utility_and_demand.ppt

Perloff Reference Chapters: Chapter 4: pp. 92-105, Chapter 5: pp. 112-128

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AgendaConsumer EquilibriumChange in EquilibriumIncome and Substitution EffectsDemandTastes and Preferences Affects on DemandConsumer Surplus

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Consumer EquilibriumConsumer equilibrium is comprised of two

concepts:The utility functionThe budget constraint

Consumer equilibrium can be defined as a consumption bundle that is feasible given a particular budget constraint and maximizes total utility.

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Consumer Equilibrium Cont.If there was no budget constraint, a

person would consume each good to the point where marginal utility of consumption for each good is zero.Why?

Given a budget constraint, the consumer maximizes total utility by consuming a bundle that is feasible.A feasible bundle is one that lies either on or

inside the budget constraint.

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Consumer Equilibrium Cont.In graphical terms, consumer equilibrium is

defined as the point where the highest utility function touches the budget constraint.

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Consumer Equilibrium ExampleSuppose we have the following utility

function:U = u(x1,x2) = x1 * x2

Where x1 is equal to the number of hotdogs consumed Where x2 is equal to the number of sodas consumed

Suppose we have the following budget constraint:I = p1*x1 + p2*x2

Where pi is equal to the price of hotdogs consumed Where pj is equal to the price of sodas consumed

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Consumer Equilibrium Example Cont.Now consider that you have a price of

hotdogs equal to $2 and a price of soda is a $1.

Also suppose that our income is $10.Examine the different indifference curves of

U = 1, U=12.5, and U = 25

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8

Con

sum

ptio

n of

sod

as

Consumption of hotdogs

U = 11

1

10

5

5

2.5

U = 12.5

U = 25

x2

x1

Consumer Equilibrium Cont.Intuitively what we have done in the graph is

equate the tradeoff from prices to the tradeoff in utility.I.e., (p2/p1) = (MU2/MU1)

Where p2 is the price of good 2 and p1 is the price of good 1

Where MU2 is the marginal utility of consuming good 2 and MU1 is the marginal utility of consuming good 1

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Consumer Equilibrium Cont.(p2/p1) = (MU2/MU1) can be rewritten as:

(MU2 / p2) = (MU1 / p1)This says that you are normalizing the change

in utility by the price of the good and then equating it to the normalized marginal utility of the other good.

Another way to look at this is to say that the marginal utility derived from the last dollar spent for each good is equal.

What happens if one side is greater than the other?

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Changes in EquilibriumThere are many things that can change

consumer equilibrium.The major two items that we will examine

that can change consumer equilibrium, ceteris paribus:IncomePrice of each good

Note: Ceteris paribus means that we hold everything else fixed.

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Mathematical Side Note on Ceteris ParibusThere are many things that enter our

utility function which we can represent with the following utility function:U = u(x1, x2, x3, …, xn)

When we say that we want to examine utility with respect to x1 and x2, ceteris paribus, what we are saying is that we hold constant the values for all other goods.

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Mathematical Side Note on Ceteris Paribus Cont.Mathematically, we can represent holding

things constant in the following two manners:U = u(x1, x2; x3, …, xn) or

U = u(x1, x2| x3, …, xn) Where it is understood using this notation that

goods x3 through xn are held at some constant level.

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Change in Equilibrium ExampleSuppose Dr. Hurley is consuming a basket of

goods that only has two items, chips and soda.

Assume for the moment that the price is held constant for chips at $1.00 and the price for soda is held constant at $1.00.

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Change in Equilibrium Example Cont.Also assume that Dr. Hurley has $10 for this

basket of goods and his utility function is represented by U = u(soda, chips) = soda * chips.

What is Dr. Hurley’s initial consumer equilibrium?

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Change in Equilibrium ExampleMathematically we can represent Dr.

Hurley’s problem as the following:Dr. Hurley’s utility function:

U = u(x1, x2) = x1 * x2

Dr. Hurley’s budget constraint: M = p1*x1 + p2*x2 10 = 1*x1 + 1*x2

Where x1 is the quantity of soda consumed

Where x2 is the quantity of chips consumed

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Change in Equilibrium Example

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Consumption of soda

1

1

10

10

5

5

U = 25

Con

sum

ptio

n of

Chi

ps

QuestionHow did Dr. Hurley know that consuming 5

chips and 5 sodas will maximize utility?He used advance math that you will learn in Ag

Bus 313?But there is another way you can find the

answer.

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Graphically Finding the Maximum Utility To find the maximum utility we can make

the following argument:We know that are maximum utility point must

lie on the budget line assuming all the consumption goods are desirable and we are non-satiated, i.e.,utility is always increasing.

This being the case we can examine the points on the budget line to see which provides the highest utility.

Once we have found the maximum utility on the budget curve, we can hold our utility fixed and draw the utility function.

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Graphically Finding the Maximum Utility

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10 = x1 + x2, U = u(x1,x2) = x1 * x2

Consumption of x1 Consumption of x2 Total Utility from consumption

0

1

2

3

4

5

6

7

8

10

9

8

7

6

5

4

3

2

0 = 0 * 10

9 = 9 * 1

16 = 2 * 8

21 = 3 * 7

24 = 4 * 6

25 = 5 * 5

24 = 6 * 4

21 = 7 * 3

16 = 8 * 2

Maximum

Change in Equilibrium Example Cont.What happens if we change the price of soda

from $1 to $2 holding the price of the chips constant.

What happens if we change the price of soda from $1 to $0.50 holding the price of the chips constant.

