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Optimal Participation in Illegitimate Market Activities: Complete Analysis of 2-Dimensional Cases

David Carfì, Angelica Pintaudi

2012
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Journal of Advanced Research in Law and Economics
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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
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... the ideas presented by him; but, 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: given by a pair P = (f, >), where the function f : T → R m is a vector payoff function defined upon a compact m-dimensional decision (time) constrain T and with values into the m-dimensional payoff space R m , for some natural number m (in our paper m is 2). So, the principal aim of this paper is to show how the D. Carfì Pareto Analysis can help to face, quantitatively, the decision problems of the Ehrlich-type in some practical cases, also because 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 the Ehrlich-decision problems (and consequently, giving a precise optimal solutions for the decisionmaker), 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. Assumption 0. A decision-maker D can participate in two different market activities, say 1 and 2; namely, an illegal one, the activity 1, and a legal one, the activity 2. Assumption 1 (the strategy space). The (universe) strategy space T of the decision maker coincides with the positive cone of the Euclidean space R 2 , that is the Cartesian square (R ≥) 2 of the real non-negative semi-line R≥ , that is the unbounded real interval [0, →[. Interpretation (of the bi-strategy). A strategy t, belonging to the strategy space T, is a pair of nonnegative real numbers (t1, t2), whose first component t1 is the time devoted to the first (illegal) activity and the second component t2 is the time devoted to the second (legal) activity. Assumption 2 (monotonicity of the returns). We assume that the returns of the two activities are increasing functions W1 and W2, both defined on the time semi-line R≥ and with value into R. Specifically, denoted by T1 and T2 the first and second Cartesian projections, respectively, of the strategy universe T (both are obviously equal to the semi-line R≥), we have, for any index i, an increasing mapping Wi : Ti → R in such a way that the returns of the activity i depends only upon the time devoted to i itself. Assumption 3 (uncertainty of illegal returns). We assume that the final returns of the first activity (the illegal one) depend upon the states of the world belonging to the unit interval of the real line [0, 1]. The final return function of the first activity is then a random variable defined on the state space S and with values into the function space F(T1,R), that is, an application L : S → F (T1, R), mapping each state of the world s of S into a function Ls : T1 → R : Ls(t1) = L(s)(t1). Interpretation (of the states of the world). Our interpretation of the state space S is the following: the state of the world s represents a degree of punishment; namely, 0 represents the state of the world in which the offender is totally safe, on the contrary 1 represents the state of the world in which the offender is fully punished. Assumption 4 (certainty of legal returns). We assume that the second activity (the legal one) is safe, in the sense that its final returns depend only on the time devoted to it. Assumption 5 (penalty function). We assume that the function Ls can be defined by the difference for any time t1 in T1, where the function F1 : T1 → R ≥ is an increasing function called penalty function. Definition (payoff random variable). We so can construct a random variable, which we shall call the payoff random variable of the decision-maker, on the set S of states of the world and with values in the function space T R ≥, namely it is the mapping X : S → T R ≥, defined by Xs(t) = W1(t1) -sF1(t1) + W2(t2),

doi:10.2478/v10257-012-0002-5
fatcat:zhxwkqea6bez3peqxaohpp24ae