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How robust is the rationality assumption in economics? A statistical test based on student grade distributions
How Robust Is the Rationality Assumption in Economics? A Statistical Test Based on Student Grade Distributions STEVEN B. CAUDILL Auburn University FRANKLIN G. MIXON, JR.* The University of Southern Mississippi
This note re-addresses the question of the rationality of economic actors in a new setting. We examine the distribution of final course grades for a “Principles of Economics Class” in an effort to draw conclusions about rational student work effort. Based on the concepts of the opportunity cost of time, the relative reward for earning various grades (based on university codes for determining grade points) and the concept of rationality, we expect to find a greater percentage of grades at the lower end of the distribution for each grade level. We find statistical evidence that there are more students at the lower end of each grade interval than at the upper end. This result is consistent with what economists have long conjectured: students are rational in the allocation of study time.
The microeconomic assumption that individual actors are rational (or behave rationally) is often the subject of scrutiny by scientists and scholars from disciplines other than economics, and is also perennially subjected to empirical testing by economists and statisticians. Recently, Carter and Irons (1991) attempted to shed light on the subject by conducting an experiment with students in economics classes. They recruited 92 subjects from four populations (freshman noneconomics majors, freshman economics majors, senior noneconomics majors, and senior economics majors) for an ultimatum– bargaining *Direct all correspondence to: Franklin G. Mixon, Jr., Department of Economics and International Business, Box 5072, The University of Southern Mississippi, Hattiesburg, Mississippi 39406-5072, Telephone: (601) 266-5083. E-mail: [email protected]. The Social Science Journal, Volume 36, Number 4, pages 665– 673. Copyright © 1999 by Elsevier Science Inc. All rights of reproduction in any form reserved. ISSN: 0362-3319.
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game (for reviews, see Kahneman, Knetsch, and Thaler, 1986; Thaler, 1988; Ochs and Roth, 1989). Students were then paired, and one in the pair asked to propose a division of $10 (with both amounts being greater than $0, and in $0.50 increments) between the two students in that pair. The second student of the pair was asked to reject the proposal, in which case neither student was monetarily rewarded, or accept the proposal, which then meant that each student in the pair received the amount specified in the proposal. As they point out, if both players are rational and self-interested, the proposer should maximize his or her own earnings in the belief that the responder will accept any proposal that generates nonzero wealth for himself or herself. Their results pointed out that “economists accept less and keep more” when participating in such an experiment (Carter and Irons, 1991, p. 173). As one might anticipate, the Carter and Irons study prompted many replies (see Gemmell, 1992; Lattimore, 1992; Rosenbluth, 1992; Kroncke and Mixon, 1993), some of which confirmed the C-I results and some of which contradicted the C-I results. In similar experimental research, (with monetary payoffs) Beil and Beard (1994) suggest that students rationally engage in criminal activity (as a rational act) when legal markets are not as remunerative and when certain activities are sequentially made illegal. Beil and Beard also provide evidence that in the vast majority of instances when student participants are segmented into groups where potential payoffs to each group are known, and where one group has an expected payoff distribution significantly greater than the other, students still behave rationally and attempt to maximize earnings (as opposed to attempting to artificially alter the experimental results at the expense of remuneration). Recently, Romer (1993) pointed out the value of class attendance in the higher educational experience in light of growing trends of absence at such institutions in the United States. He provides a plan for increasing attendance based on penalties that deduct points from students’ averages for each class lecture missed. Among a number of replies, Powell and Shughart (1994) provide econometric evidence (with class averages presented as a function of attendance and other factors) that the penalties suggested by Romer overstate the true opportunity costs of class attendance. It is believed that under such a proposal, rational students will choose to substitute along some other margin (such as reading the textbook or studying at home) if forced to attend a greater percentage of classroom lectures. The purpose of the present paper is to re-address the question of the rationality of economic actors in a new setting. We examine the distribution of final course grades (for a principles of economics course) in an effort to draw conclusions about rational student work effort. In contrast to the results of an earlier study by Gleason and Walstead (1988), we find that students are indeed rational in their allocation of study time.
