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 = 41 - 3Q_D. The inverse supply P = 17 + 5Q_S.

The government set a quota: 2.

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

$$ \begin{align*} P &= 41 - 3Q_D \\ &= 41 - 3 \times 2 \\ &= 35 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 41 - 35 \right) \times 2 }{ 2 } \\ &= \frac{ 6 \times 2 }{ 2 } \\ &= \frac{ 12 }{ 2 } \\ &= 6.0 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 35 - 27 \right) \times 2 + \frac{ \left( 27 - 17 \right) \times 2 }{ 2 } \\ &= 8 \times 2 + \frac{ 10 \times 2 }{ 2 } \\ &= 16 + \frac{ 20 }{ 2 } \\ &= 26.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 6.0 + 26.0 \\ &= 32.0 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 35 - 27 \right) \times \left( 3.0 - 2 \right) }{ 2 } \\ &= \frac{ 8 \times 1.0 }{ 2 } \\ &= 4.0 \\ \end{align*} $$