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))
Under the basic Assumption, What are the properties of the expenditure function?
1) Homogeneous of degree zero
2) Strictly increasing in u and non decreasing in pi for every i
3) Concave in p
4) Continuous in p and u
Under the basic assumption, what are the properties of the hicksian demand?
1) Homogeneous of degree zero
2) No excess utility
3) If the preference is convex, then h(p,u) is a convex set
4) If the preference is strictly convex, then h(p, u) contains a single element
5) Upper-hemicontinuous
What is the first order condition for the EMP?
p = μ* * ∇u(x*)
Why the Hicksian demand is compensated?
(p’’ - p’) * (x’’ - x’) ≤ 0
What is the Envelope Theorem?
Define f(θ) = sup {f(x,θ) | x in X}
Let x satisfy f(θ) = f(x, θ) and assume that df(x,θ)/dθ exists. Then we have
df(θ-)/dθ ≤ df(x,θ)/dθ ≤ df*(θ+)/dθ
if f(θ) is differentiable and L is differentiable with respect to θ at a saddle-point (x , λ*), we have
df(θ)/dθ = dL(x, λ*, θ)/dθ
Suppose that the Basic Assumption holds and preferences are strictly convex. For all p in P and u > u(0) we have (hicksian demand and expenditure function)
h(p,u) = ∇p e(p,u)
Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u).
Then Dp h(p,u) = ?
Dp h(p,u) = D²p e(p,u)
Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u). Dp h(p,u) is...
Dp h(p,u) is negative semidefinite and symetric
Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u).
Dph(p,u)*p = ?
Dph(p,u)*p = 0
What is the Slutsky equation?
Dp h(p,u) = Dp x(p,w) + Dw x(p,w) * x(p,w)T
By the slutsky equation when u = v(p,w) we have
S(p, w) = Dp h(p,u)
Show the Roy’s Indtity
x(p,w) = - 1/(dv(p,w)/dw) * ∇p v(p,w)
If a continuously differentiable demand function x(p,w) is generated by maximization of rational preferemce, then it must…
1) Be homogeneous of degree zero
2) Satisfy the Walras’ Law
3) Have a symmetric and negative semidefinite substitution matrix S(p,w)
1) Be homogeneous of degree zero
2) Satisfy the Walras’ Law
3) Have a symmetric and negative semidefinite substitution matrix S(p,w)
Are these conditions sufficient for rationalizability?
Yes
What are the steps to recover the rationa preference?
1) u = inf{v(p1,p2,1) | p1x1 + p2x2 = 1}
2) Avaliar se a solução é de canto ou inferior
3) Resolver o problema e substituir as soluções ótimas
