Can the Sufficient Statistic Approach Avoid the Lucas Critique?

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The sufficient statistic approach is a methodology aimed at formulating public policy recommendations. It is based on the idea that, in some cases, it is possible to address a complex policy issue using a simple formula and available empirical estimates of treatment effects. The formulas are cleverly derived with the aim of summarizing the behavior of a complex model economy when affected by a particular policy shift. The estimates of treatment effects are econometric estimates of the effect of a particular policy shift on the relevant economic outcome(s). 

Following the Palgrave Dictionary of Economics, the Lucas critique is a “criticism to econometric policy evaluation procedures that fail to recognize that optimal decision rules of economic agents vary systematically with changes in policy … the estimated regression coefficients are not invariant but will change along with agents’ decision rules in response to a new policy.”²

For example, consider a negative empirical relationship between inflation and unemployment widely known as the Phillips curve. A casual look at such empirical relationship would suggest that, given the slope of the empirical relationship, any policy pushing inflation up would also reduce unemployment. The Lucas critique says that this is not the case. In certain models, policies with the effect of permanently increasing inflation lead to changes in employers’ forecasts of inflation without lowering unemployment. This happens even though, in the model, there is a positive empirical relationship between inflation and unemployment caused by demand shocks which increase both prices and employment.

At first pass, it may seem implausible to think that simple formulas depending on just a few parameters would avoid the Lucas critique. However, the Sufficient Statistic Approach can pass this test to a large extent. There are two considerations that suggest this is the case:

• The formulas used by the sufficient statistic approach are based on the optimal response of economic agents with respect to a change in policy.
• The statistics that enter the formula should be estimated using quasi-experimental policy variation. That is, the parameters that enter the formula should take into account the empirical reaction to a shift in the policy rule.

However, there are cases for which the sufficient statistic approach requires extrapolation. This means that the policy under consideration is beyond the policy shift previously observed in the data. For example, consider using a formula to predict the top of the Laffer curve resulting from changing the top marginal tax rate on personal income. In this application, the sufficient statistic approach formulas are based on a condition for revenue maximization that holds at the top of the Laffer curve and they depend, among other things on a “behavioral elasticity” parameter that measures the reduction in income by economic agents when the tax rate on income goes up.

According to some academics, reaching the top of the Laffer curve may require top marginal rates as high as 75 percent.³ – for a blogpost describing this work click link. In contrast, the existing behavioral elasticities that may be plugged into the formula are necessarily measured around the status quo (that is, exploiting empirical policy variation around a top marginal rate of roughly 42.5 percent). Therefore, the predictions of the formulas are accurate if either:

• The top of the Laffer curve is near the status quo
• The top of the Laffer curve is away from the status quo but the elasticities that enter the formula are approximately invariant with respect to the top tax rate

To make sure either holds, researchers should bench-test their formula within one or many structural general equilibrium model(s).

In a recent working paper with Mark Huggett (available in this link),⁴ we explored the connection between quantitative macroeconomics and the sufficient statistic approach.⁵ We explored this connection in the context of setting the top marginal rate of the U.S. federal income tax system, the application mentioned above.

One of the messages of our paper is that, from the perspective of quantitative macroeconomics, the sufficient statistic approach can give useful policy guidance. We provided a formula for the revenue-maximizing tax rate which depends on three elasticity parameters and applies to a broad range of macro models. The Badel-Huggett formula embeds and extends a widely-accepted formula employed in previous work.⁶

As suggested previously, we bench tested our formula within a quantitative dynamic human capital model and found that it performs quite well in predicting the top of the Laffer curve that holds in the model economy. Our basic bench-test employs inputs to the formula directly measured within the model but assumes no prior knowledge of the location of the top of the Laffer curve.⁷

Notes and References

¹ Thanks to Bill Dupor for formulating the question in the title.

² Ljungqvist, Lars. “Lucas Critique.” The New Palgrave Dictionary of Economics. Palgrave Macmillan, 2008.

³ For example, see Badel, Alejandro. “Higher Taxes for Top Earners: Can They Really Increase Revenue?” The Regional Economist, October 2013.

⁴ See Badel, Alejandro; and Huggett, Mark. “The Sufficient Statistic Approach: Predicting the Top of the Laffer Curve,” Working Paper 2015-038A, Federal Reserve Bank of St. Louis, November 2015.

⁵ By quantitative macroeconomics, we refer to a literature that studies realistically calibrated general equilibrium dynamic economies with heterogeneous agents.

⁶ For example, see Diamond, Peter A.; and Saez, Emmanuel. “The Case for a Progressive Tax: From Basic Research to Policy Recommendations.” Journal of Economic Perspectives, 2011, Vol. 25, Issue 4, pp. 165-90.; Piketty, Thomas; and Saez, Emmanuel. “Optimal Labor Income Taxation,” a chapter in the Elsevier book Handbook of Public Economics, 2013, Vol. 5, pp. 391-474.; and Mirrlees, James et al. Dimensions of Tax Design: The Mirrlees Review, Oxford, U.K.: Oxford University Press, 2010.

⁷ On the downside, we found that commonly used econometric methods underestimate a key behavioral response when applied within the model. The key shortcoming of these methods is that they focus on short-run variation in incomes in response to a change in the top marginal rate. In human capital economies, the full response of incomes to a policy change may take several decades. Plugging an underestimated response in the formula leads to predicting a top marginal rate that is above the true top of the Laffer curve.

Additional Resources

• On the Economy: Has the Phillips Curve Relationship Broken Down?
• On the Economy: A Follow-Up to Oil Prices and Inflation Expectations
• On the Economy: Looking beyond the Unemployment Rate