Econ 710: Lec2 Consumer Choice

7/29/2019 Econ 710: Lec2 Consumer Choice

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Econ 710Lecture Slides 2

Consumer Choice, Preferences and Utility

Fei Li

University of North Carolina, Chapel Hill

August 29, 2013

Fei Li (UNC) Lecture 2 A ugust 29, 2013 1 / 30


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Consumer Theory

Consumer is the DM,

Commodities (Goods) l = 1, ...L,

Consumption Set X

contains all commodity bundles that can be conceivably consumed,reects physical constraints (cannot consume negative amounts offood, discrete goods, survival needs, cannot work more than 24 hr perday),reects institutional constraints (zero pork consumption in Muslimcountries, cannot work 1 minutes per day in a regular position, cannotbuy gun in many countries)Main example: X = RL+.



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Consumption Set Assumptions

Nonnegative: X RL+perfectly divisible good.Not important.

Closed: X is a closed setNot economically important.

Convexity: X is a convex set

any x, y2 X and 2 (0, 1), we have x+ (1 )y2 X.

Both technically and economically important.lunch in DC and NYC at the same time.



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Competitive Budget Sets

Competitive Markets

all goods can be purchased in any amount at known prices,

p = (p1, ....pL)

all prices are positive: p 0the consumer is a price taker

Walrasian (Competitive) Budget Set

Given the price vector p and the consumers income m 0, his budgetset is

B(p, m) := fx2 X : p x mg

B(p, m) is the consumers feasible set.

It reects the economic constraint.



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Competitive Budget Sets

Since we are in RL space and p 0, B(p, m) is compact (closed andbounded).

B(p, m) = B(p, m) for > 0

B(p, m) is a convex set.

if x, y2 B(p, m), for any , x+ (1 )y2 B(p, m).it rules out some relevant case: many employment contracts requireworking for no less than 8 hr per day.

Budget Hyperplane: fx2 RL+ : p x = mg

describes a family of (just aordable) consumption bundles,budget line when L = 2, and the slop p1/p2 captures the rate ofexchange between the two goods.



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Demand Correspondences

The consumers choice correspondence, dened on the set of allbudget sets, is called his (Marshallian) demand correspondence.For all (p, m),

x(p, m) B(p, m) X

If x(p, m) contains a single points for each (p, m) 2 RL++ R+, wesay it is a demand function.



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DenitionA demand correspondence x(p, m) satises Walras Law i for all(p, m) 2 RL++ R+,

p x = m for each x2 x(p, m)

Demand only on the budget hyperplane. (expends all income)

At this point, Walras law is just a property our economic intuitiontells us should be plausible.

When we get to the preference-based approach to consumer theory,we shall nd fairly weak conditions under which Walras law musthold.



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Denition

A demand correspondence x(p, m) satises Homogeneous of DegreeZero i for all (p, m) 2 RL++ R+,

x(p, m) = x(p, m) for all > 0

No framing eects since B(p, m) and B(p, m) are the same set.

No money or price illusion.



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ExampleConsider the following demand function.

x1(p, m) =p2

p1 + p2

m

p1, and x2(p, m) =

p1p1 + p2

m

p2

Does it satises (1) homogeneous of degree zero and (2) Walras law?

1 yes for all , 2 R.

2 yes only if = = 1 since

p x = [p2

p1 + p2+

p1p1 + p2

]m.



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Demand Function Comparative Statics

The comparative statics properties of a demand function x(p, m)consist of how it changes when its arguments change.

The most common comparative statics questions are about the signsof changes, e.g., is xl > 0 ifm > 0 and p = 0?

We often assume x(p, m) is dierentiable. Then the questions areabout the signs of the partial derivatives,

xl (p, m)

pkand

xl (p, m)

m

for l, k2 f1, 2, ...Kg.

Fei Li (UNC) Lecture 2 August 29, 2013 10 / 30


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Income Eects

Normal (xl/m > 0) v.s. inferior (xl/m < 0) goodsthese are local properties.we shall see when we consider preference-based demand, that evenunder standard assumptions goods can be inferior or normal.

Engel curves describe how household expenditure on particular goodsor services depends on household income.

Engels law:

It is an empirical observation in Engel (1857).It states that the poorer a family is, the larger the budget share it

spends on nourishment.Chai, Andreas, and Alessio Moneta. 2010. "Retrospectives: EngelCurves." Journal of Economic Perspectives, 24(1): 225-40.



