Problem 41 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Suppose that the demand equation for a certain commodity is \(q=200-2 p^{2}(\) for \(0 \leq p \leq 10)\). a. Express the elasticity of demand as a function of \(p\). b. Calculate the elasticity of demand when the price is \(p=6\). Interpret your answer. c. At what price is the elasticity of demand equal to 1 ?

Step-by-Step Solution

Answer

a. \[ E_d = \frac{-4p^2}{200-2p^2} \] b. \[ -1.125 \] (elastic) c. \[ p \approx 5.77 \]

Step 1: Write Down the Demand Equation

The demand equation given is \[ q = 200 - 2p^2 \]

Step 2: Understand the Elasticity of Demand Formula

Elasticity of demand (\(E_d\)) is expressed as: \[ E_d = \left(\frac{\partial q}{\partial p}\right) \left(\frac{p}{q}\right) \]

Step 3: Find the Derivative of Quantity with Respect to Price

Differentiate \( q = 200 - 2p^2\) with respect to \( p\):\[ \frac{\partial q}{\partial p} = -4p \]

Step 4: Express the Elasticity of Demand as a Function of \( p \)

Substitute \( q\) and \( \frac{\partial q}{\partial p}\) into the elasticity formula: \[ E_d = (-4p) \left(\frac{p}{200 - 2p^2}\right) = \frac{-4p^2}{200 - 2p^2} \]

Step 5: Calculate Elasticity at \( p = 6 \)

Substitute \( p = 6\) into the elasticity function:\[ E_d = \frac{-4(6^2)}{200 - 2(6^2)} = \frac{-144}{128} = -1.125 \]Interpretation: When the price is \( p = 6\), the demand is elastic since the absolute value of elasticity is greater than 1.

Step 6: Find the Price at Which Elasticity Equals 1

Set \( E_d = -1\) and solve for \( p\):\[ \frac{-4p^2}{200 - 2p^2} = -1 \]Multiply both sides by \( 200 - 2p^2\):\[ -4p^2 = -(200 - 2p^2) \]Simplify:\[ -4p^2 = -200 + 2p^2 \]Combine like terms:\[ 6p^2 = 200 \]Solve for \( p\):\[ p^2 = \frac{200}{6} = \frac{100}{3} \]\[ p = \sqrt{\frac{100}{3}} \ approx 5.77 \]

Key Concepts

Demand Equation

The demand equation relates the quantity demanded of a commodity to its price. In the given problem, the demand equation is shown as: \[ q = 200 - 2p^2 \]. This implies that for every increase in price, the quantity demanded decreases in a specific manner. Understanding the demand equation helps us predict how changes in price affect demand and is crucial in making informed economic decisions. The demand equation is foundational in economics because it illustrates the inverse relationship between price and demand. As price increases, quantity demanded typically decreases, and vice versa. By analyzing the equation, companies can optimize pricing strategies to maximize revenue.

Calculus

Calculus is an essential tool in economics, particularly for solving problems related to changes and rates of change. In this exercise, we leverage calculus to find the elasticity of demand. Calculus allows us to take derivatives, a concept used to determine how one variable changes in relation to another. In the demand equation, we use differentiation to find how the quantity demanded changes as price changes. By understanding calculus, students can gain deeper insights into dynamic economic systems and better appreciate how even small changes in variables can have significant impacts.

Differentiation

Differentiation is the process of finding the derivative of a function. In this problem, we differentiate the demand equation with respect to price to find \[ \frac{\text{d}q}{\text{d}p} \]. From \[ q = 200 - 2p^2 \], we differentiate to get \[ \frac{\text{d}q}{\text{d}p} = -4p \]. The derivative \[ \frac{\text{d}q}{\text{d}p} \] signifies the rate at which quantity demanded changes with respect to price. Calculating this is key to understanding elasticity because it quantifies how sensitive demand is to price changes. Differentiation thereby provides a precise mathematical foundation for analyzing changes in economic variables, which is invaluable for making data-driven decisions.

Elasticity

Elasticity measures how much the quantity demanded of a good responds to changes in the price of that good. It's calculated using the formula: \[ E_d = \frac{\text{d}q}{\text{d}p} \times \frac{p}{q} \]. In our exercise, we've found the elasticity function as: \[ E_d = \frac{-4p^2}{200 - 2p^2} \]. When we set \[ p=6, \] we calculated elasticity to be -1.125. This indicates that the demand is elastic because its absolute value is greater than 1. Interpreting elasticity values helps businesses understand consumer behavior. If the elasticity absolute value is greater than 1, demand is elastic, meaning consumers are sensitive to price changes. Conversely, if less than 1, the demand is inelastic, showing that consumers are less sensitive to price changes. This understanding helps companies adjust pricing strategies to either capitalize on high sensitivity or retain revenue despite price changes.