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 = 172 - 13Q_D. The inverse supply P = 19 + 4Q_S.

The government sets a $31 price ceiling.

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

Plug `P = 31` into the inverse supply function $$ \begin{align*} P &= 19 + 4 Q \\ Q &= \frac{ P - 19 }{ 4 } \\ Q &= \frac{ 31 - 19 }{ 4 } \\ Q &= 3.0 \end{align*} $$

$$ \begin{align*} CS &= \frac{ \left( 172 - 133 \right) \times 3 }{ 2 } \\ &= \frac{ 39 \times 3 }{ 2 } \\ &= \frac{ 117 }{ 2 } \\ &= 58.5 \\ \end{align*} $$

$$ \begin{align*} PS &= \left( 133 - 31 \right) \times 3 + \frac{ \left( 31 - 19 \right) \times 3 }{ 2 } \\ &= 102 \times 3 + \frac{ 12 \times 3 }{ 2 } \\ &= 306 + \frac{ 36 }{ 2 } \\ &= 324.0 \\ \end{align*} $$

$$ \begin{align*} TS &= CS + PS \\ &= 58.5 + 324.0 \\ &= 382.5 \\ \end{align*} $$

$$ \begin{align*} DWL &= \frac{ \left( 133 - 31 \right) \times \left( 9.0 - 3 \right) }{ 2 } \\ &= \frac{ 102 \times 6.0 }{ 2 } \\ &= 306.0 \\ \end{align*} $$
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