Problem 19 - Applied Calculus

Question

(a) If the demand equation is \(p q=k\) for a positive constant \(k,\) compute the elasticity of demand. (b) Explain the answer to part (a) in terms of the revenue function.

Step-by-Step Solution

Answer

(a) Elasticity is \(-1\). (b) Revenue is constant due to unit elasticity.

Step 1: Recall the Formula for Elasticity of Demand

The elasticity of demand, usually denoted as \(E\), is defined as the percentage change in quantity demanded divided by the percentage change in price. Mathematically, it is given by:\[E = \frac{dQ/Q}{dP/P} = \frac{dQ}{dP} \cdot \frac{P}{Q}.\]

Step 2: Differentiate the Demand Equation

We start with the given demand equation \(pq = k\). Solving for \(q\) gives us \(q = \frac{k}{p}\). Now, differentiate \(q\) with respect to \(p\):\[\frac{dq}{dp} = \frac{d}{dp}\left(\frac{k}{p}\right) = -\frac{k}{p^2}.\]

Step 3: Substitute into the Elasticity Formula

Using the derivative from Step 2, substitute into the elasticity formula. Since \(q = \frac{k}{p}\), the elasticity \(E\) is:\[E = \left(-\frac{k}{p^2}\right) \cdot \frac{p}{\frac{k}{p}} = \left(-\frac{k}{p^2}\right) \cdot \frac{p^2}{k} = -1.\]

Step 4: Interpret the Constant Elasticity and Revenue Function

The elasticity of demand is \(-1\), indicating that the demand is unit elastic. This means that any percentage change in price will result in an equal but opposite percentage change in quantity demanded. Consequently, the total revenue, \(R = pq\), remains constant since any price increase or decrease is exactly offset by a proportional change in quantity demanded, maintaining the constant \(k\).

Key Concepts

Understanding the Demand Equation

The demand equation is a fundamental concept in economics. It establishes a relationship between the price (\(p\)) and the quantity (\(q\)) of a product consumers are willing to buy. In our specific exercise, the demand equation is given by:

• \(pq = k\)

Here, \(k\) is a positive constant, meaning that the product of price and quantity is always the same value no matter how the price or quantity changes. This type of equation signifies inverse proportionality between price and quantity.
Understanding this relationship is key in determining how changes in price might affect the quantity demanded by consumers. In practical terms, if one increases, the other decreases, ensuring \(pq\) remains equal to \(k\).This helps businesses predict consumer behavior and make informed decisions about pricing strategies.

Explaining the Revenue Function

The revenue function is pivotal for any business looking to maximize profits. Revenue, denoted as \(R\), is the total income generated from selling a certain quantity of goods at a particular price. It is calculated as:

• \(R = pq\)

In our exercise, since the demand equation is \(pq = k\), the revenue function becomes particularly interesting.
The revenue is constant despite changes in price or quantity because any change in one is exactly balanced by a change in the other. This is an outcome of the demand being unit elastic, as we'll explain more in the next section. Constant revenue indicates that no matter how the market price fluctuates, business income remains stable at \(k\). This stability can be advantageous for businesses in planning and forecasting.

Decoding Unit Elasticity

Unit elasticity is a term used to describe a precise balance in response between price changes and quantity demanded. If demand is unit elastic, the absolute value of elasticity is exactly \(1\).
In mathematical terms, this is when:

• \(E = -1\)

For our demand equation, since \(E = -1\), it means that if there is a 1% increase in price, the quantity demanded will decrease by exactly 1%.
The importance of unit elasticity lies in its effects on revenue. As demonstrated earlier, with unit elasticity \(pq = k\), which means that any rise in price is counteracted by a proportionate fall in quantity demanded, and vice versa.
Understanding this concept is crucial for businesses as it highlights that price adjustments, under certain circumstances, won't alter the revenue, allowing firms to strategize and allocate resources efficiently.