MICROECONOMICS�

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The Theory of Consumer Choice

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(Chapter 3)

Topics in this Chapter

Consumer Behavior

Theory of consumer behavior Description of how consumers allocate incomes among different goods and services to maximize their well-being.

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Behavioral Assumption: Rationality - Consumers choose the “best” bundle of goods available to them.

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Consumer behavior is best understood in three distinct steps:

1. Budget constraints
2. Consumer preferences
3. Optimal Consumer choices

Budget Constraint�

• Denote the commodities available to the consumer by:

X₁, X₂, … , X_n

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• A consumption bundle containing x₁ units of commodity 1, x₂ units of commodity 2 and so on up to x_n units of commodity n is denoted by the vector

(x₁, x₂, … , x_n).

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• Commodity prices are (p₁, p₂, … , p_n).
• Q: When is a consumption bundle (x₁, … , x_n) affordable at given prices p₁, … , p_n?
• A: When the cost of the bundle does not exceed consumer’s (disposable) income (I):

� Budget Constraint: p₁x₁ + p₂x₂ + … + p_nx_n ≤ I�

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The Budget Constraint

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• Budget Set: the set of all affordable bundles:

(x₁, x₂, … , x_n) which satisfy

p₁x₁ + p₂x₂ + … + p_nx_n ≤ I�

• Budget line - a line that shows the combinations of goods that can be purchased at the specified prices and assuming that all of the consumer’s income is expended

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Budget line: p₁x₁ + p₂x₂ + … + p_nx_n = I

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x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₁

I /p₂

x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₂

I /p₁

Budget Constraints

• For n = 2 and x₁ on the horizontal axis, the constraint’s slope is -p₁/p₂. What does it mean?

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• Increasing x₁ by 1 must reduce x₂ by p₁/p_2.
• The constraint’s slope -p₁/p₂ is known as the Marginal Rate of Transformation (MRT): it captures the trade-off the market imposes on the consumer in terms of the amount of good 2 he has to give up to obtain more of good 1�

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x₂

x₁

+1

-p₁/p₂

Opportunity cost of an extra unit of� commodity 1 is p₁/p₂ units� foregone of commodity 2.

BUDGET CONSTRAINTS

• The Budget Line

A budget line describes the combinations of goods that can be purchased given the consumer’s income and the prices of the goods.

Line AG (which passes through points B, D, and E) shows the budget associated with an income of $80, a price of food of P_F = $1 per unit, and a price of clothing of P_C = $2 per unit.

The slope of the budget line (measured between points B and D) is −P_F/P_C = −10/20 = −1/2.

A Budget Line

(3.2)

x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₁

Just affordable

I /p₂

x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₁

Just affordable

Not affordable

I /p₂

x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₁

Affordable

Just affordable

Not affordable

I /p₂

Budget Set

x₂

x₁

Budget line is

p₁x₁ + p₂x₂ = I.

I /p₁

Budget

Set

the collection� of all affordable bundles.

I /p₂

Budget Constraint for Three Commodities

x₂

x₁

x₃

I /p₂

I /p₁

I /p₃

p₁x₁ + p₂x₂ + p₃x₃ = I

Budget Set for Three Commodities

x₂

x₁

x₃

I /p₂

I /p₁

I /p₃

Budget set

The budget constraint

• The position of the budget constraint depends on

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​

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• The income of the agent (I)

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• The price of the two goods (p₁ and p₂)

How do the budget set and budget constraint change as income I increases?

Original

budget set

x₂

x₁

Original

budget set

New affordable consumption�choices

x₂

x₁

Original and

new budget

constraints are

parallel (same

slope).

BUDGET CONSTRAINTS

• The Effects of Changes in Income and Prices

Income Changes A change in income (with prices unchanged) causes the budget line to shift parallel to the original line (L₁).

When the income of $80 (on L₁) is increased to $160, the budget line shifts outward to L₂.

If the income falls to $40, the line shifts inward to L₃.

Effects of a Change in Income on the Budget Line

How do the budget set and budget constraint change as p₁ decreases from p₁’ to p₁”?

Original

budget set

x₂

x₁

I/p₂

I/p₁’

I/p₁”

-p₁’/p₂

Original

budget set

x₂

x₁

I/p₂

I/p₁’

I/p₁”

New affordable choices

-p₁’/p₂

Original

budget set

x₂

x₁

I/p₂

I/p₁’

I/p₁”

New affordable choices

Budget constraint

pivots; slope flattens

from -p₁’/p₂ to

-p₁”/p₂

-p₁’/p₂

-p₁”/p₂

BUDGET CONSTRAINTS

• The Effects of Changes in Income and Prices

Price Changes A change in the price of one good (with income unchanged) causes the budget line to rotate about one intercept.

When the price of food falls from $1.00 to $0.50, the budget line rotates outward from L₁ to L₂.

