Optimal pricing and inventory management for a loss averse firm when facing strategic customers

Ruopeng Wang; Jinting Wang; Chang Sun

doi:10.3934/jimo.2018019

Article Contents

2018, Volume 14, Issue 4: 1521-1544. Doi: 10.3934/jimo.2018019

This issue Previous Article Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process Next Article Analysis of a dynamic premium strategy: From theoretical and marketing perspectives

Optimal pricing and inventory management for a loss averse firm when facing strategic customers

1.
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
2.
Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China

* Corresponding author: Jinting Wang
* Corresponding author: Jinting Wang

Received: January 31, 2017

Revised: June 30, 2017

Early access: January 2018

Published: October 2018

This work is supported in part by the National Natural Science Foundation of China (Grant nos. 71571014,71390334.)

Abstract Full Text(HTML) Figure(6) / Table(3) Related Papers Cited by

Abstract

This paper considers the joint inventory and pricing decision problem that a loss averse firm with reference point selling seasonal products to strategic consumers with risk preference and decreasing value. Consumers can decide whether to buy at the full price in stage 1, or to wait till stage 2 for the salvage price. They may not get the product if the product is sold out in stage 2. The firm aims to choose a base stock policy and find an optimal price to maximize its expected utility, while consumers aim to decide whether to buy or wait strategically for optimizing their payoffs. We formulate the problem as a Stackelberg game between the firm and the strategic consumers in which the firm is the leader. By deriving the rational expectation equilibrium, we find both the optimal stocking level and the full price in our model are lower than those in the classical model without strategic consumers, by which leads to a lower profit. Furthermore, it is shown that the reimbursement contract cannot alleviate the impact of strategic behavior of customers while the firm's profit can be improved by the price commitment strategy in most cases. Numerical studies are carried out to investigate the impact of strategic customer behavior and system parameters on the firm's optimal decisions.

Keywords:

Mathematics Subject Classification: Primary: 90B05, 90B60; Secondary: 91B42.

Citation:

Full Text(HTML)

Figure 1. Two-stage decision model

Download: Full-size image PowerPoint slide

Figure 2. Impact of customer's risk preference on the firm's decisions in different $\alpha$ and $\delta$

Download: Full-size image PowerPoint slide

Figure 3. Effect of decreasing rate on the firm's decision variables in different values $\alpha$ and $\lambda$

Download: Full-size image PowerPoint slide

Figure 4. The firm's expected profit changes with $\alpha$ and $\lambda$

Download: Full-size image PowerPoint slide

Figure 5. The pattern of the firm's expected profit.

Download: Full-size image PowerPoint slide

Figure 6. The pattern of the firm's expected profit

Download: Full-size image PowerPoint slide

Table 1. Classification of literature on pricing and inventory control with strategic customers

Contributions	Risk preference of Customers	Decreasing value	Loss aversion
Su & Zhang (2008)	-	-	-
Liu & Van (2008)	$\surd$	-	-
Aviv & Pazgal (2008)	-	$\surd$	-
Du, Zhang & Hua (2015)	$\surd$	$\surd$	-
This paper	$\color{red}\surd$	$\color{red}\surd$	$\color{red}\surd$

| Show Table

DownLoad: CSV

Table 2. Parameters and notations

Notation	Description
$p_{0}^{}$, $p_{\alpha}^{}$ and $p_{r}^{*}$	The full price of unit product in classical model, the model with strategic customers and the model with reimbursement contract, respectively in period 1
$Q_{0}^{}$, $Q_{\alpha}^{}$ and $Q_{r}^{*}$	The stocking quantity in classical model, the model with strategic customers and the model with reimbursement contract, respectively
$Q$, $p$	Decision variables denoting stocking quantity and full price, respectively
$D$	Nonnegative and independent random variable, which indicates customers' demand
$F(x)$	Cumulative distribution function, characterizing the demand, and tail distribution is $\overline{F}(x)]=1-F(x)$
$G(x)$	Partial expectation of random $D$, which is defined as $G(x)=\int_{0}^{x}Df(D)dD$
$s$	Salvage price in period 2
$c$	Unit procurement cost of the product to the firm
$V$	Customers' valuation for the unit production
$r$	Customers' reservation price or maximum price which the customers are willing to pay
$\xi_{r}$	The firm's belief over customers' reservation price
$\xi_{prob}$	Customers' belief from obtaining the product on the salvage market
$\delta$	The decreasing rate ($0<\delta\leq1$)
$\lambda$	Customers' risk preference ($\lambda>0$)
$\alpha$	The firm's loss aversion ($\alpha\geq1$)
$E(\cdot)$	Expectation operator
$U(\cdot)$	Utility function of the firm
$x^{+}$ and $x^{-}$	The maximum and minimal function between $0$ and $x$, respectively. $x^{+}=\max\{0, x\}$ and $x^{-}=\min\{0, x\}$

