notes on demand theory

8/8/2019 Notes on Demand Theory

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Advanced Microeconomics II: ConsumerTheory

Maria Saez Marti O ffi ce 210, IEW

tel. 044 634 37 13 e-mail: [email protected]


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This courseConsumer Theory.

Theory of the rm.

Partial Equilibrium

General Equilibrium

(Social Choice and Welfare)

Lectures (MSM) and Classes (Bernhard Ganglmair). Important thatyou participate in the Classes.


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Bibliography:

Jehle and Reny: Advanced Microeconomic Theory.

If you are interested in a more advanced book, you can read

Mas Collell et al: Microeconomic Theory


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1.1 Intoduction

The choice problem of a household

Consumers decide about

savings vs. consumption composition of consumption

composition of wealth

labor supply

type of job

number of children

education


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Decisions depend on: tastes (including time and risk preferences)

wealth (including human capital) prices

expectations about

future prices

own future preferences

future income


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Theoretical starting-point

Consumer knows set of goods, prices and income; based on them,he decides on consumption vector once and for all.

Thus the problem is simpli ed in that choices concern only consumption choices are not sequential the future is certain

Question: How does consumption vector depend on prices, income and pref-

erences?

Important: understand relation between assumptions and conclusions - not just

conclusions.


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Goal of the chapter:

formal representation of the consumers choice problem

characterization of his behavior on the market

Important: consumer theory is not an end itself, rather a buildingblock of the explanation in the economic system.


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1.2 Primitive Notions

Consumer ChoiceOverview

Four building blocks in any model of consumer choice

consumption set

feasible set

preference relation behavioral assumption


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Consumption Set

Consumption set represents the set of all alternatives, or completeconsumption plans, that the consumer can conceive - whether someof them will be achievable in practice or not

Let each commodity be measured in some in nitely divisible units.

xiR represents the number of units of good i

x = ( x1,...,x n ) is the vector containing di ff erent quantities of eachof the n commodities ( x is called a consumption bundle or a con-sumption plan)

Think of the consumption set as the entire nonnegative orthant,X = Rn+ .


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Properties of the Consumption Set

Assumption 1 The minimal requirements on the consumption set are

1. 6= X Rn+2. X is closed

3. X is convex

4. 0 X


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Feasible Set

The feasible set B represents all those alternative consumptionplans that are both conceivable and realistically obtainable giventhe consumers circumstances.

The feasible set B then is that subset of the consumption set X that remains after we have accounted for any constraints on theconsumers access to commodities due to the practical, institutional,or economic realities of the world.


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Preference Relation

A preference relation typically speci es the limits, if any, on the con-sumers ability to perceive in situations involving choice, the form of consistency or inconsistency in the consumers choices, and informationabout the consumers tastes for the di ff erent objects of choice.

Behavioral Assumption

The consumer seeks to identify and select an available alternative thatis most preferred in the light of his personal tastes.


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1.3 Preferences and Utility

Preference Relations We represent the consumers preferences by a binary relation, %,

de ned on the consumption set, X .

If x 1 % x 2, we say that x 1 is at least as good as x 2, for thisconsumer.


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Axiom 1: CompletenessFor all x 1 and x 2 in X , either x 1 % x 2 or x 2 % x 1

Axiom 2: Re exivityFor all x , in X , x % x

Axiom 3: TransitivityFor any x 1, x 2 and x 3 in X , if x 1 % x 2 and x 2 % x 3 then x 1 % x 3.

De nition 1:The binary relation % on the consumption set X is called a preferencerelation if it is complete, re exive and transitive.


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Induced Preference RelationsDe nition 2:

The binary relation on the consumption set X is de ned as fol-lows:x 1 x 2 if and only if x 1 % x 2 and x 2 x 1.The relation is called the strict preference relation induced by%, or simply the strict preference relation when % is clear. The

phrasex 1

x 2

is read, x 1

is strictly preferred tox 2

.De nition 3:

The binary relation on the consumption set X is de ned as fol-

lows:x 1 x 2 if and only if x 1 % x 2 and x 2 % x 1.The relation is called the indiff erence relation induced by %, orsimply the indi ff erence relation when % is clear. The phrase x 1 x 2

is read, x 1 is indiff erent to x 2.


