Problem on competitive equilibrium of two consumers

Question

The economy consists of two consumers, A and B. Both consumers are endowed with one unit of good 1 and one unit of good 2. Consumer A is entirely indifferent between all consumption plans. Consumer B has the utility function u(x^B₁ ; x^B₂ ) = x^B₁
x^B₂ .

(i) Find a competitive equilibrium for this economy. (Hint: guess an equilibrium price and then check that it works.)

(ii) Find a second competitive equilibrium (different from the one you found in part (i)).

(iii) Show that the equilibrium is not efficient.

Answer 1

Budget constraint for person 1:

P₁x₁ +p₂x₂ = p₁(1) + p₂(2)

Similarly  for person 2:

P₁x₁+ p₂x₂= p₁(1) + p₂(2)

Now person 1 will consume according to the MRS= Price Ratio condition and we get:

So we get:
X₁* = 1/2 (p₁₊₁)/p₁
X₂*= 1/2 (p₁₊₁)/p₂

Similarily for person 2:
X₁*= X₁* = 1/2 (p₁₊₁)/p₁
X₂*= 1/2 (p₁₊₁)/p₂

Now total x₁* in economy is 2

So p₁₊₁/p₁= 2
P₁₊₁= 2P₁
So, p₁*=1
And p₂=1 (numeraire)

So competitive equilibrium:
(1,1) – Person 1
(1,1)- Person 2

b) second competitive equilibrium

(1/3, 2/3) and( 2/3, 1/3)

c) For efficiency MRS₁= MRS₂

Now x₂/x₁= x₂/x₁
For above(Put the values 1/3, 2/3 and 2/3,1/3 in MRS condition 2 not equal to 1

So this is not efficient