However, when the price increases from $1.00 to $2.00, the line rotates inward from L₁ to L₃.

Effects of a Change in Price on the Budget Line

Shifts in Budget Lines

• Two underlying factors:
• Income Changes
• A change in income with constant prices produces a parallel shift in the budget line.
• Price Changes
• A change in the price of one good, with income and the other good’s price remaining unchanged, causes the budget line to rotate about one of the intercepts.
• Indicative of change in real or relative prices

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Example:�������

Hurley’s income: $1200

Prices: P_F = $4 per fish, P_M = $1 per mango

A. If Hurley spends all his income on fish, �how many fish does he buy?

B. If Hurley spends all his income on mangos, �how many mangos does he buy?

C. If Hurley buys 100 fish, how many mangos can he buy?

D. Plot each of the bundles from parts A – C on a graph that measures fish on the horizontal axis and mangos on the vertical, connect the dots.

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�Answers

A. $1200/$4�= 300 fish

B. $1200/$1�= 1200 mangos

C. 100 fish cost $400,�$800 left buys 800 mangos

Quantity of Fish

Quantity of Mangos

D. Hurley’s budget constraint shows the bundles he can afford.

The Slope of the Budget Constraint

THE THEORY OF CONSUMER CHOICE

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Quantity of Fish

Quantity of Mangos

From C to D,

“rise” =�–200 mangos

“run” = �+50 fish

Slope = – 4

Hurley must �give up �4 mangos �to get one fish.

Example:�

Show what happens to Hurley’s budget constraint if:

A. His income falls to $800.

B. The price of mangos rises to �P_M = $2 per mango

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Answers, part A�

Now, �Hurley �can buy

$800/$4�= 200 fish

or�$800/$1�= 800 mangos

or any combination in between.

Quantity of Fish

Quantity of Mangos

A fall in income shifts the budget constraint down.

Answers, part B

Hurley �can still buy �300 fish.

But now he �can only buy $1200/$2 = �600 mangos.

Notice: �slope is smaller, relative price of fish is now only 2 mangos.

Quantity of Fish

Quantity of Mangos

An increase in the price of one good pivots the budget constraint inward.

The Food Stamp Program

• Food stamps are coupons that can be legally exchanged only for food.
• How does a commodity-specific gift such as a food stamp alter a family’s budget constraint?

The Food Stamp Program

• Suppose m = $100, p_F = $1 and the price of “other goods” is p_G = $1.
• The budget constraint is then� F + G =100.

The Food Stamp Program

G

F

100

100

F + G = 100; before stamps.

G

F

100

100

F + G = 100: before stamps.

G

F

100

100

F + G = 100: before stamps.

Budget set after 40 food�stamps issued.

140

40

G

F

100

100

F + G = 100: before stamps.

Budget set after 40 food�stamps issued.

140

The family’s budget�set is enlarged.

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• What if food stamps can be traded on a black market for $0.50 each?

G

F

100

100

F + G = 100: before stamps.

Budget constraint after 40� food stamps issued.

140

120

Budget constraint with� black market trading.

40

G

F

100

100

F + G = 100: before stamps.

Budget constraint after 40� food stamps issued.

140

120

Black market trading� makes the budget� set larger again.

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Budget Constraints - Relative Prices

• “Numeraire” means “unit of account”.
• Suppose prices and income are measured in dollars. Say p₁=$2, p₂=$3, m = $12. Then the budget line is� 2x₁ + 3x₂ = 12.

Budget Constraints - Relative Prices

• If prices and income are measured in cents, then p₁=200, p₂=300, m=1200 and the budget line is � 200x₁ + 300x₂ = 1200,�the same as� 2x₁ + 3x₂ = 12.
• Changing the units of measurement changes neither the budget constraint nor the budget set.

Budget Constraints - Relative Prices

• The constraint for p₁=2, p₂=3, m=12� 2x₁ + 3x₂ = 12 �is also 1.x₁ + (3/2)x₂ = 6,�the constraint for p₁=1, p₂=3/2, m=6. Setting p₁=1 makes commodity 1 the numeraire and defines all prices relative to p₁; 3/2 is the price of commodity 2 relative to the price of commodity 1.

Shapes of Budget Constraints

• But what if prices are not constants?
• Ex: bulk buying discounts, or price penalties for buying “too much”?

Quantity Discounts

• Suppose p₂ is constant at $1 but that p₁=$2 for 0 ≤ x₁ ≤ 20 (for the first 20 units purchased) and p₁=$1 for x₁>20 (for the 21^st unit and beyond).

