1 The Utility-Maximizing Model Module 11. Use the utility-maximizing model to explain how consumers...

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The Utility-Maximizing Model


Module 11Module 11Module 11Module 11

Use the utility-maximizing model to explain how consumers choose goods and services.


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ObjectivesObjectivesObjectivesObjectives


Use the concept of utility to explain how the law of demand results from consumers adjusting their consumption choices to changes in prices.


Use the concept of utility to explain how the law of demand results from consumers adjusting their consumption choices to changes in prices.

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ObjectivesObjectivesObjectivesObjectives

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Objective 1

Use the utility-maximizing model to explain how consumers choose goods and services

Objective 1


Be sure to understand the difference between total utilitytotal utility and marginal utilitymarginal utility.

Recall the law of diminishing marginal law of diminishing marginal utilityutility.

Also know the terminologyterminology of this chapter, for example, terms such as “consumption bundle”, “consumer preferences.

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Objective 1


Objective 1


The goal of the utility-maximizing model is to derive the consumer’s demand curvedemand curve.

Economists assume that consumers try to allocate their limited incomes to maximize their satisfaction, a goal referred to as utility utility maximizationmaximization.

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Objective 1


Objective 1


The utility-maximizing model is used to determine the optimaloptimal amounts of goods and services a consumer will purchase given:

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Objective 1


Objective 1


The utility-maximizing model is used to determine the optimaloptimal amounts of goods and services a consumer will purchase given: knowledge of consumer’s preferencespreferences

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Objective 1


Objective 1


The utility-maximizing model is used to determine the optimaloptimal amounts of goods and services a consumer will purchase given: knowledge of consumer’s preferencespreferences the pricesprices of the goods and services

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Objective 1Use the utility-maximizing model to explain how consumers choose goods and services

Objective 1Use the utility-maximizing model to explain how consumers choose goods and services

The utility-maximizing model is used to determine the optimaloptimal amounts of goods and services a consumer will purchase given: knowledge of consumer’s preferencespreferences the pricesprices of the goods and services the consumer’s budgetbudget constraint

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Objective 1


Objective 1


The consumer’s equilibrium bundleequilibrium bundle is a combination of goods and services consumed which gives the consumer the maximum total utility, subject to a budget constraintbudget constraint or an income constraint.

The terms “equilibrium” bundle, “optimal” bundle and “utility-maximizing” bundle are used interchangeably.

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Suppose a consumer has $I to spend on two goods, X and Y. Let Px = price of good X and Py = price of good Y.

Objective 1: Using the utility-maximizing Objective 1: Using the utility-maximizing model model


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Suppose a consumer has $1 to spend on two goods, X and Y. Let Px = price of good X and Py = price of good Y.

How will the consumer allocate her $I towards these two goods so that she gets the most satisfaction?



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How will the consumer allocate her $1 towards these two goods so that she gets the most satisfaction?

The equilibrium bundle satisfies two conditions:



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How will the consumer allocate her $1 towards these two goods so that she gets the most satisfaction?


Condition 1:Condition 1: Income should be allocated so that the last dollar spent on each good yields the same amount of marginal utility.

Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model

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Condition 1:Condition 1:

Income should be allocated so that the last dollar spent on each good yields the same amount of marginal utility.

In terms of an equation, where MU = marginal utility, and P = price,


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Income should be allocated so that the last dollar spent on each good yields the same amount of marginal utility.

In terms of an equation, where MU = marginal utility, and P = price,

MU MUX Y=P PX Y


Marginal utility per Marginal utility per dollar spent on good Xdollar spent on good X

Marginal utility per Marginal utility per dollar spent on good Ydollar spent on good Y

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The consumer must spend the total income allocated to the consumption of goods and services.

Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model

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X YP X + P Y = I


The consumer must spend the total income allocated to the consumption of goods and services.

In terms of an equation:

Where = Price of good X × Quantity of good X = expenditure on good X,

and = Price of good Y × Quantity of good Y = expenditure on good Y

XP X

YP Y

Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model Objective 1: Using the utility-maximizing model

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The table below shows Kayla's utility from soup and sandwiches.

Cups of Cups of SoupSoup

Total Total UtilityUtility

Number of Number of SandwichesSandwiches


1 40 1 45

2 60 2 75

3 72 3 102

4 82 4 120

5 88 5 135

6 90 6 145

Objective 1: Using the utility-maximizing model … an example


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The price of a cup of soup is $2 and the price of a sandwich is $3.





1 40 1 45

2 60 2 75

3 72 3 102

4 82 4 120

5 88 5 135

6 90 6 145



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The price of a cup of soup is $2 and the price of a sandwich is $3.

