Bechoros, Chapter 4, Mishnah 5. The Mishnah continues to discuss the process and the laws surrounding the expert who examines a blemish on a firstborn animal. The broad rule is that one may not take a fee for assisting in the performance of a mitzvah, and this is the subject of the next Mishnah. While the actual ruling on the firstborn could be considered assisting in a mitzvah, it involves a lot of work: the animal must be laid down and tied up, and this is a great hassle. In principle, therefore, it would have been possible to charge a fee for this service.
However, a clear concern of conflict of interest arises here: an expert who takes a fee for his ruling on a firstborn might say to the animal's owner, "I will make things easier for you and rule that the firstborn has a blemish" - meaning that it is permitted to slaughter and eat it, and there is no longer a need to spend money on feeding it - all in exchange for a sum of money. Consequently, a tendency will develop to maximize profit and become known as someone who always rules leniently and decides that every firstborn has a permanent blemish.
The basic law:
Therefore, the Mishnah establishes that it is forbidden to take a fee for ruling on a blemish of a firstborn, and if he took a fee - his ruling is void, and it cannot be relied upon to slaughter the animal under the assumption that it has a permanent blemish. The concern is regarding a conflict of interest, improper behavior, and systemic corruption.
The exception:
If, in addition to being established as an expert - proficient in the laws and the entire body of knowledge surrounding rulings on a firstborn - he also has certification from a beis din that he is a recognized pious person who is above suspicion and will not be corrupted by payment, then he is permitted to take a fee subject to specific conditions.
The Mishnah brings the famous pious man, Ila, who lived in the city of Yavneh: they permitted him in Yavneh to take a fee for ruling on firstborns, because he was above all suspicion and it was not assumed that he could be corrupted. However, even for him, boundaries were set:
Fixed fee: He must charge the exact same amount whether he ruled that the animal is unblemished or he ruled that it has a blemish, and even if he ruled that it is a passing blemish, meaning a temporary blemish. He has no incentive to lean toward one side or the other, and he is not permitted to raise his fee.
The amount of the fee: Four issars for a small animal, like a goat or a sheep, and six for a large animal, like a cow. The reason is simple: laying down a cow involves more hassle than laying down and tying up a sheep, and therefore the fee is slightly higher.
One payment per animal: The Gemara says explicitly that he only takes a fee once per animal. For if he were to take a fee for a second examination as well, he would have an interest to say, "This blemish looks to me like a passing blemish, come back to me in another week and I will examine it again," and thus he would collect an additional amount. Therefore, he takes one fee per animal, even if he is required to examine it a hundred times.
This is the language of the Mishnah: "Notel arba issaros b'vehemah dakah v'sheish b'gassah, bein tamim bein ba'al mum" - whether he ruled that the animal is unblemished and cannot be slaughtered, or whether he ruled that it has a permanent blemish and is permitted to be slaughtered.
The Halachah:
The halachah is decided according to this Mishnah, insofar as the law of firstborns is applicable nowadays. The Bartenura quotes the Rambam, that this law also applies to a shochet who rules on the kashrus of an animal: a shochet is not permitted to take a higher fee for ruling that the animal is kosher as opposed to ruling that it is a treifah. Because if he would receive payment only when he rules "kosher," and would not receive payment when he rules "treif," he would have an interest to declare kosher even that which is not truly kosher. Therefore, shochtim who rule on animals must receive a fixed amount, whether they said "kosher" or whether they said "treif."
In summary: In this Mishnah, we learned that due to the concern of a conflict of interest, one does not take a fee for ruling on a blemish of a firstborn, and if one does take a fee - his ruling is void. The exception is an expert who was also established as a recognized pious person by a beis din, such as Ila of Yavneh, who was permitted to take a fixed fee - four issars for a small animal and six for a large animal - whether he ruled it was unblemished or whether he ruled it had a blemish, and only one time per animal. This principle is also applied to a shochet who rules on the kashrus of an animal, that he must take a fixed amount whether he declared it kosher or whether he declared it a treifah.