One Equation for Pricing: ψ(p*) = c

We start from one derivative you could do in your sleep and end somewhere genuinely deep. Every step in between is shown.

Part I — The hook

Picture a pricing meeting. Three strategies go up on the whiteboard.

• Cost-plus. Take your cost, add a margin, ship it. Feels safe. It is almost always wrong, and — as we'll see — wrong in a measurable direction and by a measurable amount.
• Match the competitor. Price a hair under whoever you're chasing. This treats a competitor as a strategy to be countered, when they are really just data about your buyers.
• Charge what it's worth. Find the willingness-to-pay and charge it. Closer, but "what it's worth" is not a single number — it's a distribution, and the optimal price is a specific, computable point inside it.

Two promises for the end of this post:

1. You'll know why a $1 cost increase should sometimes raise your price by more than $1 — and you'll be able to say exactly when.
2. You'll see that competitors don't need a strategy; they're just a term in the distribution of buyer values.

Everything reduces to one equation. It has your marginal cost on one side and a single function built entirely from your demand curve on the other. We'll construct that function from scratch in Part III, box the equation in Part IV — you will derive it yourself, every step shown — and then, in Part VI, run it end-to-end on a concrete product (an AI document-review SaaS — software sold as a subscription) and put dollar figures on every claim. At the very end we come back to this whiteboard and give each of the three strategies its verdict.

Part II — The detour: spectrum auctions

Why the FCC needs the same math you do

In 1994 the U.S. government had a problem that sounds nothing like your pricing page and is mathematically identical to it. The FCC — the Federal Communications Commission, the agency that licenses the airwaves — had to sell radio-spectrum licenses — the airwaves your phone uses — to companies whose true valuations it could not observe, with billions of dollars on the line. Price them too low and you hand public money to telecoms; too high and the licenses go unsold. The economists who designed the fix would later share a Nobel Prize for exactly this work. It is the same math as your pricing page.

Auction formats and their equivalences are a rabbit hole — the Vickrey auction is the usual entry point — but nothing below requires them. Auctions come back exactly once, as a one-liner near the end.

The one object we need is the distribution of buyer values. A buyer's value $v$ is drawn from a distribution $F$:

$v \sim F, \qquad F(v) = \Pr[\text{value} \le v], \qquad f(v) = F'(v).$

$F$ is just a CDF (cumulative distribution function) — the same object you'd fit from telemetry. Survey 100 prospects; then $F$ evaluated at $200k is the fraction who walk away at that price, and $1 - F$ there is the fraction who'd still buy. The density $f$ is how those buyers pile up along the value axis. Everything in this post is a manipulation of that one estimated curve.

For a gentle, engineer-friendly entry to all of this, the anchor recommendation is Roughgarden, Twenty Lectures on Algorithmic Game Theory, Lectures 2–3.

Part III — Virtual value, derived

What one lever move actually costs you

We'll build the machinery in three moves. First a baby example you can do with high-school calculus. Then the same three moves with a general $F$. Then one new object — the hazard rate — with a picture.

III.1 — The complete baby example

Assume values are spread uniformly between $0$ and $b$: everyone's willingness to pay (WTP) is equally likely to be anywhere in that range. Then

$F(p) = \frac{p}{b}, \qquad \text{demand} = 1 - F(p) = 1 - \frac{p}{b}.$

Revenue is price times the fraction who buy. The trick that keeps this at Level 1: expand the product into a polynomial so we never need the product rule.