A MODEL OF STUDENT STUDY BEHAVIOR The model we develop is based on the Beckerian (see Becker, 1965, 1976; Becker and Michael, 1973) notion of human behavior which points out that individuals and households combine time and market goods to produce more “ultimate” objects of utilitymaximization. In our model, students face time constraints and must allocate time among studying and other activities so as to maximize utility. In a Beckerian framework, students
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Figure 1. Transformation curve. combine market goods (pens, paper, computers, etc.) with study time to produce “satisfaction” from a course grade in economics (and other classes as well). The production of a grade in economics requires time spent studying. Thus, each student must allocate time between studying and all other activities. To develop a model of student study behavior, we assume that each student is maximizing utility associated with his or her grade in economics and all other goods. We assume that the student derives utility from the letter grade only, and not the score. An ‘A’ provides more utility than a ‘B’, but an 80 provides the same utility as an 89 because both are ‘B’s. The production of a grade in economics requires time spent studying. Thus, each student must allocate time between studying and all other activities. Formally, each student maximizes a utility function, U, which is given by: U ⫽ U共T o,g共T s兲兲, with ⭸U ⭸ 2U ⬎ 0, ⬍ 0. ⭸T 0 ⭸T 02
(1)
where To is time spent in other activities, Ts is time spent studying, and g represents the letter grade received in economics. We further assume that g is a monotonic nondecreasing function of study time, Ts. Also, g is not a continuous function of study time but is characterized by jump discontinuities at the letter grades, and there are diminishing returns to additional study time. The implications of these additional assumptions are clear. Students receive utility from their letter grades only, and not their numerical scores. A student can study no time and receive a grade of ‘F’. A student wishing a ‘D’ must allocate some time to study, but to receive a ‘C’, even more time must be allocated, and so on. A picture of a typical transformation curve is given in Figure 1, with “time allocated for
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Figure 2. Indifference curve. other activities,” To, on the vertical axis and course grade in economics on the horizontal axis. As the figure indicates, students experience diminishing returns to additional study time. Thus, more and more time spent in other activities must be given up to obtain higher and higher course grades. Students wish to obtain any particular grade with the least possible study time. The consequence is that the relevant part of the transformation curve is actually a set of points not coincidentally labeled D, C, B, and A. A typical “indifference curve” is presented in Figure 2. A student is better off getting an 80 than an 89 because, presumably, both are Bs but the 80 requires less time taken away from other activities. The consequence is that the relevant part of the indifference curve is actually a set of points. Those are the points not coincidentally labeled D, C, B, and A in Figure 2. In this framework, with no uncertainty, each student would maximize utility by allocating study time so as to obtain the lowest D, C, B, or A, possible. The student would study just enough to obtain a 60, a 70, an 80, or a 90. Utility maximization must lead a student to one of the points labeled D, C, B, or A in the Figures. No student would study any extra time. However, students studying to achieve a particular grade in a course face many kinds of uncertainty. One uncertainty is not knowing exactly how much time to study to achieve a desired grade. This uncertainty will cause even risk neutral students to study, on average, more than the certain case. To see why this must occur, consider the case of a student whose goal is a B in the course. With no uncertainty, this student would allocate exactly enough time to studying to obtain an 80. Now suppose the student is uncertain about exactly how much time to study to obtain a B. If the student studies, on average, the same amount of time as in the certain case, on some occasions the students will end up with an 83 and a B in the course, but other times the student will end up with a 77 and a C in the course. If the student, on average, wants to obtain a B (an 80), more time must be allocated to study than in the certain case because studying even a small amount less than necessary
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leads to a lower grade. Students will study a greater amount in the uncertain case to insure against this possibility by studying enough to obtain the desired grade on average. This problem is exacerbated by the diminishing returns to study time assumption. Studying more will, on average, have less effect than studying a similar amount less. Also note that this result holds even if the students are assumed to be risk neutral because the extra studying is motivated by concern for the average grade obtained, not the variance. The implication of this uncertainty is that we should expect to find students more heavily concentrated in the lower portion of the deciles, the low 60s (the low 70s, the low 80s, and the low 90s). In the certain case all scores would be stacked on 60, 70, 80, and 90.
