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Econ-622 · economics · Economics

Economic allocation of mental-health care

LLM status Open unverified

Statement

Original atlas identifier: H8; entry 58. Field: Health economics. Original tractability label: Easy.

Classification: Parameterized theoretical/empirical research agenda; not a certified unsolved theorem.

Definitions and assumptions

Let patients $i=1,\ldots,n$ have pretreatment observable covariates $X_i$ and latent need $\theta_i$. Treatment $k\in\{0,1,\ldots,K\}$ includes no treatment ($k=0$), with a defined intensity, duration, and provider qualification. Potential outcomes $H_i(k)$, $Y_i(k)$, and $E_i(k)$ represent health, employment-related resources, and educational outcomes over a fixed horizon. Family/network spillovers require joint assignments $\mathbf k$ and replace these outcomes by $H_i(\mathbf k)$, etc. Specify a monetary welfare map $V_i$ rather than add health, earnings, and education in incompatible units. Costs $c_i(k)$ are real treatment and opportunity costs, and $b_k$ are provider-capacity limits. Fix welfare weights $w_i\geq0$ and budget $B\geq0$.

Formal research question

Editorial formulation: the mathematical target below is not a theorem quoted from the references.

For a measurable allocation rule $\pi$ based only on admissible pretreatment information, characterize \[ \sup_{\pi}\mathbb E\sum_iw_i\left[V_i(H_i(\pi),Y_i(\pi),E_i(\pi))-V_i(H_i(0),Y_i(0),E_i(0))\right] \] subject to $\mathbb E\sum_ic_i(\pi)\leq B$ and stated provider-capacity constraints. Define whether capacity applies in expectation or for every realized assignment. Identify heterogeneous intention-to-treat effects of randomized access, then distinguish them from treatment-received effects under explicit compliance assumptions. Incorporate treatment interactions and spillovers if relevant. Report an identified set of allocation values and $\varepsilon$-optimal rules, with sensitivity to latent severity, dropout, and the monetary valuation of health and education.

Interpretation and scope

A ratio of average benefits to dollars need not maximize total welfare when treatments are indivisible, capacities bind, or beneficiaries overlap; the allocation problem is the rigorous target. Earnings can be a component of resource gain or a proxy for well-being, but should not be counted again through an education valuation already incorporating lifetime earnings. Diagnosis codes may respond to access, so stratification should use pretreatment information and an explicit measurement model. Waiting-list randomization identifies the specified offer effect, not every therapy's long-run causal return. Multiyear follow-up should include initial assignees and family outcomes, with missing-data assumptions stated. The original Easy classification describes the atlas author's judgment; identifying heterogeneous long-run welfare and spillovers is not thereby demonstrated easy.

Source audit and status

The original sources are identifiable. [S1] and [S2] concern mental-health associations/outcomes, not randomized treatment returns; [S3] is a cost-benefit analysis. Publisher migration dates do not replace [S3]'s 2007 year. These works motivate the allocation question but do not certify a universally optimal or currently unresolved rule.

References

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  1. Janet Currie and Mark Stabile (2006). Child Mental Health and Human Capital Accumulation: The Case of ADHD. Journal of Health Economics 25(6), 1094--1118. DOI: 10.1016/j.jhealeco.2006.03.001. https://www.sciencedirect.com/science/article/pii/S0167629606000282
    Locator and support limits: Abstract: ADHD symptoms and child outcomes; not randomized treatment returns.
  2. Pinka Chatterji, Margarita Alegria, and David Takeuchi (2011). Psychiatric Disorders and Labor Market Outcomes: Evidence from the National Comorbidity Survey-Replication. Journal of Health Economics 30(5), 858--868. DOI: 10.1016/j.jhealeco.2011.06.006. https://www.sciencedirect.com/science/article/pii/S0167629611000774
    Locator and support limits: Abstract: psychiatric disorders and labor outcomes under selection assumptions; not direct randomized treatment effects.
  3. Richard Layard, David Clark, Martin Knapp, and Guy Mayraz (2007). Cost-Benefit Analysis of Psychological Therapy. National Institute Economic Review 202(1), 90--98. DOI: 10.1177/0027950107086171. https://www.cambridge.org/core/journals/national-institute-economic-review/article/costbenefit-analysis-of-psychological-therapy/CBEC26AB2AB0ABC47D0C3F570E0FFA21
    Locator and support limits: Abstract and cost-benefit analysis: therapy expansion calculations. Later online migration dates do not change 2007 publication year.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-622-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-622

Economic allocation of mental-health care Hard-capacity allocation with family spillovers and identified offer values

Question and scope

The task is to allocate specified mental-health services using baseline information, with health and economic consequences, real budget costs, and provider capacity treated consistently. Symptoms and earnings associations do not by themselves identify treatment benefits. The source studies [currie,chatterji,layard] motivate economic and educational consequences and cost-benefit accounting; they do not establish the intervention coefficients used below. This attempt solves a restricted allocation problem with pairwise family spillovers, gives an explicit experimental identification design, and derives a robust policy guarantee under coefficient uncertainty. It makes no clinical recommendation or empirical claim about a treatment's effectiveness.

