# Quota

A quota restricts 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 = 147 - 12Q_D. The inverse supply is P = 17 + 1Q_S.

The government sets a quota: 4.

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

 \begin{align*} P &= 147 - 12Q_D \\ &= 147 - 12 \times 4 \\ &= 99 \end{align*} 

 \begin{align*} CS &= \frac{ \left( 147 - 99 \right) \times 4 }{ 2 } \\ &= \frac{ 48 \times 4 }{ 2 } \\ &= \frac{ 192 }{ 2 } \\ &= 96.0 \\ \end{align*} 

 \begin{align*} PS &= \left( 99 - 21 \right) \times 4 + \frac{ \left( 21 - 17 \right) \times 4 }{ 2 } \\ &= 78 \times 4 + \frac{ 4 \times 4 }{ 2 } \\ &= 312 + \frac{ 16 }{ 2 } \\ &= 320.0 \\ \end{align*} 

\begin{align*} TS &= CS + PS \\ &= 96.0 + 320.0 \\ &= 416.0 \\ \end{align*}
\begin{align*} DWL &= \frac{ \left( 99 - 21 \right) \times \left( 10.0 - 4 \right) }{ 2 } \\ &= \frac{ 78 \times 6.0 }{ 2 } \\ &= 234.0 \\ \end{align*}