| Show Table

DownLoad: CSV

Table 3. Numerical results for various systems of expected profit when $s = 2$ and $V = 15$

		$\alpha=1$			$\alpha=2$			$\alpha=3$
		$\delta=0.25$	$\delta=0.5$	$\delta=1$	$ \delta=0.25 $	$\delta=0.5$	$\delta=1$	$\delta=0.25$	$\delta=0.5$	$\delta=1$
		465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	$\Pi_{0}^{*}$
	$\lambda=0.5$	$\color{red} {\pmb {406.6204}} $	298.9421	162.4174	$\color{red}{\pmb {408.1164}} $	304.5730	175.2384	$\color{red}{\pmb {409.3841}} $	309.4728	185.7115	$\Pi_{\alpha}^{*}$
		$\color{green}{\pmb {403.3333}} $	403.3333	403.3333	$\color{green}{\pmb {402.8578}} $	402.8578	402.8578	$\color{green}{\pmb {401.5556}} $	401.5556	401.5556	$\Pi_{p}^{*}$
		465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	$\Pi_{0}^{*}$
$c=4$	$\lambda=1$	394.9592	260.6531	94.1742	395.8801	264.8910	105.3170	396.6050	268.5129	113.9793	$\Pi_{\alpha}^{*}$
		403.3333	403.3333	403.3333	402.8578	402.8578	402.8578	401.5556	401.5556	401.5556	$\Pi_{p}^{*}$
		465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	465.3846	$\Pi_{0}^{*}$
	$\lambda=2$	388.0647	235.1442	46.7179	388.5513	237.8978	54.9701	388.8571	240.1336	60.9781	$\Pi_{\alpha}^{*}$
		403.3333	403.3333	403.3333	402.8578	402.8578	402.8578	401.5556	401.5556	401.5556	$\Pi_{p}^{*}$
		384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	$\Pi_{0}^{*}$
	$\lambda=0.5$	$\color{red}{\pmb {338.1945}} $	256.3412	154.3494	$\color{red}{\pmb {340.4831}} $	264.0884	169.3020	$\color{red}{\pmb {341.9950}} $	270.0823	180.6548	$\Pi_{\alpha}^{*}$
		$\color{green}{\pmb {333.3333}} $	333.3333	333.3333	$\color{green}{\pmb {332.4099}} $	332.4099	332.4099	$\color{green}{\pmb {330}} $	330	330	$\Pi_{p}^{*}$
		384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	$\Pi_{0}^{*}$
$c=5$	$\lambda=1$	323.4039	212.4169	84.3076	324.9323	218.7253	96.7836	325.7676	223.4203	105.7881	$\Pi_{\alpha}^{*}$
		333.3333	333.3333	333.3333	332.4099	332.4099	332.4099	330	330	330	$\Pi_{p}^{*}$
		384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	384.6154	$\Pi_{0}^{*}$
	$\lambda=2$	313.9245	180.8531	36.9033	314.7191	185.1787	45.1564	314.8643	188.0670	50.7402	$\Pi_{\alpha}^{*}$
		333.3333	333.3333	333.3333	332.4099	332.4099	332.4099	330	330	330	$\Pi_{p}^{*}$
		311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	$\Pi_{0}^{*}$
