)=120-2E\). Set it equal to marginal cost: \(120-2E^*=20\), hence \(2E^*=100\) and \(E^*=50\). Revenue at this effort is \(R(50)=120(50)-50^2=6{,}000-2{,}500=3{,}500\). Cost is \(20(50)=1{,}000\), giving total surplus \(3{,}500-1{,}000=2{,}500\).
Surplus loss and check. Open-access effort is 50 units above the efficient level, and the loss in total surplus is \(2{,}500-0=2{,}500\). At \(E=50\), marginal revenue equals cost; at \(E=100\), average revenue equals cost. These checks confirm that the two conditions have been applied to the intended choices.
Common mistake avoided: Setting marginal revenue equal to cost to find open-access effort would give \(E=50\), but that is the efficient choice, not the free-entry outcome. Open access uses average revenue, which gives \(E=100\).
Rent Dissipation in the Model
Economic rent here means the part of revenue remaining after paying the opportunity cost of effort. At a positive open-access equilibrium, total revenue equals total effort cost, so this economic surplus is zero. This does not mean users receive no revenue or that their activity produces nothing. It means that the revenue is fully absorbed by the costs of the effort used to obtain it. In a fuller account, those costs may include the value of users’ time and other resources that could have been used elsewhere.
The social loss is measured relative to the best feasible effort, not by the total amount of revenue. Extra effort beyond \(E^*\) may still produce positive revenue, but its marginal contribution is less than its marginal cost. The surplus from the efficient level is therefore lost as open access drives effort farther into the diminishing-returns range. This is called rent dissipation: entry uses up the surplus that could otherwise have remained as economic rent.
Worked Example: A Nonlinear Revenue Function
In a hypothetical shellfish area, total revenue from effort is \(R(E)=200\sqrt{E}\), and each effort unit costs \(c=20\). Find the open-access and efficient levels and calculate the surplus available at the efficient level.
Method and assumptions. Apply average revenue for free entry and marginal revenue for the social choice. For \(E>0\), \(R'(E)=100/\sqrt{E}>0\) and \(R''(E)=-50/E^{3/2}<0\), so revenue is increasing and concave.
Open access. Average revenue per unit of effort is \(R(E)/E=200/\sqrt{E}\). Set this equal to 20:
Revenue at this effort is \(200\sqrt{100}=200(10)=2{,}000\), and effort cost is \(20(100)=2{,}000\). Thus open-access total surplus is zero.
Efficient effort. Marginal revenue is \(100/\sqrt{E}\). Set it equal to 20: \(100/\sqrt{E^*}=20\), so \(\sqrt{E^*}=5\) and \(E^*=25\). At \(E=25\), revenue is \(200(5)=1{,}000\), cost is \(20(25)=500\), and total surplus is \(1{,}000-500=500\).
Check. \(E_{\mathrm{OA}}=100\) exceeds \(E^*=25\). At \(E=25\), marginal revenue equals 20; at \(E=100\), average revenue equals 20. The surplus sacrificed at the open-access outcome is \(500-0=500\).
Common mistake avoided: The cost is \(20\) per effort unit, so total cost is \(20E\), not a fixed cost of 20. Subtracting only 20 from revenue would overstate surplus at both effort levels.
Inferring the Model Before Comparing Outcomes
Sometimes the revenue function is not given directly. If it is specified as \(R(E)=aE-bE^2\), with \(a>0\), \(b>0\), and \(c<a\), the two equilibrium conditions imply
The efficient level is half the open-access level in this particular quadratic model. This ratio is not a universal law; it follows from the specified revenue function and constant marginal cost. The model also gives a direct surplus comparison. At the efficient effort,
whereas open-access total surplus is zero. These expressions require the stated quadratic form and positive interior solutions; they should not be carried over automatically to other revenue functions.
Worked Example: Recovering Parameters From Resource Data
Suppose a hypothetical resource has revenue \(R(E)=aE-bE^2\), with \(a,b>0\), and effort costs \(c=24\). Revenue is observed to be 480 when \(E=12\), and the free-entry equilibrium is observed at \(E_{\mathrm{OA}}=36\). Infer \(a\) and \(b\), then find efficient effort and its surplus.