Kayla has $18 to spend on these two goods.





1 40 1 45

2 60 2 75

3 72 3 102

4 82 4 120

5 88 5 135

6 90 6 145



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Step 1:Step 1: Calculate the marginal utility per dollar spent on each good using the formula:

marginal utility per dollar = marginal utility ÷ price of the good

Note:Note: To apply this step you must have marginal utility figures and the price of the product. If you are given total utility figures, you will have to calculate the marginal utility before using the equation above.

Objective 1: … applying the utility-Objective 1: … applying the utility-maximizing conditionsmaximizing conditions

Objective 1: … applying the utility-Objective 1: … applying the utility-maximizing conditionsmaximizing conditions

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1 40 40 40÷2=20 1 45 45 45÷3=15

2 60 20 20÷2=10 2 75 30 30÷3=10

3 72 12 6 3 102 27 9

4 82 10 5 4 120 18 6

5 88 6 3 5 135 15 5

6 90 2 1 6 145 10 3.33



Begin by calculating Marginal Utility.

(1)Cups

of soup

(2)TotalUtility

(3)Marginal

Utility

(4)Marginal Utility per

dollarMU/Psoup

(5)Number of

Sandwiches

(6)TotalUtility

(7)Marginal

Utility


dollarMU/Psandwich

Price of Soup = $2 per cup Price of Sandwich = $3 per cup

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1 40 40 40÷2=20 1 45 45 45÷3=15

2 60 20 20÷2=10 2 75 30 30÷3=10

3 72 12 6 3 102 27 9

4 82 10 5 4 120 18 6

5 88 6 3 5 135 15 5

6 90 2 1 6 145 10 3.33



Begin by calculating Marginal Utility. Next, calculate Marginal Utility per dollar using the

equation above.

(1)Cups

of soup

(2)TotalUtility

(3)Marginal

Utility


dollarMU/Psoup

(5)Number of

Sandwiches

(6)TotalUtility

(7)Marginal

Utility


dollarMU/Psandwich


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Step 2:Step 2: Identify the combinations of the goods that satisfy the marginal utility per dollar rule:

soup sandwich

soup sandwich

MU MU=P P

Objective 1: Applying the utility-maximizing conditionsObjective 1: Applying the utility-maximizing conditions

(1)Cups

of soup


dollar

MU/Psoup

(5)Number of

Sandwiches


dollar

MU/Psandwich

1 40÷2=20 1 45÷3=15

2 20÷2=10 2 30÷3=10

3 6 3 9

4 5 4 6

5 3 5 5

6 1 6 3.33

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The marginal utility per dollar rule holds for these three combinations:

2 cups of soup and 2 sandwiches3 cups of soup and 4 sandwiches 4 cups of soup and 5 sandwiches


(1)Cups

of soup


dollar

MU/Psoup

(5)Number of

Sandwiches


dollar

MU/Psandwich

1 40÷2=20 1 45÷3=15

2 20÷2=10 2 30÷3=10

3 6 3 9

4 5 4 6

5 3 5 5

6 1 6 3.33

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Which of the three bundles is the optimal Which of the three bundles is the optimal bundle?bundle?


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Which of the three bundles is the optimal Which of the three bundles is the optimal bundle?bundle?

Step 3:Step 3: Identify the optimal bundle by applying Condition 2: that Kayla’s expenditure on the two goods must exhaust her

budget of $18.


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Which of the three bundles is the optimal bundle?Which of the three bundles is the optimal bundle?

Step 3:Step 3: Identify the optimal bundle by applying the condition 2: that Kayla’s expenditure on the two goods must exhaust her budget of $18.

2 cups of soup and 2 sandwiches will cost her $10 ($2×2 cups of soup + $3×2 sandwiches)


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Kayla’s equilibrium bundleequilibrium bundle is 3 cups of soup and 4 sandwiches.


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The utility maximizing model applies a key economic principle: optimal decisions are made at the margin.

Objective 1: … the utility-maximizing modelObjective 1: … the utility-maximizing modelObjective 1: … the utility-maximizing modelObjective 1: … the utility-maximizing model

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Examine the marginal utility per dollar rule again:


MU MUX Y=P PX Y

Marginal utility per Marginal utility per dollar spent on good Xdollar spent on good X

Marginal utility per Marginal utility per dollar spent on good Ydollar spent on good Y

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Examine the marginal utility per dollar rule again:

X Y

X Y

X Y Y X

X X

Y Y

MU MU=P P

MU ×P =MU ×P

MU P=MU P

Rearrange to get


ratio of ratio of marginal marginal utilitiesutilities

ratio of pricesratio of prices

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Examine the marginal utility per dollar rule:

X Y

X Y

X Y Y X

X X

Y Y

MU MU=P P

MU ×P =MU ×P

MU P=MU P

Rearrange to get

ratio of prices is also called relative pricesrelative prices


What is the meaning of relative prices?

