Duality and equilibrium prices in economics of uncertainty

Abstract

A random variable (RV) X is given aminimum selling price

$$S_U \left( X \right): = \mathop {\sup }\limits_x \left\{ {x + EU\left( {X - x} \right)} \right\}$$

((S))

and amaximum buying price

$$B_p \left( X \right): = \mathop {\inf }\limits_x \left\{ {x + EP\left( {X - x} \right)} \right\}$$

((B))

whereU(·) andP(·) are appropriate functions. These prices are derived from considerations ofstochastic optimization with recourse, and are calledrecourse certainty equivalents (RCE's) of X. Both RCE's compute the “value” of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of theCsiszár φ-divergence

$$I_\phi \left( {p,q} \right): = \sum\limits_{i = 1}^n {q_i \phi \left( {\frac{{p_i }}{{q_i }}} \right)} $$

a generalized entropy function, measuring the distance between RV's with probability vectors p and q.

The RCES _U was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCEB _P, and apply it to problems ofinventory control (where the attitude towards risk determines the stock levels and order sizes) andoptimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage.