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 = 177 - 9Q_D. The inverse supply P = 24 + 8Q_S.

The government set a quota: 8.

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

$$ \begin{align*} P &= 177 - 9Q_D \\ &= 177 - 9 \times 8 \\ &= 105 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 177 - 105 \right) \times 8 }{ 2 } \\ &= \frac{ 72 \times 8 }{ 2 } \\ &= \frac{ 576 }{ 2 } \\ &= 288.0 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 105 - 88 \right) \times 8 + \frac{ \left( 88 - 24 \right) \times 8 }{ 2 } \\ &= 17 \times 8 + \frac{ 64 \times 8 }{ 2 } \\ &= 136 + \frac{ 512 }{ 2 } \\ &= 392.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 288.0 + 392.0 \\ &= 680.0 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 105 - 88 \right) \times \left( 9.0 - 8 \right) }{ 2 } \\ &= \frac{ 17 \times 1.0 }{ 2 } \\ &= 8.5 \\ \end{align*} $$
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