Adding Government Sector
New flows are required:
- Tax: Households → Government
\[Tax = GDP \times Tax_{rate}\] - Government Spending: Government → Firms
\[Spend_{Gov} = GDP \times Spend_{rate}\] - Bond Sales: Households → Government
\[Bonds = (Spend_{Gov} - Tax) + (Debt_{Gov} \times Int_{rate})\] - Interest on Bonds: Government → Households
\[Int_{Gov} = Debt_{Gov} \times Int_{rate}\]
Differential Equations (Loanable Funds with Government)
\[\frac{dDebt_{Gov}}{dt} = Bonds\]
\[\frac{dDebt_{Firms}}{dt} = Credit\]
\[\frac{dFirms}{dt} = Consume + Credit + Spend_{Banks} + Spend_{Gov} - Wages - Int_{Firms}\]
\[\frac{dHouseholds}{dt} = Wages + Int_{Gov} + Int_{Firms} - Consume - Credit - Fee - Tax + Bonds \quad (4)\]
\[\frac{dGovernment}{dt} = Tax + Bonds - Spend_{Gov} - Int_{Gov}\]
\[\frac{dBanks}{dt} = Fee - Spend_{Banks}\]
Result: Inevitable Crisis
Running with 1% deficit (Spend = 31% GDP, Tax = 30% GDP):
- Government debt → 3,000% of GDP (over 100 years)
- Interest payments → 100% of GDP
- System becomes unsustainable
Conclusion: Under Loanable Funds, sustained deficits cause inevitable fiscal crisis.
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