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 = 110 - 5Q_D. The inverse supply P = 44 + 6Q_S.

The government sets a \$74 price ceiling.

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

Plug P = 74 into the inverse supply function \begin{align*} P &= 44 + 6 Q \\ Q &= \frac{ P - 44 }{ 6 } \\ Q &= \frac{ 74 - 44 }{ 6 } \\ Q &= 5.0 \end{align*}

\begin{align*} CS &= \frac{ \left( 110 - 85 \right) \times 5 }{ 2 } \\ &= \frac{ 25 \times 5 }{ 2 } \\ &= \frac{ 125 }{ 2 } \\ &= 62.5 \\ \end{align*}

\begin{align*} PS &= \left( 85 - 74 \right) \times 5 + \frac{ \left( 74 - 44 \right) \times 5 }{ 2 } \\ &= 11 \times 5 + \frac{ 30 \times 5 }{ 2 } \\ &= 55 + \frac{ 150 }{ 2 } \\ &= 130.0 \\ \end{align*}

\begin{align*} TS &= CS + PS \\ &= 62.5 + 130.0 \\ &= 192.5 \\ \end{align*}

\begin{align*} DWL &= \frac{ \left( 85 - 74 \right) \times \left( 6.0 - 5 \right) }{ 2 } \\ &= \frac{ 11 \times 1.0 }{ 2 } \\ &= 5.5 \\ \end{align*}