Competitive Equilibrium (Stokey-Lucas)

An allocation [(xi0), (yj0)] together with a continuous linear functional φ: S → R is a competitive equilibrium if

(E1) [(xi0), (yj0)] is feasible;
(E2) for each i, x ∈ Xi and φ(x) ≤ φ(xi0) implies ui(x) ≤ ui(xi0); and
(E3) for each j, y ∈ Yj implies φ(y) ≤ φ(yj0).

Stokey, Lucas "Recursive Methods in Economic Dynamics" p.453

Pareto Optimality (Stokey-Lucas)

Definition (Allocation)
An (I+J)-tuple [(xi),(yj)] describing the consumption xi of each consumer and the production yj of each firm is an allocation for the economy.

Definition (Feasible Allocation)
An allocation is feasible if xi ∈ Xi, all i; yj ∈ Yj, all j; and Σxi - Σyj =0.

Definition (Pareto Optimality)
An allocation [(xi), (yj)] is Pareto optimal if it is feasible and if there is no other feasible allocation [(xi'), (yj')] such that ui(xi') ≥ ui(xi), all i; ui(xi') > ui(xi), some i.


Stokey, Lucas "Recursive Methods in Economic Dynamics," p.453

Non-Negative Function

f(x) is a non-negative function defined on [a,b]

⇔

f(x) ≥0 for all x in [a,b]

Cramer-Rao Lower Bound

いかなる最尤推定量もCramer-Rao Lower Boundで与えられる推定誤差の分散より小さくなることはない。

w2.gakkai-web.net/gakkai/ieice/vol4no1pdf/vol4no1_32.pdf

Based on - synonyms

according to

as shown by

as evidenced by

Bertrand Model

Model
Two firms with the same marginal cost (MC=c), producing a homogeneous product and competing in prices.

homogeneous product → consumers purchase from cheapest firms

Bertrand's theorem
Let (p1*,p2*) be a Nash Equilibrium. Then, p1*=p2*=c.

If each firm has a different marginal cost c1, c2 (c1<c2), the Nash equilibria are (p1, p2) such that c1≤p1=p2≤c2.

http://en.wikipedia.org/wiki/Bertrand_competition

RHS/ LHS

RHS: right-hand side

LHS: left-hand side