Calculating the Efficient Public-Good Quantity

From Individual Benefits to an Efficient Quantity

For a private good, different consumers can choose different quantities. A pure public good is different: once it is provided, people can share the same quantity, whether or not they contribute to its cost. To decide how much to provide efficiently, the relevant question is how much all beneficiaries together value one more unit, compared with the resources needed to supply it.

This tutorial focuses on that calculation. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. The central operation is vertical summation: at a given quantity of the public good, add each person’s marginal benefit. The resulting aggregate marginal benefit is compared with marginal cost.

Definition: Let $Q\geq 0$ be the quantity of a pure public good. For person $i$, let $MB_i(Q)$ be that person’s marginal benefit from a small additional amount when the shared quantity is $Q$. Let $C(Q)$ be the resource cost of providing $Q$, and let $MC(Q)=C'(Q)$ where the derivative exists. In the surplus model used here, total social surplus is $$ W(Q)=\sum_{i=1}^{n}B_i(Q)-C(Q), $$ where $B_i'(Q)=MB_i(Q)$. An efficient quantity maximizes $W(Q)$ over feasible quantities.

This calculation treats the marginal-benefit schedules as willingness to pay in a common monetary unit and uses the unweighted sum of benefits. It assumes the relevant benefits and costs are included, and that the surplus comparison is appropriate—for example, preferences can be represented with money-metric benefits without income-effect complications. The quantity is shared by all individuals; we do not add their quantities together. Distributional weights, financing rules, and who pays are separate questions.

The Efficient-Quantity Rule

At an interior efficient quantity where the schedules are differentiable, a small increase in provision changes social surplus by the aggregate marginal benefit less marginal cost. An efficient interior choice therefore satisfies the Samuelson condition: the sum of individual marginal benefits equals marginal cost.

\sum_{i=1}^{n} MB_i(Q^*)=MC(Q^*)

The superscript $^{*}$ denotes an efficient quantity. The equality is a candidate condition, not a complete check by itself. If marginal benefits are nonincreasing and marginal cost is nondecreasing, then the difference between aggregate marginal benefit and marginal cost is nonincreasing. A crossing from positive to negative identifies a global maximum; if the schedules are strictly sloped, that maximum is unique. At a boundary, a kink, or a zero-benefit cutoff, check the relevant one-sided values and feasible endpoints rather than forcing a derivative equation outside its valid range.

Write each person’s schedule.
Use the same quantity $Q$ and monetary units for every marginal-benefit schedule. Record any range restriction or cutoff.

Vertically sum benefits.
At a given $Q$, calculate $AMB(Q)=\sum_i MB_i(Q)$. Do not add quantities or average the schedules.

Compare with marginal cost.
Solve $AMB(Q)=MC(Q)$ on the interval where the formulas apply, then check the solution, cutoffs, and boundaries.

Why the Condition Works

The condition follows from how a small change in provision affects total surplus. For a small increase $\Delta Q$, each beneficiary receives an additional benefit of approximately $MB_i(Q)\Delta Q$. The total additional benefit is approximately $\sum_i MB_i(Q)\Delta Q$. The added resource cost is approximately $MC(Q)\Delta Q$. Thus the approximate change in surplus is:

\Delta W \approx \left[\sum_{i=1}^{n}MB_i(Q)-MC(Q)\right]\Delta Q

If aggregate marginal benefit exceeds marginal cost, a small increase raises surplus. If it is below marginal cost, a small decrease raises surplus. An interior maximum occurs where neither direction improves surplus, giving equality. More formally, differentiating the surplus function gives $W'(Q)=\sum_i MB_i(Q)-MC(Q)$. When this derivative changes from positive to negative, surplus rises up to the crossing and falls afterward. This argument also explains why a solution must be checked against schedule cutoffs: the marginal-benefit formula can change at a cutoff.

Worked Calculations

Worked Example: Two Different Linear Valuations

A neighborhood is considering a shared warning beacon. In a hypothetical surplus model, the marginal benefits per unit of beacon capacity are $MB_1(Q)=26-2Q$ dollars and $MB_2(Q)=14-Q$ dollars for $Q\geq0$. Marginal cost is $MC(Q)=4+Q$ dollars. Find the efficient quantity and the maximum surplus. Both benefit schedules are positive at the candidate quantity.

Method. Vertically sum the two marginal-benefit schedules, equate the sum with marginal cost, then integrate the schedules to verify the surplus. Since the schedules are decreasing and marginal cost is increasing, a valid crossing is the unique maximum.

The aggregate marginal benefit is $AMB(Q)=(26-2Q)+(14-Q)=40-3Q$. Set it equal to marginal cost:

40-3Q=4+Q,\qquad 36=4Q,\qquad Q^*=9.

