questions regarding exchange

[Simran.A]

Hi. i'm getting confused .. If possible please suggest me the answer.
the question goes as -

give examples of the following using edge worth box.
1. an exchange economy with no competitive equilibrium relative to particular initial endowments.

2. an exchange economy with an infinite number of competitive equilibria relative to particular initial endowments.

3. a Pareto optimum which cannot be sustained as a competitive equilibrium.

4. a competitive equilibrium which is not Pareto optimal.

[Amit Goyal]

I will use the following notations to specify the economy
Set of individuals, N = {U, V},
Two goods, X and Y
Utility function of U, u(x, y)
Utility function of V, v(x, y)
Endowment is, E = (e(u), e(v)) where e(u) is endowment vector of U and e(v) is endowment vector of V.

1. an exchange economy with no competitive equilibrium relative to particular initial endowments.
u(x, y) = max{x, y}
v(x, y) = min{x, y}
E = ((2, 0), (0, 2))

2. an exchange economy with an infinite number of competitive equilibria relative to particular initial endowments.
u(x, y) = min{x, y}
v(x, y) = min{x, y}
E = ((2, 0), (0, 2))

3. a Pareto optimum which cannot be sustained as a competitive equilibrium.
u(x, y) = max{x, y}
v(x, y) = min{x, y}
E = ((2, 0), (0, 2))
E is Pareto Optimal but cannot be sustained as a competitive equilibrium.

4. a competitive equilibrium which is not Pareto optimal.
u(x, y) = 0
v(x, y) = x + y
E = ((2, 0), (0, 2))
E is a competitive equilibrium at prices (1, 1) but is not Pareto optimal.

[Simran.A]

thank you sir :)

[maahi]

can u explain the rationale behind this answer .