Tax on Consumers

Consumers pay the government a tax per item purchased on the market.

Example

The government taxes banana consumers $4 per banana.

The inverse demand is `P = 14 - Q_D`.

The inverse supply is `P = 2 + Q_S`.

With a $4 tax there are 4 million bananas on the market.

Consumers pay $10 per banana.

Producers get $6 per banana.

Consumer surplus is `CS = \frac{\left( 14 - 10 \right) \times 4}{2} = 8`.

Producer surplus is `PS = \frac{\left( 6 - 2 \right) \times 4}{2} = 8`.

The Government Revenue is `G = 4 \times 4 = 16`.

Total Surplus is equal to `TS = CS + PS +G = 8 + 8 + 16 = 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 = 189 - 7Q_D. The inverse supply P = 59 + 6Q_S.

The government taxes producers $78.

What is the market quantity? What price do consumers pay? What price do producers get? Calculate the Consumer Surplus, the Producer surplus, the Government Revenue, Total Surplus, and the Dead Weight Loss.

The new demand curve is `P = 111 - 7 Q`:

$$ P = 189 - 78 - 7 Q $$ $$ P = 111 - 7 Q $$

The new demand intersects the supply at `Q=4`:

\begin{align*} 59 + 6 Q &= 111 - 7 Q \\ 7 Q + 6 Q &= 111 - 59 \\ 13 Q &= 52 \\ Q &= \frac{ 52 }{ 13 } \\ Q &= 4\\ \end{align*}

Plug the market quantity Q=4 into the original demand curve

\begin{align*} P &= 189 - 7 \times 4 \\ P &= 161 \end{align*}

Plug the market quantity Q=4 into the original supply curve

\begin{align*} P &= 59 + 6 \times 4 \\ P &= 83 \end{align*}

$$ \begin{align*} CS &= \frac{ \left( 189 - 161 \right) \times 4 }{ 2 } \\ &= \frac{ 28 \times 4 }{ 2 } \\ &= \frac{ 112 }{ 2 } \\ &= 56.0 \\ \end{align*} $$

$$ \begin{align*} PS &= \frac{ \left( 83 - 59 \right) \times 4 }{ 2 } \\ &= \frac{ 24 \times 4 }{ 2 } \\ &= \frac{ 96 }{ 2 } \\ &= 48.0 \\ \end{align*} $$

$$ GR = \text{quantity} \times \text{tax} = 4 \times 78 = 312 $$

$$ TS = CS + PS + GR = 56.0 + 48.0 + 312 = 416.0 $$

$$ \begin{align*} DWL &= \frac{ \left( 161 - 83 \right) \times \left( 10.0 - 4 \right) }{ 2 } \\ &= \frac{ 78 \times 6.0 }{ 2 } \\ &= 234.0 \\ \end{align*} $$