Optimal basket and the maximum utility

Question

Given two goods (where more is better), and I am consuming at a point that maximizes my utility (i.e on the budget line, with an indifference curve tangential to the budget line).

When the prices of the two goods change to an unknown, and yet I consume where my utility is the same maximum utility as before, can I conclude that the price of this new basket is the same as before?

Logically, I think it does. But on the other hand, I'm thinking about possibilities where price of good A increases, and price of good B decreases by the same amount, which leaves me stuck.

Answer 1

Given the circumstances and model you describe, consuming the entire budget will give higher utility than consuming less than the entire budget.

You can therefore assume that the optimal basket will have the same total price as before the price change.

Answer 2

If the price ratio of the new basket is the same as the original basket, it means the budget line pivots about the original consumption bundle. In this case , quantity of goods consumed is same but prices change. Therefore, price of new basket changes.

However, if price ratio of new basket is not the same as before. It means that the budget line changes such that it is still tangential to the original indifference curve. ( there can be many different ways to draw a tangent to a curve )

In this case, it is not possible to tell whether the price of the original basket changes as both quantities and price changes for the new basket.

Just my thoughts :)