  Pricing decisions are often made when market information is still poor. In
turn, existing theoretical models often reason about the response of optimal
prices to changing market characteristics without exploiting all available
information about the demand distribution. Our aim is to develop a theory for
the optimization and systematic comparison of prices between different
instances of the same market under various forms of knowledge about the
corresponding demand distributions. We revisit the classic problem of monopoly
pricing under demand uncertainty in a vertical market with an upstream supplier
and multiple forms of downstream competition between arbitrary symmetric
retailers. In all cases, demand uncertainty falls to the supplier who acts
first and sets a uniform price before the retailers observe the realized demand
and place their orders. Our main methodological contribution is that we express
the price elasticity of expected demand in terms of the mean residual demand
(MRD) function of the demand distribution. This leads to a closed form
characterization of the points of unitary elasticity that maximize the
supplier's profits and the derivation of a mild unimodality condition for the
supplier's objective function that generalizes the widely used increasing
generalized failure rate (IGFR) condition. A direct implication is that optimal
prices between different markets can be ordered if the markets can be
stochastically ordered according to their MRD functions or equivalently, their
elasticities. Using the above, we develop a systematic framework to compare
optimal prices between different market instances via the rich theory of
stochastic orders. This leads to comparative statics that challenge previously
established economic insights about the effects of market size, demand
transformations and demand variability on monopolistic prices.