Quantity Discounts

• Suppose p₂ is constant at $1 but that p₁=$2 for 0 ≤ x₁ ≤ 20 (for the first 20 units purchased) and p₁=$1 for x₁>20 (for the 21^st unit and beyond). Then the constraint’s slope is� - 2, for 0 ≤ x₁ ≤ 20�-p₁/p₂ = � - 1, for x₁ > 20�and the constraint is

{

m = $100

50

100

20

Slope = - 2 / 1 = - 2� (p₁=2, p₂=1)

Slope = - 1/ 1 = - 1� (p₁=1, p₂=1)

80

x₂

x₁

m = $100

50

100

20

Slope = - 2 / 1 = - 2� (p₁=2, p₂=1)

Slope = - 1/ 1 = - 1� (p₁=1, p₂=1)

80

x₂

x₁

m = $100

50

100

20

80

x₂

x₁

Budget Set

Budget Constraint

Shapes of Budget Constraints - One Price Negative

• Commodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p₁ = - $2. p₂ = $1. Income, other than from accepting commodity 1, is m = $10.
• Then the constraint is� - 2x₁ + x₂ = 10 or x₂ = 2x₁ + 10.

Shapes of Budget Constraints - One Price Negative

10

Budget constraint’s slope is

-p₁/p₂ = -(-2)/1 = +2

x₂

x₁

x₂ = 2x₁ + 10

10

x₂

x₁

Budget set is� all bundles for� which x₁ ≥ 0,�x₂ ≥ 0 and

x₂ ≤ 2x₁ + 10.

More General Choice Sets

• Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints.
• A bundle is available only if it meets every constraint.

More General Choice Sets

Food

Other Goods

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10

At least 10 units of food

must be eaten to survive

More General Choice Sets

Food

Other Goods

10

Budget Set

Choice is also budget�constrained.

More General Choice Sets

Food

Other Goods

10

Choice is further restricted by a time constraint.

More General Choice Sets

Food

Other Goods

10

More General Choice Sets

Food

Other Goods

10

More General Choice Sets

Food

Other Goods

10

The choice set is the

intersection of all of

the constraint sets.

CONSUMER PREFERENCES

• Some Basic Assumptions about Preferences

• Completeness: Preferences are assumed to be complete. In other words, consumers can compare and rank all possible baskets. Thus, for any two market baskets A and B, a consumer will prefer A to B, will prefer B to A, or will be indifferent between the two. By indifferent we mean that a person will be equally satisfied with either basket.��Note that these preferences ignore costs. A consumer might prefer steak to hamburger but buy hamburger because it is cheaper.

Given the choice between 2 bundles of goods a consumer either

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Prefers bundle A to bundle B: A  B.

Prefers bundle B to bundle A: A  B.

Is indifferent between the two: A ~ B.

CONSUMER PREFERENCES

• Some Basic Assumptions about Preferences

• Transitivity: Preferences are transitive. Transitivity means that if a consumer prefers basket A to basket B and basket B to basket C, then the consumer also prefers A to C. Transitivity is normally regarded as necessary for consumer consistency.

• More is better than less (monotonicity): Goods are assumed to be desirable—i.e., to be good. Consequently, consumers always prefer more of any good to less. In addition, consumers are never satisfied or satiated; more is always better, even if just a little better. This assumption is made for pedagogic reasons; namely, it simplifies the graphical analysis. Of course, some goods, such as air pollution, may be undesirable, and consumers will always prefer less. We ignore these “bads” in the context of our immediate discussion.

CONSUMER PREFERENCES

• Indifference Curves

Describing Individual Preferences

Because more of each good is preferred to less, we can compare market baskets in the shaded areas. Basket A is clearly preferred to basket G, while E is clearly preferred to A.

However, A cannot be compared with B, D, or H without additional information.

CONSUMER PREFERENCES

• Indifference Curves

The indifference curve U₁ that passes through market basket A shows all baskets that give the consumer the same level of satisfaction as does market basket A; these include baskets B and D. �

An Indifference Curve

● indifference curve Curve representing all combinations of market baskets that provide a consumer with the same level of satisfaction.

Our consumer prefers basket E, which lies above U₁, to A, but prefers A to H or G, which lie below U₁.

CONSUMER PREFERENCES

• Indifference Maps

An indifference map is a set of indifference curves that describes a person's preferences.

An Indifference Map

● indifference map Graph containing a set of indifference curves showing the market baskets among which a consumer is indifferent.

Any market basket on indifference curve U₃, such as basket A, is preferred to any basket on curve U₂ (e.g., basket B), which in turn is preferred to any basket on U₁, such as D.

More Is Better!

• More Is Better Property
• Bundles that have at least as much of every good and more of some good are preferred to other bundles.
• Bundle B is preferred to A since B contains at least as much of good Y and strictly more of good X.
• Bundle B is also preferred to C since B contains at least as much of good X and strictly more of good Y.
• More generally, all bundles on IC_III are preferred to bundles on IC_II or IC_I. And all bundles on IC_II are preferred to IC_I.

