test score compensation, moral hazard, and student reaction to high-scorer disincentives in...
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![Page 1: TEST SCORE COMPENSATION, MORAL HAZARD, AND STUDENT REACTION TO HIGH-SCORER DISINCENTIVES IN ECONOMICS PRINCIPLES Johnnie B. Linn III Concord University](https://reader036.vdocuments.us/reader036/viewer/2022062421/56649d2a5503460f949feaae/html5/thumbnails/1.jpg)
TEST SCORE COMPENSATION, MORAL HAZARD, AND STUDENT
REACTION TO HIGH-SCORER DISINCENTIVES IN
ECONOMICS PRINCIPLES
Johnnie B. Linn III
Concord University
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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.
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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%.
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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
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THE TRANSFORMATION
bmxy
.77
,77
xh
xkhb
xh
km
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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?
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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)
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WHAT ABOUT GUESSING?
• We assume for now that there is no guessing.
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THE UTILITY FUNCTION AND PARAMETERS
],)),(([ eexyUU
.
,
e
U
U
e
y
U
U
y
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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 .
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THE FIRST-ORDER CONDITION
,0
)(
e
U
e
Uexy
.)( exy
EFFORT
SCORE
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SCORE DISTRIBUTION, SPECIAL SCENARIOS
• Same utility function, different abilities.
• Same abilities, different utility functions.
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DIFFERENT ABILITIES
EFFORT
SCORE
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DIFFERENT UTILITY FUNCTIONS
EFFORT
SCORE
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“INCOME” AND SUBSTITUTION EFFECTS OF THE CURVE
EFFORT
SCORE
T1
b2
T2
O
77
U2 U1
U3
T3
b3
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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.
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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.
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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.
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INCENTIVES FOR THE HIGHEST SCORER
EFFORT
SCORE
k
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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.
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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.
.77
,77
xh
xkhb
xh
km
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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.
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UTILITY FUNCTION WITH GUESSING
nnNcnxyUU )),()((
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FIRST-ORDER CONDITION WITH GUESSING
))(( cnxy
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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.
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EFFECT OF GUESSING ON INCENTIVES
Effort
Score
T
A
D C
B
O N
U2
U1
1 E
F
U3
A’
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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.
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THE ENVELOPE THEOREM
0
U Udm db
m b
0
y ydm db
m b
dbx
dm
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FOR THE AVERAGE SCORING STUDENT
db dm
y b m
ln( ) ln( )y b m C
y bx
m
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THE NULL HYPOTHESIS
• a plot of the ratio of the logarithm arguments should have no slope.
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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.
(1 )y m ph x b
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RESULT ON THE ENVELOPE THEOREM
((1 ) ) 0xd ph m db mpxdh
((1 ) ) ((1 ) )
db dhx mpx
d ph m d ph m
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APPROXIMATION
dbx
dm
db dm
y b m
ln( ) ln( ) ( 1) ln( )y b m m C
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THE DATA (MICROECONOMICS)
14
141
4
1411
2 343 32
32
223
324
0
10
20
30
40
50
60
70
1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00
m
b
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THE DATA (MACROECONOMICS)
4
1
2
3
4
1
33
4
1
2
41
23
4
1
23
41
23
41
23
1
2
34
2
0
10
20
30
40
50
60
70
1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00
m
b
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COLOR CODE FOR DATA
• Red represents the spring term of academic year 2002-2003 (second edition of text, exams from database)
• White represents the academic year 2003-2004 (second edition of text, exams by instructor)
• Green represents the fall term of the academic year 2004-2005. (third edition of text, exams from database)
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FIRST EXAM EXCLUDED, MICROECONOMICS
423
322
2 3233 4 32
44
4
4
0
10
20
30
40
50
60
70
1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00
m
b
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FIRST EXAM EXCLUDED, MACROECONOMICS
2
4 3
2
32
4
324
324
3 2
4
24
33
4
3
2
4
0
10
20
30
40
50
60
70
1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00
m
b
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THE TEST RESULTS
• Microeconomics: Results are not significant.• Macroeconomics: Results are significant.
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TEST STATISTICS
Course Sample Size
Coefficient Constant t for coefficient
Microeconomics 16 -0.020 (0.205)
2.66 (.226)
-0.099
Macroeconomics 23 0.46 (0.106)
2.16 (0.113)
4.39
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TEST RESULTS, MICROECONOMICS
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
ln(m )
ln((y-b)/m)
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TEST RESULTS, MACROECONOMICS
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
ln(m )
ln((y-b)/m)
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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.
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MAJORS OF HIGHEST SCORERS
Micro Highest Scorers Business Majors Non-Business or Unknown Total Two of Three Exams 3 0 6 One of Three Exams 6 4 10
Macro Highest Scorers Business Majors Non-Business or Unknown Total Two of Three Exams 2 2 8 One of Three Exams 8 7 15
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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.
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PRELIMINARY RESULTS OF CHANGE, MICROECONOMICS
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6 0.8 1 1.2 1.4 1.6
ln(m )
ln((y-b)/m)
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PRELIMINARY RESULTS OF CHANGE, MACROECONOMICS
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6 0.8 1 1.2 1.4
ln(m )
ln((y-b)/m)