Acl

input 1 Figure 53.6(d)

5.C.1 If there is a production plan y € Y with p-y > 0, then, by using ay e Y with a large a > 1, it is possible to attain any sufficiently large profit level. Hence ir[p) - If, on the contrary, p*y ^ 0 for all y e Y, then 7r(p) ^ 0. Thus we have either 7r(p) = + ® or 7r(p) ^ 0.

5.C.2 Let p » 0, p' » 01 a € [0,1], and y € yiap + (1 - cOp1), then p-y * 7i(p) and p1-y ^ nip'). Thus,«

(ap + (I - a)p')-y = ap-y + {1 - a)p?'y ^ anip) + (1 - <x;7r(pf). Since (ap + (1 - a}p')*y *= Tr(ap + (1 - a)p'),

5.C.3 The homogeneity of dO in q is implied by that of z(0. We shall thus prove this latter homogeneity only. Let w » 0, q ^ 0, and a > 0. Let z e z(w,q). Since /[-) is homogeneous of degree one, f{az) = afiz) £ aq. For every z' € \ if f[zy) ± aq, then /{a *z') ~ a ^fiz) £ q. Thus, by z €

z(w,q), w* (a lz) ^ wz. Hence wz' 21 w-(az). Thus az e ziw.aq). So aziw, c z(w,aq). By applying this inclusion to a"1 in place of a and aq in place of q, we obtain a ^(w.aq) c z(w,a ^aq)), or z(w,aq) c az(w,q) and thus conclude that ziw,aq) = a2(w,q).

We next prove property (viii). Let w e q - 0, q' ^ 0, and a e

[0,1], Let 2 € 2(wfq) and zr € z(w,q'}. .Then jf(z) q, f(z') z q\ c(wfq) = w- z, and c(w,q') = w-z\ Hence ac(w,q) + (1 - a)c(w,q') = a(w-z) + (1 - a)(w-zT) = w-(az + (2 - <x)z'}. Since /(■) is concave,

/(az + (1 - a)z') > a/(z) + (1 - a)f(z') * aq + (1 - a)q\ Thus w*(az + [1 - a)z') c(w, aq + (1 - a)q'). That is, oc(w, q) + (1 - a)c(w, q') 2: c(wt aq + (1 - a)q').

5.C.4 (First printing errata: When there are multiple outputs, the function f[z) need not be well defined because it is conceivably possible to produce different combinations of outputs from a single combination of inputs. Assuming that the first L - M commodities are inputs and the last M commodities are outputs, we should thus understand the set {z £ 0- fiz) ^ q}

as (z € P^ (- z, q) € Y)J For each q i 0, define ■

Y(q) = {z € {- z, q) € Y> = {z e /lz) £ q>.

Then c(-,q) is the support function of Y(q) for every q. Hence property (ii)

follows from the discussion of Section 3.F. Moreover, according to Exercise

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