	$\lambda=0.5$	$\color{red}{\pmb {275.9692}} $	214.7582	138.5999	$\color{red}{\pmb {278.5443}} $	223.2811	153.8583	$\color{red}{\pmb {279.5104}} $	228.8967	164.4633	$\Pi_{\alpha}^{*}$
		$\color{green}{\pmb {270}} $	270	270	$\color{green}{\pmb {268.5185}} $	268.5185	268.5185	$\color{green}{\pmb {264.8633}} $	264.8633	264.8633	$\Pi_{p}^{*}$
		311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	$\Pi_{0}^{*}$
$c=6$	$\lambda=1$	259.6189	169.7678	71.4952	261.4923	177.1413	83.8399	261.8907	181.7852	92.0999	$\Pi_{\alpha}^{*}$
		270	270	270	268.5185	268.5185	268.5185	264.8633	264.8633	264.8633	$\Pi_{p}^{*}$
		311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	311.5384	$\Pi_{0}^{*}$
	$\lambda=2$	248.2615	135.5270	27.4710	249.1956	140.7354	34.7699	248.7501	143.5081	39.4236	$\Pi_{\alpha}^{*}$
		270	270	270	268.5185	268.5185	268.5185	264.8633	264.8633	264.8633	$\Pi_{p}^{*}$
		96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	$\Pi_{0}^{*}$
	$\lambda=0.5$	$\color{red}{\pmb {32.8174}} $	29.5585	24.7304	$\color{red}{\pmb {31.7544}} $	$\color{red}{\pmb {29.7348}} $	26.4622	$\color{red}{\pmb {29.1689}} $	$\color{red}{\pmb {27.8619}} $	$\color{red}{\pmb {25.6263}} $	$\Pi_{\alpha}^{*}$
		$\color{green}{\pmb {30}} $	30	30	$\color{green}{\pmb {28.2369}} $	$\color{green}{\pmb {28.2369}} $	28.2369	$\color{green}{\pmb {25.4313}} $	$\color{green}{\pmb {25.4313}} $	$\color{green}{\pmb {25.4313}} $	$\Pi_{p}^{*}$
		96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	$\Pi_{0}^{*}$
$c=12$	$\lambda=1$	27.4769	17.5997	8.61672	27.3621	19.2783	10.8693	$\color{red}{\pmb {25.6164}} $	19.0687	11.674	$\Pi_{\alpha}^{*}$
		30	30	30	28.2369	28.2369	28.2369	$\color{green}{\pmb {25.4313}} $	25.4313	25.4313	$\Pi_{p}^{*}$
		96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	96.1538	$\Pi_{0}^{*}$
	$\lambda=2$	20.6633	6.42676	0.8171	20.8133	7.6716	1.2734	19.6287	7.9764	1.5628	$\Pi_{\alpha}^{*}$
		30	30	30	28.2369	28.2369	28.2369	25.4313	25.4313	25.4313	$\Pi_{p}^{*}$
Note: The expected profits are the classical inventory model, the proposed model and the model under price commitment strategy in turn. We mark by red and green color when the expected profit of our model is larger than that of under price commitment strategy model.

