week 5 consumer theory (jehle and reny,...

Week 5 Consumer Theory (Jehle and Reny, Ch.2)

Week 5Consumer Theory

(Jehle and Reny, Ch.2)

Serçin �ahin

Y�ld�z Technical University

23 October 2012


Duality

Expenditure and Consumer Preferences

Choose (p0, u0) ∈ Rn++ × R+, and evaluate E there to obtain

the number E (p0, u0).And use this number to construct the (closed) 'half space' in

the consumption set:


Duality


Now choose di�erent prices p1, keep u0 �xed, and construct

the set,


Duality


Imagine proceeding like this for all prices p� 0 and forming

the in�nite intersection,


Duality


Theorem 2.1

Constructing a Utility Function From an Expenditure

Function

Let E : Rn++ × R+ → R+ satisfy properties 1 through 7 of an

expenditure function given in Theorem 1.7.

Let A(u) be as

Then the function u : Rn+ → R+ given by

is increasing, unbounded above and quasiconcave.


Duality


Theorem 2.2


Duality

Convexity and Monotonicity

Let e(p, u) be the expenditure function generated by u(x).

Consider the utility function, generated by e(·), call it w(x),

regardless of whether or not u(x) is quasiconcave or increasing,

w(x) will be both quasiconcave and increasing.

Then u(x) and w(x) need not coincide.


Duality

Convexity and Monotonicity


Duality

Indirect Utility and Consumer Preferences


Duality


Because v(p, y) is homogeneous of degree zero in (p, y), wehave

v(p,p.x) = v(p/(p.x), 1)

whenever p.x > 0.

Consequently, if x� 0 and p∗ � 0 minimises v(p,p.x) forp ∈ Rn

++, then p̂ ≡ p∗/(p∗.x)� 0 minimises v(p, 1) forp ∈ n

++ such that p.x = 1.

Moreover, v(p∗,p∗.x) = v(p̂, 1).

Thus, we may rewrite (T.1) as


Duality



Integrability

Theorem 2.5: Budget Balancedness and Symmetry

Imply Homogeneity

If x(p, y) satis�es budget balancedness and its Slutsky matrix

is symmetric, then it is homogeneous of degree zero in p and y .

Thus if x(p, y) is a utility maximiser's system of demandfunctions, we may summarise the implications for observablebehaviour in the following three items alone:

Budget Balancedness: p.x(p, y) = yNegative Semide�niteness: The associated Slutsky matrix

s(p, y) must be negative semide�nite.

Symmetry: s(p, y) must be symmetric.


Integrability

Theorem 2.6: Integrability TheoremA continuously di�erentiable function x : Rn+1

++ → Rn+ is the

demand function generated by some increasing, quasiconcaveutility function if it satis�es

budget balancedness,

symmetry, and

negative semide�niteness.


Choice Under Uncertainty

The individual will be assumed to have a preference relation

over gambles.

Let A = {a1, ..., an} denote a �nite set of outcomes.

A simple gamble assigns a probability, pi , to each of the

outcomes ai , in A. We denote the simple gamble by

Then GS , the set of simple gambles (on A) is given by



Gambles whose prizes are themselves gambles are called

compound gambles.

Let G denote the set of all gambles, both simple and

compound.

So if g is any gamble in G, then

for some k ≥ 1 and some gambles g i ∈ G, where the g i 's

might be compound gambles, simple gambles, or outcomes.



Axioms of Choice Under Uncertainty

Axiom 1: Completeness.

For any two gambles, g and g′in G, either g � g

′or g

′ � g .

Axiom 2: Transitivity.

For any three gambles g , g′, g

′′in G, if g � g

′and g

′ � g′′,

then g � g′′.

Axiom 3: Continuity.

For any gamble g in G, there is some probability α ∈ [0, 1],such that




Axiom 4: Monotonicity.

For all probabilities α, β ∈ [0, 1],

if and only if α ≥ β.Axiom 5: Substitution.

are in G, and if hi ∼ g i for every i , then h ∼ g .




For any gamble g ∈ G, if pi denotes the e�ective probability

assigned to ai by g , then we say that g induces the simple

gamble

Axiom 6: Reduction to Simple Gambles.

For any gamble g ∈ G, if (p1 ◦ a1, ..., pn ◦ an) is the simple

gamble induced by g , then



Von Neumann-Morgenstern Utility

Suppose that u : G → R is a utility function representing � on

G.So for every g ∈ G, u(g) denotes the utility number assigned

to the gamble g . In particular, for every i , u assigns the

number u(ai ) to the degenerate gamble (1 ◦ ai ), in which the

outcome ai occurs with certainty.

We will refer to u(ai ) as simply the utility of the outcome ai .




Expected Utility Property

The utility function u : G → R has the expected utility

property if, for every g ∈ G,

where (p1 ◦ a1, ..., pn ◦ an) is the simple gamble induced by g .

If an individual's preferences are represented by a utility

function with the expected utility property, and if that person

always chooses his most preferred alternative available, then

that individual will choose one gamble over another if and only

if the expected utility property of the one exceeds that of the

other.

Consequently, such an individual is an expected utility

maximiser.



Risk Aversion

The expected value of the simple gamble g o�ering wi with

probability pi is given by

Now suppose the agent is given a choice between accepting

the gamble g on the one hand or receiving with certainty the

expected value of g on the other.

If u(·) is the agent's VNM utility function, we can evaluate

these two alternatives as follows:



Risk Aversion

week 5 consumer theory (jehle and reny,...

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