Transcript Totals, Averages, and Marginals Part 1

Totals,
Averages, &
Marginals
Part 1:
Grades
& Utility
Example Set 1:
Grades
A
Suppose you take five courses this semester
(all worth the same number of credits).
Let
A=4
B=3
C=2
D=1
F = 0.
Your grades: A B B C C, or
4 3 3 2 2.
TG = Total Grade = Grade1 + Grade2
+ Grade 3 + Grade 4 + Grade 5
= 4 + 3 + 3 + 2 + 2
= 14.
5 courses: TG = 14
Average Grade (AG)
AG = TG / N
= 14 / 5
= 2.80
Marginal Grade (MG)
=
=
change in Total Grade
change in # of courses
DTG/DN
tells how much your new grade, or set of
grades, is contributing per course.
TG was 14 from 5 courses. Suppose you take
a summer course and you get a B (3).
What is the new Total Grade?
TG = sum of grades = 14 + 3 = 17.
What is your Marginal Grade?
MG = DTG/DN
= (17-14)/(6-5)
= 3/1 = 3
Suppose you took no summer courses.
1 semester: 5 courses, TG = 14, AG = 2.80
You take 5 more courses this semester.
New grades: B C C C D (3 2 2 2 1)
What is your Total Grade for the two semesters?
TG = 14 + (3 + 2 + 2 + 2 + 1)
= 14 + 10 = 24
1 semester:
5 courses
2 semesters: 10 courses
TG = 14
TG = 24
What is MG for the additional semester?
MG = DTG/DN
= (24 - 14) / (10 - 5)
= 10 / 5 = 2.00
2 semesters: 10 courses
TG = 24
What is your new Average Grade?
AG = TG / N
= 24 / 10
= 2.4
1 semester: TG = 14 AG = 2.8
2 semesters: TG = 24 AG = 2.4 MG = 2.00
Was your Marginal Grade above, below, or
equal to your old Average Grade?
below (2.00 < 2.80)
Did your Average Grade go up, down, or stay
the same?
It went down (from 2.80 to 2.40), because
the marginal pulled it down.
2 semesters: TG = 24
3rd semester grades for 5 new courses:
B B C C C or 3 3 2 2 2.
What is your Total Grade for the 3 semesters?
TG = 24 + (3 + 3 + 2 + 2 + 2)
= 24 + 12
= 36
2 semesters: 10 courses, TG = 24
3 semesters: 15 courses, TG = 36
What is your MG for the 3rd semester?
MG = DTG / DN
= (36 - 24) / (15 - 10)
= 12 / 5
= 2.40
2 semesters: 10 courses, TG = 24, AG=2.40
3 semesters: 15 courses, TG = 36, MG=2.40
What is your new Average Grade?
AG = TG / N
= 36 / 15 = 2.40
Was your Marginal Grade above, below, or
equal to your old Average Grade?
equal (2.40 = 2.40)
So the AG stayed the same.
An Illustration of the
Relationship between the
Marginal and the Average
marginal > average:
average
marginal
average
marginal < average:
average
average
marginal
marginal = avg: avg stays same
average
marginal
Relation between
Marginal and Average
Marginal > Average: Average
Marginal < Average: Average
Marginal = Average: Average stays
the same
Example Set 2:
Utility or Satisfaction
from eating pizza
What is utility?
satisfaction, happiness, well-being, or
benefit
The amount of utility that you get from
having a piece of pizza is the amount
satisfaction or happiness that you get from
that pizza.
What is a util?
Sometimes the term util is used to measure
the quantity of satisfaction that you get from
something.
A slice of pizza might be worth 100 utils of
happiness or satisfaction to you.
You are hungry and you are about to eat a slice
of pizza. This slice gives you 100 utils of
satisfaction.
What is the marginal utility of this slice of pizza?
MU = Marginal Utility
= change in Total Utility = DTU
change in # of slices
DN
= (100 - 0) / (1 - 0)
= 100 / 1 = 100
1 slice: TU = 100
What is the average utility from pizza?
AU = Average Utility
= TU / N
= 100 / 1 = 100
Suppose you ate a second slice of pizza today.
That was good too, but not as good as the first
slice. Suppose the second slice brought your
total utility from pizza from 100 utils to 190
utils.
What is the marginal utility of the second slice?
MU = Marginal Utility = DTU / DN
= (190 - 100)/(2 - 1)
= 90/1 = 90
2 slices:
TU = 190
What is the average utility of the two slices?
AU = Average Utility = TU / N
= 190 / 2
= 95
If a third slice of pizza brings your total utility
from pizza to 270 utils from 190, what is the
marginal utility of the third slice?
MU = Marginal Utility
= DTU / DN
= (270 - 190) / (3 - 2)
= 80 / 1 = 80
Diminishing Marginal Utility:
Marginal utility is falling
(even though Total Utility may still be rising).
You are getting less and less satisfaction
from each additional piece of pizza. Your
total satisfaction is still increasing, however.
3 slices:
TU = 270
What is the average utility from three slices of
pizza?
AU = Average Utility = TU / N
= 270 / 3
= 90
You have a big appetite. You've eaten 10
slices of pizza. Your total utility is 550 utils.
You're at the point where you would feel no
better or worse off if you had another slice.
Your total utility from eating 11 slices is 550
utils also. What is the marginal utility of the
11th slice?
MU = DTU/DN
= (550 - 550) / (11 -10)
=0/1=0
11 slices: TU = 550
What is the average utility of the 11 slices of
pizza?
AU = Average Utility
= TU / N
= 550 / 11
= 50
You're considering eating a 12th slice of
pizza. You're feeling like someone would
have to pay you to make you eat another slice.
Suppose your total utility from pizza would
drop from 550 utils to 540 utils.
What is the marginal utility of the 12th slice
of pizza?
MU = DTU/DN
= (540 - 550) / (12 - 11)
= - 10 / 1 = - 10
Your marginal utility is negative because you
are getting dissatisfaction from the additional
slice. That 12th slice made you worse off.
Your total utility is positive but falling.
Note
When marginal utility is negative, total utility
is getting smaller.
You’re adding negative numbers to a positive
total.
When marginal utility is positive, total utility
is getting larger.
You’re adding positive numbers to a positive
total.
slice Marginal
number Utility
0
1
2
3
4
5
6
7
8
9
10
11
12
--100
90
80
70
60
50
40
30
20
10
0
-10
Total
Utility
0
100
190
270
340
400
450
490
520
540
550
550
540
slice Marginal
number Utility
0
1
2
3
4
5
6
7
8
9
10
11
12
--100
90
80
70
60
50
40
30
20
10
0
-10
Total
Utility
0
100
190
270
340
400
450
490
520
540
550
550
540
MU is falling but
still positive until
slice 11.
TU is rising (at a
slower & slower
rate) until slice 11.
At slice 12,
MU becomes
negative & TU
starts to decline.
Marginal Utility: Pizza Example
utils
100
0 1
MU
11
slice
Total Utility: Pizza Example
utils
550
TU
0
11
slice