Gemara

R. Yonason learns Shelichus for Kodshim from "each man will take Seh l'Veis Avos (a lamb for his household)".

Question: We need the verse for R. Yitzchak's law, that a man can acquire on behalf of others, but a minor cannot!

Berachos 20b (Beraisa): A tradition from Sinai teaches that a son may bless for (and exempt) his father, and a wife may bless for her husband.

Inference (Rava): Women are obligated mid'Oraisa, therefore they can exempt men. If they were obligated only mid'Rabanan, they could not exempt men!

Objection: Surely, a minor is exempt mid'Oraisa, yet a son can bless for his father! We must say that the man's obligation to say Birkas ha'Mazon is only mid'Rabanan (he did not eat to satiation. There is no proof from the Beraisa.)

Megilah 19b (Mishnah): A minor may not read the Megilah (to exempt adults);

R. Yehudah permits.

Rishonim

Rambam (Hilchos Berachos 5:15,16): If Shimon and Levi ate and Levi does not know how to bless Birkas ha'Mazon, Shimon blesses out loud and Levi answers Amen. Similarly, a son may bless for his father if the father was not satiated, and he is obligated only mid'Rabanan. Therefore, a minor can be Motzi him.

Rebuttal (Ra'avad): This is like Rav Avira. We hold that the Torah obligated Birkas ha'Mazon for a k'Zayis or a k'Beitzah, even if he was not satiated. One who was not satiated may be Motzi others who were satiated!

Kesef Mishneh: The Ra'avad learns from Berachos 48a, which says that one can be Motzi others if he ate a k'Zayis of grain. It did not specify that they were not satiated! Rather, one who eats even a k'Zayis is obligated mid'Oraisa. One is Yotzei through his son when he repeats the words. The Rambam rules like Rav Avira, for no one disagrees. He explains Berachos 48a to teach that one cannot be Motzi others, even if they ate a Shi'ur mid'Rabanan, unless he ate a k'Zayis of grain. One is Yotzei through listening to his son and answering Amen. In a Teshuvah, the Rambam said that we follow what is explicit (a minor cannot be Motzi a man who was satiated), since there are ways to explain 'one who ate a k'Zayis can be Motzi.' Perhaps the Rambam holds like Rashi, Tosfos and the Rosh, who say that letter of the law, due to Arvus (mutual responsibility of Yisre'elim for each other) a man can be Motzi others even if he did not eat. Chachamim enacted not to do so for Berachos for food, unless he himself eats. Women and children have no Arvus for others, so they cannot be Motzi one who is liable mid'Oraisa. However, the Teshuvah connotes that even a man cannot be Motzi a man with a greater obligation than himself.

Teshuvas Rosh (4:21): A minor cannot exempt others in Birkos Kri'as Shema. Even if he reached the age of Chinuch, he is obligated only for Chinuch.

Question (Tosfos Pesachim 88a): "Seh l'Veis Avos" teaches that a man appoints his minor children on Korban Pesach. One opinion says that this is mid'Rabanan. If so, why may we feed a minor? Mid'Oraisa, he was not counted on the Seh!

Answer #1 (Tosfos): It is a Mitzvah of Chinuch for him to eat, so it is permitted.

Answer #2 (Ran Nedarim 36a DH Omar R. Zeira): The Torah forbids only adults, who could be appointed on a Pesach, to eat without appointment.

Minchas Chinuch (6 DH v'Noheges): If a minor was appointed on a Pesach and became Bar Mitzvah on Pesach night, according to Tosfos he may not eat, for now he has no Mitzvah of Chinuch. Perhaps the Ran permits, for he was unable to be appointed. Or, perhaps he needs a proper appointment.

Tosfos (Eruvin 96b DH Dilma): Even though a blind person is exempt from Hagadah, he is obligated mid'Rabanan. Therefore, Rav Yosef was able to exempt his household. However, perhaps even one for whom it is Reshus may exempt others for a Mitzvah mid'Rabanan. We find that a minor can bless Birkas ha'Mazon for his father, and have one of the seven Aliyos on Shabbos, and exempt others in Kri'as ha'Megilah (according to R. Yehudah).