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22Consumption of soda

2.5

10

10

5

5

U = 50

Con

sum

ptio

n of

Chi

ps

U = 25U = 12.5

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Observations on Changing EquilibriumCoincidently, the consumption of chips did

not change.This is a property of the function we used, it is

not always true.

The line/curve that connected all three equilibrium points is considered a price consumption curve.This curve relates the quantity of chips and

soda consumed when changing the price of soda.

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Observations on Changing Equilibrium Cont.As price went down for soda, more was

consumed and when price went up for soda less was consumed.

There are two effects at work when price changes:The income effectThe substitution effect

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Income and Substitution EffectsSubstitution Effect

It is the change in the quantity consumed due to a change in the price of the good, while holding other prices for goods constant and utility constant.

Income EffectIt is the change in the quantity consumed due

to a relative increase in a change of income while holding prices constant.

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Graphical ExampleAssume that the price for soda has

decreased.To find the substitution effect graphically,

we examine what quantities would be consumed if the consumer had to stay on her original indifference curve facing the new prices.This is equivalent to taking a parallel line to the

new budget line and setting it tangent, i.e., just touching at one point, to the old utility level.

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Income Effect

Substitution Effect

Qua

ntit

y of

Chi

ps

Quantity of Soda

Original consumption level

New consumption level

I1

I2

Original Budget line

New Budget Line

Notes on Income and Substitution EffectsThe total effect of a price change is the

summation of the substitution and income effect.

The substitution and income effect can work in opposite direction of each other.

The income effect defines whether the good is an inferior good or a normal good.

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Inferior and Normal GoodsA normal good can be defined as a good

whose consumption has a positive correlation with the income effect.

An inferior good can be defined as a good whose consumption has a negative correlation with the income effect.

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Deriving DemandBy changing the price

of soda and examining the new equilibrium point, we can derive the demand curve for soda for an individual.

Summarizing the changing equilibrium example gives the following demand schedule for soda:

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Price of Soda Quantity Demanded of Soda

$0.50

$1.00

$2.00

10

5

2.5

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Price of Soda

Quantity of Soda

$0.50

$1.00

$2.00

2.5 5 10

•

•

•

Deriving Demand Cont.Now suppose we

change the price for soda on a continuous basis.Instead of points

on the graph, you would begin to see a curve like the following:

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P

Q

Demand curve for soda

Notes on DemandWe have seen that using the idea of a budget

constraint and utility function, we can derive a person’s demand schedule or curve.A demand schedule is a table that shows the

relationship between the quantity demanded of a good and its corresponding price.

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Notes on Demand Cont.When we derived demand, we only change

the price of the good we were investigating and the change in the quantity demanded for the good.Prices of the other good(s) and income were

held fixed.Any other variable that might affect the utility

function were held fixed also.

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Notes on Demand Cont.We can mathematically represent demand

for good i as the following:D(pi) = d(price of good i| price of all other

goods and income)D(pi) = d(pi| p1, p2, …, pi-1, pi+1, …, pn, M)

Where d(·) is a functional relationship that maps prices to quantities.

pi is the price of good i pj for j = 1,2, …, i-1, i+1, …, n are the prices of all

other good except good i M is the persons income.

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Other Demand DeterminantsBeside the price of the good, there are three

other major items that affect the demand curve:Income (M)Prices of other goods (pj)Tastes and preferences

This can either show up as a variable in the demand function or it can change the function altogether.

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How Income Affects DemandRemember that an increase in income shifts

the budget curve out, while a decrease in income shifts the budget curve in.

Does an increase in income imply that you will always increase demand for a good?No. It depends on whether the good is a

inferior or normal good.

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Inferior and Normal Goods RevisitedA good can be classified as a normal good if

the consumption for it has a positive correlation with income.I.e., when income increases, you consume

more of the good and when income decreases you consume less.

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Inferior and Normal Goods Revisited Cont.A good can be classified as an inferior good if

the consumption for it has a negative correlation with income.I.e., when income increases, you consume less

of the good and when income decreases you consume more.

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Inferior and Normal Goods Revisited Cont.A good can be both a normal good and an

inferior good.It all depends on where you are on the level of

consumption of the good and your income. Suppose you have $10 to use for buying food each

week. You might try living off spaghetti because you cannot afford steak.

What happens when your income doubles, you might find yourself eating more spaghetti and still no steak.

What happens if you have $100 to spend, you might begin to eat less spaghetti and start consuming steak.

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Engel’s CurveEngel’s curve tells you what happens to your

consumption of a good as you change your level of income.

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How Price Changes of Other Goods Change DemandThe demand curve for a particular good may

shift if the price of another good changes.How the demand curve shifts will depend on

whether the goods are substitutes, complements, or have no correlation.

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Substitute GoodGood j is said to be a substitute of good i if an

increase in the price of good j causes you to consume more of good i.

Good j is also said to be a substitute of good i if a decrease in the price of good j causes you to consume less of good i.I.e., the demand for product i is positively

correlated with the price of product j.

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Complementary GoodGood j is said to be a complement of good

i if an increase in the price of good j causes you to consume less of good i.

Good j is also said to be a complement of good i if a decrease in the price of good j causes you to consume more of good i.I.e., the demand for product i is negatively

correlated with the price of product j.

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Other Items that Affect the Demand CurveComposition of the PopulationAttitudes toward Nutrition and HealthFood SafetyLifestylesTechnological ForcesAdvertising

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Consumer SurplusConsumer surplus is a measure of the

difference between the amount of money a person was willing to pay to buy a quantity of good and the actual price they paid.

This measure is used as a tool in policy analysis.

Consumer surplus is represented graphically as the area underneath the demand curve above the price paid for the goods.

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P

Q

p = 5

q = 5

Consumer Surplus