A feasible offer policy and a monetary objective

There are $n$ patients with recorded pretreatment characteristics. The intervention is one fully specified service package, including duration and provider quality; the alternative is the specified usual-care option. A binary decision $s_i$ reserves and offers one service slot to patient $i$. Reservation costs a real amount $c>0$ even if declined, and an accepted offer consumes at most that one slot. A hard budget $B\ge0$ and an integer provider-slot limit $L\ge0$ therefore permit \[ |S|\le K:=\min\{n,\lfloor B/c\rfloor,L\},\qquad S=\{i:s_i=1\}. \] This is a conservative hard-capacity design, not an assumption that expected attendance can replace a pathwise capacity constraint.

Conditional on the recorded baseline information, suppose the expected monetary welfare increment is \[ F(S)=\sum_{i\in S}v_i+\sum_{\{i,j\}\in E:\ i,j\in S}b_{ij}, \tag{1} \] where $(\{1,\ldots,n\},E)$ is a known forest. Coefficients can have either sign. They include declared distributional weights and monetized health, earnings and education consequences, with education benefits not counted again when already included through earnings. Any reservation-resource cost included in welfare is subtracted once in $v_i$, as well as being a feasibility constraint. Nonattendance and substitution away from other care are already part of the offer effect. Formula (1) is a conditional mean restriction on the entire joint assignment; it allows family spillovers but excludes higher-order and nonforest interactions.

Theorem (Exact hard-budget allocation on a forest): For rational coefficients in (1), an optimal deterministic offer set under $|S|\le K$ can be computed in $O(nK^2)$ arithmetic operations when $K\ge1$, with polynomial bit complexity. The case $K=0$ has the unique feasible set $S=\varnothing$. The algorithm is valid for positive or negative interaction coefficients and recovers an actual feasible set, not merely fractional inclusion probabilities.
Proof: Root each tree. For a vertex $v$, let $f_v(s,k)$ be the maximum welfare from its rooted subtree, including all internal edge terms, conditional on $v$ having selection state $s\in\{0,1\}$ and exactly $k$ selected vertices in that subtree. Impossible states have value minus infinity. Before its children are processed, initialize $f_v(0,0)=0$ and $f_v(1,1)=v_v$, with all other states impossible. If a child $u$ has been processed, combine it with the previously processed part by the recurrence \[ f_v^{\rm new}(s,k)= \max_{\substack{t\in\{0,1\}\\k_1+k_2=k}} \{f_v^{\rm old}(s,k_1)+f_u(t,k_2)+b_{vu}st\}. \tag{2} \] Only counts at most $K$ need be kept. A forest has no edge between two different child subtrees, so conditional on $s,t$ and their counts, (2) includes every welfare term exactly once. Conversely every feasible subtree set decomposes into exactly such states. Induction from leaves proves that every table entry is the stated optimum. At each root maximize over its selection state; convolve the component tables in the same way, with no connecting edge. Finally maximize over total counts $0,\ldots,K$.

There are $O(n)$ child and component combinations, each requiring $O(K^2)$ comparisons and additions up to fixed factors for the two states. Record maximizing choices to reconstruct the selected set. Every finite value is the sum of at most $n+|E|$ input coefficients; rational numerator and denominator lengths remain polynomial in total input length. Thus the arithmetic implementation also has polynomial bit complexity. ∎

The forest condition is an explicit structural restriction, not a claim that real family and provider networks are always acyclic. Even on this restricted class, ranking individual benefits can fail. With three patients, $K=2$, individual values $(3,2,2)$ and a single edge $\{2,3\}$ of value $4$, selecting the two highest individual values gives welfare $5$, while selecting patients 2 and 3 gives $8$. The dynamic program evaluates the complementarity instead of treating it as noise.

A design that identifies the coefficients

For identification, suppose independent replicates of the same baseline-defined cluster are available. Across replicates, offers $Z_i$ are mutually independent Bernoulli variables with known probabilities $\pi_i\in(0,1)$. The research design has enough reserved research capacity to implement its assignments; it need not obey the smaller deployment limit $K$. Let the observed monetary cluster outcome satisfy \[ {\mathbb{E}}[Y\mid Z]=\beta_0+\sum_i v_iZ_i+\sum_{\{i,j\}\in E}b_{ij}Z_iZ_j. \tag{3} \] If real reservation costs are known, they can first be subtracted from $Y$ to match the coefficients in (1). Set $\sigma_i^2=\pi_i(1-\pi_i)>0$.
Theorem (Factorial moment identification with spillovers): Under (3) and integrability of $Y$, the observable joint law identifies every interaction and individual coefficient through \[ b_{ij}=\frac{{\mathbb{E}}[Y(Z_i-\pi_i)(Z_j-\pi_j)]}{\sigma_i^2\sigma_j^2}, \qquad v_i=\frac{{\operatorname{Cov}}(Y,Z_i)}{\sigma_i^2}-\sum_{j:\{i,j\}\in E}b_{ij}\pi_j. \tag{4} \] These are offer-effect coefficients; (4) does not by itself identify effects of actual treatment receipt.
Proof: Write $Y$ as the conditional mean (3) plus a residual with zero conditional mean. Multiplication by any bounded function of $Z$ makes the residual expectation vanish. For the first expression, every constant or single-variable term in (3) has at least one independent centered factor with mean zero. Any pair term other than $b_{ij}Z_iZ_j$ also leaves at least one of $Z_i-\pi_i$, $Z_j-\pi_j$ unpaired and independent. The surviving expectation is $b_{ij}{\mathbb{E}}[Z_i(Z_i-\pi_i)]{\mathbb{E}}[Z_j(Z_j-\pi_j)]=b_{ij}\sigma_i^2\sigma_j^2$. Similarly multiplying (3) by $Z_i-\pi_i$ leaves $v_i\sigma_i^2$ and the terms $b_{ij}\sigma_i^2\pi_j$ from its neighbors. Rearranging gives the second formula. Neither calculation conditions on attendance or substitutes it for randomized offers, proving the final distinction. ∎

