Productive Allocative Efficiency (Competitive Equilibrium) I am doing an Intermediate Microeconomics class... in a 2*2*2*2 economy, I know that MRS (marginal rate of substitution) is supposed to be equal to MRTS (marginal rate of technical substitution) in order for it to be a competitive equilibrium. Why is this? I understand that supply is supposed to be equal to demand... 
Thank you!
Equilibrium conditions will require, among other things,: 

1.) $MRS_{a,b}^i=MRS_{a,b}^j \forall [i,j] \in N$ , $N$ the set of agents

2.) $MRTS_{x,y}^i=MRTS_{x,y}^j \forall [i,j] \in J$ , $J$ the set of firms

3.) $MRT=MRS$ (Assuming that 1,2 hold) 

And I note that 1,2 are efficiency conditions. 3 is efficiency in the output market. 

Why? Suppose 1,2 hold at:


$MRS^i_{y,x}=3$ 
$MRT_{y,x}=2$ 


Obviously, condition 3 does not hold. 

Can you see why this cannot be an equilibrium outcome? Think about what should happen in the economy and the forces that will drive this economy toward a state s.t. condition 3 holds. 



So:

$MRT_{x,y}=\frac{MC_x}{MC_y} \equiv \frac{P_x}{P_y} = MRS$

where I assume 1 holds so MRS is same for all agents.