The Adequacy of Full-Cost-Based Pricing Heuristics

We investigate the performance of a full-cost heuristic in a service setting. In our model, a service firm determines the amount of capacity, a price, and a price discount each period. Based upon the price, a stochastic number of customers will place service orders. If too many orders arrive in a period, the firm will offer a price discount to those customers willing to back order and accept service the next period. Even though the model is fairly simple, the optimal pricing, price discount, and capacity rules are complex and require extensive calculations. We examine how closely three distinct heuristics approximate the optimal performance. The best-performing heuristic is a fullcost pricing rule based upon a constrained version of the firm’s optimization program. It consists of setting a price using full costs plus an adjustment based upon the nonlinear elasticity of demand. In 500 random simulations our full-cost heuristic obtains 99.5 percent of the optimal performance. Preliminary analysis suggests that a modified fullcost heuristic may continue to do well in settings where interim demand information arrives after the capacity choice, but before the pricing choice. However, a modified fullcost heuristic may not perform well when capacity lasts several periods. INTRODUCTION The argument over whether to include capacity (fixed cost) charges into product costs is one of the longest running debates in management accounting (Church 1911; Williams et al. ([1921] 1990); Vatter 1945; Zimmerman 1979; Shim and Sudit 1995). The classic exchange of views (Church 1915; Gantt 1915) is that if managers set prices using only variable costs, a proxy for marginal costs, then they will generate low prices and high sales. However, if the price is too low, then the firm will not recover its fixed costs and it will lose money in the long run. Conversely, if the managers set prices using full costs (variable plus unit fixed costs), then every order will cover its portion of fixed costs. However, the higher prices may reduce orders to the point where the firm again loses money in the long run. This discussion suggests that a manager would be able to choose between direct and fullcosting methods if she understood how product prices interact with customer demand and the firm’s capacity. A more detailed business model that captures the marketing and capacity trade-offs would show which alternative was preferred. We thank our two anonymous reviewers, Robert Göx, Ramji Balakrishnan, seminar participants at the 2000 Management Accounting Research Conference, the 2000 European Accounting Association Conference, Baruch College of CUNY, Santa Clara University, Rutgers University, UCLA, and The University of Texas at Dallas for insightful comments. JMAR Volume Fourteen 2002 34 Journal of Management Accounting Research, 2002 However, with a sophisticated model of the firm’s operations, the manager could directly choose the optimal prices and capacity. Given this complete model and analysis, the choice of full or variable costing becomes irrelevant because the cost system does not affect any of the manager’s decisions. Although this is a telling point, managers can rarely generate robust, solvable models of their markets and operations. Pricing and capacity decisions are very complex and determining the optimal solution may be prohibitively expensive. An alternative is to identify simple rules (heuristics) that generate close to the optimal solution (Banker et al. 2002; Göx 2002; Hansen and Magee 1993). Prior literature on the efficacy of heuristics (reviewed in Balakrishnan and Sivaramakrishnan [2002]) has concentrated on full-cost heuristics because full-cost prices are commonly used by managers (Shim and Sudit 1995; Govindarajan and Anthony 1983). We investigate the performance of three simple heuristics, including one fullcost heuristic, in a model where costs, prices, and capacity choices are intertwined. We model a service company that determines both the amount of capacity and the price each period. Based upon the price, a stochastic number of customers will place service orders. If too many orders arrive in a period, the manager will offer a price discount, set at the start of the period, to those customers willing to back order and accept service in the next period. Even though the model is fairly simple, the optimal pricing, discount, and capacity rules are determined simultaneously via a system of nonlinear equations and require extensive calculations. Three heuristics naturally emerge from the limiting behavior of the model. They incorporate either the limiting equation for the optimal price (full-cost heuristic), the limiting equation for the optimal capacity (balanced-capacity heuristic), or both limiting equations (basic heuristic). All three heuristics can be implemented recursively by solving comparatively simple equations for the price, discount, and capacity. We evaluate the performance of these three heuristics relative to the optimal solution and determine the size of the expected loss for each. The results of this simulation horse race are striking. The full-cost-based heuristic wins and achieves a median performance of 99.5 percent of optimal profits in a simulation with 500 observations vs. 94.0 percent (94.1 percent) for the balanced-capacity heuristic (basic heuristic). Our results provide evidence in support of reporting unit costs for pricing decisions. Our paper is distinguished from prior work in several ways. First, we investigate a service setting, which complements the prior work that focuses exclusively on manufacturing. Second, we allow the manager to select a price discount, a method of shifting future capacity into the present period. As section six demonstrates, this discount is related to the concept of soft capacity in manufacturing, additional capacity purchased at a premium to augment existing capacity (e.g., overtime). Depending on the parameters, the price discount captures the cases of no soft capacity allowed, unlimited soft capacity permitted, and a new case in which the amount of soft capacity is endogenously determined by the manager. Finally, unlike prior simulations, our analysis does not restrict the parameters to fall into one of the two polar soft capacity cases (no soft capacity allowed, unlimited soft capacity permitted). Our simulations cut across several cases and, therefore, our full-cost heuristic is robust to the form of capacity. Once we complete our basic analysis, we provide preliminary evidence about the robustness of our results. Our preliminary analysis suggests that a full-cost heuristic may continue to do well in a setting where interim-demand information arrives after the capacity choice, but before the pricing decision. However, our Banker and Hansen 35 preliminary evidence also suggests that a full-cost heuristic does not perform well in situations where capacity lasts several periods. The organization of this paper is as follows. The second section presents