Repeated clusters with a common coefficient vector are a strong transport and homogeneity assumption. With heterogeneous baselines, (3) must be interpreted within adequately supported strata or replaced by a justified conditional model. A single realized family does not reveal the expectations in (4). Generalizing from experimental reservation and provider quality to a scaled service is another empirical task.

Robust allocation under joint coefficient uncertainty

Let the uncertainty set be a nonempty rectangular box \[ |v_i-\widehat v_i|\le r_i,\qquad |b_e-\widehat b_e|\le r_e,\qquad r_i,r_e\ge0. \] Nature selects one coefficient array from this common box after the deterministic policy is fixed. Write $F_-$ for (1) with every coefficient at its lower endpoint, and let $\widehat S$ maximize $F_-$ under the hard limit.
Theorem (Exact robust rule and a simultaneous-error guarantee): The set $\widehat S$, computed by the preceding dynamic program, maximizes worst-case welfare over all feasible deterministic sets. On every event where the true coefficient array lies in the box, if $S^*$ maximizes its true welfare, then \[ F(S^*)-F(\widehat S) \le 2\sum_{i\in S^*}r_i+ 2\sum_{e\subseteq S^*}r_e. \tag{5} \] If all vertex radii are at most $r_V$ and edge radii at most $r_E$, the right side is at most $2Kr_V+2(K-1)_+r_E$. Thus a simultaneous coverage probability $1-\alpha$ for the box implies the same probability for this policy-value guarantee, despite selecting the policy from those estimates.
Proof: All coefficients in (1) multiply zero-one nonnegative indicators. For every fixed set, nature minimizes its value by selecting all included coefficients at their lower endpoints. The same lower-endpoint array lies in the box and minimizes every set simultaneously. Hence worst-case optimization is exactly maximization of $F_-$. On the stated coverage event, $F(\widehat S)\ge F_-(\widehat S)\ge F_-(S^*)$. Subtracting yields $F(S^*)-F(\widehat S)\le F(S^*)-F_-(S^*)$. Each included true coefficient exceeds its lower endpoint by at most twice its radius, proving (5). An induced subgraph of a forest on $k$ vertices has at most $(k-1)_+$ edges, giving the uniform bound. Since the inequalities hold for every parameter array in the box, no independence between policy selection and the estimates is needed. ∎

The box is a declared ambiguity set. Correlated coefficient restrictions can make it conservative; lower endpoints need not be jointly feasible in a narrower identified set. Likewise, confidence radii must account for the cluster design, monetary-outcome bounds or tails, and simultaneous estimation. The theorem does not manufacture such coverage from point estimates.

Proposition (Expected capacity is a different allocation problem): Replacing the hard constraint by ${\mathbb{E}}|S|\le K$ can strictly increase achievable expected welfare, even in a forest with two patients.
Proof: Take two patients with $v_1=v_2=0$, $b_{12}=1$, and $K=1$. Every hard-feasible set has welfare zero. A lottery selecting both patients with probability $1/2$ and neither otherwise has expected size one and expected welfare $1/2$. It violates the physical slot limit on half its realizations, proving both the strict difference and the reason it cannot implement the hard-capacity optimum. ∎

The remaining work is substantive empirical identification: choosing a defensible service package and usual-care baseline, measuring effects and spillovers over the relevant horizon, monetizing outcomes without duplication, treating dropout, and establishing transport to the target patients and provider capacity. The allocation theorem solves the stated forest offer model once those inputs are supplied; it does not resolve arbitrary treatment portfolios, latent clinical need or unknown network structure.

Completion Estimate

61%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves hard-budget offer allocation on a forest, identifies pairwise spillover coefficients through factorial moments, and proves robust regret guarantees under simultaneous uncertainty. The full mental-health allocation target still requires heterogeneous service and usual-care effects, credible monetary valuations, dropout and severity measurement, transport to patients and providers, and treatment portfolios beyond the known binary forest model.

References

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