# 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 = 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*} 

\begin{align*} TS &= CS + PS \\ &= 122.5 + 192.5 \\ &= 315.0 \\ \end{align*}
\begin{align*} DWL &= \frac{ \left( 122 - 112 \right) \times \left( 8.0 - 7 \right) }{ 2 } \\ &= \frac{ 10 \times 1.0 }{ 2 } \\ &= 5.0 \\ \end{align*}