Problem on competitive equilibrium of two consumers

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(xB1 ; xB2 ) = xB1 xB2 .

(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.

E

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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

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