Consider the reduced-form GDP (behavioral) equation:
(1) Yt = At + b ΔDt - r Dt-1
where:
a) Yt is GDP in period t from 1 to T,
b) At is “autonomous” (Kaleckian) gross accumulation (C + I + X − M),
c) ΔDt ≡ Dt - Dt-1 is new government debt issuance (debt financing),
d) b is the fiscal multiplier on debt-financed spending,
e). r Dt-1 are interest payments on the existing debt stock, and
f) r (viewed as a constant here) is the average interest rate on the government debt.
Equation (1) is basic and applies to any economy, capitalist or socialist, rich or poor.
Taking the one-period (forward) difference, ΔXt ≡ Xt - Xt-1, and noting the second difference Δ2Dt ≡ ΔDt - ΔDt-1, we get:
ΔYt = ΔAt + b [(Dt - Dt-1) - (Dt-1 - Dt-2)] - r (Dt-1 - Dt-2)
(2) ΔYt = ΔAt + b Δ2Dt - r ΔDt-1
Divide both sides of equation (2) by Yt to express the per-period growth rates:
(3) gt ≡ (ΔYt) / (Yt) = [ΔAt + b Δ2Dt - r ΔDt-1] / (Yt)
where gt = ΔY / Y is the one-period GDP growth rates.
The Domar condition of debt sustainability
In his article “Burden of Debt and National Income” (1944), Evsey Domar shows that if we define the debt-to-GDP ratio dt ≡ Dt / Yt, its law of motion (discrete-time analogue) is:
Δdt = (ΔDt) / (Yt) - dt-1 (ΔYt) / (Yt) = (st + r Dt-1) / (Yt) - gt dt-1
(4) Δdt = st / Yt + (r - gt) dt-1,
where st ≡ ΔDt - r Dt-1 is called the primary deficit.
Setting Δdt = 0 for a steady debt-to-GDP ratio gives:
d* = (s / Y) / (g - r),
which is finite (and positive) only if g > r.
Conversely, if g < r then Δd > 0 and the debt/GDP ratio explodes. Hence the growth rate of GDP must exceed the interest rate for public debt to be sustainable.
Recasting equation (3) as a Domar-style inequality
From equation (3), debt sustainability (g > r) becomes:
(5) (ΔAt + b Δ2Dt - r ΔDt-1)/(Yt) > r ==> ΔAt + b Δ2Dt > r (Yt + ΔDt-1).
To obtain a growth rate above the borrowing rate r, the net injection of autonomous demand growth (ΔA) plus the acceleration of debt-financed spending (b Δ2D) must cover the interest drain on past debt: r ΔDt-1.
In summary, Domar’s condition is about nominal GDP growth vs. nominal interest rates;
Domar (1944) shows that if:
dt = (Dt) / (Yt)
is the debt‐to‐GDP ratio, its approximate law of motion is:
(7) Δdt ≈ (rnominal - gnominal) dt-1 + (primary deficitt) / (Yt).
In steady state (no change in d), we need to satisfy:
gnominal > rnominal
to prevent Δdt > 0 from becoming uncontrolled by monetary or fiscal policy..
References
Evsey Domar, "The 'Burden of the Debt' and the National Income," American Economic Review (December 1944): https://www.jstor.org/stable/1807397
Gross accumulation is a concept from Michał Kalecki, Theory of Economic Dynamics (2nd edition), George Allen & Unwin, 1965.