Starr's 1969 paper and contemporary economics Shapley–Folkman lemma

kenneth arrow (1972 nobel laureate) helped ross m. starr study non-convex economies.

previous publications on non-convexity , economics collected in annotated bibliography kenneth arrow. gave bibliography starr, undergraduate enrolled in arrow s (graduate) advanced mathematical-economics course. in term-paper, starr studied general equilibria of artificial economy in non-convex preferences replaced convex hulls. in convexified economy, @ each price, aggregate demand sum of convex hulls of consumers demands. starr s ideas interested mathematicians lloyd shapley , jon folkman, proved eponymous lemma , theorem in private correspondence , reported starr s published paper of 1969.

in 1969 publication, starr applied shapley–folkman–starr theorem. starr proved convexified economy has general equilibria can closely approximated quasi-equilibria of original economy, when number of agents exceeds dimension of goods: concretely, starr proved there exists @ least 1 quasi-equilibrium of prices popt following properties:

for each quasi-equilibrium s prices popt, consumers can choose optimal baskets (maximally preferred , meeting budget constraints).
at quasi-equilibrium prices popt in convexified economy, every s market in equilibrium: supply equals demand.
for each quasi-equilibrium, prices clear markets original economy: upper bound on distance between set of equilibria of convexified economy , set of quasi-equilibria of original economy followed starr s corollary shapley–folkman theorem.

starr established that

in aggregate, discrepancy between allocation in fictitious economy generated [taking convex hulls of of consumption and production sets] , allocation in real economy bounded in way independent of number of economic agents. therefore, average agent experiences deviation intended actions vanishes in significance number of agents goes infinity .

following starr s 1969 paper, shapley–folkman–starr results have been used in economic theory. roger guesnerie summarized economic implications: key results obtained under convexity assumption remain (approximately) relevant in circumstances convexity fails. example, in economies large consumption side, preference nonconvexities not destroy standard results . derivation of these results in general form has been 1 of major achievements of postwar economic theory , wrote guesnerie. topic of non-convex sets in economics has been studied many nobel laureates: arrow (1972), robert aumann (2005), gérard debreu (1983), tjalling koopmans (1975), paul krugman (2008), , paul samuelson (1970); complementary topic of convex sets in economics has been emphasized these laureates, along leonid hurwicz, leonid kantorovich (1975), , robert solow (1987). shapley–folkman–starr results have been featured in economics literature: in microeconomics, in general-equilibrium theory, in public economics (including market failures), in game theory, in mathematical economics, , in applied mathematics (for economists). shapley–folkman–starr results have influenced economics research using measure , integration theory.