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 = 70 - 3Q_D. The inverse supply P = 30 + 2Q_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.

$$ \begin{align*} P &= 70 - 3Q_D \\ &= 70 - 3 \times 1 \\ &= 67 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 70 - 67 \right) \times 1 }{ 2 } \\ &= \frac{ 3 \times 1 }{ 2 } \\ &= \frac{ 3 }{ 2 } \\ &= 1.5 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 67 - 32 \right) \times 1 + \frac{ \left( 32 - 30 \right) \times 1 }{ 2 } \\ &= 35 \times 1 + \frac{ 2 \times 1 }{ 2 } \\ &= 35 + \frac{ 2 }{ 2 } \\ &= 36.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 1.5 + 36.0 \\ &= 37.5 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 67 - 32 \right) \times \left( 8.0 - 1 \right) }{ 2 } \\ &= \frac{ 35 \times 7.0 }{ 2 } \\ &= 122.5 \\ \end{align*} $$
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