Editor's note: Richard Smallwood is president of Applied Decision Analysis, Menlo Park, California. This article is adapted from a presentation made at the 1991 Sawtooth Software Conference.

One of the dramatic lessons of the collapse of the communist economic system in the Soviet Union and Eastern Europe is the importance of the marketplace as the final judge of pricing decisions. In the expanded international economy resulting from these economic and political upheavals, pricing decisions will become even more important. Understanding the sensitivity of the market to different pricing policies is essential to successful participation in the high stakes game of international marketing.

To the manufacturer with multiple products in a complex market, setting the prices of all the products in the portfolio can be an imposing task, particularly if the products compete with one another. Moreover, any adjustments in the price of the product portfolio will undoubtedly cause changes to the competitor's prices. In the face of such complexity, it is not surprising that many companies choose to use ad hoc methods and to set each product as a separate decision.

There are of course analytical aids that can help with these complex decisions. To use them, we need to understand how the sales of a product, or portfolio of products, will depend on the prices of the products. The specification of this demand function, particularly as it depends on the interactions among many products, is often the most difficult part of an analytical approach to deriving an optimal pricing policy.

This article will demonstrate how conjoint analysis can help in the specification of the multi-product demand function. This leads to the development of techniques for deriving the optimal pricing policy for a portfolio of products within a competitive environment.

Finding the demand function

Let's imagine a situation in which our client has one or more products competing for sales with competitor's products in the market. Suppose further that we have conducted a conjoint analysis survey over a sample of customers in this market. This will produce for each respondent in the survey the following:

  • The utility for price, ui(p), where the index i refers to the ith respondent.
  • The utility of all the other attributes for each product in the market; let's call this utility for the ith respondent and the jth product uij
  • The number of customers represented by each respondent; wi will denote this "weight" for the ith respondent.

Notice that the first two quantities are measurements resulting from the conjoint analysis survey, while the third is determined by the sampling plan used to recruit respondents.

With these three quantities it is possible to estimate a demand function over the products in the market. To do so we need an additional model to describe how respondents' ultimate choice of products depends on the measured utilities. The two most common models for this purpose are the logit and probit choice models. If we use the logit model, the demand function for the products in the market becomes:

Equation 1
D(k) = ΣwIExp[ui(pk)+uik]/ΣExp[ui(pj)+ uij]

Where D(k) is the total demand for the kth product, Pk is its price, and Exp[.] denotes the exponential function.

The purpose of this equation is to illustrate the intuitive notion that the utilities resulting from a properly administered conjoint analysis can be used to form a demand function over the products within a market.

Estimating sensitivities to price

The availability of an analytic demand function is a powerful tool for pricing analysis. For example, with the demand function it is now possible to calculate each product's elasticity with respect to price. Since the elasticity of demand to price is just the percentage change in demand per percentage change in price, the elasticity, E(k), for the kth product is just:

E(k) = pk[dD(k)/dpk]/D(k)

where the quantity in brackets is the derivative of the demand for the kth product with respect to its price. This can be calculated analytically from Eq. 1 above, and so it is possible to provide price elasticities as a direct output of the conjoint analysis.

Price elasticities are an important source of information about how a market is likely to react to changes in price. They can be used to determine which products will have large changes in demand and which will have relatively small changes for a fixed percentage change in price.

This idea can be taken one step further. Suppose that our client has several products in the market and wants to estimate how changing the price of one will affect the demand of another. This cross-pricing effect can be represented as a cross-elasticity, defined as the percentage change in the demand of one product per percentage change in the price of another. If E(klj) is the cross-elasticity of the demand of the kth product to the price of the jth product, then we have:

E(ljk) = pj[dD(k)/dpj]/D(k)

where the bracketed term is the derivative of the demand of the kth product with respect to the price of the jth product. These cross-elasticities can be calculated analytically from the demand function in Eq. 1 and so can be reported as a direct output of the conjoint analysis study.

In practice it is often more intuitively appealing to report out the derivatives, dD(k)/dpj, themselves rather than the cross-elasticities. The derivatives represent he change in demand for a change in the price of a single product, and so have the following physical interpretation. Suppose we raise the price of one product and observe that its demand has decreased by some number of units. The derivatives of demand with respect to its price illustrate how the product's losses will be reallocated by the market to the other products.

Finding the optimal price

The availability of a multi-product demand function along with elasticities and cross-elasticities raises the possibility of calculating an optimal price. Let's consider the single product situation first. If the variable cost of the product is c, then the total variable profits for the product are:

Equation 2
V = D(p-c)

and so we want to know the price that maximizes this quantity. The change in variable profits per change in price is just:

Equation 3
DV/dp = [dD/dp](p-c)+D

and this quantity can be calculated analytically from the demand function in Eq. 1.