$R(p) = p\left(1 - \frac{p}{b}\right) = p - \frac{p^2}{b}.$

Differentiate term by term — no product rule, nothing skipped:

$R'(p) = 1 - \frac{2p}{b}.$

Set it to zero:

$R'(p) = 0 \;\Rightarrow\; p^* = \frac{b}{2} \quad (\text{zero cost}).$

Now add a marginal cost $c$. Profit is $(p - c)$ per sale times the fraction who buy. Expand again:

$\pi(p) = (p - c)\left(1 - \frac{p}{b}\right) = p - \frac{p^2}{b} - c + \frac{cp}{b},$

$\pi'(p) = 1 - \frac{2p}{b} + \frac{c}{b} = 0 \;\Rightarrow\; p^* = \frac{b + c}{2}.$

There's a second route that needs no calculus at all. Demand is a straight line from $(0, b)$ down to full participation at price zero; revenue is the area of the rectangle tucked under that line whose upper corner sits at your chosen price. That rectangle is largest when the corner sits at the midpoint of the demand line — again $p^* = b/2$. Two independent routes to the same answer is the strongest reassurance a Level-1 reader can get.

You just did the entire theory. Everything from here is the same three moves — write revenue, differentiate, set to zero — with a general $F$ in place of a straight line.

III.2 — The general derivation

Drop the straight-line assumption. Revenue for any distribution is

$R(p) = p\,\bigl(1 - F(p)\bigr).$

Differentiating needs the product rule this time. The fast-lane result:

$R'(p) = \bigl(1 - F(p)\bigr) - p\,f(p).$

Now the one clean move that produces the hero of this post. At the revenue peak $R'(p) = 0$; divide through by $f(p)$ — a density, so it's positive and dividing is safe — and rearrange:

$p - \frac{1 - F(p)}{f(p)} = 0.$

The left-hand side earns a name. Define the virtual value:

$\psi(v) \equiv v - \frac{1 - F(v)}{f(v)}.$

(The auction-theory literature — including Roughgarden and the Brown AGT notes — usually writes this as $\varphi$; it's the same object.)

Read it as a bias-corrected value: the naive value $v$, minus a tax you pay on account of everyone whose value sits above $v$. Revenue climbs as you cut price exactly while $\psi$ is still negative, and peaks where $\psi$ crosses zero. In Part IV, cost will do something very small: it will raise that crossing from $0$ to $c$.

III.3 — The hazard rate

The correction term $\tfrac{1 - F(v)}{f(v)}$ is worth its own object. Its reciprocal is the hazard rate:

$h(v) = \frac{f(v)}{1 - F(v)}, \qquad \psi(v) = v - \frac{1}{h(v)}.$

A bridge for anyone who's done reliability engineering. This is the exact same hazard rate as in failure analysis — just replace "time to failure" with "willingness to pay." A constant hazard rate is the memoryless exponential distribution, the way constant-failure-rate components have exponential lifetimes.

What it means for pricing. A fat tail (lots of high-value buyers far out to the right) means a small $h$, which means a big correction $1/h$, which pushes $\psi$ far below the raw value $v$. Fat tails are where naive pricing goes most wrong.

Part IV — Where cost enters

Cost sets the intersection, not the price

Cost has been waiting in the wings. Watch how little it does.

IV.1 — The one-line solve

Profit for a general $F$ is

$\pi(p) = (p - c)\bigl(1 - F(p)\bigr).$

Same two moves as III.2 — the product rule (now with $u = p - c$, still $u' = 1$), then divide by $f(p)$:

$\begin{aligned} \pi'(p) = \bigl(1 - F(p)\bigr) - (p - c)\,f(p) &= 0 \\ \Longrightarrow\quad p - \frac{1 - F(p)}{f(p)} &= c. \end{aligned}$

which is one line:

$\boxed{\;\psi(p^*) = c\;}$

That's the whole post. Demand fixes the entire shape of $\psi$; cost is a single horizontal line; the optimal price is where they cross.

IV.2 — Sanity checks

An engineer will poke the edges, so let's beat them to it.

• $c \to 0$: the price does not go to zero. In the uniform case $p^* \to b/2$, not $0$. Pricing is not cost recovery — even a free-to-produce product has an optimal price set entirely by demand.
• Point-mass $F$ (all buyers at one value $v_0$): argue directly. If everyone values it at exactly $v_0$, charge $v_0$; cost only decides whether to sell at all. So $p^* = v_0$ whenever $v_0 \ge c$.
  Footnote — the formula is undefined here, and that's fine

  At a point mass, $f$ is a Dirac delta, so $\tfrac{1 - F}{f}$ is a $0/\infty$-ish object and $\psi$ is formally undefined. Take a narrow-uniform limit around $v_0$ and $p^* \to v_0$, agreeing with the direct argument. The formula doesn't contradict the edge case; it just declines to be evaluated exactly on it.

