# 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 = 117 - 5Q_D. The inverse supply is P = 57 + 5Q_S.

The government sets 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 &= 117 - 5Q_D \\ &= 117 - 5 \times 1 \\ &= 112 \end{align*} 

 \begin{align*} CS &= \frac{ \left( 117 - 112 \right) \times 1 }{ 2 } \\ &= \frac{ 5 \times 1 }{ 2 } \\ &= \frac{ 5 }{ 2 } \\ &= 2.5 \\ \end{align*} 

 \begin{align*} PS &= \left( 112 - 62 \right) \times 1 + \frac{ \left( 62 - 57 \right) \times 1 }{ 2 } \\ &= 50 \times 1 + \frac{ 5 \times 1 }{ 2 } \\ &= 50 + \frac{ 5 }{ 2 } \\ &= 52.5 \\ \end{align*} 

\begin{align*} TS &= CS + PS \\ &= 2.5 + 52.5 \\ &= 55.0 \\ \end{align*}
\begin{align*} DWL &= \frac{ \left( 112 - 62 \right) \times \left( 6.0 - 1 \right) }{ 2 } \\ &= \frac{ 50 \times 5.0 }{ 2 } \\ &= 125.0 \\ \end{align*}