What is the ruling on someone who vowed to slaughter a calf for a specific amount, then the price of the calf increased? Should he slaughter a calf worth the original monetary value, or should he slaughter a calf of a higher value?

The phrasing you used does not constitute a valid vow (Nathr), because a vow is only binding if it is expressed in words that signify commitment. Even if you had used words that signify commitment but added "in sha' Allah" (God willing), you would not be obligated to fulfill the vow. Therefore, nothing is binding upon you. You have the choice: if you wish, you may slaughter something from what was mentioned, and whatever is more beneficial for the poor is preferable. However, if you do not slaughter anything, there is no sin upon you.