I.

II.

III.

Good Y

Good X

A

C

B

CONSUMER PREFERENCES

• Indifference Maps

If indifference curves U₁ and U₂ intersect, one of the assumptions of consumer theory is violated.

Indifference Curves Cannot Intersect

According to this diagram, the consumer should be indifferent among market baskets A, B, and D. Yet B should be preferred to D because B has more of both goods.

Marginal Rate of Substitution

• “MRS”
• A measure of a consumer’s willingness to trade one good for another

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• The maximum number of one good the consumer is willing to give up to obtain one more of another good

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• Depends upon the initial bundle

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• Equal to the slope of an indifference curve

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CONSUMER PREFERENCES

• The Marginal Rate of Substitution

The magnitude of the slope of an indifference curve measures the consumer’s marginal rate of substitution (MRS) between two goods.

The Marginal Rate of Substitution

In this figure, the MRS between clothing (C) and food (F) falls from 6 (between A and B) to 4 (between B and D) to 2 (between D and E) to 1 (between E and G).

Convexity The decline in the MRS reflects a diminishing marginal rate of substitution. When the MRS diminishes along an indifference curve, the curve is convex.

● marginal rate of substitution (MRS) Maximum amount of a good that a consumer is willing to give up in order to obtain one additional unit of another good.

Convexity

x₂

y₂

x₁

y₁

x

y

Preferences are strictly convex� when all mixtures z � are strictly� preferred to their� component � bundles x and y.

z

Non-Convex Preferences

x₂

y₂

x₁

y₁

z

Better

The mixture z�is less preferred

than x or y.

More Non-Convex Preferences

x₂

y₂

x₁

y₁

z

Better

The mixture z�is less preferred

than x or y.

Categories of Goods

• “Good” – when more is preferred to less
• “Bad” – when less is preferred to more
• “Neuter” – when the consumer does not care about a particular good
• Perfect Substitutes – when a consumer is willing to substitute one good for another at some constant rate and remain equally well off
• Perfect Complements – when goods must be consumed in a precise combination in order for the consumer to remain equally well off

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CONSUMER PREFERENCES

• Perfect Substitutes and Perfect Complements

In (a), Bob views orange juice and apple juice as perfect substitutes: He is always indifferent between a glass of one and a glass of the other.

Perfect Substitutes and Perfect Complements

In (b), Jane views left shoes and right shoes as perfect complements: An additional left shoe gives her no extra satisfaction unless she also

obtains the matching right shoe.

Utility Function

Definition:�A utility function U(x) is a mapping assigning a unique number to every possible bundle x, such that more preferred bundles get assigned larger numbers.

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• A utility function U(x) represents preferences if and only if:

x’ ≻ x” ⬄ U(x’) > U(x”)�� x’ ~ x” ⬄ U(x’) = U(x”)

• Utility is an ordinal (i.e. ordering) concept.
• Ex: if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.

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�

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• Example: Consider the bundles (4,1), (2,3) and (2,2).

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• Suppose (2,3) ≻ (4,1) ~ (2,2)
• Assign to these bundles any numbers that preserve the preference ordering
• Ex: U(2,3) = 6 > U(4,1) = U(2,2) = 4

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��

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U ≡ 6

U ≡ 4

x₁

x₂

p

U(2,3) = 6

U(2,2) = 4 �U(4,1) = 4

3D plot of consumption & utility levels for 3 bundles

x₁

x₂

Utility

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

x₂

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

Utility Functions & Indiff. Curves

x₁

x₂

Preferences, Utility, and Indifference Curves�

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• The fundamental construct describing consumer behavior are preferences.
• Preferences can be represented using a utility function:
• Theorem: A preference relation that is complete,transitive and continuous can be represented by a continuous utility function
• The complete map of all indifference curves is equivalent to the utility function it represents.

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​

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Perfect Substitutes Utility Functions�

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Consider�� U(x₁,x₂) = x₁ + x₂.

�What do the indifference curves for this utility function look like?

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​

​

Perfect Substitutes

5

5

9

9

13

13

x₁

x₂

x₁ + x₂ = 5

x₁ + x₂ = 9

x₁ + x₂ = 13

U(x₁,x₂) = x₁ + x₂.

Perfect Complements Utility Functions�

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Consider:�� W(x₁,x₂) = min{x₁,x₂}.��What do the indifference curves for this utility function look like?

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​

​

Perfect Complements

x₂

x₁

45^o

min{x₁,x₂} = 8

3

5

8

3

5

8

min{x₁,x₂} = 5

min{x₁,x₂} = 3

W(x₁,x₂) = min{x₁,x₂}

Quasi-linear Utility�

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A utility function of the form�� U(x₁,x₂) = f(x₁) + x₂��is linear in just x₂ and is called quasi-linear.

