Question

1. (20 points) suppose the market demand function is given by (d(p_d) = 60 - p_d) and (s(p_s) = 2p_s).

(a) (5 points) find the market equilibrium without taxes (that is, when (p_d = p_s)).
(b) (5 points) suppose now the supplier is required to pay a per unit tax of (t = 3). draw a graph to show the change in the supply curve. find the equilibrium quantity and prices, (q^), (p_s^), and (p_d^*). how much of the per unit tax is passed on to the consumer? (hint: in this case, the vertical axis is the demand price.)
(c) (5 points) suppose now the demander is required to pay a per unit tax of (t = 3). draw a graph to show the change in the demand curve. find the equilibrium quantity and prices, (q^), (p_s^), and (p_d^*). (hint: in this case, the vertical axis is the supply price.) compare your answers in parts b and c.
(d) (5 points) calculate the change in consumers surplus, change in producers surplus, tax revenue, and deadweight loss as a result of the policy change. illustrate them on a graph.

Explanation:

Equilibrate without taxes

Using the Market Equilibrium knowledge point

$$
LATEXBLOCK0
$$

Analyze supplier tax

With a per-unit tax \(t = 3\) on suppliers, the relation between consumer price \(P_D\) and producer price \(P_S\) is \(P_D = P_S + 3\).
The supply function is \(q = 2P_S\). Expressing this in terms of \(P_D\):

$$ q = 2(P_D - 3) = 2P_D - 6 $$

Equating demand and the new supply:

$$
LATEXBLOCK1
$$

The tax passed on to the consumer is:

$$ \Delta P_D = P_D^* - P^* = 22 - 20 = 2 $$

Analyze demander tax

With a per-unit tax \(t = 3\) on demanders, the relation is \(P_D = P_S + 3\).
The demand function is \(q = 60 - P_D\). Expressing this in terms of \(P_S\):

$$ q = 60 - (P_S + 3) = 57 - P_S $$

Equating the new demand and supply:

$$
LATEXBLOCK2
$$

Comparing parts (b) and (c), the equilibrium quantity \(q^* = 38\), buyer price \(P_D^* = 22\), and seller price \(P_S^* = 19\) are identical. This demonstrates tax incidence equivalence.

Calculate welfare changes

The changes in consumer surplus (\(\Delta CS\)), producer surplus (\(\Delta PS\)), tax revenue (\(TR\)), and deadweight loss (\(DWL\)) are:

$$
LATEXBLOCK3
$$

Alternatively, \(DWL = \frac{1}{2} \times t \times (q^_{\text{old}} - q^_{\text{new}}) = \frac{1}{2} \times 3 \times (40 - 38) = 3\).

Answer:

Question 4(a)

The market equilibrium without taxes is:

• Equilibrium quantity: \(q^* = 40\)
• Equilibrium price: \(P^* = 20\) (where \(P_D = P_S = 20\))

Question 4(b)

With a supplier tax of \(t = 3\):

• Equilibrium quantity: \(q^* = 38\)
• Seller price: \(P_S^* = 19\)
• Buyer price: \(P_D^* = 22\)
• Tax passed on to the consumer: \(2\) per unit (since \(P_D^*\) rose from \(20\) to \(22\))

Question 4(c)

With a demander tax of \(t = 3\):

• Equilibrium quantity: \(q^* = 38\)
• Seller price: \(P_S^* = 19\)
• Buyer price: \(P_D^* = 22\)
• Comparison: The economic incidence is identical to part (b). The statutory burden of the tax does not affect the final equilibrium prices and quantity.

Question 4(d)

The welfare changes resulting from the tax are:

• Change in Consumer Surplus: \(\Delta CS = -78\)
• Change in Producer Surplus: \(\Delta PS = -39\)
• Tax Revenue: \(TR = 114\)
• Deadweight Loss: \(DWL = 3\)