Exercise 5.c.3 from microeconomics theory by Andreu Mas-Colell: Establish properties: viii: if f(?) is homogeneous degree one and iX: if f(?) is concave, then c(?) is convex function of q.

Proposition 5.C.2: Suppose that c(w, q) is the cost function of a single-output technology Y with production function f( ) and that z(w, q) is the associated 8. Note. however, that the first-order conditions are sufficient for a solution to the CMP as long as the set (2: S(=) 2 q) is convex. Thus, the key condition for the sufficiency of the first-order conditions of the CMP is the quasiconcavity of f(). This is an important fact because the quasiconcavity of f(. ) is compatible with increasing returns to scale (see Example 5.C.I). SECTION 5. C: PROFIT MAXIMIZATION AND COST MINIMIZATION 141 conditional factor demand correspondence. Assume also that Y is closed and satisfies the free disposal property. Then (i) c(.) is homogeneous of degree one in w and nondecreasing in q. (ii) c( .) is a concave function of w. (iii) If the sets {z 2 0: f(z) > q) are convex for every q, then Y = {(-z, q): w.z> c(w, q) for all w >> 0;. (iv) z( .) is homogeneous of degree zero in w. (v) If the set {z 2 0: f(z) > q) is convex, then z(w, q) is a convex set. Moreover, if {z 2 0: f(z) > q) is a strictly convex set, then z(w, q) is single-valued. (vi) (Shepard's lemma) If z(w,q) consists of a single point, then c(.) is differentiable with respect to w at w and V,c(w, q) = z(w, q). (vii) If z( .) is differentiable at w, then Dwz(w, q) = Dwc(w, q) is a symmetric and negative semidefinite matrix with D,z(w, q) w = 0. (viii) if f( .) is homogeneous of degree one (i.e., exhibits constant returns to scale), then c(.) and z(.) are homogeneous of degree one in q. (ix) If f( .) is concave, then c(.) is a convex function of q (in particular, marginal costs are nondecreasing in q)