DUAL REPRESENTATIONS OF STRONGLY MONOTONIC UTILITY FUNCTIONS

We present theorems that establish dualities, i.e., bijections, be- tween speci¯ed sets of direct utility functions, indirect utility functions and expenditure functions. The substantive properties characterizing the speci¯ed set of direct utility functions are strong monotonicity, upper semicontinuity and quasi-concavity. Our results are strictly in- termediate between two classes of analogous results in the literature. We also provide applications that use all the three classes of duality results.Direct utility function, indirect utility function, ex-penditure function, duality, strong monotonicity

Strong Duality for a Multiple-Good Monopolist

We characterize optimal mechanisms for the multiple-good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure $\mu$ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand-bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for $n$ independent uniform items each supported on $[c,c+1]$ is a grand-bundling mechanism, as long as $c$ is sufficiently large, extending Pavlov's result for $2$ items [Pavlov'11]. At the same time, our characterization also implies that, for all $c$ and for all sufficiently large $n$, the optimal mechanism for $n$ independent uniform items supported on $[c,c+1]$ is not a grand bundling mechanism

On the super replication price of unbounded claims

In an incomplete market the price of a claim f in general cannot be uniquely identified by no arbitrage arguments. However, the ``classical'' super replication price is a sensible indicator of the (maximum selling) value of the claim. When f satisfies certain pointwise conditions (e.g., f is bounded from below), the super replication price is equal to sup_QE_Q[f], where Q varies on the whole set of pricing measures. Unfortunately, this price is often too high: a typical situation is here discussed in the examples. We thus define the less expensive weak super replication price and we relax the requirements on f by asking just for ``enough'' integrability conditions. By building up a proper duality theory, we show its economic meaning and its relation with the investor's preferences. Indeed, it turns out that the weak super replication price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the class of pricing measures with finite generalized entropy (i.e., E[\Phi (\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Compatibility between pricing rules and risk measures: the CCVaR

Research partially supported by “RD Sistemas SA”, “Comunidad Autónoma de Madrid” (Spain), Grant s-0505/tic/000230, and “MEyC” (Spain), Grant SEJ2006-15401-C0