Dynamics of Government Debt

I hope I don't make Nick Rowe [1], Scott Sumner, and their fellow Monetarists too angry by assuming that central banks can only monetize government debt, but I think this analysis is still relevant since central banks usually refrain from trading assets other than government bonds.

Anyway, on to the post. Imagine a world in which the fiscal authority never issues any debt. In this world, monetary policy would be equivalent to fiscal policy. Every deficit is funded by seigniorage, so either the central bank gets to target some nominal variable or the fiscal authority gets to set the inflation rate. To see how this works, consider eliminating government bonds (and other assets, should they be present) from the governments budget constraint. This gives

$$ (1) \: M_t + P_t \tau_t = M_{t-1} $$

where $ M_t $ is the money supply, $ P_t $ is the price level, and $ \tau_t $ is the treasury's surplus. Assuming the money demand function simplest money demand function possible, $ M_t = L(P_t) = P_t $ and expressing the constraint in real terms gives

$$ (2) \: \pi_t = -\tau_t $$ ($ \pi_t $ is the rate of inflation)

This world has the unfortunate problem of either being ultra-FTPL (fiscal authority determines the inflation rate) or just plain weird (I don't know what else to call a world where the central bank chooses the fiscal authority's surplus/deficit). Aside from the obvious difficulty of Sargent and Wallace's [2] game of chicken, the problem of a serious conflict of interests arises. What if the optimal fiscal policy is austerity, but the optimal monetary policy involves a high rate of inflation and vice versa? [3] Proposition 1: If that situation can arise, then a non-zero level of government debt is optimal. 

Essentially, government debt allows the monetary and fiscal authority to have contradicting policies at any given point in time. So long as there is government debt, the central bank can always control inflation (I think, but I need to look into the FTPL under an exogenous inflation rate or a money growth rule) and the treasury can always control the surplus.

Let's assume the the level of government debt must be positive [4]. Given this constraint, the central bank can at most monetize 100% of current government debt, essentially imposing a maximum rate of inflation that the central bank can achieve. Proposition 2: The ideal level of government debt is whatever is required for the central bank to achieve its nominal target at any point in time. [5] So, if government debt levels are not sufficiently high (or government debt is not growing quickly enough), then the central bank won't be able to attain its goals.

For complete monetary freedom in my model. government debt needs no upper limit, but having infinitely large government debt is not optimal for obvious fiscal reasons. Ideally, the real value of government debt should not be so high debt servicing costs on the part of the fiscal authority demand constant high primary surpluses. Proposition 3: In order to minimize the burden of high real debt levels, nominal debt should grow at a rate consistent with the central bank's nominal target.

Combining Propositions 2 & 3, we get Proposition 4: The level of government debt should always be high enough for the monetary authority to achieve its nominal target and should grow at the minimum rate required for said nominal target to be achieved.

[1] Nick Rowe, 2015. "Why can't central banks monetise something else?" 

Worthwhile Canadian Initiative.

[2] Thomas J. Sargent & Neil Wallace, 1981. "Some unpleasant monetarist arithmetic,"
Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.

[3] Of course, the optimality of the policies is pretty unnecessary, the problem still arises if the the central bank and the treasury want to pursue opposite policies. Maybe the fiscal authority is being stupid, and the monetary authority chooses to offset its actions, for example.

[4] This assumption is theoretically weak, but given that I don't know of many governments that are net creditors, I think it is acceptable for my current purposes.

[5] David Andolfatto sort of touched on this in his post "Understanding Lowflation"