We analyse the implications of optimal taxation for the stochastic behaviour of debt. We show that when a government pursues an optimal fiscal policy under complete markets, the value of debt has the same or less persistence than other variables in the economy and it declines in response to shocks that cause the deficit to increase. By contrast, under incomplete markets debt shows more persistence than other variables and it increases in response to shocks that cause a higher deficit. Data for US government debt reveals diametrically opposite results from those of complete markets and is much more supportive of bond market incompleteness.