Government Debt Bias

Abstract

Almost all governments issue large stocks of debt. Optimal taxation theory typically concludes that they should hold large stocks of assets. To reconcile facts and theory, we introduce two simple modifications into an otherwise standard optimal taxation model with commitment: government impatience and continuous debt limits. Two results are obtained. First, positive government debt is optimal for even minimal government impatience. Second, the optimality of negative government debt disappears even without impatience if discrete debt limits are replaced by very wide but continuous debt limits. We go on to quantify the implications for debt, interest rates and business cycle dynamics.

Appendix: The Non-stochastic Steady State

We drop time subscripts to denote steady-state values of variables. The non-stochastic steady state of the economy is given by the system of five equations (11), (16), (19), (20) and (21) determining the variables $c, \ell , b, \eta$ and $\lambda$. Equations (19), (20) and (21) become

$$\eta =\frac{1}{c}\left( 1+\lambda \frac{1-\ell -\kappa \ell }{1-\ell } \right) +\lambda b\frac{1}{c^{2}}\left( \beta -\frac{1}{\gamma }\right) ,$$

(A1)

$$\eta =\frac{\kappa (1-\ell +\lambda )}{(1-\ell )^{2}}-\lambda \frac{1}{c} \frac{\phi }{4}\left( \frac{b}{\ell }\right) ^{2},$$

(A2)

$$2\frac{\phi }{4}\frac{b}{\ell }=\beta (1-\gamma )$$

(A3)

where we have combined (19) with (16). Consider the case of $\lambda =0$. In that case, we would have $\eta =1/c$ and $\eta =\kappa /(1-\ell )$. Then by the consumer’s first-order condition (4), it would have to be true that $\tau =0$. Such a case would only be possible in the first-best, which is only achievable if the government can accumulate a sufficient amount of assets to finance fiscal spending without any distortionary taxation. This is however ruled out in our model by condition (A3), which makes the steady-state debt stock positive. We can therefore rule out $\lambda =0$. The remaining steady-state conditions are

$$c+g=\ell ,$$

(A4)

$$1-\frac{\kappa \ell }{1-\ell }=\frac{1}{c}\left( (1-\beta )b+\frac{\phi }{4} \frac{b^{2}}{\ell }\right) .$$

(A5)

In a first step, the steady-state values b, c and l can be solved from (A3), (A4) and (A5). In a second step, the remaining equations (A1) and (A2) then determine $\lambda$ and $\eta$. The first step results in the quadratic equation for $\ell$

$$\left[ \kappa +1-\varphi \right] \ell ^{2}-\left[ 1+(\kappa +1)g-\varphi \right] \ell +g=0,$$

(A6)

where

$$\varphi =\frac{2\beta (1-\gamma )}{\phi }\left( 1-\frac{\beta (1+\gamma )}{2} \right) .$$

(A7)

There are therefore two possible solutions for steady-state labor, and by (A4) also for steady-state consumption. The roots of equation (A6) are given by

$$\ell _{1,2}=\frac{1+(\kappa +1)g-\varphi \pm \sqrt{(1+(\kappa +1)g-\varphi )^{2}-4g(\kappa +1-\varphi )}}{2(\kappa +1-\varphi )}.$$

(A8)

While both roots are positive for our parameterization, the smaller root implies a level of consumption very close to zero ($c=0.0002$) and a much lower welfare than the larger root. The smaller root can therefore be ruled out. This means that even for large fluctuations around the steady state, the use of a perturbation method that approximates the solution around the superior steady state remains appropriate.