5 - A simple model

Summary

In this chapter we outline a simple formal model to capture rigorously the possibility of expansionary fiscal consolidation, using the mechanism discussed in the earlier chapters. We start with the goods market equilibrium condition, which we presented as equation (1), where we have made specific assumptions about the determinants of consumption and investment:

Here, Y_t is national income, r_t is the real long-term interest rate, G_t is government spending, T_t is taxation and I_t is investment, all at time t. Φ_t is intended to capture the effects of future variables on current consumption; for example, changes in future expected taxation, due to a fiscal consolidation policy, can impact current consumption through their effect on this term. The equation is effectively an IS curve, where the relevant interest rate is the long-term rate; the only other way it differs from the standard textbook IS curve is the presence of the Φ_t term in the equation.

However, instead of adding an LM curve to determine overall macroeconomic equilibrium, we add an equation for the long-term nominal interest rate derived from (14) which we write as follows:

If we know the current short-term and long-term interest rates, as well as the current stock of government debt, the long-term interest rate next period can be determined, and so forth. It follows that, if the time path of both debt and the short-term interest rate is known, we should be able to work out the time path of the long-term interest rate. For simplicity we assume that agents have static expectations of the short-term interest rate, and that the stock of government debt follows a time path such that it reaches a steady-state level at time t + n, which means that we can write the following expression for the long-term interest rate at time t:

which is the solution to equation (16). This states that the long-term interest rate equals the current short-term rate plus a term that depends positively on the future time path of government debt between now and the relevant date in the future. We would stress that it is assumed that the short-term interest rate is expected to be constant; otherwise, expected future short rates will affect the long-term rate as well.