HYPOTHESIS AND STATISTICAL RESULTS The statistical test employed here is based on the shape of the distribution function of final averages in a college economics course. The data come from an introductory microeconomics class at a large, southern university. The class grades were based on the usual 90-80-70-60 grade allocation, and the raw data consisted of final averages of students in the course. Only those scores above 60 were retained for this estimation. Scores below 60 were omitted because they could have been obtained due to absence and some students might have known early in the quarter that they were going to fail and rationally chose to do little work thereafter. Also, the exam scores were curved according to the following scheme. If the raw average on an exam was less than 72, points were added to every student’s score so that the adjusted mean would be 72. If the raw average exceeded 72, no adjustment was made. For example, on an exam in which the raw score was 65, seven points would be added to each student’s grade. If the raw average had been 75, no points would be given. This adjustment scheme resulted in two students completing the course with averages in excess of 100. To be consistent with the theory, these students should have had scores in the low 90s, but that might have been difficult for them to achieve because the course might have been so easy for them that they were studying very little (we recognize that negative study time is not possible). These students should not be considered irrational by our narrow definition because they studied too much. In the empirical work their scores were counted as 99s. With these adjustments, we based our analysis on 104 final grade averages. What we examine is the distribution of grades within each grading interval. For this reason, a grade of 61 is no different from a 71 or an 81 or a 91. If we denote the final grade average by G, what we wish to examine is the remainder when the final grade average is divided by 10. If we denote this remainder by X, in mathematical terms: X ⫽ G共mod10兲.
(2)
For example, the grades 61, 71, 81, and 91, X will have the value of one. Sixty-two, 72, 82, and 92 have the value of two, and so on. To simplify the analysis we change the scale of these measures by dividing by 10 so that the raw data fall between zero and one; that is, we define x ⫽ X/10. Our analysis is based on the distribution of this transformed variable, x, which must lie in the unit interval.
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At first glance, the distribution of x seems to be consistent with our conjecture. More scores fall in the lower third of the interval than in the upper third. Specifically, 39.2% of the scores fall in the lower third of the interval (0 – 0.333), 32.4% fall in the middle third of the interval (0.333– 0.667), and 28.4% fall in the upper third of the interval (0.667–1). These data suggest that disproportionately more students are studying to obtain low As, Bs, Cs, and Ds. Although this evidence is consistent with our hypothesis, stronger evidence can be obtained by using a statistical test. To test the hypothesis that disproportionately more students receive low As, Bs, Cs, and Ds, we examine the empirical distribution function of the variable, x. The null hypothesis in this test is that the distribution of grades is uniform. If the grade distribution is uniform, the variable, x, will have a density function given by: f共x兲 ⫽ 1
0ⱕxⱕ1
⫽0
elsewhere.
(3)
The associated distribution function, F, is a line through the origin with a slope equal one or the unit interval. That is: F共x兲 ⫽ 1
0ⱕxⱕ1
0
x⬍0
1
x ⬎ 1.
(4)
A test of the grade distribution can be obtained if one recalls that the uniform distribution is a special case of a power distribution. The density function of the power distribution we wish to examine is given by: f共x兲 ⫽ x ⫺1 0
0ⱕxⱕ1
elsewhere.
(5)
Notice that if ⫽ 1, the density in (4) is that of the uniform. The distribution function of this density, upon which our statistical test is based, is given by: F共x兲 ⫽ x
0ⱕxⱕ1
⫽0
x⬍0
⫽1
x ⬎ 1.
(6)
Again, if ⫽ 1, F(x) ⫽ x and the linear distribution function of the uniform is recognized. If ⬎ 1, the distribution function is concave upward, indicating that a disproportionate number of the observations fall in the upper third of the interval. On the other hand, if ⬍ 1, the distribution function is convex indicating that a disproportionate number of the observations fall in the lower third of the interval (which is our alternative hypothesis).
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Table 1. Estimation Results Model 1 Model 2 Model 3 Model 4
Intercept 0.003 (0.12)* — ⫺0.022 (0.68) —
Theta 0.948 (46.81) 0.947 (69.23) 0.917 (40.74) 0.928 (61.98)
R2 0.96
Test(H0: ⫽ 1) 2.55
Nobs 104
—
3.91
104
0.96
3.71
77
—
4.82
77
*Absolute values of t-ratios in parentheses.
