Question

The demand and supply functions for a commodity are \(D(p)=10-2p\) and \(S(p)=-2+p\), respectively (}p in USD\textbf{). If the government sets a price ceiling of USD 3 per unit, then the increase in consumer surplus (USD) is _______ (round off to two decimal places).

Hint:

Under a binding price ceiling, quantity traded equals {supply} at the ceiling price. Compute CS as “area under demand up to traded quantity minus price $\times$ quantity”.

Answer 1

Correct Answer: 0.71

Step 1: Competitive equilibrium without ceiling
Set \(D(p)=S(p)\Rightarrow 10-2p=-2+p \Rightarrow p^\ast=4\). Quantity \(q^\ast=D(4)=2\).
Demand inverse: \(Q=10-2p \Rightarrow p(Q)=5-\tfrac{Q}{2}\).
Initial consumer surplus (CS) is the triangle under demand above price \(4\) up to \(Q=2\):
\(\text{CS}_0=\tfrac{1}{2}\times(5-4)\times 2=1\). Step 2: With price ceiling \(p_c=3\)
Traded quantity is supply-limited: \(Q_c=\min\{D(3),S(3)\}=\min\{4,1\}=1\).
Consumer surplus at the ceiling: \[ \text{CS}_c=\int_{0}^{1}\!\left(5-\frac{Q}{2}\right)dQ - 3 . 1 = \left[5Q-\frac{Q^2}{4}\right]_{0}^{1}-3 = 4.75-3=1.75 . \] Step 3: Increase in CS
\(\Delta \text{CS}=\text{CS}_c-\text{CS}_0=1.75-1=0.75\ \text{USD}\).