​

Ex: U(x₁,x₂) = 2x₁^1/2 + x₂.

​

​

Quasi-linear Indifference Curves

x₂

x₁

Each curve is a vertically shifted copy of the others.

Cobb-Douglas Utility Function�

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A utility function of the form�� U(x₁,x₂) = x₁^a x₂^b��with a > 0 and b > 0 is called a Cobb-Douglas utility function.

​

Ex: U(x₁,x₂) = x₁^1/2 x₂^1/2 (a = b = 1/2)� V(x₁,x₂) = x₁ x₂³ (a = 1, b = 3)

​

​

Cobb-Douglas Indifference Curves

x₂

x₁

All curves are hyperbolic,�asymptotic to, but never�touching either axis.

Marginal Utility�

​

​

​

​

Definition: The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes while holding the consumption of all other commodities fixed

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The marginal utility is the partial derivative of the utility function with respect to consumption of commodity i

​

​

​

​

Marginal Utilities

So, if U(x₁,x₂) = x₁^1/2 x₂² then

Marginal Utilities and Marginal Rates-of-Substitution

The general equation for an indifference curve is� U(x₁,x₂) ≡ k, a constant.

�Totally differentiating this identity gives���

Marginal Utilities and Marginal Rate of Substitution

This is the MRS:

MRS Example:

Suppose U(x₁,x₂) = x₁x₂. Then��

so

MRS Example

MRS(1,8) = - 8/1 = -8� MRS(6,6) = - 6/6 = -1.

x₁

x₂

8

6

1

6

U = 8

U = 36

U(x₁,x₂) = x₁x₂;

The optimal consumer choice

• This requires bringing in the two elements of the theory
• The indifference curves, which show how agents rank the different bundles
• The budget constraint, which shows which bundles are affordable, and which are not

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• Both of these are defined over the “consumption space”

The optimal consumer choice

The optimal consumer choice

Recall, the slope of an indifference curve is:

Slide 116

Further, the slope of the budget line is:

The optimal consumer choice

• Therefore, it can be said that satisfaction is maximized where:

Chapter 3: Consumer Behavior

Slide 117

The optimal consumer choice

• The optimal bundle is on the tangency between the budget constraint and the indifference curve.

​

• This means that for the optimal bundle the slope of the IC is equal to the slope of the budget constraint

​

MRS = ratio of prices

​

The optimal consumer choice

119

Quantity �of Fish

Quantity �of Mangos

​

​

1200

600

300

150

A is the optimum: �the point on the budget constraint that touches the highest possible indifference curve.

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

​

​

​

​

Hurley can afford C and D, �but A is on a higher indifference curve.

The optimal consumer choice

120

Quantity �of Fish

Quantity �of Mangos

​

​

1200

600

300

150

At the optimum,

slope of the indifference curve equals �slope of the budget constraint:

MRS = P_F/P_M

Consumer Optimization Problem

• Utility Maximization:�Maximize U(x₁,x₂) subject to p₁x₁ + p₂x₂ ≤ m
• The choice variables are the quantities x₁ and x₂
• The parameters of this problem are:
• Prices p₁ and p₂
• Income m
• The utility function U
• The solution to this problem will be denoted by (x₁*,x₂*) and is called the consumer’s Ordinary Demand.

Consumer Optimization Problem

x₁

x₂

x₁*

x₂*

(x₁*,x₂*) is the most�preferred affordable�bundle.

Interior Solutions

• Interior Solutions
• Solutions in which the Ordinary Demand for both goods is positive: x₁* > 0 and x₂* > 0
• Typically occur when preferences are monotonic and strictly convex
• Solving for an interior solution:
• Occurs at a point where one of the indifference curves is tangent to the budget line, i.e. equal slopes MRS=MRT
• Occurs at a point ON the budget line, i.e. p₁x₁ + p₂x₂ = m
• These are two equations in two unknowns
• The Solution is Ordinary Demand:
• x₁*(p₁,p₂,m)
• x₂*(p₁,p₂,m)
• Note that the solution is a function of the parameters. If the parameters change, then so will the optimal bundle.

​

​

�

Interior Solutions – Cobb-Douglas

Interior Solutions – Cobb-Douglas

x₁

x₂

OPTIMAL CONSUMER CHOICE

• Corner Solutions

When the consumer’s marginal rate of substitution is not equal to the price ratio for all levels of consumption, a corner solution arises. The consumer maximizes satisfaction by consuming only one of the two goods.

Given budget line AB, the highest level of satisfaction is achieved at B on indifference curve U₁, where the MRS (of ice cream for frozen yogurt) is greater than the ratio of the price of ice cream to the price of frozen yogurt.

A Corner Solution

● corner solution Situation in which the marginal rate of substitution of one good for another in a chosen market basket is not equal to the slope of the budget line.

