Econ 500Fall, 2012

Li, HaoUBC

Lecture 1. Consumer Theory: Basics

Part I. Consumer Theory

Rational individual choice

1. Preferences

Primitives of the choice problem:

• Choice set : X ⊆ RL+. A choice x = (x1, . . . , xL) ∈ X is a consumption bundle (vector).

xi ≥ 0 is quantity of good i consumed, i = 1, . . . , L. Feasible set is X. Perfect divisibility.

• Preferences : defined by a binary relation % on X. Given x,y ∈ X, either “x % y” or not.

Nothing about preference intensity.

1

Strict preference relation ≻: derived from %, by

x ≻ y if x % y but not y % x.

Indifference relation ∼: derived from %, by

x ∼ y if x % y and y % x.

Note: x % y implies that either x ≻ y or x ∼ y, but not both.

Indifference sets: the indifference set containing x is {y | y ∼ x}. Indifferent curve for L = 2:

x1

x2

b

b

x

y

x and y on the same curve indicates x ∼ y

2

Preference Axioms:

(1) Completeness : for any x,y ∈ X, either x % y or y % x (or both). Completeness implies

reflexivity : x % x for all x ∈ X. Incomplete preferences?

(2) Transitivity : for any x,y, z ∈ X, if x % y and y % z, then x % z. Transitivity of % implies

the same for ≻ and ∼. Inconsistent preferences: choice from {x,y, z} when x % y % z ≻ x?

We will always assume (1) and (2). We often make additional assumptions (3)-(5).

3

(3a) Monotonicity : if x ≫ y, meaning xi > yi for every good i, then x ≻ y. Holds if the

consumer strictly prefers receiving more of all goods.

x1

x2

b

b y

x

Violation of monotone preferences

(3b) Strong monotonicity : if x ≥ y, meaning xi ≥ yi for all goods i, but x 6= y, then x ≻ y.

x1

x2

Perfect complements: monotone but not strongly monotone

(3c) Local non-satiation: given any x ∈ X and δ > 0, ∃y ∈ X such that ‖x−y‖ < δ and y ≻ x.

Rules out bliss points. Weaker than either (3a) or (3b).

4

(4a) Convexity : if x % y then tx+(1−t)y % y for all t ∈ [0, 1]. Implies convexity of {y | y % x}

(and diminishing marginal rates of substitution; see later). Examples of convex and non-

convex monotone preferences with two goods:

x1

x2

b

b

bx

y

tx+ (1− t)y

x1

x2

b

b

b

x

y

tx+ (1− t)y

(4b) Strict convexity : if x % y and x 6= y then tx+ (1− t)y ≻ y for all t ∈ (0, 1).

x1

x2

Perfect substitutes: convex but not strictly convex

5

(5) Continuity : for all x ∈ X, the sets {y | y % x} and {y | x % y} are closed (i.e. contain

their boundaries). Continuity is necessary to draw indifference curves.

An example of preferences that are not continuous: lexicographic preferences for L = 2. Suppose

that x % y if x1 > y1 or if x1 = y1 and x2 ≥ y2. Good 1 dominates good 2 in terms of preference

in the sense that the consumer is never willing to give up a small amount of good 1 for any

amount of good 2. The set {y | y % x} is not closed:

x1

x2

bx Indifference sets are individual points.{y | y % x}

6

2. Utility representations

Representation: a function u : X −→ R represents % if ∀x,y ∈ X,

u(x) ≥ u(y) if and only if x % y.

Difficult to work with preference relations. Utility function allows us to find the most preferred

choice by looking for the largest value of u(·).

Under what conditions on % does it have a representation? Representation for lexicographic

preferences?

Proposition (Existence of utility representation). Suppose that % is complete, transitive, con-

tinuous, and monotone. Then there exists a continuous utility function representing %.

7

Proof of existence: sketch. Fix any price vector p = (p1, . . . , pL) ≫ 0. For each x ∈ X, define

u(x) as

u(x) = min{y|y∼x}

p · y,

where p · y = p1y1 + · · · + pLyL. Completeness, transitivity, and continuity ensure that this is

well-defined.

The level sets of u(·) coincide with the indifference curves of %. If x ∼ y, then u(x) = u(y)

by transitivity. Conversely, if u(x) = u(y), then x ∼ y. Otherwise, say if x ≻ y, then by

monotonicity and continuity there is some z such that x ≥ z and z ∼ y. This implies u(x) >

u(z) = u(y), a contradiction.

Monotonicity ensures that u(·) orders the indifference curves in the right way.

8

Proof of existence: detail

• u is well-defined.

Solution to minimization problem exists for any x: constraint set {y | y ∼ x} is closed by

continuity, and bounded by monotonicity; the objective function is continuous.

u(x) is unique for x. (The solution may not be unique.)

• x ∼ y if and only if u(x) = u(y).

Only if: by transitivity, x ∼ y implies the two minimization problems are identical.

If: suppose u(x) = u(y) but x ≻ y; let u(x) = p ·x′; by continuity, {t ∈ [0, 1] | tx′ % y} and

{t ∈ [0, 1] | y % tx′} are both closed and have a unique intersection t′ ∈ [0, 1); let z = t′x′;

then x′ ≥ z and z ∼ y, implying u(y) = u(z) ≤ p · z < p · x′ = u(x), a contradiction.

• x ≻ y if and only if u(x) > u(y).

Suppose x ≻ y but u(x) < u(y); same contradiction as for u(x) = u(y) but x ≻ y.

