Lectures 4-6 - The Theodry of Consumer Choice

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The Theory of Consumer Choice

• Every day people in a market economy make decisions. • Should you buy a Coke for lunch, a bottle of tea, or

just drink water?

• Should you purchase a laptop computer or stick to your old desktop?

• Because your financial resources are limited you cannot buy everything you want.

The Theory of Consumer Choice

• You therefore consider the prices of the various goods offered for sale and buy a bundle of goods that best suits your needs and desires.

• One of the most important Principles of Economics is that people face trade-offs.

• When a consumer buys more of one good, he can afford less of other goods.

• The theory of consumer choice examines how consumers facing these trade-offs make decisions and how they respond to changes.

Budget Constraint

• Most people would like to increase the quantity of the goods they consume however their spending is constrained by their income.

• For studying the link between income and consumption, we simplify matters and assume that that the consumer can buy only two goods: meal (Y) and Jazz club ticket (X).

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Budget Constraint

• Suppose the consumer has an income of $1000 and he spends his entire income on meals and Jazz club tickets.

• meals go for a fixed price of $20 per meal and Jazz club tickets are $10 a ticket.

• Suppose (for simplicity) the consumer uses all his/her money on these two goods (has no saving).

Budget Constraint

Num. of meals

Num. of Jazz tickets

Spending on meals

Spending on Jazz tickets

Total Spending

50 0 $ 1000 $ 0 $1000

40 20 $ 800 $ 200 $1000

30 40 $ 600 $ 400 $1000

20 60 $ 400 $ 600 $1000

10 80 $ 200 $ 800 $1000

0 100 0 $1000 $1000

Few budget combinations :

The budget constraint shows the various bundles of goods that consumer can buy for a given income.

Consumer’s budget constraintGraphical presentation

Num. of meals


Spending on meals


Total Spending

50 0 $ 1000 $ 0 $1000

40 20 $ 800 $ 200 $1000

30 40 $ 600 $ 400 $1000

20 60 $ 400 $ 600 $1000

10 80 $ 200 $ 800 $1000

0 100 0 $1000 $1000

6040200 80 100

0

10

20

30

40

50 Budget constraint

Jazz club tickets

Meals

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The slope of the Budget Constraint (line)

6040200 80 100

0

10

20

30

40

50

Meals

Jazz club tickets

Y

X

MaxXP

MX

0Y

YP

MY

0XMax

Y

X

X

Y

P

P

PM

PM

Slope Relative price ratio

Budget Constraint

• The slope of the budget constraint measures the rate at which the consumer can trade one good for the other, so it is the exchange rate between the goods given market determined prices.

• The slope of the budget constraint equals the relative price of the two goods.

• meal costs twice as much as a ticket to Jazz club.

• The opportunity cost (or the exchange rate) of one meal is 2 Jazz club tickets.

Budget Constraint

• The budget constraint’s slope of 0.5 (or│-0.5│) reflects the trade-off the market (via prices) is offering the consumer :1 meal for 2 Jazz club tickets.

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The equation of the Budget constraint

1000$_10$20$ TicketsJazzMeals

1000$1020 XY

Let Y represent the number of meals and X represent the number of Jazz club tickets.

In general, the budget constraint can be written

MYPXP YX Where Px = the price per unit X; Py = the price per unit Y; M= the consumer income

Budget Constraints Change When Prices Rise of Fall

• Suppose the restaurant is offering the same meal at half price (20/2=$10).

• How would the budget constraint change ?


• New budget combinations

Num. of meals


Spending on meals


Total Spending

100 0 $ 1000 $ 0 $1000

80 20 $ 800 $ 200 $1000

60 40 $ 600 $ 400 $1000

40 60 $ 400 $ 600 $1000

20 80 $ 200 $ 800 $1000

0 100 0 $1000 $1000

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The Effect of a Decrease in Price on Budget Constraint

6040200 80 100

0

20

40

60

80

100 New Budget constraintNum. of meals


Spending on meals


Total Spending

100 0 $ 1000 $ 0 $1000

80 20 $ 800 $ 200 $1000

60 40 $ 600 $ 400 $1000

40 60 $ 400 $ 600 $1000

20 80 $ 200 $ 800 $1000

0 100 0 $1000 $1000

Meals

Jazz club tickets

Original Budget constraint

50

New CombinationsPrice of meals decreases

Budget Constraints Change

• The opportunity cost of a meal is now only one Jazz club ticket.

