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ECON 500

ECON 500 –Microeconomic Theory

Consumer Theory

ECON 500

Consumer Theory

A theory of how consumers allocate incomes among different goods and

services to maximize their well-being.

Part I Preferences and Utility

Part II Constraints and Choice

Part III Comparative Statics

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Part I. Preferences and Utility

Axioms of Rational Choice

I. Completeness

II. Transitivity

III. Continuity

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I. Completeness

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 A

to B but buy only B because it is cheaper.

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I. Completeness

II. Transitivity

If a consumer prefers basket A to basket B and basket B to basket C, then

the consumer also prefers A to C. Transitivity ensures that the individual’s

choices are internally consistent.

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I. Completeness

II. Transitivity

III. Continuity

If A is preferred to B, then situations suitably “close to” A must also be

preferred to B. Enables the analysis of individuals’ responses to relatively

small changes in income and prices

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Utility

Assuming completeness, transitivity, and continuity implies that people

are able to rank all possible situations from the least desirable to the most.

This ordinal relationship can be represented by what we call a utility

function:

If A is preferred to B, then the utility assigned to A exceeds the utility

assigned to B:

U(A) > U(B)

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Utility

Individuals’ preferences are assumed to be represented by a utility

function of the form U(x1, x2, . . . , xn , Z )

Where x1, x2,…, xn are the quantities of each of n goods that might be

consumed in a period, and Z is everything else.

Ceteris Paribus assumption allows us to drop Z.

This function is unique only up to an order-preserving (monotone)

transformation

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Utility

The ordinal nature of utility and the non-uniqueness of utility functions

makes them:

Useful in analyzing only the relative desirability of choices

No so useful in analyzing absolute utility gains across bundles

Not so useful in making interpersonal comparisons

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Utility

Arguments of utility functions

U(W) = utility an individual receives from real wealth (W)

U(c,h) = utility from consumption (c) and leisure (h)

U(c1,c2) = utility from consumption in two different periods

Two-good utility function U(x,y)

More of any particular xi during some period is preferred to less

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Alternative Market Bundles

Bundle Units of Food Units of Clothing

A 20 30

B 10 50

D 40 20

E 30 40

G 10 20

H 10 40

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Indifference Curve:

Shows a set of consumption bundles that provide the consumer

with the same level of utility.

The consumer ranks these bundles equally, in other words the consumer is

indifferent about consuming any bundle on the indifference curve

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Marginal rate of substitution, MRS is measured as the negative of the

slope of an indifference curve (U1) at some point

MRS reflects the individual’s willingness to trade y for x

and it changes as x and y change

1

U U

dyMRS

dx

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MRS=?

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Indifference Curves and Transitivity

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The transitivity axiom implies that indifference curves cannot intersect.

There is a single indifference curve that passes through every point in the

goods space, in other words every bundle is associated with a unique

utility level.

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Convexity and Indifference Curves

An alternative way of stating the principle of a diminishing marginal rate

of substitution uses the mathematical notion of a convex set.

A set of points is said to be convex if any two points within the set can be

joined by a straight line that is contained completely within the set.

The assumption of a diminishing MRS is equivalent to the assumption that

all combinations of x and y that are preferred or indifferent to a particular

combination x*, y* form a convex set.

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Convexity of the indifference curve implies diminishing MRS

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Convexity and Balance in Consumption

If indifference curves are convex (if they obey the assumption of a

diminishing MRS), then the line joining any two points that are indifferent

will contain points preferred to either of the initial combinations.

Intuitively, balanced bundles are preferred to unbalanced ones.

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Suppose the utility function s of the form

U =

An indifference curve for this function that set of combinations of x

and y for which utility has the value 10 is:

10 =

Rearranging:

100=x·y

y=100/x

MRS = -dy/dx(along U1)=100/x2

As x rises, MRS falls

When x = 5, MRS = 4

When x = 20, MRS = 0.25

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MRS and Marginal Utilities

The total differential of a utility function U ( x, y )

rearranging:

MRS of x for y is equal to the ratio of the marginal utility of x (∂U/∂x)

to the marginal utility of y (∂U/∂y).

