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Economics 101—Lecture 3The Basic Model o Consumer Choice I

George J. Mailath

January 20, 2011

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Consumer Preerences

A consumption bundle:

x = (x 1, x 2, x 3, . . . , x n )

where x i is the amount o good i consumed (e.g., books,

pizzas, holidays, ancy dinners, operas, movies).

In many examples, n = 2 (easier to draw pictures), and

then write (x , y ) or a consumption bundle.

I consumer (Bruce) preers a bundle x to another bundlex , write x x .

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“Rational” Choice

Plausible assumptions on behavior:

1 preerences are complete: Bruce can compare any two

bundles, i.e., or all x and x , either x x , or x x , or

both (i.e., Bruce is indierent between x and x , written

x x ). I Bruce strictly preers x to x (i.e., x x but not

x x ), then write x x .

2 preerences are transitive: I x x and x x , then

x x .

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Representing Preerences

Suppose choice set is {x 1,

x 2,

x 2} and x 1 x 2 x 3. (Forexample, x 1 is Tristan und Isolde at the Metropolitan

Opera, x 2 is a day in the Philadelphia Museum o Art, and

x 3 is Swan Lake.)

I Bruce must choose rom {x 1, x 3}, he chooses x 1.

I Bruce must choose rom {x 2, x 3}, he chooses x 2.

The utility assignment U (x 1) = 3, U (x 2) = 2, and U (x 3) = 1

represents Bruce’s preerences, in the sense that more

preerred bundles are assigned a higher utility index.

The representing utility unction is not unique: V (x 1) = 10,

V (x 2) = 1, and V (x 3) = 0 represents the same preerences;

as does U 2 (i.e., utility is ordinal, not cardinal).

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Comparing Utility

Suppose Bruce and Sheila have the same preerences

over {x 1,

x 2,

x 3

} given by x 1

x 2

x 3

.Suppose Bruce reports a utility o 3 rom the bundle x 1,

while Sheila reports a utility o 6. Does this mean Sheila

likes the bundle x 1 twice as much as Bruce?

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Representing Choice

Bruce’s choice o x 1 rom {x 1, x 2, x 3} can be represented as

max

x ∈{x

1,

x

2,

x

3

}

U (x ),

while his choice o x 2 rom {x 2, x 3} can be represented as

maxx ∈{x 2,x 3}

U (x ).

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Consumption bundles o goods

In general, we will consider consumption bundlesx = (x 1, x 2, . . . , x n ) that satisy

x i ≥ 0, or all i .

Preerences will be represented by a utility unction

U (x ) = U (x 1, x 2, . . . , x n ).

Examples:

1 U (x 1, x 2) = x 1 + x 2.

2 U (x 1, x 2) = x α1 x β2 , where α, β > 0 are constants.

3 U (x 1, x 2) = x α1 + x 2.

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Indierence Curves

Suppose consumption bundles are in R2+.

Indierence curve through x = (x 1, x 2) is the locus o points x

yielding the same utility, i.e., is the locus o bundles x that the

consumer fnds indierent to x :

U (x 1, x 2) = U (x ) = u .

Implicitly defnes a unction x 2 = f (x 1, u ) by

U (x 1, f (x 1, u )) = u .

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Indierence Curves

I U (x ) = x 1 + x 2, then f (x 1, u ) = u − x 1.

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Indierence Curves

I U (x ) = x 1x 2, then f (x 1, u ) = u /x 1.

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Indierence Curves

I U (x ) = x α + x 2, then f (x 1, u ) = u − x α1 .

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Better Than Sets and Quasiconvexity

A set X is convex i or all x ∈ X , x ∈ X , and all λ ∈ (0, 1),

λx + (1− λ)x ∈ X .

For any bundle x , the set o bundles that the consumer weakly

preers to x is the weakly better than x set.

The utility unction U is quasiconcave i, or all x , the weakly

better than x set is convex.

We say preerences are convex i there is a quasiconcave

representing utility unction, or equivalently, i all weakly better

than sets are convex.

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Marginal Rates o Substitution

Slope o an indierence curve: recall indierence curve through

the bundle x

with utilityˉu is the unction x 2 = f (x 1

,

ˉu ) satisying

U (x 1, f (x 1, u )) = u .

What is the slope o the indierence curve?

Dierentiating, ∂U

∂x 1+

∂U

∂x 2

∂f

∂x 1= 0,

so∂f

∂x 1

= −∂U /∂x 1

∂U /∂x 2

= −U 1(x )

U 2(x )

.

This is the (negative o) the MRS.

Note that∂f

∂x 1=

dx 2dx 1

U (x )=u

.

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Diminishing MRS

A unction f (x 1) is convex i or all x 1, x 1, and all λ ∈ (0, 1),

f (λx 1 + (1− λ)x 1) ≤ λf (x 1) + (1− λ)f (x 1).

f is convex i f (x 1) ≥ 0 or all x 1. (This is the opposite o

concave.)

U quasiconcave implies diminishing MRS, i.e., the indierence

curves are convex unctions.

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Example: Quasilinear utilityU (x 1, x 2) = u (x 1) + x 2, or some increasing u .

dx 2/dx 1 = −u and so d 2x 2/dx 21 = −u .

What i u (x 1) = x α1 ?

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Example: perect complements

U (x 1, x 2) = min{αx 1, βx 2}, or some α, β > 0.

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Example: Cobb Douglas

U (x 1, x 2) = x α1 x β2 , or some α, β > 0.

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Example: Perect substitutes

U (x 1, x 2) = αx 1 + βx 2, or some α, β > 0.

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