If the quantity demanded of good X increases from 2300 units to 2700 units when the price of related good Y rises from Rs 45 to Rs 55, what is the arc cross elasticity of demand between X and Y?

Introduction / Context:
Cross elasticity of demand measures how the quantity demanded of one good responds to changes in the price of another related good. It is positive for substitute goods and negative for complementary goods. The arc method of elasticity uses average values of price and quantity to compute elasticity over a finite change, rather than at a point. This question provides data on how the quantity of good X changes when the price of good Y changes and asks for the arc cross elasticity of demand between the two goods.

Given Data / Assumptions:
• Initial quantity of good X demanded, Q1, is 2300 units. • New quantity of good X demanded, Q2, is 2700 units. • Initial price of good Y, P1, is Rs 45. • New price of good Y, P2, is Rs 55. • We must compute arc cross elasticity of demand using the arc elasticity formula.

Concept / Approach:
Arc cross elasticity of demand is calculated using the formula: Exy = percentage change in quantity of X divided by percentage change in price of Y, where the percentage changes are computed using average values. That is, percentage change in quantity equals change in quantity divided by average quantity, and percentage change in price equals change in price divided by average price. The ratio of these two percentages gives the arc cross elasticity. A positive result indicates that X and Y are substitutes, while a negative result indicates that they are complements.

Step-by-Step Solution:
Step 1: Compute the change in quantity of good X. Change in quantity equals Q2 minus Q1, so 2700 minus 2300 equals 400 units. Step 2: Compute the average quantity of X. Average quantity equals (Q1 plus Q2) divided by 2, which is (2300 plus 2700) divided by 2. This sum is 5000 and the average is 2500. Step 3: Compute the change in price of good Y. Change in price equals P2 minus P1, so 55 minus 45 equals 10 rupees. Step 4: Compute the average price of Y. Average price equals (P1 plus P2) divided by 2, which is (45 plus 55) divided by 2. This sum is 100 and the average is 50. Step 5: Calculate the percentage change in quantity using the arc formula. That is change in quantity divided by average quantity, so 400 divided by 2500, which equals 0.16. Step 6: Calculate the percentage change in price using the arc formula. That is change in price divided by average price, so 10 divided by 50, which equals 0.2. Step 7: Compute arc cross elasticity by dividing 0.16 by 0.2. This equals 0.8. Step 8: Therefore the arc cross elasticity of demand between X and Y equals 0.8, which is positive and indicates substitute goods.

Verification / Alternative check:
To verify, note that both numerator and denominator represent fractional changes. A 400 unit change out of an average of 2500 is 16 per cent, while a 10 rupee change out of an average of 50 is 20 per cent. Dividing 16 per cent by 20 per cent again gives 0.8. The sign is positive because both price of Y and quantity of X move in the same direction, confirming that the goods are substitutes and that the computed elasticity value makes sense.

Why Other Options Are Wrong:
An elasticity of 4 would require the percentage change in quantity to be much larger than the percentage change in price, which is not supported by the given data.

1.25 would correspond to 25 per cent change in quantity for a 20 per cent change in price, but we only have a 16 per cent change in quantity, so this option does not match the calculations.
Zero point two five would arise if we mistakenly inverted the ratio or made an error in computing either the change in quantity or change in price and their averages. It understates the responsiveness implied by the numbers.

Common Pitfalls:
Students sometimes use the initial quantity and price instead of averages, which gives a slightly different value and is not the arc elasticity as defined in many exam syllabi. Another common mistake is to divide the percentage change in price by the percentage change in quantity, reversing the formula. Confusing own price elasticity with cross elasticity is also common. Focusing on the systematic steps of calculating changes, averages and then the ratio helps avoid these errors.

Final Answer:
The arc cross elasticity of demand between goods X and Y is 0.8, indicating that they are substitute goods with moderate responsiveness.