the basic model, and the third section presents the complex, optimal solution. Section four provides the limiting solution as the number of customers increases. The fifth section creates heuristics using this benchmark, and section six presents simulation results. We recast our model into a manufacturing setting in section seven. Section eight discusses several extensions, and section nine presents our conclusions. THE BASIC MODEL We model a generic service company in which the manager determines the service capacity at the start of each period.1 The manager then creates and publicizes a price list that describes the service and its price. Customers read the price list and queue up to purchase the service. If too many customers arrive during the period, the manager offers customers a predetermined price discount if they are willing to wait and accept service the next period. For example, our service company could be a barbershop. The manager’s capacity decision is the number of barbers to schedule during a day. The price of a haircut is posted in the window. If too many customers arrive at one time, the manager may offer the customers a preprinted discount coupon if they will come back for a haircut the next day. Returning customers are always served first. Figure 1 shows the timeline for our model. The model reflects a representative period in an infinite horizon game with no discounting and, hence, the time subscript is suppressed. At the start of the period the manager chooses the service capacity level, y. Capacity consists of two parts: the amount committed to fill the prior period’s backorders, b–1, and the amount that can be devoted to new orders in the current period, x. The manager selects the capacity level before knowing the current period demand for his product. The cost of capacity is linear with parameter k, so that the total cost of capacity is ky. Next, the manager sets the current period price, p > 0 and the price discount, d ≥ 0, to induce customers to back order when all service capacity is committed. Both the price and price discount are “sticky” and, once set, cannot be changed until the start of the next period (Blinder et al. 1998; Wolman 2000).2 To simplify the analysis, all customers order one unit of service that requires one unit of capacity and generates variable costs, v > 0.3 Performing the service requires a short amount of time and all sales begun in a period are completed in that period. 1 For simplicity, we assume that there is no upper bound on the amount of potential service capacity. 2 If the manager were allowed to adjust the optimal price throughout the period, then the optimal price would vary with the number of customers who have already ordered, the number of potential customers remaining who could order, and the amount of remaining current capacity available. We conjecture that the price would vary for every customer until capacity was filled. However, once current capacity is assigned, we believe that the manager would offer the same price discount to all remaining customers. The difference is driven by the distinction between constrained current capacity and unconstrained future capacity. The manager can always purchase enough capacity in the following period to fill back orders. Therefore, the manager sets the price discount by making the same trade-off between price and probability of ordering for every customer. 3 Our model assumes that all capacity costs are proportional to the service-level capacity. Adding in fixed capacity costs that would not vary with the service capacity or the actual service level is straightforward. We would just subtract a constant from the objective functions throughout the paper. This additional, constant, term would not affect any of the manager’s choices or the performance of any heuristic. However, we would have to add the additional condition that the firm would shut down if its profits were insufficient to cover their fixed capacity costs. 36 Journal of Management Accounting Research, 2002 Price discounts are an important element of our model. Understanding how they work requires a detailed description of how customers decide to order and back order. There are m potential customers who arrive sequentially during the period. Customers are risk-neutral and choose between purchasing and not purchasing the manager’s service. The utility of not purchasing the service is set equal to 0. Each customer i’s utility for the manager’s service is ui, where ui is independently and identically drawn from the probability density f(ui); f(ui) > 0 for all ui. Clearly, customer i will choose to order from the manager if her utility is greater than the stated price, ui ≥ p. Given these assumptions, the probability a customer will order is (1 – F(p)). If a customer places an order and capacity is available, then the manager fills the order and the customer pays price p. If insufficient orders arrive during the period, there will be unused service capacity. If a customer places an order when all capacity has been committed, then the manager asks customers to back order, i.e., to return the next period for service. There is no systematic pattern in the manager’s selection of the ordering customers who are not served, and customers do not act strategically in deciding when to place an order.4 As a consequence, the distribution of back-ordered customers is a truncated version of the original ordering distribution. All customers asked to back order experience a decline in their utility of a > 0.5,6 This decline reflects the cost of returning to the service facility plus the customer’s 4 We rule out the following behavior. Some travelers with tight budgets and spare time book highvolume flights during the holidays. They choose their flights to maximize the probability of being bumped and thereby obtaining free tickets. 5 If back ordering produced no disutility, the manager would never acquire any initial capacity and would back order all customer orders. The manager would acquire exactly enough capacity to satisfy the actual demand and thereby avoid the risk of having excess capacity. Similarly, if customers could credibly communicate their utility to the manager, the manager could pay overcapacity customers just enough to ensure they would return the following period. The manager would acquire precisely enough capacity to satisfy current orders and avoid the risk of excess capacity. 6 We implicitly assume that the customer’s annoyance at being back ordered is not related to the size of their utility for the service. FIGURE 1 The Timeline Manager chooses Manager sets Backorders level of service 1 b− filled. capacity, x. price, p, and discount, d. … __|________________|_______________|_____________>____ Orders placed Current customer If orders, s, by s of m orders are filled exceed available customers up to available capacity, customers capacity, x – b–1. may backorder, b0, obtaining discount, d. ____>____|_______________|__________________| … rd r, 0, capacity, x. price, p, and discount, d.