Using a menu of geographic pricing plans: a theoretical investigation.

In a menu plan of geographic pricing, the retailer allows a customer to choose either a free on board origin (F.O.B. origin) plan, or a uniform delivered plan. In this article, we develop an approach to construct the best menu plan and compare its profitability and market coverage with that of a uniform delivered plan and an F.O.B. origin plan when these are used alone. It is shown that a menu plan can generate at least as much profit as either an F.O.B. origin or a uniform delivered plan used by itself under any demand condition. However, whether the menu plan can outperform the F.O.B. origin and the uniform delivered plans depends on the specific demand conditions. When demand is linear in price and delivery cost is uniformly distributed, it is shown that the menu plan is substantially more profitable than the F.O.B. origin and the uniform delivered plans and generates at least 96% of the contribution of an optimal plan which sets a profit maximizing price for individual customers based on cost. On the other hand, when demand is a constant elasticity function of price, the menu plan is found to be only marginally superior to the F.O.B. origin plan which generates more profit than the uniform delivered plan.

1. INTRODUCTION

Even when customers buy the same product or service, a seller may incur different levels of cost to serve different customers. In a retailing context, for instance, the cost of delivering products may vary because customers may reside at different distances from the store location. A key decision that a retailer has to make in such situations is to adopt an appropriate pricing plan that incorporates the variations in the cost of delivery over the market served. This issue of setting prices when the cost of serving customers varies due to their locations has long been recognized as important by both academic researchers and industry practitioners, resulting in a body of work commonly referred to as the geographic or spatial pricing literature (see Greenhut and Ohta 1975 for an overview).

Early work in the spatial pricing area has focused on developing an optimal pricing plan (or spatial monopoly) that perfectly discriminates among customers based on their geographic locations. The optimal plan generates the highest contribution among all types of pricing plans. However, implementing the optimal plan requires a retailer to have precise information about the costs of delivery and price sensitivity for each of its customers at different locations. Because of the practical difficulties associated with collecting or monitoring customer cost information, retailers often adopt simpler pricing plans, namely the Free On Board Origin (F.O.B. origin) and the Uniform Delivered plans.

In the F.O.B. origin plan, the delivered price to the customer is a base price (e.g., price of the product) plus the exact cost of delivery. This plan can be implemented simply by letting the customer bear the entire responsibility for transportation. In the uniform delivered plan, the retailer designates a geographical area that it wants to serve and charges all customers within that area the same delivered price regardless of the actual location of the customers. For example, retailers selling durables such as appliances and furniture, and mail order computer companies make "free" deliveries (i.e., charge all customers the same delivered price) as long as these customers are located within designated areas. Regulated industries, such as the US Postal Service, are often mandated to serve all customers and charge them a uniform price. While these two simpler plans are rarely optimal, the ease with which they can be implemented makes them attractive. Recognizing the prevalent use of these plans, academic research has examined how these two plans perform compared to each other as well as relative to the optimal plan (Beckmann and Ingene, 1976; Beckmann, 1976).

Past research on the F.O.B. origin and the uniform delivered plans have generally assumed that one of these two plans is implemented exclusively. However, there can be a third type of geographic pricing plan, labeled here as a menu plan, where the retailer lets the customer choose either an F.O.B. origin plan or a uniform delivered plan. Consider an example of a menu plan in its simplest form. A furniture store offers its products at a regular price and offers "free" delivery to customers located within a certain distance from the store. In addition, customers regardless of location are offered a discount from the regular price if they arrange their own transportation. Thus, customers located within the area where "free" delivery is offered have two alternatives to choose from. They can either pay the regular price and receive "free" delivery, or pay the exact cost of transportation and receive a discount from the regular price. Note that the first option is a uniform delivered plan and the latter option is an F.O.B. origin plan where the base price is the regular price minus the discount.

Even casual observations indicate that some retailers, notably appliance and furniture stores, permit their customers the flexibility of choosing between an F.O.B. origin plan and a uniform delivered plan. From a strategic standpoint, an appealing feature of the menu plan is that unlike under the optimal plan, the retailer here does not have to monitor the costs of delivery for the different customers. Instead, customers actually reveal their true cost types by their decision to accept either the F.O.B origin plan or the uniform delivered plan. For example, customers located near the store (i.e., low delivery cost customers) in their self interest will accept the F.O.B. origin plan and are, therefore, not required to subsidize customers with higher costs of delivery. Another advantage of the menu plan is that it can serve a larger market area by including customers with very high costs of delivery, who may have been excluded under a uniform delivered plan.

Furlong and Slotsve (1983) examined the menu plan described above when demand is uniformly distributed over an infinite line and is a linear function of price at every location. Their results include the finding that a firm can obtain a greater profit by offering a menu plan instead of either an F.O.B. origin plan or a uniform delivered plan alone. In this article, we examine the menu plan under more general conditions and try to answer the following questions: (1) How to construct a menu plan under a given demand condition? (2) When is it more profitable to use a menu plan rather than adopt either the F.O.B. origin plan or the uniform delivered plan? (3) How does the profit from the menu plan compare with that from the optimal plan?

