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Econ-488 · economics · Economics

Targeted sanitation subsidies with neighborhood spillovers

LLM status Open unverified

Statement

ECON-063 | R | Source-backed agenda; exact formalization is editorial

Definitions and formal target

Household $i$ chooses sanitation adoption $a_i\in\{0,1\}$ under subsidy $s_i$. Health outcome $Y_i(a_i,e_i)$ depends on own adoption and neighborhood exposure $e_i=\sum_jw_{ij}a_j$, with known exposure weights. A targeting rule maps baseline covariates to subsidies.

For budget $B$, determine \[ s^\star\in\arg\max_s\sum_i\mathbb E[Y_i(a_i(s),e_i(s))] \quad\text{s.t.}\quad\sum_i\mathbb E[s_i a_i(s)]\leq B. \] Identify private adoption responses and external health benefits sufficiently to compare targeted with uniform subsidies.

Clarifications and required assumptions

Declare $w_{ij}\geq0$, $w_{ii}=0$, and normalization if $e_i$ is an adoption share. This exposure mapping is an assumption about transmission; it is not automatically sufficient for all sanitation spillovers. Define households' utility, costs, information and adoption response or a selected adoption equilibrium. $Y$ must be oriented so larger values mean better health in the maximization. Include program administration and other fiscal costs in the budget. The vector $s$ uses baseline covariates and cannot choose unobserved individual potential gains without a justified targeting model.

Variants and changes of scope

  • Fiscal subsidy expenditure and social resource cost differ because subsidies are transfers.
  • Household, neighborhood and network saturation designs identify different spillover contrasts.
  • Multiple equilibria, maintenance and sustained use require more than initial adoption indicators.

Audit conclusion and source relationship

[S1] and [S2] support sanitation targeting, constraints and community externalities. The exposure map and welfare optimization are editorial and must be empirically justified.

References and exact source roles

[S1] Britta Augsburg, Andrew Foster, Molly Lipscomb, Anna Tompsett, and Bansi Malde (2025). Sanitation Infrastructure. VoxDevLit 16(1), July 2025.

Verified locator / source role: Primary publisher page checked for review identity, authors, version and background. The specific published agenda is corroborated in [S2]; the exact formal target is editorial.

https://voxdev.org/voxdevlit/sanitation-infrastructure

[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.

Verified locator / source role: Topic heading: Sanitation Infrastructure. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-488-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-488

Sanitation subsidies with adoption cascades: minimum implementation budgets and identified policy values

Problem ID: I18-484. Status: unresolved; restricted research attempt.

Target and the role of sustained adoption

The target maximizes aggregate health under a subsidy budget when sanitation adoption affects neighbors. We solve a nontrivial implementation subproblem: the minimum budget that guarantees complete adoption under a specified least-equilibrium selector. We also give a finite-policy identification design and an exact example where uniform subsidies are strictly dominated. These results do not identify population health benefits or justify universal subsidies without local evidence.

Augsburg et al. [A] review sanitation coverage, maintenance and externalities. The primary publisher summary, especially its discussion of coverage, quality, maintenance and subsidy design, was inspected. Our thresholds and graph model are declared restrictions; they are not empirical coefficients from that review. Interpret adoption as sustained safe use over the evaluation horizon, including a prespecified maintenance package, rather than mere latrine ownership.

There are $n$ baseline households and known weights $w_{ij}\geq0$ with $w_{ii}=0$. If exposure is a share require $\sum_jw_{ij}\leq1$. A household adopts when its incremental utility is nonnegative: \[ s_i-k_i+\eta_i\sum_jw_{ij}a_j\geq0,\qquad \eta_i\geq0. \tag{1} \] Here $k_i$ is its observed net private cost before neighbor benefits and $s_i\geq0$ is a subsidy paid only on adoption. All quantities in (1) are money equivalents using household-specific utility normalization fixed before policy. For this implementation theorem, $k_i,\eta_i,w_{ij}$ are known baseline covariates. When they contain unobserved types, an implementable targeting rule cannot condition on them and this theorem is only an information benchmark.

Define $F_s(S)=\{i:s_i+\eta_i\sum_{j\in S}w_{ij}\geq k_i\}$. Start at $S_0=\varnothing$ and iterate $S_{t+1}=F_s(S_t)$. Because $F_s$ is increasing, this sequence is increasing, stabilizes after at most $n+1$ updates, and gives the least fixed point $S(s)$. To see leastness, if $T=F_s(T)$, induction from $\varnothing\subseteq T$ gives $S_t\subseteq T$ for every $t$. This proves existence and specifies the equilibrium selection, including adoption at indifference.

