Pareto efficiency and Competitive Equilibrium

[Sia]

Agent a utility function min(x,2y) endowment (4,0)
Agent b utility function min(2x+y,x+2y) endowment  (0,3)

Find out:

1- Pareto optimal allocation

2- Competitive equilibrium prices and allocations

3-Utility possibility frontier

 Will the pareto efficient allocations be the y2 axis?
Please help me with a detailed answer.

[Amit Goyal]

Pareto Optimal Allocations will be points on the line yA = xA/2.
Competitive Equilibrium prices are p(x) = 1 p(y) = 2,  (xA=2, yA = 1), (xB=2, yB = 2).
Now figure out the UPF by plotting Edgeworth Box.

[Asd1995]

Sir,

Wouldn't some x(b)=y(b) also not satisfy pareto optimality? For any point to the right of the intersection of y(a)=x(a)/2 and x(b)=y(b) and on the first line distributes utilities in the same manner as any point between the two lines on the same y coordinate? As in, the indifference curves coincide for a set of points.

[Asd1995]

Moreover, if x(b)=y(b), the only way to make person 2 better off is to move on the above line, for if we stray, we're giving him some x or y which doesn't affect his minimum so can be used for person 1s welfare. But if we do move on this line, we're taking away both x and y from the first person. Similarly, if we remain on the same indifference curve of b, we're giving him extra Without changing his utility, so we're taking something away from 1, which can't increase his utility. Hence this is a pareto optimap point.