Problem 71 - Calculus Single Variable

Question

If \(q(p)\) is the demand for a product at price \(p-\) that is, the number of units of the product that are sold at price \(p-\) then \(E(p)=-q^{\prime}(p) \cdot p / q(p)\) is called the elasticity of demand for the product at price \(p\). Suppose that \(a\) and \(b\) are positive constants. a. What is \(E(p)\) if \(q(p)=a-b p ?\) b. What is \(E(p)\) if \(q(p)=a / p^{b} ?\)

Step-by-Step Solution

Answer

a. \( E(p) = \frac{bp}{a-bp} \); b. \( E(p) = b \).

Step 1: Understand Elasticity of Demand

The elasticity of demand, denoted as \( E(p) \), measures how the quantity demanded changes in response to a change in price. It is given by the formula \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \), where \( q'(p) \) is the derivative of the demand function \( q(p) \).

Step 2: Differentiate the Demand Function for Part a

For \( q(p) = a - bp \), find the derivative \( q'(p) \). The derivative with respect to \( p \) is \( q'(p) = -b \).

Step 3: Substitute into the Elasticity Formula for Part a

Substitute \( q'(p) = -b \) and \( q(p) = a - bp \) into the elasticity formula: \[ E(p) = -(-b) \cdot \frac{p}{a-bp} = \frac{bp}{a-bp} \]

Step 4: Differentiate the Demand Function for Part b

For \( q(p) = \frac{a}{p^b} \), find the derivative \( q'(p) \). Use the power rule: \( q'(p) = -\frac{ab}{p^{b+1}} \).

Step 5: Substitute into the Elasticity Formula for Part b

Substitute \( q'(p) = -\frac{ab}{p^{b+1}} \) and \( q(p) = \frac{a}{p^b} \) into the elasticity formula: \[ E(p) = -\left(-\frac{ab}{p^{b+1}}\right) \cdot \frac{p}{\frac{a}{p^b}} = b \]

Step 6: Conclusion

For part a, \( E(p) = \frac{bp}{a-bp} \). For part b, \( E(p) = b \).

Key Concepts

Demand Function

A demand function is an equation that expresses the relationship between the price of a product and the quantity of that product consumers are willing and able to purchase. It essentially answers the question, "How many products will people buy at this price?" In the given exercise, the demand function is denoted by \( q(p) \), where \( q \) represents the quantity demanded and \( p \) stands for the price of the product.

Let's break down the examples provided in the exercise. In the first case, \( q(p) = a - bp \), we have a simple linear demand function. The variable \( a \) is the intercept, indicating the quantity demanded when the price is zero. The parameter \( b \) represents the slope, showing the rate at which demand decreases as price increases.

In the second example, \( q(p) = \frac{a}{p^b} \), we have a non-linear demand relationship. This formula suggests an inverse relationship between price and demand. As the price \( p \) increases, the denominator in the fraction becomes larger, therefore reducing the overall quantity demanded. This type of function can model situations where products are viewed as necessities, and demand drops sharply with price increases.

Derivative

In mathematics, the derivative of a function represents the rate at which the function value changes as its input changes. For demand functions, the derivative is crucial as it helps to determine how sensitive quantity demanded is to changes in price.

For a linear demand function, like \( q(p) = a - bp \), the derivative \( q'(p) \) simplifies to \(-b\). This indicates a constant rate of change in quantity demanded per unit change in price. The negative sign indicates that as price increases, the quantity demanded decreases, a common scenario in economic theory.

When it comes to the demand function \( q(p) = \frac{a}{p^b} \), finding the derivative involves using the power rule. This demand function's derivative, \( q'(p) = -\frac{ab}{p^{b+1}} \), illustrates that the rate of change in demand is not constant. It becomes complex as price changes due to the exponent \( b+1 \) in the denominator. The derivative shows how demand responsiveness varies across different price levels, essential for understanding and predicting consumer behavior.

Calculus

Calculus is a branch of mathematics focused on change, using derivatives and integrals as primary tools. It's the mathematical backbone for analyzing situations involving continuous change, such as demand in economics.

In this exercise, calculus is used to determine the elasticity of demand by differentiating demand functions. Differentiation, a part of calculus, allows us to find the derivative of the demand function, indicating how demand changes with small price variations.

Understanding these changes through derivatives is important for calculating elasticity, a metric that portrays how strong the relationship between price and demand is. With the derivative, students can substitute it into the elasticity formula, guiding them to an understanding of how sensitive demand is about price changes.

Essentially, calculus provides the tools to break down, analyze, and apply these mathematical relationships in real-world economic cases, helping businesses and economists forecast consumer behavior and strategize accordingly.

Price Elasticity

Price elasticity of demand is a measure that shows how much the quantity demanded of a product responds to a change in price. It helps in understanding the consumer's sensitivity to price changes. The given formula for elasticity is \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \).

In simple terms, this formula calculates the percentage change in quantity demanded relative to a one-percent change in price. A higher elasticity value implies consumers are more responsive to price changes, while a lower value suggests insensitivity.

For the linear demand function \( q(p) = a - bp \), substituting the variables into the elasticity formula gives \( E(p) = \frac{bp}{a-bp} \). If the elasticity is greater than 1, demand is said to be elastic; if less than 1, it is inelastic. This value tells us how a small change in price impacts sales, guiding pricing strategies.

On the other hand, with the demand \( q(p) = \frac{a}{p^b} \), elasticity comes out as \( b \), which is constant and independent of price \( p \). This suggests a proportional response in demand to price changes, providing a straightforward metric for businesses to anticipate demand shifts at different pricing points.