For example, if ⫽ 2 the density function given in (5) becomes: f共x兲 ⫽ 2x 0
0ⱕxⱕ1
(7)
elsewhere.
This linear density function with a positive slope of two has only 1/9 of the observations in the lower third of the unit interval, but 5/9 in the upper third of the unit interval. A simple test of our conjecture about student study effort is a test of whether is less than 1. The null and alternative hypotheses are: H 0: ⫽ 1 H 1: ⬍ 1.
(8)
To test this hypothesis, the data on grades in the form of the variable, x, are used to construct the empirical distribution function. The empirical distribution function, F(x), is the fraction of observations below the value, x. With F as the dependent variable and x as the lone independent variable, the model we wish to estimate is given by (6) above: F共x兲 ⫽ x .
(9)
This model can be estimated by OLS if the log transformation is used. Then, the empirical model becomes: 1nF ⫽ lnx.
(10)
The results from estimating several versions of this simple model are given in Table 1. The first row in Table 1 contains the results of estimating this model with an intercept. The R2 of the model is a high 0.96. Although the estimated intercept is not significantly different from zero, the coefficient of is highly significant and estimated to be 0.948 that is less than one. The t-ratio for testing the null hypothesis that ⫽ 1 is rejected in favor of the alternative that ⬍ 1 at the usual levels of significance. The associated t-ratio is 2.55.
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The relationship in (10) suggests that the model should be estimated without an intercept. The results of estimating this model are given in row 2 of Table 1. The estimated coefficient of is again less than one and highly significant. The null hypothesis that ⫽ 1 can again be rejected in favor of the alternative (t-ratio of 3.91). Although these findings do support our claims of rational study behavior by students, the results may be weakened by our inclusion of students in the 60 to 70 interval in our sample. The reason is that students pay an enormous price for falling below the 60 cutoff, much more so than falling below any other cutoff. The difference between a 59 and a 61 is much greater than, for example, the difference between a 79 and an 81. Students may devote extra time to study to avoid failing the course with all the consequences. For this reason we re-estimated the model– omitting students with grades in the 60 to 70 interval. The results continue to support our hypothesis of rational student behavior and are even stronger than before. The null hypothesis that ⫽ 1 is rejected in favor of the alternative in both cases. In the model with an intercept, the t-ratio is 3.71 and in the model without an intercept the t-ratio is 4.82. These results can be found in rows 3 and 4 of Table 1.
CONCLUSIONS This study has examined the issue of whether or not agents are rational in a new setting. We statistically test course grade distributions in a microeconomics principles class to determine whether there are disproportionately more students at the lower end of a grade interval than at the upper end. Based on the concepts of the opportunity cost of time, the relative reward for earning various grades (based on university codes for determining grade points), and the concept of rationality, we expect to find a greater percentage of grades at the lower end of the distribution. We find statistical evidence that there are more students at the lower end of each grade interval than at the upper end. This result is consistent with what economists have long conjectured: students are rational in the allocation of study time—that is, they do not “over study” in their efforts to achieve a particular grade. Acknowledgment: The authors thank an anonymous referee for helpful comments.
REFERENCES Becker, G. S. (1965). A Theory of the Allocation of Time. Economic Journal, 75: 493–517. Becker, G. S. (1976). The Economic Approach to Human Behavior. Chicago: University of Chicago Press. Becker, G. S. and R. T. Michael. (1973). On the New Theory of Consumer Behavior. Swedish Journal of Economics, 75: 378 –395. Beil, R. O., Jr. and T. R. Beard. (1994). Do People Rely on the Self-Interested Maximization of Others? Management Science, 40: 250 –262. Carter, J. R. and M. D. Irons. (1991). Are Economists Different, and if so, Why? Journal of Economic Perspectives, 5: 171–177. Gemmell, N. (1992). Are Economists Different? Journal of Economic Perspectives, 6: 202–203. Gleason, J. P. and W. B. Walstead. (1988). An Empirical Test of Student Study Time. Journal of Economic Education, 19: 315–21.
