Inference

The inference layer combines the stage 1 markups, the stage 2 CES parameters, and observed quantities and prices into ROI metrics. Results are produced for each organization-project pair and each markup method, with a yearly breakdown.

Metrics

With estimated contribution value V(i,j) and investment I(i,j):

Metric	Definition
Investment	`I(i,j) = Σ_t Σ_k p(i,j,k,t) · x(i,j,k,t)`
Contribution value	`V(i,j) = μ(i,j) · PY(i,j)`, with the markup floored at break-even (see below)
Net value	`V(i,j) − I(i,j)`
ROI	`(V − I) / I`; undefined if `I = 0` (reported as 0 in the model output, shown blank on the ROI page)
BCR	`V / I`; undefined if `I = 0` (reported as 0 in the model output, shown blank on the ROI page)
Break-even markup	`I / PY` when `PY > 0`
Sign check	markup > 1 and net value ≥ 0, or markup ≤ 1 and net value ≤ 0

BCR = ROI + 1 by construction. Investment is summed over all years of the pair and all contribution types, including types that were not given positive weight in the CES estimate.

A pair appears only if it has both a markup and CES parameters.

Computing PY

PY(i,j) is the cost-minimizing expenditure on the pair's contribution portfolio at observed prices: the P · Y of The model, evaluated on the observed data. How it is computed depends on the pair.

Stage 2 estimates α against scale-normalized quantities x / x̄(k). Inference works on raw quantities, so the shares must be rescaled. The cost-minimization condition (★) holds in both units:

normalized:   (x(k)/x̄(k)) / (x(0)/x̄(0))  =  (α(k)/α(0)) · (p(0)/p(k))^σ
raw:          x(k) / x(0)                 =  (α̃(k)/α̃(0)) · (p(0)/p(k))^σ

Matching the two gives

α̃(k) = α(k) · x̄(k)

Only types with α(k) > 0 are kept as active. P · Y does not change when every share is multiplied by the same constant, so the rescaled shares need not be renormalized. A year in which only some types are present can use the pair's α̃ for the types present directly.

Simple branch

Pairs with one active type, with an undefined σ, or with σ = 1 use, in each year t,

PY(i,j,t) = Σ_k p(i,j,k,t) · x(i,j,k,t)

so PY(i,j) = Σ_t PY(i,j,t) = I(i,j).

CES branch

For all other pairs, value is computed year by year and then summed. In year t, let K(t) be the types present that year, with quantity x(k,t) (summed over the year) and price p(k,t) (quantity-weighted, so that p · x reproduces the year's spending exactly). Then

P(i,j,t)  = [ Σ_{k in K(t)}  α̃(k) · p(k,t)^(1−σ) ] ^ ( 1/(1−σ) )
Y(i,j,t)  = [ Σ_{k in K(t)}  α̃(k)^(1/σ) · x(k,t)^((σ−1)/σ) ] ^ ( σ/(σ−1) )
PY(i,j,t) = P(i,j,t) · Y(i,j,t)

PY(i,j)   = Σ_t PY(i,j,t)

Details:

A year with a single type reduces to p · x, the year's spending.
A year contributes zero value if either bracketed sum is not positive.
By CES duality PY(i,j,t) ≤ Σ_k p(k,t) · x(k,t) in every year, with equality when that year's mix satisfies the first-order conditions for the pair's α and σ.

Value and investment are therefore both sums over the same years, so ROI is comparable across pairs with different numbers of active years.

Participation floor

An organization that keeps contributing values participation at least at its cost: μ(i,j) · PY(i,j) ≥ I(i,j). The calibrated markup μ_raw comes from ranking pairs by participation and does not condition on allocative efficiency, so it can fall below the break-even markup I / PY. Inference enforces the constraint:

μ(i,j)  = max( μ_raw(i,j),  I(i,j) / PY(i,j) )
V(i,j)  = μ(i,j) · PY(i,j)          (so V ≥ I)

A pair whose raw markup is raised is floored. A floored pair has V = I, so its net value and ROI are exactly 0 and its BCR is 1. A floored pair's ROI of 0 is a lower bound: the pair contributes at least at break-even. It is not a point estimate.

If PY(i,j) = 0 there is no finite break-even markup, so the markup stays at μ_raw, but positive investment still floors the return at cost, so net value and ROI are again exactly 0.

Consequences:

ROI is never negative. Every pair has V ≥ I.
Only pairs whose raw markup would contradict the revealed-preference assumption are changed. The ranking and calibration of every other pair are untouched.
The identity below uses the floored markup for pairs with positive PY.

Allocative efficiency

By CES duality, PY(i,j) ≤ I(i,j), with equality when the observed mix matches the fitted allocation. Define allocative efficiency as η(i,j) = PY(i,j) / I(i,j), which lies in [0, 1]. For pairs with positive PY,

BCR(i,j) = μ(i,j) × η(i,j)

so the benefit-cost ratio is the markup, discounted by how far the observed allocation is from the cost-minimizing one.

Annualization

The yearly breakdown splits each pair's totals across years in proportion to expenditure:

ω(i,j,t) = Σ_k p(i,j,k,t) · x(i,j,k,t) / I(i,j)

V(i,j,t)   = ω(i,j,t) · V(i,j)
I(i,j,t)   = ω(i,j,t) · I(i,j)
net(i,j,t) = ω(i,j,t) · net(i,j)

so the ROI in every year of a pair equals the pair's overall ROI. Annual figures show when the money was spent and how the total is spread. They are not a separate estimate of value per year.

Research methodology: additional constructs

The following appear in the research methodology and are not in the product. The participation floor and allocative efficiency, which the research methodology also describes, are implemented and covered above.

Mechanism analysis

Once ROI is computed, associations with external characteristics (industry, project category and maturity, organization size, geography) can be explored with a model of the form ROI = f(drivers), for example with a Shapley decomposition. This is correlational: organizations select into projects, and some mechanisms are unobserved.

Counterfactual analysis

Given α, σ, and μ, the methodology evaluates alternative portfolios (for example, 10% more code, doubling governance participation, or dropping a membership) by recomputing value at the counterfactual quantities, with μ and prices held constant, and comparing it to actual value.