Understanding Price Elasticity of Demand Calculation

This problem requires us to use the concept of price elasticity of demand to find the new price of good X after a change in quantity demanded, given the initial conditions and the elasticity coefficient.

Given Information:

• Initial Quantity Demanded (\(Q_1\)): 50 units
• Initial Price (\(P_1\)): 10 per unit
• Change in Quantity Demanded (\(\Delta Q\)): +20 units (Quantity demanded rises)
• New Quantity Demanded (\(Q_2\)): \(Q_1 + \Delta Q = 50 + 20 = 70\) units
• Coefficient of Elasticity of Demand (\(E_d\)): 1 (Unity)
• We need to find the New Price (\(P_2\)).

Key Concept: Price Elasticity of Demand

Price elasticity of demand (\(E_d\)) measures the responsiveness of the quantity demanded of a good to a change in its price. It is calculated as the ratio of the percentage change in quantity demanded to the percentage change in price.

The formula for price elasticity of demand using the initial values is:

\(E_d = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}\)

Or, expressed mathematically:

\(E_d = \frac{\frac{\Delta Q}{Q_1} \times 100}{\frac{\Delta P}{P_1} \times 100} = \frac{\Delta Q}{Q_1} \times \frac{P_1}{\Delta P}\)

Where:

• \(\Delta Q\) is the change in quantity demanded (\(Q_2 - Q_1\)).
• \(Q_1\) is the initial quantity demanded.
• \(\Delta P\) is the change in price (\(P_2 - P_1\)).
• \(P_1\) is the initial price.

Step-by-Step Calculation of New Price

We are given \(E_d = 1\), \(Q_1 = 50\), \(P_1 = 10\), and \(\Delta Q = +20\).

Substitute these values into the elasticity formula:

\(1 = \frac{+20}{50} \times \frac{10}{\Delta P}\)

Simplify the equation:

\(1 = \frac{200}{50 \times \Delta P}\)

\(1 = \frac{4}{\Delta P}\)

Now, solve for \(\Delta P\):

\(\Delta P = 4\)

According to the Law of Demand, when the quantity demanded of a good increases (rises), its price must have decreased. Since \(\Delta Q\) is positive (+20), \(\Delta P\) must be negative. The price elasticity coefficient is usually taken as an absolute value, so the calculated value of 4 represents the magnitude of the price change, but the direction is negative.

So, the actual change in price is \(\Delta P = -4\).

Now, we can find the new price (\(P_2\)) using the relationship:

\(\Delta P = P_2 - P_1\)

\(-4 = P_2 - 10\)

Add 10 to both sides to isolate \(P_2\):

\(P_2 = 10 - 4\)

\(P_2 = 6\)

Thus, the new price of good X is 6 per unit.

Let's verify the calculation:

• Initial: \(P_1 = 10, Q_1 = 50\)
• New: \(P_2 = 6, Q_2 = 70\)
• \(\Delta P = 6 - 10 = -4\)
• \(\Delta Q = 70 - 50 = +20\)

Calculate \(E_d\):

\(E_d = \left| \frac{\Delta Q}{Q_1} \times \frac{P_1}{\Delta P} \right| = \left| \frac{+20}{50} \times \frac{10}{-4} \right| = \left| \frac{2}{5} \times \frac{10}{-4} \right| = \left| \frac{20}{-20} \right| = |-1| = 1\)

The calculated elasticity matches the given elasticity of unity, confirming our result.

Conclusion on New Price Calculation

Based on the initial demand conditions and a unity price elasticity of demand, when the quantity demanded of good X rises by 20 units, the new price must be 6 per unit to satisfy the elasticity condition.

Revision Table: Key Concepts in Price Elasticity

Additional Information: Types of Price Elasticity of Demand

The value of the price elasticity of demand coefficient helps classify the demand for a good:

• Perfectly Inelastic Demand (\(E_d = 0\)): Quantity demanded does not change at all when the price changes. Demand curve is vertical.
• Inelastic Demand (\(0 < E_d < 1\)): Quantity demanded changes by a smaller percentage than the price change. Consumers are relatively unresponsive to price changes.
• Unity Elastic Demand (\(E_d = 1\)): Quantity demanded changes by exactly the same percentage as the price change. Total revenue remains constant when the price changes.
• Elastic Demand (\(E_d > 1\)): Quantity demanded changes by a larger percentage than the price change. Consumers are relatively responsive to price changes.
• Perfectly Elastic Demand (\(E_d = \infty\)): Quantity demanded changes infinitely with a tiny change in price. Demand curve is horizontal. This is a theoretical extreme.

Understanding elasticity is crucial for businesses in setting prices and for governments in setting taxes or subsidies, as it impacts total revenue and market outcomes.