Finding a Good Price: A Tutorial (2000)

ABSTRACT This tutorial presents a very simple-to-use profit-maximizing model for small businesses to determine the best price to set on a product. The assumptions are explained and tested. Examples are given to show how a restaurant owner or a catalog merchandiser might use this model to help select a price that would improve profits. For situations when the model cannot be directly applied, the model still provides insights as to reasoning that is appropriate when selecting and adjusting prices. INTRODUCTION The task of assigning a price to a product can be simple; put the same price on the product as the closest competitor does, or assign a certain markup over cost. But such simple methods do not insure that the best price will be chosen, or even that a good price for that product will be chosen. And when the product is reasonably differentiated from its competition, there are very few good rules to help the decision-maker select a good, profitable price. Good managerial reasoning, with some good intuition, and maybe even luck, will always be needed for good pricing decisions; however, a very simple mathematical model could help to make that job somewhat easier. This tutorial will describe how to use a profit-maximizing pricing model, giving information about its derivation, underlying assumptions, limitations and contributions. Here is the pricing model that will always give the price that maximizes profits, as long as the assumptions are met: P = V/2 + Z/2 where: P is the Price that will maximize profits V is the unit Variable cost Z is the Zero-Sale Price (price where sales would go to zero) Assumptions: The product has a linear, downward sloping demand function. Appendix A shows exactly how this formula is derived. Appendix B offers two examples to illustrate that the model does give the exact price to maximize profits. Appendix C contains some background information on the model's history. APPLYING THE MODEL Consider the situation illustrated below in Figure 1. Sales will decrease at a uniform rate as the price is raised from $0 to $50 (This meets the requirement of a downward sloping demand function that is in the form of a straight line). Because sales decrease by a total of 5000 units as price increases by $50, there is a decrease of 100 units for each $1 price increase. The point where sales would go to zero (Zero-Sale Price) is $50. If the product cost is $20 (unit variable cost), then the profit-maximizing price is given by the model to be $35. P = V/2 + Z/2 = $20/2 + $50/2 = $10 + $25 = $35.00 Sales at that price will be 1500 units. (The price was lowered $15 from the Zero-Sale Price of $50, and each $1 decrease resulted in additional sales of 100 units.) Note: Selecting a price of either $34.00 or $36.00 will result in exactly $ 100 less Net Revenue in either case. Or, there would be $100 less Profit realized by raising or lowering the price by $ 1. If the cost of this product should increase from $20 to $25 (and the demand function remains as it is), then the most profitable price would increase to $37.50. (P = V/2 + Z/2 = $25/2 + $50/2 = $12.50 + $25) It is interesting to note that the product's price should only be raised half as much as the increase in cost. The figures now are as follows: Sales: 1250 units (Raising the price $2.50 will decrease sales by 250 units.) Gross Revenue: $46,875 Variable Costs: $31,250 Net Revenue: $15,625 (The increased costs reduce profits by $6875.) But, if the price is raised a full $5.00 to $40.00, matching the cost increase, then the resulting figures are: Sales: 1000 units (Sales go down by an additional 250 units.) Gross Revenue: $40,000 Variable Costs: $25,000 Net Revenue: $ 15,000 (This results in an additional loss of $625.) Similarly, if the cost is reduced by $5, then the selling price should be reduced by half that amount. …