Public debt dynamics: the effects of austerity, inflation, and growth shocks

Abstract

We study the impact of macroeconomic shocks on US public debt dynamics using a VAR with debt feedback. Following a primary balance, or austerity, shock, the debt ratio initially declines but at a cost of lower growth. The debt ratio then rises to its pre-shock path, suggesting the austerity shock could be self-defeating. An inflation shock reduces the debt ratio initially, while a positive growth shock unambiguously lowers debt. Our specification, properly incorporating the debt equation, produces different debt impulse responses and forecasts from VAR models either excluding debt or including debt linearly.

Additional derivations

We define the decomposition of the debt impulse response, $d^{IR}$, in terms of the contribution of each macroeconomic aggregate as follows:

$$\begin{aligned} d_t^{IR} = d_t^s - d_t^n = pb_t^* +i_t^* -\pi _t^* -g_t^*, \end{aligned}$$

where s and n stand for “shock” and “no-shock” debt paths. Using debt dynamics equation (2) in the text and approximating the nonlinear component, the components of the decomposition at time t are:

$$\begin{aligned} pb_t^*&= \left( pb_t^s-pb_t^n\right) + \left( 1+i_t^s-\pi _t^s-g_t^s\right) pb_{t-1}^*\\ i_t^*&= \left( i_t^s-i_t^n\right) d_{t-1}^n + \left( 1+i_t^s-\pi _t^s-g_t^s\right) i_{t-1}^*\\ \pi _t^*&= \left( \pi _t^s-\pi _t^n\right) d_{t-1}^n + \left( 1+i_t^s-\pi _t^s-g_t^s\right) \pi _{t-1}^*\\ g_t^*&= \left( g_t^s-g_t^n\right) d_{t-1}^n + + \left( 1+i_t^s-\pi _t^s-g_t^s\right) g_{t-1}^* \end{aligned}$$

The first term in each equation indicates the difference between “shock” and “no-shock” paths of the components scaled by the previous “no-shock” debt ratio. The second term is the adjusted previous value of the component. Thus, the debt impulse response decomposition is:

$$\begin{aligned} d_t^{IR} = \Delta ^{s/n}pb_t + \bigl (\Delta ^{s/n} i_t-\Delta ^{s/n}\pi _t-\Delta ^{s/n}g_t\bigr ) d_{t-1}^n + \bigl (1+i_t^s-\pi _t^s-g_t^s\bigr ) d_{t-1}^{IR}\,, \end{aligned}$$

where $\Delta ^{s/n}$ stands for the difference between “shock” and “no-shock” paths. Note also that the last term disappears in the initial period, $t=1$, as the previous (before shock, $t=0$) debt ratio is same.