Key Issues in Pricing Decisions
Under ordinary competitive market conditions, the primary objectives of pricing policies are to set prices that maximize revenue, or more to the point, to set prices that maximize profits.
This simple statement of objectives hides the full complexity of pricing decisions. Complexities include these issues.
- What are the main factors that determine customer reaction to prices? Such factors can include be sensitivity to quality, reliability of supply, social status (mostly relevant for consumer products), and obviously costs.
- How do your prices interact with competitors’ prices? In efficient commodity markets (where competing products are good substitutes for one another), your revenue can climb rapidly or decline sharply in response to small changes in your prices or competitors’ prices.
- How much does the past history of pricing affect the market response to current prices? For example, how will market reaction to a price of $100 differ if the price in the recent past was $80 or $120?
- How do prices interact with access to distribution channels? In some markets, distribution channels have enough power that the biggest challenge for suppliers is gaining widespread distribution, from which a large market share will follow.
- How should longer-term strategic factors affect pricing? Some products and services have high growth rates or otherwise have long-term potential that greatly exceeds their current revenue and profits. Prices of such products should be set by a different set of strategic principles.
The approach we shall take encompasses the first four issues in a broad pricing framework. The strategic issues in the last bullet cannot be resolved by simple market testing and quantitative analysis, and we plan to discuss them in another article.
Assumptions Behind the Models
ModelSheet’ offers two price testing and price elasticity spreadsheet applications that address these questions in a wide range of situations. The first application answers these questions for one product in isolation, and the second one answers these questions for dozens of interacting products in the same market.
Our approach to pricing decisions is based on several assumptions.
- Revenues and unit sales for one product vary in a smooth fashion over a range of prices. The relationship is smooth but need not be a straight line.
- More generally, revenue and sales units for related products vary in a smooth fashion over a range of prices for all the related products.
- Your products can compete with one another, and with competitors’ products, as good substitutes or marginal substitutes for one another.
- You know the sales units for each product in a broad market at one set of prices for the products. (In the application, these are referred to as reference prices and reference sales units.)
- You can test different prices in different sub-markets in a way that enables you can deduce reaction of the entire market to price changes. Many factors can complicate the situation so it may not fit our simple interpretation. Examples: People may be able to get different prices in nearly towns, and if the price difference is great enough to cause people to tell their friends and to travel a short distance to get lower prices. Participants may know that the price changes are temporary (for example if they know that a price test is being conducted), which may change their behavior. Different test markets may have sharply different reactions to prices (for example, very affluent and poor neighborhoods), which means you cannot treat the responses to price changes in submarkets as representative of the whole market.
If your situation fits these assumptions reasonably well, then you can probably apply our pricing models and get reasonable estimates of optimal prices.
The Pricing Analysis Procedure
Under these assumptions, the pricing analysis process is relatively simple to implement. Just follow these steps.
- Measure the sales units (called the reference sales units) in the larger market at a know reference prices under normal market conditions.
- Define sub-markets that each have behavior that approximates that of the entire market, and avoid the problems discussed above that can complicate interpretation of test results. If you are testing price interactions of N products, you need at least N * (N + 3)/2 test markets, and preferably twice that number. Record the number of sales units in each of the test markets at the reference price under normal market conditions.
- Choose test prices for the related products in each test market. The test prices should be higher and lower than the reference price for each product in some test markets, and the changes in prices for each product should be independent of one another in at least a few of the test markets.
- Run the pricing tests and record the sales units in each test market.
- Choose a price range for each product over which you want to predict sales units and revenue for each product in the total market.
- Enter into the ModelSheet spreadsheet application: (a) the reference price for each product, (b) reference sales units for each product for the total market, (c) sales units for each product in each test market at reference prices, (d) sales units for each product in each test market at test prices, and (e) the price range over which you want sales predictions (and the specific prices at which you want predictions).
The Price Testing and Price Elasticity model will return predictions of sales units and revenue for the prices at which you requested predictions. You will not necessarily get a price that maximizes revenues. For example if you test prices between $100 and $150, and the revenue keeps increasing up to a price of $200, then the analysis will not include the price that maximizes revenue.
Prices that Maximize Profits
For simplicity, we omitted above the steps in the analysis needed to determine prices that optimize profits. To do this, you must enter into the model the reference cost of the reference sales units for each product, and a cost elasticity factor for each product. (If costs increase in direct proportion to sales units, then cost elasticity equals 1. If cost is fixed regardless of sales units, then cost elasticity is zero. Cost elasticity is usually a number between 0.5 and 1.0.) The model will then predict contribution margins (revenue less costs) for each product at each combination of product prices at which you requested a prediction.
If you include competing products in the analysis, you can specify which products are yours and which are not. The model will produce sales and profit predictions for all products in the analysis, and for your products alone.
In Summary
The Price Testing and Price Elasticity models reduce a very complex process to a quantitative analysis that is easy to use and that can help you to make better pricing decisions. Your real-world situation may contain other, subtle factors that require expert guidance to take into account in your pricing study. As always when using analytical methods in business, listen carefully to experienced field experts who are not analysts; they may know about some factors that the analysis does not take into account.
For more information about the Price Testing and Price Elasticity models, see
Appendix: How the Pricing Models Works
This section is more technical, and you can skip it if you like.
The models answer the question, “If I raise (lower) prices by x%, by what percentage will sales units change?” The answer to this question is known in economics as “price elasticity”, or more precisely the “elasticity of sales units with respect to price”. The technical description of price elasticity is:
elasticity of sales units with respect to price = (change in units/ reference units) / (change in price / reference price) = (percentage change in sales units) / percentage change in price)
You can also define elasticity of revenue with respect to price, which equals 1+ elasticity of sales units with respect to price. We will not use this in the analysis.
Use notations:
- Q = ln(sales units/reference sales units), where “ln” is the natural logarithm base ‘e‘
- P = ln(price/reference price)
- ε (Greek epsilon) = elasticity of sales units with respect to price.
The defining equation of price elasticity is
ε = d ln(Q) / d ln(P)
where ‘d’ means take the derivative (as in calculus). The textbook case assumes that ε is a constant, so that Q and P have a linear relationship.
Q = ε * P
This is a serious limitation, because you can’t do a good job of identifying prices that yield maxima of revenue or profit with a relationship that is (log-) linear. So our models allow ε to change linearly with P, so that the basic relationship is
Q = ε[1] * P + ε[2] * P^2
so: ε = ε[1] + 2 * ε[2] * P
This yields a (log-) quadratic relationship between sales units and price, so the model is much more effective at solving for the prices that maximize revenue or profit.
If you are studying just one product, then this is the end of the story. If you are studying the market interactions of N products (N>1), then Q and P become vectors with one component for each product, and ε becomes a matrix that describes the effect of changes in the price of product j on the sales units of product i.
Q[i] = ε[1, i] * P[i] + ε[2, i, j] * P[j]^2 (sum over j = 1, …, N)
ε[i] = ε[1, i] + 2 * ε[2, i , j] * P[j] (sum over j = 1, …, N)
The model uses quadratic regressions (generalizations of linear regressions) to compute the best estimates of the elasticity coefficients ε[1, i] and ε[2, i , j] (where i and j = 1, …, N).
