Abstract
We extend stochastic newsvendor games with information lag by including dynamic retail prices, and we characterize their equilibria. We show that the equilibrium wholesale price is a nonincreasing function of the demand, while the retailer's output increases with demand until we recover the usual equilibrium. In particular, it is then optimal for retailer and wholesaler to have demand at least equal to some threshold level, beyond which the retailer's output tends to an upper bound which is absent in fixed retail price models. When demand is given by a delayed Ornstein-Uhlenbeck process and price is an affine function of output, we numerically compute the equilibrium output and we show that the lagged case can be seen as a smoothing of the no lag case.
Highlights
The newsvendor problem is a two-staged game in which a wholesaler sells products to a retailer, and the retailer sells to consumers in a market according to a demand constraint
Stochastic differential Stackelberg games have been studied via the maximum principle, and applied to continuous time newsvendor problems with exogenously fixed retail prices in [2], where demand is modeled by an Ito-Levy process, and information about the demand is lagged
In Theorem 3.1 we present necessary and sufficient conditions for optimality of controls based on the stochastic maximum principle in the delayed case with δ > 0, by allowing retail prices to depend on the quantity of products sold between wholesaler and retailer
Summary
The newsvendor problem is a two-staged game in which a wholesaler sells products to a retailer, and the retailer sells to consumers in a market according to a demand constraint. Stochastic differential Stackelberg games have been studied via the maximum principle, and applied to continuous time newsvendor problems with exogenously fixed retail prices in [2], where demand is modeled by an Ito-Levy process, and information about the demand is lagged. In this framework, the retailer can only sell a quantity limited to a level Dt 0 of products at time t ∈ [δ, T ], with δ > 0. Unlike in the Cournot setup [5], we do not observe a collapse from multiple equilibria to a single equilibrium
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