Why Personalize the Price of a Public Good?
A public good is consumed jointly: providing a quantity \(Q\) makes that quantity available to each person, rather than dividing it among them. As a result, the people who benefit can value an extra unit differently even though they receive the same unit. A single market price cannot generally represent those different marginal valuations while giving everyone the same quantity.
Lindahl pricing addresses this by assigning each person an individualized marginal price. At an efficient quantity, each person chooses that shared quantity at their own price, while the individual prices together equal the marginal cost of supplying it. Thus, the relevant comparison is not whether one person's price equals the full cost. It is whether the personalized prices add up to the cost and each person’s marginal benefit matches their own price.
Here \(MB_i(Q)\) is person \(i\)'s marginal benefit from the public good at quantity \(Q\), and \(MC(Q)\) is the marginal cost of supplying it. The conditions above describe an interior solution: the chosen quantity is positive, marginal benefits and cost are defined there, and people can choose quantities rather than being forced to consume a fixed amount. At a boundary such as \(Q=0\), or where preferences are not differentiable, the relevant conditions are inequalities or other optimality conditions rather than these equalities.
How the Conditions Fit Together
Suppose each person has quasi-linear utility \(u_i(Q)-t_i\), where \(u_i(Q)\) is their benefit from the shared quantity and \(t_i\) is their payment. In the basic constant-price model, person \(i\) pays \(p_iQ\). Choosing \(Q\) to maximize \(u_i(Q)-p_iQ\) gives the interior first-order condition \(u_i'(Q)=p_i\). Since \(u_i'(Q)=MB_i(Q)\), the person's choice condition is \(MB_i(Q)=p_i\).
To see why the prices must add to marginal cost, consider a supplier with cost \(C(Q)\). Efficient provision maximizes the total surplus \(\sum_i u_i(Q)-C(Q)\). Differentiating with respect to the one shared quantity gives \(\sum_i u_i'(Q)-C'(Q)=0\) at an interior maximum. Thus \(\sum_i MB_i(Q)=MC(Q)\). Substituting each person's Lindahl condition, \(MB_i(Q)=p_i\), gives \(\sum_i p_i=MC(Q)\). The individualized prices are therefore a way to make private choices consistent with the efficient total valuation.
This is the public-good counterpart to vertical summation: at any fixed quantity, add the individuals' marginal benefits, not their quantities. The earlier material on public-good valuation and the Samuelson condition establishes that efficiency comparison; Lindahl pricing asks how personalized marginal prices can support the resulting shared quantity.
Finding the Quantity and Personalized Prices
Use the same quantity \(Q\) for every person because the public good is shared.
Set \(\sum_i MB_i(Q)=MC(Q)\) and solve for the efficient quantity, checking that it is feasible and interior.
At the shared efficient quantity, set \(p_i=MB_i(Q^*)\) separately for each person.
Check that each person's price matches their marginal benefit and that the prices sum to marginal cost.
Worked Example: Two People and Constant Marginal Cost
A hypothetical town is choosing the number \(Q\) of public warning beacons. Residents A and B have marginal benefits \(MB_A(Q)=18-2Q\) and \(MB_B(Q)=12-Q\), measured in dollars per beacon. The marginal cost is a constant \(MC=6\) dollars per beacon, and the quantity is continuous. Find the efficient quantity and the Lindahl prices. Check total cost recovery.
Method. Add the marginal benefits and equate their sum to marginal cost:
Combining terms gives \(30-3Q=6\). Subtract 6 from both sides: \(24-3Q=0\). Hence \(3Q=24\), so \(Q^*=8\) beacons. This is feasible and interior.
Evaluate each marginal benefit at \(Q=8\):
The prices are \(2\) dollars per beacon for A and \(4\) dollars per beacon for B. They support the same quantity because A's marginal benefit at 8 equals 2, and B's equals 4. Their sum is \(2+4=6\), exactly the marginal cost. With constant marginal cost and no fixed cost, their payments are \(2(8)=16\) dollars and \(4(8)=32\) dollars. Together they pay \(16+32=48\) dollars, equal to the cost \(6(8)=48\) dollars.
Common mistake avoided: assigning both residents half the marginal cost, or \(3\) dollars, would not support a shared quantity. At price 3, A's marginal-benefit condition gives \(18-2Q=3\), so \(Q=7.5\); B's gives \(12-Q=3\), so \(Q=9\). They would choose different quantities. The correction is to use distinct personalized prices, \(2\) and \(4\), at the shared efficient quantity.
Marginal Prices Are Not Always Total Payments
The simple constant-cost example makes the accounting transparent: if \(C(Q)=cQ\), then constant personalized prices whose sum is \(c\) generate total payments \(\sum_i p_iQ=cQ=C(Q)\). With nonlinear cost, however, the sum of marginal prices at the final quantity equals marginal cost, not necessarily average cost. Multiplying those final marginal prices by the entire quantity may therefore fail to recover total cost.
This distinction matters when the model allows the personalized marginal price to vary with the quantity supplied. In that case, a person's total payment is the accumulated price over the quantity, not automatically the final marginal price multiplied by all units. A financing rule must be specified and checked separately. Fixed costs also require a funding arrangement beyond marginal prices: marginal prices alone do not necessarily cover a fixed cost.