Exact minimum budget for complete adoption

Let $h_i\geq0$ be real program administration expenditure incurred if a strictly positive subsidy is offered to household $i$. Program cost is \[ C(s)=\sum_i s_i{\mathbf 1}\{i\in S(s)\} +\sum_i h_i{\mathbf 1}\{s_i>0\}. \] For a predecessor set $S$ not containing $i$, put \[ d_i(S)=\left[k_i-\eta_i\sum_{j\in S}w_{ij}\right]_+, \qquad c_i(S)=d_i(S)+h_i{\mathbf 1}\{d_i(S)>0\}. \tag{2} \]
Theorem (An ordering formula and finite exact algorithm): The smallest cost among all nonnegative subsidies whose least equilibrium has every household adopt is attained and equals \[ B_{\rm all}=\min_{\pi}\sum_{r=1}^n c_{\pi(r)}(\{\pi(1),\ldots,\pi(r-1)\}), \tag{3} \] where $\pi$ ranges over permutations. It is computed by \[ V(\varnothing)=0,\qquad V(S)=\min_{i\in S}\{V(S\setminus\{i\})+c_i(S\setminus\{i\})\}, \qquad B_{\rm all}=V(\{1,\ldots,n\}). \tag{4} \] With predecessor weight sums precomputed, (4) uses at most $n2^{n-1}$ candidate comparisons. With rational inputs, additions, comparisons and positive-part operations can be performed exactly; the number of states is exponential, so this is not a polynomial-time claim.
Proof: Fix a permutation and pay household $\pi(r)$ subsidy $d_{\pi(r)}(\{\pi(1),\ldots,\pi(r-1)\})$. Inductively every household in the first $r-1$ positions eventually adopts. Equation (1) then makes household $\pi(r)$ willing to adopt, whether or not it already adopted earlier. The least-equilibrium iteration is increasing, so all households eventually adopt. Its fiscal cost is exactly the permutation's sum in (3).

Conversely suppose $s$ yields complete adoption. Order households by the iteration in which they first adopt, breaking ties arbitrarily. At its first-adoption iteration, household $i$ satisfies (1) using only households from earlier iterations. This set is contained in its permutation predecessors. Nonnegative weights imply $s_i\geq d_i(\text{predecessors})$. If that lower bound is positive, $s_i>0$ and its administration cost is incurred; if it is zero, $c_i=0$. Thus $s_i+h_i{\mathbf 1}\{s_i>0\}\geq c_i(\text{predecessors})$. Summing proves that every implementing subsidy costs at least the minimum in (3). There are finitely many permutations, so one attains that minimum and the first construction attains it as a subsidy vector.

For (4), any ordering of $S$ has a last household $i$. Its contribution is $c_i(S\setminus\{i\})$; earlier contributions depend only on their own predecessors and have optimum $V(S\setminus\{i\})$. Minimizing over $i$ proves the recurrence by induction on $|S|$. The candidate count is $\sum_S|S|=n2^{n-1}$. Every arithmetic operation preserves rationality; bit lengths grow at most by the bit lengths of finitely many input sums, but the state count remains exponential. ∎

This is an implementation result about a specified adoption game. If the budget is below $B_{\rm all}$, complete adoption is impossible under these primitives even if the all-adopt profile is another equilibrium at zero subsidy. If $B\geq B_{\rm all}$ and health is coordinatewise increasing in adoption and exposure, complete adoption maximizes health within this model: every other adoption profile is coordinatewise smaller and yields no larger health for each household. That conclusion is about the stated health objective, not net social surplus.

Targeted and uniform subsidies need not be equivalent

Take two households, $k_1=k_2=1$, $\eta_1=\eta_2=1$, $w_{12}=w_{21}=1$ and $h_1=h_2=0$. A targeted vector $(1,0)$ first induces household 1 and then household 2, costing one. Theorem Theorem (An ordering formula and finite exact algorithm) also shows one is the minimum. Under a uniform subsidy $s<1$, neither adopts in the first iteration and the least equilibrium remains empty; under $s\geq1$, both adopt and expenditure is at least two. With budget one, targeting achieves full adoption and every feasible uniform subsidy yields zero adoption. Baseline household identifiers must be allowable covariates for the asymmetric targeting rule; anonymity constraints would change this comparison.

There is also no automatic diminishing-returns property. With three households, a third household requiring two initially adopting neighbors can generate a larger cascade when a second seed is added than when the first seed is added. More explicitly, let a voucher force selected seeds 1 and 2 to adopt, let unseeded 1 and 2 never adopt, and let household 3 adopt exactly when both 1 and 2 adopt. The number of adopters is $0,1,1,3$ for seed sets $\varnothing,\{1\},\{2\},\{1,2\}$. The marginal gain from adding 2 rises from 1 to 2. A greedy guarantee requiring submodularity therefore cannot be imported into this threshold model without further assumptions.

