# 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 = 298 - 18Q_D. The inverse supply P = 98 + 2Q_S.

The government sets a \$102 price ceiling.

What is the market quantity? Calculate the Consumer Surplus, the Producer surplus, Total Surplus, and the Dead Weight Loss.

Plug P = 102 into the inverse supply function \begin{align*} P &= 98 + 2 Q \\ Q &= \frac{ P - 98 }{ 2 } \\ Q &= \frac{ 102 - 98 }{ 2 } \\ Q &= 2.0 \end{align*}

\begin{align*} CS &= \frac{ \left( 298 - 262 \right) \times 2 }{ 2 } \\ &= \frac{ 36 \times 2 }{ 2 } \\ &= \frac{ 72 }{ 2 } \\ &= 36.0 \\ \end{align*}

\begin{align*} PS &= \left( 262 - 102 \right) \times 2 + \frac{ \left( 102 - 98 \right) \times 2 }{ 2 } \\ &= 160 \times 2 + \frac{ 4 \times 2 }{ 2 } \\ &= 320 + \frac{ 8 }{ 2 } \\ &= 324.0 \\ \end{align*}

\begin{align*} TS &= CS + PS \\ &= 36.0 + 324.0 \\ &= 360.0 \\ \end{align*}

\begin{align*} DWL &= \frac{ \left( 262 - 102 \right) \times \left( 10.0 - 2 \right) }{ 2 } \\ &= \frac{ 160 \times 8.0 }{ 2 } \\ &= 640.0 \\ \end{align*}