OPTIMAL PARTICIPATION IN ILLEGITIMATE MARKET ACTIVITIES: COMPLETE ANALYSIS OF 2-DIMENSIONAL CASES

In this paper we consider the quantitative decision problem to allocate a certain amount of time upon two possible market activities, specifically a legal one and an illegal one. The problem was considered in literature by Isaac Ehrlich, in his seminal paper 'Participation in Illegitimate Activities: A Theoretical and Empirical Investigation', published in The Journal of Political Economy in 1973. The mathematical model we propose and use is essentially a formal mathematical translation of the ideas presented by Ehrlich, but, on the other hand, our approach will allow to apply efficiently and quantitatively the Ehrlich qualitative model. We model an Ehrlich decision problem as a pair P = (f, >), where the function f: T ? Rm is a vector payoff function defined upon a compact m-dimensional decision time-constrain T and with values into the m-dimensional payoff space Rm, for some natural number m (in our paper m is 2). Finally, we apply the Complete Analysis of a Differentiable Decision Problem and part of the Complete Analysis of a Differentiable normal-form Game, recently introduced in literature by David Carfì, to examine exhaustively an example of Ehrlich decision problem.