Two firm two consumer economy efficiency (two goods)

[Asd1995]

For pareto efficiency (given everything is convex, returns to scale are <=1)

1. MRS of all consumers=Marginal rate of transformation (slope of production possibilites frontier)=p(x)/p(y)

2. MRTS (Rate of technical substitution) for all producers=wage rate/rental rate

Is it necessary that all conditions in 1 and 2 are equal among themselves too? As in, is MRS=MRTS=...==...?

Also, for 2nd welfare theorem to hold, do we require the PPF to be a convex set, or each individual production functions to be convex set?

[Daffodils]

Mrs= mrts
And I'm not sure but I think individual production function should be convex as well

[Asd1995]

https://economics.stackexchange.com/questions/5385/does-a-general-equilibrium-here-require-pareto-optimality/15165#15165

Take a look at this. The answer shows that MRS (=4 )may not equal MRTS for firms (=w/r).
I'm confused. Could you resolve this for me?

[Asd1995]

Also sounds awkward because we are equating producer profit maximisation with consumer preferences

[Daffodils]

I'll get back to u if I get a clue