13 Measuring Demand Under Rivalry
Where the parameters come from, and why one price is no longer enough
Every number in the last two chapters was handed to you. The appeal, both price sensitivities, both directions of switching, the costs. That was deliberate, so the machinery would be visible while it ran, and it is not a position you can make a decision from.
This chapter is about where those numbers actually come from. The good news is that nothing you learned about demand experiments is wrong. The complication is that adding a second firm adds a second price, and a second price changes what an experiment has to accomplish.
Two Prices, Not One
When demand depended on your price alone, an experiment answered one question: how does quantity respond as your price moves?
Under rivalry, quantity responds to two prices, and the responses are different things.
The own-price effect is the familiar one. Raise your price and some customers buy less or leave. That is \(\mathsf{b_i}\), and everything in designing experiments about eliciting it still applies.
The cross-price effect is new. Raise the rival’s price and some of their customers come to you. That is \(\mathsf{d_i}\), and no experiment that varies only your own price will ever reveal it.
Measure only the first and you have a demand curve that will mislead you the moment the other firm moves. Measure only switching, without price sensitivity, and you know customers are mobile without knowing what it costs you when they move.
And there is a third thing, which the last chapter argued is the one most often missed. Your rival has an equation too, and \(\mathsf{d_j}\) — how many of your customers leave for them when you raise price — is a separate quantity that your own customers’ answers can tell you about only if you ask.
That is three measurements, not one, and only the first is familiar.
The Rival’s Price Is a Treatment
The structural change to experimental design is small to state and easy to get wrong.
Under monopoly the rival’s price does not exist. Under rivalry it is not background context either. It is a variable you must deliberately manipulate, in the same way and for the same reason you manipulate your own.
The sequence of a good demand experiment does not change. You still screen for the right population, establish context, ground the problem and the solution, evaluate appeal, and elicit price or quantity. What changes is that the price question now has to describe two offers rather than one, and the rival’s offer has to be described consistently, framed neutrally, and priced differently across respondents.
That framing burden is heavier than it sounds. Describe the competitor warmly and you will measure more switching than exists; describe them dismissively and you will measure less. Both errors look like data. Under rivalry, neutrality in how you present the other firm is not politeness, it is measurement.
What a stated-preference experiment cannot settle
Asking people what they would do when a competitor changes price measures intention under a described scenario, not behavior in a market. Cross-price effects are especially exposed to this, because switching is exactly the kind of decision people find easy to imagine and harder to execute against habit, convenience, and inertia.
Treat estimated substitution as an upper bound on how mobile your customers really are, and prefer a smaller number when the evidence is thin. The checks that decide whether a curve is safe to use apply here with more force than it did under monopoly, not less.
Identification, Not Realism
Here is the requirement that decides whether an experiment works at all, and it is the one founders most often violate while being careful about everything else.
To separate own-price sensitivity from substitution, the two prices have to move independently. Your price must change while the rival’s holds still. The rival’s must change while yours holds still. And some respondents must see both change together.
If every scenario moves both prices in step, the data cannot tell the two effects apart. Not because the sample is too small or the statistics too crude, but because the information simply is not in there. No amount of analysis recovers a distinction the design did not create.
This is where a sensible instinct does real damage. Founders build scenarios that feel realistic, and in a real market prices tend to move together — everyone’s costs rise, everyone raises price. Reproduce that faithfully in your design and you will produce a beautifully realistic experiment that cannot answer your question.
The goal of experimental design is not realism. It is identification.
A respondent may never encounter the price pair you show them. That is acceptable and often necessary. What is not acceptable is a set of scenarios from which the parameters cannot be recovered.
The obvious way to guarantee independent variation is to show every combination of prices, and that fails for a different reason: the combinations multiply fast, and a respondent asked twenty pricing questions stops thinking somewhere around the eighth. Structured designs exist that get identification from a fraction of the full grid, and the method layer for a market with a rival has the mechanics. The principle is what matters here. Competitive demand has to be identified on purpose. It does not turn up on its own.
