16 Measuring Demand Under Rivalry

Substitution, price sensitivity, and experimental design in competitive markets

In the previous chapters on competition, we analyzed profit by assuming a demand structure.

We showed how profit becomes a surface defined by two prices.
We derived best-response functions.
We identified equilibrium.

But all of those results depend on the shape of demand.

Until now, demand was given. In practice, it must be measured.

Under rivalry, demand is no longer one-dimensional. It depends on both your price and your rival’s price.

\[ \mathsf{Q_i = f(P_i, P_j)}\]

That single change — adding a second price — transforms how experiments must be designed and how demand must be estimated.

This chapter extends the experimental discipline developed earlier to competitive settings.
The goal is not to add complexity.
The goal is to measure substitution directly.

Once competitive demand is estimated, cost and scale can be layered on top to evaluate profit under rivalry. That step comes next.

First, we must measure the structure that makes competition possible.

16.1 Demand Under Rivalry Is Two-Dimensional

When demand depended only on your price, estimation focused on one question:

How does quantity change when I change price?

Under rivalry, there are two distinct effects.

Own-price effect

When you raise your price, some customers reduce quantity or leave.
This is price sensitivity — the familiar slope discipline.

Cross-price effect

When the rival raises their price, some customers shift toward you. This is substitution.

Both must be measured.

If you only estimate how demand responds to your own price, you do not understand competition. If you only estimate switching without price discipline, you do not understand margin.

Rivalry is the joint presence of both.

Importantly, the principles of good experimentation do not change.

You still must:

Define the unit clearly.
Define the period clearly.
Frame the problem carefully.
Avoid measurement error.
Validate responses.

Nothing about data discipline disappears under competition.
But something becomes more fragile.

Dimensionality Increases Fragility

When demand is one-dimensional, errors distort one slope.
When demand is two-dimensional, errors distort interaction.

If units differ across products, substitution is mismeasured.
If time periods differ, cross-price sensitivity becomes noise.
If framing favors one product over another, estimated advantage becomes artificial.

Under rivalry, sloppiness compounds.

The experimental principles developed earlier do not change.
They become more important.

Move from Optimization to Strategic Interaction

With one-dimensional demand, the central question was:

What price maximizes profit?

With two-dimensional demand, the question becomes:

What price maximizes profit given that someone else is also choosing a price?

This is strategic interaction. But strategic interaction is not abstract theory. It rests on measurable substitution and measurable price sensitivity.

Before we solve for equilibrium profit, we must measure those forces directly.

16.2 Designing Experiments Under Rivalry

To estimate competitive demand, both prices must vary.
Your price must vary.
The rival’s price must vary.

Without variation in both, substitution cannot be identified.

If the rival’s price is held fixed, you can estimate own-price sensitivity — but not cross-price effects.
If your price does not vary independently of the rival’s, the two effects become confounded.

Competitive demand requires deliberate variation in two dimensions.

Rival Price Is Now an Experimental Variable

Under competition, the rival’s price is not background context. It is a treatment variable.

This does not require abandoning prior survey discipline.

The same six-step structure applies:

Screen the right population.
Establish context.
Ground the problem and solution.
Evaluate appeal.
Elicit price or quantity.
Gather customer characteristics.

The only structural extension is this:
Price questions must now incorporate the rival explicitly.

The rival’s offer must be described consistently.
Its framing must not bias responses.
Its price must vary systematically.

Careless framing can artificially inflate substitution or suppress it.
Under rivalry, neutrality matters more.

Identification Requires Independent Variation

To estimate own-price and cross-price effects separately, price variation must be structured so that:

Your price changes while the rival’s does not.
The rival’s price changes while yours does not.
Both change in combination.

If prices move together in every scenario, the data cannot distinguish substitution from own-price discipline.

The goal of experimental design is not realism.
It is identification.

Respondents may not encounter every price combination in reality.
But without structured variation, the resulting demand estimates will not support competitive analysis.

The Cost of Combinatorial Explosion

When two prices vary, the number of possible combinations grows quickly.
Naively enumerating all price pairs creates respondent fatigue and noisy data.
Under rivalry, experimental compression becomes necessary.
Structured design — rather than exhaustive enumeration — allows substitution to be estimated without overwhelming respondents.

The detailed mechanics of these designs are provided in Toolkit 16.

Here, the principle is simple:
Competitive demand must be intentionally identified.
It does not emerge accidentally.

16.3 Estimating Cross-Price Effects

Design creates variation.
Estimation translates that variation into structure.

Under rivalry, demand is no longer estimated as:

\[ \mathsf{Q = f(P)} \]

It is estimated as:

\[ \mathsf{Q_i = f(P_i, P_j)} \]

At minimum, estimation must identify two forces:

Own-price sensitivity — how quantity changes when your price changes.
Cross-price substitution — how quantity shifts when the rival’s price changes.

A simple linear specification illustrates the structure:

\[ \mathsf{Q_i = a_i - b_i P_i + d_{ij} P_j} \]

Each parameter carries strategic meaning:

\(\mathsf{a_i}\) — baseline demand (intercept)
\(\mathsf{b_i}\) — own-price sensitivity (slope discipline)
\(\mathsf{d_{ij}}\) — cross-price substitution (switching intensity)

The goal of estimation is not statistical elegance.
It is structural interpretation.