Question (Tosfos Berachos 16a DH v'R. Yehudah): If the minor reached the age of Chinuch, all should agree that he can be Motzi adults, like we find regarding Birkas ha'Mazon (when the adult is obligated only mid'Rabanan)! If the minor did not reach the age of Chinuch, all should agree that he cannot be Motzi adults!

Answer #1 (Tosfos): Chachamim hold that he can be Motzi adults only when his Chiyuv is mid'Rabanan for only one reason, i.e. he is a minor. Megilah is mid'Rabanan even for adults, therefore there are two reasons why it is mid'Rabanan for a minor. Therefore, he cannot exempt adults.

Answer #2 (Ramban Kidushin 31a DH va'Ani): The minor is not obligated even mid'Rabanan. His father has a Mitzvah to be Mechanech him.

Milchamos Hash-m (Berachos 20b): The Yerushalmi says that a son can bless for his father, i.e. the father repeats the words after him. The primary text of the Bavli did not ask about a son. It understands that we discuss an adult son.

Turei Even (Megilah 24a DH Eino): Even though a minor is not obligated even mid'Rabanan (the Mitzvah of Chinuch is upon his father), R. Yehudah holds that Reshus (an optional Mitzvah) can exempt one who is obligated.

Poskim

Shulchan Aruch (OC 186:2): A minor is obligated mid'Rabanan (to say Birkas ha'Mazon), to train him in Mitzvos. We say that a son may bless for his father, i.e. when the father did not eat to satiation, and is obligated only mid'Rabanan.

Mishnah Berurah (5,6): This is only when the father does not know how to bless. A minor cannot exempt a woman, lest she is obligated mid'Oraisa.

Mishnah Berurah (7): If the minor was not satiated, some say that he cannot be Motzi even an adult obligated mid'Rabanan. One should be stringent.

Kaf ha'Chayim (10): Some say that if a minor has no father, Chinuch applies to Beis Din. Others disagree. One should not be Yotzei from such a minor.

R. Akiva Eiger (1): If a child ate to satiation during the day, and became Bar Mitzvah at night before the food was digested, perhaps he must bless Birkas ha'Mazon mid'Oraisa. Or, perhaps he is exempt, for when he ate, the Torah did not obligate him to bless. If he is liable, even if he blessed during the day, this does not fulfill his mid'Oraisa obligation later. Chachmas Adam (153:1) asked about an Onen who ate and buried his Mes before the food was digested. There he was always obligated, just Aninus overrode other Mitzvos.

Magen Avraham (267:1): The Mordechai (Megilah 798) says that the opinion that Tosefes Shabbos is mid'Rabanan agree that one is Yotzei with Kidush that he said before dark, since the Chiyuv mid'Oraisa will come later. Why is this different than a minor, who cannot exempt an adult obligated mid'Oraisa in Birkas ha'Mazon, even though later the child will be obligated mid'Oraisa?

Note: Perhaps Birkas ha'Mazon is different, for the Chiyuv does not come by itself (he must eat), except for the case that R. Akiva Eiger asked about.

R. Akiva Eiger (267:2): (If one slaughtered for a minor on Pesach Rishon, and he became Bar Mitzvah before Pesach Sheni, he is exempt from Pesach Sheni.) The Kesef Mishnah (Hilchos Korban Pesach 5:7, citing Ri Korkus, explains this is because a verse allows including a minor on Korban Pesach. This) is unlike the Mordechai. (According to the Mordechai, in any case the minor was Yotzei, for he will be obligated later!)

Note: Perhaps Pesach Sheni is different than Kidush, for it is a new Mitzvah.

Or Some'ach (Korban Pesach 5:7 b'Sof): No one requires a minor who becomes Bar Mitzvah during the festival to delay bringing Olas Re'iyah and Chagigah until he matures. Rather, he was Yotzei with what he brought with Klal Yisrael when he was a minor, because he was obligated due to Chinuch.