9

Proof of continuity

Indirect proof: apply theory of maximum

Direct proof:

Fix any x; let {xn}∞n=1 be a sequence converging to x; let {yn} be corresponding sequence

such that u(xn) = p · yn (which implies yn ∼ xn); let {ynk} be a subsequence of {yn}

converging to some y (which implies y ∼ x).

Claim: u(x) = p ·y. Suppose not; then y ∼ x implies that there is some z such that z ∼ x

but p · z < p · y; let {znk} be any sequence converging to z such that znk

∼ xnk; then

p · znk< p · ynk

for sufficiently great k, a contradiction.

Continuity of u at x follows immediately, because u(x) is unique (though y may not be).

10

An illustration of the proof with L = 2 and “standard” indifference curves. Suppose p1 = p2 = 1.

0

1

2

3

0 1 2 3 x1

x2

b

b b

x

yz

u(x) = 2 = u(y)

u(z) = 3

The above utility representation is known as the money-metric utility for % at prices p.

Difference price vectors p result in different representations of the same %.

11

Utility representations are purely ordinal. If u(·) represents some %, then for any increasing

function f : R −→ R, the function f ◦ u represents % too. A property of some utility function u

is ordinal if it is invariant to such monotone transformation.

• Diminishing marginal utility (concavity) is not ordinal. For L = 1, the function u(x) =√x

represents monotone preferences for one good, but so does u(x) = x2.

• Monotonicity is ordinal. For L = 1, if u(·) is increasing, then preferences are monotone,

and it is necessary and sufficient for any other utility representation to be increasing.

• Quasi-concavity is ordinal. Utility function u(·) is quasi-concave if ∀x,y ∈ X with u(x) ≥

u(y) and t ∈ [0, 1],

u(tx+ (1− t)y) ≥ u(y).

Concavity implies quasi-concavity but the reverse is not true. Strict quasi-concavity, quasi-

convexity are defined similarly. Quasi-concavity of u corresponds to convexity of the pref-

erence that u represents.

12

3. Marginal rate of substitution

Indifferent set (indifference curve for L = 2). With u representing %, indifference set through x:

{y | u(y) = u(x)}.

Marginal rate of substitution. For L = 2, MRS at x = (x1, x2) is the absolute value of the slope

of indifference curve at x. Each indifference curve is defined by u(x1, x2) = k for some fixed k.

MRS is obtained by totally differentiating u(x1, x2) = k.

Convexity of preferences, or equivalently, quasi-concavity of the utility function, implies dimin-

ishing marginal rate of substitution.

x1

x2

b x

MRS(x1, x2) = −dx2

dx1

=∂u(x1,x2)/∂x1

∂u(x1,x2)/∂x2

=MU1(x1,x2)MU2(x1,x2)

.

13

More generally, with L goods,

MRSij(x) =MUi(x)

MUj(x).

14

4. Important classes of preferences

Homothetic preferences: % is homothetic if

x % y =⇒ αx % αy ∀α > 0.

This means that indifference curves are just magnifications of each other, so knowing one indif-

ference curve is enough to know all of them.

x1

x2

b

b

b

b

x

y

αy

αx

If x ∼ y then αx ∼ αy

15

Example (i) Cobb-Douglas utility:

u(x1, . . . , xL) = kxβ1

1 · · · xβL

L for k > 0, β1, . . . , βL > 0

Example (ii) Constant elasticity of substitution (CES) utility:

u(x1, . . . , xL) =(

k1xρ1 + · · ·+ kLx

ρL

)1/ρfor ρ > 0, k1, . . . , kL > 0

Example (iii) Perfect complements:

u(x1, . . . , xL) = min{k1x1, . . . , kLxL} for k1, . . . , kL > 0

Example (iv) Perfect substitutes:

u(x1, . . . , xL) = k1x1 + · · ·+ kLxL for k1, . . . , kL > 0

16

Quasilinear preferences: % is quasilinear in good 1 if, for e1 = (1, 0, . . . , 0)

x % y =⇒ x+ αe1 % y + αe1 ∀α > 0,

Utility representation

u(x1, . . . , xL) = x1 + v(x2, . . . , xL).

For L = 2, this means that MU1 and MU2 do not depend on x1, so changing x1 while keeping x2

fixed gives the same MRS. Knowing the shape of one indifference curve allows us to construct

the entire indifference map.

x1

x2

b

b

b

b

Indifference curves are horizontal shifts of each other.

17

5. The consumer problem

The consumer problem is

maxx∈X

u(x) s.t. p · x ≤ w.

Unit prices p = (p1, . . . , pL) ≫ 0. Consumer wealth w ≥ 0.

Feasible set: {x | p · x ≤ w}, which is closed, bounded, convex and contains the zero vector.

The solution to the consumer problem gives, for each p and w, a vector x = (x1, . . . , xL) indicating

the optimal choice, which is unique if preferences are strictly convex and monotone.

Marshallian demand function: solution to the consumer problem, as a function x(p, w).

18

Example (Cobb-Douglas) Consider the utility function u(x1, x2) = xα1xβ2 , where α, β > 0.

(1) Form the Lagrangian

L(x1, x2, λ) = xα1xβ2 + λ(p1x1 + p2x2 − w).

(2) Derive the first-order conditions:

αxα−11 x

β2 + λp1 = 0; βxα1x

β−12 + λp2 = 0; p1x1 + p2x2 − w = 0.

(3) Solve the three first order conditions:

x1 =αw

(α+ β)p1; x2 =

βw

(α + β)p2.

(4) Verify: second order condition; dropped non-negativity constraints; binding constraint.

19