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: this 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) and the mathematical model we propose and use is essentially a formal mathematical translation of the ideas presented by him. On the other hand, our approach will allow to apply efficiently and quantitatively the Ehrlich qualitative model. Specifically, in this original paper, we apply the Complete Pareto Analysis of a differentiable decision problem, recently introduced in literature by David Carfì, to examine exhaustively the above Ehrlich-kind decision problem. An Ehrlich-kind decision problem is given by a pair P = (f, >), where the function f : T → E is a vector payoff function defined upon a compact m-dimensional decision (time) constrain T and with values into the m-dimensional payoff space E, for some natural number m (in our paper m is 2). So, the principal aim of this paper is to show how Carfì's Pareto Analysis can help to face, quantitatively, the decision problems of the Ehrlich-type in some practical cases; also, the computational aspects were not considered by Ehrlich. Our methodologies and approaches permit (in principle), by giving a total quantitative view of the possible payoff space of Ehrlich-decision problems (and consequently, giving a precise optimal solutions for the decision- maker), to perform quantitative econometric verifications, in order to test the payoff functions chosen in the various Ehrlich models. In particular, we apply our mathematical methodology to determine the topological boundary of the payoff space of a decision problem, for finding optimal strategies in the participation in such legal and illegitimate market activities. The theoretical framework is clarified and applied by an example.
