# Stimulating equations

January 27, 2009 | 1 Comment | Joshua Gans

Chicago’s Kevin Murphy has generated a ton of discussion regarding his framework for analysing spending stimulus packages. Here is the video (starts at around 17 minutes in) and it is best viewed with these accompanying slides. Brad de Long summarises the model here.

Murphy argues that government spending will be good if the following inequality is satisfied:

f(1-λ) > α+d

where f is the fraction of the output produced using “idle” resources, λ is the relative value of “idle” resources, α is the inefficiency of government spending and d is the deadweight cost per dollar of revenue from the taxation required to pay for the spending. This equation neatly divides up into left and right. The right hand side has all of the costs of extra government spending (inefficiency relative to private spending plus the costs of taxation) while the left hand side has the benefits (the level of unemployed resources multiplied by the benefits of employing them).

The model here suffers from the dangers of simplicity which I will get to in a second. But it does neatly tell us why the case for a government spending stimulus is stronger now than a year ago. Our expectation is that f is or is going to be relatively high soon. So projects that might not have satisfied this test a year ago (like a big bang broadband infrastructure investment) may well satisfy it now.

But we also need to be cautious in reducing everything to this particular equation. Here are the reasons:

• Margins versus averages: this is an equation that allows us to evaluate — project by project — whether government spending is worthwhile and not whether the entire stimulus package is good or not. Unless you believe this equation can never be satisfied for any government spending, there exist projects that just missed out on a tick last year that will get over the line this year. And, in the Australian context, it would not surprise me if \$20 billion of stimulus now satisfied this. Roughly, Australian governments spend \$400 billion per annum. So the notion that a 5 percent jump in spending is warranted when facing the sort of recession we appear to be facing (and without monetary policy to work effectively) is not necessarily crazy. (In the US, total federal government spending is roughly \$5200 billion and so they are talking of stimulus packages of around 20 percent).
• The temporal dimension: this equation makes it look like we should evaluate whether to do a particular project at all. However, when we are talking about stimulus packages the projects we should be looking to be part of it are projects which we were likely to do in the future but it is better to bring forward and do now. So the decision choice is often between doing it later and doing it now. Infrastructure, public works, environmental policy all are candidates for this whereas handouts to people and industry are much harder cases to make.
• Endogeneity: all of these parameters are not given but are potentially related to one another. I won’t go into it here but, as with all models, this one might be useful in framing the debate but it is far from the final word — up or down. That said, it is better than nothing.

One Response to “Stimulating equations”

1. derrida derider on January 27th, 2009 4:46 pm

Just seems to me like a version of the old saw “you can fit an awful lot of Harberger triangles into an Okun gap”.