Question

Find the Walrasian Equilibrium price and allocations for all the economies

(a) Utilities are u1(x1,x2) = x2√x1, u2(x1,x2) = x1√x2. Endowments are i. e1 = (10,0), e2 = (0,10); ii. e1 = (5,5), e2 = (5,5).

(b) Utilities are u1(x1,x2) = x1 +√x2, u2(x1,x2) = x1 + lnx2. The endowment is e1 = (100,0),e2 = (0,100).

(c) Utilities are u1(x1,x2,x3) = x1(x2 + 1) and u2(x1,x2,x3) = x3(x2 + 1). The endowment is e1 = (5,5,5) and e2 = (5,5,5).

(d) Utilities are u1(x1,x2) = min(x1,x2), u2(x1,x2) = min(x1,x2). The endowment is e1 = (3,6) and e2 = (7,4)

(e) Utilities are u1(x1,x2) = max(x1,x2), u2(x1,x2) = min(x1,x2). The endowment is e1 = (5,7), e2 = (25,3)

Answer 1

a)

i) Let the price of y be numeraire.

Consumers' Problems:

For both consumers, at equilibrium, MRS is equal to price ratio:

Substituting the values of x and y in the consumers' budget constraints:

Demands for goods x and y by both consumers are:

Total demand for x and y is equal to the total endowment:

Therefore, competitive equilibrium allocations are:

ii) For new endowments:

Substituting the values of x and y in the consumers' budget constraints from the previous part:

Demands for goods x and y by both consumers are:

Total demand for x and y is equal to the total endowment:

Therefore, competitive equilibrium allocations are: