Question

Consider a competitive market where the demand and supply functions are given by \( q^d = 12 - 2P \) and \( q^s = 4P \), respectively. The tax rate per unit of output that maximizes the tax yield (revenue) is _________ (in integer).

Hint:

To maximize tax revenue, differentiate the tax revenue function with respect to the tax rate and set it equal to zero.

Answer 1

Correct Answer: 3

The tax revenue is given by the product of the tax per unit of output \( t \) and the quantity sold \( q \). For maximum tax revenue, we first need to determine the equilibrium price and quantity in the presence of the tax. The demand and supply functions with the tax \( t \) are: \[ q^d = 12 - 2P, \quad q^s = 4(P - t) \] At equilibrium, \( q^d = q^s \), so: \[ 12 - 2P = 4(P - t) \] Simplifying the equation: \[ 12 - 2P = 4P - 4t \] \[ 12 + 4t = 6P \] \[ P = \frac{12 + 4t}{6} \] Substitute this value of \( P \) into the demand function to find \( q \): \[ q = 12 - 2\left(\frac{12 + 4t}{6}\right) \] Simplifying: \[ q = 12 - \frac{24 + 8t}{6} = 12 - 4 - \frac{4t}{3} = 8 - \frac{4t}{3} \] Tax revenue \( R(t) \) is: \[ R(t) = t \times q = t \left( 8 - \frac{4t}{3} \right) \] Now, differentiate \( R(t) \) with respect to \( t \) and set it equal to zero to find the value of \( t \) that maximizes the tax revenue: \[ R'(t) = 8 - \frac{8t}{3} \] Setting \( R'(t) = 0 \): \[ 8 - \frac{8t}{3} = 0 \] \[ \frac{8t}{3} = 8 \quad \Rightarrow \quad t = 3 \] Thus, the tax rate that maximizes the tax yield is \( t = 3 \).