Price Floor

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

Example

The government sets a price floor to $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 = 86 - 7Q_D. The inverse supply P = 46 + 13Q_S.

The government sets a $79 price floor.

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

Plug `P = 79` into the inverse demand function $$ \begin{align*} P &= 86 - 7 Q \\ Q &= \frac{ 86 - P }{ 7 } \\ Q &= \frac{ 86 - 79 }{ 7 } \\ Q &= 1.0 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 86 - 79 \right) \times 1 }{ 2 } \\ &= \frac{ 7 \times 1 }{ 2 } \\ &= \frac{ 7 }{ 2 } \\ &= 3.5 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 79 - 59 \right) \times 1 + \frac{ \left( 59 - 46 \right) \times 1 }{ 2 } \\ &= 20 \times 1 + \frac{ 13 \times 1 }{ 2 } \\ &= 20 + \frac{ 13 }{ 2 } \\ &= 26.5 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 3.5 + 26.5 \\ &= 30.0 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 79 - 59 \right) \times \left( 2.0 - 1 \right) }{ 2 } \\ &= \frac{ 20 \times 1.0 }{ 2 } \\ &= 10.0 \\ \end{align*} $$