Perfect complement preferences in an exchange economy
Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$.



Competitive Equilibrium is the price $(p_x, p_y=1)$ satisfying the following conditions:


Budget Requirement: $p_xx_J+ y_J = 2p_x + 2$ and $p_xx_D+ y_D = 4p_x + 1$
Equilibrium Conditions: $\displaystyle\frac{2y_J}{x_J} = p_x$ and $y_D=\displaystyle\frac{x_D}{2}$


Solving the system of equations we get equilibrium price vector as:
$(p_x, p_y) = (1,1)$
and the equilibrium allocation is
$((x_J, y_J), (x_D, y_D)) = \displaystyle \left(\left(\frac{8}{3}, \frac{4}{3}\right), \left(\frac{10}{3}, \frac{5}{3}\right)\right)$