A stochastic newsvendor game with dynamic retail prices

Ido Polak; Nicolas Privault

doi:10.3934/jimo.2017072

Article Contents

2018, Volume 14, Issue 2: 731-742. Doi: 10.3934/jimo.2017072

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A stochastic newsvendor game with dynamic retail prices

Ido Polak and
Nicolas Privault^,

School of Physical and Mathematical Sciences Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link Singapore 637371

* Corresponding author: Nicolas Privault
* Corresponding author: Nicolas Privault

Received: October 31, 2016

Revised: February 28, 2017

Early access: August 2017

Published: March 2018

This research was supported by Singapore MOE Tier 1 Grant MOE2015-T1-2-130 RG122/15

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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.

Keywords:

Mathematics Subject Classification: Primary: 91A05; Secondary: 91A10, 91B24, 93E20.

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Figure 1. Retailer output $d \longmapsto q^*_t(d,w) = \psi_d (w)$ in (38) with $w=0.5$

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Figure 2. Wholesaler profit $w\longmapsto wq^*_t(d,w) = w \psi_d (w)$ with $d=0.2 < a/(4b)$

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Figure 3. Wholesaler profit $w\longmapsto wq^*_t(d,w) = w \psi_d (w)$ with $d=0.35>a/(4b)$

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Figure 4. Wholesaler vs retailer profits (46) and (47) as functions of $d_t$ with $\delta=0$

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