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Two firm two consumer economy efficiency (two goods)

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Two firm two consumer economy efficiency (two goods)

Asd1995
286 posts
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?
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Re: Two firm two consumer economy efficiency (two goods)

Daffodils
93 posts
Mrs= mrts
And I'm not sure but I think individual production function should be convex as well
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Re: Two firm two consumer economy efficiency (two goods)

Asd1995
286 posts
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?
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Re: Two firm two consumer economy efficiency (two goods)

Asd1995
286 posts
In reply to this post by Daffodils
Also sounds awkward because we are equating producer profit maximisation with consumer preferences
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Re: Two firm two consumer economy efficiency (two goods)

Daffodils
93 posts
In reply to this post by Asd1995
I'll get back to u if I get a clue