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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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Inaugurato nel 1967 nel Palazzo delle Scienze all’Eur con l’obiettivo di  dotare Roma di un museo archeologico dell’età postclassica e di promuovere la ricerca su un periodo strategico per lo studio della trasformazione del mondo antico, il Museo espone materiali databili tra il IV ed il XIV secolo provenienti per la maggior parte da Roma e dall’Italia centrale.