At today’s macro conference hosted by John Taylor, George Hall presented his paper with Tom Sargent titled “Three world wars: Fiscal-monetary consequences” at today’s Taylor conference. James Bullard asked Hall and Sargent to compare their analysis of taxation and spending with the Barro theory. In their paper, Hall and Sargent say

“After World War I and World War II, tax revenues remained elevated, letting the government run a primary surplus for many years. These patterns are consistent with the tax-smoothing responses to temporary government spending surges called for in [Barro, 1979]”

This is the standard story told in macro courses and seminars for 43 years. I recently told a student to read Barro(1979) because it is part of the literature relevant for his thesis. I figured I should also read it because I suspected that he might find it difficult to understand the style of a 1979 JPE paper.

I apparently had never read it carefully, and was familiar only with the special case where the loss function was quadratic. There is nothing wrong with Barro’s analysis of that case. However, as I was reading the paper, I realized something was wrong. Barro assumes a general loss function for tax revenue and specifies a simple dynamic programming problem to study choices between taxation and borrowing but he never solves it. Instead he uses a certainty equivalent argument: Compute the decision rule when government spending is known and use it as an approximation of the decision rule for the stochastic case. This implies that debt will follow something like a random walk for any specification of the loss function.

I generally do not like models that imply random walk dynamics because such models are often not structurally stable; that is, a small change in the model elements will produce dynamics very different form random walk dynamics. After computing a perturbation solution (up to fourth order in the standard deviation of spending shocks) I discovered that Barro’s “solution” was correct for the quadratic loss function assumption but otherwise almost always wrong. Not just wrong in the sense of numerical errors but wrong in the qualitative results. Instead of debt following a random walk, the optimal policy implies that debt should be reduced and that the government should accumulate assets to the level where interest on the assets covers most government expenditures.

I was pleased with my discovery, but a few days ago I made another discovery. In 1995, Dehejia and Rowe published “The Laffer and Precautionary Taxation: A Rationale for Paying Down the National Debt” Australian Economic Papers which makes all these points, and exposits them in a much nicer way than I had.

The key point is that Barro implicitly assumed that the loss function was quadratic, which implies that there is no upper limit on how much revenue can be raised. This is inconsistent with basic tax theory which tells us that there is an upper bound on potential tax revenue. How can a society produce tax revenue which is ten times its national income? It can’t but the quadratic assumption for the loss function allows this to happen. Random walk dynamics imply that this will happen with significantly positive probability under the standard assumption that government expenditures are exogenous. Dehejia and Rowe gives a very nice exposition of the problem using ideas from precautionary savings to argue that the Barro model in general implies precautionary taxation.

Macroeconomists love quadratic specifications for payoff functions, even when such assumptions violate simple microeconomic logic. Macroeconomists’ devotion to the Barro (1979) analysis and rejection of Dehejia and Rowe’s analysis in their 1995 paper is just one example of how their desire to keep things simple produces results that are not robust.

I have made nasty comments about economics training at Columbia University. I should remind readers of my blog that my negative comment was only about a Columbia professor. I am happy to report that Vivek Dehejia started this work when he was a PhD student at Columbia.