Question

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

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

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, \(P = 100-Q = 100-100 = 0\).

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, \(P = 100 - 90 = 10\).

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\

Key concepts

Demand Curve

In any market, the demand curve helps us understand how the price and quantity demanded of a good or service are related. It shows how many units of a good consumers want to buy at each price point. For Uber's ride-sharing services, the demand curve can vary depending on the time of day or week.
For example, the given demand curves in the exercise "weekdays: \(P = 50 - Q\)" and "surge hours: \(P = 100 - Q\)" indicate that as the quantity demanded \(Q\) increases, the price \(P\) decreases.

• On weekdays, demand is lower, so prices start at 50 and decrease with more rides.
• During surge hours, prices are higher at 100 and also decrease but from a higher point due to increased demand.

Using these equations, Uber can predict how many rides they will be able to sell at different price points, which is crucial for setting optimal pricing strategies.

Marginal Cost

Marginal cost (MC) is the additional cost incurred by producing one more unit of a good or service. It's an important concept for businesses as it impacts how they price their products. In our case, Uber’s marginal cost was initially zero, implying no additional cost for adding more rides. However, when considering a marginal cost of 10, things change.

• When MC is zero, the lowest price Uber can charge while still maximizing profits is the price where the demand curve hits the x-axis, meaning the highest quantity of rides.
• When MC is 10, Uber needs to find a balance where charges are at least equivalent to the marginal cost, ensuring that every additional ride contributes to profit.

This subtle shift in MC from zero to 10 significantly impacts the pricing strategy and quantity of rides Uber targets, as the price must now cover the marginal cost to maximize profit.

Profit Maximization

Profit maximization is crucial for any business because it means finding the price and quantity that allow for the highest possible profit. For Uber, understanding this concept is key to their pricing strategy.

• During weekdays, with \(MC = 0\), Uber sets prices where \(MC = MR\) (Marginal Revenue), which results in a price of zero, but maximizes rides.
• During surge hours, with \(MC = 0\) and demand higher, they can command a larger quantity but still at price zero based on the equation.
• When the marginal cost is set at 10, the calculation changes. Here, the profit-maximizing price needs to accommodate the marginal cost, aiming for maximal difference between price and cost. This leads to a price of 10 and maximizing rides at lower quantities than when MC was 0.

Profit maximization not only involves finding the right balance between demand and cost but also strategically increasing revenue, especially during surge hours when demand peaks.