Question

Below are given the prices in two different months for a product and the corresponding quantities demanded. But we do not know whether the price rose from $\mathrm{\$}.80$ to $\mathrm{\$}1.00$ or fell from $\mathrm{\$}1.00$ to $\mathrm{\$}.80.$ Show how each assumption will give a different answer for elasticity of demand and how using average values will alleviate this problem. $$\begin{array}{|ll|}\hline \text{ Price }& \text{ Quantity demanded }\\ \mathrm{\$}1.00& 4000\\ \mathrm{\$}.80& 5000\\ \hline\end{array}$$

Short answer

Using the midpoint method and average values, the elasticity of demand is found to be -1 regardless of the price change assumption, either increasing from $0.80 to $1.00 or decreasing from $1.00 to $0.80. This shows that using average values alleviates the problem of having different answers for elasticity of demand based on the given assumptions.

Step-by-step solution

Assumption 1: Price increases from $0.80 to $1.00

In this case, the price rises from $0.80 to $1.00, and the quantity demanded decreases from 5000 units to 4000 units. Let's calculate the elasticity of demand using the midpoint method formula: Elasticity of demand = $\frac{\mathrm{\%}\text{ change in quantity demanded}}{\mathrm{\%}\text{ change in price}}$ To find the percentage changes, we first find the average quantity demanded and the average price: $$\text{Average Quantity}=\frac{Q_2+Q_1}{2}=\frac{4000+5000}{2}=4500$$ $$\text{Average Price}=\frac{P_2+P_1}{2}=\frac{1.00+0.80}{2}=0.90$$ Now, we find the percentage changes: $$\mathrm{\%}\text{ change in quantity demanded}=\frac{Q_2-Q_1}{\text{Average Quantity}}\times 100=\frac{4000-5000}{4500}\times 100=-22.22\mathrm{\%}$$ $$\mathrm{\%}\text{ change in price}=\frac{P_2-P_1}{\text{Average Price}}\times 100=\frac{1.00-0.80}{0.90}\times 100=22.22\mathrm{\%}$$ Finally, we calculate the elasticity of demand: $$\text{Elasticity of demand}=\frac{-22.22\mathrm{\%}}{22.22\mathrm{\%}}=-1$$

Assumption 2: Price decreases from $1.00 to $0.80

In this case, the price decreases from $1.00 to $0.80, and the quantity demanded increases from 4000 units to 5000 units. Note that the average quantity demanded and the average price remain the same as in the previous case. Thus, we just need to find the percentage changes: $$\mathrm{\%}\text{ change in quantity demanded}=\frac{Q_2-Q_1}{\text{Average Quantity}}\times 100=\frac{5000-4000}{4500}\times 100=22.22\mathrm{\%}$$ $$\mathrm{\%}\text{ change in price}=\frac{P_2-P_1}{\text{Average Price}}\times 100=\frac{0.80-1.00}{0.90}\times 100=-22.22\mathrm{\%}$$ Finally, we calculate the elasticity of demand: $$\text{Elasticity of demand}=\frac{22.22\mathrm{\%}}{-22.22\mathrm{\%}}=-1$$

Conclusion

Using the midpoint method and average values for price and quantity demanded, we find that the elasticity of demand is -1, regardless of the assumption made for the price change. This demonstrates that using average values alleviates the problem of getting different answers for elasticity of demand depending on the price change assumption.

Key concepts

Midpoint Method

The midpoint method is a useful tool when calculating elasticity of demand. It helps us address the ambiguity that arises with price changes. Typically, when the price of a product changes, the quantity demanded also shifts. But what happens when we analyze both an increase and a decrease in price separately? The midpoint method provides a consistent way to determine the elasticity of demand by using the average between initial and final quantities and prices.
This technique is especially important because it eliminates discrepancies in calculations that might occur if we only focus on a one-sided price adjustment.

• To apply the midpoint method, we calculate the percentage change in quantity demanded and the price using their respective averages, not just the starting values.
• This ensures that the elasticity derived remains consistent whether the price increases or decreases between two points.

Understanding this concept is crucial for accurate economic analysis.

Price Change

When analyzing elasticity of demand, price change is a central factor. It's essential to know whether the price of a product has increased or decreased to apply the correct calculations. With the example problem, two scenarios are explored: one where the price rises from $0.80 to $1.00, and another where it decreases from $1.00 to $0.80.
The direction of the price change affects how we interpret the elasticity of demand. Without a proper method like the midpoint method, analyzing price changes could lead to different elasticity results based solely on the sequence of events.

• A price increase might lead to a quantity demand decrease, and vice versa, demonstrating the basic law of demand.
• However, using average values can standardize results, consequently minimizing potential confusion about elasticity findings.

So, understanding price change and its impact is crucial for calculating demand elasticity accurately.

Quantity Demanded

Quantity demanded refers to how much of a product consumers are willing and able to purchase at a given price. In this exercise, it's the key variable that changes in response to price adjustments.
As expected by the law of demand, when prices drop, the quantity demanded usually increases. Conversely, when prices go up, the demand tends to drop. In the given exercise, at a price of $0.80, 5000 units were demanded. But at a price of $1.00, units demanded fell to 4000.

• This shift showcases the sensitivity of demand to price changes—an insight that elasticity measurements aim to quantify.
• Elasticity of demand reflects how dramatically quantity demanded responds to changes in price, providing insights into consumer behavior.

Accurately capturing this response is pivotal for businesses and economists alike, as it influences production, pricing strategies, and market analysis.

Average Values

Average values play a pivotal role when using the midpoint method. They ensure that calculations of elasticity of demand reflect an unbiased analysis of changes. Instead of relying on only the initial or final figures, average values offer a balanced perspective.
In the given problem, the average price is calculated as $\text{Average Price}=\frac{1.00+0.80}{2}=0.90$, and the average quantity demanded as $\text{Average Quantity}=\frac{4000+5000}{2}=4500$. These average figures are then used to compute the percentage changes for elasticity.

• This approach makes the elasticity calculation symmetric, regardless of the direction of price change, ensuring consistent results.
• It also eliminates the potential bias that could arise if only initial or final values were used, providing a more accurate representation of true market reactions.

Thus, average values are indispensable in correctly applying the midpoint method to elasticity of demand.