A careful analysis of demand for Bubbles in Strasburg, North Dakota, reveals a strange segmentation in the market. (Recall Bubbles is the beverage which produces an unexplained craving for Lawrence Welk's music. It is produced by the process Q = min{ , W}, where R is the number of pulverized Lawrence Welk records and W is gallons of North Dakota well water. P_R = $1, P_W = $4.) If demand for Bubbles by senior citizens is described by while demand by those under 65 years old is Q_y = 50P^-⁴, how should Bubbles be priced to maximize profits?

Question

Quizplus · Accepted Answer

To maximize profits, we need to set marginal revenue equal to marginal cost for each market segment. For senior citizens: Marginal revenue (MR) = 11ea6c55_6e49_9fa9_9aaf_a4c8f34702ce_TB6162_11 Marginal cost (MC) = P_R = $1 (since they only need 1 pulverized Lawrence Welk record to satisfy their demand) Setting MR = MC: 11ea6c55_6e49_9fa9_9aaf_a4c8f34702ce_TB6162_11 = 1 ea6c55_6e49_9fa9_9aaf_a4c8f34702ce = ln(1/11) ea6c55_6e49_9fa9_9aaf_a4c8f34702ce = -2.397895 Therefore, the price for senior citizens should be: P = e^ea6c55_6e49_9fa9_9aaf_a4c8f34702ce * MC P = e^-2.397895 * $1 P ≈ $0.0905 However, since we cannot charge less than the minimum cost of production, we will price Bubbles at the minimum possible price: $1. For those under 65 years old: Marginal revenue (MR) = Q_y = 50P^-⁴ Marginal cost (MC) = P_W = $4 (since they need at least 4 gallons of well water to satisfy their demand) Setting MR = MC: 50P^-⁴ = 4 Solving for P: 50P - 50P^2 - 150P + 150P^2 = 4 -100P^2 + 50P + 4 = 0 Using the quadratic formula, we get: P ≈ $0.1067 or $0.038 Since we cannot charge less than the cost of production, we will price Bubbles at $4 for those under 65 years old. Therefore, the optimal pricing strategy is to charge $1 for senior citizens and $4 for those under 65 years old.

A Careful Analysis of Demand for Bubbles in Strasburg, North