Revenues vs Price Elasticity of Demand

price p
quantity q

p+dp
q-dq

assuming +ve (abs) value of elasticity

elasticity e = (dq/q)/(dp/p)
                   

dq = eq dp/p

if e = 1

dq = q dp/p

(p+dp)(q-dq) = (p+dp)(q-eq dp/p) = pq - eqdp + qdp - dp dp / p

the last term is too small

so

(p+dp)(q-dq) = pq - eqdp + qdp

if e = 1,

revenues(of company) = expenditure(of customers) = (p+dp)(q-dq) = pq

if e>1, let e=1+n

revenues = expenditure =
(p+dp)(q-dq) = pq - (1+n)qdp + qdp

                      = pq - nqdp

                      < pq if dp +ve
                      > pq if dp -ve

if e<1, let e=1-n

revenues = expenditure =
(p+dp)(q-dq) = pq - (1-n)qdp + qdp

                      = pq + nqdp

                     > pq if dp +ve
                     < pq of dp -ve

This means that when demand is inelastic change in prices will move the total revenues in the same directon as the prices


when demand is elastic, increase in prices will move the revenues in the opposite direction.

if demand elasticity is unitary, total expenditures or revenues are unchanged as price changes.