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AU / Economics / ECON 2020 / How can a person maximize utility?

# How can a person maximize utility? Description

##### Description: all the notes, including graphs, { 2/24 - 3/21 } and a review for the third exam
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Reviews

Intro to econ (ch. 1-2)

## How can a person maximize utility? Definition Formula

Week 6

2/24/16

 consumer behavior

o why do we buy stuff?

 To maximize utility:

∙ The satisfaction a consumer obtains from the

consumption of a good or service

∙    Utility facts:

o Measured in utils

o We CAN compare the utils assigned to multiple

goods by one person

o We CANNOT compare the utils assigned to a

good by multiple people

 people have different preferences, the

assessments differ

 Total vs. Marginal utility

## When do people stop buying stuff? Don't forget about the age old question of What are magnetic fields and pulsars for?

∙ Total utility – the total satisfaction a person derives

form consuming some specific quantity

o TU increases as Qd increases (together)

o TU = Σ MU

∙ Marginal Utility – the additional utility a consumer

derives from an additional unit of a good

o Ex) date to all you can eat pizza buffet at Cici’s

Quantity

1

2

3

4

5

6

TU

15

28

39

48

55

60

MU

15

13

11

9

7

5

## What determines which stuff people buy first? o How many utils did each slice of pizza add to

my overall happiness? We also discuss several other topics like What makes brand attitude important?

 0 units of something, 0 utils (nothing

gives you nothing)

 MU = ΔTU

 Law of Diminishing MU – as Qd rises, MU

falls (opposite)

∙ As MU falls, willingness to pay falls

as well

∙ The second slice isn’t going to add

as much happiness as the first

o When do we stop buying stuff?

 When we run out of moneyyyyyyy

∙ Budget line/constraint – the line that shows the

different consumption bundles a consumer can

purchase with a specific money income

o Consumption bundle – The combination of

goods and services consumed by an individual

Intro to econ (ch. 1-2)

Definition Formula

o the price of a consumption bundle cannot

exceed the consumer’s total income

o Income = (Qx * Px) + (Qy * Py)

o Ex) assume a consumer with \$24 to spend is

choosing between \$4 hot dogs and \$2 chicken

fingers

 The consumer can purchase all bundles

on or to the left of the budget line If you want to learn more check out Where can lipases be found?  If the consumer is at a bundle on the

budget line, the only way to consumer

more of one good is to give up some of

the other

 OC of x = max Qy / max Qx = Px / Py

OC of HD = 12/6 = 2C

OC of C = 6/12 = ½ HD

∙ Quantity: y / x

o Giving up the other good

∙ Price: x / y

o Giving up the \$ you would’ve

used to buy the other good

o Factors that change the budget line

 Income – a change in income causes a

shift of the budget line

 Price of y – a change in price causes a

rotation of the budget line

Week 7

2/29/16

 Price of x – a change in price causes a  We also discuss several other topics like When does us conduct federal census?

rotation of the budget line

Intro to econ (ch. 1-2)

Definition Formula o What do we buy first? We also discuss several other topics like How can i compute using euler's method?

 The good that gives us the most:

∙ Marginal utility per dollar = MU / P

∙ Vacation vs. Cheeseburger

o MU of vacation = 50,000 utils

o MU of cheeseburger = 20 utils

o P of vacation = \$5,000

o P of cheeseburger = \$1

o MUv / Pv = 50,000 / 5,000 = 10

o MUc / Pc = 20 / 1 = 20

∙ Utility-Maximization Rule (aka Consumer Equilibrium)

o For utility maximization, the consumer must

get the same amount of utility from the last

dollar spent on each good

o Basically, TU is maximized when:

 All income is spent

 MUx / Px = MUy / Py

∙ Utility-Maximizing (Optimal) Consumption

o Optimal Consumption Bundle – the bundle that  If you want to learn more check out Which cultures support independent sleeping for babies?

maximizes total utility

o Ex) Find the optimal consumption bundle using:

 Consumer income = \$24

 Px = \$3

Intro to econ (ch. 1-2)