Corner Solutions

• Corner Solutions
• Solutions in which the Ordinary Demand for at least one of the two goods is equal to 0: x₁* = 0 or x₂* = 0 (or both)
• Typically occur when preferences violate monotonicity or strict convexity
• Solving for a corner solution:
• We can no longer use the system of equations from the interior solution case
• In most cases we can obtain the solution graphically, but there is no general rule that always works
• Example: Perfect Substitutes
• Maximize U(x₁,x₂) = x₁ + x₂ subject to p₁x₁ + p₂x₂ ≤ m

​

​

​

�

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Slope = -p₁/p₂ with p₁ > p₂.

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Slope = -p₁/p₂ with p₁ > p₂.

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Slope = -p₁/p₂ with p₁ > p₂.

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Slope = -p₁/p₂ with p₁ < p₂.

Corner Solutions

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

MRS = -1

Slope = -p₁/p₂ with p₁ = p₂.

Examples of Corner Solutions -- the Perfect Substitutes Case

x₁

x₂

All the bundles in the �constraint are equally the� most preferred affordable� when p₁ = p₂.

Examples of Corner Solutions -- the Non-Convex Preferences Case

x₁

x₂

Better

Examples of Corner Solutions -- the Non-Convex Preferences Case

x₁

x₂

Examples of Corner Solutions -- the Non-Convex Preferences Case

x₁

x₂

Which is the most preferred�affordable bundle?

Examples of Corner Solutions -- the Non-Convex Preferences Case

x₁

x₂

The most preferred�affordable bundle

Examples of Corner Solutions -- the Non-Convex Preferences Case

x₁

x₂

The most preferred�affordable bundle

Notice that the “tangency solution”�is not the most preferred affordable�bundle.

Other Solutions

• Non-standard Interior Solutions
• Perfect Complements
• Maximize U(x₁,x₂) = min{ax₁,x₂} subject to
• p₁x₁ + p₂x₂ ≤ m
• In this case we will have an interior solution, but the standard condition MRS=MRT will not work.
• This occurs because the slope of the Indifference Curves is not well defined at all points.

​

​

​

�

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁

x₂

MRS = 0

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁

x₂

MRS = -

∞

MRS = 0

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁

x₂

MRS = -

∞

MRS = 0

MRS is undefined

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

Which is the most�preferred affordable bundle?

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

The most preferred

affordable bundle

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁*

x₂*

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁*

x₂*

(a) p₁x₁* + p₂x₂* = m

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

x₁*

x₂*

(a) p₁x₁* + p₂x₂* = m�(b) x₂* = ax₁*

Other Solutions

x₁

x₂

U(x₁,x₂) = min{ax₁,x₂}

x₂ = ax₁

The Mathematics behind Consumer Choice*

Explain the mathematics behind consumer choice.

155

The Mathematics Behind Consumer Choice

• Preferences of the Consumer:
• The slope of an indifference curve (MRS) equals (minus) the ratio of the marginal utilities.
• The Budget Constraint:
• The slope of the budget line equals the negative of the price ratio.
• The Consumer’s Choice:
• To maximize utility, the ratio of marginal utilities equals the ratio of prices
• MRS = slope of the budget line
• The consumer’s choice must lie on the budget line.

159

Preferences of the Consumer

160

The Budget Constraint

161

The Consumer’s Choice

162

MICROECONOMICS�

​

​

Individual and Market Demand

​

Chapter 4

Topics in this Chapter

Properties of Demand Functions

​

• In the previous chapter we learned how to derive Ordinary Demand functions:
• x₁*(p₁,p₂,m) and x₂*(p₁,p₂,m)
• These functions give us the quantity demanded for both goods as functions of the parameters p₁,p₂,m�
• “Comparative statics” analysis of ordinary demand functions -- the study of how ordinary demands x₁*(p₁,p₂,m) and x₂*(p₁,p₂,m) change as prices p₁, p₂ and income m change.
• Comparative means that we want to compare the level of demand before and after one of those variables changes
• Static refers to the fact that we are only comparing one equilibrium outcome to another; we are not interested in the adjustment process involved

​

Own-price Changes�

​

• Q: How does x₁*(p₁,p₂,m) change as p₁ changes, holding p₂ and m constant?
• The relationship between the quantity demanded of a good x₁* and the price of that good p₁ is known as the “Demand Curve”
• We can derive the demand curve from the demand function
• To achieve this we need to hold the other parameters constant

​

​

• Suppose only p₁ increases, from p₁’ to p₁’’ and then to p₁’’’

​

​

​

��

​

x₁

x₂

p₁ = p₁’

Fixed p₂ and m.

p₁x₁ + p₂x₂ = m

Own-Price Changes

x₁

x₂

p₁= p₁’’

p₁ = p₁’

Fixed p₂ and m.

