To show quasi-convexity, assume that v(p,w)≤ v and v(p’,w′)≤y. For any a E [O, 1] consider the price wealth pair )=(ap+(1-a)p,a+(1-a)n) Assume that v(p, w)is not quasi convex L e. there exists an x, such that ap·x+(1-a)p·x≤1+(1-a) but u(x)>y If u(x)>y, then x must not have been affordable at the old budget sets (otherwise it would have been chosen and would have yielded higher utility), which implies that p·x> w and p'°x>wTo show quasi-convexity, assume that vp,w  v and vp ,w   v . For any   0, 1 consider the price wealth pair p ,w  p  1  p ,w  1  w  Assume that vp,w is not quasi convex, i.e. there exists an x, such that p  x  1  p  x  w  1  w , but ux  v. If ux  v, then x must not have been affordable at the old budget sets (otherwise it would have been chosen and would have yielded higher utility), which implies that p  x  w and p  x  w