I'm trying to solve the following problem on general equilibrium:
Consider an economy with two individuals with utility functions $u^A(x^A,y^A) = \min \{ x^A, y^A \}$ and $u^B(x^B,y^B) = \min \{ x^B, y^B \}$, with initial endowments $w^A = (1,0)$ and $w^B = (0,1)$. First find the set of Pareto efficient (PE) allocations, then the set of core allocations in the unreplicated economy, $C_1$, and finally in the twofold replica, $C_2$.
My take:
Pareto Set = $\{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$.
Because $I = 2$ (the number of agents is 2), then Contract Curve = Core. Here the Contract Curve is the same as the Pareto Set. Then:
$C_1 = \{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$.
In the twofold replica now I've got two agents of each type and total endowments are $w^A = (2,0)$, $w^B = (0,2)$. Because of equal treatment in the core, $x^A_1 = x^A_2$ and $y^A_1 = y^B_2$, the same with the type-B individuals. So nothing change and the core is the same?
Shouldn't the core shrinks as the economy enlarges?
