Question: 1. provide a careful proof:
(a) Prove that a firm's cost function, c(w,y), is concave in w.
(b) Recall the lexicographic preference ordering on X⊂R_+^2: for any x,y∈X,x?y iff either
(i) x_1>y_1or
(ii) x_1=y_1 and x_2≥y_2. Prove that the lexicographic ordering is complete and transitive.
(c) Consider an n-good exchange economy(?^i,e^i )_(i∈Ι) where Ι = {1, ..., I}. Assume that each ?^i satisfies local non-satiation and can be represented by a utility function ui. Prove that any Walrasian equilibrium allocation (WEA) of this economy is Pareto efficient.