The Fundamental Identity
Sectoral Balances
The change in Treasury equity:
\[\frac{dTreasury_{Equity}}{dt} = Tax - (Spend_{Gov} + Int_{Gov}) \quad (7)\]
The change in Private sector equity (Banks + Firms + Households):
\[\frac{dPrivate_{Equity}}{dt} = (Wages - Consume) + (Consume + Spend_{Gov} - Int_{Firms} - Wages) + (Int_{Firms} + Int_{Gov} - Tax)\]
Terms cancel to leave:
\[\frac{dPrivate_{Equity}}{dt} = Spend_{Gov} + Int_{Gov} - Tax \quad (8)\]
The Key Insight
Comparing equations (7) and (8):
\[\boxed{\frac{dPrivate_{Equity}}{dt} = -\frac{dTreasury_{Equity}}{dt}}\]
Government deficit = Private sector surplus (by accounting identity)
In the BOMD model: - Government “debt” IS private sector wealth - Government deficit ADDS to private sector equity - This is not a theory—it’s an accounting identity
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