Suppose, where X=soup and Y=sandwich, then Kayla

must give up 2/3 of a sandwich (Y) for I more cup of soup (X).

The relative price is the opportunity costopportunity cost of consuming 1 more unit of X or the market dictated rate of exchangemarket dictated rate of exchange between good X and good Y.

X

Y

P 2=P 3

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X Y

X Y

X Y Y X

X X

Y Y

MU MU=P P

MU ×P =MU ×P

MU P=MU P

Rearrange to get


Ratio of marginal utilitiesalso called the marginal marginal rate of substitutionrate of substitution


What is the ratio of marginal utilities?

The ratio of marginal utilities is also called the marginal rate marginal rate of substitutionof substitution.

It shows the rate at which the consumer is willing to give up some of good Y (sandwich) for an additional unit of good X (soup). In other words, it is a measure of the consumer’s subjectivesubjective orpersonalpersonal rate of exchange.

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X Y

X Y

X Y Y X

X X

Y Y

MU MU=P P

MU ×P =MU ×P

MU P=MU P

Rearrange to get


ratio of marginal utilitiesalso called the marginal marginal rate of substitutionrate of substitution


In equilibrium, the consumer’s subjective rate of exchange (ratio of marginal utilities) equals the rate of exchange required by the market (ratio of prices).

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Objective 2Use the concept of utility to explain how the law of demand results …

Objective 2Use the concept of utility to explain how the law of demand results …

An individual’s utility-maximizing choices lead to a demand curve.

Recall, that the demand curve shows the quantities demanded at alternative prices

The utility-maximizing consumer adjusts her consumption choices to changes in prices.

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To derive Kayla’s demand for sandwiches curve, we must change the price of sandwiches and observe what happens to her quantity demanded of sandwiches, holding all else constant.

Objective 2Objective 2Use the utility maximizing model to derive Use the utility maximizing model to derive

a demand curvea demand curve



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To derive Kayla’s demand for sandwiches curve, we must change the price of sandwiches and observe what happens to her quantity demanded of sandwiches, holding all else constant.

We already have one price-quantity combination: At a price of $3, Kayla’s optimal quantity was 4 sandwiches.





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To construct a demand curve we need at least one other price-quantity combination.

Objective 2: … deriving a demand curve.Objective 2: … deriving a demand curve.Objective 2: … deriving a demand curve.Objective 2: … deriving a demand curve.

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Suppose the price of sandwiches rises to $4.00. How would Kayla’s quantity demanded of sandwiches change?


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Suppose the price of sandwiches rises to $4.00. How would Kayla’s quantity demanded of sandwiches change?

Since the price of one good has changed we have to recalculate the marginal utility per dollar for that good.


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marginal utility per dollar = marginal utility ÷ price of the goodmarginal utility per dollar = marginal utility ÷ price of the good


(1)Cups

of soup

(2)TotalUtility

(3)Marginal

Utility

(4)Marginal Utility

per dollar

MU/Psoup

(5)Number of

Sandwiches

(6)TotalUtility

(7)Marginal

Utility

(8)Marginal Utility

per dollar

MU/Psandwich1 40 40 40÷2=20 1 45 45 45÷4=11.25

2 60 20 20÷2=10 2 75 30 30÷4=7.5

3 72 12 6 3 102 27 6.75

4 82 10 5 4 120 18 4.50

5 88 6 3 5 135 15 3.75

6 90 2 1 6 145 10 2.5


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How to determine the optimal combination of soup and sandwiches in the case where the rule of equal marginal utility per dollar does not hold?


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Apply the principle of marginal analysis. Ask the question: what is the first item Kayla should buy?


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Apply the principle of marginal analysis. Ask the question: what is the first item Kayla should buy?

(1)Cups

of soup

(4)Marginal Utility

per dollar

MU/Psoup

(5)Number of

sandwiches

(8)Marginal Utility

per dollar

MU/Psandwich

1 40÷2=20 1 45÷4=10.8

2 20÷2=10 2 30÷4=7.5

3 6 3 6.75

4 5 4 4.50

5 3 5 3.75

6 1 6 2.5

Objective 2: … deriving a demand curveObjective 2: … deriving a demand curveObjective 2: … deriving a demand curveObjective 2: … deriving a demand curve

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Obviously, the item that gives her the highest marginal utility per dollar spent. What is the 2nd item? … and so forth until her budget is exhausted.