• When the price of a single good changes, more than just the quantity demanded of that good may be affected because the new relative price ratio has change.

• The equation of the budget constraint changed From: 10X+20Y=1000 to 10X+10Y=1000 .


• Suppose the Jazz club ticket has gone up to $20 (from $10) and the price of meal hasn’t changed (stay at $ 20).

• How would the budget constraint change ?

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Budget Constraints Change When Prices Rise

Num. of meals


Spending on meals


Total Spending

50 0 $ 1000 $ 0 $1000

40 10 $ 800 $ 200 $1000

30 20 $ 600 $ 400 $1000

20 30 $ 400 $ 600 $1000

10 40 $ 200 $ 800 $1000

0 50 0 $1000 $1000

The Effect of an Increase in Price on Budget Constraint

6040200 80 100

0

10

20

30

40

50

Meals

Jazz club tickets

Y

X

Original Budget constraint

Num. of meals


Spending on meals


Total Spending

50 0 $ 1000 $ 0 $1000

40 20 $ 800 $ 200 $1000

30 40 $ 600 $ 400 $1000

20 60 $ 400 $ 600 $1000

10 80 $ 200 $ 800 $1000

0 100 0 $1000 $1000

50

New Budget constraint


• How would the budget constraint change if instead of changing the meals price, the income increases from $ 1000 to $ 2000 ?

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Budget Constraint Change(i) Income Increase

12080400 160 200

0

20

40

60

80

100Original Budget constraint (M0) New Budget

constraint (M1)

Meals

Jazz club tickets

001

yx PPMM


• How would the budget constraint changes if instead of changing the meals price, the income decreases from $ 1000 to $ 500 ?

Budget Constraint Change(ii) Income Decrease

6040200 80 100

0

10

20

30

40

50New Budget constraint (M1) Original Budget

constraint (M0)

Jazz club tickets50

25

Meals

001

yx PPMM

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Utility Function &Indifference Curves

• Our goal is to see how consumers make choices.

• The budget constraint is one piece of the analysis which is basically exogenous to the consumer coming from the market. However, consumers choices depend not only on the budget constraint he/she faces but also on his/her own preferences regarding the two goods.

Utility Function &Indifference Curves

• Consumers preferences allow them to choose among different bundles of goods X and Y. Each X and Y bundle provides a certain level of utility (level of “happiness”) which we call U.

• There may exist many combinations of X and Y among which consumers are indifferent.

• Combining all different X and Y bundles among which the consumer is indifferent will result in a continuous graph called an indifference curve which represents combinations (X,Y) all representing a given level of utility.

Utility Function – The concept of utility

• Why do people consume?

• To satisfy their wants/desires.

• Utility function indicates the level of enjoyment or preference attached by consumers to market bundle.

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Cardinal Utility & Ordinal Utility

• There are two kind of measurement : (i) Cardinal(ii) Ordinal.

• Cardinal utility is measurable.

• Ordinal utility is only an ordinal measure.

Examples for Cardinal

• Examples for cardinal measurement can be :

• High

• Weight

• Age

• Distance

• Grades

Examples for Ordinal

• Examples for ordinal measure can be :• Beauty (you can declare who looks better and

therefore you can rank them. However, you cannot measure the size/intensity of beauty).

• Knowledge (you can claim who knows more about life, but you cannot measure it).

• Tastes (you can prefer pizza to meat but you cannot measure by how much).

Utility is an ORDINAL measure

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Utility Mountain

U

Y X

Max U

U0

U1

U2

Utility Mountain

Y

X

U0

U1

U2

Umax

A1

A2

A3A4

I II

IIIIV

Red arrow represents preference direction

Indifference Curves

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U0

The consumer is indifferent betweenCombination (X1, Y1) and (X2, Y2), both producing the same level of utility U0

An indifference curve shows a set of consumption bundles among which the individual is indifferent.

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Indifference Curves - Example

• Lets say that Tom is indifferent between eating 5 meals and visiting the Jazz club for five times.

• He is also indifferent with the option of eating 2 meal and visiting the Jazz club for two time and so on…

• Assume that the indifferent bundles of Tom are : (5,0),(2,2),(0,5), we can draw the indifference curve of Tom.

Tom’s Indifference Curve

3210 4 5

0

1

2

3

4

5Tom’s Indifference Curve

B

C

A

U1

Meals

Jazz club tickets


• Tom is indifferent among combinations A, B and C because they are all on the same curve.

