A two‐period pricing model with intertemporal and horizontal reference price effects

• DOI: http://doi.org/10.1111/itor.12613
• Published date: 01 May 2021
• Author: Lu Zhang,Yong Zha,Tingting Zhang,Chuanyong Xu
• Date: 01 May 2021

Intl. Trans. in Op. Res. 28 (2021) 1417–1440

DOI: 10.1111/itor.12613

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

A two-period pricing model with intertemporal and horizontal

reference price effects

Yong Zha , Lu Zhang, Chuanyong Xu∗and Tingting Zhang

School of Management, University of Science and Technology of China, No.96 Jinzhai Rd., Hefei Anhui 230026,

P.R. China

E-mail: zhabeer@ustc.edu.cn [Zha]; zlu@mail.ustc.edu.cn [L. Zhang]; xcy@ustc.edu.cn [Xu];

ustcztt@mail.ustc.edu.cn [T. Zhang]

Received 19 March 2018; receivedin revised form 29 October 2018; accepted 31 October 2018

Abstract

Consumers tend to compare the current price with historical prices of the same brand and selling prices of

other brands, when they make purchase decisions. The intertemporal and horizontal reference price effect

(RPE), formed by the historical price and the competitor’s price, respectively, should be taken into account

when developing optimal pricing strategies over several periods and in a competitive environment. This paper

considers a two-period pricing problem with two competing sellers, incorporating both types of RPE. We

first develop a duopoly game to study the impacts and interactions of RPE of different types. Then we study

a practice of price commitment when one firm gives up dynamic pricing. We find different types of RPEs

have distinct impacts. The intertemporal RPE (IRPE) leads to a Hi-Lo pricing strategy, while the horizontal

RPE (HRPE) drives the selling prices of both periods downward. The IRPEis weakened by the HRPE, while

the HRPE is weakened (intensified) in the first (second) period as the IRPE becomes stronger. Also we find

price commitment is not beneficial. On the contrary, both firms may be worse off if one firm makes price

commitment. Furthermore, if one firm decides to make price commitment he should announce the selling

price well in advance.

Keywords: reference price effect; horizontal reference price; intertemporal reference price; price commitment; price

competition

1. Introduction

In practice, reference prices areoften used in retail environments. For example, retailersoften ter m a

price promotion as “previous $x, now $y” or “$x, 50% off.” In other situations, retailers refer to the

selling price elsewhere, such as “$x elsewhere, choose us, save $y,” where the target of comparison

is often their competitors. Referring to (high) historical prices or competitors’ prices usually makes

∗Corresponding author.

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International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

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1418 Y. Zha et al. / Intl. Trans. in Op. Res.28 (2021) 1417–1440

the current deal more appealing. The effect is often called the reference price effect (RPE), which

relates consumer perception of gains (loss) to high (low) reference price.

Consumers tend to compare the current price with historical prices of the same brand and

selling prices of other brands, when they make purchase decisions. In fact, historical prices and

competitors’ prices represent two types of reference prices of interest. Historical prices represent

temporal influence of firm decision while competitors’ prices represent horizontal influence of

competitors’ decision. As a result, we refer to historical prices as intertemporal reference prices

and competitors’ prices as horizontal reference prices. In reality, a seller inevitably faces these two

types of reference prices as long as the seller is in a competitive environment. To fully understand

the effects of reference prices, these two types of reference prices should be incorporated into the

pricing model.

However, these two types of reference prices should not be treated in the same way. Historical

prices affect decision of current prices but not vice versa since history is not reversible. However,

the decision of current prices interacts with competitors’ prices since competition is reciprocal. In

other words, the competitors can observe these promotions, and take relevant action. That is, the

horizontal reference price should not be treated as exogenous when making pricing decision.

In extant pricing models, researchers typically use an aggregated reference price by taking a

weighted average (Briesch et al., 1997) when there are multiple sources of reference prices. This is

plausible only if the reference prices are exogenous. It does not sufficewhen making pricing decision

over several periods and in a competitive environment. In such a setting, reference prices should be

treated as endogenous and the interactions between references prices should be considered. Then,

what is the impact of intertemporal and horizontal reference price? How do they interact with each

other?

In an effort to better understand the impact of intertemporal and horizontal RPE (HRPE), we

build a two-period pricing model incorporating both types of reference price. We begin with a

duopoly setting in which two competing firms sell similar products in two periods. The demand of

each firm is affected by both types of RPE. We investigate the optimal pricing decision of the firms

in the duopoly setting. We find that in equilibrium, the two types of RPE have distinct impacts on

the pricing decision of the firms. The intertemporal RPE (IRPE) induces firms to set a higher price

in the first period and then a lower price in the latter period to stimulate demand. The price structure

is always of Hi-Lo type, and the reference gap is increasing in the strength of IRPE. However, the

HRPE drives the price of both periods downward. The stronger the HRPE is, the lower the prices

are. In addition, the impact of referenceprice of one type is affected by the other. With the existence

of IRPE, the competition is alleviated in the first period but intensified in the second period. While

with the existence of HRPE, the reference price gap is reducedand the base price decreases, implying

the impact of IRPE is weakened by HRPE.

We then investigate the features of price discount in equilibrium and find that the price discount

in the second period will not be greater than 40%, and will always be lower than the case without

HRPE. Moreover, the demand and profit in the second period is always increasing in the strength

of IRPE. With the existence of HRPE, the demand will be greater as well.

Further, we discuss a marketing practice where one firm commits to a constant price and the

other varies selling price over time. We find that the price of the firm who makes price commitment

is lower than that of his competitor in the first period and is higher in the second period. Also, his

profit is lower than his competitor. That is, the firm who commits hurts himself, while benefits the

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2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies