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Chapter 10: Problem 19

In some cities, Uber has a monopoly on ride-sharing services. In one town, the demand curve on weekdays is given by the following equation: \(P=50-Q\) However, during weekend nights, or surge hours, the demand for rides increases dramatically and the new demand curve is: \(P=100-Q\). Assume that marginal \(\operatorname{cost}\) is zero. a. Determine the profit-maximizing price during weekdays and during surge hours. b. Determine the profit-maximizing price during weekdays and during surge hours if \(\mathrm{MC}=10\) in stead of zero. c. Draw a graph showing the demand, marginal revenue, and marginal cost curves during surge hours from part (b), indicating the profit-maximizing price and quantity. Determine Uber's profit and the deadweight loss during surge hours, and show them on the graph.

Short Answer

Expert verified
a) The profit-maximizing price during weekdays and surge hours with zero MC is 0. b) With MC = 10, the profit-maximizing price during weekdays and surge hours is 10. c) The graph depicts the demand curve, MR and MC = 10 during surge hours. The profit for Uber is 2025 and the deadweight loss during surge hours is 1012.5.

Step by step solution

01

Determine the profit-maximizing price during weekdays and during surge hours

Since marginal cost (MC) is zero, the profit-maximizing quantity where MC equals to demand is where \(P=MC\), thus we can equate the demand equation to zero. \1. For weekdays demand curve: \(50 - Q = 0\) would yield \(Q = 50\), substituting this back into demand equation, \(P = 50 - 50 = 0\). \2. For surge hours demand curve: \(100 - Q = 0\), would yield \(Q = 100\), substituting this back into surge hours demand equation, \(P = 100-Q = 100-100 = 0\).
02

Determine the profit-maximizing price during weekdays and during surge hours if MC = 10

Now, if MC = 10, we need to find the quantity (Q) where MC equals to demand (P). \1. For weekdays: solve for \(Q\) in \(50 - Q = 10\), gives us \(Q = 50 - 10 = 40\), then substitute this back into demand equation, \(P = 50 - 40 = 10\).\2. For surge hours: solve for \(Q\) in \(100 - Q = 10\), gives us \(Q = 100 - 10 = 90\), then substitute this into surge hours demand equation gives, \(P = 100 - 90 = 10\).
03

Graphing and derive profit and loss

To draw the graph for MC = 10 during surge hours, the y-axis is the price (P) and the x-axis is the quantity (Q). Draw the demand curve (\(P = 100 - Q\)), MC curve (MC = 10) and MR curve (\(MR=100 - 2Q\)). \For quantity, set \(MC=MR\), solving \(10 = 100 - 2Q\) gives \(Q = 45\). Substitute \(Q = 45\) into demand equation to get profit-maximizing price \(P = 100 - 45 = 55\).\To calculate the Uber's profit: it’s the area of the rectangle with height \(P-MC\) and width \(Q\), which is \((55-10) * 45 = 2025\).\Deadweight loss is the area of triangle formed by MC, demand curve and Q quantity, which is \((1/2) * (100 - 55) * (90 - 45) = 1012.5\

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