Goods consumed

Budget = $18

1st cup of soup $18-$2 =$16

1st sandwich $16-$4 =$12

2nd cup of soup $12-$2 =$10

2nd sandwich $10-$4=$6

3rd sandwich $6-$4 =$2

3rd cup of soup $2-$2 =$0

(1)Cups

of soup

(4)

Marginal Utility per dollar

MU/Psoup

(5)Number of

Sandwiches

(8)


MU/Psandwich

1 40÷2 = 20 1 45÷4 = 10.8

2 20÷2 = 10 2 30÷4 = 7.5

3 6 3 6.75

4 5 4 4.50

5 3 5 3.75

6 1 6 2.5


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Obviously, the item that gives her the highest marginal utility per dollar spent. What is the 2nd item? … and so forth until her budget is exhausted.

Goods consumed

Budget = $18

1st cup of soup $18-$2 =$16

1st sandwich $16-$4 =$12

2nd cup of soup $12-$2 =$10

2nd sandwich $10-$4=$6

3rd sandwich $6-$4 =$2

3rd cup of soup $2-$2 =$0

Her utility maximizing bundleutility maximizing bundle is 3 cups of soup and 3 sandwiches.

(1)Cups

of soup

(4)


MU/Psoup

(5)Number of

Sandwiches

(8)


MU/Psandwich

1 40÷2 = 20 1 45÷4 = 10.8

2 20÷2 = 10 2 30÷4 = 7.5

3 6 3 6.75

4 5 4 4.50

5 3 5 3.75

6 1 6 2.5


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Now that we have two price-quantity combination points on Kayla’s demand for sandwiches curve we can trace her demand curve.


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The resulting demand curve is downward-sloping. It obeys the law of demand.law of demand.


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The utility-maximizing choices lead to a demand demand curve.curve. Each price-quantity combination on a demand curve is a utility-maximizing quantityutility-maximizing quantity, given the price.

Key PointsKey PointsKey PointsKey Points

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The utility-maximizing choices lead to a demand demand curve.curve. Each price-quantity combination on a demand curve is a utility-maximizing quantityutility-maximizing quantity, given the price.

If people seek to maximize utility, then the law of law of demanddemand follows.

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When price changes, there a substitutionsubstitution effect and an incomeincome effect on the quantity of the good demanded.

The substitution effectsubstitution effect of a price change is the change in quantity demanded that results from a change in price, making the good more of less expensive relative to other goods (that can substitute for it), holding purchasing power constant.

The income effectincome effect of a price change is the change in quantity demanded that results from the effect of the price change on the consumer’s purchasing power, all else constant.

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When price changes, there a substitutionsubstitution effect and an incomeincome effect on the quantity of sandwiches demanded.

This movement along This movement along the curve is caused by the curve is caused by a change in the price a change in the price of sandwiches and is of sandwiches and is made up of two made up of two effects.effects.

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Initially, when the price of soup = $2 and the price of sandwich = $3, Kayla’s equilibrium bundle was 3 cups of soup and 4 sandwiches.

Objective 2: … how a consumer adjusts to a Objective 2: … how a consumer adjusts to a price changeprice change

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The ratio of marginal utility to price was the same

for soup and for sandwiches.


soup sandwichMU MU=$2 $3

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soup sandwichMU MU=$2 $3


The ratio of marginal utility to price was the same

for soup and for sandwiches.

When a consumer is in equilibrium, she is maximizing utility.


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When the price of sandwiches rises to $4, the ratios of marginal utility to price no longer hold with equality.


soup sandwichMU MU

$2 $4

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soup sandwich>MU MU$2 $4

A dollar spent on soup gives Kayla more A dollar spent on soup gives Kayla more utility than a dollar spent on sandwichesutility than a dollar spent on sandwiches


When the price of sandwiches rises to $4, the ratios of marginal utility to price no longer hold with equality.

We now have:

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To restore equilibrium,equilibrium, Kayla buys more soup and fewer sandwiches, subject to her budget constraint.




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In my example, given her budget, Kayla buys fewer sandwiches but is not able to increase her soup consumption.




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In my example, given her budget, Kayla buys fewer sandwiches but is not able to increase her soup consumption.

Note that when the price of sandwiches rises, quantity

demanded falls – a result consistent with the law of law of demand.demand.

Objective 2: … how a consumer adjusts to a price Objective 2: … how a consumer adjusts to a price changechange



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End of Module 11End of Module 11End of Module 11End of Module 11

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