Suppose that Neptune Music has the copyright to the latest CD of the heavy Iron Band. The market demand curve for the CD is Q=800-100p, where Q represents quantity demanded in thousands and p represents the price in dollars. Production requires a fixed cost of $100,000 and a constant marginal cost of $2 per unit.
a. What price will maximize profits?
b. At that price, what will be the sales?
c. What is the Maximum Profit?
d. Calulate the Lerner Index at the profit-maximizing scale of production.
e. Supppose the fixed cost rises to $200,000. How would this affect the profit maximizing price?
a.50
b.400
c.20000
d.0.5
e.
a. To maximize profits, we need to find the price that will make marginal revenue (MR) equal to marginal cost (MC).
First, let's find the MR:
MR = dTR/dQ
Given the demand function Q = 800 - 100p, we can determine total revenue (TR) as:
TR = Q * p
TR = (800 - 100p) * p
TR = 800p - 100p^2
Taking the derivative of TR with respect to Q, we get:
MR = 800 - 200p
Setting MR equal to MC, we have:
800 - 200p = 2
Simplifying the equation:
200p = 798
p = 3.99
Therefore, the price that will maximize profits is $3.99.
b. To find the sales at this price, we substitute p = 3.99 into the demand function:
Q = 800 - 100 * 3.99
Q ≈ 400.8
The sales at the profit-maximizing price will be approximately 400,800 CDs.
c. To calculate the maximum profit, we need to determine total cost (TC) and total revenue.
FC (fixed cost) = $100,000
MC (marginal cost) = $2
TC = FC + (MC * Q)
TC = $100,000 + ($2 * 400.8)
TC ≈ $901,600
TR = Q * p
TR = 400.8 * 3.99
TR ≈ $1,599.20
Profit = TR - TC
Profit ≈ $1,599.20 - $901,600
Profit ≈ -$900,000.80
Therefore, the maximum profit is approximately -$900,000.80. (Looks like they may need to reconsider their pricing strategy!)
d. The Lerner Index is calculated as (P - MC) / P, where P is the profit-maximizing price and MC is the marginal cost.
Using the profit-maximizing price p = $3.99 and MC = $2, we can calculate the Lerner Index:
Lerner Index = (3.99 - 2) / 3.99
Lerner Index ≈ 0.4987
Therefore, the Lerner Index at the profit-maximizing scale of production is approximately 0.4987.
e. If the fixed cost rises to $200,000, it would have no direct effect on the profit-maximizing price. The profit-maximizing price is determined by the equality of MR and MC, not affected by the fixed cost.
To answer these questions, we need to follow a step-by-step approach:
Step 1: Find the profit function.
Step 2: Differentiate the profit function to find the marginal profit.
Step 3: Set the marginal profit equal to zero to find the profit-maximizing price.
Step 4: Use the profit-maximizing price to calculate the corresponding quantity and sales.
Step 5: Calculate the maximum profit.
Step 6: Calculate the Lerner Index.
Step 7: Analyze the effect of an increase in fixed costs on the profit-maximizing price.
Let's begin:
Step 1: Find the profit function.
The profit function is given by the equation:
Profit = Total Revenue - Total Cost
Total Revenue = Price × Quantity = p × Q
Total Cost = Fixed Cost + (Marginal Cost × Quantity)
Using the given information:
Fixed Cost (FC) = $100,000
Marginal Cost (MC) = $2 per unit
Profit = p × Q - (FC + MC × Q)
Profit = (p - MC) × Q - FC
Step 2: Differentiate the profit function to find the marginal profit.
To find the marginal profit, differentiate the profit function with respect to quantity (Q):
Marginal Profit (MP) = ∂(Profit) / ∂(Q)
MP = (p - MC) + Q × (-MC)
Step 3: Set the marginal profit equal to zero to find the profit-maximizing price.
To find the price that maximizes profit, we set the marginal profit equal to zero:
0 = (p - MC) + Q × (-MC)
Solving for p:
p = MC - Q × (-MC)
Step 4: Use the profit-maximizing price to calculate the corresponding quantity and sales.
To find the quantity demanded at the profit-maximizing price (Q*), substitute the price (p) back into the demand curve:
Q* = 800 - 100p
Substituting p into Q*, we get:
Q* = 800 - 100(MC - Q* × (-MC))
Simplifying the equation:
Q* = 800 - 100MC + Q* × 100MC
Solving for Q*:
Q* - Q* × 100MC = 800 - 100MC
Q* × (1 - 100MC) = 800 - 100MC
Q* = (800 - 100MC) / (1 - 100MC)
The corresponding sales would be the quantity demanded (Q*) at the profit-maximizing price (p*):
Sales = p* × Q*
Step 5: Calculate the maximum profit.
To find the maximum profit, substitute the profit-maximizing price (p*) and quantity (Q*) into the profit function:
Profit = (p* - MC) × Q* - FC
Step 6: Calculate the Lerner Index.
The Lerner Index measures market power and is calculated using the formula:
Lerner Index = (p* - MC) / p*
Step 7: Analyze the effect of an increase in fixed costs on the profit-maximizing price.
To determine how an increase in fixed costs affects the profit-maximizing price, we need to repeat steps 1-5 with the new fixed cost value of $200,000.
Please let me know which step you would like to start with, or if you would like me to proceed with all the steps.
To answer these questions, we need to analyze the profit-maximizing output level and price for Neptune Music. Here are the steps to find the solutions:
a. What price will maximize profits?
To find the price that maximizes profits, we need to determine the quantity at which marginal cost (MC) equals marginal revenue (MR). In this case, MC is constant at $2 per unit.
1. Start by finding the revenue function (TR) by multiplying the quantity (Q) by the price (p): TR = Q * p.
In this case, TR = (800 - 100p) * p, where Q is in thousands and p is in dollars.
2. Next, find the marginal revenue function (MR) by differentiating the revenue function with respect to quantity (Q):
MR = d(TR)/dQ = d((800 - 100p) * p)/dQ.
3. Set the marginal revenue (MR) equal to the marginal cost (MC) to find the profit-maximizing quantity:
MR = MC
(800 - 100p) + p * d(800 - 100p)/dQ = $2.
Solve this equation to find the value of p that maximizes profits.
b. At that price, what will be the sales?
Once we have the price (p) from the previous step, substitute it into the demand equation to find the corresponding quantity demanded (Q).
c. What is the Maximum Profit?
To find the maximum profit, we need to calculate the total revenue (TR), total cost (TC), and subtract TC from TR. Total cost consists of both fixed cost (FC) and variable cost (VC).
1. Total Cost (TC) = Fixed Cost (FC) + (Variable Cost per unit * Quantity)
In this case, FC = $100,000 and Variable Cost per unit = $2.
2. Total revenue (TR) = Quantity * Price
Use the price obtained in step a and multiply it by the corresponding quantity (Q) to get TR.
3. Maximum Profit = TR - TC
d. Calculate the Lerner Index at the profit-maximizing scale of production.
The Lerner Index represents the degree of market power a firm has. It can be calculated using the following formula:
Lerner Index = (P - MC) / P
where P is the price obtained in step a, and MC is the marginal cost, which is constant at $2.
e. Suppose the fixed cost rises to $200,000. How would this affect the profit-maximizing price?
Repeat steps a, b, c above using the new fixed cost of $200,000 instead of $100,000, and compare the profit-maximizing price with the previous result to see the effect of the increase in fixed cost on the price.
By following these steps, you should be able to find the answers to the specific questions about Neptune Music and the heavy Iron Band CD.