• The slope at any point on an indifference curve equals the rate at which the consumer is willing to substitute one good for the other and stay at the same utility level.

• This rate is called Marginal rate of substitutions (MRS).

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Marginal Rate of Substitution

• The slope of the indifference curve at any point is called the marginal rate of substitution (MRS)

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

1UUdX

dYMRS

Diminishing Marginal Utility

• The more you have of a good the less of utility you get when you add additional (marginal) unit and hence the less you are willing to give up from the other good to get an additional unit of the good whose utility had decreased. Thus, the MRS diminishes the more you have of a good. This is expressed graphically by the convex indifference curve.

• Example: the first mug of bear contributes more to ones utility level than the subsequent one and so forth.

Marginal Rate of Substitution

• MRS changes as X and Y change• reflecting the individual’s willingness to trade Y

for X.

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

At (X1, Y1), the indifference curve is steeper.The person would be willing to give up moreY to gain additional units of X

At (X2, Y2), the indifference curve is flatter. The person would be willing to give up less Y to gain additional units of X

X3

Y3

1

2

3 321 MRSMRSMRS

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Tom’s Indifference Curve -Continue

• We already know that Tom is indifferent between (5,0),(2,2),(0,5).

• Tom is also indifferent among the following combinations : (8,0), (3,3), (0,8).


3210 4 5

0

1

2

3

4

5

B

C

A

…… 8

A’

B’

C’

8

U1 U2

Points on a higher indifference curve are preferred to points on a lower indifference curve.

Transitivity: A’ is preferred to A and A is indifferent to B then A’ is preferred to B. Generally, if a P b and b P c then a P c.

Properties of Indifference Curves

• Indifference curves cannot cross each other.

Quantity of X

Quantity of Y

U1

U2

A

BC

An individual is indifferent between A and C (bothare on U1). Also, the individual is indifferent betweenB and C (both are on U2). From transitivity it should bethat A is indifferent to B.

But B is preferred to Abecause B contains more

X and Y than A

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Properties of Indifference Curves

• Indifference curves do not cross

• Indifference curves are convex to the origin.

• When goods are easy to substitute for each other, the indifference curves are less convex.

• When the goods are hard to substitute the indifference curves are very convex.

• Straight-line indifference curves represents perfect substitutes.

• Fixed proportion indifference curves represents perfect complements.

Perfect Substitutes and Perfect Complements

• Perfect Substitutes

utility = U(X,Y) = X + Y

• Perfect Substitutes

utility = U(X,Y) = X + Y

Quantity of X(Dime)

Quantity of Y(Nickels)

U1U2

U3

The indifference curves will be linear.The MRS will be constant along the indifference curve.

Perfect Substitutes and Perfect Complements

• Perfect Complements (no substitution)

utility = U(X,Y) = min (X, Y)

Quantity of X(right shoe)

Quantity of Y(Left shoe) The indifference curves will be

L-shaped. Only by choosing more of the two goods together can utility be increased.

U1

U2

U3

0 1 2 3

1

2

3

4

4

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Less Extreme Cases: Close Substitutes and Close Complements

Quantity of Coke

Quantity of Pepsi

Indifference curves for close substitutes are not very convex

Indifference curves for close substitutes are not very convex

Quantity of hot dogs

Quantity of hot

dog buns

Indifference curves for

close complements

are very convex

Indifference curves for

close complements

are very convex

Optimization: What the Consumer Chooses

• For understanding consumer choices we need to know :

• The consumer budget constraint (how much he can afford to spend, objective).

• The consumer’s preferences (what he wants to spend it on, subjective).

• Returning to our meals and Jazz club example, our consumer would like to maximize utility subject to his/her budget constraint.

The Consumer’s optimal Choices

Budget constraint

C

D

A

B

U1

U2

U3

Meals

Jazz Tickets

A is the optimum: the point on the budget constraint that touches the highest possible indifference curve (level of utility).

Tom prefers B to A, but he cannot afford B.

Tom can afford Cand D, but A is on a higher indifference curve.

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The Consumer’s optimal Choices

Meals

Jazz club tickets

Consumer optimization is

another example of “thinking at the

margin.”

Consumer optimization is

another example of “thinking at the

margin.”