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Kahneman, D., J. L. Knetsch, and R. H. Thaler. (1986). Fairness and the Assumptions of Economics. Journal of Business, 59: 285–300. Kroncke, C. O., Jr. and F. G. Mixon, Jr. (1993). Are Economists Different? An Empirical Note. The Social Science Journal, 30: 341–345. Lattimore, R. (1992). Are Economists Different? Journal of Economic Perspectives, 6: 199 – 201. Ochs, J. and A. E. Roth. (1989). An Experimental Study of Sequential Bargaining. American Economic Review, 79: 355–384. Powell, W. A. and W. F. Shughart, II. (1994). Should Class Attendance be Mandatory? Journal of Economic Perspectives, 8: 208 –210. Romer, D. (1993). Do Students go to Class? Should They? Journal of Economic Perspectives, 7: 167–174. Rosenbluth, G. (1992). Are Economists Different? Journal of Economic Perspectives, 6: 201–202. Thaler, R. H. (1988). The Ultimatum Game. Journal of Economic Perspectives, 2: 195–206.
This note re-addresses the question of the rationality of economic actors in a new setting. We examine the distribution of final course grades for a “Principles of Economics Class” in an effort to draw conclusions about rational student work effort. Based on the concepts of the opportunity cost of time, the relative reward for earning various grades (based on university codes for determining grade points) and the concept of rationality, we expect to find a greater percentage of grades at the lower end of the distribution for each grade level. We find statistical evidence that there are more students at the lower end of each grade interval than at the upper end. This result is consistent with what economists have long conjectured: students are rational in the allocation of study time.
The microeconomic assumption that individual actors are rational (or behave rationally) is often the subject of scrutiny by scientists and scholars from disciplines other than economics, and is also perennially subjected to empirical testing by economists and statisticians. Recently, Carter and Irons (1991) attempted to shed light on the subject by conducting an experiment with students in economics classes. They recruited 92 subjects from four populations (freshman noneconomics majors, freshman economics majors, senior noneconomics majors, and senior economics majors) for an ultimatum– bargaining *Direct all correspondence to: Franklin G. Mixon, Jr., Department of Economics and International Business, Box 5072, The University of Southern Mississippi, Hattiesburg, Mississippi 39406-5072, Telephone: (601) 266-5083. E-mail: [email protected]. The Social Science Journal, Volume 36, Number 4, pages 665– 673. Copyright © 1999 by Elsevier Science Inc. All rights of reproduction in any form reserved. ISSN: 0362-3319.
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game (for reviews, see Kahneman, Knetsch, and Thaler, 1986; Thaler, 1988; Ochs and Roth, 1989). Students were then paired, and one in the pair asked to propose a division of $10 (with both amounts being greater than $0, and in $0.50 increments) between the two students in that pair. The second student of the pair was asked to reject the proposal, in which case neither student was monetarily rewarded, or accept the proposal, which then meant that each student in the pair received the amount specified in the proposal. As they point out, if both players are rational and self-interested, the proposer should maximize his or her own earnings in the belief that the responder will accept any proposal that generates nonzero wealth for himself or herself. Their results pointed out that “economists accept less and keep more” when participating in such an experiment (Carter and Irons, 1991, p. 173). As one might anticipate, the Carter and Irons study prompted many replies (see Gemmell, 1992; Lattimore, 1992; Rosenbluth, 1992; Kroncke and Mixon, 1993), some of which confirmed the C-I results and some of which contradicted the C-I results. In similar experimental research, (with monetary payoffs) Beil and Beard (1994) suggest that students rationally engage in criminal activity (as a rational act) when legal markets are not as remunerative and when certain activities are sequentially made illegal. Beil and Beard also provide evidence that in the vast majority of instances when student participants are segmented into groups where potential payoffs to each group are known, and where one group has an expected payoff distribution significantly greater than the other, students still behave rationally and attempt to maximize earnings (as opposed to attempting to artificially alter the experimental results at the expense of remuneration). Recently, Romer (1993) pointed out the value of class attendance in the higher educational experience in light of growing trends of absence at such institutions in the United States. He provides a plan for increasing attendance based on penalties that deduct points from students’ averages for each class lecture missed. Among a number of replies, Powell and Shughart (1994) provide econometric evidence (with class averages presented as a function of attendance and other factors) that the penalties suggested by Romer overstate the true opportunity costs of class attendance. It is believed that under such a proposal, rational students will choose to substitute along some other margin (such as reading the textbook or studying at home) if forced to attend a greater percentage of classroom lectures. The purpose of the present paper is to re-address the question of the rationality of economic actors in a new setting. We examine the distribution of final course grades (for a principles of economics course) in an effort to draw conclusions about rational student work effort. In contrast to the results of an earlier study by Gleason and Walstead (1988), we find that students are indeed rational in their allocation of study time.
