Neat Little Elasticity Trick with Proof

Our Managerial Economics professor was going over elasticity and mentioned the formula for quantity as it relates to price as a linear demand relationship (for simplification):

P = yQ + b

Note: For price elasticity for a product, y is usually negative (the more something costs, the less people will buy)

In a linear demand relationship, he was showing us a trick for approximating the elasticity at a given price and quantity. Just like my lecture on terminal value (in which we were told something worked without really understanding why) I wanted to demonstrate the proof (as Roger Martin says: "We shouldn't use models we don't understand").

So how can we simplify the formula for elasticity, given a linear demand curve?

Well:

Elasticity is defined as % change of Quantity / % change in Price.

Unlike the CFA which calculates % change with averages in the denominator:
% change = (first - last) / average

Rotman MBAs calculate % change as
% change = (first - last) / first

Either way, the trick still works:

Elasticity = % change Q / % change P

% change Q = (Q0 - Q1) / Q0
% change P = (P0 - P1) / P0

However: recall P = yQ + b
Therefore:
% change P = [(yQ0 + b) - (yQ1 + b)] / P0
= (yQ0 - yQ1) / P0
= y(Q0-Q1) / P0
= y (change in Q) / P0

Therefore Elasticity
= [(Q0 - Q1)/Q0]/[(P0 - P1)/P0]
= [(Q0 - Q1)*P0]/[(P0 - P1)*Q]
= (change in Q * P) / (y * change in Q * Q)
= P / yQ

QED
Elasticity can be accurately approximated by P/yQ for any given point Q, P on the linear demand relationship.

Note: Because the Q1 and P1 terms cancel out of equation completely, there is no calculus required as there is no theoretical second offset point required to approximate elasticity.