## Derivation of the simplest demand curve that has an increasing price...

 Author Message samuel Junior Member         Group: Forum Members Posts: 25, Visits: 1 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 thenE(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 * PdQ/Q = (- a/P – b) dPlnQ = - a * ln P – b * P + C, where C is the constant of integrationlnQ + ln(P^a) = - b * P + Cln(Q*P^a) = - b*P + CExponentiating both sides givesQ*(P^a) = C*exp(-b*P), where C is a new constantThe final demand equation is thenQ = C*(P^-a) * exp(-b*P)This equation gives the “correct” behavior at P = 0 and P = infinity.Limit (P->0) Q(P) = infinityLimit (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 curveQ(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)
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