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Where the money peaks

Revenue peaks exactly where elasticity is 1. Profit peaks somewhere else entirely — and that gap is where most pricing mistakes live.

How much do they care? left one number unexplained. On our demand curve, elasticity hits exactly 1.00 at a price of $30 — and at that price, adding a dollar moves revenue by essentially nothing.

That is not a coincidence. It is the definition, seen from the other side.

Why revenue peaks at elasticity 1

Every price rise does two things at once, and elasticity is the exchange rate between them:

  • Elasticity below 1. You lose proportionally fewer customers than the price gains. Revenue rises. Keep going.
  • Elasticity above 1. You lose proportionally more customers than the price gains. Revenue falls. Come back.
  • Elasticity exactly 1. The two effects are the same size. Revenue has stopped changing — which is what being at the top means.

So the revenue peak is not a place you find by guessing prices. It is the one price where a customer's willingness to leave exactly balances your gain from those who stay.

Now put the costs back

Revenue is not the goal, and a company optimising it is optimising the wrong thing. Each of our units costs $10 to make, so what we actually keep is:

profit=(p−c)×q=(p−10)(120−2p)\text{profit} = (p - c) \times q = (p - 10)(120 - 2p)

Two hills, from the same curve. Drag the price across both — and then drag the cost.

The revenue hill and the profit hill are not the same hill

05001,0001,5002,0001021.2532.543.7555Price ($)Dollars per month
RevenueProfit
Units sold
60
Revenue
$1,800
Profit
$1,200
Elasticity here
1.00
Cost does not just lower the profit hill — it moves its summit. The profit peak sits at $30 plus half the unit cost.

Set the cost back to $10 and compare the two candidate prices:

Charging $35 sells ten fewer units, takes $50 less revenue — and keeps $50 more.

$30

Units
60
Revenue
$1,800
Profit
$1,200
the best revenue

$35

Units
50
Revenue
$1,750
Profit
$1,250
the best profit

Every one of those ten lost customers was paying $30 for something that cost $10, so each was worth having. Losing them is still right, because the other fifty are now paying $5 more each. Fifty times $5 is $250; ten times $20 is $200.

The general result, worth remembering: cost pushes the best price up. On this curve the profit peak sits at $30 plus half the unit cost — drag the cost slider to $20 and watch the coral summit settle at $40. A company that prices off revenue is systematically too cheap, and the more expensive its product is to make, the more too-cheap it is.

The same answer, counted one unit at a time

There is a second route to $35 that needs no curve at all — just What does one more cost?'s question, asked repeatedly. To sell one more unit you must shade the price for everyone, so the extra revenue from unit number n is less than its price. Each unit costs $10 to make. Make it while the extra revenue beats the extra $10.

Extra revenue falls as you push volume. The last unit worth making is the 50th — which prices at $35.

48

Price needed
$36.00
Total revenue
$1,728
Extra revenue from this unit
—
vs $10 cost

49

Price needed
$35.50
Total revenue
$1,739.50
Extra revenue from this unit
$11.50
vs $10 cost
make it

50

Price needed
$35.00
Total revenue
$1,750
Extra revenue from this unit
$10.50
vs $10 cost
make it

51

Price needed
$34.50
Total revenue
$1,759.50
Extra revenue from this unit
$9.50
vs $10 cost
stop

52

Price needed
$34.00
Total revenue
$1,768
Extra revenue from this unit
$8.50
vs $10 cost
stop

Stop where the extra revenue meets the extra cost, and you land on exactly the price the profit hill picked. Two methods, no calculus, same answer — which is the point: the peak of a curve and the marginal rule are the same statement.

What to take into the case

Revenue peaks at elasticity 1

In one line
above it, price rises cost you revenue; below it, they earn it

Profit peaks above the revenue peak

In one line
by half the unit cost, on this curve

Never optimise revenue

In one line
the revenue-maximising price is a known, predictable mistake

Margin decides, not volume

In one line
ten lost customers can be cheaper than the discount that would keep them

You now have every tool the last session needs: marginal cost, opportunity cost, diminishing returns, elasticity, and the gap between revenue and profit. Time to use all five at once.

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