Question 12.21: For the non-linear demand function p = 1,800 − 0.6q² and the......
For the non-linear demand function p = 1,800 − 0.6q² and the corresponding marginal revenue function MR = 1,800 − 1.8q² , use definite integrals to find:
(i) TR when q is 10;
(ii) the change in TR when q increases from 10 to 20;
(iii) consumer surplus when q is 10
(i) TR when q is 10 will be
\int_{0}^{10} MR dq = \int_{0}^{10}{(1,800 – 1.8q^2)dq}
= \left[1,800q − 0.6q^3\right]_0^{10}
= 18,000 − 600 = £17,400
(ii) The change in TR when q increases from 10 to 20 will be
\int_{10}^{20} MR dq = \int_{10}^{20}{(1,800 – 1.8q^2)dq}
= \left[1,800q − 0.6q^3\right]_{10}^{20}
= (36,000 − 4,800) − (18,000 − 600) = £13,800
(iii) Consumer surplus when q is 10 will be the definite integral of the demand function minus total revenue actually spent by consumers. This integral will be
\int_{0}^{10}{(1,800 – 0.6q^2)dq} = \left[1,800q − 0.2q^3\right]_0^{10} = 18,000 − 200 = £17,800
and
TR = pq = 1,800q − 0.6q³ = 18,000 − 600 = £17,400
Therefore consumer surplus = £17,800 − £17,400 = £400