If the pope is infallible, then why doesn’t he elect his own successor himself? Wouldn’t that be the best solution?
Answer: Please read attentively what is written above in the text introducing the Theme “The Church”, part III on specific Catholic teaching regarding the Church and its teaching structures. In the “Catechism of the Catholic Church” (2005 Libreria Editrice Vaticana, Città del Vaticano, or Pattloch Verlag, Munich), we find the following statements relevant to answering these questions:
[What is the Pope’s ministry?]
882. The Pope, Bishop of Rome and Peter's successor, is the perpetual and visible source and foundation of the unity both of the bishops and of the whole company of the faithful. For the Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered.
[What are the duties of the college of bishops?]
883. The college or body of bishops has no authority unless united with the Roman Pontiff, Peter's successor, as its head. As such, this college has supreme and full authority over the universal Church; but this power cannot be exercised without the agreement of the Roman Pontiff.
[The teaching office]
888. Bishops, with priests as co-workers, have as their first task to preach the Gospel of God to all men, in keeping with the Lord's command. They are heralds of faith, who draw new disciples to Christ; they are authentic teachers of the apostolic faith endowed with the authority of Christ.
889. In order to preserve the Church in the purity of the faith handed on by the apostles, Christ who is the Truth willed to confer on her a share in his own infallibility. By a supernatural sense of faith the People of God, under the guidance of the Church's living Magisterium, unfailingly adheres to this faith.
[Nature and role of the Magisterium]
890. The mission of the Magisterium is linked to the definitve nature of the covenant established by God with his people in Christ. It is the Magisterium’s task to preserve God’s people from deviations and defections and to guarantee them the objective possibility of professing the true faith without erre. Thus the pastoral duty of the Magsterium is aimed at seeing to it that the People of God abides in the truth that liberates. To fulfil this service, Christ endowed the Church’s shepherds with the charism of infallibility in matters of faith and morals. The exercise of this charism takes several forms:
[When is the Magisterium infallible?]
891. The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith – he proclaims by a definitive act a doctrine pertaining to faith or morals. The infallibility promised to the Church is also present in the body of bishops when, together with Peter's successor, they exercise the supreme Magisterium, above all in an Ecumenical Council. When the Church through its supreme Magisterium proposes a doctrine for belief as being divinely revealed, and as the teaching of Christ, the definitions must be adhered to with the obedience of faith. This infallibility extends as far as the deposit of divine Revelation itself (Catechism of the Catholic Church).”
The texts illuminate what the Church understands by the infallibility of the papal Magisterium and what not. For the Roman Pontiff to determine his successor would contravene the applicable rules under cannon law, and it would be by no means an infallible decision.