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The problem provided asks to find the equilibrium quantity after a government imposes a fixed tax of $4 on each good. The given supply and demand functions are:

\[
P = -3Q_D + 48 \quad \text{(Demand function)}
\]
\[
P = \frac{1}{2} Q_S + 23 \quad \text{(Supply function)}
\]

The tax affects the price that suppliers receive, so the supply function is adjusted for the tax. The new price the supplier receives is $ P - 4 $ because $4 is taken as tax for each good.

### Step-by-step solution:

1. **Modify the supply equation for the tax:**
   The supply function becomes:
   \[
   P - 4 = \frac{1}{2} Q_S + 23
   \]
   Simplifying this equation:
   \[
   P = \frac{1}{2} Q_S + 27 \quad \text{(New supply equation after tax)}
   \]

2. **Find the equilibrium:**
   At equilibrium, the quantity demanded equals the quantity supplied, and thus the price $ P $ from both equations should be the same. Set the modified supply function equal to the demand function:
   \[
   -3Q + 48 = \frac{1}{2}Q + 27
   \]
   Solve for $ Q $:
   \[
   -3Q - \frac{1}{2}Q = 27 - 48
   \]
   \[
   -\frac{6}{2}Q - \frac{1}{2}Q = -21
   \]
   \[
   -\frac{7}{2}Q = -21
   \]
   Multiply both sides by $ -\frac{2}{7} $:
   \[
   Q = \frac{-21 \times -2}{7} = 6
   \]

Thus, the equilibrium quantity is **6 units**.

Would you like a breakdown of the calculations, or do you have any specific questions?

Here are 5 related questions to expand your understanding:
1. What happens to the equilibrium price after the tax is imposed?
2. How would an increase in the tax amount affect the equilibrium quantity?
3. What if a subsidy of $4 was given instead of a tax—how would this change the equilibrium?
4. Can you derive the new price at equilibrium using the quantity found?
5. How would changes in demand elasticity influence the effect of the tax?

**Tip:** Always adjust the supply function when a tax or subsidy is applied, as it affects the price suppliers receive, not the price consumers pay.

Find the equilibrium quantity if the government imposes a fixed tax of $4 on each good. The demand and supply functions of a good are given by: P = -3Q_D + 48 (Demand function), P = (1/2)Q_S + 23 (Supply function).

Solve for the equilibrium quantity given a supply and demand problem where a $4 tax is imposed. The functions are linear, and the tax impacts the supply side. Learn how to adjust the supply function and solve for equilibrium.

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