I ‘ve been bogged down with my PhD courses lately and this blog has not seen the share of posts that it was accustomed to. In any case, since I ‘m looking back into our Macro 1 course (which is based mainly on the Romer and Blanchard books, both quite informative and good to have) I ‘d like to touch on the math of the Ricardian Equivalence and the Government Budget Constraint.
As is well known, the Ricardian Equivalence (RE) states that only the quantity of government purchases, not the division of the financing of those purchases between taxes and bonds, affects the economy. This is based on the household budget constraint of a Ramsey model where the government budget constraint (with equality) is applied. What the latter suggests is that the government must run primary surpluses large enough in present value to offset its initial debt.
Usually the government budget constraint is taken at face value by most economists while Ricardian Equivalence is suggested that it might not be (completely) valid in practice due to facts such as short-term horizons, liquidity constraints, different interest rates faced by the government and households and the Zero Level Bound. Nevertheless, I ‘m not so sure that a lot of people would disagree with the RE model in its original form (a very nice reference for both issues is chapter 12 of Romer).
In this post I would like to support the proposition that mathematically both the RE and the government budget constraint require the assumption that, at the infinite horizon limit, private household net financial wealth is zero. Households only gain utility from consumption so, at the limit, they will consume all financial wealth they hold which suggests that the present value of government primary surpluses should be equal to government debt.
Obviously this only works in a world where money is neutral and ultimately is only used as a transactions medium and not as a store of value to hedge an uncertain future. The notion of «safe assets» does not really enter into the realm of the Ramsey model where the representative agent either has perfect foresight or faces stochastic risk (and not genuine uncertainty).
As Philip Arestis elegantly states, «all economic agents with their rational expectations are perfectly creditworthy. All IOUs in the economy can, and would, be accepted in exchange. There is thus not need for a specific monetary asset. All fixed-interest financial assets are identical so that there is a single rate of interest in any period.»
Once this assumption is relaxed and the private sector requires a safe store of value not only to facilitate wealth maintenance but even the daily flow of transactions (with government debt securities playing a crucial role in monetary policy operations and the repo markets), the government budget constraint breaks down while an increase in the supply of government debt can even be stabilizing (see for example the relevant Treasury programs during the 2008 credit crisis) and is required in order for the private sector to maintain its wealth in a safe store of value, free from shocks that might emerge in an uncertain world.
Further, mathematical details can be found in the attached (small) pdf document.