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Cobb-Douglas Utility

U(x,y) = xy

Where and are positive constants

The relative sizes of and indicate the relative importance of the goods

We can normalize + = 1 by an invariant monotonic transformation

U(x,y) = xy1-

Where =/(+) and 1-=/(+)

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Perfect substitutes

U(x,y) = x + y

Where and are positive constants

The MRS will be constant along the linear indifference curves

Perfect complements

U(x,y) = min (x, y)

Where and are positive parameters

L-shaped indifference curves

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CES Utility (constant elasticity of substitution)

U(x,y) = x/ + y/

when 1, 0 and

U(x,y) = ln x + ln y

when = 0

Perfect substitutes = 1

Cobb-Douglas = 0

Perfect complements = -

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General Cobb-Douglas Form

U(x,y) = xy

MRS depends on only the ratio of the amounts of the two goods, and not

their total quantities. This type of utility functions are called homothetic.

Slopes of the curves depend only on the ratio y=x, not on how far the

curve is from the origin. Hence, we can study the behavior of an

individual who has homothetic preferences by looking only at one

indifference curve or at a few nearby curves without fearing that our

results would change dramatically at very different levels of utility.

1

1

U x x y yMRS

U y x y x

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Many Goods

Suppose utility is a function of n goods given by

U(x1, x2,…, xn)

Indifference surface in n dimensions is defined by

U(x1, x2,…, xn)=k

MRS is then given by

1

21 2

1 22

1 1 2( , ,..., )

( , ,..., )

( , ,..., )n

x n

x nU x x x k

U x x xdxMRS

dx U x x x

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Part II. Constraints and Choice

Basic Model of Consumer Choice: Utility Maximization

This model assumes that individuals who are constrained by limited

incomes will behave as if they are using their purchasing power in such a

way as to achieve the highest utility possible.

That is, individuals are assumed to behave as if they maximize utility

subject to a budget constraint.

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Basic Model of Consumer Choice: Utility Maximization

Two Concerns:

Do individuals make the “lightning calculations” required for utility

maximization?

Is the economic model of choice is extremely selfish?

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To maximize utility, his or her preferences and a fixed amount of income

to spend an individual will buy those quantities of goods:

a) That exhaust his or her total income

b) and for which the MRS is equal to the rate at which

the goods can be traded one for the other in the marketplace

MRS (of x for y) = the ratio of the price of x to the price of y (px/py)

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Additional Assumptions

I = Budget to allocate between good x and good y

px = price of good x

py = price of good y

Budget constraint:

pxx + pyy ≤ I

Slope = -px/py

If all of I is spent on good x(y), I/px(y) units of good x(y) can be bought

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Conditions (Necessary) for a maximum

Point of tangency between the budget constraint

and the indifference curve:

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The tangency rule is necessary but not sufficient

Unless we assume that MRS is diminishing

If MRS is diminishing, then indifference curves are strictly convex and

first order conditions are necessary AND sufficient

If MRS is not diminishing, one must check second-order conditions to

ensure that we are at a maximum

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Interior vs. Corner Solutions

If the rate at which x can be traded for y in the market (px/py) is lower

(higher) than the MRS of x for y for all possible bundles, individuals may

end up maximizing utility by choosing to consume only x (y).

At the optimal point the budget constraint is flatter (steeper) than the

indifference curve

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In the case of n-goods, consumer maximizes the utility

subject to:

To find the maximum s.t. the constraint we set up the Lagrangian:

Taking the partial derivatives of Lagrangian w.r.t x1,x2,…,xn, and λ, and

setting them equal to zero yields the n+1 necessary conditions for an

interior maximum.

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First order conditions:

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Rearranging the first order conditions yields that

for any two goods xi and xj

where the left hand side is the ratio of the marginal utilities, which have

derived to be the Marginal Rate of Substitution between xi and xj

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Interpreting the Lagrange Multiplier

or

At the utility-maximizing point, each good purchased should yield the

same marginal utility per dollar spent on that good.

