Precautionary taxation vs tax-smoothing - on paying down the debt

Bob Murphy is arguing with Steve Landsburg over whether the debt/GDP ratio should be (slowly, eventually) reduced. So I have to join in. Plus, (with my Carleton colleague Vivek Dehejia) I actually published a paper once on this very topic (unfortunately not available online).

(Just to forestall some comments, this is an argument about paying down the debt over the long run, and not about paying down debt in the middle of a recession).

Steve's argument is based on tax-smoothing. Planning to lower the debt/GDP ratio over time would mean that you would plan to have lower tax rates at some future time (when the debt/GDP ratio is lower) than today. But this would violate Ramsey's principle of optimal taxation, according to which the tax rates on two goods which have the same elasticities of supply and demand should be the same. To minimise deadweight losses for a given amount of tax revenue, we want to equalise the marginal deadweight loss per dollar of tax revenue, not just across two different goods, but across two different time periods. This Barro/Pigou/Ramsey argument for tax-smoothing implies that (provided we expect the future economy to be just a scaled-up version of the present) we should plan to keep the debt/GDP ratio at whatever it is right now. (Barro wrote a paper on this once, but I can't find it.)

I'm going to make one very small and very reasonable change to Steve's (implicit) assumptions. Assume that the future is uncertain. There is some degree of uncertainty over future government spending, or future GDP growth. So Steve's intertemporal first order condition now becomes "Current marginal deadweight cost per dollar of revenue equals expected future marginal deadweight cost per dollar of revenue". (All I have done is added the word "expected").?

Let "t" be the tax rate, and MDWL(t) be the function that represents Marginal DeadWeight Loss per extra dollar of tax revenue as a function of the tax rate t. Then:

Current MDWL(t) = Expected future MDWL(t)

But this implies current t = expected future t only if MDWL(t) is a linear function (that is also time-invariant because everything scales up). In a world where the future is certain, or where the MDWL function is linear, Steve would be right. We should plan to have future tax rates equal to current tax rates, and so plan to keep the debt/GDP ratio equal to whatever it is right now.

But any reasonable marginal deadweight cost function (like in Steve's quadratic example) will be concave convex in tax rates (or is it convex?, I always get them muddled). So we apply Jensen's Inequality, which tells us that MDWL(t)=Expected future MDWL(t) implies that current t > expected future t. We should plan to have lower tax rates in the future, which means we should plan to have a declining debt/GDP ratio.

The intuition is that slowly paying down the debt is like buying insurance against an uncertain fiscal future, because the benefits of a good surprise aren't as big as the costs of a bad surprise. It's like precautionary saving, only applied to the government debt.

Math appendix: Steve assumes R=At-Bt² where R is tax revenue and t is tax rate. I think that means the area of the deadweight loss triangle is DWL=(1/2)Bt² . We want to find the Marginal Deadweight Loss function, which is defined as MDWL(t) = dDWL/dR = (dDWL/dt).(dt/dR) = Bt/(A-2Bt). And I think that function is increasing at an increasing rate in t (positive second derivative) provided you are on the good side of the Laffer curve, and so is concave convex (or convex, whatever).

Source: http://worthwhile.typepad.com/worthwhile_canadian_initi/2012/11/precautionary-taxation-vs-tax-smoothing-on-paying-down-the-debt.html

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