Why Public-Good Efficiency Uses a Sum
For a private good, one person’s consumption does not normally determine how much another person consumes. A pure public good is different: it is nonrival, so one person’s use does not reduce another’s, and nonexcludable, so people cannot readily be kept from benefiting. If a community provides a street-lighting system, for example, several residents can benefit from the same provision. The key efficiency question is therefore not whether each resident individually values one more unit as much as it costs. It is whether the residents’ combined marginal benefit matches the resources used to provide it.
This is the Samuelson condition. It is a marginal condition: it compares the total value individuals place on a small increase in the public good with the opportunity cost of producing that increase. It does not, by itself, identify a particular quantity; it states the condition an interior efficient quantity must satisfy.
Suppose producing \(G\) uses \(C(G)\) units of the private numeraire, with \(C\) differentiable. Then \(C'(G)\) is the marginal resource cost: the additional private resources required for a small increase in public-good provision. More generally, if the economy’s production possibilities trade the private good for \(G\), the marginal rate of transformation (MRT) is the amount of private good that must be forgone to produce one more unit of \(G\). In this setting, \(MRT=C'(G)\).
Deriving the Condition
A useful derivation uses a social-planner problem. The planner chooses each person’s private consumption and the common public-good quantity, subject to the economy’s resource constraint. To represent a particular supported Pareto-efficient allocation, maximize a weighted sum of utilities, \(\sum_i \alpha_i U_i(x_i,G)\), where each welfare weight \(\alpha_i\) is positive. This weighted-sum representation is not automatic for every Pareto-efficient allocation: it requires suitable convexity and regularity assumptions, such as a convex attainable utility set. The derivation below applies to solutions of this planner problem.
Let total available resources be \(Y\), so feasibility requires \(\sum_i x_i+C(G)\leq Y\). At an interior solution with a binding resource constraint, form the Lagrangian with multiplier \(\lambda\) on that constraint. The first-order conditions for private consumption and public-good provision are:
The first set says that, after applying the welfare weights, the marginal social value of private consumption is equalized across people and to the shadow value of resources. Divide the public-good condition by \(\lambda\). From the private-consumption conditions, \(\alpha_i/\lambda=1/U_{ix}\). Substituting this for each person gives:
Each ratio on the middle expression is that individual’s \(MRS_i\). Therefore \(\sum_i MRS_i=C'(G)\), or equivalently the sum equals the MRT. If utilities are concave and the feasible set is convex, the weighted objective is concave and these first-order conditions characterize a maximum, subject to the usual qualification for boundaries. Without appropriate convexity, a first-order condition need not establish a global maximum.
The sum appears because an extra unit of the public good is enjoyed by every individual, while it is produced only once. Each individual’s \(MRS_i\) measures that person’s marginal willingness to trade private resources for the shared increase. Adding these marginal valuations gives the total marginal benefit in private-good units. This is vertical summation: for a fixed \(G\), add the individuals’ marginal valuations rather than treating them as quantities demanded by separate consumers.
Worked Applications
Worked Example: Checking the Condition at a Feasible Allocation
Two residents have utilities \(U_1(x_1,G)=x_1+18\ln(G+1)\) and \(U_2(x_2,G)=x_2+12\ln(G+1)\). The public good costs \(C(G)=G^2/2\). At \(G=5\), the proposed private consumptions are \(x_1=9\) and \(x_2=6\), and total resources are \(Y=27.5\). Check feasibility and determine whether the interior Samuelson condition holds.
Method. Differentiate each utility to find its marginal rate of substitution, add the two rates, and compare the sum with marginal cost.
For individual 1, \(U_{1G}=18/(G+1)\) and \(U_{1x}=1\). Thus \(MRS_1=18/(G+1)\), which at \(G=5\) is \(18/6=3\). For individual 2, \(U_{2G}=12/(G+1)\) and \(U_{2x}=1\). Thus \(MRS_2=12/(G+1)=12/6=2\) at \(G=5\). The total marginal valuation is \(3+2=5\).
The marginal cost is \(C'(G)=G\), so \(C'(5)=5\). The two sides are equal: \(MRS_1+MRS_2=5=MRT\). The proposed allocation satisfies the interior marginal condition.
Check resource feasibility separately. Public-good cost is \(C(5)=5^2/2=25/2=12.5\). Total use is \(x_1+x_2+C(5)=9+6+12.5=27.5=Y\), so the allocation is feasible and the resource constraint binds. Common mistake avoided: equality of marginal benefits and cost does not replace the separate check that the proposed allocation satisfies the resource constraint.
Worked Example: Adding Marginal Valuations, Not Averaging Them
Two households have utilities \(U_i(x_i,G)=\sqrt{x_iG}\). At a proposed interior allocation, \(x_1=8\), \(x_2=12\), and \(G=4\). Producing the public good costs \(C(G)=3G\). Find each household’s marginal rate of substitution, compare the total with marginal cost, and interpret the result.
Method. Differentiate utility with respect to both goods, form the ratio for each household, and vertically sum those ratios.
For either household, \(U_{ix}=\tfrac12\sqrt{G/x_i}\) and \(U_{iG}=\tfrac12\sqrt{x_i/G}\). Dividing gives \(MRS_i=x_i/G\). Therefore \(MRS_1=8/4=2\), and \(MRS_2=12/4=3\). The total marginal valuation is \(2+3=5\).