A MODEL OF STUDENT STUDY BEHAVIOR The model we develop is based on the Beckerian (see Becker, 1965, 1976; Becker and Michael, 1973) notion of human behavior which points out that individuals and households combine time and market goods to produce more “ultimate” objects of utilitymaximization. In our model, students face time constraints and must allocate time among studying and other activities so as to maximize utility. In a Beckerian framework, students
Rationality Assumption in Economics
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Figure 1. Transformation curve. combine market goods (pens, paper, computers, etc.) with study time to produce “satisfaction” from a course grade in economics (and other classes as well). The production of a grade in economics requires time spent studying. Thus, each student must allocate time between studying and all other activities. To develop a model of student study behavior, we assume that each student is maximizing utility associated with his or her grade in economics and all other goods. We assume that the student derives utility from the letter grade only, and not the score. An ‘A’ provides more utility than a ‘B’, but an 80 provides the same utility as an 89 because both are ‘B’s. The production of a grade in economics requires time spent studying. Thus, each student must allocate time between studying and all other activities. Formally, each student maximizes a utility function, U, which is given by: U ⫽ U共T o,g共T s兲兲, with ⭸U ⭸ 2U ⬎ 0, ⬍ 0. ⭸T 0 ⭸T 02
(1)
where To is time spent in other activities, Ts is time spent studying, and g represents the letter grade received in economics. We further assume that g is a monotonic nondecreasing function of study time, Ts. Also, g is not a continuous function of study time but is characterized by jump discontinuities at the letter grades, and there are diminishing returns to additional study time. The implications of these additional assumptions are clear. Students receive utility from their letter grades only, and not their numerical scores. A student can study no time and receive a grade of ‘F’. A student wishing a ‘D’ must allocate some time to study, but to receive a ‘C’, even more time must be allocated, and so on. A picture of a typical transformation curve is given in Figure 1, with “time allocated for
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Figure 2. Indifference curve. other activities,” To, on the vertical axis and course grade in economics on the horizontal axis. As the figure indicates, students experience diminishing returns to additional study time. Thus, more and more time spent in other activities must be given up to obtain higher and higher course grades. Students wish to obtain any particular grade with the least possible study time. The consequence is that the relevant part of the transformation curve is actually a set of points not coincidentally labeled D, C, B, and A. A typical “indifference curve” is presented in Figure 2. A student is better off getting an 80 than an 89 because, presumably, both are Bs but the 80 requires less time taken away from other activities. The consequence is that the relevant part of the indifference curve is actually a set of points. Those are the points not coincidentally labeled D, C, B, and A in Figure 2. In this framework, with no uncertainty, each student would maximize utility by allocating study time so as to obtain the lowest D, C, B, or A, possible. The student would study just enough to obtain a 60, a 70, an 80, or a 90. Utility maximization must lead a student to one of the points labeled D, C, B, or A in the Figures. No student would study any extra time. However, students studying to achieve a particular grade in a course face many kinds of uncertainty. One uncertainty is not knowing exactly how much time to study to achieve a desired grade. This uncertainty will cause even risk neutral students to study, on average, more than the certain case. To see why this must occur, consider the case of a student whose goal is a B in the course. With no uncertainty, this student would allocate exactly enough time to studying to obtain an 80. Now suppose the student is uncertain about exactly how much time to study to obtain a B. If the student studies, on average, the same amount of time as in the certain case, on some occasions the students will end up with an 83 and a B in the course, but other times the student will end up with a 77 and a C in the course. If the student, on average, wants to obtain a B (an 80), more time must be allocated to study than in the certain case because studying even a small amount less than necessary
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leads to a lower grade. Students will study a greater amount in the uncertain case to insure against this possibility by studying enough to obtain the desired grade on average. This problem is exacerbated by the diminishing returns to study time assumption. Studying more will, on average, have less effect than studying a similar amount less. Also note that this result holds even if the students are assumed to be risk neutral because the extra studying is motivated by concern for the average grade obtained, not the variance. The implication of this uncertainty is that we should expect to find students more heavily concentrated in the lower portion of the deciles, the low 60s (the low 70s, the low 80s, and the low 90s). In the certain case all scores would be stacked on 60, 70, 80, and 90.
