please some tell me how to solve mathematically such questions?

[PRACHI 214]

Consider an exchange economy with agents A and B and goods x and y. A’s endowment
is (0, 1) (i.e., no good x and 1 unit of good y) and B’s endowment is (2, 0)
(i.e., 2 units of good x and no good y). The agents can consume only nonnegative
amounts of x and y.
4. Suppose A lexicographically prefers x to y and B considers x and y to be perfect
substitutes, i.e., between bundles (x, y) and (x
0
, y0
), she strictly prefers (x, y) if and
only if x + y > x0 + y
0
.
The competitive equilibrium allocation for this economy is
(A) A gets (0,1) and B gets (2,0)
(B) A gets (2,0) and B gets (0,1)
(C) A gets (3/2,0) and B gets (1/2,1)
(D) * A gets (1,0) and B gets (1,1)

6. Now suppose A lexicographically prefers y to x and B considers x and y to be perfect
substitutes.
The set of all possible competitive equilibrium prices consists of all px > 0 and py > 0
such that
(A) px/py = 1
(B) px/py ≥ 1
(C) * px/py ≤ 1
(D) px/py >

[Asd1995]

Each agent maximizes utility w.r.t budget constraint(Write these out. The budget constraint is px*x+py*y=px*(endowment of x)+py*(endowment of y)). From this, after optimizing, you get the demand function of both agents. Now use those to put total demand of each good=total supply of each good. Solve for prices. In such a setup you can always assume price of any one good=1 to make the algebra easier.