Question

Sketch a demand curve that is unit elastic for a price change between \(\$ 9\) and \(\$ 11 .\) Assume that the quantity demanded is 110 when price is \(\$ 9 .\) You'll have to determine the quantity demanded when price is \(\$ 11 .\)

Short answer

The quantity demanded when the price is $11 is approximately 86 units. Connect points (110, $9) and (86, $11) to sketch the demand curve.

Step-by-step solution

Calculate the percentage change in price

The percentage change in price is calculated as the change in price divided by the original price. In this case, the change in price is \( $11 - $9 = $2 \), and the original price is \( $9 \). So, the percentage change in price is \( \frac{$2}{$9} = 0.222 \), or 22.2%.

Calculate the new quantity demanded

With unit elastic demand, the percentage change in quantity demanded is equal to the percentage change in price. So if the price has increased by 22.2%, the quantity demanded will decrease by 22.2%. Therefore, the new quantity demanded can be calculated as the original quantity minus 22.2% of the original quantity: \n\( 110 - (0.222 \times 110) = 85.58 \). This will round to 86 units.

Sketch the demand curve

Plot the two points \((110, $9)\) and \((86, $11)\) on a graph. Draw a straight line joining the two points. This line represents the demand curve that is unit elastic for a price change between $9 and $11.

Key concepts

Elasticity of Demand

Elasticity of demand is a measure of how sensitive the quantity demanded of a good is to a change in its price. Generally, when the price of a good rises, consumers buy less of it and vice versa when the price falls. However, how much less (or more) consumers will buy depends on the goods elasticity.

Elastic goods are those where a small change in price leads to a significant change in quantity demanded. In contrast, inelastic goods see little change in quantity demanded even with a large price shift. There's also unit elastic demand, which is the focus of our exercise, where the percentage change in quantity demanded is exactly equal to the percentage change in price. Understanding this concept is crucial for businesses when setting prices and for economists when analyzing market responses to price fluctuations.

Price Elasticity of Demand

Price elasticity of demand quantifies the elasticity concept. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. The formula looks like this:
$$ \text{Price Elasticity of Demand} = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} $$.

When this ratio is equal to 1, we say the demand is unit elastic. In the example we're working with, the demand is unit elastic, meaning a 1% increase in price leads to a 1% decrease in quantity demanded and vice versa.

Demand Curve Graphing

Graphing a demand curve visualizes the relationship between the price of an item and the quantity demanded at those prices. When plotting a demand curve on a graph, the price is typically on the vertical (Y) axis and quantity on the horizontal (X) axis. A downward sloping curve illustrates the inverse relationship between price and demand.

In our exercise, to graph a demand curve that is unit elastic, we start by plotting the known point where the price is $9, and the quantity demanded is 110. We then calculate the new quantity demanded after a price increase, plot the second point, and draw a straight line through the points to depict the demand curve.

Quantity Demanded Calculation

The quantity demanded calculation is straightforward when we are given the percentage change in quantity that results from a price change. In the given exercise, because demand is unit elastic, the percentage change in quantity demanded is equal to the percentage change in price. Here's how the math works out:
$$ \text{New Quantity} = \text{Original Quantity} - (\text{Percentage Change in Price} \times \text{Original Quantity}) $$.

Substituting the numbers from our problem: $$ 110 - (0.222 \times 110) = 85.58 $$. After rounding, we find the new quantity demanded to be 86 units. It's essential to grasp this calculation, as it's used to predict consumer behavior and to make informed business decisions concerning pricing strategies.