THE PRICE ELASTICITY

How sharply quantity answers price — and why total revenue peaks exactly where the answer is one-for-one. Elasticity E = (dQ/Q)/(dP/P) = (dQ/dP)·(P/Q) is negative down a demand curve; revenue R = P·Q crests at |E| = 1. Rendered, not quoted. These are models of markets, not investment advice.

source Alfred Marshall, Principles of Economics (1890), Book III Ch. 4 “The Elasticity of Wants” — marxists.org archive (public-domain transcription; no publisher DOI — AMBER).

Blue Team · builds & defends
3

THE MODEL

Point elasticity is the log-slope of demand:

E = (dQ/dP)·(P/Q)

Down a demand curve dQ/dP < 0, so E is negative. By magnitude:

• |E| > 1 — elastic (quantity answers hard)

• |E| < 1 — inelastic (quantity barely moves)

• |E| = 1 — unit-elastic (one-for-one)

Linear demand Q = a − bP runs |E| from ∞ (choke price) to 0 (free), passing through 1 at the midpoint P = a/2b. Constant-elasticity Q = cP−e holds E = −e everywhere.

5

THE LINEAGE

Marshall 1890 gave the responsiveness a number. That number is what sets tax incidence in the-supply-and-demand: the inelastic side of a market carries the heavier share of any tax, because it cannot flee the price. Same E, next sphere.

7

THE WITNESS

Live re-check of every identity below. Green = the engine still proves what it claims; it flips red the instant the Red-Team tamper (6) touches the elasticity formula.

re-checking…

The Machine
4

DATA IN in ↓

Linear demand: a = 100, b = 2 → Q = 100 − 2P

Constant-E demand: c = 1000, e = 1.5 → Q = 1000·P−1.5

Tolerance for the numeric root: 1e−6

↓ ↓ ↓
0

THE PANEL LIT

Demand curve (violet), revenue R=P·Q (pink). The vertical line marks the unit-elastic midpoint where revenue crests.

Choke price P=a/b = 50 · revenue-max P=a/2b = 25 · |E| there = 1 · max R = 1250

↓ ↓ ↓
8

DATA OUT out ↓

Proven: dR/dP = 0 lands exactly on |E| = 1 (to 1e−6). A price cut raises revenue where elastic, lowers it where inelastic. Constant-E demand holds E = −e at every price.

booting…

Red Team · attacks & breaks
1

THE ADVERSARY WALL

Elasticity is a local number, not a curve's label. “Demand is inelastic” is only true at a price. The same straight line is elastic up top and inelastic down low.

The clean formulas assume a smooth, single-valued demand curve, rational agents and ceteris paribus AMBER — idealizations, not laws of nature. Real demand shifts, kinks, and remembers.

Any elasticity read off future prices is a forecast AMBER, not advice.

2

THE GRAVEYARD

“Cutting the price always raises revenue.”
→ Only when demand is elastic (|E|>1). In the inelastic region a cut loses revenue.
“Elasticity is just the slope dQ/dP.”
→ Slope is unit-bound; elasticity divides by P/Q to be unitless. Drop that factor and the revenue identity collapses (see 6).
“Revenue peaks where quantity peaks.”
→ Quantity peaks at P=0 (R=0). Revenue peaks at the midpoint, |E|=1.
6

THE TAMPER

Strip the P/Q factor so E → dQ/dP (a bare slope). The revenue-max–at–unit-elasticity identity breaks, and the Witness (7) catches it live.

state: clean