Fixed Tax Rate

Abstract

In section 1, we started from the premise that the government fixes the deficit ratio. Instead, in section 2, we shall suppose that the government fixes the tax rate. To get a feeling of the basic idea, regard a simple model of budget dynamics. The government buys goods and services in a certain proportion to national income G = gY with purchase ratio g = const. In addition the government collects a tax at a flat rate t = const on both factor income and debt income T = t(Y + rD). The budget deficit is defined as the excess of government purchases and public interest over tax revenue B = G + rD − T. The budget deficit in turn augments public debt $\dot{D}$ = B. From this follows $\dot{D}$ = gY + rD − t(Y + rD). Now it is helpful to perform the analysis in per capita terms. Take the time derivative of public debt per head $\dot{d}$ = $\dot{D}$ / N − D$\dot{N}$ / N² and observe $\dot{D}$ = gY + rD − t(Y + rD). Moreover let labour grow at the natural rate $\dot{N}$ = nN with n = const. This yields:

$$\dot{d}=gy+rd-t(y+rd)-nd$$

(1)

Here g, n, r, t and y are exogenous. In the steady state, public debt per head stops to change $\dot{d}$ = 0. Put this into (1) and regroup:

$$d=\frac{gy-ty}{n-(1-t)r}$$

(2)

To simplify matters, the investigation will be restricted to the case t < g. Correspondingly the evaluation of (2) gives rise to two distinct subcases. If the net interest rate stays below the natural rate, then public debt per head will be positive. The other way round, if the net interest rate surpasses the natural rate, then public debt per head will be negative, which seems to be somewhat like a paradox. Finally check for stability. Differentiate (1) for d to accomplish d$\dot{d}$/dd = (1− t) r − n. That means, as long as the net interest rate is less than the natural rate, the steady state will be stable. Yet as soon as the net interest rate is greater than the natural rate, the steady state will be unstable. In the real world, however, the interest rate will be endogenous, relying on the deeper parameters of the model. A tax cut, for instance, forces the government to borrow money. This displaces investment and capital, thus bidding up the interest rate.