Problem 2: You are a rock concert producer planning a rock concert. The demand schedule for tickets is P = 75 -0.005X where X is the number of people attending the concert. The marginal revenue function is MR = 75-0.01X.
The fixed cost of putting on the concert is $150,000. The marginal cost per person attending is zero. The capacity of the concert hall is 5,000. What price should you charge to maximize profits? How many people will attend the concert and what is the value of your profit or loss? What price should you charge to maximize attendance?
Set MR = MC... in this case 75 - .01x = 0. You'll find the profit-maximizing quantity to be 7500 and the profit-maximizing price to be $37.50. However, there aren't 7500 seats available; there are only 5000. Therefore, substitute 5000 for X in the original inverse demand function. This will yeild P = 50. This is the profit-maximizing price with a quantity of 5000. The rest should fall in place. Be sure you agree with my work as I'm only a grad student!!!
Do you not Multiple the .01 by 2
MR=75-.02x
No, MR = a + 2bQ or 75 + 2(-0.005X) which gives you MR = 75 + .001X. I think you're on the right track though b/c you multiply the original inverse demand function (-0.005X) by 2 to get MR...
Actually, let's go through the correct process to find the profit-maximizing price, quantity, and profit.
To find the profit-maximizing price, we need to set the marginal revenue (MR) equal to the marginal cost (MC). In this case, the marginal cost per person attending is zero, as stated in the problem.
So, we have the equation:
MR = MC
75 - 0.01X = 0
Solving for X gives us:
X = 7500
Now we have the quantity that maximizes profit, but we need to check if it is feasible given the concert hall capacity. As stated in the problem, the concert hall can only accommodate 5000 people. Therefore, we need to substitute 5000 for X in the original demand function to determine the price:
P = 75 - 0.005X
P = 75 - 0.005(5000)
P = 75 - 25
P = 50
So, the profit-maximizing price is $50 with a quantity of 5000.
To calculate the value of the profit or loss, we need to subtract the total costs from the total revenue. The fixed cost of putting on the concert is $150,000.
Total revenue = Price * Quantity = $50 * 5000 = $250,000
Total costs = Fixed cost = $150,000
Profit = Total revenue - Total costs = $250,000 - $150,000 = $100,000
Therefore, the value of the profit is $100,000.
Now, to find the price that maximizes attendance, we can simply refer to the original demand function:
P = 75 - 0.005X
To maximize attendance, we need to set the price as low as possible. Since the capacity of the concert hall is 5000, we can substitute 5000 for X:
P = 75 - 0.005(5000)
P = 75 - 25
P = 50
So, the price that maximizes attendance is $50.
I hope this clears up any confusion and helps you understand the correct approach to solving this problem.