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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