The demand and supply functions of a commodity are \(p = 20 - x\) and \(p = 2 + 2x\) respectively, where p is the price (in Rs.) per unit and x is the quantity. Which of the following statements are correct?
(A) The equilibrium quantity is 6 units.
(B) The equilibrium price is Rs. 12.
(C) The consumer surplus at the equilibrium is Rs. 18.
(D) The producer surplus at the equilibrium is Rs. 18.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
At the equilibrium, the demand price equals the supply price:
\(20 - x = 2 + 2x\)
\(3x = 18\), so \(x_0 = 6\)
\(p_0 = 20 - 6 = 14\), and \(p_0 x_0 = 84\)
The consumer surplus:
\( \displaystyle CS = \int_{0}^{6}(20 - x)\,dx - 84 = (120 - 18) - 84 = 18 \)
The producer surplus:
\( \displaystyle PS = 84 - \int_{0}^{6}(2 + 2x)\,dx = 84 - (12 + 36) = 36 \)
So the equilibrium is 6 units at Rs. 14, the consumer surplus is Rs. 18 and the producer surplus is Rs. 36.
Checking the statements
(A) The equilibrium quantity is 6 units. — Correct. \(20 - 6 = 2 + 12 = 14\).
(B) The equilibrium price is Rs. 12. — Incorrect. The equilibrium price is \(20 - 6 = 14\).
(C) The consumer surplus at the equilibrium is Rs. 18. — Correct. 102 − 84 = 18.
(D) The producer surplus at the equilibrium is Rs. 18. — Incorrect. It is 84 − 48 = 36.