Three profit shares on arm revised — GOS/VA, π = r/R, and shp = YD_c/YD

Why the income-side share (0.34/0.40) sits below both the production share (0.45/0.49) and the Sraffa–Pivetti standard ratio (0.42), and where the surplus goes in between

Written: 2026-09-28 · Read from: output/scenarios/arms/revised/runs/baseline.RDS (t = 100).

The revised arm has three distinct “profit shares”, and they are not the same object. R1 (the money-wage numéraire) removed the units bug that made shp and π incomparable; what remains is the pure redistribution gap between the production account and the income account.

1. The three measures

region	GOS/VA (production)	pi = r/R (standard)	shp = YD_c/YD (income)
Z1	0.4497	NA	0.341
Z2	0.4929	NA	0.402
global	NA	0.4227	NA

At t = 100: \(R_{\max}\) = 0.6036 (with \(D = \iota\, d'\)), \(r\) = 0.2551 (Sraffa–Pivetti, \(r = r_b^w + e\)), so \(\pi_{\rm std} = r/R\) = 0.4227. That is a price-system ratio — the profit share implied by the uniform rate on the standard commodity — not an income share.

2. GOS → YD_c: where the surplus goes

The production surplus GOS = VA − WB is not what reaches capitalists as disposable income. The largest deductions are depreciation (a cost retained before profit is distributed) and retained profit; the rest is the interest/dividend/tax reshuffle.

	region	GOS	minus_AF	minus_retained	minus_DIV	residual	equals_YDC
Z1_va	Z1	398.158	-154.869	-9.434	-224.826	-171.825	180.853
Z2_va	Z2	1555.553	-557.070	9.434	-965.668	-785.314	827.564

Read GOS = AF + DIV + retained + YDC + residual. AF (depreciation) is the single largest leak — it is part of gross surplus but never reaches any household as income, because the quantity side netted it out before dividends were struck. The residual column is the interest/tax reshuffle that this table has not yet separated (bank and central-bank interest, capital-income tax, and cross-border dividend payments), and it is the part a full SFC trace would close to zero by identity.

3. Why shp sits below GOS/VA and π_std

Three channels, in order of size:

1. Depreciation AF. It is inside GOS (gross of it) but inside YD_c (net of it), so it lowers shp relative to GOS/VA by AF/VA — roughly 0.175 (Z1) and 0.177 (Z2) of value added.
2. The denominator. shp divides by disposable income YD, not VA; the two differ by taxes net of transfers and depreciation, so even an unchanged YD_c moves shp.
3. The interest/dividend reshuffle. YD_c includes dividends and interest received by households but not interest paid to banks, the central bank, or abroad, nor capital-income tax — all of which are part of GOS. This is the residual column above.

π_std = r/R = r format(pi_std, digits=4) is different again: it is the standard-commodity ratio the uniform rate implies, not an income share. The hand-off’s R1 finding stands: once wages and prices share a numéraire, the remaining shp vs GOS/VA gap is pure redistribution, and closing it is a stock-flow identity exercise, not a redefinition of π.

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Sources: output/scenarios/arms/revised/runs/baseline.RDS · model/code/MVP_model_2026.R (household income 1.2–1.4, firm income 2.4–2.6) · model/code/sraffian_block_2026.R (world_money_rate, capital_composition_D).