Non-convex preferences Shapley–Folkman lemma

when consumer s preferences have concavities, consumer may jump between 2 separate optimal baskets.

however, if preference set non-convex, prices determine budget-line supports 2 separate optimal-baskets. example, can imagine that, zoos, lion costs as eagle, , further zoo s budget suffices 1 eagle or 1 lion. can suppose zoo-keeper views either animal equally valuable. in case, zoo purchase either 1 lion or 1 eagle. of course, contemporary zoo-keeper not want purchase half of eagle , half of lion (or griffin)! thus, zoo-keeper s preferences non-convex: zoo-keeper prefers having either animal having strictly convex combination of both.

when consumer s preference set non-convex, (for prices) consumer s demand not connected; disconnected demand implies discontinuous behavior consumer, discussed harold hotelling:

if indifference curves purchases thought of possessing wavy character, convex origin in regions , concave in others, forced conclusion portions convex origin can regarded possessing importance, since others unobservable. can detected discontinuities may occur in demand variation in price-ratios, leading abrupt jumping of point of tangency across chasm when straight line rotated. but, while such discontinuities may reveal existence of chasms, can never measure depth. concave portions of indifference curves , many-dimensional generalizations, if exist, must forever remain in unmeasurable obscurity.

the difficulties of studying non-convex preferences emphasized herman wold , again paul samuelson, wrote non-convexities shrouded in eternal darkness ... , according to diewert.

nonetheless, non-convex preferences illuminated from 1959 to 1961 sequence of papers in journal of political economy (jpe). main contributors farrell, bator, koopmans, , rothenberg. in particular, rothenberg s paper discussed approximate convexity of sums of non-convex sets. these jpe-papers stimulated paper lloyd shapley , martin shubik, considered convexified consumer-preferences , introduced concept of approximate equilibrium . jpe-papers , shapley–shubik paper influenced notion of quasi-equilibria , due robert aumann.