Discussion Paper

No. 2018-50 | June 15, 2018
Dynamic pricing for inventories with reference price effects


This article presents a dynamic pricing model of a retailer selling an inventory, accounting for consumer behavior. The authors propose an optimal control model, maximizing the intertemporal profit with consumers sensitive to the selling price and to a reference price. The optimal dynamic pricing policy is solved with Pontryagin’s maximum principle with a structural (general) demand function. They obtain an original pricing rule, which explicitly accounts for the impact of price and inventory on future profits. The dynamics of price do not have to imitate the dynamics of the reference price. Instead, the dynamics of price are tied to opposing effects linked to this reference price. The authors also discuss managerial implications with regards to behavioral pricing policies.

JEL Classification:

C61, D03, D40, M21, M37


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Cite As

Régis Chenavaz and Corina Paraschiv (2018). Dynamic pricing for inventories with reference price effects. Economics Discussion Papers, No 2018-50, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2018-50

Comments and Questions

Anonymous - Referee Report 1
July 23, 2018 - 11:21

Review of “Dynamic Pricing for Inventories with Reference Price Effects” submitted to Economics E-Journal, manuscript number: Economics MS 2678.

The paper studies the problem of dynamically pricing perishable items in the presence of reference effects under a general demand model. The methodology is a continuous time monopoly with deterministic ...[more]

... demand and a finite time horizon. Demand intensities are time homogeneous and depend on a firm’s current price as well as a reference price, which is based on the firm’s past prices. Inventory holding costs and discounting is taken into account; inventory replenishment is excluded.

The introduction reads well and motivates the objective of the paper. The primary objective of the paper is to investigate the impact of reference price effects under a general demand model. Based on Pontryagin’s maximum principle necessary optimality conditions for open-loop controls are derived in order to infer structural properties of optimal price paths.

In my assessment, the paper deals with an interesting and relevant topic: dynamic pricing with customer behavioral effects. Although the analysis remains only theoretical, I think there is some merit in the paper. However, I see two weaknesses: (i) due to the complexity of the model, general useful structural properties can hardly be derived, and (ii) the presented approach is not constructive, i.e., it remains unclear how to compute optimal prices.

Hence, the authors should add numerical results to their model in order to (i) verify that (under certain conditions) the interplay between prices and reference prices follows some rules and (ii) to show a way how to actually derive prices in practical applications.

I have the following critical comments/questions:

(1) Could the authors provide a numerical example? Can the solution of the model be derived using approximation techniques?

(2) Is there an incentive to start with extremely high prices (e.g., 1 million) to build a potential in order to generate demand by reducing prices afterwards?

(3) Can the model be generalized by time-dependent demand?

(4) Proposition 1 and 2 are interesting. However, they hardly allow to infer how the optimal open-loop price path should look like as we have a complex interplay of (i) discounting, (ii) holding costs, (iii) time-to-go, (iv) price impact, and (v) reference price effect.
Is there a relation to corresponding optimality conditions in the literature (for models without reference effect), e.g.,

Stiglitz, J.,E. (1976) Monopoly and the rate of extraction of exhaustible resources. The American Economic Review 66(4), 655–660

Cao, P., Li,J., Yan,H. (2012). Optimal dynamic pricing of inventories with stochastic demand and discounted criterion, European Journal of Operational Research 217, 580–588.

Further, can the necessary optimality conditions be used to actively construct optimal open-loop solutions as done in, cf. (13),
Schlosser, R. (2015). Dynamic Pricing with Time-Dependent Elasticities, Journal of Revenue and Pricing Management 14 (5), 365-383

(4) Is it possible to analyze feedback controls by looking at optimality conditions for open-loop controls for given states (time t, reference price r)? This way, it may be possible to derive structural properties regarding single state variables.

(5) In the conclusion the authors discuss further extensions of the model, such as marketing expenses or promotion activities. There is related work that studies structural properties of pricing and advertising controls under general demand models, e.g.,

Schlosser, R. (2016). Joint Stochastic Dynamic Pricing and Advertising with Time-Dependent Demand, Journal of Economic Dynamics and Control 73, 439-452.

In my opinion, the author’s model with reference price effects is interesting. However, the current presentation of the results of the paper has to be improved. The authors should describe how optimized prices can be derived.

Minor comments:

page line comment

2 -12 I think at the end of the sentence should be a “.”

4 -8 Is it ensured that I(t) may not be negative?

7 7 Shouldn’t be t finite, i.e., 0<t<T?

8 -7 I think the reference “equation (2)“ is wrong

Régis Chenavaz - Reply to Reviewer 1
August 01, 2018 - 10:28

Please see the attached file.