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Price Eects

A demand function xl (p, m) satises the Law of Demand ixl/pl < 0.

We shall see that preference-based demand, even under standard

assumptions, may not satisfy the Law of Demand.Good l is called a Gien good if its demand curve slopes upxl/pl > 0.

Good k is a (gross) substitute for good l i xl/pk > 0.

Good k is a (gross) complement for good l i xl/pk < 0.



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Preference-Based Demand Correspondences

We start from the choice itself, and some of its desirable properties.

Question:

Where the demand function x(p, m) comes from?

Answer:A consumers rational choice.

Basic problem in the preference-based Consumer theory:

to choose a feasible bundle x2 B(p, m) .



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Preference-Based Demand Correspondences

Assume that for a rational preference relation ,

x(p, m) = C(B(p, m),)

In other words, the demand results from a rational choice.

Since B(p, m) = B(p, m), we have an immediate result.

Theorem

Preference-based demand correspondences are homogenous of degree zero

in (p,

m).Without loss of any generality, we can normalize the price of a good to beone.



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Restrictions on the Consumers Preference

Denition

A preference on X is

locally non-satiated (NS) if any open neighborhood of any x2 Xcontains a bundle y s.t. y x.

monotone (M) if x y) x y.

strongly monotone (SM) if x y and x6= y implies x y.

Immediately, we have

(SM) ) (M) ) (NS).



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Restrictions on the Consumers Preference

More implications:

TheoremIf satises NS, then x(p, m) = C(B(p, m),) satises Walras law.

Proof.

Suppose not. x(p, m) p< m. Then 9 > 0, such that 8x0 2 N [x(p, m)], x0 p< m. By (NS), there is a x0 x, which is acontradiction!

Lemma

Suppose a rational preference is monotone and it is represented by u.Then u is nondecreasing.

Proof.

Exercise.


C i i


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Continuity

Denition

A preference on X is continuous i for any sequence jxn j with xn ! xand xn y for all n, we have x y.a

aThe denition is dierent from Denition 3.C.1 in MWG.

Example (Lexicographic preferences)

xn = (1/n, 0) and y = (0, 1). For each n, we have xn y. But

y = (0, 1) (0, 0) = limn!xn.


C i i


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Continuity

Lemma

The following statements are equivalent:

1 A preference is continuous,

2 the upper and lower contour sets, fy2 X : y xg andfy2 X : x yg, are closed.

3 the strict upper and lower contour sets, fy2 X : y xg andfy2 X : x yg, are open.

4 if x y, then neighborhoods, Nx exist s.t. x0 y for all x0 2 Nx.

Intuition on BB.Proof.

Key: NWT (y) = fxjx yg and BT (y) = fxjx yg.


U ili R i Th


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Utility Representation Theorem

Theorem (Monotone Representation Theorem)

A continuous rational preference on RL+ that is monotone is

representable by a continuous function u.

When L = 1, trivial!

Suppose L = 2, see the graph on BB for intuition.


P f t ti


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Proof: construction

Let x2 RL+. Dene two subsets ofRL+:

A+(x) := f 2 R+ : e xg, and A(x) := f 2 R+ : x eg

where e = (1, 1....1).

by monotonicity A+(x) is nonempty. Since (0, ...0) 2 A(x), A(x)

is nonempty too.By continuity, both sets are closed.

By completeness, A+(x)S

A(x) = R+,

Since R+ is connected, A+(x)

TA(x) 6= ?. (otherwise, one of

them is open)(x) = A+(x)

TA(x) is a singleton. (if there are two: ,, then

, but then by monotonicity = )

so there exists a unique (x) s.t. (x)e x.


P f R t ti


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Proof: Representation

()) Suppose u(x) u(y). There are two cases:

1 If they are equal, then x (x)e y, so x y.2 If u(x) > u(y). then u(x)e u(y)e by monotonicity. Since

x u(x)e and y u(y)e, x y.(() Suppose x y. Then

u(x)e x y u(y)e.

Hence u(x)e u(y)e. By monotonicity, we must have u(x) u(y).


Proof: Continuity


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Proof: Continuity

Recall that u is continuous if for any a 2 R, the sets u1(a,) andu1(, a) are open in the domain of u.

Since u(a, ...a) = a, these sets are strict contour sets:

u1((a,)) = fx : u(x) > u(a)g = fx : x ag

and

u1((, a) = fx : u(x) < u(a)g = fx : a xg

Hence, the proof is complete.