p₁x₁ + p₂x₂ = m

x₁

x₂

p₁= p₁’’

p₁= p₁’’’

Fixed p₂ and m.

p₁ = p₁’

p₁x₁ + p₂x₂ = m

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’)

Own-Price Changes

p₁ = p₁’

Fixed p₂ and m.

x₁*(p₁’)

p₁

x₁*(p₁’)

p₁’

x₁*

Own-Price Changes

Fixed p₂ and m.

p₁ = p₁’

x₁*(p₁’)

p₁

x₁*(p₁’)

p₁’

p₁ = p₁’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

p₁’

p₁ = p₁’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’)

p₁’

p₁’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’)

p₁’

p₁’’

p₁ = p₁’’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’)

p₁’

p₁’’

p₁ = p₁’’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’’)

x₁*(p₁’’)

p₁’

p₁’’

p₁’’’

x₁*

Own-Price Changes

Fixed p₂ and m.

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’’)

x₁*(p₁’’)

p₁’

p₁’’

p₁’’’

x₁*

Own-Price Changes

�demand curve�for commodity 1

Fixed p₂ and m.

A reduction in the price of food, with income and the price of clothing fixed, causes this consumer to choose a different market basket.

In (a), the baskets that maximize utility for various prices of food (point A, $2; B, $1; D, $0.50) trace out the price-consumption curve.

Part (b) gives the demand curve, which relates the price of food to the quantity demanded. (Points E, G, and H correspond to points A, B, and D,

respectively).

Derivation of the Individual Consumer’s Demand Curve

Own-price Changes�

​

• The Demand Curve plots for each price p₁ the corresponding value of x₁*(p₁,p₂,m) while holding p₂ and m constant at some specified values

​

• Another way to summarize the same information is via the so-called Price Offer Curve (your textbook calls it the price-consumption curve):
• The curve containing all the utility-maximizing bundles traced out as p₁ changes, with p₂ and m constant, is the p₁- price offer curve.

​

​

��

​

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*(p₁’)

x₁*(p₁’’’)

x₁*(p₁’’)

p₁’

p₁’’

p₁’’’

x₁*

Own-Price Changes

�demand curve�for commodity 1

p₁ price� offer� curve

Fixed p₂ and m.

Ordinary and Giffen Goods�

Giffen Good

Fixed p₂ and m.

x₁

x₂

Giffen Good

Fixed p₂ and y.

x₁

x₂

p₁ price offer

curve

Giffen Good

Fixed p₂ and y.

x₁

x₂

p₁ price offer

curve

x₁*

Demand curve has� a positively

sloped part

Good 1 is�Giffen

⇔

p₁

Inverse Demand Curve�

Own-Price Changes

p₁

x₁*

p₁’

Given p₁’, what quantity is�demanded of commodity 1?

Own-Price Changes

p₁

x₁*

p₁’

Given p₁’, what quantity is�demanded of commodity 1?�Answer: x₁’ units.

x₁’

Own-Price Changes

p₁

x₁*

x₁’

Given p₁’, what quantity is�demanded of commodity 1?�Answer: x₁’ units.

The inverse question is:�Given x₁’ units are� demanded, what is the� price of� commodity 1?

Own-Price Changes

p₁

x₁*

p₁’

x₁’

Given p₁’, what quantity is�demanded of commodity 1?�Answer: x₁’ units.

The inverse question is:�Given x₁’ units are� demanded, what is the� price of� commodity 1?

Answer: p₁’

Cobb-Douglas�

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

Cobb-Douglas

Fixed p₂ and m.

x₁*(p₁’’’)

x₁*(p₁’)

x₁*(p₁’’)

p₁

x₁*

Cobb-Douglas

�demand curve�for commodity 1� is

Fixed p₂ and m.

Cobb-Douglas�

Perfect Complements�

p₁

x₁*

�demand curve�for commodity 1� is

Fixed p₂ and m.

Own-Price Changes

x₁

x₂

p₁’

p₁’’

p₁’’’

m/p₂

Perfect Substitutes�

Fixed p₂ and m.

Own-Price Changes

x₂

x₁

p₁

x₁*

Fixed p₂ and .

p₁< p₂

​

p₁ = p₂

p₁ > p₂

​

p₁ price� offer

curve

�demand curve�for commodity 1

Income Changes

An increase in income, with the prices of all goods fixed, causes consumers to alter their choice of market baskets.

In part (a), the baskets that maximize consumer satisfaction for various incomes (point A, $10; B, $20; D, $30) trace out the income-consumption curve.

The shift to the right of the demand curve in response to the increases in income is shown in part (b). (Points E, G, and H correspond to points A, B, and D, respectively.)

Income Changes�

​

• Q: How does x₁*(p₁,p₂,m) change as m changes, holding p₁ and p₂ constant?

