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Chapter 5: Problem 19

Currently, 1800 people ride a certain commuter train each day and pay $$\$ 4$$ for a ticket. The number of people \(q\) willing to ride the train at price \(p\) is \(q=600(5-\sqrt{p})\). The railroad would like to increase its revenue. (a) Is demand elastic or inelastic at \(p=4 ?\) (b) Should the price of a ticket be raised or lowered?

Short Answer

Expert verified
a) Demand is inelastic at \( p=4 \). b) The price should be raised.

Step by step solution

01

Understand the Demand Function

The demand function given is: \( q = 600(5 - \sqrt{p}) \) where \( q \) is the quantity of people willing to ride the train at a price \( p \).
02

Evaluate the Demand at Given Price

Substitute \( p = 4 \) into the demand function: \( q = 600(5 - \sqrt{4}) \) \( q = 600(5 - 2) \) \( q = 600 \times 3 \) \( q = 1800 \) This confirms the current ridership of 1800 people.
03

Calculate the Price Elasticity of Demand

The formula for price elasticity of demand (E) is: \( E = \frac{dq}{dp} \times \frac{p}{q} \) First, find the derivative \( \frac{dq}{dp} \). The demand function is \( q = 600(5 - \sqrt{p}) \). The derivative with respect to \( p \) is: \( \frac{dq}{dp} = 600 \times \frac{d}{dp}(5 - \sqrt{p}) \) \( \frac{dq}{dp} = 600 \times -\frac{1}{2\sqrt{p}} \) \( \frac{dq}{dp} = -300 / \sqrt{p} \) At \( p = 4 \), \( \frac{dq}{dp} = -300 / 2 \) \( \frac{dq}{dp} = -150 \).
04

Compute the Elasticity at \( p=4 \)

Using the elasticity formula \( E = \frac{dq}{dp} \times \frac{p}{q} \), substitute the values: \( E = -150 \times \frac{4}{1800} \) \( E = - \frac{600}{1800} \) \( E = - \frac{1}{3} \) Since \( |E| < 1 \), the demand is inelastic.
05

Determine if the Price Should be Raised or Lowered

Since the demand is inelastic at \( p=4 \), increasing the price will lead to an increase in revenue. Therefore, the price should be raised.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Function
The demand function helps us understand the relationship between the price of a product or service and the quantity demanded by consumers. In this exercise, the demand function is given by:
\( q = 600(5 - \sqrt{p}) \).
Here, \( q \) represents the number of people willing to ride the train, and \( p \) is the price of a ticket.

To find out how many people will ride the train at a specific ticket price, we simply substitute the price \( p \) into the equation and solve for \( q \). This function shows that as the price \( p \) increases, the quantity \( q \) of people willing to ride decreases.
Key points to remember about demand function:
  • The demand function can often be expressed as \( q = f(p) \), where \( f(p) \) denotes the functional relationship between price and quantity demanded.
  • In our example, as the price decreases, more people are willing to ride the train, indicating a typical downward-sloping demand curve.
  • It helps businesses understand how changes in price could impact the number of customers or sales.
Price Elasticity
Price elasticity of demand measures the responsiveness of the quantity demanded to a change in price. Mathematically, it is defined as:
\( E = \frac{dq}{dp} \times \frac{p}{q} \).

This tells us how much the quantity demanded changes with a one percent change in price. Based on the steps in the solution:

  • First, we find the derivative of the demand function with respect to price, \( \frac{dq}{dp} \). For our function, this is calculated to be \( \frac{dq}{dp} = -300 / \sqrt{p} \).
  • We then use this derivative to calculate elasticity at the specific price point. Substituting \( p = 4 \), we find \( \frac{dq}{dp} = -150 \).
  • The final elasticity is computed as \( E = - \frac{1}{3} \), indicating inelastic demand since \( |E| < 1 \).
When demand is inelastic, it means that consumers are not very responsive to price changes. In other words, a change in price leads to a smaller proportional change in the quantity demanded.
Revenue Optimization
Revenue optimization involves determining the best price to maximize total revenue. The price elasticity of demand plays a crucial role in this decision.

From the exercise, we know the demand at \( p = 4 \) is inelastic.
This implies that increasing the price will result in an increase in total revenue because the loss in quantity demanded will be less than proportionate to the corresponding increase in price.

Steps to optimize revenue:
  • Evaluate the elasticity of demand at different price points.
  • If demand is inelastic (\( |E| < 1 \)), raising prices will typically increase revenue.
  • If demand is elastic (\( |E| > 1 \)), lowering prices will usually increase revenue.
By understanding the elasticity, businesses can set prices that align with their revenue goals. In our example, since the demand is inelastic at \( p = 4 \), increasing the ticket price is the recommended strategy to boost revenue.

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