Elasticity Of Demand: Is It Elastic Or Inelastic?

Hey guys, ever wondered how a change in price actually affects how much stuff people buy? It’s not always a simple one-to-one relationship, and that’s where the concept of elasticity of demand comes in. Basically, it tells us how sensitive the quantity demanded is to a change in price. For us math and economics nerds, understanding this can be super helpful in predicting market behavior and making smart business decisions. Today, we're diving deep into a specific scenario involving the demand function $D(p)=\sqrt{150-3 p}$ and figuring out the elasticity of demand at a price of $31$. We’ll break down what the elasticity value means and determine if, at this particular price, the demand is inelastic, elastic, or unitary. So, grab your calculators, and let's crunch some numbers!

Understanding the Elasticity of Demand Formula

Before we jump into our specific problem, let's make sure we're all on the same page with what elasticity of demand actually is and how we calculate it. In simple terms, it measures the responsiveness of the quantity demanded of a good or service to a change in its price. Think about it: if the price of your favorite coffee goes up by 10%, do you buy way less, or do you barely notice and keep buying the same amount? That difference in reaction is elasticity. Economists use a specific formula to quantify this. The price elasticity of demand (often denoted by $E_d$ or $\epsilon$) is calculated as the percentage change in quantity demanded divided by the percentage change in price. Mathematically, this is expressed as:

$E_d = \frac{\% \text{ change in Quantity Demanded}}{\% \text{ change in Price}}$

However, we often use a more convenient formula when we have a demand function $D(p)$. This formula involves the derivative of the demand function with respect to price. For a demand function $D(p)$, the elasticity of demand at a price $p$ is given by:

$E_d = \frac{p}{D(p)} \cdot D'(p)$

Here, $D'(p)$ represents the derivative of the demand function $D(p)$ with respect to $p$. This formula is super handy because it allows us to calculate elasticity at any given price point directly from the demand function. It essentially tells us the ratio of the relative change in quantity to the relative change in price at a specific point on the demand curve.

Now, what do the results of this calculation tell us? We categorize the elasticity into three main types:

• Elastic Demand ( $|E_d| > 1$ ): When the elasticity is greater than 1 in absolute value, it means that the percentage change in quantity demanded is larger than the percentage change in price. So, if the price goes up by 10%, the quantity demanded drops by more than 10%. This usually happens with goods that have many substitutes or are considered luxuries.
• Inelastic Demand ( $|E_d| < 1$ ): If the elasticity is less than 1 in absolute value, the percentage change in quantity demanded is smaller than the percentage change in price. A 10% price increase leads to a drop in quantity demanded of less than 10%. Necessities or goods with few substitutes tend to have inelastic demand.
• Unitary Demand ( $|E_d| = 1$ ): When the elasticity is exactly 1 in absolute value, the percentage change in quantity demanded is equal to the percentage change in price. A 10% price increase leads to a 10% decrease in quantity demanded. This is a critical point where total revenue is maximized.

Understanding these categories is key to interpreting our final answer. So, let's get back to our specific problem and see where our demand function falls!

Calculating Elasticity for $D(p)=\sqrt{150-3 p}$ at $p=31$

Alright guys, let's get down to business with our specific demand function: $D(p)=\sqrt{150-3 p}$. We need to find the elasticity of demand at a price of $p=31$. To do this, we'll use the formula we just discussed: $E_d = \frac{p}{D(p)} \cdot D'(p)$.

First things first, we need to calculate the quantity demanded at $p=31$. Let's plug $31$ into our demand function:

$D(31) = \sqrt{150 - 3 \cdot 31}$ $D(31) = \sqrt{150 - 93}$ $D(31) = \sqrt{57}$

So, at a price of $31$, the quantity demanded is $\sqrt{57}$. This is our $D(p)$ value for the elasticity formula.

Next up, we need to find the derivative of our demand function, $D'(p)$. Our demand function is $D(p) = (150 - 3p)^{1/2}$. Using the chain rule for differentiation:

$D'(p) = \frac{1}{2}(150 - 3p)^{-1/2} \cdot \frac{d}{dp}(150 - 3p)$ $D'(p) = \frac{1}{2}(150 - 3p)^{-1/2} \cdot (-3)$ $D'(p) = \frac{-3}{2\sqrt{150 - 3p}}$

Now that we have the derivative, we need to evaluate it at our specific price, $p=31$. We already know that $150 - 3p = 150 - 3(31) = 57$ when $p=31$. So, let's plug that into $D'(p)$:

