Question

The equation for a demand curve is P = 48 – 3Q. What is the elasticity in moving from a quantity of 5 to a quantity of 6?

Short answer

First, we determine the price values at each quantity using the demand curve equation: \(P_5 = 48 - 3(5) = 33\) and \(P_6 = 48 - 3(6) = 30\). The percentage change in quantity and price are \(\% \Delta Q = \frac {6 - 5}{5} \cdot 100 \% = 20 \%\) and \(\% \Delta P = \frac {30 - 33}{33} \cdot 100 \% = -9.09 \%\), respectively. Finally, we calculate the elasticity: Elasticity = \(\frac{20 \%}{-9.09 \%} = -2.20\).

Step-by-step solution

Determine the price values at each quantity

We will use the demand curve equation P = 48 - 3Q to calculate the price at each of the given quantities: Price at quantity of 5: \(P_5 = 48 - 3(5)\) Price at quantity of 6: \(P_6 = 48 - 3(6)\) Calculate the price values.

Calculate the percentage change in quantity and price

Calculate the percentage change in quantity: \(\% \Delta Q = \frac {Q_{new} - Q_{old}}{Q_{old}} \cdot 100 \% = \frac {6 - 5}{5} \cdot 100 \%\) Calculate the percentage change in price: \(\% \Delta P = \frac {P_{new} - P_{old}}{P_{old}} \cdot 100 \% = \frac {P_6 - P_5}{P_5} \cdot 100 \%\)

Calculate the elasticity

Using the formula for elasticity: Elasticity = \(\frac{% \Delta Q}{% \Delta P}\) Plug in the percentage change in quantity and price calculations from Step 2 to find the elasticity in moving from a quantity of 5 to a quantity of 6.

Key concepts

Demand Curve Equation

Understanding the demand curve equation is fundamental in economics to analyze consumer behavior. It represents the relationship between the price of a good and the quantity demanded by consumers. In the exercise, the demand curve is given by the equation P = 48 – 3Q, where P is the price of the good and Q is the quantity demanded.

Let's break this down: The equation is a simple linear function where the coefficient of Q (-3 in this instance) represents the slope of the demand curve. This slope tells us how the demand for a product changes as its price changes. Specifically, a negative slope indicates that a higher price results in a lower quantity demanded, which aligns with the law of demand in economics.

For example, using the given demand equation, if the quantity demanded increases from 5 to 6 units, we can plug these values into the equation to find the corresponding prices. We find that P_5 and P_6 are 48 - 3(5) and 48 - 3(6) respectively. This demonstrates how the demand curve equation is used to calculate different price points based on varying quantities.

Percentage Change Calculation

The concept of percentage change is central to various fields, including business and economics, as it allows us to measure the relative change of a variable over time or between different states. The percentage change formula is: % Change = (New Value - Original Value) / Original Value * 100%.

In economics, this formula helps us understand the magnitude of change in price and quantity demanded or supplied. It's crucial when assessing the price elasticity of demand, which relies on the percentage changes in price and quantity. Let's apply this to the textbook example: to move from a quantity of 5 to 6, we have a percentage change in quantity (% ∆ Q) calculated as ((6 - 5) / 5) * 100%, which results in 20%.

Similarly, if the calculated price at quantity 5 (P_5) was \(33 and at quantity 6 (P_6) was \)30, the percentage change in price (% ∆ P) would be ((30 - 33) / 33) * 100%, or approximately -9.09%. These calculations show how much the quantity and price have changed in percentage terms, playing a pivotal role in further elasticity calculations.

Economics Elasticity

Elasticity in economics is a measure of how sensitive the quantity demanded or supplied of a good is to a change in one of its determinants, such as price. Price elasticity of demand, specifically, shows how much the quantity demanded of a good responds to a change in its price, with a higher elasticity indicating a more significant response.

To calculate price elasticity of demand, we use the formula Elasticity = % Change in Quantity Demanded / % Change in Price. This ratio explores the responsiveness of consumers to price changes of a product. If the elasticity is greater than 1, the demand is considered elastic, meaning consumers are highly responsive to price changes. Conversely, if it's less than 1, the demand is inelastic, and consumers aren't as reactive to price changes.

With the provided example from the exercise, after calculating the percentage change in quantity (20%) and the percentage change in price (-9.09%), you can then plug these values into the elasticity formula to determine the responsiveness of demand when quantity moves from 5 to 6. This comprehensive approach to understanding elasticity provides valuable insights into consumer purchasing patterns and helps businesses make informed pricing decisions.