# Price Ceiling

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

## Example

The government sets a price ceiling to $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 = 188 - 10Q_D. The inverse supply P = 53 + 5Q_S.

The government sets a \$63 price ceiling.

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

Plug P = 63 into the inverse supply function \begin{align*} P &= 53 + 5 Q \\ Q &= \frac{ P - 53 }{ 5 } \\ Q &= \frac{ 63 - 53 }{ 5 } \\ Q &= 2.0 \end{align*}
\begin{align*} CS &= \frac{ \left( 188 - 168 \right) \times 2 }{ 2 } \\ &= \frac{ 20 \times 2 }{ 2 } \\ &= \frac{ 40 }{ 2 } \\ &= 20.0 \\ \end{align*}
\begin{align*} PS &= \left( 168 - 63 \right) \times 2 + \frac{ \left( 63 - 53 \right) \times 2 }{ 2 } \\ &= 105 \times 2 + \frac{ 10 \times 2 }{ 2 } \\ &= 210 + \frac{ 20 }{ 2 } \\ &= 220.0 \\ \end{align*}
\begin{align*} TS &= CS + PS \\ &= 20.0 + 220.0 \\ &= 240.0 \\ \end{align*}
\begin{align*} DWL &= \frac{ \left( 168 - 63 \right) \times \left( 9.0 - 2 \right) }{ 2 } \\ &= \frac{ 105 \times 7.0 }{ 2 } \\ &= 367.5 \\ \end{align*}