Price Ceiling

A Price Ceiling is the maximum price allowed on the market.

Example

The government sets a price ceiling to $4.

The inverse demand is `P = 14 - Q_D` and the inverse supply is `P = 2 + Q_S`.

After the price ceiling, there are `Q=2` bananas sold at $4.

Consumer surplus is `CS = \left( 12 - 4 \right) \times 2 + \frac{\left( 14 - 12 \right) \times 2}{2} = 16 + 2 = 18`

Producer surplus is `PS = \frac{\left( 4 - 2 \right) \times 2}{2} = 2`

Total Surplus is equal to `TS = CS + PS = 18 + 2 = 20`

The Dead weight loss is equal to `DWL = \frac{\left( 12 - 4 \right) \times \left( 6 - 2 \right)}{2} = 16`

Question

The inverse demand for bananas is P = 241 - 6Q_D. The inverse supply P = 81 + 14Q_S.

The government sets a $123 price ceiling.

What is the market quantity? Calculate the Consumer Surplus, the Producer surplus, Total Surplus, and the Dead Weight Loss.

Plug `P = 123` into the inverse supply function $$ \begin{align*} P &= 81 + 14 Q \\ Q &= \frac{ P - 81 }{ 14 } \\ Q &= \frac{ 123 - 81 }{ 14 } \\ Q &= 3.0 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 241 - 223 \right) \times 3 }{ 2 } \\ &= \frac{ 18 \times 3 }{ 2 } \\ &= \frac{ 54 }{ 2 } \\ &= 27.0 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 223 - 123 \right) \times 3 + \frac{ \left( 123 - 81 \right) \times 3 }{ 2 } \\ &= 100 \times 3 + \frac{ 42 \times 3 }{ 2 } \\ &= 300 + \frac{ 126 }{ 2 } \\ &= 363.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 27.0 + 363.0 \\ &= 390.0 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 223 - 123 \right) \times \left( 8.0 - 3 \right) }{ 2 } \\ &= \frac{ 100 \times 5.0 }{ 2 } \\ &= 250.0 \\ \end{align*} $$
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