Calculate the equilibrium price and quantity of the market for standard-quality headphones. Also calculate the quantity supplied by each firm as well as the profit of each firm.

Steps to Solve

1. Find Equilibrium Price and Quantity

   To find the equilibrium price and quantity, set the demand and supply equations equal to each other: $$ 150 - 2P = 3P - 15 $$

   Rearranging gives: $$ 150 + 15 = 3P + 2P $$ $$ 165 = 5P $$ $$ P = 33 $$

   Now, substitute $P$ back into either the demand or supply equation to find quantity $Q$: $$ Q = D(33) = 150 - 2(33) = 150 - 66 = 84 $$

2. Calculate Quantity Supplied by Each Firm

   To find quantity supplied by each firm, we need to determine how many firms are in the market. First, find the market supply function when $P = 33$. From the supply equation: $$ S(33) = 3(33) - 15 = 99 - 15 = 84 $$

   Thus, the total quantity supplied ($Q = 84$) matches the equilibrium quantity.

3. Assume the number of firms

   Let’s denote the number of firms as $n$. If each firm produces $q$, then: $$ n \cdot q = 84 $$

4. Find Individual Firm's Supply Quantity

   Since the firm sets its marginal cost equal to the price at equilibrium: $$ MC(q) = P $$ $$ 4q + 5 = 33 $$

   Rearranging gives: $$ 4q = 33 - 5 $$ $$ 4q = 28 $$ $$ q = 7 $$

5. Determine Number of Firms

   Using the equation from Step 3 ($n \cdot 7 = 84$): $$ n = \frac{84}{7} = 12 $$

6. Calculate Profit for Each Firm

   The profit ($\pi$) can be calculated as: $$ \pi = (P - AC) \cdot q $$ First, calculate the average cost (AC): $$ AC = \frac{C(q)}{q} = \frac{2q^2 + 5q}{q} = 2q + 5 $$

   Substitute $q = 7$: $$ AC = 2(7) + 5 = 14 + 5 = 19 $$

   Now substitute into the profit formula: $$ \pi = (33 - 19) \cdot 7 = 14 \cdot 7 = 98 $$