Final answer:
When the prices of horseback riding lessons and baseball lessons change while keeping Gabriel's income constant, the budget line will shift inward or outward depending on the direction of the price change.
Step-by-step explanation:
When the prices of horseback riding lessons and baseball lessons change while keeping Gabriel's income constant, the budget line will shift inward or outward depending on the direction of the price change. If the price of horseback riding lessons decreases, the budget line will shift outward, allowing Gabriel to afford more of both activities. If the price of baseball lessons decreases, the budget line will shift inward, reducing Gabriel's ability to afford both activities.
The change in prices will also affect the relative prices of horseback riding lessons and baseball lessons. If the price of horseback riding lessons decreases, Gabriel's opportunity cost of horseback riding lessons will decrease, making them relatively cheaper compared to baseball lessons. Conversely, if the price of baseball lessons decreases, Gabriel's opportunity cost of baseball lessons will decrease, making them relatively cheaper compared to horseback riding lessons.
By analyzing the new budget line and Gabriel's indifference curves, we can determine the new optimal point for Gabriel, which represents his preferred bundle of horseback riding lessons and baseball lessons given the new prices.