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Cobb-Douglas Example (α + β = 1)

Lagrangian

FOCs

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From the first two FOCs

Rearranging

Substituting in the budget constraint

Solving for optimal consumption bundle

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Indirect Utility Function

We can solve for the optimum consumption bundles from the FOCs as a

function of prices and income

We can substitute the optimum bundles back into the utility function to

find maximum utility attainable given prices and income

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Lump Sum Principle

Lump sum income taxes or subsidies are superior to good-specific taxes or

subsidies in terms of their impact on consumer well-being.

The intuition is that the lump sum transfers leave the individual free to

decide how to allocate the final income across various good.

Good-specific taxes or subsidies not only impact purchasing power but

also distort choices by artificially distorting prices.

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Duality

Utility maximization subject to a budget constraint is equivalent to

expenditure minimization subject to a utility target. This duality allows us

to employ the expenditure minimization approach in certain cases where

it is more useful due to the direct observability of expenditures.

The dual problem is to choose x1, …, xn to minimize

subject to

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Part III. Comparative Statics

Demand functions for the two goods case are

In this formulation prices and income are “exogenous” to this process, i.e.

the individual has no control at this stage of the analysis.

Changes in the parameters will shift the budget constraint and cause this

person to make different choices.

Our focus is analyzing the partial derivatives ∂x/∂I and ∂x/∂px as well as

some useful properties of these demand functions.

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Homogeneity

Functions that obey this property are said to be

homogeneous of degree zero.

If we were to double all prices and income simultaneously (or multiply

them all by any positive constant), then the optimal quantities demanded

would remain unchanged.

Changing all prices and income changes only the units by which we count,

not the “real” quantity of goods demanded.

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Changes in Income

As a person’s purchasing power rises, it is natural to expect that the

quantity of each good purchased will also increase.

Changes in income shift the budget lines in a parallel fashion, reflecting

that the relative prices of x and y remain unchanged. Because the ratio

px/py stays constant, the utility-maximizing conditions also require that the

MRS stay constant as the individual moves to higher levels of satisfaction.

However, the point of tangency might or might not be on an upward

sloping expansion path with increasing consumption levels.

If ∂xi/∂I=0 over some range of income variation then the good is a normal

(or “noninferior”) good in that range. A good xi for which ∂xi/∂I < 0 over

some range of income changes is an inferior good in that range.

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Changes in Prices

Changing a price involves changing not only the intercepts of the budget

constraint but also its slope. Therefore, when a price changes, two

analytically different effects come into play.

One of these is a substitution effect: even if the individual were to stay on

the same indifference curve, consumption patterns would be changed so as

to equate the MRS to the new price ratio.

A second effect, the income effect, arises because a price change

necessarily changes an individual’s “real” income. The individual cannot

stay on the initial indifference curve and must move to a new one.

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In the case of inferior goods, income and substitution effects work in

opposite directions, and the combined result of a price change is

indeterminate.

A fall in price, will always cause an individual to tend to consume more of

a good because of the substitution effect. But if the good is inferior, the

increase in purchasing power caused by the price decline may cause less

of the good to be bought.

Unlike the situation for normal goods, it is not possible here to

predict even the direction of the effect of a change in px on the quantity of

x consumed and if the income effect of a price change is strong enough,

the change in price and the resulting change in the quantity demanded

could actually move in the same direction, which is called Giffen’s

Paradox

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Individual’s (Uncompensated) Demand Curve

The demand curve looks at the relationship between x and px while

holding py, I, preferences and all other determinants constant.

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Compensated Demand Curve

A compensated demand curve shows the relationship between the price of

a good and the quantity purchased on the assumption that other prices and

utility are held constant.

It therefore illustrates only substitution effects and isolates the analysis

from income effects. The choice between compensated and

uncompensated demand curves depends on the application at hand.

Mathematically, the curve is a two-dimensional representation of the

compensated demand function:

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Suppose the utility function is of the form

Then the Marshallian demand functions are

The indirect utility function is then

Rearranging and substituting into the demand function reveals

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Relationship between Compensated and Uncompensated Demand

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