Problem 33

Question

A company sells two products whose demand functions are given by $x_{1}=500-3 p_{1}$ and $x_{2}=750-2.4 p_{2}$ So, the total revenue is given by $R=x_{1} p_{1}+x_{2} p_{2}$ Estimate the average revenue if the price $p_{1}$ varies between $\$ 50$ and $\$ 75$ and the price $p_{2}$ varies between $\$ 100$ and $\$ 150$.

Step-by-Step Solution

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Answer

The average revenue when $p_{1}$ is between \$50 and \$75 and $p_{2}$ is between \$100 and \$150 can be found by applying these values into the revenue function. After calculating, we get the total minimum and maximum revenue. Finally, the average revenue is the mean of these two values.

1Step 1: Determine The Quantity Sold

First, calculate $x_{1}$ and $x_{2}$ using the demand functions given by $x_{1}=500-3 p_{1}$ and $x_{2}=750-2.4 p_{2}$, independently for $p_{1} = \$50$, \$75 and $p_{2}= \$100$, \$150.

2Step 2: Calculate Individual Revenues

Then, calculate the revenues for both the products using the revenue function formula $R=x_{1} p_{1}+x_{2} p_{2}$. Do this independently for each price point to get the revenue range for product 1 and product 2.

3Step 3: Determine Total Revenue

Next, find the total revenue by summing up the revenues examined in the previous step. To find the range of total revenue, take both minimum and maximum revenues.

4Step 4: Calculate Average Revenue

Lastly, average revenue can be found by summing up the minimum and maximum of total revenue, then divide by 2.

Key Concepts

Demand FunctionAverage RevenueTotal Revenue

Demand Function

A demand function illustrates the relationship between the price of a product and the quantity demanded by consumers. In simple terms, it shows how much of a product people are willing to buy at different prices. In the given exercise, we have two demand functions:

For product 1: $x_{1}=500-3 p_{1}$
For product 2: $x_{2}=750-2.4 p_{2}$

These equations tell us that as the price $p_1$ increases, the demand $x_1$ decreases for product 1. Similarly, as the price $p_2$ increases, the demand $x_2$ decreases for product 2. The constants 500 and 750 indicate the quantity demanded when the prices are zero. Meanwhile, the coefficients -3 and -2.4 show how sensitive demand is to a change in price. This concept is crucial because understanding how price affects demand helps businesses make informed decisions about pricing strategies.

Average Revenue

Average revenue is a metric that helps companies understand how much revenue they generate per unit sold. It is calculated by dividing the total revenue by the total quantity sold. In this problem, to determine the average revenue, we need to first compute the total revenue for different price levels.Let's say we calculate revenue for two price points. To find the average revenue, simply take the total revenue at the lowest price and the total revenue at the highest price and average them. This is done with the formula:\[ \text{Average Revenue} = \frac{\text{Minimum Total Revenue} + \text{Maximum Total Revenue}}{2} \]By finding the average revenue, a business can obtain a general idea of their earning performance per unit within the specified price ranges. This information can be pivotal, especially when planning marketing strategies or adjusting prices in a competitive market.

Total Revenue

Total revenue represents the complete earnings a company makes from sales. To calculate this, multiply the quantity sold by the price for each product and then add them together. In this scenario, the total revenue formula is given by:\[ R = x_{1} p_{1} + x_{2} p_{2} \]Here,

$x_{1}$ and $x_{2}$ are the quantities demanded based on their respective prices $p_{1}$ and $p_{2}$.
The company earns revenue for product 1 by selling $x_1$ units at a price $p_1$, and for product 2 by selling $x_2$ units at a price $p_2$.

In the exercise, finding total revenue involves calculating it at different price points to determine a range. Knowing this range helps businesses set realistic financial goals and assess the impact of different pricing strategies or market conditions on their overall earnings.

Problem 32

Problem 33

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