The rest of the paper is organized as follows. We first present the model assumptions and describe the menu plan formally. Next, we analyze the menu plan under two demand conditions: (1) linear demand and (2) constant elasticity demand. The results highlight the fact that the nature of the demand function determines whether a menu plan should be implemented. Finally, we summarize the findings and identify directions for future research. A glossary of the terms used is provided in Table 1.(1)

 
TABLE 1 
 
                       Glossary of Terms Used 
 
1. Costs 
c                     Retailer's cost of acquiring one 
                        unit of the product 
x                     Cost of delivering a unit of the product 
R                     Highest delivery cost where uniform 
                        delivered plan is offered 
2. Price 
P(x) 
[P.sup.*](x)          Delivered price/unit paid by customers with 
                        delivery cost x 
[P.sup.*](x)          Optimal price/unit paid by customers with 
                        delivery cost x 
([P.sub.U])[P.sub.u]  (Best) uniform delivered price/unit 
(P.sub.F])[P.sub.f]   (Best) base price/unit under an F.O.B. 
                         origin plan 
3. Demand 
q(P(x))                 Demand rate for customers with delivered 
                         price/unit P(x) 
Q                       Total demand 
q(P) = a - bP           Linear demand rate function 
q(P) = [AP.sup.-[Eta]]  Constant elasticity demand rate function 
[Epsilon](P)            Price elasticity of demand at price P 
 
4. Market 
f(x)                    Market concentration (of demand) at 
                          delivery cost/unit x 
L                       Highest level of delivery cost/unit for 
                          uniform distribution. It is also called 
                          market width. 
(a/b - c)               Under linear demand, this is the highest 
                          delivery cost/unit for which a 
                          profit can be generated. This is also 
                          called the range of feasible costs. 
K = (a/b- c)/L          Inverse of spatial dispersion (linear 
                          demand) 
 
5. Profitability 
[[Pi].sub.x](P(x))     Profit rate for selling product to customers 
                         with delivery cost x 
[Pi]                   Total profit from a pricing plan 
[[Pi].sup.*]           Total profit from the optimal pricing plan 
[[Pi].sub.u] 
  ([[Pi].sub.U)        Total profit from any (best) uniform 
                         delivered plan 
[[Pi].sub.f] 
  [[Pi].sub.F])        Total profit from any (best) F.O.B. origin 
                         plan 
[[Pi].sub.m] 
  ([[Pi].sub.M])       Total profit from any (best) menu plan 
[Psi]ser = 1 - 
  [[Pi].sub.M]/ 
  [[Pi].sup.*]         Total sub-optimality of best menu plan 
[Psi]ser               Sub-optimality of best menu plan from 
                        reduced market served 
[Psi]sh                Sub-optimality of best menu plan 
                         due to shape 

2. PROBLEM FORMULATION & MODEL ASSUMPTIONS

2.1. Costs

As usual, the total cost that a retailer incurs to serve its customers can be expressed as the sum of fixed and variable costs. The fixed costs consist of administrative and selling overheads for the retailer, including the cost of implementing a pricing plan. As none of the plans under consideration (F.O.B. origin, uniform delivered, and menu) requires extensive monitoring of the costs of shipment to customers, the costs of implementing these three plans should be minimal. Accordingly, the fixed costs for all the three plans are about the same and, therefore, we assume without loss of generality that the fixed cost is zero.

The variable cost of serving a customer at a given location has two components: (1) the cost the retailer incurs to acquire one unit of the product from the manufacturer or distributor; and (2) the actual cost of delivering the product to the customer destination. Therefore, for any given customer, the total variable cost per unit of the product can be expressed as (c + x) where c is the retailer's cost to acquire one unit of the product and x is the delivery cost per unit, x [is greater than or equal to] 0, for any customer. We make the simplifying assumption that the delivery cost is the same regardless of whether the customer or the retailer arranges the delivery, and place customers in groups based on delivery cost rather than geographic location. Customers with unit variable cost (c + x) are called "customers at delivery cost level x." Note that the difference in the total variable cost across customers arises only from the variation in the cost of delivering the product to customers dispersed spatially. Also, since fixed costs are assumed to be zero, the contribution to profit equals profit itself.

2.2. Pricing Plans

We assume that the delivered price, i.e., the total price paid by the customer for purchasing a unit of the product, depends only on the variable cost of serving the customer and can therefore be expressed as a function of x. We call this function the pricing plan and denote it by P(x). To illustrate pricing plans, we now briefly discuss the F.O.B. origin and the uniform delivered plans. The menu plan (which lets the customer choose either an F.O.B. origin plan or a uniform delivered plan) and the optimal plan which is used as a bench-mark to evaluate profitability are introduced later.

The F.O.B. origin plan charges the customer a base price (i.e., the price of the product) plus x. That is, P(x) = ([P.sub.f] + x) where [P.sub.f] is the base price. The uniform delivered plan charges the same price regardless of x and can be expressed as P(x) = [P.sub.u], where [P.sub.u] is the uniform delivered price.

Market Served by a Pricing Plan. In designing a pricing plan, the retailer has to decide whether it should serve customers at all locations (i.e., all levels of delivery cost), or serve only a limited market instead. Note that the contribution the retailer obtains from selling a unit of the product to a customer at delivery cost level x is P(x) - (c + x). If P(x) < (c + x), the retailer receives a negative contribution by selling its product to customers at delivery cost x. Therefore, a retailer …