Problem 63 - Applied Calculus

Question

If the demand curve is a line, we can write \(p=b+m q\), where \(p\) is the price of the product, \(q\) is the quantity sold at that price, and \(b\) and \(m\) are constants. (a) Write the revenue as a function of quantity sold. (b) Find the marginal revenue function.

Step-by-Step Solution

Answer

(a) \( R(q) = bq + mq^2 \); (b) Marginal revenue is \( b + 2mq \).

Step 1: Express Revenue Function

Revenue, denoted as \( R \), is calculated as the product of price \( p \) and quantity \( q \). Given that \( p = b + mq \), we have: \[ R(q) = p \times q = (b + mq) \times q = bq + mq^2 \]. So, the revenue function in terms of quantity \( q \) is \( R(q) = bq + mq^2 \).

Step 2: Differentiate to Find Marginal Revenue

To find the marginal revenue, we differentiate the revenue function \( R(q) = bq + mq^2 \) with respect to \( q \). Applying differentiation: \( \frac{dR}{dq} = \frac{d}{dq}(bq + mq^2) = b + 2mq \). Therefore, the marginal revenue function is \( MR = b + 2mq \).

Key Concepts

Revenue Function

The revenue function is a crucial concept in economics as it represents the total income a firm receives from selling its product at a given price and quantity. To calculate revenue, you multiply the price of the product by the quantity sold. For a linear demand curve where the price is expressed as a function of quantity, such as \(p = b + mq\), the revenue can be expressed in more specific terms. By substituting this demand equation into the revenue equation \(R = p imes q\), you get \(R(q) = (b + mq) imes q = bq + mq^2\). This function, \(R(q) = bq + mq^2\), shows how revenue changes with quantity sold, where \(bq\) represents the linear component and \(mq^2\) captures the quadratic component, reflecting how changes in quantity can have a non-linear effect on revenue. Understanding this formula helps in predicting how different sales volumes impact total earnings.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a product and the quantity demanded by consumers. In the given exercise, the demand curve is characterized by a linear equation \(p = b + mq\). Here, \(p\) stands for price, while \(q\) represents the quantity. The constants \(b\) and \(m\) influence the curve's slope and position.

• \(b\) is typically the intercept, indicating the price when no items are purchased.
• \(m\) reflects the rate of price change as more units are sold, often indicating how price varies with consumer demand.

This linear representation provides a simplified view of market behavior, which helps businesses forecast sales and adjust prices accordingly. Understanding demand curves is essential for setting appropriate pricing strategies to maximize profit.

Differentiation

Differentiation is a mathematical technique used to find how a function changes as its input changes. In this context, it's used to determine the marginal revenue by differentiating the revenue function with respect to quantity, \(q\). Given the revenue function \(R(q) = bq + mq^2\), applying differentiation gives us \(\frac{dR}{dq} = b + 2mq\). This expression is known as the marginal revenue, indicating how the revenue is expected to change with each additional unit sold. Differentiation helps in optimizing revenues and profits, as knowing the marginal revenue guides decision-makers.

• It shows whether increasing production will increase or decrease revenue.
• Makes it easier to find the quantity level at which revenue or profit is maximized.

This technique forms a foundation for many economic and business calculations, helping in resource allocation and strategy formulation.

Price-Quantity Relationship

The price-quantity relationship is central to understanding how market dynamics function. It describes how the price of goods affects the quantity that consumers will purchase, and vice versa. In the given context, this relationship is expressed through the equation \(p = b + mq\). \

• A positive \(m\): Suggests an upward sloping curve, indicating that higher prices lead to higher demand, which is less common in typical consumer behavior.
• A negative \(m\): More realistic, implying that price drop results in increased demand.

This relationship informs businesses about the impact of their pricing strategies on sales volume. By altering prices, companies can influence consumer purchasing and manage inventory more effectively. This knowledge equips marketers with tools to encourage buying behavior and achieve sales targets. Understanding this relationship helps forecast how changes in pricing or market conditions can affect demand, ensuring strategic business decisions are well-informed.