At $Q=9$, person 1’s marginal benefit is $26-2(9)=8$ dollars and person 2’s is $14-9=5$ dollars. Their sum is $8+5=13$ dollars. Marginal cost is $4+9=13$ dollars, so the equality checks.

For total benefits, integrate each marginal-benefit schedule from zero to $Q$: $B_1(Q)=26Q-Q^2$ and $B_2(Q)=14Q-\tfrac12Q^2$. Thus $B_1(9)=234-81=153$ dollars and $B_2(9)=126-40.5=85.5$ dollars, for total benefits of $238.5$ dollars.

Integrating marginal cost gives $C(Q)=4Q+\tfrac12Q^2$, so $C(9)=36+40.5=76.5$ dollars. Maximum surplus is $238.5-76.5=162.0$ dollars. As a final check, the marginal-benefit difference is $AMB-MC=36-4Q$, positive below 9 and negative above 9.

Common mistake avoided: Adding marginal benefits at the solution without evaluating each schedule at the same quantity can produce a false equality. Here both schedules are evaluated at $Q=9$, then summed.

Worked Example: A Marginal-Benefit Cutoff

A hypothetical community project has two beneficiaries. Person A’s marginal benefit is $MB_A(Q)=20-Q$ for $0\leq Q\leq20$. Person B’s marginal benefit is $MB_B(Q)=12-2Q$ for $0\leq Q\leq6$, and is zero for $Q>6$: B receives no additional benefit beyond six units. Marginal cost is $MC(Q)=6+Q$. Find the efficient quantity and surplus.

Method. Solve separately on either side of B’s cutoff. The cutoff matters because the aggregate schedule changes there.

For $0\leq Q\leq6$, both people have positive marginal benefits, so $AMB(Q)=(20-Q)+(12-2Q)=32-3Q$. Equating this to $6+Q$ gives $32-3Q=6+Q$, hence $Q=6.5$. This candidate is outside the interval used to derive the equation, so it cannot be accepted as the solution on that interval.

For $Q>6$, B’s marginal benefit is zero and $AMB(Q)=20-Q$. The relevant equation is now $20-Q=6+Q$, giving $Q^*=7$. At $Q=7$, A’s marginal benefit is $20-7=13$, B’s is zero, and marginal cost is $6+7=13$. The schedules are decreasing and cost is increasing; moreover, aggregate marginal benefit exceeds cost below 7 and is below cost above 7.

To verify total surplus, total benefit for A at 7 is $B_A(7)=20(7)-\tfrac12(7^2)=140-24.5=115.5$ dollars. B’s benefit stops increasing after six units: $B_B(7)=\int_0^6(12-2Q)\,dQ=72-36=36$ dollars. Total benefit is $115.5+36=151.5$ dollars. Cost is $C(7)=6(7)+\tfrac12(7^2)=42+24.5=66.5$ dollars. Therefore, surplus is $151.5-66.5=85.0$ dollars.

Common mistake avoided: Extending B’s positive-benefit formula beyond $Q=6$ would incorrectly count marginal benefits that the problem says are zero. Always check that a candidate lies in the interval for its formula.

Worked Example: A Kink in Marginal Cost

Three groups value a hypothetical shared flood-monitoring system. Their marginal-benefit schedules are $MB_1(Q)=24-2Q$, $MB_2(Q)=16-Q$, and $MB_3(Q)=10-\tfrac12Q$. Marginal cost is $MC(Q)=8+Q$ for $0\leq Q\leq6$, and $MC(Q)=14+2(Q-6)$ for $Q>6$. Find the efficient quantity and check total surplus.

Method. Sum all three schedules, then solve using the marginal-cost expression that applies on the resulting interval. The cost schedule is continuous at six, but its slope changes there.

The aggregate marginal benefit is $AMB(Q)=(24-2Q)+(16-Q)+(10-\tfrac12Q)=50-\tfrac72Q$. On the first cost interval, the candidate equation is $50-\tfrac72Q=8+Q$, which gives $\tfrac{63}{2}Q=42$, or $Q=\tfrac43$. This value is within the first interval—but it would be a valid answer only if the first-interval equation had been set up correctly. Rechecking the algebra, $50-8=42$ and $\tfrac72Q+Q=\tfrac92Q$, so $Q=42/(\tfrac92)=\tfrac{28}{3}$, which is not in the first interval.

