Suppose that Neptune Music has the copyright to the latest CD of the heavy Iron Band. The market demand schedule for the CD is:
Q = 800 – 100P.
Symbolically, Q represents quantity demanded measured in thousands of CDs and P represents the price in dollars. Production requires a fixed cost of $100,000 and a constant marginal cost of $2 per CD produced.
1.
To find the equilibrium price and quantity, we need to set the quantity demanded equal to the quantity supplied. The quantity demanded is represented by the equation Q = 800 - 100P, and the quantity supplied is equal to the quantity produced.
To find the quantity supplied, we need to determine at what price the heavy Iron Band is willing to produce CDs. The fixed cost is $100,000, and the marginal cost is $2 per CD produced. This means that the heavy Iron Band will produce CDs as long as the price is greater than or equal to the marginal cost ($2).
2. The heavy Iron Band will start producing CDs when the price is greater than or equal to the marginal cost ($2). So the minimum price at which they will produce any CDs is $2.
3. To find the quantity supplied at a given price, we can calculate the difference between the price and the marginal cost, and then divide the fixed cost by this difference. This will give us the number of CDs needed to cover the fixed cost. Since the marginal cost is $2 and the fixed cost is $100,000, the difference is (P - $2) and the number of CDs is $100,000 / (P - $2).
4. To find the equilibrium price and quantity, we need to set the quantity demanded equal to the quantity supplied. So now we can set up the equation:
800 - 100P = $100,000 / (P - $2).
5. To solve this equation, we can multiply both sides by (P - $2) to eliminate the denominator:
(800 - 100P) * (P - $2) = $100,000.
6. We can expand the left side of the equation:
800P - 100P^2 - $1600 + 200P = $100,000.
7. Combining like terms, we get:
-100P^2 + 1000P - $1600 = $100,000.
8. Rearranging the equation, we get:
-100P^2 + 1000P - $101,600 = 0.
9. We can now use the quadratic formula to solve for P:
P = (-1000 ± sqrt(1000^2 - 4 * (-100) * (-$101,600))) / (2 * (-100)).
10. Calculating in the square root, we get:
P = (-1000 ± sqrt(1,000,000 + $40,640)) / (-200).
11. Simplifying further, we get:
P = (-1000 ± sqrt(1,040,640)) / (-200).
12. Taking the positive square root gives us:
P = (-1000 + sqrt(1,040,640)) / (-200).
13. We can now calculate the value using a calculator or online tool to get:
P ≈ $30.
14. Now that we have the equilibrium price, we can substitute it back into the quantity demanded equation to find the equilibrium quantity:
Q = 800 - 100P,
Q = 800 - 100($30),
Q = 800 - $3000,
Q = -2200.
15. Since we are measuring quantity in thousands of CDs, the negative value indicates that there is no equilibrium quantity at this price. It is possible that the heavy Iron Band will not produce any CDs if the price is $30.
Therefore, the equilibrium price is approximately $30, but there is no equilibrium quantity at this price.