U1

At the optimum, slope of the indifference curve equals slope of the budget constraint:

MRSX,Y = PX/PY

marginal value of Jazz

ticket (in terms of

Meals)

price of Jazz ticket (in terms of Meals)

The Effects of an Increase in Income

Quantity of Jazz Club tickets

Quantity of Meals

An increase in income shifts the budget constraint outward.

If both goods are “normal,” Tom buys more of each.

AB

Inferior vs. normal goods

• An increase in income increases the quantity demanded of normal goods and reduces the quantity demanded of inferior goods.

• Suppose Jazz club ticket is a normal good but meals are an inferior good.

• What is the effect of increase in the income ?

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Inferior vs. normal goods

Quantity of Jazz club tickets

Quantity of meals

If meals are inferior, the new optimum will contain fewer meals .

AB

How changes in Prices Affect the Consumer’s Choices ?

6040200 80 100

0

20

40

60

80

100

Meals

Jazz club tickets

Suppose the price of Meal decrease to $10. What will happen ?

Budget’s constraint shifts outward and changes slope.

Tom buys more Jazz tickets and less meals (B).

A

B

Income and Substitution Effects

• A fall in the price of meals (PY) has two effects on Tom’s optimal consumption of both goods.

• Income effectA fall in PY boosts the purchasing power of Tom’s income, allows him to buy more Jazz club tickets and more meals, but since meals are considered inferior he will buy less meals and more Jazz tickets.

• Substitution effect A fall in PY makes Jazz club tickets more expensive relative to meals, causes Tom to buy fewer Jazz club tickets & more meals.

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Income and Substitution Effects

Good Income Effect Substitution Effect

Total Effect

meals Consumer is richer, but since meals are considered inferior he will buy less meals and with theadditional extra income will buy more jazz tickets.

meals are relatively cheaper, so consumer buys more meals.

Income and substitution effects act in the opposite direction, but income effect > substitution effect therefor consumer buys less meals.

Deriving the Demand Curve

• What happen to the quantity demanded of meals when the price of a meal changes and the price of Jazz tickets stays constant ?

Deriving the Demand Curve

mealsquantity

Jazz club tickets

30

mealsPrice divided by Jazz ticket price

mealsquantity

30

$2

For price $2, the demand is 30 mealsFor price $1, the

budget constraint outward

Consumer buys more meals

And for price $1, the demand is 60 meals

$1

60Demand Curve

60

D

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Do All Demand Curves Slope Downward ?

• Normally, when the price of a good rises, people buy less of it.

• This phenomenon is called the law of demand.

• Do all goods obey the Law of Demand?

• Suppose the goods are potatoes and meat,and potatoes are an inferior good.


• If price of potatoes decreases,

• substitution effect: buy more potatoes

• income effect: buy less potatoes

• If income effect > substitution effect, then potatoes are a Giffen good, a good for which a decrease in price lowers the quantity demanded.


Quantity of Potatoes

Quantity of Meat

AB

The decrease in the price of potatoes rotates the budget constraint outward….

New budget constraint

Initial budget constraint

….. Which decreases potato consumption if potatoes are a Giffen good

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Demand Theory: A Mathematical Treatment

• Maximize U(X,Y) subject to the constraint that all income is spent on the two goods

• PxX + PyY = Income (I)

• Use technique of constrained optimization (Lagrangian Method):• Describes the conditions of utility maximization

Lagrangian Method

• Used to maximize or minimize a function subject to a constraint

• Lagrangian is the function to be maximized or minimized

• λ = lagrangian multiplier• Take the utility function to be maximized and

subtract the lagrangain multiplier (λ ) multiplied by the constraint as a sum equal to zero

Lagrangian Method

• Step 1 : write the lagrangian for the problem:• U(X, Y) – λ (PxX + PyY – I)

• If we choose values of X that satisfy the budget constraint, the sum of the last term will be zero

• Differentiate this function three times with respect to X, Y and λ and equate them to zero

• This will give us the three necessary conditions for maximization

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Lagrangian Method

• We will end up with the following three conditions:• MUx – λPx = 0• MUy – λPy = 0• PxX + PyY – I = 0

• What do these mean? • MUx = λPx: Marginal Utility from consuming one

more X = a multiple (λ) of its price• MUy = λPy: Marginal Utility….

