Grade Inflation and Stunted Effort in a Curved Economics Course

To protect his teaching evaluations, an economics professor uses the following exam curve: if the class average falls below a known target, $m$, then all students will receive an equal number of free points so as to bring the mean up to $m$. If the average is above $m$ then there is no curve; curved grades above $100\%$ will never be truncated to $100\%$ in the gradebook. The $n$ students in the course all have Cobb-Douglas preferences over the grade-leisure plane; effort corresponds exactly to earned (uncurved) grades in a $1:1$ fashion. The elasticity of each student's utility with respect to his grade is his ability parameter, or relative preference for a high score. I find, classify, and give complete formulas for all the pure Nash equilibria of my own game, which my students have been playing for some eight semesters. The game is supermodular, featuring strategic complementarities, negative spillovers, and nonsmooth payoffs that generate non-convexities in the reaction correspondence. The $n+2$ types of equilibria are totally ordered with respect to effort and Pareto preference, and the lowest $n+1$ of these types are totally ordered in grade-leisure space. In addition to the no-curve ("try-hard") and curved interior equilibria, we have the "$k$-don't care" equilibria, whereby the $k$ lowest-ability students are no-shows. As the class size becomes infinite in the curved interior equilibrium, all students increase their leisure time by a fixed percentage, i.e., $14\%$, in response to the disincentive, which amplifies any pre-existing ability differences. All students' grades inflate by this same (endogenous) factor, say, $1.14$ times what they would have been under the correct standard.