Definition Formula

 Py = \$2

Qx

TUx

MUx

MUx / Px

Qy

TUy

MUy

MUy / Py

1

39

39

13 = 39 / 3

1

30

30

15 = 30 / 2

2

75

36 = 75­39

12 = 36 / 3

2

58

28 = 58 ­ 30

14 = 28 / 2

3

108

33 = 108­75

11 = 33 / 3

3

84

26

13

4

135

27

9

4

108

24

12

5

159

24

8

5

128

20

10

6

180

21

7

6

146

18

9

7

198

18

6

7

162

16

8

13, 12, 9, 8 satisfies all conditions of optimal bundle (use for condition # 1,  not in formula)

BUT have to make sure you’re spending \$24 to say it’s the optimal bundle Unit of x gives 13 units per util, y gives 15 units per util

o Possible Optimal Bundles

 Use the income formula on possible

optimal bundles to determine if all \$ was

spent

(Income = (Qx * Px) + (Qy * Py)

 1x and 3y = (1 * 3) + (3 * 2) = \$9

∙ 1 unit of x for \$3, 3 units of y for \$2

= \$9

 2x and 4y = (2 * 3) + (4 * 2) = \$14

 4x and 6y = (4 * 3) + (6 * 2) = \$24

 Optimal = 4 units of x and 6 units of y

 TU of bundle = 135 + 146 = 281 utils

o Graphing Optimal Consumption

 What happens if prices change?

∙ Rewrite the entire graph OR

∙ Use a graph

 Indifference curve – a line that shows the

consumption bundles that yield the same

amount of total utility Intro to econ (ch. 1-2) Definition Formula

∙ Properties of (most) IC’s:

o downward sloping

o farther from the origin

represents a greater level of

TU

o never cross

o bowed inward

3/2/16

∙ Calculating the slope of IC’s

o Slope = ΔQy / ΔQx

o Along an IC: ΔTux + ΔTUy = 0

Can also be written as

 Mux * Δ Qx + MUy *

ΔQy = 0 OR

 Mux * ΔQx = -MUy *

ΔQy

o Dividing both sides by ΔQx

and by

-MUy: ΔQy / ΔQx = - Mux /

MUy

∙ Marginal Rate of Substitution

o Ratio of the marginal utility of one good to the

marginal utility of another

o MRS = Mux / MUy Intro to econ (ch. 1-2) Definition Formula

o Big MUx / tiny MUy = big MRS

o Tiny MUx / big MUy = tiny MRS

∙ Principle of Diminishing MRS

o The more of a good X a person consumes in

proportion to good Y, the less Y the consumer

is willing to substitute for X; MRS decreases

as Qx increases

o The MRS changes along an IC because of

diminishing marginal utility

o At 1, the consumer would be willing to give up

lots of Y to get another X

o At 2, y is scarce, so the consumer will only give

up very little to get another X Graphing optimal consumption

(graph)

add the budget line to the IC graph

which is the optimal bundle?

Intro to econ (ch. 1-2)

Definition Formula

At bundle 2, slope of the IC = slope of the BL

∙ Relative Price

o The ratio of the price of one good to the price

of the other

o RP = Px / Py

o Slope (of the budget line) = -Px / Py

o So, we can find the optimal budget by setting

MRS = RP (MUx / MUy = Px / Py)

o Rules:

 At the optimal consumption bundle, MRS

= RP

Bundle

Qx

MUx

Qy

MUy

MRS

A

2

2500

60

100

25

B

3

2000

40

200

10

C

5

1000

24

500

2

D

6

750

20

750

1

 If Px = \$50 and Py = \$5, what is the

optimal bundle?

