viewof bi = Inputs.range([1.2, 3.0], {value: 2.0, step: 0.05,
label: "Your price sensitivity (bᵢ)"})
viewof di = Inputs.range([0.2, 2.6], {value: 1.1, step: 0.05,
label: "Customers you gain when they raise price (dᵢ)"})
viewof dj = Inputs.range([0.2, 2.6], {value: 1.1, step: 0.05,
label: "Customers you lose when you raise price (dⱼ)"})12 How Differentiation Preserves Margin
Two firms, two equations, and the price where both stop moving
The last chapter ended in the worst case: identical products, customers who care only about price, and undercutting that runs until margin is gone. That case is real and worth fearing, and it is not where most markets live.
In most markets customers notice differences. Some prefer your brand, some trust your delivery, some are three blocks closer to you, and some simply do not check the other price.1
It Takes Two Equations
Here is where most treatments go wrong, and it is worth being careful because the error is easy to miss.
Your demand under rivalry gets written like this:
\[ \mathsf{q_i = a_i - b_i\,p_i + d_i\,p_j} \]
Three terms, each doing a distinct job. \(\mathsf{a_i}\) is how many would buy at a price of zero, the sheer size of your appeal. \(\mathsf{b_i}\) is how fast you lose customers when you raise your price. And \(\mathsf{d_i}\) is how many arrive when they raise theirs.
That equation is correct and it is half the system. Your rival has one too:
\[ \mathsf{q_j = a_j - b_j\,p_j + d_j\,p_i} \]
Look at what the subscripts are doing before moving on. \(\mathsf{d_i}\) is how many of their customers come to you when they raise price. \(\mathsf{d_j}\) is how many of yours leave for them when you raise yours. These are two different questions about two different groups of people, and nothing whatever requires them to be equal.
\(\mathsf{d_i}\) and \(\mathsf{d_j}\) — the two directions of switching. \(\mathsf{d_i}\) is how many you gain when the rival raises price; \(\mathsf{d_j}\) is how many you lose when you raise yours. Writing only one of them describes half the market.
Write only your own equation and you have said how many of theirs you get, while saying nothing about how many of yours you lose. Those are the two questions any real founder is actually asking, and one equation can only answer one of them.
This is not a technicality. Hold the total amount of switching in the market constant and change only how it splits between the two directions, and the consequences are enormous.
| You gain (dᵢ) | You lose (dⱼ) | Your price | Your profit |
|---|---|---|---|
| 0.3 | 1.9 | $853 | $325,646 |
| 0.6 | 1.6 | $936 | $546,602 |
| 1.1 | 1.1 | $1,055 | $911,819 |
| 1.6 | 0.6 | $1,139 | $1,203,877 |
| 1.9 | 0.3 | $1,170 | $1,317,419 |
Read the first and last rows. Same appeal, same price sensitivity, same costs, and the same total amount of switching in the market. The only difference is which way the customers flow. Your profit runs from about $326,000 to about $1.3 million, a swing of nearly a million dollars, and every row has an identical-looking demand equation of your own.
That is what one equation hides. Substitution is not a quantity. It is a direction, and it has two of them.
Your Best Response
Take the rival’s price as given for a moment. You are back to a single profit curve with a single peak, which is what the last chapter’s slider was showing.
Doing that at every possible rival price traces the ridge, and the ridge has a formula:
\[ \mathsf{p_i^*(p_j) = \frac{a_i + b_i c_i + d_i\,p_j}{2\,b_i}} \]
For the Curious — where that comes from
Profit is price above cost times quantity, less the commitment:
\[ \mathsf{\pi_i = (p_i - c_i)(a_i - b_i p_i + d_i p_j) - f_i} \]
Multiply out and the \(\mathsf{p_i}\) terms are \(\mathsf{-b_i p_i^2}\) and \(\mathsf{(a_i + b_i c_i + d_i p_j)\,p_i}\). That is a downward parabola in your own price, so it has one peak, and the peak sits where the slope is zero:
\[ \mathsf{a_i - 2 b_i p_i + b_i c_i + d_i p_j = 0} \]
Solving for \(\mathsf{p_i}\) gives the best-response function above. Everything in the numerator pushes your price up: a larger appeal, a higher unit cost to cover, a more expensive rival. The \(\mathsf{2b_i}\) underneath is what pulls it back down, which is why price sensitivity is the parameter that disciplines you.
