Industry: Government and Public Sector Product: ModelRisk Application: Education funding adequacy under uncertainty
A state funds its public schools through a fixed per-pupil formula — a base amount plus categorical add-ons, paid out across 60 districts on a statutory schedule. The policy question is not how to split a discretionary pot; the formula is locked. The question is adequacy: does the funding each district receives actually clear what it costs to educate a pupil to standard in that district? Plug the projected enrolments and a central cost estimate into the formula and the deterministic answer looks tolerable — 16 of 60 districts come out below their adequacy threshold, a known shortlist the legislature can budget a top-up for.
That point estimate is built on two numbers that will not hold: next year's enrolment and next year's cost of an adequate education. Both are uncertain, and both move every district at once. Re-run the formula as a Monte Carlo simulation, with enrolment as a truncated Normal around each projection and adequacy cost as a right-skewed LogNormal, and the count of underfunded districts stops being 16 and becomes a distribution: a median of 19, a P10 of 5, and a P90 of 44 — nearly three-quarters of the state. In a 1-in-10 bad year, 152,000 pupils sit in districts that fall below adequacy. The deterministic shortlist of 16 was not wrong so much as blind to its own range.
Adequacy is a comparison of two per-pupil dollar figures: what the formula pays, and what an adequate education costs. The formula side is fixed by statute. The cost side is not. The adequacy cost per pupil is bounded below at zero, unbounded above, and right-skewed — teacher-salary settlements, special-education caseloads, and facilities costs all carry long upper tails — so it is carried as a LogNormal calibrated to each district's cost level. Enrolment is a truncated Normal around each district's projection, never falling below a floor of 100 pupils. Crucially, enrolment feeds the cost side too: when a district's enrolment comes in below projection, its fixed plant and staffing overhead is spread over fewer pupils, raising the adequacy cost per pupil — the real squeeze that small and shrinking districts feel.
The structural choice that prevents a false sense of security is the shared driver. Costs do not rise in one district and fall in another at random — a state-wide cost-inflation surprise (a labour-market shock, a benefits-cost jump) lifts the adequacy bar in every district at once. The model imposes this with a shared cost factor entering all 60 districts, and the simulation verifies it: per-pupil gaps across districts correlate at about 0.64. That positive correlation is what keeps the count of underfunded districts from collapsing to a narrow band — when costs run hot, they run hot everywhere, and districts cross the adequacy line in clusters rather than one at a time.
Pooled across all districts and trials, the per-pupil gap — formula funding minus adequacy cost — has a mean of about $452, a P50 of $563, and a long left tail: the P10 is –$1,734 per pupil, meaning the worst tenth of district-years run more than seventeen hundred dollars short on every pupil. Overall, a district-year falls below adequacy 37% of the time. The single deterministic answer — a $470 average gap — sits comfortably positive and tells policymakers nothing about the districts on the wrong side of zero.
The number that drives the appropriation is not the average gap but the count of districts below adequacy, because each one is a separate equity problem and a separate top-up line.
The count is a distribution with a median of 19 districts, a P10 of 5, and a P90 of 44 — against the deterministic point estimate of 16. The deterministic figure lands near the median, which is exactly why it is so misleading: it looks like a stable planning number while concealing that a single bad cost year could put 44 of 60 districts below adequacy. The pupils affected scale with it — a median of 65,000 pupils in underfunded districts, rising to 152,000 at the P90. Budgeting a top-up against "16 districts" provisions for a year that the simulation says occurs only at the optimistic end.
With enrolment and cost both uncertain, the question is which one moves the underfunded count — so the state knows whether to manage demographics or costs.
The tornado is unambiguous: cost dominates. The mean adequacy cost level (±14.4 districts) and the state-wide cost-inflation surprise (±14.2 districts) are the two largest swings, with cross-district cost dispersion (±11.1) third. The state-wide enrolment surprise moves the count by only about 1.6 districts — and inversely, since higher enrolment dilutes fixed overhead and reduces underfunding. The message for the state is clear: adequacy is a cost-control problem far more than an enrolment-forecasting one. Effort spent pinning down next year's salary and benefits costs buys an order of magnitude more certainty about the underfunded count than effort spent refining enrolment projections.
If costs are the binding constraint, the natural policy lever is a per-pupil top-up to the base formula — and the simulation prices exactly how much is needed to hit an adequacy target.
At the current formula, a district clears adequacy 63% of the time across the state — but only 30% of the time in the highest-cost quartile. Sweeping a per-pupil top-up shows the cost of fixing it: roughly $1,746 per pupil added to the base lifts the all-districts clear-rate to 90%, while the highest-cost-quartile curve does not reach 90% even at a $2,500 top-up. That gap between the two curves is the heart of the equity problem — a flat top-up that satisfies the average district still leaves the most expensive districts short, which is precisely the case for a cost-weighted rather than a flat formula adjustment.
The state stopped funding to the formula's point answer and started provisioning against the distribution:
Monte Carlo turns funding-formula design from "which districts are short?" into "what is the distribution of districts that fall below adequacy, and what does it cost to close the gap for the ones that need it most?" — and on this formula the difference between those questions was the difference between 16 districts and the 44 a bad cost year could leave behind.