week 5 consumer theory (jehle and reny,...
TRANSCRIPT
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
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
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:
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Expenditure and Consumer Preferences
Now choose di�erent prices p1, keep u0 �xed, and construct
the set,
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Expenditure and Consumer Preferences
Imagine proceeding like this for all prices p� 0 and forming
the in�nite intersection,
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Expenditure and Consumer Preferences
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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Expenditure and Consumer Preferences
Theorem 2.2
Week 5 Consumer Theory (Jehle and Reny, Ch.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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Convexity and Monotonicity
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Convexity and Monotonicity
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Convexity and Monotonicity
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Indirect Utility and Consumer Preferences
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Indirect Utility and Consumer Preferences
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
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Duality
Indirect Utility and Consumer Preferences
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
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
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
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
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Axioms of Choice Under Uncertainty
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 .
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Axioms of Choice Under Uncertainty
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
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
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 .
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Von Neumann-Morgenstern Utility
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.
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Von Neumann-Morgenstern Utility
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
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:
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Risk Aversion
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Risk Aversion
Week 5 Consumer Theory (Jehle and Reny, Ch.2)
Choice Under Uncertainty
Risk Aversion