Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6810415, 9 pages

https://doi.org/10.1155/2017/6810415

## Mathematical Modeling for Risk Averse Firm Facing Loss Averse Customer’s Stochastic Uncertainty

College of Business Administration, Hongik University, Seoul, Republic of Korea

Correspondence should be addressed to Jinpyo Lee

Received 24 January 2017; Revised 18 March 2017; Accepted 26 March 2017; Published 11 April 2017

Academic Editor: Huanqing Wang

Copyright © 2017 Seungbeom Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To optimize the firm’s profit during a finite planning horizon, a dynamic programming model is used to make joint pricing and inventory replenishment decision assuming that customers are loss averse and the firm is risk averse. We model the loss averse customer’s demand using the multinomial choice model. In this choice model, we consider the acquisition and transition utilities widely used by a mental accounting theory which also incorporate the reference price and actual price. Then, we show that there is an optimal inventory policy which is base-stock policy depending on the accumulated wealth in each period.

#### 1. Introduction

Joint control of inventory and price has long been widely used for many firms such as Amazon, Dell, and J. C. Penny [1]. However, as mentioned in [2], traditional inventory control models are mainly concerned with the properties of replenishment policies to optimize the expected total profit or cost during a planning horizon. We can say that the traditional models are good strategies for the risk neutral inventory decision maker who is insensitive to profit or cost variations. However, not all inventory decision makers are risk neutral but frequently risk averse, in which the risk averse inventory decision maker would prefer a certainty equivalent to taking the bet and possibly receiving nothing, where the certainty equivalent is defined as the amount that the decision maker would accept instead of the bet.

For a better operational decision and a successful marketing campaign, a firm’s inventory decision makers should consider customer’s behavior corresponding to the price set by the firm, carefully. Customer’s behavior significantly influences firm’s revenue so that also the firm’s pricing and replenishment decisions are deeply influenced. The firm’s decision makers should construct a good operational and marketing strategy. When you see repeat-purchase markets, consumers have expectation for the price, which is known as reference prices in prospect theory. Customers perceive fluctuating prices as discounts or overcharges relative to the reference prices formed by the previous prices. Moreover, this perception affects demand and thus firm’s profit. For example, while a price discount might have a positive impact on sales on the short-run, the discounted price might result in the installation of a low price in consumers memory, eroding price expectations and willingness to pay and thereby negatively affecting profitability on the long run. It is important for a firm to understand () how consumers expectations for price and decisions for purchasing are affected by its pricing policy and history and () how prices should be set over time to optimize its utility. So, a firm needs to incorporate the behavior of loss averse customers into its strategy, to whom losses loom larger than gains. Since loss averse customers, to whom the disutility of a loss is greater than the utility of an equivalent gain, prudently consider the tradeoff between the perceived reference price from the previous prices and the current price when purchasing products, an unfavorable price is seen as a loss. So, it can significantly reduce customers’ willingness to purchase and finally influence the reduction of retail sales.

So, in this paper, we consider a multiperiod inventory control model in which a risk averse firm faces loss averse customer’s uncertain demand and makes an inventory replenishment and pricing decision by maximizing the firm’s expected utility.

#### 2. Literature Review

We will go over the literature separately to compare with our research. First, the literature on the customer’s behavior will be reviewed with respect to the loss aversion. Second, the literature on the firm’s behavior will be reviewed with respect to the risk neutral utility. Then, finally, the literature on the firm’s behavior will be reviewed with respect to the risk aversion.

