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) (link here thanks to Keshav Srinivasan).
(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. Update: found 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).
Worthwhile Canadian Initiative 
Sorry, that was my unclear point. I am completely baffled by MMT and never contribute to threads on them.
It is an example of this engineer knowing his limits. I prefer policy threads.
Nick,
“The government has a monopoly on force, and that’s what enables it to tax. It doesn’t need money to tax. It could tax us 10% of the apples we trade, so we pay taxes in apples.”
A government does not have a monopoly on force of will. There are plenty of historical examples of governments being toppled by their own citizenry by force.
A government has a monopoly on its system of laws. What enables it to tax is people accepting the need for legal proceedings (an impartial 3rd party). And that third party does not work for free. Because lawyers (like anyone else) do not like to be paid in an asset that loses value over time (or rots like apples), the government requires that taxes be paid in its own currency so that it can regulate the value of that currency with respect to all other goods.
Of course this is the ideal case. Most governments also get involved in international diplomacy, social insurance, and a lot of other things.
I heartily recommend Noah Smith’s blog Noahpinion.
He has the most wonderful list of EconoTrolls (with pictures!).
http://noahpinionblog.blogspot.ca/2012/09/econotrolls-illustrated-bestiary.html
Find your own picture, I know I’m on there. 🙂
Yeah,
I was kind of perturbed that Friedman, Keynes, Minsky, Marx, Ludwig von Mises were all included but Fisher is absent. And so maybe this is how Fisherites see themselves:
And this is how the world sees them:
Nick,
Sorry that’s not what I was asking. I understand what dead weight loss is. I see you qualified diminishing marginal returns with short run. Is the short run defined as the amount of time required to increase capital? I’ll do my own internet study since it’s not the subject of this thread and I haven’t paid tuition here and don’t want to spend too much time highlighting my ignorance. I’m asking why diminishing marginal returns rather than what is it (think Sraffa). It just doesn’t jibe with my perception of reality. It seems like the exception rather than the rule unless you assume that full capacity utilization is the normal course of operation (I have other questions too but will stop here). Cheers.
Rev: There are two distinctions:
1. Diminishing marginal product (holding some inputs fixed and varying other inputs), vs diminishing returns to scale (varying all inputs together.
2. At the firm-level vs at the economy-wide level.
Take agriculture, for example. An individual farm might have diminishing marginal product of labour (holding land fixed) but constant returns to scale (varying land and labour together). But at the level of the whole economy, different types of land will be in fixed supply, so the supply curves for different types of food could be upward-sloping. The economy-wide PPF will not be a straight line, even if there are constant returns to scale, because some land (and labour) has a comparative advantage in producing some goods, while other land (and labour) has a comparative advantage in producing others, so supply curves will slope up at the economy-wide level.
If you think there are increasing returns to scale at the level of the firm, you simply ditch perfect competition and replace it with (say) monopolistic competition. I usually think in terms of monopolistic competition, with perfect competition an OK simplification for some purposes (it’s just a limiting case anyway).
You give MDWL(t) = dDWL/dR = (dDWL/dt).(dt/dR) = Bt/(A-2Bt). This = 0 at t = 0, goes to +infinity at t = A/2B and then reappears at -infinity and climbs to -1/2 at t = infinity. At t = A/2B,the loss = 1/2 Bt2 = 1/2 B * (A/2B)2 = A**2/8B. assuming I haven’t made a math error somewhere, what does this behavior mean?.
I sympathize with Frank, because real tax policy seems always to be concerned more with externalities to your model. If this weren’t the case we wouldn’t have distortions like tax expenditures. These should all have a net deadweight loss, and their justification would be positive externalities. I don’t know whether one should wish it were true or fear it (since it would be off to the races). Marijuana taxes, carbon taxes, Obamacare penalty taxes, estate taxes, carried interest – the US congress has only two solutions to any problem – make it a crime or distort the tax system (other than for things like declaring national strawberry week).
Peter N: I haven’t checked your math, but it sounds right. As you increase t, and approach the top of the Laffer curve (where R is at a maximum), the MDWL should approach infinity. That’s because dR/dt approaches zero (and then goes negative), while dDWL/dt keeps on increasing.
This is very like an optimal prudent leverage VAR calculation that you’d see with a bank, if you assume that the government spends as much as is prudently possible (that is what is either necessary or has a positive net ROI). This isn’t government as we know it, but it fits your model in not questioning the inherent value of government spending and only looking at the result. It doesn’t, however, include Ricardian equivalence overtly, though it could possibly sneak in through the ROI calculation. As a nonbeliever in Ricardian equivalence voodoo, I don’t much care.
The hardest part of optimal debt is obviously dealing with risk and uncertainty. You have ugly low probability high risk events with arguably low NPV (sort of like the question of how much you should pay to prevent a 10 meter sea level rise 150 years from now?), but God help you if you stumble into one.
There’s also the problem that a government’s debts serve as peoples’ safe assets and low information cost collateral (Triffin dilemma, social security trust fund…).
Peter N,
Because of the way credit markets are structured, the private markets bear both bankruptcy risk(cash flow is less than cost of servicing debt) and solvency risk (present value of assets is less than present value of liabilities) where the government bears neither.
A truly “safe” asset would work to offset one of those risks. Government bonds are safe in the sense that repayment is guaranteed. But because the interest rate paid by the government is always less than that paid by private borrowers, private borrowers are disadvantaged by the legal requirement to pay taxes.
Government equity could fit the bill as a “safe” asset from a solvency point of view. While the return on investment is not guaranteed to any one buyer, the net present value of the asset is know with a high degree of certainty.
The article above discusses whether the debt to GDP ratio should be reduced and if so,how quickly? Before answering that question I think you have to ask, why should the federal government sell debt at all?
Nick: You’re exactly right of course. Thanks for posting this.