Interpreting the Coefficients

If \(\mathsf{b_i}\) is large, demand falls sharply when price rises.
Margin is fragile.

If \(\mathsf{d_{ij}}\) is large, customers shift strongly when the rival changes price.
Competition is intense.

If \(\mathsf{d_{ij}}\) is small, customers are relatively insulated from rival pricing.
Competition is softened.

These are not abstract numbers.
They describe measurable strategic pressure.

Functional Form and Flexibility

Linear demand is useful for intuition and interpretation.

In practice, competitive demand may also be estimated using:

Exponential specifications
Logistic or sigmoid forms
Other nonlinear models

Closed-form solutions for equilibrium are not required.
Once demand is estimated, competitive profit can be computed numerically.
The competition analytics app performs this optimization after parameters are estimated.

What matters here is not algebraic solvability.
What matters is whether the data credibly identify:

Own-price discipline
Cross-price substitution

Without both, competitive profit cannot be diagnosed.

From Estimation to Strategy

Under monopoly, estimation revealed price sensitivity.
Under rivalry, estimation reveals interaction.

It tells us:

How strongly customers react to our price.
How strongly they react to the rival’s price.
How tightly the two firms are linked through substitution.

Only after these forces are measured can we layer cost and scale on top to evaluate competitive profit.

That step follows next.

16.4 From Estimated Demand to Competitive Profit

Demand estimation gives us structure.

But structure alone does not answer the central question:

Is this worth doing?

To answer that, estimated demand must be translated into profit under rivalry.

The logic mirrors the monopoly case, with one critical difference.

Under monopoly:

Estimate demand.
Add cost.
Optimize price.
Evaluate profit.

Under rivalry:

Estimate competitive demand.
Add cost.
Solve mutual optimization.
Evaluate equilibrium profit.

The difference is interaction.

Step 1: Construct the Profit Function

Once demand is estimated, profit for firm ( i ) is:

\[ \mathsf{\pi_i(P_i, P_j) = (P_i - c_i)\, Q_i(P_i, P_j) - f_i} \]

Where:

\(\mathsf{c_i}\) is variable cost.
\(\mathsf{f_i}\) is fixed cost.
\(\mathsf{Q_i(P_i, P_j)}\) is the estimated demand function.

Nothing about cost changes under rivalry.
What changes is that profit now depends on two prices.

Step 2: Solve for Mutual Best Response

Each firm chooses price to maximize its own profit, taking the other’s price as given. This produces a best-response function for each firm.

Equilibrium occurs where both firms are simultaneously choosing their optimal price.

Closed-form solutions may exist in simple linear cases.
In practice, numerical optimization is sufficient.
The competition app implements this step once demand and cost are specified.

The key point is that equilibrium price is not chosen in isolation.
It is determined by strategic interaction embedded in the demand estimates.

Step 3: Evaluate Equilibrium Profit

Once equilibrium prices are identified, equilibrium quantities and profits follow directly from the estimated demand and cost structure.

At this stage, feasibility can be evaluated.
Profit may survive.
Profit may be fragile.
Profit may collapse toward cost.

But those outcomes are no longer assumed. They are measured.

The Analytics Spine Restored

The full competitive analytics sequence is now visible:

Design the experiment.
Estimate competitive demand.
Add cost.
Solve for equilibrium.
Diagnose profit durability.

Competition does not require new economic philosophy.
It requires measuring substitution with the same discipline used to measure price sensitivity.

Once competitive demand is estimated, interaction is no longer abstract theory.
It is embedded in the parameters.

Equilibrium price is a consequence.
Profit is a consequence.
Fragility is measurable.

16.5 Diagnosing Structural Position

Once competitive demand is estimated, structural position becomes visible.

Competitive advantage is not inferred from revenue.
It is not inferred from market size.
It is not inferred from narrative strength.

It is inferred from parameter differences.

If two firms have identical demand and cost parameters, equilibrium outcomes will be symmetric.
Prices will match.
Margins will match.
Profits will move together.

Symmetry produces shared discipline.
Asymmetry produces advantage.

Structural position can be diagnosed along four measurable dimensions:

Baseline demand (\(\mathsf{a}\)) — Who begins with stronger underlying customer pull?
Own-price sensitivity (\(\mathsf{b}\)) — Who faces steeper margin discipline?
Cross-price substitution (\(\mathsf{d}\)) — Who is more insulated from rival pricing?
Unit variable cost (\(\mathsf{c}\)) — Who can profit at lower price levels?

These parameters are not stories.
They are empirical estimates.

Small differences in price sensitivity or substitution can produce large differences in equilibrium profit.
Minor cost advantages can become decisive under competitive pressure.

Under rivalry, advantage does not appear as a slogan.
It appears as parameter asymmetry.
Measured demand reveals whether competition is tight or softened, fragile or durable.

In the next chapter, we layer cost and scale onto these estimated demand structures and solve for competitive profit directly.

Competitive advantage is measured, not declared.