# Price Ceiling

A Price Ceiling is the maximum price allowed on the market.

## 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*}