∙ Solution: MRS = RP

MRS = MUx / MUy

RP = Px / Py = 50 / 5 = 10

Optimal bundle = B

3/4/16

∙ Special Indifference Curves

o Perfect substitutes – goods for which the marginal rate f  substitution is constant, no matter how much of each is

consumed

 Ex) a consumer has \$55 to spend on bundles containing  Exxon and Chevron gasoline

 Px = \$2.75

 Pc = \$2.50

Intro to econ (ch. 1-2)

Definition Formula  Optimal bundle = Income / low P

 0 units of X and 22 units of C

 The point that has the highest indifference curve is at 22 C  and 0 X

 You don’t have to buy equal amount of each good,

consumer always buys the cheaper of the 2 goods

o Perfect complements – goods that a consumer will consume in  the same ratio regardless of their relative price

 ex) a consumer has \$24 to spend on bundles containing  ham and bread

 Ph = \$6

 Pb = \$2  Optimal bundle = Income / (Px + Py)

 3 units of H and 3 units of B

ch. 9

∙ Costs and Profits

Intro to econ (ch. 1-2)

Definition Formula

o Profit (Π)

 Π = Total Revenue – Total Cost

(TR = P * Q)

 TC can be defined in multiple ways:

∙ Accounting Π = TR – Explicit Cost

∙ Economic Π = TR – Economic Cost

 Economic cost includes implicit cost, which for a producer  is the forgone income from employing resources in their

next-best use

 Ex) Find Accounting and Economic Profits

∙ TR = \$15,000

∙ Explicit cost = \$11,000

∙ Implicit cost = \$4,000

Solution:

∙ Accounting Π = 15000 – 11000 = \$4,000

∙ Economic Π = 15000 – (11000 + 4000) = \$0

 Normal profit

∙ The first is doing just as well as it could in another

industry

∙ Accounting profit = Implicit Cost

∙ Economic profit = \$0

Week 8

3/7/16

 Time in production

o Short run – an amount of time insufficient to allow plant capacity  to vary

 Firms may shutdown production, but they cannot exit the  market

 Plant capacity – the size of the building and amount of

capital equipment

 Costs

∙ Fixed costs

o Cannot be varied in the SR

o Do not change as output change

o Still exist when output falls to zero

o Usually associated with capital

∙ Variable costs – costs that change as output changes;

goes to 0 during shut down

∙ Total Cost = Total Fixed Cost + Total Variable Cost

∙ Average Fixed Cost = TFC / Q

∙ Average Variable Cost = TVC / Q

∙ Average Total Cost = TC / Q = AFC + AVC

Intro to econ (ch. 1-2)

Definition Formula

∙ Marginal Cost = ΔTC / ΔQ

∙ Marginal cost – the additional cost associated with a

1 unit increase in Q

Q

TFC

TVC

TC

MC

AFC

AVC

ATC

0

240

0

240

­­­

­­­

­­­

­­­

1

240

50

290

50

240

50

290

2

240

90

330

40

120

45

165

3

240

120

360

30

80

40

120

4

240

160

400

40

60

40

100

5

240

240

480

80

48

48

96

∙ What is Π for a firm with these costs selling 5 units

for ap rice of \$100?

Π = TR – TC

Π = (5 * 100) – 480 = \$20

() ∙ ATC and AVC must cross MC at their minimum

3/9/16

o Long run – all costs are variable; entry and exit are possible Intro to econ (ch. 1-2) Definition Formula

 economies of scale/increasing returns to scale – as q rises,  LRATC falls

 diseconomies of scale/decreasing returns to scale – as q  rises, LRATC rises

o change in LR costs o factors that shift the LR cost curves

 input prices – input prices and cost curves move together  regulation/taxes – taxes and costs move together

 technology – as technology improves, costs fall (opposite)  Law of Diminishing Marginal Returns

o As successive units of a variable resource are added to a fixed  resource, the marginal product of the variable resource

eventually decreases

o LDMR explains why the short run cost curves eventually increase as q increases

o Only applies to SR (no fixed resources in LR)

Intro to econ (ch. 1-2)

Definition Formula

o Formulas:

 Marginal product of labor – the additional output produced  by one more unit of a variable input

∙ MPL= ΔQ / ΔL

 Total product = Q = Σ MPL

 MC = Wage / MPL

K

L

W

Q

MPL

MC = W / MPL

1

0

\$5

0

­­­

­­­

1

1

\$5

10

10

\$0.50

1

2

\$5

30

10

\$0.25

1

3

\$5

40

10

\$0.50

1

4

\$5

44

4

\$1.25

1

5

\$5

46

2

\$2.50

o As returns (MPL) diminish, marginal cost increases

Review!!!!!