Lagrangian Method

• If we combine the first two equations (the third is the budget constraint), we get:• λ = MUx/Px = MUy/Py

• This is the equal marginal principal we already studied - MUx/MUy=Px/Py

• To optimize (maximize utility subject to a budget constraint), the consumer MUST GET THE SAME UTILITY FROM THE LAST DOLLAR SPENT ON BOTH X AND Y

• In other words, the ratio of the marginal utilities is equal to the ratio of the prices.

Marginal Utility of Income

• λ = MU of income, or marginal utility of adding one dollar to the budget

• We will see in an example how this works, but for now:• If λ = 1/100

• Then if Income increases by $1, Utility will increase by 1/100

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Example: Cobb-Douglas Utility Function

• U(X, Y) = XaY1-a

• We can express this function as linear in logs: alog(X) + (1-a)log(Y)

• These two are equivalent in that they yield identical demand functions for X and Y

Lagrangian Set-up

• alog(X) + (1-a)logY – λ(PxX +PyY – I)• Differentiating with respect to X, Y and λ, and

setting equal to zero gives three necessary conditions for a maximum

• X: a/X – λPx = 0• Y: (1-a)/Y – λPy = 0• λ: PxX + PyY – I = 0• Solve for PxX and PyY and substitute into the

third equation

Lagrangian Set-up

• Solving for PxX and PyY gives:• PxX = a/λ

• PyY = (1-a)/λ

• Now: substituting these back into the budget constraint gives:• a/λ + (1-a)/λ – I = 0

• And solving for λ gives: λ = 1/Income (I)

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Lagrangian

• If λ = 1/I then we can use λ as a function of Income to solve for X and Y using the two original conditions

• Recall:• PxX = a/λ and PyY = (1-a)/λ

• Now: PxX = a/(1/I) = Ia

• And: PyY = (1-a)I

• So: X = Ia/Px and Y = I(1-a)/Py

Lagrangian

• Notice that the demand for X is dependent on Income and the price of X, while the demand for Y is dependent on Income and the price of Y

• Demand for X, Y, NOT dependent on the price of the other good

• Cross-price elasticity is equal to zero

Meaning of Lagrangian Multiplier

• λ = Marginal Utility of an additional dollar of Income

• If λ = 1/100, then if income increases by $1, utility should increase by 1/100

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Welfare and decomposition

• When there is a change in the relative prices the consumer may loss or gain depending on whether price has increased or decreased.

• The gain or loss is expressed by the consumer moving to a higher or lower utility level.

• Thus, we would like to compensate (positively or negatively) the consumer such that he will regain his original level of utility.

Welfare and decomposition

• We will demonstrate the two major ways by which we can compensate the consumer:

1. Hicks decomposition. 2. Slutsky decomposition.

Hicks decomposition

Y

X

A

Suppose that the price of X increase (price of Y unchanged)

B

Results : A B Total EffectC

Now we add income at the new prices up to where we reach the original level of utility (point c).

Substitution effect

Income effect

AY

BY

BX AX

U1

U0

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Slutsky decomposition

Y

X

A

Suppose that the price of X increase (price of Y unchanged)

B

Results : A BC

Now we add income at the new prices up to where we reach the original basket (point A).

Substitution effect

Income effect

AY

BY

BX AX

U0

U1

At the new budget line consumer can reach yet a higher level of utility u2 and he’s now at point C

U2

CY

CX

Comparing Slutsky to Hicks

Y

X

A

Hicks Slutsky

bias

U0

U1

Summary

• A consumer’s budget constraint shows the possible combinations of different goods she can buy given her income and the prices of the goods.

• The slope of the budget constraint equals the relative price ratio of the goods.

• A consumer’s indifference curves represents his/her preferences.

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Summary

• An indifference curve shows the various bundles of goods that make the consumer equally happy (constant level of utility).

• Points on higher indifference curves are preferred to points on lower indifference.

• The slope of an indifference curve is the consumer’s MRS.

Summary

• The consumer optimizes by choosing the point on his/her budget constraint that lies on the highest indifference curve. At this point, the marginal rate of substitution equals the relative price of the two goods.

• When the price of a good falls, the impact on the consumer’s choices can be broken down into two effects, an income effect and a substitution effect.

Summary

• The income effect is the change in consumption that arises because a lower price makes the consumer better off. It is represented by a movement from a lower indifference curve to a higher one.

• The substitution effect is the change that arises because a price change encourages greater consumption of the good that has become relatively cheaper. It is represented by a .

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Lectures 4-6 - The Theodry of Consumer Choice