Derivation of the simplest demand curve that has an increasing elasticity:

For any demand curve, the price elasticity is negative. The simplest equation showing an increase in the absolute value of the price elasticity is then

E(P) = -a – b*P, where a, b > 0.

The definition of elasticity is E = (dQ/dP)*(P/Q).

Equating these gives

(dQ/dP)*(P/Q) = - a – b * P

dQ/Q = (- a/P – b) dP

lnQ = - a * ln P – b * P + C, where C is the constant of integration

lnQ + ln(P^a) = - b * P + C

ln(Q*P^a) = - b*P + C

Exponentiating both sides gives

Q*(P^a) = C*exp(-b*P), where C is a new constant

The final demand equation is then

Q = C*(P^-a) * exp(-b*P)

This equation gives the “correct” behavior at P = 0 and P = infinity.

Limit (P->0) Q(P) = infinity

Limit (P->infinity) Q(P) = 0.

The graph of it looks similar to a hyperbola, as we would expect.

We check the elasticity of this demand curve

Q(P) = C * P^(-a) * exp(-b * P).

E = (dQ/dP)*(P/Q)

E = C * P^(-a) * exp(-b * P) * (-b) + ( - a * C * P^(-a – 1) * exp(-b * P) * (P/ (C * P^(-a) * exp(-b * P)))

E = -b * P – a * P * P^(-1)

E = - a – b * P


(The attached file has better formatting)