8 of 16 - + Automatic Zoom Topic III. Producer Theory (2) (3) r(p, w) is convex in (p, w), i.e., let (p",w") = (tp+ (1-t) p',tw+(1-t)w') for te [0,1], then T(p" , W) Str(p, w) + (1-t)(p' ,w) Proof: Let x" be the solution to (4.1) at (p", w"). T(p" , w") = max{[tp + (1-t) p']f(x) -[tw + (1-t)w') ] . x} =[tp + (1-t) p']f(x") -[tw + (1-t) w')].x" =t[pf (x") - w.x" ] + (1 -t)[p'f(x") - w.x" ] 4