# 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!

• I don't think that MRS = MRTS is a condition for a competitive equilibrium. Are you sure you are not confusing with an equilibrium of pure exchange (cf. Edgeworth's box) where TMS1 (marginal rate of substitution of person 1) = TMS2 (marginal rate of substitution of person 2) ? – Alexis L. May 12 '16 at 22:46
• This is for a 2x2x2x2 economy where there is both production and consumption. So, it is correct. Just wanted to get a clearer understanding of why. – TheEconomist May 13 '16 at 7:09
• I think you're a little off the mark. – 123 May 13 '16 at 19:46

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.