Example
The government sets a price ceiling of $4.
The inverse demand is `P = 14 - Q_D` and the inverse supply is `P = 2 + Q_S`.
After the price ceiling, there are `Q=2` bananas sold at $4.
Consumer surplus is `CS = \left( 12 - 4 \right) \times 2 + \frac{\left( 14 - 12 \right) \times 2}{2} = 16 + 2 = 18`
Producer surplus is `PS = \frac{\left( 4 - 2 \right) \times 2}{2} = 2`.
Total Surplus is equal to `TS = CS + PS = 18 + 2 = 20`.
The Dead weight loss is equal to `DWL = \frac{\left( 12 - 4 \right) \times \left( 6 - 2 \right)}{2} = 16`.
Question
The inverse demand for bananas is `P = 116 - 18Q_D`. The inverse supply `P = 16 + 2Q_S`.
The government sets a $22 price ceiling.
What is the market quantity? Calculate the Consumer Surplus, the Producer surplus, Total Surplus, and the Dead Weight Loss.
Plug `P = 22` into the inverse supply function $$ \begin{align*} P &= 16 + 2 Q \\ Q &= \frac{ P - 16 }{ 2 } \\ Q &= \frac{ 22 - 16 }{ 2 } \\ Q &= 3.0 \end{align*} $$
$$ \begin{align*} CS &= \frac{ \left( 116 - 62 \right) \times 3 }{ 2 } \\ &= \frac{ 54 \times 3 }{ 2 } \\ &= \frac{ 162 }{ 2 } \\ &= 81.0 \\ \end{align*} $$
$$ \begin{align*} PS &= \left( 62 - 22 \right) \times 3 + \frac{ \left( 22 - 16 \right) \times 3 }{ 2 } \\ &= 40 \times 3 + \frac{ 6 \times 3 }{ 2 } \\ &= 120 + \frac{ 18 }{ 2 } \\ &= 129.0 \\ \end{align*} $$
$$ \begin{align*} TS &= CS + PS \\ &= 81.0 + 129.0 \\ &= 210.0 \\ \end{align*} $$
$$ \begin{align*} DWL &= \frac{ \left( 62 - 22 \right) \times \left( 5.0 - 3 \right) }{ 2 } \\ &= \frac{ 40 \times 2.0 }{ 2 } \\ &= 40.0 \\ \end{align*} $$