💰 Microeconomics Exercise: Market Equilibrium with a 10% Ad Valorem Tax

Question:
The demand function for a good is given by:

$$q_D = 100 – 2p$$

The supply function is:

$$q_S = 20 + 2p$$

Where:

• $p$ is the price per unit (in dollars)
• $q$ is the quantity demanded or supplied

In the past, no tax was imposed on this good.
Now, a 10% value-based tax is levied on consumers (i.e., the price paid by consumers is 10% higher than what producers receive).

Calculate the equilibrium quantity before and after the tax is imposed.

✅ Solution

Step 1: Initial Equilibrium (No Tax)

Set demand equal to supply:

$$100 – 2p = 20 + 2p$$

Solve for $p$:

$$100 – 20 = 2p + 2p \Rightarrow 80 = 4p \Rightarrow p = 20$$

Substitute back to find equilibrium quantity:

$$q = 100 – 2(20) = 60$$

🔹 Equilibrium Price (No Tax): $20
🔹 Equilibrium Quantity (No Tax): 60 units

Step 2: New Equilibrium with 10% Tax on Consumers

Let:

• $p_S$ = price received by sellers (net price)
• $p_C = p_S \cdot 1.10$ = price paid by consumers

We adjust the demand function because consumers now face a higher price:

$$q_D = 100 – 2p_C = 100 – 2(1.10p_S) = 100 – 2.2p_S$$

Supply remains unchanged:

$$q_S = 20 + 2p_S$$

Set $q_D = q_S$ to find new equilibrium:

$$100 – 2.2p_S = 20 + 2p_S$$

Solve:

$$80 = 4.2p_S \Rightarrow p_S = \frac{80}{4.2} \approx 19.05$$

Now find consumer price and new quantity:

$$p_C = 1.10 \cdot 19.05 \approx 20.96$$ $$q = 20 + 2 \cdot 19.05 = 20 + 38.10 = \mathbf{58.1 \text{ units}}$$

🎯 Final Answer

📚 Interpretation

After the 10% ad valorem tax on consumers:

• Quantity traded in the market decreases slightly (from 60 → 58.1),
• The burden is shared: producers receive less, and consumers pay more.

This demonstrates how taxes distort markets, even if demand and supply are both linear.