The shape of that expression is worth more than its derivation. Your best price rises when theirs does, and the rate is \(\mathsf{d_i / 2b_i}\) — the ratio of how many you gain from their price rise to how sharply your own price costs you. With the numbers from the last chapter that ratio is about a quarter, which is why you follow them partway and never all the way.
And notice the limiting case. If \(\mathsf{d_i}\) is zero, the rival’s price falls out of the formula entirely and you price as though alone. Perfect differentiation is not rivalry at all; it is two monopolies that happen to share a market.
Where the Two Best Responses Meet
Your rival has a best-response function of the same shape, pointing the other way. Each of you wants to sit on your own line, and there is exactly one price pair where both of you do.
That intersection is the equilibrium. Neither firm can improve its profit by moving alone: if you raise your price you leave your own best-response line, and so do they. Game theory has a name for that condition and the next part of the book will use it, but the idea needs no vocabulary. It is the only place where two firms optimizing at the same time both stop moving.
It is not an agreement, and it is emphatically not collusion. Neither firm has communicated with the other or intends anything cooperative. It is simply where two independent, self-interested calculations run out of moves.
Below, both lines are drawn. The sliders change your structure only, and the rival keeps responding as they always would.
Lower your price sensitivity and your line swings outward: at every rival price you can charge more, and the intersection slides up and to the right. Raise the customers you gain and the same thing happens more gently. Raise the customers you lose and it works against you, because that slider belongs to their equation rather than yours.
Push far enough and the two lines stop crossing inside the picture. That is not a drawing error. The equilibrium is still there, and the readout will still name a price for it; it has simply run off the top of the window, and the prices and profits it reports have stopped describing any market you would recognize. Whether an equilibrium can vanish outright is a separate question, and a real one.
What this model cannot tell you
The best-response lines only cross when differentiation is strong enough relative to switching — formally when \(\mathsf{(d_i/2b_i)(d_j/2b_j) < 1}\). Past that point the arithmetic has no equilibrium to report, which is the model’s way of saying that products this substitutable do not settle at a price. That is the price war of the last chapter, arriving as a mathematical fact rather than a story.
A second boundary is easier to walk into. It is tempting to treat \(\mathsf{d}\) as a differentiation dial and turn it up for both firms at once, but doing that while holding \(\mathsf{a}\) and \(\mathsf{b}\) fixed quietly enlarges the whole market, because at equal prices demand becomes \(\mathsf{a - (b - d)p}\). Turn the dial far enough and the model reports prices and profits that no market has ever produced. That is the instrument rather than the world.
Both are reasons to use this model the way an entrepreneur actually needs it: not to ask what would happen if the whole industry were more differentiated, but to ask what happens if you are more differentiated than the firm across the street.
What Differentiation Actually Is
Which brings us to the question the equations were built to answer.
Differentiation is not one number. It is a claim about customers, and it shows up in the demand system in more than one place, doing different work in each.
It shows up as \(\mathsf{b_i}\), and this is the one people underrate. Your own-price sensitivity says how many customers you lose when you raise your price and the rival stands still. Low \(\mathsf{b_i}\) means people who want you specifically, and it buys the one thing every other advantage struggles to deliver: pricing power. Tune each lever until it opens the same profit gap over a rival, and the routes are visibly different. Extra appeal and extra switching both get you there mostly by selling more. Lower price sensitivity gets you there by charging appreciably more.
