Classical Deand Theory Flashcards
What is Monotonicity?
if y»_space; x implies x ≻ y
What is strong Monotonicity
y ≥ x and y ≠ x implies y ≻ x
What is weak monotonicity?
y ≥ x implies y ≽ x
What is Local Nonsatiation
For every x and ϵ > 0 então existe y tal que ||x - y|| < ϵ and y ≻ x
What is convexity?
y ≽ x and z ≽ x implies α*y + (1-α) z ≽ x for all α in [0 ,1]
What is strict convexity?
y ≽ x and z ≽ x and y ≠ z implies α*y + (1-α) z ≻ x for all α in [0 ,1]
When a monotonne preference is homothetic?
If x ~ y implica em αx ~ αy
What is a quasilinear preference?
1) x + αe1 ≻ x for all α in R+
2) x ~ y implies x + αe1 ~ y + αe1
A Lexicograph preference is rational, strogly monotne and strictly convex but does not admit a utiliy fubction
True
When a preference is continuous?
If it is a closed subset of X x X: for every convergent sequence (xn, yn) -> (x, y) such thar xn≽ yn for all n, we have x ≽ y
If ≽ has a continuous utility representation, then…
≽ is continuous
If ≽ is a continuous raational preference then…
There exists a continuous utility representation
When a correspondence is Upper-Hemicontinuous?
If for every open set V such that V ∩ G(θ) ≠ ∅, there exists an open set U ∋ θ such that G(θ’) ∩ V ≠ ∅ for all θ’
When G is Upper-Hemicontinuous?
1) θn -> θ, xn ∈ G(θn) and xn -> x imply G(θ)
2) G(A) is bounded for every compact A
G is lower-hemicontinuous if…
θn - θ and x in G(θ) implies that there exists a sequence xn - > x with xn in G(θn) for all sufficiently large n
What is the basic assumption?
u(.) is a continuous utility function for locally nonsatitates preference
Under the Basic assumption the Walras emand satisfies:
1) Homogeneity of degree zero
2) Walras’ Law
3) If the preference is convex then x(p,w) is convex
4) If the preference is strictly convex then x(p, w) is a singleton
5) Upper-hemicontinuous
∂u(x*)/∂xi =…
λp
MRS = ?
pi/pk
What are the properties of the indirect utility under the basic assumption?
1) Homogeneous of degree zero
2) Strictly increasing in w and non increasing in pi for any i
3) Quasi convex: The set {p,w) \in P x W | v(p,w) ≤ ubarr} is convex
4) Continuous
If x* is optimal in UMP at wealth w, then x* is optimal in the EMP when the utility requirement is u = u(x*). Moreover
e(p, u(x*)) = w
If x* is optimal in the EMP when the utility requirement is u > u(0) then x* is optimal in the UMP at wealth w = p * x*
V(p, p * x*) = u
h(p, u) = … (Walrasian)
x(p, e(p, u))
x(p,w) = … (Hicksian)
h(p, v(p, w))