Optimal pricing strategies under co-existence of price-takers and bargainers in a supply chain

Abstract

We investigate how the co-existence of two types of customers, price-takers, and bargainers, influences the pricing decisions in a supply chain. We consider a stylized supply chain that includes one manufacturer and one retailer, and we characterize the optimal prices of the retailer and the manufacturer. We further discuss the effects of the fraction of the bargainers in the customer population and the relative bargaining power of the bargainers on these optimal prices. Our results show that, given the wholesale price, the lowest price at which the retailer is willing to sell (ie, cut-off price) increases with the relative bargaining power of the bargainers. Both posted and cut-off prices increase in the fraction of the bargainers in the customer population. Moreover, depending on the type of negotiation cost, the variations of both prices will vary. In equilibrium, both posted and cut-off prices do not monotonically increase with the fraction of the bargainers in the customer population. When the maximum reservation price of the customers is low, and/or the negotiation costs are high, and/or the relationship between the bargainer's negotiation cost and reservation price is high, the retailer may reduce both posted and cut-off prices as the fraction of the bargainers increases.

Keywords:

• game theory
• pricing
• supply chain management
• negotiation

Acknowledgements

The authors thank the editor and the referees for helpful comments and suggestions. This work was supported by NSC grant NSC-97-2410-H-002-213 in Taiwan.

Notes

1 ‘Shoppers haggle for deals from retailers’, The Associated Press, December 2008. Source: http://www.msnbc.msn.com/id/28351832/ns/business-small-business.

2 Our model assumes that the cost of negotiation incurred by the bargainer is based on the opportunity cost each bargainer values. With different reservation price, each bargainer tends to have different cost of negotiation for the time and effort spent on negotiation. One may simplify our model by considering a case where the cost of negotiation is identical for each bargainer. Such assumption is a special case of our model by setting α=0.

3 As aforementioned discussion, the cost of negotiation incurred by the retailer, c_r, is implicitly considered in p_m, where p_m⩾w+c_r, as the retailer will not choose the cut-off price below w+c_r. The case under which the retailer must sell to bargainers who pay at least the retailer's cost, w+c_r, is a special case of our model.

4 We assume b is finite to reflect the fact that customers will not pay unreasonably high price for a product. In addition, one may envision a situation where the lower bound of the customer reservation price is positive, instead of zero. Adding such a feature to our model will not change qualitative results and derived managerial insights, but only raise computational complexity.

5 As the proof reveals, the results in Propositions 1 and 2 hold under general conditions that do not require F being uniform distribution.

6 Note that a function f: R^m → R is supermodular if f(x∧y)+f(x∨y)⩾f(x)+f(y) for x, y∈R^m, where x∧y:=(min(x₁, y₁), min(x₂, y₂),…, min(x_m, y_m)) and x∨y:=(max(x₁, y₁), max(x₂, y₂),…,max(x_m, y_m)).

7 When the customer's reservation price follows a uniform distribution over the interval (0, b), one can show that the manufacturer's profit, Π_M(w, p^*(w), p_m^*(w)) is concave in w, and thus the optimal wholesale price, w^*, can be uniquely determined.

8 Notice that these 2430 combinations show similar patterns for the following analysis; thereby, the figures and the associated parameters in Section 5 are depicted for demonstration.

9 Based on our negotiation outcome, bargainers with reservation price r⩾(p−βp_m+(1−β)c_e)/((1−β)(1−α)) pay at p and those with r∈[(p_m+c_e)/(1−α), (p−βp_m+(1−β)c_e)/((1−β)(1−α))) pay at (1−α)(1−β)r+βp_m−(1−β)c_e. The expected selling price to bargainers is while the expected selling price to price-takers is simply p. For q=0.3, we have p=740.88 and p_m=652.48, and hence, the expected selling price to bargainers and price-takers are 679.98 and 740.88, respectively. Same logic applies for q=0.7.

10 Notice from Equation (4) that there may exist a case in which the optimal prices are given by p^*(w)=p_m^*(w)=w+c_r, and thus the profit from the bargainers is equal to zero. In this case, the retailer's best strategy is to allow only the posted pricing (no cut-off price decision) and the profit function is given by Π_R(p, p_m, w)=Π_R(p, w)=a(p−w)F̄(p), in which case Lemma 1 gives the same result. Thus, we focus only on the case w+c_r<p_m^*(w)⩽p^*(w).