Why Public-Good Valuation Uses a Different Sum
For a private good, different consumers can choose different quantities. At a given price, market demand adds those quantities across consumers. A nonrival public good is different: everyone can benefit from the same amount provided. If a community receives \(Q\) units of flood-warning coverage, for example, each resident experiences that same provision level, even if residents value it differently. To describe the group’s willingness to pay for another unit, we add their marginal values at that common quantity.
This operation is called vertical summation. It produces an aggregate marginal-benefit curve: at each quantity, the curve gives the sum of the individuals’ marginal willingness to pay for that unit. The group’s total willingness to pay through a specified quantity is the area under this aggregate curve, provided the individual monetary valuations can be added.
The setup assumes a common quantity of the nonrival good is available to all individuals and that their dollar valuations are expressed on a comparable basis. A marginal benefit may depend on income and other circumstances; the schedule used in a calculation is understood to hold those circumstances fixed. If a stated schedule becomes negative, ordinary willingness-to-pay demand is usually truncated at zero: a person is not treated as paying to receive additional units. The curve can therefore change formula when someone’s marginal benefit reaches zero.
Constructing the Aggregate Curve
To construct the curve, choose a quantity and add the marginal willingness to pay of every person who still values an additional unit at that quantity. Repeat across quantities. If individual schedules are linear, the aggregate schedule is linear between the quantities where an individual’s curve reaches zero. Those points are breakpoints; beyond a breakpoint, that person no longer contributes to the sum.
Record marginal willingness to pay as a function of the shared quantity, and find where each schedule reaches zero.
Use the choke quantities to divide the quantity axis into intervals on which the set of contributing individuals is unchanged.
Sum the active individuals’ marginal benefits at the same \(Q\), and check that the formulas agree at each breakpoint.
Calculate the area under the aggregate curve, splitting the integral at every breakpoint it crosses.
The reason the sum is vertical can be seen from a small increase \(dQ\) in provision. Individual \(i\)’s willingness to pay for that increment is approximately \(MB_i(Q)dQ\). Adding across people gives \(\sum_i MB_i(Q)dQ\). Dividing by the common increment \(dQ\) gives the aggregate marginal benefit:
Accumulating those marginal values from zero to \(Q\) gives the group’s total willingness to pay. For continuous schedules this is an integral; for a piecewise schedule, it is the sum of the integrals over its separate intervals:
This is an aggregation rule, not a conclusion about how much of the good should be provided. It describes total marginal valuation at each quantity. Determining an efficient provision level requires comparing aggregate benefits with the relevant costs, a separate question.
Worked Example: Summing Two Residents’ Marginal Benefits
A hypothetical town evaluates a nonrival drainage-monitoring service. Resident A’s marginal willingness to pay is \(MB_A(Q)=20-2Q\), and resident B’s is \(MB_B(Q)=12-Q\), in dollars per additional unit. Both schedules are used only while positive. Find the aggregate marginal benefit and the group’s total willingness to pay through \(Q=8\).
Method and cutoffs. Set each schedule equal to zero. Resident A’s choke quantity is \(20-2Q=0\), so \(Q=10\). Resident B’s is \(12-Q=0\), so \(Q=12\). Both residents contribute from \(Q=0\) through \(Q=10\); between \(Q=10\) and \(Q=12\), only B contributes.
Because \(Q=8\) is below both choke quantities, the aggregate marginal benefit throughout the requested interval is \(MB_G(Q)=(20-2Q)+(12-Q)=32-3Q\). At \(Q=8\), it equals \(32-24=8\) dollars per unit. This also equals A’s \(20-16=4\) plus B’s \(12-8=4\).
Total willingness to pay. Integrate the aggregate schedule from zero to eight:
Check. A’s total is \(\int_0^8(20-2Q)\,dQ=160-64=96\) dollars. B’s is \(\int_0^8(12-Q)\,dQ=96-32=64\) dollars. Their sum is \(96+64=160\) dollars, as obtained from the aggregate curve. The method avoids the mistake of adding quantities at a common price, which would be appropriate for private-good demands, not this shared public-good quantity.
Piecewise Curves and Total Valuation
When one person’s marginal benefit reaches zero, the aggregate curve’s slope can change because that person drops out of the sum. The total valuation still comes from the area under the curve, but the integral must be split at that point. A useful check is to calculate total willingness to pay both by integrating the aggregate curve and by adding each individual’s area. The two methods should agree.
Worked Example: A Breakpoint Before the Requested Quantity
Suppose three residents value a shared wildfire-alert system according to \(MB_1(Q)=16-2Q\), \(MB_2(Q)=10-Q\), and \(MB_3(Q)=6-2Q\), in dollars per unit while positive. Find the aggregate marginal-benefit schedule and total willingness to pay through \(Q=4\).
Method and cutoffs. The choke quantities are \(16-2Q=0\), or \(Q=8\), for resident 1; \(10-Q=0\), or \(Q=10\), for resident 2; and \(6-2Q=0\), or \(Q=3\), for resident 3. Thus, all three schedules contribute through \(Q=3\). After \(Q=3\), only residents 1 and 2 contribute.
For \(0\leq Q\leq3\), add all three schedules: \(MB_G(Q)=16-2Q+10-Q+6-2Q=32-5Q\). For \(3<Q\leq4\), resident 3 contributes zero, so \(MB_G(Q)=16-2Q+10-Q=26-3Q\). The formulas agree at the breakpoint: \(32-5(3)=17\) and \(26-3(3)=17\).