​

• The relationship between the quantity demanded of a good x₁* and the level of income m is known as the “Engel Curve”

​

• We can derive the Engel Curve from the demand function

​

• The curve containing all the utility-maximizing bundles traced out as m changes, with p₁ and p₂ constant, is the income offer curve (income-consumption curve)

​

​

​

​

​

��

​

Income Changes

Fixed p₁ and p₂.

m’ < m’’< m’’’

x₁’’’

x₁’’

x₁’

x₂’’’

x₂’’

x₂’

Income�offer curve

x₁*

x₂*

m

m

x₁^’’’

x₁’’

x₁’

x₂’’’

x₂’’

x₂’

m’

m’’

m’’’

m’

m’’

m’’’

Engel�curve;

good 2

Engel�curve;

good 1

Cobb-Douglas�

Cobb-Douglas

m

m

x₁*

x₂*

Engel curve�for good 1

Engel curve�for good 2

Perfect Complements�

Perfect Substitutes�

Perfect Substitutes

m

m

x₁*

x₂*

0

Engel curve�for good 1

Engel curve�for good 2

Homothetic Preferences�

​

• Is it a coincidence that all the Engel curves we have seen so far are straight lines?
• No, Engel Curves are straight lines when preferences are Homothetic
• Definition: Preferences are called Homothetic if whenever
• (x₁,x₂) ~ (y₁,y₂)
• It also follows that (kx₁,kx₂) ~ (ky₁,ky₂) for any positive number k

​

• Interpretation: if I am indifferent between two bundles, then after scaling both bundles by a factor k (making them “k times larger”), then I am still indifferent between the new bundles

​

• When Preferences are Homothetic the consumer’s indifference curves have the same slope along each line from the origin�

​

​

​

​

��

​

Income Changes: Quasilinear Utility (non-Homothetic)

x₁*

x₂*

m

m

x₁

~

Engel�curve

for�good 2

Engel�curve

for�good 1

x₂

x₁

Income Effects�

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

Income�offer curve

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

x₁*

m

Engel curve�for good 1

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior

x₂

x₁

x₁*

x₂*

m

m

Engel curve�for good 2

Engel curve�for good 1

Income Changes: Inferior Goods

An increase in a person’s income can lead to less consumption of one of the two goods being purchased.

Here, hamburger, though a normal good between A and B, becomes an inferior good when the income-consumption curve bends backward between B and C.

Cross-price Effects�

Cross-price Effects

A perfect-complements example:

so

Therefore commodity 1 is a gross�complement for commodity 2.

Cross-price Effects

A Cobb-Douglas example:

so

Therefore commodity 2 is neither a gross�complement nor a gross substitute for�commodity 1.

Income and Substitution Effects of a Price Change

• Income effect – a change in a consumer’s real purchasing power brought about by a change in the price of a good.

​

• Substitution effect – an incentive to increase consumption of a good whose price falls, at the expense of other, now relatively more expensive, goods.

233

INCOME AND SUBSTITUTION EFFECTS NORMAL GOOD

A fall in the price of a good has two effects:

​

• Consumers will tend to buy more of the good that has become cheaper and less of those goods that are now relatively more expensive.

​

• Because one of the goods is now cheaper, consumers enjoy an increase in real purchasing power.

INCOME AND SUBSTITUTION EFFECTS NORMAL GOOD

A decrease in the price of food has both an income effect and a substitution effect.

The consumer is initially at A, on budget line RS. When the price of food falls, consumption increases by F₁F₂ as the consumer moves to B.

The substitution effect F₁E (associated with a move from A to D) changes the relative prices of food and clothing but keeps utility (satisfaction) constant.

The income effect EF₂ (associated with a move from D to B) keeps relative prices constant but increases purchasing power.

Food is a normal good because the income effect EF₂ is positive.

INCOME AND SUBSTITUTION EFFECTS NORMAL GOOD

• substitution effect Change in consumption of a good associated with a change in its price, with the level of utility held constant.

• income effect Change in consumption of a good resulting from an increase in purchasing power, with relative prices held constant.

Total Effect (F₁F₂) = Substitution Effect (F₁E) + Income Effect (EF₂)

The total effect of a change in price is given theoretically by the sum of the substitution effect and the income effect:

Income and Substitution Effects: Inferior Goods

​

Inferior Goods = Negative Income effects

​

​

Two possibilities:

• Substitution effect > income effect
• Demand curve slopes downward

​

• Income effect > substitution effect
• Demand curve slopes upward
• Giffen Good

237

INCOME AND SUBSTITUTION EFFECTS: INFERIOR GOOD

The consumer is initially at A on budget line RS.

With a decrease in the price of food, the consumer moves to B.

The resulting change in food purchased can be broken down into a substitution effect, F₁E (associated with a move from A to D), and an income effect, EF₂ (associated with a move from D to B).