Example
The government restricts the number of banana available on the market to 4 millions.
The inverse demand is `P = 14 - Q_D` and the inverse supply is `P = 2 + Q_S`.
After the quota, there are 4 millions bananas sold at $10.
Consumer surplus is `CS = \frac{\left( 14 - 10 \right) \times 4}{2} = 8`.
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 = 8 + 24 = 32`.
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 = 157 - 5Q_D`. The inverse supply is `P = 77 + 5Q_S`.
The government sets a quota: 7.
What is the price consumers pay? Calculate the Consumer Surplus, the Producer surplus, Total Surplus, and the Dead Weight Loss.
$$ $$ \begin{align*} P &= 157 - 5Q_D \\ &= 157 - 5 \times 7 \\ &= 122 \end{align*} $$ $$
$$ $$ \begin{align*} CS &= \frac{ \left( 157 - 122 \right) \times 7 }{ 2 } \\ &= \frac{ 35 \times 7 }{ 2 } \\ &= \frac{ 245 }{ 2 } \\ &= 122.5 \\ \end{align*} $$ $$
$$ $$ \begin{align*} PS &= \left( 122 - 112 \right) \times 7 + \frac{ \left( 112 - 77 \right) \times 7 }{ 2 } \\ &= 10 \times 7 + \frac{ 35 \times 7 }{ 2 } \\ &= 70 + \frac{ 245 }{ 2 } \\ &= 192.5 \\ \end{align*} $$ $$