# THE PRICE LEVEL: THEORETICAL CONSIDERATIONS 2

We want to express the budget constraint in terms of total government liabilities, M + B, and to scale the fiscal variables on GDP. We do this to facilitate policy discussions and empirical applications. After some tedious algebra, the budget constraint becomes

(2) says that the ratio of total government liabilities to GDP has to be equal to the ratio of the primary surplus (now inclusive of central bank transfers) to GDP plus the discounted value of the ratio of next period’s liabilities to GDP; the discount factor is the ratio of the real growth in GDP to the real interest rate. Finally, we want to simplify our notation by replacing (2) with

Wj is the liabilities to GDP ratio, Sj is the surplus to GDP ratio, and a} is the discount factor. It should be kept in mind that Sj includes central bank transfers (or seignorage).

We will follow Woodford’s (1995) development of the theory and focus on the government’s present value budget constraint. Iterating equation (3) forward from the current period, t, and taking expectations on information available in period t, we obtain the present value constraint

where П| ‘ак= 1. The two expressions in (4) are equivalent ways of writing the constraint.

Following Hamilton and Flavin (1986), a growing literature has tried to test one or the other of these constraints empirically; this literature interprets the results as a test of government solvency. By contrast, the new theory of price determination treats (4) as an equilibrium condition that must be satisfied. In our empirical application, it is part of the maintained hypothesis. The fundamental question here is: how does (4) get satisfied, and how do we solve the model?

There are a number of possibilities. For example, there may be an endogenous fiscal policy that makes the sequence {Sj} satisfy (4), no matter what values the discount factors, {aj}, or the initial liabilities to GDP ratio, wt, take in equilibrium. Another possibility is that the sequence {Sj} is independent of the level of the debt. Then, the discount factors, {otj}, and/or the initial liabilities to GDP ratio, w„ have to fluctuate in equilibrium to satisfy (4). How can wt = (Mt + B^/P^ move to satisfy (4)? Nominal liabilities are fixed at the beginning of the period, but a “jump” in nominal income can generate a change in the ratio. Here is where the assumption of nominal public sector liabilities is important. This would not be a theory of price determination without it; all of the onus of the adjustment to equilibrium in (4) would be on the discount factors or real income.