Question

Consider the following demand–supply model, where
Demand function: \( P = Q^2 - 12Q + 35 \)
Supply function: \( 4P - 3Q = 0 \)
The stable market equilibrium price-quantity combination will be

• A. \( (P^, Q^) = (3, 4) \)
• B. \( (P^, Q^) = \left( \frac{105}{16}, \frac{70}{8} \right) \)
• C. \( (P^, Q^) = (6, 8) \)
• D. \( (P^, Q^) = (14, 3) \)

Hint:

In market equilibrium, the quantity demanded equals the quantity supplied. To find equilibrium, set the demand and supply functions equal and solve for the price and quantity.

Answer 1

The Correct Option is A

To find the market equilibrium, set the demand function equal to the supply function. First, solve the supply function for \( P \): \[ 4P = 3Q \quad \Rightarrow \quad P = \frac{3Q}{4}. \] Now, substitute \( P = \frac{3Q}{4} \) into the demand function: \[ \frac{3Q}{4} = Q^2 - 12Q + 35. \] Multiply through by 4 to eliminate the denominator: \[ 3Q = 4Q^2 - 48Q + 140. \] Rearrange the equation: \[ 4Q^2 - 51Q + 140 = 0. \] Solving this quadratic equation using the quadratic formula: \[ Q = \frac{-(-51) \pm \sqrt{(-51)^2 - 4(4)(140)}}{2(4)} = \frac{51 \pm \sqrt{2601 - 2240}}{8} = \frac{51 \pm \sqrt{361}}{8} = \frac{51 \pm 19}{8}. \] Thus, \( Q = \frac{51 + 19}{8} = 8 \) or \( Q = \frac{51 - 19}{8} = 4 \). Since \( Q = 4 \) is the only feasible solution, substitute this back into the supply function to find \( P \): \[ P = \frac{3(4)}{4} = 3. \] Thus, the equilibrium price and quantity are \( P^ = 3 \) and \( Q^ = 4 \). Final Answer: \boxed{(P^, Q^) = (3, 4)}