Identifying a policy menu despite endogenous adoption

Suppose $C$ independent neighborhoods are sampled from the declared population. Cross-neighborhood transmission is absent by assumption; all within-neighborhood spillovers may be arbitrary. A finite menu of rules $\mathcal Z$ maps baseline covariates to complete subsidy vectors. Assign rule $Z_c=z$ with known probability $\pi_z(X_c)>0$ independently of all potential neighborhood outcomes conditional on baseline $X_c$. For rule $z$, let $H_c(z)=\sum_{i\in c}Y_i(z)$ and let $G_c(z)$ include subsidy and administration expenditure. Actual adoption, compliance and equilibrium selection are part of the potential policy outcomes.
Proposition (Policy values and a finite-sample precision bound): With consistency and integrability, ${\mathbb{E}} H_c(z)$ and ${\mathbb{E}} G_c(z)$ are identified by the corresponding inverse-probability means. If $0\leq H_c(z)\leq\bar H$ with $\bar H>0$, and $\pi_z(X_c)\geq\underline\pi>0$, independent identically distributed neighborhoods give, for every $t>0$, \[ \Pr\left\{\max_{z\in\mathcal Z}|\widehat H(z)-{\mathbb{E}} H(z)|>t\right\} \leq 2|\mathcal Z|\exp\left(-\frac{2C t^2\underline\pi^2}{\bar H^2}\right), \quad \widehat H(z)=\frac1C\sum_c\frac{{\mathbf 1}\{Z_c=z\}H_c}{\pi_z(X_c)}. \tag{5} \] The analogous expenditure statement holds with a declared bound $\bar G$.
Proof: Conditional assignment independence and consistency imply ${\mathbb{E}}[{\mathbf 1}\{Z=z\}H/\pi_z(X)\mid X,H(z)]=H(z)$, proving identification. For completeness, a random variable $V$ in an interval of length $b$ satisfies ${\mathbb{E}} e^{\lambda(V-{\mathbb{E}} V)}\leq e^{\lambda^2b^2/8}$: convexity bounds its exponential by that of a two-endpoint variable with the same mean, and the log moment-generating function of that two-point variable has second derivative equal to a variance bounded by $b^2/4$; integrating twice from zero proves the inequality. Independence and Markov's inequality then bound the upper tail of a mean of $C$ such variables by $\inf_{\lambda>0}\exp(-\lambda Ct+C\lambda^2b^2/8) =\exp(-2Ct^2/b^2)$. Use $b=\bar H/\underline\pi$, repeat for the lower tail, and take the union bound over the finite menu. Expenditure is identical. ∎

Simultaneous upper expenditure bounds can restrict the empirically chosen menu to rules certifiably within budget. Randomized subsidy assignment identifies these policy values even though adoption is endogenous. It does not, without additional restrictions, identify the controlled health function $Y_i(a,e)$ or separate private effects from spillovers: assignment changes adoption, exposure and possibly maintenance together. Richer saturation or directly implemented adoption designs and an exposure-map validation are needed for extrapolation outside the tested menu.

Scope and accounting

The exact ordering theorem closes a full-adoption budgeting subproblem; the general optimal partial-coverage problem, heterogeneous latent costs, and sustained-use dynamics remain open here. The theorem's budget is a fiscal constraint. Subsidies are transfers to households, whereas administration and sanitation construction/maintenance consume resources. A welfare objective must subtract the latter once, include any fiscal distortion with an explicit valuation, and state how health is monetized. No face-value subsidy subtraction is justified in an equal-weight combined resource account. A practical next step is to estimate a baseline-observable threshold model, assess equilibrium selection with repeated neighborhood experiments, and compare its counterfactual recommendations with directly identified policy-menu values.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves minimum-budget full adoption in a least-equilibrium threshold game and identifies tested subsidy-rule values with finite-sample bounds. The full health-maximizing targeting problem still requires optimal partial coverage, latent-cost restrictions, validated transmission and sustained use, and identification separating private benefits from spillovers.

References

  • [A] B. Augsburg, A. Foster, M. Lipscomb, A. Tompsett and B. Malde (2025), Sanitation Infrastructure, VoxDevLit 16(1), July. Primary publisher policy summary inspected, including coverage, quality, maintenance and subsidy-design discussion. https://voxdev.org/voxdevlit/sanitation-infrastructure.

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