• Non-monotone $\psi$: if $\psi$ isn't increasing, the crossing $\psi(p^*) = c$ may not be the whole story. One sentence for now; the Appendix pays it off.

IV.3 — Four distributions

Every row below is re-derivable by hand with the III.2 recipe.

For the closed-form rows, $F$ and $f$ are explicit — $F(v) = 1 - e^{-\lambda v}$ for the exponential and $F(v) = 1 - (v_{\min}/v)^\alpha$ for the Pareto — so the table is an exercise set with answers, not an appeal to authority.

The last column is the pass-through rate: the fraction of a $1 cost increase that lands on the price. It runs $\tfrac12$ / $1$ / $>1$ across the rows.

Why can pass-through exceed 1 — why raise price by more than a dollar when cost goes up a dollar? Because of the fat tail. Raising price on a Pareto population sheds proportionally few buyers (the ones who remain are far out in the tail and barely flinch), so the optimal response to a cost shock is to retreat further up the tail and serve fewer, richer buyers. That's luxury-market behavior — and here it's derived, not asserted.

IV.4 — The pass-through theorem

How does the optimal price move when cost moves? Differentiate $\psi(p^*) = c$. Fast lane:

$\frac{dp^*}{dc} = \frac{1}{\psi'(p^*)}.$

The theorem also tells you when pass-through exceeds 1: whenever $0 < \psi'(p^*) < 1$, i.e. when $\psi$ is flat — exactly the fat-tail regime.

IV.5 — Code

Numeric $\psi$ plus a root-find works for any $F$, including the log-normal that has no closed form — same three lines.

import numpy as np
from scipy.stats import lognorm
from scipy.optimize import brentq

def psi(v, F, f):
    return v - (1 - F(v)) / f(v)

# Example: log-normal WTP, median $180k, sigma = 0.7
s, scale = 0.7, 180_000
F = lambda v: lognorm.cdf(v, s, scale=scale)
f = lambda v: lognorm.pdf(v, s, scale=scale)

c = 68_000
p_star = brentq(lambda p: psi(p, F, f) - c, 1.0, 5_000_000)
print(f"optimal price ≈ ${p_star:,.0f}")

Run this on your own WTP survey data Monday morning. (We'll use exactly this snippet — and its output — in Part VI.)

IV.6 — Decoder ring: the Lerner index

You now own the result from every seat at the table, so here is the economist's dialect for it.

The same object, seen from three seats at the table:

IV.7 — The auction reframing

Here's the one auction line we promised. Myerson's rule (Myerson, 1981): allocate to the bidder with the highest $\psi(v_i)$, provided $\psi \ge c$, else don't sell. In an auction, cost is a reserve price; in posted pricing, it's a floor. Same math, same role — and it closes the loop on Part II: this is literally what the FCC's economists were computing. Their reserve price is your cost line; their bidders' virtual values are your $\psi$. Roughgarden's Lecture 3 is the gentler entry.

Part V — Where competitors enter

Competitors are demand, not strategy

A competitor changes what a buyer's alternative is worth, and that's all. Let a buyer's best outside option be $o = u_C - p_C$ (the utility of the competitor's product minus its price). The buyer's net value for you is $\tilde v = u - o$. Refit the distribution on $\tilde v$ instead of $v$ — call it $\tilde F$ — and the pricing equation is untouched:

$\psi_{\tilde F}(p^*) = c.$

The equation is the interface; competitors, costs, and demand estimates are implementations behind it. Separation of concerns, in a pricing model.

