* For economies with fractional reserve-generated fiat money*, balancing the budget is characterized by an exponential growth D(t) ≈ D

_{0}(1 + r)

^{t}of any initial debt D

_{0}subjected to interest r as a function of time t due to the compound interest;

*. At the same time, besides default, this increasing debt can only be reduced by the following five mostly linear, measures:*

**a fact known since antiquity**(i) more income or revenue I (in the case of sovereign debt: higher taxation or higher tax base);

(ii) less spending S;

(iii) increase of borrowing L;

(iv) acquisition of external resources, and

(v) inflation; that is, devaluation of money.

Whereas (i), (ii) and (iv) without inflation are essentially measures contributing linearly (or polynomially) to the acquisition or compensation of debt, inflation also grows exponentially with time t at some (supposedly constant) rate f ≥ 1; that is, the value of an initial debt D_{0}, without interest (r = 0), in terms of the initial values, gets reduced to F(t) = D_{0}/f^{t}. Conversely, the capacity of an economy to compensate debt will increase with compound inflation: for instance, the initial income or revenue I will, through adaptions, usually increase exponentially with time in an inflationary regime by If^{t}.

Because these are the only possibilities, we can consider such economies as closed systems (with respect to money flows), characterized by the (continuity) equation

If^{t} + S + L ≈ D_{0}(1+r)^{t}, or

L ≈ D_{0}(1 + r)^{t} − If^{t} − S.

Let us concentrate on sovereign debt and briefly discuss the fiscal, social and political options. With regards to the five ways to compensate debt the following assumptions will be made: First, in non-despotic forms of governments (e.g., representative democracies and constitutional monarchies), increases of taxation, related to (i), as well as spending cuts, related to (ii), are very unpopular, and can thus be enforced only in very limited, that is polynomial, forms.

Second, the acquisition of external resources, related to (iv), are often blocked for various obvious reasons; including military strategy limitations, and lack of opportunities. We shall therefore disregard the acquisition of external resources entirely and set A = 0.

As a consequence, without inflation (i.e., for f = 1), the increase of debt

L ≈ D_{0}(1 + r)^{t} − I − S

grows exponentially. This is only “felt” after trespassing a quasi-linear region for which, due to a Taylor expansion around t = 0, D(t) = D_{0}(1 + r)^{t} ≈ D_{0} + D_{0}rt.

So, under the political and social assumptions made, compound debt without inflation is unsustainable. Furthermore, inflation, with all its inconvenient consequences and re-appropriation, seems inevitable for the continuous existence of economies based on fractional reserve generated fiat money; at least in the long run.