Total willingness to pay. Since the requested quantity crosses the breakpoint, integrate separately:
Check. Resident 1’s total through \(Q=4\) is \(16(4)-4^2=48\) dollars. Resident 2’s is \(10(4)-\frac12(4^2)=32\) dollars. Resident 3 values only the first three units, giving \(6(3)-3^2=9\) dollars. The individual totals sum to \(48+32+9=89\) dollars. The common mistake is to continue adding resident 3’s negative marginal benefit after \(Q=3\); truncating at the choke quantity gives the correct total.
Why Private-Good Demand Is Summed Horizontally
For a private good, the quantity purchased by one person is not automatically consumed by everyone else. At a given market price, each consumer chooses an individual quantity. Aggregate market demand therefore adds those quantities while holding price fixed. This is horizontal summation. It answers, “How much will all consumers together demand at this price?” Public-good vertical summation instead holds the shared quantity fixed and adds marginal willingness to pay.
| Feature | Nonrival public good | Private good |
|---|---|---|
| What is common when adding? | The provision quantity \(Q\) | The market price \(P\) |
| What is added? | Individuals’ marginal willingness to pay | Individuals’ demanded quantities |
| Aggregation name | Vertical summation | Horizontal summation |
| Typical result | Aggregate marginal-benefit schedule | Market demand schedule |
Worked Example: Horizontal Summation for a Private Good
Two customers have private-good demands \(P=24-q_A\) and \(P=18-2q_B\), where \(P\) is dollars per unit and quantities are nonnegative. Find total market demand at \(P=12\), and give the inverse aggregate demand over the relevant ranges.
Method. At a fixed price, solve each individual demand for quantity. Customer A demands \(q_A=24-P\), so at \(P=12\), \(q_A=24-12=12\). Customer B demands \(q_B=(18-P)/2\) when \(P\leq18\), so at \(P=12\), \(q_B=(18-12)/2=3\). The market quantity is \(Q=12+3=15\) units.
To obtain the aggregate inverse demand, note that customer B’s demand is zero at prices above \(18\), while customer A’s demand reaches zero at \(24\). For \(18<P\leq24\), only A buys, so \(Q=24-P\), or \(P=24-Q\). This corresponds to \(0\leq Q<6\), since at \(P=18\), A demands six units.
For \(P\leq18\), both customers buy. Their combined quantity is \(Q=(24-P)+(18-P)/2=33-\frac32P\). Solving for price gives \(P=22-\frac23Q\), which applies for \(Q\geq6\) while the demands remain nonnegative. At \(P=12\), this formula gives \(P=22-\frac23(15)=12\), confirming the calculation. The common mistake is to add inverse-demand prices at a fixed quantity; for private goods, add quantities at a fixed price.
Interpretation and Common Exam Traps
Aggregate willingness to pay is a monetary measure of the group’s valuation of provision increments. It is not the amount any single person pays, and it does not imply that everyone has the same preferences. A high marginal value for one resident can coexist with a low value for another; vertical summation preserves that heterogeneity by adding values at each shared quantity.
- Summing at the wrong coordinate. For a public good, hold quantity fixed and add marginal benefits. For a private good, hold price fixed and add quantities.
- Ignoring a choke quantity. Once a person’s marginal willingness to pay reaches zero, do not include a negative value in the ordinary aggregate valuation curve.
- Integrating across a breakpoint with one formula. If the set of contributing individuals changes, split the area calculation at that quantity.
- Confusing marginal and total values. \(MB_G(Q)\) is a rate in dollars per unit; total willingness to pay through \(Q\) is the area under that schedule and is measured in dollars.
- Interpreting the aggregate curve as a provision decision. The curve reports aggregate marginal valuation. It does not alone determine the quantity to provide.
Check Your Understanding
For each problem, identify the appropriate aggregation method before calculating. Assume stated marginal benefits are truncated at zero when they reach their choke quantities.
- A shared public service has \(MB_A(Q)=18-3Q\) and \(MB_B(Q)=10-Q\). Find aggregate marginal benefit at \(Q=2\) and total group willingness to pay through \(Q=2\). (Answer: \(20\) dollars per unit; \(48\) dollars total.)
- For a nonrival good, three groups have \(MB_1(Q)=16-2Q\), \(MB_2(Q)=10-Q\), and \(MB_3(Q)=6-2Q\). Give the aggregate marginal-benefit formula on \(0\leq Q\leq4\), stating the breakpoint, and find total willingness to pay through \(Q=4\). (Answer: \(32-5Q\) for \(0\leq Q\leq3\), then \(26-3Q\) for \(3<Q\leq4\); breakpoint \(Q=3\); total \(89\) dollars.)
- For a private good, two consumers have demands \(P=30-2q_1\) and \(P=18-q_2\). At \(P=10\), find each quantity and total market quantity. (Answer: \(q_1=10\), \(q_2=8\), and \(Q=18\) units.)
- A shared public good has \(MB_A(Q)=25-5Q\), \(MB_B(Q)=15-3Q\), and \(MB_C(Q)=6-2Q\). Find the aggregate marginal benefit at \(Q=3\) and total group willingness to pay through \(Q=4\), accounting for any schedule that has reached zero. (Answer: \(16\) dollars per unit at \(Q=3\); \(105\) dollars total through \(Q=4\).)