Debrue Theorem


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Debrue Theorem

Monotonicity is not necessary for the utility representation but makes theproof intuitive. In general, we have

Theorem (Debreus Representation Theorem)

1 For any a and b> a in R, a continuous rational preference on aconnected set X RL is representable by a continuous functionu : X! [a, b].

2 If a continuous function u represents a preference, then is

continuous.


Debrue Theorem: Proof


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Debrue Theorem: Proof

The proof of the rst part is mathematically technical and not veryeconomically based. See Rubinstein (2013) for the proof of most of thetheorem. We provide the proof of the second part here.

Proof.Suppose the preference is not continuous. There exists a sequence

fxng ! x, and xn y for any n but y x. Since is represented byu(), u(xn ) u(y) and u(x) < u(y). By the continuity of u(),9 > 0, s.t. 8x0 2 N (x), u(x0) < u(y) or y x0. Contradiction!


Dierentiability


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Dierentiability

We often assume the utility function is dierentiable for tractability.

A classical non-dierentiable utility is Leontief utility function,

min fx1, x2g .

However, there is no generally accepted justication for thedierentiability of utility functions.

See Rubinstein (2013) for some justications.


Homotheticity


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Homotheticity

Denition

A preference is homothetic i x y implies that for any 0,x y.

Theorem

A rational preference is continuous, monotone, and homothetic i they can

be represented by a continuous and homogenous utility function:u(x) = u(x).

Proof.

\

)

00 For any x, there exists a(

x)

, s.t. x (

,

, ....)

. We letu(x) = (x). Since the preference is homothetic,u(x) (, , ...). So u(x) = = u(x). \ (00, obvious.

Notice that u may not be a linear function. For example,

u(x1,

x2) = x

1 x

1

2 for 2 (0,

1).Fei Li (UNC) Lecture 2 August 29, 2013 26 / 30

Separable Utility


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Separable Utility

In many applications, we always assume the consumers preference isseparable.

Example

1 In Macro, U(c1, c2, ....ct) =

t=1tu(ct) where 2 (0, 1),

2 In labor, U(c, h) = u(c) + v(h)

3 In IO and auction theory, U(q, m) = u(q) + m

It captures the idea that agents preference over dierent goods are

somehow independent.What implicit assumptions we made by assuming separable utility?


Separable Utility


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Separable Utility

Consider a two-goods world.

DenitionA preference on X satises double cancellation property i(x1, x2) (y1, y2) and (y1, z2) (z1, x2) implies that (x1, z2) (z1, y2).

Lemma

Suppose is represented by U(x1, x2). U(x1, x2) = u1(x1) + u2(x2) onlyif satises double cancellation property.

Proof.

Since (x1, x2) (y1, y2), (y1, z2) (z1, x2) and U = u1 + u2 represents, we must have u1(x1) + u2(x2) u1(y1) + u2(y2), andu1(y1) + u2(z2) u1(z1) + u2(x2). Adding two inequality together yieldsu1(x1) + u2(z2) u1(z1) + u2(y2), so (x1, z2) (z1, y2).


Quasi-linear Utility


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Quasi linear Utility

TheoremA rational preference on X = R2 satises the following three conditions

1 (t is valuable) (t, y) (t0, y) i t t0,

2 (compensation is possible) For every y, y 2 R, there exists some

t2 R s.t. (0, y) (t, y

).3 (no wealth eects) if (t, y) (t0, y0) then for all d2 R,

(t+ d, y) (t0 + d, y0).

if and only if it can be represented by a utility function of the form

u(t, y) = t+ v(y).

In application, the rst good is interpreted as money, wealth or numerategood.


Proof


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Proof

the proof of if part is simply to verify three conditions, so we only show

the Only If part. Fix a y,

by condition (2), dene a function v(y) s.t. (0, y) (v(y), y).

by condition (3), for any (t, y), (t0, y0), (t, y) (t+ v(y), y) and(t0, y0) (t0 + v(y0), y).

by transitivity, (t, y) (t0, y0) i(t, y) (t+ v(y), y) (t0, y0) (t0 + v(y0), y).

by condition (1), this is equivalent to

t+ v(y) t0 + v(y0)

so (t, y) (t0, y0) i t+ v(y) t0 + v(y0), and the proof iscomplete.


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