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 = min{R/2, W}, where R is pulverized Lawrence Welk records and W is gallons of North Dakota well water.P_R = P_W = $1.Demand for Bubbles is Q = 576P^-²A^0.5.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.

Question

Quizplus · Accepted Answer

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(F1F1F1S1F1F1F10R/2)

R/2 = 2Q/F1F1F1S1F1F1F10

R/2 = 20(F1F1F1S1F1F1F10Q)

Substituting into the cost function:

C(Q) = 20(F1F1F1S1F1F1F10Q) + WQ

C(Q) = 20(F1F1F1S1F1F1F10Q)

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(F1F1F1S1F1F1F110Q) = 0

0.441 - 20(F1F1F1S1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.

An Obscure Inventor in Strasburg, North Dakota, Has a Monopoly