There have been lots of research papers regarding the customers’ irrational behavior since Barbara L. Fredrickson and Daniel Kahneman won the Nobel prize for their works on the prospect theory. Reference [3] shows that the decision makers are not rational and do not follow the expected utility theory and develop an alternative model, called prospect theory. In prospect theory, outcomes are valued as gains or losses relative to a current reference point instead of final levels of wealth and suggest that the utility of an equivalent gain is less than the disutility of a loss, which is referred to as loss aversion. Also, they present the concept of certainty effect which contributes to risk aversion over gains and to risk seeking over losses. Reference [4] mentions that consumer’s choice is affected by the brands’ position related to reference points with multiple attributes and that consumers keep their weight on losses from a reference point more than gains in the same amount, which is loss aversion. They develop a Multinomial Logit formulation which incorporates a reference-dependent choice model. Reference [10] addresses a behavioral decision bias in the newsvendor ordering problem: orders for low-profit products were higher than the expected profit-maximizing quantities, while orders for high-profit products were lower than the expected profit-maximizing quantities. They show that any of risk aversion, risk seeking preferences, prospect theory preferences, loss aversion, waste aversion, stockout aversion, or undervaluing opportunity costs cannot explain the bias pattern of ordering decision, but a preference of ex-post inventory error reduction and the anchoring heuristic might explain the bias pattern of ordering decision. Reference [11] proposes a behavioral theory to see the actual ordering decision in multilocation newsvendor problem. They assume that there are psychological costs for stockouts or leftovers and then show that decision makers psychological disutility for stockpots is less strong than that for leftovers. They test whether the pull-to-center bias exists in a multilocation newsvendor problem. Reference [14] proposes a dynamic pricing model based on the peak-end rule and reference price, where loss averse consumers make a purchasing decision depending on the lowest price and the most recent price. Here, as defined in [12], the peak-end rule is a psychological heuristic in which people’s experience is evaluated largely based on how to feel at its peak (its lowest price point) and at its end (its most recent price), rather than based on the summation or average of every experience (past prices). Reference [13] shows that consumer’s loss aversion behavior could result in higher prices and profits when consumer’s valuation is higher enough than his/her search costs and the proportion of consumers with positive search costs is in an intermediate range. Also, they show that when forward-looking firms incorporate the negative effect of price promotions on future profits, the equilibrium range of price promotions may actually increase.

Second, we will see some traditional research papers on the risk neutral firm. Traditionally, many research literatures consider a model in which the firm is risk neutral and the customer is not loss averse. Actually, the demand from the customer is just affected by the list price set by the firm and is nonincreasing in the price. Reference [5] examines a newsvendor problem with risk neutral profit in which replenishment and selling price are decided simultaneously. References [6, 7, 15, 16] address the simultaneous decision problem of pricing and inventory replenishment in the face of demand uncertainty of which distributions depend on the price set by the risk neutral firm. References [8, 9] address an inventory policy and a pricing strategy maximizing risk neutral expected profit given that the demand function is decreasing just in the price set by the firm.

Finally, we will see the literatures on the risk averse firm. The literature on the risk averse inventory control model is quite limited. Reference [17] considers a tradeoff between the stochastic profit’s expected value and its standard deviation to hedge the undesirable uncertainty in stochastic profit, where a degree of risk aversion is reflected by the multiplication of some constant to the standard deviation. Reference [18] examines the effects of risk aversion in the newsboy problem in which comparative-static effects of changes in the various prices and costs are related to the newsboy’s risk aversion. Reference [19] addresses an inventory model in which the objective is to optimize the expected exponential utility of the present value of net profits over time to incorporate the effects of sensitivity to risk. Reference [20] considers a newsvendor model in which a risk averse retailer faces uncertain demand and makes ordering quantity decisions and pricing decision with the objective of optimizing expected risk averse utility. In their model, the distribution of demand is a function of the price set by the risk averse retailer. Reference [2] incorporates risk aversion in multiperiod inventory models that coordinate inventory and pricing strategies.

The dynamic control model is utilized in a wide range of industries [21, 22] and its use is also prevalent in the control of inventory systems [23]. Reference [24] investigates the problem of adaptive tracking control for a class of switched stochastic nonlinear systems in nonstrict-feedback form with unknown nonsymmetric actuator dead-zone and arbitrary switching. Reference [19] formulates the dynamic programming models to solve multiperiod stochastic inventory problems with exponential utility function.

As reviewed above and summarized in Table 1, to the best of our knowledge, there is no research for a model combining the loss averse customer and risk averse firm simultaneously. So, it is pretty much new and will fill the research gap in the behavioral inventory control model.