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Sets in X derived from the Preference RelationsDe nition 4:

Let x 0 be any point in the consumption set, X . Relative to any suchpoint, we can de ne the following subsets of X :

1. % x 0 nx | x X, x % x 0o, called the at least as good as set.

2. - x 0 nx | x X, x 0 % x o, called the no better than set.3. x 0 nx | x X, x 0 x o, called the worse than set.4. x 0 nx | x X, x x 0o, called the indiff erence set.


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Hypothetical preferences satisfying Axioms 1, 2 and 3


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Further Assumptions on PreferencesAxiom 4: Continuity

For all x Rn+ , the at least as good as set, % ( x ), and the no

better than set - ( x ), are closed in Rn+ .

Axiom 5: Local Nonsatiation

For all x 0 Rn+ , and for all > 0, there exists some x B x 0Rn+ such that x x 0; where B x0 is the neighborhood of x 0with radius .

Axiom 5: Strict Monotonicity For all x 0, x 1 Rn+ , if x 0 x 1 then x 0 % x 1, while if x 0 x 1,

then x 0 x 1.


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Hypothetical preferences satisfying Axioms 1, 2, 3, 4 and 5


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Hypothetical preferences satisfying Axioms 1, 2, 3, 4 and 5


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Axiom 6: Convexity

If x 1 % x 0, then t x 1 + (1 t ) x 0 % x 0 for all t (0 , 1) .

Axiom 6: Strict Convexity

If x 1 6= x 0 and x 1 % x 0, then t x 1 + (1 t ) x 0 x 0 for all t (0 , 1) .


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Hypothetical preferences satisfying Axioms 1, 2, 3, 4, 5 and 6 or 6


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Marginal Rate of Substitution

When X = R2+ , the (absolute value of the) slope of an indi ff erencecurve is called the marginal rate of substitution .

If preferences are strictly monotonic, any form of convexity requires

the indi ff erence curves to be at least weakly convex-shaped relativeto the origin.


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The Utility Function

De nition 5:

A real valued function u : Rn+ R is called a utility function represent-ing the preference relation %, if for all x 0, x 1 Rn+ ,

u x 0 u x 1 x 0 % x 1.

Note:

It can be shown that any binary relation that is complete, transi-

tive, and continuous can be represented by a continuous real-valuedutility function.

Here we will take a detailed look at a slightly less general result.


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Representation of Preferences

Theorem 1:If the binary relation % is complete, transitive, continuous, and strictly monotonic, there exists a continuous real-valued function, u : Rn+ Rwhich represents %.

Interpretation :The result frees us to represent preferences either in terms of the prim-

itive set-theoretic preference relation or in terms of a numerical repre-sentation, a continuous utility function.


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Sketch of Proof: Let the relation % be complete, transitive, continuous, and strictly

monotonic. Let e (1, ..., 1) Rn+ be a vector of ones, andconsider the mapping u: Rn

+ R de ned so that the following

condition is satis ed:

u ( x ) e x (1)

Remaining issues:

First, does there always exist a number u ( x ) satisfying (1)?

Second, is it uniquely determined, so that u ( x ) is a well-de nedfunction?

Third, is the utility function u: Rn+ R representing % continuous?


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Constructing the mapping


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Monotone Transformations

Theorem 2:

Let % be a preference relation on Rn+ and suppose u (x ) is a utility

function that represents it. Then v ( x ) also represents % if and only if v (x ) = f (u ( x )) for every x , where f : R R is strictly increasing onthe set of values taken on by u.


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Relation between Preferences and Utility Func-tions

Theorem 3:

Let % be represented by u : Rn+ R. Then:

1. u ( x ) is strictly increasing if and only if % is strictly monotonic.

2. u ( x ) is quasiconcave if and only if % is convex.

3. u ( x ) is strictly quasiconcave if and only if % is strictly convex.


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Diff

erentiable Utility Functions marginal utility of good i = u ( x ) /x i

For the case of two goods, we de ned the marginal rate of substi-

tution as the absolute value of the slope of an indi ff erence curve.

Let x2 = f (x1) be the function describing the indi ff erence curve inthe ( x1, x 2) plane.

Therefore MRS 12 x11, x 12 | f 0x11 |= f 0x11 (2)

Suppose u ( x ) /x i > 0 for almost all bundels x , and all i =1,...,n. Then

MRS 12 x 1 =u x 1 /x 1u x 1 /x 2

.

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