Assume that demand for product A can be expressed as QA = 500 ¨C 5PA + 3PB and demand for
product B can be expressed as QB = 300 ¨C 2PB + PA. Currently, market prices and quantities for
these goods are PA, = 5, PB = 2, QA = 481, and QB = 301.
a. Suppose the price of product B increases to 3. What happens to the quantity demanded of
both products?
b. Calculate the arc cross©\elasticity between product A and product B using prices for product
B of 2 and 3.
c. Are these goods substitutes or complements?
To answer these questions, we need to substitute the given values into the demand equations and compare the resulting quantities. Let's go through each question step by step.
a. To find out what happens to the quantity demanded of both products when the price of product B increases to 3, we need to substitute PB = 3 into the demand equations and calculate the new quantities.
For product A:
QA = 500 - 5PA + 3PB
= 500 - 5(5) + 3(3)
= 500 - 25 + 9
= 484
For product B:
QB = 300 - 2PB + PA
= 300 - 2(3) + 5
= 300 - 6 + 5
= 299
Therefore, the quantity demanded of product A decreases to 484 and the quantity demanded of product B decreases to 299 when the price of product B increases to 3.
b. To calculate the arc cross-elasticity between product A and product B, we need to use the formula:
Arc Cross-Elasticity = ((QA2 - QA1) / ((QA1 + QA2)/2)) / ((PB2 - PB1) / ((PB1 + PB2)/2))
where QA1 and QA2 are the quantities demanded of product A, and PB1 and PB2 are the prices of product B.
Given the prices for product B as 2 and 3, and the initial and final quantities demanded for product A (481 and 484), we can plug these values into the formula:
Arc Cross-Elasticity = ((484 - 481) / ((481 + 484)/2)) / ((3 - 2) / ((2 + 3)/2))
= (3 / (965/2)) / (1 / (5/2))
= (3 / 482.5) / (1 / 2.5)
= 0.0124
Therefore, the arc cross-elasticity between product A and product B, using prices for product B of 2 and 3, is 0.0124.
c. To determine if these goods are substitutes or complements, we can look at the sign of the cross-elasticity. A positive cross-elasticity indicates that the goods are substitutes, while a negative cross-elasticity indicates that the goods are complements.
From the previous calculation, we found that the arc cross-elasticity is positive (0.0124). Therefore, product A and product B are substitutes, meaning that an increase in the price of one product leads to an increase in the quantity demanded of the other product.