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	S. Anily and R. Hassin , Pricing, replenishment, and timing of selling in a market with heterogeneous customers, International Journal of Production Economics, 145 (2013) , 672-682.
	Y. Aviv and A. Pazgal , Optimal pricing of seasonal products in the presence of forward-looking consumers, Manuf. Serv. Oper. Manag., 10 (2008) , 339-359.
	O. Baron , M. Hu , S. Najafi-Asadolahi and Q. Qian , Newsvendor selling to loss-averse consumers with stochastic reference points, Manuf. Serv. Oper. Manag., 17 (2015) , 456-469.
	O. Besbes and I. Lobel , Intertemporal price discrimination: Structure and computation of optimal policies, Management Science, 61 (2015) , 92-110.
	J. I. Bulow , Durable-goods monopolist, J. Political Econom, 90 (1982) , 314-332.
	G. P. Cachon and R. Swinney , Purchasing, pricing, and quick response in the presence of strategic consumer behavior, Management Science, 55 (2009) , 497-511.
	G. P. Cachon and R. Swinney , The value of fast fasion: Quick response, enhanced design, and strategic consumer behavior, Management Science, 57 (2011) , 778-795.
	R. H. Coase , Durability and monopoly, J. Law Econom., 15 (1972) , 143-149.
	J. R. Correa , R. Montoya and C. Thraves , Contingent preannounced pricing policies with strategic consumers, Operations Research, 64 (2016) , 251-272. doi: 10.1287/opre.2015.1452.
	J. Du , J. Zhang and G. Hua , Pricing and inventory management in the presence of strategic customer with risk preference and decreasing value, International Journal of Production Economics, 164 (2015) , 160-166.
	L. Eeckhoudt , C. Gollier and H. Schlesinger , The risk-averse (and prudent) newsboy, Management Science, 41 (1995) , 786-794.
	X. D. He and X. Y. Zhou , Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science, 57 (2011) , 315-331.
	X. D. He and X. Y. Zhou , Myopic loss aversion, reference point, and money illusion, Quant. Finance, 14 (2014) , 1541-1554. doi: 10.1080/14697688.2014.917805.
	P. Heidhues and B. Koszegi, The Impact of Consumer Loss Averse on Pricing, Woking paper, University of California, Berkeley, 2005.
	P. Heidhues and B. Koszegi , Competition and price variation when consumers are loss averse, Amer. Econom. Rev., 98 (2008) , 1245-1268.
	P. Heidhues and B. Koszegi , Regular prices and sales, Theor. Econom., 9 (2014) , 217-251. doi: 10.3982/TE1274.
	D. Kahneman and A. Tversky , Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979) , 263-291.
	B. Keren and J. S. Pliskin , A benchmark solution for the risk-averse newsvendor problem, Eur. J. Oper. Res., 37 (2006) , 1463-1650.
	V. Kobberling and P. P. Wakker , An index of loss aversion, J. of Economic Theory, 122 (2005) , 119-131. doi: 10.1016/j.jet.2004.03.009.
	B. Koszegi and M. Rabin , A model of reference-dependent preferences, Quart. J. Econ., 121 (2006) , 1133-1165.
	J. Li , N. Granedos and S. Netessine , Are consumers strategic? structural estimation from the air travel industry, Management Science, 60 (2014) , 2114-2137.
	Q. Liu and G. J. van Ryzin , Strategic capacity rationing to induce early purchases, Management Science, 54 (2008) , 1115-1131.
	X. Y. Long and J. Nasiry , Prospect theory explain newsvendor behavior: The role of reference points, Management Science, 60 (2014) , 1057-1062.
	J. F. Muth , Rational expectations and the theory of price movements, Econometrica, 29 (1961) , 315-335.
	M. Nagarajan and S. Shechter , Prospect theory and the newsvendor problem, Management Science, 60 (2014) , 1057-1062.
	M. Pervin , S. K. Roy and G. W. Weber , Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 253 (2016) , 1-24.
	M. Pervin , S. K. Roy and G. C. Mahata , An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016) , 243-251.
	M. Pervin , S. K. Roy and G. W. Weber , A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017) , 21-50. doi: 10.3934/naco.2017002.
	Porteus and L. Evan, Foundations of Stochastic Inventory Theory, Stanford University Press, 2002.
	P. Ray and M. Jenamani , Mean-variance analysis of souring decision under disruption risk, Eur. J. Oper. Res., 250 (2015) , 679-689. doi: 10.1016/j.ejor.2015.09.028.
	J. Rubio-Herrero , M. Baykal-Gursoy and A. Jaskiewicz , A price setting newsvendor problem under mean-variance criteria, Eur. J. Oper. Res., 247 (2015) , 575-587. doi: 10.1016/j.ejor.2015.06.006.
	Z. J. Shen and X. Su , Customer behavior modeling in revenue management and auctions: A review and new research opportunities, Production and Operations Management, 16 (2007) , 713-728.
	Y. Shi , X. Y. Cui , J. Yao and D. Li , Dynamic trading with reference point adaptation and loss aversion, Operations Research, 63 (2015) , 789-806. doi: 10.1287/opre.2015.1399.
	X. Su , Intertemporal pricing with strategic customer behavior, Management Science, 53 (2007) , 726-741.
	X. Su and F. Zhang , Strategic customer behavior, commitment, and supply chain performance, Management Science, 54 (2008) , 1759-1773.
	N. L. Stokey , Rational expectation and durable goods pricing, The Bell Journal of Economics, 12 (1981) , 112-128. doi: 10.2307/3003511.
	C. X. Wang , S. Webster and N. Surech , Would a risk-averse newsvendor order less at a highter selling price?, Eur. J. Oper. Res., 196 (2009) , 544-553. doi: 10.1016/j.ejor.2008.04.002.
	J. Wu , J. Li , S. Y. Wang and T. C. E. Cheng , Mean-variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009) , 724-730.
	R. Yin , Y. Aviv , A. Pazgal and C. S. Tang , Optimal markdown pricing: Implications of inventory display formats in the presence of strategic customers, Management Science, 55 (2009) , 1391-1408.
	P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill/Irwin, 2000.