$D'(31) = \frac{-3}{2\sqrt{57}}$

Perfect! We have all the components we need: $p=31$, $D(31)=\sqrt{57}$, and $D'(31)=\frac{-3}{2\sqrt{57}}$. Now, let's plug these values into the elasticity formula:

$E_d = \frac{p}{D(p)} \cdot D'(p)$ $E_d = \frac{31}{\sqrt{57}} \cdot \frac{-3}{2\sqrt{57}}$

Let's simplify this expression:

$E_d = \frac{31 \cdot (-3)}{\sqrt{57} \cdot 2\sqrt{57}}$ $E_d = \frac{-93}{2 \cdot 57}$ $E_d = \frac{-93}{114}$

To make it easier to compare with our elasticity categories, let's find the absolute value of $E_d$:

$|E_d| = \left|\frac{-93}{114}\right| = \frac{93}{114}$

Now, we just need to simplify the fraction or convert it to a decimal to compare it with 1. Let's simplify:

Both 93 and 114 are divisible by 3: $93 \div 3 = 31$ $114 \div 3 = 38$

So, $|E_d| = \frac{31}{38}$.

As a decimal, $\frac{31}{38}$ is approximately $0.8158$.

Now we can determine the nature of the demand at this price.

Interpreting the Results: Inelastic, Elastic, or Unitary?

So, guys, we’ve done the math, and we found that the absolute value of our elasticity of demand at a price of $31$ for the function $D(p)=\sqrt{150-3 p}$ is $|E_d| = \frac{31}{38}$, which is approximately $0.8158$. Now comes the crucial part: interpreting what this number actually means for the demand.

Remember our categories for elasticity? We have:

• Elastic Demand: $|E_d| > 1$
• Inelastic Demand: $|E_d| < 1$
• Unitary Demand: $|E_d| = 1$

Our calculated value, $|E_d| \approx 0.8158$, is less than 1. What does this signify? It means that at a price of $31$, the demand for this product is inelastic.

What Does Inelastic Demand Mean?

When demand is inelastic, it signifies that the quantity demanded is relatively unresponsive to changes in price. In our case, a percentage change in price will lead to a smaller percentage change in the quantity demanded. For instance, if the price were to increase by, say, 10%, the quantity demanded would decrease by less than 10%. Conversely, if the price were to decrease by 10%, the quantity demanded would increase by less than 10%.

This characteristic is typical for goods or services that are considered necessities, have few close substitutes, or represent a small portion of a consumer's budget. For example, essential medications, gasoline (in the short term), or basic food staples often exhibit inelastic demand. Consumers will continue to buy these items even if the price goes up because they truly need them or don't have many alternatives.

Implications for Businesses

For a business selling a product with inelastic demand, this is generally good news in terms of pricing power. If they need to increase prices, they can do so with less fear of significantly losing customers. The revenue generated would likely increase because the revenue gained from the higher price per unit would outweigh the loss in quantity sold. Conversely, lowering prices might not significantly boost sales volume enough to compensate for the reduced price per unit, potentially leading to lower total revenue.

So, for our specific scenario with $D(p)=\sqrt{150-3 p}$ at $p=31$, the demand is inelastic. This means that if the price were to fluctuate slightly around $31$, the quantity purchased wouldn't change drastically. This is a key insight for any economic analysis or business strategy involving this particular product at this price point.

Conclusion: Our Demand is Inelastic!

Alright, so to wrap things up, we took our demand function $D(p)=\sqrt{150-3 p}$ and investigated the elasticity of demand at a price of $31$. We followed the steps, calculated the quantity demanded at that price, found the derivative of the demand function, and then plugged everything into the elasticity formula. The result we got was $E_d = \frac{-93}{114}$, which simplifies to $E_d \approx -0.8158$.

When we look at the absolute value, $|E_d| \approx 0.8158$, we see that it is less than 1. This clearly places our demand in the inelastic category. What does this mean in plain English, guys? It means that at a price of $31$, consumers are not super sensitive to price changes. If the price goes up a bit, they won't cut back on buying the product by a lot. If the price goes down a bit, they won't rush to buy a whole lot more. Demand is relatively stable in response to price fluctuations around this point.

This is a really important concept in economics and business. Understanding whether demand is elastic or inelastic helps businesses make crucial decisions about pricing strategies. For a product with inelastic demand, like the one we analyzed at $p=31$, a company might feel more confident about raising prices because they likely won't lose a significant number of customers, and their total revenue might even increase. On the other hand, for products with elastic demand (where $|E_d| > 1$), a price increase could lead to a substantial drop in sales and revenue.

So, to directly answer the question: At a price of $31$, we would say the demand is A. Inelastic. Keep this stuff in mind as you navigate the fascinating world of economics and consumer behavior!