All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
In principle, a person must perform Hajj by himself and not appoint another person to perform Hajj on his behalf. However, if a person is unable to perform Hajj by himself, then it becomes permissible for him to appoint someone else to perform Hajj on his behalf as the Prophet said to the woman who told him that her father cannot sit firmly on the riding animal due to his old age and she asked him whether she is permitted to perform Hajj on his behalf, and he () replied: “Yes.” [Al-Bukhari]
The criterion of inability is when there is an unbearable hardship due to a chronic disease or old age and the like. An-Nawawi when defining the person who is physically unable to perform Hajj and who is permitted to appoint someone to perform Hajj on his behalf, said: “If a person has a disease and it is expected that he will be cured from it, then he is not considered physically unable to perform Hajj and it is not permissible for him to appoint someone else to perform Hajj on his behalf in his life. There is no difference of opinion among the scholars on this issue... However, if he is unable to perform Hajj by himself in a way that is not expected to end due to old age, infirmity, a sickness that is not expected to be cured, or that he is too old to be able to sit firmly on the riding animal except with great difficulty, …then this is the person who is physically unable to perform Hajj.”
Therefore, if a person is unable to bend his leg but he is able to perform Hajj by himself even with a bearable hardship, then it is not sufficient for him to appoint someone else to perform Hajj on his behalf, and he must perform Hajj by himself. However, if he knows that Hajj is difficult for him due to that disability, and its difficulty is usually unbearable, and it is not expected for him to be cured of this disability, then in this case, it is not obligatory for him to perform Hajj by himself. In case he has enough money for Hajj, then he is obligated to appoint someone else to perform Hajj on his behalf.”
Nonetheless, if one does not have enough money to be considered financially able to perform Hajj and his son offered to pay him the expenses of Hajj does Hajj become obligatory on him? There are two different opinions amomg the scholars regarding this: the first opinion is that it becomes obligatory on him, and this is the view of the Hanbali School of jurisprudence, and the second opinion is that it does not become obligatory on him and this is the view of the Shaafi’i School of jurisprudence.
Hence, if the old person whom you referred to in the question has enough money to perform Hajj with, then it becomes an obligation on him to perform Hajj whether he gathered it from retirement pension or from the money given to him by his son or by someone else.
Allaah Knows best.