TEST SCORE COMPENSATION, MORAL HAZARD, AND STUDENT REACTION TO HIGH-SCORER DISINCENTIVES IN ECONOMICS PRINCIPLES Johnnie B. Linn III Concord University

THE PROBLEM To what extent the “curve” used for exam scores generates disincentives among the better scorers, and to what extent these incentives are anticipated by the students in general.

HOW THE CURVE IS COMPUTED 1.If the mean score in terms of percentage of questions answered correctly is less than 77%, the percentage score of all test takers is increased by the difference of 77% and the mean score percentage. 2.If Step 1 results in a percentage value of the highest score greater than 100%, a linear transformation of the raw scores is made so that the mean is 77% and the maximum is 100%.

THE VARIABLES x untransformed score y transformed score k maximum allowed transformed score h highest untransformed score m linear coefficient of transformation b constant term of transformation

THE TRANSFORMATION

THE MODEL What curve will be “selected” by the students? In what ways will the students’ performance be affected by the curve that they expect to be given?

ASSUMPTIONS Test takers maximize utility functions whose arguments are test score and effort. Effort on the margin by any one student has negligible effect on the mean score. Effort on the margin by any one student has negligible impact on the expected score transformation (this assumption will be relaxed later)

WHAT ABOUT GUESSING? We assume for now that there is no guessing.

THE UTILITY FUNCTION AND PARAMETERS

MEANING OF THE PARAMETERS All students have a positive benefit, , from higher test scores. Students for whom leisure is a normal good have positive values of . Students who enjoy the challenge of taking test questions will have negative values of .

THE FIRST-ORDER CONDITION

SCORE DISTRIBUTION, SPECIAL SCENARIOS Same utility function, different abilities. Same abilities, different utility functions.

DIFFERENT ABILITIES

DIFFERENT UTILITY FUNCTIONS

“INCOME” AND SUBSTITUTION EFFECTS OF THE CURVE

MORAL HAZARD Without the curve, the student selects T1. With an “income-compensated” curve, the student would select T2. The mean-scoring student would expect an overcompensation of income--an increase in the constant term of the transformation from Ob2 to Ob3--to raise the selected score to 77 at T3.

MORAL HAZARD, CONTINUED For most students, for whom leisure is a normal good, the income and substitution effects offset each other. For students with negative , the income and substitution effects reinforce each other. If there is variance in , the curve enhances the effect of that variance on the raw scores.

WHAT ABOUT THE HIGHEST SCORER? for the highest-ranking student we must abandon the assumption made earlier that performance on the margin has a negligible effect on m and b. the value of y’ for the highest-ranking student is zero for values of x greater than the expected raw score of the second- highest ranking student.

INCENTIVES FOR THE HIGHEST SCORER

CLASS SIZE AND THE HIGHEST SCORER For small class size, likelihood that the high scorer has a negative  is less, hence, the possibility of high-scorer moral hazard is greater. For large class size, probability of negative  is greater and expected second-highest score is greater, hence, less moral hazard for highest scorer; in the limit, none.

HIGHEST SCORER’S IMPACT ON THE CURVE Effect on b is negligible, since h appears both in numerator and denominator Effect on m is not negligible, since h appears only in the denominator.

WHAT ABOUT GUESSING? likelihood c of a choice being correct. The expected opportunity cost of attempting to answer a question is c times the value of the question. the net marginal product of attempting a question is (1 - c ) times the value of the question.

UTILITY FUNCTION WITH GUESSING

FIRST-ORDER CONDITION WITH GUESSING

EFFECTS OF GUESSING ON INCENTIVES for a given number of questions attempted, an higher value of c will be associated with a higher value of m. no penalty for guessing will result in less effort.

EFFECT OF GUESSING ON INCENTIVES

DO STUDENTS ANTICIPATE THE HIGHEST SCORE? If students pay no attention to what h will be when they select m and b, the observed values of m and b will conform to what is expected when the envelope theorem is applied to a utility function whose arguments are only m and b.

THE ENVELOPE THEOREM

FOR THE AVERAGE SCORING STUDENT

THE NULL HYPOTHESIS a plot of the ratio of the logarithm arguments should have no slope.

WHICH SIGN FOR THE ONE- TAILED TEST? Ratio of the two logarithm arguments should have a positive slope. Students expect a negative correlation between h and m. Transformation as viewed by students is the equation on the right.

RESULT ON THE ENVELOPE THEOREM

APPROXIMATION

THE DATA (MICROECONOMICS)

THE DATA (MACROECONOMICS)

COLOR CODE FOR DATA Red represents the spring term of academic year (second edition of text, exams from database) White represents the academic year (second edition of text, exams by instructor) Green represents the fall term of the academic year (third edition of text, exams from database)

FIRST EXAM EXCLUDED, MICROECONOMICS

FIRST EXAM EXCLUDED, MACROECONOMICS

THE TEST RESULTS Microeconomics: Results are not significant. Macroeconomics: Results are significant.

TEST STATISTICS

TEST RESULTS, MICROECONOMICS

TEST RESULTS, MACROECONOMICS

WHY THE DIFFERENCE? High scorers in macroeconomics exhibit disincentives but high scorers in microeconomics do not do so. Or, the significant results for macroeconomics are due to some other unidentified factor, perhaps related to the difference in difficulty of the material.

MAJORS OF HIGHEST SCORERS

WHAT TO DO ABOUT THE CURVE? allow the highest score to curve to a value greater than 100%. High scorers can “bank” points. High scorer moral hazard will be reduced. If moral hazard is present for macroeconomics, effect of change will be greater than that for microeconomics.

PRELIMINARY RESULTS OF CHANGE, MICROECONOMICS

PRELIMINARY RESULTS OF CHANGE, MACROECONOMICS