Method. Divide the revenue observation by effort to obtain average revenue, then use the open-access condition. Solve the resulting equations before applying the social marginal condition.
At \(E=12\), average revenue is \(480/12=40\). For the quadratic function, average revenue is \(a-bE\), so \(a-12b=40\). At the open-access level, average revenue must equal cost: \(a-36b=24\). Subtracting the second equation from the first gives \(24b=16\), hence \(b=16/24=2/3\). Substitution into \(a-12b=40\) gives \(a-8=40\), so \(a=48\).
Verify open access. At \(E=36\), average revenue is \(48-(2/3)(36)=48-24=24\), equal to cost. Directly, revenue is \(48(36)-(2/3)(36^2)=1{,}728-864=864\), and cost is \(24(36)=864\).
Find efficiency and surplus. Marginal revenue is \(R'(E)=48-(4/3)E\). Setting it equal to 24 gives \((4/3)E^*=24\), so \(E^*=18\). Revenue is \(R(18)=48(18)-(2/3)(18^2)=864-216=648\); cost is \(24(18)=432\). Efficient surplus is \(648-432=216\). Open-access surplus is zero, so the surplus loss is 216.
Check. \(18<36\); average revenue at 36 equals cost, and marginal revenue at 18 equals cost. The inferred parameters also reproduce the given revenue observation: \(48(12)-(2/3)(144)=576-96=480\).
Common Exam Traps and Interpretation
- Using marginal instead of average revenue for entry. Free entry is stopped by zero average profit per effort unit, \(R(E)/E=c\). The condition \(R'(E)=c\) identifies the interior efficient choice.
- Confusing zero rent with zero revenue. At open access, total revenue equals total effort cost in this model. Both can be positive even though total economic surplus is zero.
- Subtracting total effort cost incorrectly. If cost is \(c\) per unit, total cost is \(cE\); the surplus calculation is \(R(E)-cE\).
- Treating the quadratic half-effort result as general. \(E^*=E_{\mathrm{OA}}/2\) depends on \(R(E)=aE-bE^2\) and constant marginal cost. For a different revenue function, derive both conditions again.
- Ignoring boundaries or assumptions. The positive interior solutions require revenue and cost parameters that support them. If no positive effort can cover its cost, the relevant choice may be zero; if revenue is not concave, a first-order condition alone may not identify the global maximum.
The comparison matters because a privately attractive entry decision need not maximize aggregate surplus. This simple model isolates the mechanism: open-access users respond to average returns, while efficient resource use depends on marginal returns. The result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, use the stated model and distinguish the free-entry condition from the social efficiency condition.
- A hypothetical resource has \(R(E)=84E-E^2\) and effort cost \(c=20\) per unit. Find \(E_{\mathrm{OA}}\), \(E^*\), and efficient total surplus. (Answer: \(E_{\mathrm{OA}}=64\), \(E^*=32\), efficient surplus \(=1{,}024\).)
- For a resource with \(R(E)=150\sqrt{E}\) and \(c=30\), calculate both effort levels and the efficient surplus. (Answer: \(E_{\mathrm{OA}}=25\), \(E^*=6.25\), efficient surplus \(=187.5\).)
- Revenue is \(R(E)=100E-2E^2\) and effort costs \(c=28\) per unit. Find the open-access level, efficient level, and surplus loss at open access. (Answer: \(E_{\mathrm{OA}}=36\), \(E^*=18\), surplus loss \(=648\).)
- A quadratic resource has \(R(E)=aE-bE^2\), \(a,b>0\), and cost \(c=20\). Revenue at effort 10 is 360, and open-access effort is 30. Infer \(a\) and \(b\), then find efficient effort and efficient surplus. (Answer: \(a=44\), \(b=0.8\), \(E^*=15\), efficient surplus \(=180\).)
- For \(R(E)=90E-E^2\) with effort cost \(c=18\), calculate open-access revenue and cost, efficient total surplus, and the surplus loss under open access. (Answer: \(E_{\mathrm{OA}}=72\), open-access revenue and cost are each \(1{,}296\), \(E^*=36\), efficient surplus and surplus loss are each \(1{,}296\).)