GDP Growth by Debt Acceleration

Consider the reduced-form GDP (behavioral) equation:

(1)      Y_t = A_t + b ΔD_t - r D_t-1

where:

a)  Y_t is GDP in period t from 1 to T,

b)  A_t is “autonomous” (Kaleckian) gross accumulation (C + I + X − M),

c)  ΔD_t ≡ D_t - D_t-1 is new government debt issuance (debt financing),

d)  b is the fiscal multiplier on debt-financed spending,

e). r D_t-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, ΔX_t ≡ X_t - X_t-1, and noting the second difference Δ²D_t ≡ ΔD_t - ΔD_t-1, we get:

        ΔY_t = ΔA_t + b [(D_t - D_t-1) - (D_t-1 - D_t-2)] - r (D_t-1 - D_t-2)

(2)    ΔY_t  = ΔA_t + b Δ²D_t - r ΔD_t-1

Divide both sides of equation (2) by Y_t to express the per-period growth rates:

(3)     g_t ≡ (ΔY_t) / (Y_t) = [ΔA_t + b Δ²D_t - r ΔD_t-1] / (Y_t)

where g_t = ΔY / Y is the one-period GDP growth rates.

Appendix A: 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 d_t ≡ D_t / Y_t, its law of motion (discrete-time analogue) is:

         Δd_t = (ΔD_t) / (Y_t) - d_t-1 (ΔY_t) / (Y_t) = (s_t + r D_t-1) / (Y_t) - g_t d_t-1

 (4)    Δd_t = s_t / Y_t + (r - g_t) d_t-1,

where s_t ≡ ΔD_t - r D_t-1 is called the primary deficit.

Setting Δd_t = 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.

Appendix B: Recasting Equation (3) as a Domar-Style Inequality
From equation (3), debt sustainability (g > r) becomes:

(5)     (ΔA_t + b Δ²D_t - r ΔD_t-1)/(Y_t) > r ==> ΔA_t + b Δ²D_t > r (Y_t + ΔD_t-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 ΔD_t-1.

In summary, Domar’s condition is about nominal GDP growth vs. nominal interest rates;

Domar (1944) shows that if:

d_t = (D_t) / (Y_t)

is the debt‐to‐GDP ratio, its approximate law of motion is:

(7)     Δd_t ≈ (r_nominal - g_nominal) d_t-1 + (primary deficit_t) / (Y_t).

In steady state (no change in d), we need to satisfy:

g_nominal > r_nominal

to prevent Δd_t > 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.