Review Class Two

Outlines  A typical consumer will satisfy himself as much as possible with limited resources.  How to describe “limited resources”?  Budget constraint or feasible set  How to describe “satisfaction”, or say, “desire”?  Preference

Outlines  The central definition of the consumer theory is demand.  Three factors of demand are price, income and preference.  Demand is the feasible, conditional and optimal choice.  The task facing to us is to attain the demand, so we should grasp the analysis of preference.

Outlines  Two key definitions of preference:  Consumption set  Preference relation  From Preference to Utility function

Consumption set  Def.1 Consumption set is the set which consists of all consumption bundles, or say, consumption plans which the typical consumer desires, regardless whether they are feasible or not.  Four properties a consumption set should satisfy.  Consumption set is the basis of the whole theory of preference.

Preference relation  Def.2 If the binary relation defined on the consumption set satisfies Axiom 1,2 and 3, then this binary relation is said to be the Preference relation. Axiom 1 Completeness : x is w.p. to y or y is w.p. to x for any pair of x and y.

Preference relation Axiom 2 Reflexivity: x is w.p. to x for any bundle x. Axiom 3 Transitivity: If x is w.p. to y and y is w.p. to z, then x is w.p. to z. Rationality is defined as Axiom 1,2 and 3,that is to say, a rational consumer is able to make a choice and the choices he makes are consistent.

Strict preference relation and indifference relation  Def.3 If the binary relation defined on the consumption set satisfies and then we say,which is called the strict preference relation. Def.4 If the binary relation defined on the consumption set satisfies and then we say,which is called the indifference relation,

Indifference curves  Def. 5 An indifference curve is a set of consumption bundles with the same desire level of a representative consumer.  Now we can conclude that for a specific preference, there’s a unique shape of indifference curves corresponding to it.  We can use a cluster of IC to describe a specific preference.

Examples  Perfect substitutes and perfect complements. Goods, bads, and neutrals. Satiation.

Well-behaved preference  Def.6 A given preference is called well-behaved preference if it satisfies Axiom 1,2,3,4 and 5.  Axiom 4monotonic (meaning more is better)  Axiom 5convex (meaning average are preferred to extremes).

Utility function  Utility function is a way to describe preference.  Def. 7 A mapping u: is called the utility function which stands for the preference relation if the mapping satisfies u ( x ) ≥ u ( y ) if and only if bundle x is w.p. to bundle y.

Utility function  If we want to use a continuous U.F. to describe the given preference,the case must satisfy Axiom1,2,3 and the assumption of Continuity.

Positive Monotonic transformation  Lemma: Ordinal utility holds that the size of the utility difference between any two consumption bundles doesn’t matter.  So what we care is only the ordinal represented by the amount of utility function.  Positive Monotonic transformation of the utility function represents the same preference as the original utility function.

Positive Monotonic transformation  For a given preference, there’s at least one utility function to describe it.

Relation between U.F. and IC: 1 Draw a diagonal line and label each indifference curve with how far it is from the origin.

Relation between U.F. and IC:  2 The indifference curves are the projections of contours of u = u ( x 1, x 2 ).

2 A utility function  0 x2x2 x1x1  Take a slice at given utility level  Project down to get contours U(x1,x2)U(x1,x2) U(x1,x2)U(x1,x2) The indifference curve

Relation among Preference, IC and U.F.  对于一种特定的偏好，可以用至少一个效用函数 进行刻画，把偏好关系转化为函数关系来讨论， 反过来，对于一个特定的效用函数形式，只能够 描述一种特定的偏好；  无差异曲线既可以从效用函数得到，也可以从定 义得到，但根本依据是定义。给定特定的效用函 数形式，可以划出无数组无差异曲线束，然而无 差异曲线束的形状却是唯一的；  一种特定的偏好与一种特定的无差异曲线的形状 一一对应。

 Quasilinear preferences: All indifference curves are vertically (or horizontally) shifted copies of a single one, for example u (x 1, x 2 ) = v (x 1 ) + x 2.

How to find the parameter?  u (x 1, x 2 ) = ax 1 + bx 2 (perfect substitutes);  u (x 1, x 2 ) = min{ax 1, bx 2 } (perfect complements).

Thank you!