Measuring λ: a first estimate for France

In The circles of interest I proposed treating the decay of human concern across circles as a measurable object: fit w(d) ≈ e^−λd to revealed resource allocations and read off λ, a parochialism coefficient. This note runs the fit once, crudely, for France — to show the method computes, not to settle it.

1. Setup

Assign circle distances: self d=0, family d=1, community d=2, compatriot strangers d=3, distant foreigners d=4. Let w(d) be the revealed per-capita weight — how much of a marginal euro a person at distance d effectively commands, relative to the self. Two regimes must be fitted separately, because they answer different questions: voluntary allocations (what individuals do freely) reveal the private gradient; compulsory allocations (what the tax-and-transfer state enforces) reveal the political settlement.

2. The anchors

Four order-of-magnitude anchors, all already used in the parent essay (French data where available; each is the crude midpoint of a documented range):

1. Family (d=1), voluntary: household income is substantially pooled — call the marginal weight of a family member ~0.5 of self.
2. Compatriot strangers (d=3), voluntary: declared charitable giving in France runs on the order of 0.1–0.5% of income, spread across distant beneficiaries — a marginal weight around 0.005.
3. Distant foreigners (d=4), voluntary: the international share of private giving is roughly a tenth of that — weight ~0.0005.
4. Compulsory: compulsory levies near 45% of GDP force an effective compatriot weight around 0.5; official development aid at ~0.5% of GNI implies a foreigner weight near 0.005.

3. The fits

Voluntary: a least-squares line through ln w at d = 0, 1, 3, 4 gives λ_voluntary ≈ 2.0 per ring — concern decays about seven-fold per circle. The fit is decent because the decay really is close to geometric: 1 → 0.5 → (~0.05) → 0.005 → 0.0005.

Compulsory: the same procedure fails informatively. Inside the border, the state forces the compatriot weight up to ~0.5, which corresponds to λ_forced ≈ 0.23 per ring — nine times flatter than the private gradient. But between d=3 and d=4 the weight collapses from 0.5 to 0.005: a single-step drop of ln(100) ≈ 4.6, more than double the total voluntary decay over the same step. No single exponential fits; the compulsory curve is not a slope but a plateau with a cliff — which is exactly the shape the parent essay predicted qualitatively, now with numbers on both parts.

One sentence of interpretation: the French state buys a ~9× flattening of human parochialism inside its border, and pays for it by concentrating all of the suppressed decay into one 100-fold cliff at the frontier. Politics, in this frame, is the machine that converts a smooth private gradient into a step function.

4. What would change my mind

1. Better anchors. Household budget surveys, tax incidence studies and giving registries can replace midpoints with distributions — the d=2 community ring, skipped here for lack of a clean proxy, needs one (local giving, associative time, communal budgets).
2. Cross-country λ. The same four anchors exist for every OECD country. If the framework is right, the US should show a steeper forced interior (smaller state) with a similar cliff, and the Nordics a flatter interior; if measured λ does not order countries the way welfare-state size does, the model is in trouble.
3. Time and shocks. λ should move measurably in wars, pandemics and disasters (the parent essay's natural experiments). A λ that never moves is a λ that explains nothing.
4. The non-concentric residuals. Remittances alone (kin abroad ≫ neighbors at home) show the exponential is only the radial baseline — the interesting sociology lives in the deviations.