If only some buyers face that competitor, $\tilde F$ becomes a mixture of shifted and unshifted populations — and mixtures are exactly where $\psi$ can go non-monotone. That's the second time this post has pointed at the Appendix; mixtures are where the ironing machinery earns its keep.

Notice that nothing new happened mathematically in this whole section. That's the point — a deliberate breather after the IV.4 spike. The profundity here is conceptual, not technical.

One aside for completeness: genuine strategic competition (Bertrand, Cournot) shows up when competitors react to you. This post lives in the world where competitor prices are exogenous inputs to $\tilde F$.

Part VI — Case study: pricing an AI document-review SaaS

Doing this for real

VI.1 — Anchoring WTP

A permit review that a human does in $h_{\text{hrs}}$ hours at wage $w$, which our tool automates at capture rate $\alpha$, is worth roughly

$v_{\text{per-permit}} = \alpha\, h_{\text{hrs}}\, w + \beta\, D\, R,$

where the second term is the delay side: $\beta$ is the fraction of permits where faster review actually avoids a delay, $D$ the days saved, and $R$ the applicant's cost per day of delay. With $\alpha \approx 0.4$, $h_{\text{hrs}} \approx 10$, and $w$ around $60/hr, the labour term alone anchors about $240 per permit. Total engineering value scales with volume: $V_{\text{eng}} = q \cdot v_{\text{per-permit}}$.

One honest wedge before the bands below: a public-sector buyer does not pay engineering value dollar-for-dollar. Budget caps, procurement friction, and status-quo bias take a haircut, so realized willingness-to-pay is a fraction of $V_{\text{eng}}$. (Multiply it out for a mid city — $8{,}000 \times 240 \approx 1.9\text{M}$ dollars of modeled value against a WTP band topping out at $500k — and you can see the haircut. Treat the $240/permit as an upper anchor, and the bands as what survives procurement.)

VI.2 — Segments

VI.3 — Cost model

$C(q) = c_v\, q + c_f, \qquad c_v \approx \$6, \quad c_f \approx \$20\text{k}.$

For a mid-city deployment at $q = 8000$, that's $C = 6 \cdot 8000 + 20000 = 68{,}000$ — about $68k per year.

VI.4 — Worked example

Take the mid segment and model value as $V \sim \text{Uniform}[50\text{k}, 500\text{k}]$ (in dollars), with cost $c = 68\text{k}$.

$p^* = \frac{\$500\text{k} + \$68\text{k}}{2} = \$284\text{k}.$

Participation is $1 - F(p^*) = \tfrac{500 - 284}{500 - 50} \approx 48\%$. The margin on a customer you win is $p^* - c \approx 216\text{k}$ ($216k); weighting by the 48% chance that a given prospect buys at this price, the expected profit per prospect is about $104k.

VI.5 — Cost-sensitivity sweep

At the base case (cost $68k), cost-plus charges $95k against the optimal $284k — about $190k left on the table, a 3× underprice. That surplus doesn't vanish — it goes to customers, not because they earned it, but because the seller measured the wrong quantity. This is the figure the TL;DR promised to compute — and there it is. The effect only sharpens as the product gets cheaper to run: the gap widens toward $214k (a 5× underprice) at low cost and narrows to $115k (1.5×) at high cost — because naive cost-plus tracks cost while the optimal price barely moves. (That last part is the pass-through theorem in action: each $1 of cost shifts $p^*$ by only $0.50, since $\psi' = 2$.)

VI.6 — How wrong can your F be?

Here's the question every engineer has been holding since Part II: $F$ comes from a survey — how much does $p^*$ care if I fitted the wrong curve? Answer it by comparison, using the IV.5 code on the same survey data:

• Uniform fit over $[50\text{k}, 500\text{k}]$ (dollars) → $p^*$ = $284k.
• Log-normal fit, median $180k, $\sigma = 0.7$ → $p^*$ ≈ $224k.
• Log-normal fit, median $180k, $\sigma = 1.0$ (heavier tail) → $p^*$ ≈ $343k.

