Asked by Landon Busse on May 27, 2024
Verified
An obscure inventor in Strasburg, North Dakota, has a monopoly on a new beverage called Bubbles, which produces an unexplained craving for Lawrence Welk music.Bubbles is produced by the following process: Q minR/2, W, where R is pulverized Lawrence Welk records and W is gallons of North Dakota well water.PR PW $1.Demand for Bubbles is Q 576P2A0.5.If the advertising budget for Bubbles is $81, the profit-maximizing quantity of Bubbles is
A) 0.
B) 36.
C) 432.
D) 144.
E) 140.
Monopoly
A market structure characterized by a single seller, selling a unique product in the market. In a monopoly, the seller faces no competition, as he is the sole seller of goods with no close substitute.
Bubbles
In economics, a market condition characterized by rapid escalation of asset prices followed by a contraction.
- Interpret demand functions and calculate profit-maximizing quantities.
- Understand the role of advertising budget in maximizing profits for monopolistic products.
Verified Answer
AW
Aidan WagnerJun 01, 2024
Final Answer :
D
Explanation :
To find the profit-maximizing quantity of Bubbles, we need to use the profit function:
P(Q) = PQ – C(Q)
where P(Q) is the profit function, Q is the quantity of Bubbles sold, P is the price of Bubbles, and C(Q) is the cost function.
We can find the price of Bubbles by setting the quantity demanded equal to 576 and solving for P:
576 = 576P^2/(P+0.5)
Simplifying:
P^2 + 0.5P - 1 = 0
Using the quadratic formula:
P = (-0.5 ± sqrt(0.5^2 + 4))/2
P = 0.441 or P = -0.941
We can ignore the negative price and choose P = 0.441.
Next, we need to find the cost function. Using the production function, we can write the cost function as:
C(Q) = R/2 + WQ
where R is the cost of pulverized Lawrence Welk records and W is the cost of North Dakota well water (assumed to be negligible). We can express R in terms of Q using the production function:
Q = 10(R/2)
R/2 = 2Q/
R/2 = 20(Q)
Substituting into the cost function:
C(Q) = 20(Q) + WQ
C(Q) = 20(Q)
To find the profit-maximizing quantity, we need to take the derivative of the profit function with respect to Q and set it equal to zero:
P'(Q) = P - C'(Q) = 0
Substituting in the price and cost functions:
P - 20(F1F1F110Q) = 0
0.441 - 20(F1F1F110Q) = 0
Solving for Q:
Q = 0.02205
Multiplying by 576 to get the actual quantity demanded:
Q* = 12.68
Rounding down to the nearest integer, the profit-maximizing quantity of Bubbles is 12.
The advertising budget is irrelevant to the profit-maximizing quantity, so the answer is D) 12.
P(Q) = PQ – C(Q)
where P(Q) is the profit function, Q is the quantity of Bubbles sold, P is the price of Bubbles, and C(Q) is the cost function.
We can find the price of Bubbles by setting the quantity demanded equal to 576 and solving for P:
576 = 576P^2/(P+0.5)
Simplifying:
P^2 + 0.5P - 1 = 0
Using the quadratic formula:
P = (-0.5 ± sqrt(0.5^2 + 4))/2
P = 0.441 or P = -0.941
We can ignore the negative price and choose P = 0.441.
Next, we need to find the cost function. Using the production function, we can write the cost function as:
C(Q) = R/2 + WQ
where R is the cost of pulverized Lawrence Welk records and W is the cost of North Dakota well water (assumed to be negligible). We can express R in terms of Q using the production function:
Q = 10(R/2)
R/2 = 2Q/
R/2 = 20(Q)
Substituting into the cost function:
C(Q) = 20(Q) + WQ
C(Q) = 20(Q)
To find the profit-maximizing quantity, we need to take the derivative of the profit function with respect to Q and set it equal to zero:
P'(Q) = P - C'(Q) = 0
Substituting in the price and cost functions:
P - 20(F1F1F110Q) = 0
0.441 - 20(F1F1F110Q) = 0
Solving for Q:
Q = 0.02205
Multiplying by 576 to get the actual quantity demanded:
Q* = 12.68
Rounding down to the nearest integer, the profit-maximizing quantity of Bubbles is 12.
The advertising budget is irrelevant to the profit-maximizing quantity, so the answer is D) 12.
Learning Objectives
- Interpret demand functions and calculate profit-maximizing quantities.
- Understand the role of advertising budget in maximizing profits for monopolistic products.