It shows up as the split between \(\mathsf{d_i}\) and \(\mathsf{d_j}\), and the split is the part that is yours. Not the total switching in the market, which you do not control, but its direction, which you influence with everything you do. A firm customers defect to rather than from is a differentiated firm, and the table earlier put a number on what that asymmetry is worth.
It shows up as \(\mathsf{a_i}\), the size of your appeal. Real, and the least distinctive of the three, because a bigger \(\mathsf{a}\) is closer to being better known than to being harder to leave.
What differentiation is not is a story about your product. It is a set of claims about behavior, and every one of them is measurable: how much do you lose when you raise your price, how much do you gain when they raise theirs, and how much do you lose when they cut. Those are three different experiments, and they are what the next chapter is about.
Before you accept an equilibrium
The equilibrium is a prediction about what two firms will do, produced from six parameters and an assumption that both of them optimize. Check it against the market you can see.
- Does the rival’s predicted price resemble what the rival charges today? A wide gap means one of three things: the system is mis-estimated, the rival is not optimizing, or the firm you named is not the one your customers compare you to.
- Do the two switching numbers match what you observe? Read \(\mathsf{d_i}\) and \(\mathsf{d_j}\) back in words — this many come to me when they raise price, this many leave when I do — and say whether that is the market you know.
- Does the crossing sit at a price you would actually charge? Lines that meet far outside your range have not found a price war; they have found the edge of where this model describes anything.
If you cannot answer all three, what you have is a solved system rather than a competitive position.
Reading Your Position
The practical shift in this chapter is where you evaluate everything you already know how to evaluate.
Feasibility is no longer about finding some price where profit is positive. It is about whether profit survives at the intersection — the price that emerges once the other firm has finished responding. A venture can show a healthy profit curve in isolation and sit at an equilibrium that does not clear its commitments.
Equilibrium price is disciplined rather than chosen. It rises with your unit cost, rises with your appeal, and falls with your price sensitivity, and none of those is a decision you make on the day. They are structural, which is exactly why the next two chapters are about measuring them and then about which of them are durable.
The one that should worry you is \(\mathsf{b_i}\). A small disadvantage in price sensitivity compounds through the whole system: it lowers your best-response line, which lowers the intersection, which lowers your margin at the price you end up charging. Being slightly more replaceable than your rival is not a slightly worse position.
Ask yourself — which direction do customers flow?
Name your closest real competitor. Not a category, a firm.
Now answer two separate questions and resist the urge to merge them. If they raised their price ten percent tomorrow, how many of their customers would come to you? And if you raised yours ten percent, how many of yours would go to them?
Most people find one of those numbers is much easier to guess than the other, and most people find the honest answers are not equal. Whichever direction is larger is the direction your market is actually differentiated in, and it may not be the direction you have been telling yourself.
Then the harder one. If you raised your price ten percent and your rival did nothing at all, how much of your volume would you keep? That number is \(\mathsf{b_i}\) in plain language, and it is the closest thing to a measure of whether your differentiation is real.
The move: Differentiation is not what makes your product different. It is what makes your customers slow to leave when you charge more, and it has to be checked in both directions — what you gain when they raise price, and what you lose when you raise yours.
One thing should be nagging at you by now. Every number in this chapter was handed to you. The appeal, both price sensitivities, both directions of switching, the costs — all supplied, so the machinery would be visible while it worked.
Not one of them is knowable about your own venture without going and finding out. And a demand system you guessed at produces an equilibrium you guessed at, carrying all the confidence of arithmetic and none of the standing.
The observation that a small difference is enough to stop the collapse goes back to Hotelling (1929), whose customers preferred the nearer of two sellers along a road. Location was the difference in his case; anything customers care about does the same work. When a rival drops a dollar, you lose a few of them rather than all of them. That partial loyalty is the whole subject of this chapter, because it is what stands between you and the collapse.↩︎