Suppose an economy with two consumers with the following utility functions:

U^1(x1^11, x2^1) =x1^1+ 4x₂^1

U^2(x₁^2, x₂^2) =x₁^2+ 2x₂^2

And, they are endowed with

e^1=(4,12)e^2=(8,8)

Suppose a firm exist in this economy which can only produce goodx^2using goodx^1as input with production function:

x2= f(x1) = x1(R+)

Assume that each agent has equal share in the profits.

1. Compute a competitive equilibrium (if any exist) which has no (extra) production of goodx2i.e. availability of goodx2remains equal to the endowment

2. Compute a competitive equilibrium (if any exist) which has positive production of goodx2i.e. availability ofx2exceeds endowment and extra availability is produced in equilibrium.