Worked Example: Nonlinear Cost and Cost Recovery
A hypothetical shared flood-monitoring system has cost \(C(Q)=Q^2\) dollars, so \(MC(Q)=2Q\). Two residents have marginal benefits \(MB_1(Q)=20-Q\) and \(MB_2(Q)=16-Q\). Find the efficient quantity and final marginal Lindahl prices. Then show one price-schedule arrangement that recovers total cost.
Method. Equate summed marginal benefits to marginal cost:
Thus \(36-2Q=2Q\), so \(36=4Q\), and \(Q^*=9\). The individual marginal prices at this quantity are \(p_1=20-9=11\) dollars and \(p_2=16-9=7\) dollars. Their sum is \(11+7=18\), which equals \(MC(9)=2(9)=18\).
The system's total cost is \(C(9)=9^2=81\) dollars. If the final marginal prices were incorrectly charged as constant prices on all nine units, payments would be \(11(9)=99\) dollars and \(7(9)=63\) dollars, totalling \(162\) dollars. That exceeds the actual cost of \(81\) dollars. The marginal-price condition is correct, but treating final marginal prices as constant average prices is not.
One cost-recovering arrangement uses quantity-dependent marginal prices \(p_1(q)=\frac{11}{9}q\) and \(p_2(q)=\frac{7}{9}q\), for \(0\leq q\leq9\). At each \(q\), these prices sum to \(2q=MC(q)\), and at \(q=9\) they are 11 and 7. Total payments are the accumulated marginal prices:
The payments sum to \(49.5+31.5=81\) dollars, equal to total cost. The residents' choices also align with \(Q=9\): their marginal benefits there are 11 and 7, equal to their respective marginal prices at 9. Because each marginal-benefit function declines with \(q\) and each price schedule increases, each resident's marginal benefit minus their marginal price changes from positive to negative at 9, making it their maximizing quantity. This example shows why a complete implementation must specify both marginal prices and how total payments are collected.
Personalized Prices, Information, and Limits
A Lindahl arrangement requires information about each person's marginal-benefit schedule and the supplier's marginal cost. Without those inputs, a planner cannot calculate the efficient shared quantity and the corresponding personalized prices. Preferences are not directly observable from the quantity consumed, because everyone receives the same public-good quantity. A person who benefits from a larger quantity may have an incentive to report a lower valuation if doing so reduces their assigned price.
Consequently, calculated Lindahl prices describe a way to align incentives when the relevant valuations and costs are known; they do not by themselves solve the practical problem of obtaining truthful information. Implementation also needs a rule for assigning and enforcing payments, and any fixed cost or other financing requirement must be accounted for. The model is most useful as a benchmark for understanding how individualized marginal charges could support efficient provision.
- One shared quantity: all individuals evaluate the same \(Q\), rather than choosing separate quantities of the public good.
- Different marginal prices: each person's price reflects their own marginal benefit at the efficient quantity.
- Vertical adding-up: the personalized marginal prices sum to marginal cost at that quantity.
- Separate financing check: marginal cost pricing does not automatically settle questions about fixed costs, total payments, or information gathering.
Check Your Understanding
For each problem, assume the stated marginal-benefit functions apply over the relevant range, and use an interior solution unless the calculations show otherwise.
- Two residents value a shared signal network according to \(MB_1(Q)=15-Q\) and \(MB_2(Q)=9-0.5Q\). Marginal cost is constant at 6 dollars per unit. Find the efficient quantity and both Lindahl prices. (Answer: \(Q^*=12\); \(p_1=3\), \(p_2=3\) dollars per unit.)
- Three users have marginal benefits \(MB_1(Q)=22-2Q\), \(MB_2(Q)=16-Q\), and \(MB_3(Q)=10-0.5Q\). Marginal cost is 12 dollars per unit. Find the efficient quantity and all three prices, checking their sum. (Answer: \(Q^*=\frac{72}{7}\); \(p_1=\frac{10}{7}\), \(p_2=\frac{40}{7}\), and \(p_3=\frac{34}{7}\) dollars per unit; their sum is 12.)
- A shared facility has constant marginal cost of 15 dollars per unit and no fixed cost. Two residents have \(MB_1(Q)=25-2Q\) and \(MB_2(Q)=17-Q\). Find the quantity, prices, and total payments; verify cost recovery. (Answer: \(Q^*=9\); \(p_1=7\), \(p_2=8\); payments are 63 and 72 dollars, totalling 135 dollars, equal to cost.)
- A public service has cost \(C(Q)=Q^2\), with marginal benefits \(MB_1(Q)=18-Q\) and \(MB_2(Q)=14-Q\). Find the efficient quantity and final marginal prices. If those final prices are incorrectly charged as constant prices on all units, calculate total payments and compare them with cost. (Answer: \(Q^*=8\); final marginal prices are 10 and 6 dollars per unit; constant-price payments total \(10(8)+6(8)=128\) dollars, while cost is \(8^2=64\) dollars.)
- For the facility in Question 4, state one type of information needed to calculate the Lindahl arrangement and explain why the final marginal prices alone do not show total cost recovery. (Answer: The planner needs each person's marginal-benefit schedule and the marginal-cost schedule. Final marginal prices need not equal average cost, so total payments depend on the specified payment schedules or financing rule.)