Finding the optimal price is just a question of finding the price that causes the expression in Eq. 3 to equal zero. There are many sophisticated techniques in the arsenal of operations research for solving this type of problem.

The problem becomes more interesting and somewhat more difficult if we seek to optimize the prices of a portfolio of products. In this case the variable profit expression in Eq. 2 must be expanded to include the sum over all the products. Techniques are equally available for solving the multi-product version of Eq. 3 to find the optimal portfolio of prices.

Including competitive effects

The discussion in the preceding section assumed that the competition will make no changes to its prices in response to our attempts to optimize prices. This is, of course, not realistic. To include competitive effects in the optimization of prices requires some explicit statement about how the competition will respond. For example, if we raise our prices by 10% will the competition raise theirs by 10%, or by 5%, or not at all?

The best approach is to build a simple model of how the competition will respond. This competitive response model can then be incorporated into the optimization of prices. If there is uncertainty about the competitive response, then this can be included in the model so that a probabilistic competition model results.

Once the competitive response model has been constructed, sensitivity analyses can be conducted over the uncertain elements of the model. In most cases, there will be a few key factors about competitive responses that significantly affect the optimal price settings. Once identified, attention can be focused on resolving the uncertainty about these key factors.

Limitations of the approach

The optimization scheme proposed above is based on several simplifying assumptions, each of which can be relaxed at the expense of increased complexity. Some of these assumptions are discussed below:

Logit choice model
The logit model in Eq. 1 has the obvious advantage of analytical simplicity. Other choice models such as the multivariate probit could be used instead, although the probit model in particular requires considerably more complex numerical computations.

Preference model
Regardless of the form used, the demand function of Eq. 1 only describes customer preferences for products rather than actual sales. It is quite feasible to add a separate module to the above structure that describes those aspects of the market not contained in simple preferences. Examples of issues that might be included are the distribution network. manufacturing capacity constraints, sales force coverage and effectiveness, and customer reluctance to change.

Linear cost model
The formulation of the optimal pricing problem above assumes in Eq. 2 and 3 that the incremental cost of each additional product sold is constant. If there are significant economies of scale over the range of demand considered in the demand function, then the cost model can be modified to include these effects.

Homogeneous market
The formulation of the demand function in Eq. 1 assumes implicitly that the size of the market to be distributed among the products is constant. As the prices of products change it is reasonable to expect that customers will enter or leave the market in response to the overall price level. To include this effect requires a separate model of the total market size and its dependence on overall price levels.

Designing the survey

If the utilities from a conjoint analysis survey are to be used to estimate optimal prices, care must be taken that the utilities adequately represent customer attitudes toward product prices. There are four issues that require particular attention:

Form of the price utility
The utility for price, uj(p) in Eq. I, can in general be either discrete or continuous valued. Continuous valued price utilities are analytically more tractable and allow the analyst to test the appropriateness and different non-linear functional forms. It is even possible to let different respondents within the same study have different functional forms for their price utilities depending on their answers to the trade-off questions.

Choosing price differentials
Each price trade-off question requires a difference in the prices used for each side. If this price differential is too small, respondents will always choose the higher cost side thus producing unrealistically low price sensitivities. Similarly, a price differential that is too large will cause respondents to choose the lower cost side. The problem is complicated by the differences among respondents; what is too large for one respondent may be too small for another. If a computer-based interview is used, the computer can alter the price differential based on previous answers by the respondent. For paper/pencil conjoint analysis, pilot test results can be used to adjust price differentials. These alterations of price differentials are much easier to accomplish if the price attribute is continuous valued.

Adapting the price range to the individual
In some pricing studies individuals will have very different ideas about the range of prices that is appropriate for the product. For example, in choosing a personal computer one person may be thinking about a small computer in the $1000-2000 range while another may want a more powerful unit in the $4000-6000 range. To make the interview credible the respondent must trade-off prices that are realistic. A computer-based interview can accomplish this easily with a few questions at the beginning of the interview. For paper/pencil studies it is sometimes possible to establish a reference price at the beginning of the interview and then use phrases such as "Reference price + $200" in the trade-off questions.

Adapting pricing units to the individual
In some situations different respondents may think of price in different units. The most common example of this occurs for expensive items in which some respondents may base their purchase decision on the total price while others may use the equivalent monthly payment. For computer-based interviewing this can be handled easily with a few questions prior to the trade-offs. It is even possible to include financing attributes such as down payment and loan length, if that is important. For paper/pencil studies this can be handled by asking one or two questions during recruiting and then sending different questionnaires depending on the responses.

Conclusions

Conjoint analysis is a data collection and processing technique that can bring some of the simple ideas and techniques of microeconomics to the market planner. Specifically it allows for the estimation of a multi-product demand function in which price is an explicit attribute. When combined with the concepts and tools of modeling and optimization, it can yield new insights into the complex portfolio pricing decisions facing modern corporations.