Three defensible fits of one survey span roughly $224k–$343k — a ±21% band around the uniform answer. That band is your model-uncertainty error bar, and notice it is wider than the entire cost-sensitivity sweep above: for this product, fitting $F$ carefully matters more than measuring $c$ precisely. The direction of the error depends on where the fit puts its mass — a median-$180k log-normal concentrates buyers lower than the uniform (price drops), while fattening its tail to $\sigma = 1$ pushes the optimum up past both. The pass-through machinery is doing double duty here: it's also your error-propagation tool. A steep $\psi$ means $p^*$ is insensitive both to a mis-estimated cost and to a mis-fitted tail.

One honesty note: stated preference overstates WTP — surveys are optimistic. Treat a survey-fitted $p^*$ as an upper anchor and update from real win/loss data. (A full treatment of that is another post.)

VI.7 — Menu design teaser

Why three tiers? A menu lets buyers sort themselves — the same virtual-value machinery applied to types rather than prices. Tier-price ratios come from virtual-type ratios, not cost ratios. That's Part 2. (Mussa–Rosen, 1978.)

Part VII — The meeting, revisited

Back to the whiteboard from Part I. Each strategy now gets its verdict, in the machinery's own words.

• Cost-plus measures the wrong quantity. Cost enters the problem exactly once — as the height of a horizontal line — and in our sweep, pricing off it instead of off demand left $115k–$214k per customer per year on the table.
• Match the competitor confuses an input with a strategy. A competitor's price is a term inside $\tilde F$; when they cut $100, your optimal response was a computable $50 — the pass-through rate — not imitation.
• Charge what it's worth had the right instinct and the wrong object. "Worth" is a distribution, not a number, and the chargeable point inside it is wherever $\psi$ crosses $c$.

Which is to say: three take-home theorems, each stated twice — symbols, then meeting-ready words.

1. Pricing. $\psi(p^*) = c$ — "Demand fixes the curve; cost picks the intersection."
2. Pass-through. $\dfrac{dp^*}{dc} = \dfrac{1}{\psi'(p^*)}$ — "How much of a cost shock reaches customers depends only on the slope of $\psi$."
3. Competitor neutrality. $\psi_{\tilde F}(p^*) = c$ — "Competitors change the distribution, never the equation."

One equation — $\psi(p^*) = c$ — and everything else (cost, competitors, distribution shape) enters through its two sides.

Appendix — for the very curious

Nothing below is needed for anything above. Two threads planted earlier (the non-monotone $\psi$ in the IV.2 sanity checks, the mixture caveat in Part V) get paid off here.

A.1 — Regularity and ironing

$\psi$ can be non-monotone when $f$ has bumps — precisely the segment mixtures of Part V. The fix is ironing: replace $\psi$ with its concave envelope over the offending region. Practically, some buyer types end up pooled at one price. See Myerson (1981), §5, or Hartline's Mechanism Design and Approximation (free draft, CS-friendly), chapter 3.

A.2 — Multi-dimensional types

When a buyer's type has two dimensions at once — say volume and per-document value — the problem gets genuinely hard (Rochet–Choné, 1998). For our document-review example the two dimensions correlate strongly, so a one-dimensional type index usually suffices.

Reading list

• Brown CSCI 1440/2440 (Algorithmic Game Theory) — where I first met this machinery; its Myerson lecture derives the same $\psi$ (there $\varphi$) through the incentive-constraints route.
• Bulow & Roberts (1989), "The Simple Economics of Optimal Auctions" — the marginal-revenue derivation this post follows, and the natural bridge for readers arriving from an AGT background.
• Roughgarden, Twenty Lectures on Algorithmic Game Theory, Lec. 2–5 — anchor.
• Myerson (1981), "Optimal Auction Design."
• Hartline, Mechanism Design and Approximation — free draft; ironing.
• Wikipedia: Vickrey auction, failure rate, Lerner index, price elasticity, Bertrand, Cournot.
• Mussa–Rosen (1978) — the Part 2 menu-design teaser.