HYPOTHESIS AND STATISTICAL RESULTS The statistical test employed here is based on the shape of the distribution function of final averages in a college economics course. The data come from an introductory microeconomics class at a large, southern university. The class grades were based on the usual 90-80-70-60 grade allocation, and the raw data consisted of final averages of students in the course. Only those scores above 60 were retained for this estimation. Scores below 60 were omitted because they could have been obtained due to absence and some students might have known early in the quarter that they were going to fail and rationally chose to do little work thereafter. Also, the exam scores were curved according to the following scheme. If the raw average on an exam was less than 72, points were added to every student’s score so that the adjusted mean would be 72. If the raw average exceeded 72, no adjustment was made. For example, on an exam in which the raw score was 65, seven points would be added to each student’s grade. If the raw average had been 75, no points would be given. This adjustment scheme resulted in two students completing the course with averages in excess of 100. To be consistent with the theory, these students should have had scores in the low 90s, but that might have been difficult for them to achieve because the course might have been so easy for them that they were studying very little (we recognize that negative study time is not possible). These students should not be considered irrational by our narrow definition because they studied too much. In the empirical work their scores were counted as 99s. With these adjustments, we based our analysis on 104 final grade averages. What we examine is the distribution of grades within each grading interval. For this reason, a grade of 61 is no different from a 71 or an 81 or a 91. If we denote the final grade average by G, what we wish to examine is the remainder when the final grade average is divided by 10. If we denote this remainder by X, in mathematical terms: X ⫽ G共mod10兲.
(2)
For example, the grades 61, 71, 81, and 91, X will have the value of one. Sixty-two, 72, 82, and 92 have the value of two, and so on. To simplify the analysis we change the scale of these measures by dividing by 10 so that the raw data fall between zero and one; that is, we define x ⫽ X/10. Our analysis is based on the distribution of this transformed variable, x, which must lie in the unit interval.
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At first glance, the distribution of x seems to be consistent with our conjecture. More scores fall in the lower third of the interval than in the upper third. Specifically, 39.2% of the scores fall in the lower third of the interval (0 – 0.333), 32.4% fall in the middle third of the interval (0.333– 0.667), and 28.4% fall in the upper third of the interval (0.667–1). These data suggest that disproportionately more students are studying to obtain low As, Bs, Cs, and Ds. Although this evidence is consistent with our hypothesis, stronger evidence can be obtained by using a statistical test. To test the hypothesis that disproportionately more students receive low As, Bs, Cs, and Ds, we examine the empirical distribution function of the variable, x. The null hypothesis in this test is that the distribution of grades is uniform. If the grade distribution is uniform, the variable, x, will have a density function given by: f共x兲 ⫽ 1
0ⱕxⱕ1
⫽0
elsewhere.
(3)
The associated distribution function, F, is a line through the origin with a slope equal one or the unit interval. That is: F共x兲 ⫽ 1
0ⱕxⱕ1
0
x⬍0
1
x ⬎ 1.
(4)
A test of the grade distribution can be obtained if one recalls that the uniform distribution is a special case of a power distribution. The density function of the power distribution we wish to examine is given by: f共x兲 ⫽ x ⫺1 0
0ⱕxⱕ1
elsewhere.
