Classical Deand Theory

Question 1

What is Monotonicity?

Answer 1

if y&raquo_space; x implies x ≻ y

Question 2

What is strong Monotonicity

Answer 2

y ≥ x and y ≠ x implies y ≻ x

Question 3

What is weak monotonicity?

Answer 3

y ≥ x implies y ≽ x

Question 4

What is Local Nonsatiation

Answer 4

For every x and ϵ > 0 então existe y tal que ||x - y|| < ϵ and y ≻ x

Question 5

What is convexity?

Answer 5

y ≽ x and z ≽ x implies α*y + (1-α) z ≽ x for all α in [0 ,1]

Question 6

What is strict convexity?

Answer 6

y ≽ x and z ≽ x and y ≠ z implies α*y + (1-α) z ≻ x for all α in [0 ,1]

Question 7

When a monotonne preference is homothetic?

Answer 7

If x ~ y implica em αx ~ αy

Question 8

What is a quasilinear preference?

Answer 8

1) x + αe1 ≻ x for all α in R+

2) x ~ y implies x + αe1 ~ y + αe1

Question 9

A Lexicograph preference is rational, strogly monotne and strictly convex but does not admit a utiliy fubction

Answer 9

True

Question 10

When a preference is continuous?

Answer 10

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

Question 11

If ≽ has a continuous utility representation, then…

Answer 11

≽ is continuous

Question 12

If ≽ is a continuous raational preference then…

Answer 12

There exists a continuous utility representation

Question 13

When a correspondence is Upper-Hemicontinuous?

Answer 13

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’

Question 14

When G is Upper-Hemicontinuous?

Answer 14

1) θn -> θ, xn ∈ G(θn) and xn -> x imply G(θ)

2) G(A) is bounded for every compact A

Question 15

G is lower-hemicontinuous if…

Answer 15

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

Question 16

What is the basic assumption?

Answer 16

u(.) is a continuous utility function for locally nonsatitates preference

Question 17

Under the Basic assumption the Walras emand satisfies:

Answer 17

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

Question 18

∂u(x*)/∂xi =…

Answer 18

λp

Question 19

MRS = ?

Answer 19

pi/pk

Question 20

What are the properties of the indirect utility under the basic assumption?

Answer 20

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

Question 21

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

Answer 21

e(p, u(x*)) = w

Question 22

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*

Answer 22

V(p, p * x*) = u

Question 23

h(p, u) = … (Walrasian)

Answer 23

x(p, e(p, u))

Question 24

x(p,w) = … (Hicksian)

Answer 24

h(p, v(p, w))

Question 25

Under the basic Assumption, What are the properties of the expenditure function?

Answer 25

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

Question 26

Under the basic assumption, what are the properties of the hicksian demand?

Answer 26

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

Question 27

What is the first order condition for the EMP?

Answer 27

p = μ* * ∇u(x*)

Question 28

Why the Hicksian demand is compensated?

Answer 28

(p’’ - p’) * (x’’ - x’) ≤ 0

Question 29

What is the Envelope Theorem?

Answer 29

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

Question 30

if f(θ) is differentiable and L is differentiable with respect to θ at a saddle-point (x , λ*), we have

Answer 30

df(θ)/dθ = dL(x, λ*, θ)/dθ

Question 31

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)

Answer 31

h(p,u) = ∇p e(p,u)

Question 32

Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u).
Then Dp h(p,u) = ?

Answer 32

Dp h(p,u) = D²p e(p,u)

Question 33

Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u).
Dp h(p,u) is...

Answer 33

Dp h(p,u) is negative semidefinite and symetric

Question 34

Suppose that the basic assumption holds and that h(.,u) is a continuou and differentuable function at (p,u).
Dph(p,u)*p = ?

Answer 34

Dph(p,u)*p = 0

Question 35

What is the Slutsky equation?

Answer 35

Dp h(p,u) = Dp x(p,w) + Dw x(p,w) * x(p,w)T

Question 36

By the slutsky equation when u = v(p,w) we have

Answer 36

S(p, w) = Dp h(p,u)

Question 37

Show the Roy’s Indtity

Answer 37

x(p,w) = - 1/(dv(p,w)/dw) * ∇p v(p,w)

Question 38

If a continuously differentiable demand function x(p,w) is generated by maximization of rational preferemce, then it must…

Answer 38

1) Be homogeneous of degree zero
2) Satisfy the Walras’ Law
3) Have a symmetric and negative semidefinite substitution matrix S(p,w)

Question 39

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?

Answer 39

Yes

Question 40

What are the steps to recover the rationa preference?

Answer 40

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