1. a) what is the opportunity cost of a movie ticket?

Opp cost of x = max Qy / max Qx = 12/4 = 3 a) If the consumer has \$36 along BL1, what is the price of a movie  ticket?

income = Px * Ox + Py * Qy

36 = 4Px + 0

Px = 9

Intro to econ (ch. 1-2)

Definition Formula

b) What would a movement from BL1 to BL2 represent?

a decrease in income from \$36 to \$27

2. if MUx / Px < MUy / Py, then how should the consumer alter the Qx and Qy to reach optimal consumption?

Optimal: MUx / Px = MUy / Py

MUx must increase and MUy must decrease

Buy less X and more Y

3. a) find the optimal consumption bundle

Qx

TUx

MUx

MUx / Px

Qy

TUy

MUy

MYy / Py

1

18

18

9

1

32

32

8

2

30

12

6

2

56

24

6

3

40

10

5

3

72

16

4

4

48

8

4

4

80

8

2

Px = \$2, Py = \$4, income - \$20

#1 – 2 X and 2 Y = (2*2) + (2*4) = 4 + 8 = \$12 {wrong}

#2 – 4 X and 3 Y = (4*2) + (3*4) = 8 +12 = \$20 {correct}

b) find the total utility derived from the entire purchase

Px = \$2, Py = \$4, income = \$20

TU of 4X = 48

TU of 3Y = 72

TU = 48+72 = 120 utils

c) Graph the budget line and draw an IC tangent at the optimal bundle, label points 1 and 2 Intro to econ (ch. 1-2)

Definition Formula

d) what is the opportunity cost of one unit of good Y?

max Qx / Max Qy = 10/5 = 2 or Py / Px = 4/2 = 2

e) which point (1 or 2) has the higher MRS?

point 2

f) which point (1 or 2) yields a higher level of total utility?

Neither, they are the same on IC

4. a) using the RP, find the optimal bundle if Px = \$4 and Py = \$2

Bundle

Qx

MUx

Qy

MUy

MRS

A

2

160

30

40

4

B

3

120

20

60

2

C

4

80

12

80

1

D

5

60

8

120

½

RP = 4/2 = 2

Optimal bundle = B

b) Find the optimal bundle if the price of Y rises to \$8

RP = 4/8 = ½

Optimal bundle = D

5. a) a consumer considers X and Y perfect substitutes, if Px = \$4 and Py  = \$8 and income = \$48, how many units of each good will be in the  consumer’s optimal bundle?

Optimal bundle = income / low P

Optimal = 48 / 4 = 12

12 units of X and 0 units of Y

b) if the consumer instead considers the goods perfect complements,  then how many units of each good will be in the bundle?

Optimal bundle = income / (Px + Py)

Optimal = 48 / (4+8) = 4 {quantities of each good}

4 units of X and 4 units of Y

6. a) complete the table (more will be filled in on exam)

Q

TFC

TVC

TC

MC

AFC

AVC

ATC

0

60

0

60

­­­

­­­

­­­

­­­

1

60

40

100

40

60

40

100

2

60

60

120

20

30

30

60

3

60

90

150

30

20

30

50

4

60

140

200

50

15

35

50

5

60

240

300

100

12

48

60

Intro to econ (ch. 1-2)

Definition Formula

b) Find Π if Q = 5 units and P = \$80

Π = TR – TC = (5*80) – 300 = 400 – 300 = \$100

7. complete the table

K

L

W

Q

MPL

MC = W/MPL

1

0

12

0

­­­

­­­

1

1

12

2

2

\$6

1

2

12

6

4

\$3

1

3

12

9

3

\$4

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