Marginal cost is \(C'(G)=3\), so the MRT is 3. Since \(5>3\), the interior Samuelson equality does not hold at this allocation: the combined marginal willingness to give up private resources exceeds the marginal resources required. Under the stated smooth, interior setting, this allocation fails the necessary marginal condition for efficiency.
Common mistake avoided: averaging the individual rates would give \(2.5\), which is not the public-good marginal benefit. Both households benefit from the same additional unit, so their valuations are added, not averaged.
Worked Example: Marginal Valuation Depends on Private Consumption
Two individuals have utilities \(U_1(x_1,G)=\ln x_1+4\ln G\) and \(U_2(x_2,G)=2\ln x_2+3\ln G\). A production technology has resource cost \(C(G)=\tfrac98G^2\). At \(x_1=6\), \(x_2=8\), and \(G=4\), check feasibility if \(Y=32\), then test the Samuelson condition.
Method. Compute the marginal utilities at the specified allocation. In these preferences, unlike the previous example, each individual’s MRS depends on their private consumption as well as on \(G\).
For individual 1, \(U_{1G}=4/G\) and \(U_{1x}=1/x_1\), so \(MRS_1=(4/G)/(1/x_1)=4x_1/G\). At the given allocation, \(MRS_1=4(6)/4=6\).
For individual 2, \(U_{2G}=3/G\) and \(U_{2x}=2/x_2\), so \(MRS_2=(3/G)/(2/x_2)=3x_2/(2G)\). Thus \(MRS_2=3(8)/(2(4))=24/8=3\). Their total marginal valuation is \(6+3=9\).
Marginal cost is \(C'(G)=\tfrac94G\). At \(G=4\), it is \((9/4)(4)=9\), so the sum of MRS values equals the MRT. For the feasibility check, \(C(4)=(9/8)(16)=18\); total resource use is \(6+8+18=32=Y\). The allocation is feasible and satisfies the interior marginal condition. Common mistake avoided: using only the coefficients on \(\ln G\) misses the effect of private consumption on each person’s MRS.
When the Equality Needs Qualification
The Samuelson condition is a marginal efficiency result, not a rule that every observed provision level must satisfy as an equality. Its standard form assumes a smooth choice strictly inside the feasible range. If provision is constrained to be nonnegative and the best choice is \(G=0\), for example, an equality need not hold there. With diminishing marginal benefits and a differentiable cost function, a zero-provision corner can be optimal when the sum of marginal valuations at zero is no greater than marginal cost at zero. At a maximum permitted quantity, the corresponding one-sided comparison is reversed.
The condition is also specific to a public good that provides the same quantity to all individuals. For a private good, one ordinarily compares an individual’s marginal willingness to pay with the relevant market price; summing individual MRS values is not the same efficiency condition. Nor should a tax or contribution paid by one person automatically be treated as the social resource cost: transfers redistribute purchasing power, whereas the MRT measures the real opportunity cost of producing the good.
For exam problems, identify the shared public-good quantity, calculate each person’s MRS at the same allocation, add those values, and compare the total with the MRT in the same units. The left side is measured in units of private good per extra unit of public good; the right side must use those same units. If preferences, costs, or constraints place the choice at a boundary, state the relevant corner qualification rather than forcing an interior equality.
Check Your Understanding
For each problem, calculate the individual marginal rates of substitution at the stated allocation, then use the appropriate marginal-cost comparison.
- Two residents have utilities \(U_1=x_1+10\ln(G+1)\) and \(U_2=x_2+5\ln(G+1)\). The cost is \(C(G)=G^2/2\). At \(G=4\), find both MRS values, their sum, and marginal cost. Does the interior equality hold? (Answer: \(MRS_1=2\), \(MRS_2=1\), sum \(=3\), marginal cost \(=4\); no, total marginal valuation is 1 below marginal cost.)
- Two individuals have \(U_i=\sqrt{x_iG}\). At \(x_1=9\), \(x_2=15\), and \(G=6\), the cost is \(C(G)=G^2/3\). Calculate the total MRS and MRT and state whether the condition holds. (Answer: \(MRS_1=9/6=1.5\), \(MRS_2=15/6=2.5\), sum \(=4\); \(MRT=C'(6)=2(6)/3=4\), so the condition holds.)
- Utilities are \(U_1=\ln x_1+6\ln G\) and \(U_2=2\ln x_2+3\ln G\). At \(x_1=4\), \(x_2=12\), \(G=3\), cost is \(C(G)=7G^2/3\), and \(Y=37\). Check feasibility and the Samuelson condition. (Answer: \(MRS_1=6(4)/3=8\), \(MRS_2=3(12)/(2(3))=6\), sum \(=14\); \(C'(3)=14\). Cost is \(21\), so total resource use is \(4+12+21=37\); both checks hold.)
- Three individuals at a candidate interior allocation have MRS values of 1.2, 2.3, and 1.5 units of private good per unit of public good. The MRT is 4.5 in the same units. Add the marginal valuations and assess the condition. (Answer: the sum is \(1.2+2.3+1.5=5.0\), which exceeds the MRT by \(0.5\); the interior equality does not hold.)
- At a zero-provision boundary, two individuals’ marginal valuations are 2 and 3 units of private good per unit of public good. Marginal cost at zero is 6, and marginal benefits decline as provision rises while marginal cost does not decline. Is equality required at the boundary, and is increasing provision locally justified? (Answer: equality is not required at the boundary; the marginal valuation sum is \(5<6\), so a small increase has marginal cost greater than marginal benefit and is not locally justified.)