Budget Allocation Across Diminishing-Return Projects

Why Marginal Returns Guide Allocation

Suppose a fixed budget can be distributed among several projects. Each project produces a return that depends on how much funding it receives. If returns have diminishing marginal gains, putting another dollar into an already well-funded project may be less valuable than putting it into a less-funded one.

The allocation problem is to choose nonnegative funding amounts that use the budget and maximize total return. The key principle is to compare marginal returns: at an optimum, projects receiving positive funding should have equal marginal return. A project receiving no funding follows a related inequality rather than an equality.

Model and Marginal-Return Conditions

Assume each \(f_i\) is differentiable, increasing, and concave on the relevant nonnegative funding range. Concavity means that marginal returns do not increase as funding rises; when \(f_i\) is twice differentiable, a sufficient condition is \(f_i''(x_i)\le 0\). If \(f_i''(x_i)<0\) throughout, the return is strictly concave. The feasible allocations form a closed, bounded simplex, so continuous returns attain a maximum there.

For a project with positive funding, a small transfer of money from it to another positively funded project must not improve the total return. This gives the marginal balancing rule. A project at zero funding cannot be reduced further, so its condition is an inequality.

The common marginal return \(\lambda\) is the rate of return from a small increase in funding to any project already receiving a positive allocation. The boundary inequality says that a project left unfunded would not offer a higher initial marginal return than the funded projects.

Why the Conditions Prove a Global Maximum

Concavity supplies the justification. For a differentiable concave function \(f_i\), its graph lies below the tangent line at any point \(a\) in its domain:

Apply this inequality with \(a=x_i^*\), and compare the candidate allocation with any other feasible allocation \(y=(y_1,\ldots,y_n)\). If \(x_i^*>0\), the candidate conditions give \(f_i'(x_i^*)=\lambda\). If \(x_i^*=0\), then \(y_i\ge 0\), so \(f_i'(0)y_i\le\lambda y_i\). Summing the tangent inequalities therefore gives:

Thus no feasible allocation has a larger total return, which proves global optimality. If the functions are strictly concave, the tangent inequality is strict whenever a coordinate changes, so a different feasible allocation cannot tie the candidate. That proves uniqueness.

For an interior optimum, where every project receives positive funding, the boundary case does not arise: the marginal returns of all projects must simply be equal. When solving, however, one must check whether the resulting allocation is indeed positive. If it is not, the zero-allocation condition must be used.

Worked Example: Logarithmic Returns with All Projects Funded

When a Project Receives No Funding

Equal-marginal equations can produce a negative proposed allocation. That does not mean the real allocation should be negative; it signals that the all-projects-positive assumption is invalid. A project at zero funding is handled by comparing its marginal return at zero with the common marginal return of the funded projects.

Worked Example: Balancing Square-Root Returns

A Common Error: Ignoring the Boundary

A frequent mistake is to solve equal-marginal equations for every project and accept the result without checking whether all allocations are nonnegative. In the logarithmic example with budget \(2\), the all-project calculation produced \(x_3=-3/8\). That value cannot represent funding. The error is not in the algebra: it is in applying an interior condition to a boundary optimum.

The correction is to set the infeasible project's allocation to zero, solve for the funded projects using the budget, and then check the boundary inequality. In that example, the third project's marginal return at zero was \(1\), which is below the funded projects' common marginal return \(7/4\). Therefore, directing some funding to project three would lower the total return.

Using the Rule in Practice

For logarithmic returns \(f_i(x_i)=a_i\ln(1+x_i)\), with \(a_i>0\), the marginal return is \(a_i/(1+x_i)\). If a proposed solution has a set \(S\) of positively funded projects, equal marginal returns imply \(x_i=a_i/\lambda-1\) for \(i\in S\). The budget constraint determines \(\lambda\), while each proposed allocation must be positive. Every project outside \(S\) must satisfy \(a_i\le\lambda\), its marginal return at zero being \(a_i\). These checks prevent an algebraic solution from being mistaken for a feasible optimum.

The same logic applies to other differentiable concave return functions: solve for allocations by equating the marginal returns of funded projects, respect nonnegativity, and verify the inequality for each unfunded project. Concavity makes these conditions powerful because they certify a global optimum rather than just a local candidate.

This approach is useful wherever resources are divided among alternatives that yield diminishing gains. It provides a systematic way to decide not only how much to allocate to each active project, but also whether some projects should receive nothing. The essential check is always the same: funded projects balance at a common marginal return, while an unfunded project must not offer a larger marginal return at the boundary.