# Quota

A quota restrict the quantity available on the market.

## 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 = 106 - 7Q_D. The inverse supply P = 16 + 11Q_S.

The government set a quota: 1.

What is the price consumers pay? Calculate the Consumer Surplus, the Producer surplus, Total Surplus, and the Dead Weight Loss.

\begin{align*} P &= 106 - 7Q_D \\ &= 106 - 7 \times 1 \\ &= 99 \end{align*}
\begin{align*} CS &= \frac{ \left( 106 - 99 \right) \times 1 }{ 2 } \\ &= \frac{ 7 \times 1 }{ 2 } \\ &= \frac{ 7 }{ 2 } \\ &= 3.5 \\ \end{align*}
\begin{align*} PS &= \left( 99 - 27 \right) \times 1 + \frac{ \left( 27 - 16 \right) \times 1 }{ 2 } \\ &= 72 \times 1 + \frac{ 11 \times 1 }{ 2 } \\ &= 72 + \frac{ 11 }{ 2 } \\ &= 77.5 \\ \end{align*}
\begin{align*} TS &= CS + PS \\ &= 3.5 + 77.5 \\ &= 81.0 \\ \end{align*}
\begin{align*} DWL &= \frac{ \left( 99 - 27 \right) \times \left( 5.0 - 1 \right) }{ 2 } \\ &= \frac{ 72 \times 4.0 }{ 2 } \\ &= 144.0 \\ \end{align*}