objectives: use the utility-maximizing model to explain how consumers choose goods and services. ...

Objectives:Objectives:

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

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

Objectives:Objectives:

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

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

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Module 11: The Utility-Maximizing ModelModule 11: The Utility-Maximizing ModelModule 11: The Utility-Maximizing ModelModule 11: The Utility-Maximizing Model

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Objective 1

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

Objective 1

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

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

The utility-maximizing model is used to determine the optimal 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

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 constraint or an income constraint.

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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.

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

The equilibrium bundle satisfies two conditions:

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,

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

MU MUX Y=P PX Y

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

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.

Cups of Soup

Total Utility

Number of Sandwiches

Total Utility

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 Objective 1: Using the utility-maximizing model ...an examplemodel ...an exampleObjective 1: Using the utility-maximizing Objective 1: Using the utility-maximizing model ...an examplemodel ...an example

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X YP X + P Y = I

Condition 2:Condition 2:

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

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

XP X

YP Y

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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

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

*If you are given total utility figures, you will have to calculate the marginal utility before using the equation above. For example, see columns 3 and 7.

(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)


MU/Psandwich

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

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

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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


(1)Cups of

soup

(4)


MU/Psoup

(5)Number of

Sandwiches

(8)


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

(4)


MU/Psoup

(5)Number of

Sandwiches

(8)


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 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)



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.

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 prices is also called relative prices

ratio of marginal utilitiesalso called the marginal rate of substitution

In equilibrium, the consumer’s personal rate of exchange equals the rate of exchange required by the market.

Objective 1: ..more on the utility-maximizing model..Objective 1: ..more on the utility-maximizing model..

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

a demand curvea demand curve

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.

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.

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

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

Price of Soup = $2 per cup Price of Sandwich = $4 per cup

(1)Cups

of soup

(2)TotalUtility

(3)Marginal

Utility

(4)


MU/Psoup

(5)Number of

Sandwiches

(6)TotalUtility

(7)Marginal

Utility

(8)


MU/Psandwich

1 40 40 40÷2=20 1 45 45 45÷4=10.8

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?

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

(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

Objective 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

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

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

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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.

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


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Some key points:Some key points:

The utility-maximizing choices lead to a demand curve.demand 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 demandlaw of demand follows.

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

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soup sandwichMU MU=$2 $3

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.

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.

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

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When the price of sandwiches rises to $4, the ratios of MU to price no longer hold with equality. We now have:


soup sandwich>MU MU$2 $3

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

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soup sandwich>MU MU$2 $3

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

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 demand.law of demand.


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

objectives: use the utility-maximizing model to explain how consumers choose goods and services. ...

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