(5)
Notice that if ⫽ 1, the density in (4) is that of the uniform. The distribution function of this density, upon which our statistical test is based, is given by: F共x兲 ⫽ x
0ⱕxⱕ1
⫽0
x⬍0
⫽1
x ⬎ 1.
(6)
Again, if ⫽ 1, F(x) ⫽ x and the linear distribution function of the uniform is recognized. If ⬎ 1, the distribution function is concave upward, indicating that a disproportionate number of the observations fall in the upper third of the interval. On the other hand, if ⬍ 1, the distribution function is convex indicating that a disproportionate number of the observations fall in the lower third of the interval (which is our alternative hypothesis).
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Table 1. Estimation Results Model 1 Model 2 Model 3 Model 4
Intercept 0.003 (0.12)* — ⫺0.022 (0.68) —
Theta 0.948 (46.81) 0.947 (69.23) 0.917 (40.74) 0.928 (61.98)
R2 0.96
Test(H0: ⫽ 1) 2.55
Nobs 104
—
3.91
104
0.96
3.71
77
—
4.82
77
*Absolute values of t-ratios in parentheses.
For example, if ⫽ 2 the density function given in (5) becomes: f共x兲 ⫽ 2x 0
0ⱕxⱕ1
(7)
elsewhere.
This linear density function with a positive slope of two has only 1/9 of the observations in the lower third of the unit interval, but 5/9 in the upper third of the unit interval. A simple test of our conjecture about student study effort is a test of whether is less than 1. The null and alternative hypotheses are: H 0: ⫽ 1 H 1: ⬍ 1.
(8)
To test this hypothesis, the data on grades in the form of the variable, x, are used to construct the empirical distribution function. The empirical distribution function, F(x), is the fraction of observations below the value, x. With F as the dependent variable and x as the lone independent variable, the model we wish to estimate is given by (6) above: F共x兲 ⫽ x .
(9)
This model can be estimated by OLS if the log transformation is used. Then, the empirical model becomes: 1nF ⫽ lnx.
(10)
The results from estimating several versions of this simple model are given in Table 1. The first row in Table 1 contains the results of estimating this model with an intercept. The R2 of the model is a high 0.96. Although the estimated intercept is not significantly different from zero, the coefficient of is highly significant and estimated to be 0.948 that is less than one. The t-ratio for testing the null hypothesis that ⫽ 1 is rejected in favor of the alternative that ⬍ 1 at the usual levels of significance. The associated t-ratio is 2.55.
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The relationship in (10) suggests that the model should be estimated without an intercept. The results of estimating this model are given in row 2 of Table 1. The estimated coefficient of is again less than one and highly significant. The null hypothesis that ⫽ 1 can again be rejected in favor of the alternative (t-ratio of 3.91). Although these findings do support our claims of rational study behavior by students, the results may be weakened by our inclusion of students in the 60 to 70 interval in our sample. The reason is that students pay an enormous price for falling below the 60 cutoff, much more so than falling below any other cutoff. The difference between a 59 and a 61 is much greater than, for example, the difference between a 79 and an 81. Students may devote extra time to study to avoid failing the course with all the consequences. For this reason we re-estimated the model– omitting students with grades in the 60 to 70 interval. The results continue to support our hypothesis of rational student behavior and are even stronger than before. The null hypothesis that ⫽ 1 is rejected in favor of the alternative in both cases. In the model with an intercept, the t-ratio is 3.71 and in the model without an intercept the t-ratio is 4.82. These results can be found in rows 3 and 4 of Table 1.
CONCLUSIONS This study has examined the issue of whether or not agents are rational in a new setting. We statistically test course grade distributions in a microeconomics principles class to determine whether there are disproportionately more students at the lower end of a grade interval than at the upper end. Based on the concepts of the opportunity cost of time, the relative reward for earning various grades (based on university codes for determining grade points), and the concept of rationality, we expect to find a greater percentage of grades at the lower end of the distribution. We find statistical evidence that there are more students at the lower end of each grade interval than at the upper end. This result is consistent with what economists have long conjectured: students are rational in the allocation of study time—that is, they do not “over study” in their efforts to achieve a particular grade. Acknowledgment: The authors thank an anonymous referee for helpful comments.
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