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)