Since that candidate exceeds six, use the second marginal-cost expression: $MC(Q)=14+2(Q-6)=2Q+2$. Then $50-\tfrac72Q=2Q+2$, so $48=\tfrac{11}{2}Q$, giving $Q^*=\tfrac{96}{11}$, approximately $8.727$ units. The candidate lies above six, as required. At this quantity, $AMB=50-\tfrac72(\tfrac{96}{11})=\tfrac{214}{11}$ dollars, and $MC=2(\tfrac{96}{11})+2=\tfrac{214}{11}$ dollars.

For a surplus check, total benefit is $B(Q)=50Q-\tfrac74Q^2$. On the second cost interval, integrate marginal cost from six and add cost up to six: $C(6)=8(6)+\tfrac12(6^2)=66$, and $C(Q)=66+\int_6^Q(2t+2)\,dt=Q^2+2Q+18$. At $Q=\tfrac{96}{11}$, total benefit is $\tfrac{36672}{121}$ dollars and cost is $\tfrac{13506}{121}$ dollars. Surplus is therefore $\tfrac{23166}{121}$, or $191.455$ dollars to three decimal places. Aggregate marginal benefit is decreasing while marginal cost is increasing on both intervals, so the crossing gives the global maximum.

Common mistake avoided: A candidate from one piece of a cost schedule is not automatically valid. Check its interval, and if it falls outside, solve with the other piece rather than reporting it.

Common Calculation Traps

The arithmetic is usually short; the main risks are applying the right operation to the wrong schedules or range. Keep the following checks in view:

Do not horizontally add quantities. Everyone receives the same $Q$. At that quantity, add marginal benefits vertically: $AMB(Q)=\sum_i MB_i(Q)$.
Evaluate each schedule at the same quantity. The sum is not obtained by mixing values from different quantities or by averaging the beneficiaries’ marginal benefits.
Respect units. If benefits are dollars per unit and costs are thousands of dollars per unit, convert one side before equating them.
Check the formula’s domain. A linear schedule may be specified only up to a cutoff, or marginal cost may change at a kink.
Verify the direction of the surplus change. Equality alone does not establish a maximum if schedules are nonmonotonic or the candidate is at a boundary. Compare the marginal-benefit and marginal-cost values on either side, or evaluate feasible alternatives.
Keep efficiency distinct from payment. The efficient quantity identifies the total provision that maximizes the stated surplus. It does not by itself specify how the cost is divided among beneficiaries.

Key takeaway: For a shared public-good quantity, calculate each person’s marginal benefit at that quantity and add the values. The efficient interior quantity equates this aggregate marginal benefit with marginal cost, subject to checking cutoffs, kinks, feasibility, and the direction of the surplus change.

Check Your Understanding

For each problem, use the stated marginal-benefit and cost schedules, check that the result lies in the applicable range, and verify the marginal equality.

A two-person community has $MB_1(Q)=18-2Q$, $MB_2(Q)=12-Q$, and $MC(Q)=6+Q$. Find the efficient quantity and state each person’s marginal benefit and marginal cost there. (Answer: $Q^*=6$; marginal benefits are 6 and 6, summing to marginal cost 12.)
A shared service has $MB_1(Q)=21-3Q$, $MB_2(Q)=9-Q$, and $MC(Q)=5+Q$. Find the efficient quantity and aggregate marginal benefit. (Answer: $Q^*=5$; aggregate marginal benefit is $6+4=10$, equal to marginal cost 10.)
Person A has $MB_A(Q)=18-Q$. Person B has $MB_B(Q)=12-2Q$ through $Q=6$ and zero thereafter. If $MC(Q)=6+Q$, find the efficient quantity and identify which benefit schedule is zero there. (Answer: $Q^*=6$; B’s marginal benefit is zero at the cutoff and A’s is 12, equal to marginal cost 12.)
Three beneficiaries have $MB_1(Q)=30-2Q$, $MB_2(Q)=18-Q$, and $MB_3(Q)=12-\tfrac12Q$. Marginal cost is $MC(Q)=12+Q$ for $Q\leq8$, and $MC(Q)=20+2(Q-8)$ for $Q>8$. Find the efficient quantity and determine which cost interval applies. (Answer: $Q^*=\tfrac{40}{3}\approx13.333$; the second interval applies, and aggregate marginal benefit and marginal cost both equal $\tfrac{56}{3}$.)
Aggregate marginal benefit is $36-2Q$. Marginal cost is $4+Q$ for $Q\leq8$, and $12+2(Q-8)$ for $Q>8$. Find the efficient quantity and explain why solving with the first cost formula alone is insufficient. (Answer: $Q^*=10$; the first-formula candidate is $Q=\tfrac{32}{3}>8$, so use the second interval, where both schedules equal 16 at $Q=10$.)

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