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 = 123 - 13Q_D. The inverse supply P = 95 + 1Q_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.

$$ \begin{align*} P &= 123 - 13Q_D \\ &= 123 - 13 \times 1 \\ &= 110 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 123 - 110 \right) \times 1 }{ 2 } \\ &= \frac{ 13 \times 1 }{ 2 } \\ &= \frac{ 13 }{ 2 } \\ &= 6.5 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 110 - 96 \right) \times 1 + \frac{ \left( 96 - 95 \right) \times 1 }{ 2 } \\ &= 14 \times 1 + \frac{ 1 \times 1 }{ 2 } \\ &= 14 + \frac{ 1 }{ 2 } \\ &= 14.5 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 6.5 + 14.5 \\ &= 21.0 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 110 - 96 \right) \times \left( 2.0 - 1 \right) }{ 2 } \\ &= \frac{ 14 \times 1.0 }{ 2 } \\ &= 7.0 \\ \end{align*} $$
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