# Price Floor

The Price Floor is the minimum price chosen by the government.

## Example

The government sets a price floor of $10. The inverse demand is P = 14 - Q_D and the inverse supply is P = 2 + Q_S. After the price floor, there are 4 millions bananas sold at$10.

Consumer surplus is CS = \frac{\left( 14 - 10 \right) \times 4}{2} = 6.

Producer surplus is PS = \left( 10 - 6 \right) \times 4 + \frac{\left( 6 - 2 \right) \times 4}{2} = 16 + 8 = 24.

Total Surplus is equal to TS = CS + PS = 6 + 24 = 30.

The Dead weight loss is equal to DWL = \frac{\left( 10 - 6 \right) \times \left( 6 - 4 \right)}{2} = 4.

### Question

The inverse demand for bananas is P = 123 - 14Q_D. The inverse supply is P = 21 + 3Q_S.

The government sets a \$67 price floor.

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

Plug P = 67 into the inverse demand function \begin{align*} P &= 123 - 14 Q \\ Q &= \frac{ 123 - P }{ 14 } \\ Q &= \frac{ 123 - 67 }{ 14 } \\ Q &= 4.0 \end{align*}

\begin{align*} CS &= \frac{ \left( 123 - 67 \right) \times 4 }{ 2 } \\ &= \frac{ 56 \times 4 }{ 2 } \\ &= \frac{ 224 }{ 2 } \\ &= 112.0 \\ \end{align*}

\begin{align*} PS &= \left( 67 - 33 \right) \times 4 + \frac{ \left( 33 - 21 \right) \times 4 }{ 2 } \\ &= 34 \times 4 + \frac{ 12 \times 4 }{ 2 } \\ &= 136 + \frac{ 48 }{ 2 } \\ &= 160.0 \\ \end{align*}

\begin{align*} TS &= CS + PS \\ &= 112.0 + 160.0 \\ &= 272.0 \\ \end{align*}

\begin{align*} DWL &= \frac{ \left( 67 - 33 \right) \times \left( 6.0 - 4 \right) }{ 2 } \\ &= \frac{ 34 \times 2.0 }{ 2 } \\ &= 34.0 \\ \end{align*}