Reading the Coefficients
Estimation gives you a fitted system rather than a single curve:
\[ \mathsf{q_i = a_i - b_i\,p_i + d_i\,p_j} \] \[ \mathsf{q_j = a_j - b_j\,p_j + d_j\,p_i} \]
Six numbers, and each one is a claim about behavior you can sanity-check against what you know.
A large \(\mathsf{b_i}\) says your customers leave quickly when you raise price. Your margin is fragile and you have less pricing room than the raw appeal suggests. A small \(\mathsf{b_i}\) says the opposite, and it is the parameter the last chapter argued is worth the most.
A large \(\mathsf{d_i}\) says you capture a lot when the rival raises price, which means you are a close substitute for them. A small one means your customers are largely insulated from what they do.
And the pair \(\mathsf{d_i}\) against \(\mathsf{d_j}\) is the comparison worth making first, because it is the one a single equation cannot show you. If they are close to equal, you and your rival are symmetric substitutes and neither has an edge in the flow. If they diverge, the firm on the favorable side has an advantage that will show up in equilibrium whether or not anyone has noticed it.
Linear demand is a convenience rather than a commitment. Exponential, logistic, and other shapes can be fitted the same way, and the same caution from estimating demand governs the choice: prefer the shape whose implied behavior matches what you can see, not the shape with the better fit statistic. Equilibrium does not need a closed-form solution. Once the system is estimated, the competition app will find where the best responses cross numerically, and so will any competent AI you hand the parameters to.
Where These Numbers Come From
One reframing before you go and measure anything, because it changes what you do with the result.
It is tempting to treat these six parameters as facts about a market — conditions you discover, the way you would discover a population or a regulation. They are not. They are consequences of decisions somebody made, most of them yours.
Your appeal \(\mathsf{a_i}\) follows from product design, positioning, and who you chose to sell to. Your price sensitivity \(\mathsf{b_i}\) follows from how clearly the value is understood, how distinctive the offering is, and whether your customers have an easy alternative in mind. The substitution terms follow from how differentiated you actually are and from which segment you targeted, which is why two firms selling similar things to different people can have very different \(\mathsf{d}\) values.
You do not choose your equilibrium price. You choose the parameters that produce it, mostly long before the price is ever charged, through decisions that did not look like pricing decisions at the time.
That is what makes measurement worth the trouble. A parameter you inherited is a constraint. A parameter you produced is a design choice you can revisit, and knowing which of the six are weak tells you which decision to reopen.
Before you trust a competitive demand estimate
Do not carry these parameters into an equilibrium calculation until you can answer four questions.
- Did both prices move independently? Point to scenarios where yours changed and theirs did not, and scenarios where the reverse happened. If you cannot, your own-price and cross-price effects are not separately identified whatever the software reported.
- Was the rival described neutrally? Read your own scenario text as though you worked for the competitor. If it would annoy you, your substitution estimate is biased and you know which way.
- Do both directions make sense together? Compare \(\mathsf{d_i}\) and \(\mathsf{d_j}\) and say out loud what their difference claims about real customers. A large asymmetry is a strong claim and needs a reason you can name.
- Does the fitted system behave at the prices you care about? Check quantity at your candidate price and at the rival’s likely price. If either is negative or absurd, the fit is being extrapolated past the evidence.
If you cannot answer all four, what you have is a set of coefficients rather than a measurement.
Ask yourself — could my experiment tell the two effects apart?
Sketch the pricing scenarios you would actually put in front of people. Four or five of them, with both prices written down.
Now look only at the pairs of numbers. Do your price and the rival’s price move together in every scenario? If they do, you have designed an experiment that will produce clean-looking estimates of something other than what you wanted, and you will not find out until the numbers disagree with the market.
Then ask the question behind it. If a competitor cut price by fifteen percent next month, how much of your volume would go? You are going to answer that number one way or another — from evidence, or from optimism, on the day it happens.
The move: Vary both prices independently or you cannot separate your own price sensitivity from substitution. Realistic scenarios where prices move together produce estimates that cannot be told apart.
With demand measured on both sides, cost and scale go on top exactly as they did before rivalry existed, and the equilibrium follows from the parameters rather than from anybody’s judgment about where prices ought to land. What remains is the harder question of whether the position those parameters describe is one worth having.