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 = 108 - 5Q_D`. The inverse supply is `P = 45 + 2Q_S`.

The government sets a $93 price floor.

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

Plug `P = 93` into the inverse demand function $$ \begin{align*} P &= 108 - 5 Q \\ Q &= \frac{ 108 - P }{ 5 } \\ Q &= \frac{ 108 - 93 }{ 5 } \\ Q &= 3.0 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 108 - 93 \right) \times 3 }{ 2 } \\ &= \frac{ 15 \times 3 }{ 2 } \\ &= \frac{ 45 }{ 2 } \\ &= 22.5 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 93 - 51 \right) \times 3 + \frac{ \left( 51 - 45 \right) \times 3 }{ 2 } \\ &= 42 \times 3 + \frac{ 6 \times 3 }{ 2 } \\ &= 126 + \frac{ 18 }{ 2 } \\ &= 135.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 22.5 + 135.0 \\ &= 157.5 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 93 - 51 \right) \times \left( 9.0 - 3 \right) }{ 2 } \\ &= \frac{ 42 \times 6.0 }{ 2 } \\ &= 126.0 \\ \end{align*} $$