We thank our discussant, Bill Easterly, as well as Michele Boldrin, Tom Holmes, Ed Prescott, Ken Rogoff, and John Tilton for their comments. We thank Sanghoon Lee, Casey Otsu, and Mauro Rodrigues for their able research assistance. Finally, we especially thank Alan Stockman, who has long served as a mentor, guide, and inspiration to many of us. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research.

This publication primarily presents economic research aimed at improving policymaking by the Federal Reserve System and other governmental authorities.

Multiple-choice exams are frequently used as an efficient and objective method to assess learning, but they are more vulnerable to answer copying than tests based on open questions. Several statistical tests (known as indices in the literature) have been proposed to detect cheating; however, to the best of our knowledge, they all lack mathematical support that guarantees optimality in any sense. We partially fill this void by deriving the uniformly most powerful (UMP) test under the assumption that the response distribution is known. In practice, however, we must estimate a behavioral model that yields a response distribution for each question. As an application, we calculate the empirical type I and type II error rates for several indices that assume different behavioral models using simulations based on real data from 12 nationwide multiplechoice exams taken by fifth and ninth graders in Colombia. We find that the most powerful index among those studied, subject to the restriction of preserving the type I error, is one based on the work of Wollack and is superior to the index developed by Wesolowsky.

Risk adjustment is vital in health policy design. Risk adjustment defines the annual capitation payments to health insurers and is a key determinant of insolvency risk for health insurers. In this study we compare the current risk adjustment formula used by Colombia's Ministry of Health and Social Protection against alternative specifications that adjust for additional factors. We show that the current risk adjustment formula, which conditions on demographic factors and their interactions, can only predict 30% of total health expenditures in the upper quintile of the expenditure distribution. We also show the government's formula can improve significantly by conditioning ex ante on measures indicators of 29 long-term diseases. We contribute to the risk adjustment literature by estimating machine learning based models and showing non-parametric methodologies (e.g., boosted trees models) outperform linear regressions even when fitted in a smaller set of regressors.

We offer a new proof that the equilibrium manifold (under complete markets) identifies individual demands globally. Moreover, under observation of only a subset of the equilibrium manifold, we find domains on which aggregate and individual demands are identifiable. Our argument avoids the assumption of Balasko (2004) requiring the observation of the complete manifold.