In the Republic, Socrates argues that justice pays.

Justice is necessary for the ability to make choices that result in the best life and most happiness. The parts of a just soul are in "harmony." Reason rules, spirit is reason's ally, and appetite is held in line. In this way, the three parts of a just soul are organized so that action is in accordance with the knowledge of the good. This knowledge belongs to reason. In this part of the soul, there is the knowledge that existence in the body is temporary, that the body and its needs are distractions, and that the good most of all resides in "contemplation" (θεωρία).

Crater Plato
Crater Plato, north of the Mare Imbrium ("Sea of Rains")

Injustice is never more profitable than Injustice

Glaucon ask Socrates to show that just life is better, and he sets a very high bar. He asks Socrates to show that a just human being who suffers what are popularly understood as great misfortunes is still better off, even if the comparison is to an unjust human being who suffers none of these so-called misfortunes but instead is showered in what are popularly understood as the good things in life.

"We must take away his reputation, for a reputation for justice would bring him honor and rewards, so that it wouldn't be clear whether he is just for sake of justice itself or for those honors and rewards. We must strip him of everything except justice and make his situation the opposite of the unjust person's. Though he does no injustice, he must have the greatest reputation for it, so that his justice may be tested. ... [T]hose who praise injustice at the expense of justice will say that a just person in such circumstances is worse because he will be whipped, stretched on a rack, chained, blinded with fire, and, at the end, when he has suffered every kind of ill, he'll be impaled..." (Republic II.361b-362a).

"Socrates, I think that from this point on our inquiry looks ridiculous now that justice and injustice have been shown to be as we have described them. Even if one has every kind of food and drink, lots of money, and every sort of power to rule, life is thought not to be lived if the nature of the body is ruined. So if someone can do whatever he wishes, except what will free him from vice and injustice and make him have justice and virtue (δικαιοσύνην δὲ καὶ ἀρετὴν κτήσεται), how can it be worth living if the nature by which we live is ruined and in turmoil (Republic IV.445a-b)?

Despite Glaucon's suggestion that further inquiry would be "ridiculous," it is not at all clear that Socrates has shown that the just life is better. Part of the reason is that Socrates has not explained in any real detail what happiness is. He thinks that "contemplation" (θεωρία) makes a substantial contribution. This much seems clear, but Socrates never explains how this contribution is weighed in comparison with other things traditionally thought to contribute to or detract from happiness.

Nor is it clear that it follows that injustice is never more profitable than justice. If the least happy just life is happier than the happiest unjust life, then injustice is not more profitable for someone who has always lived a just life. His will have a less happy life if he resorts to injustice. It is less clear, though, that it follows that a person who is living an unjust life must have a more happy life if he resorts to justice and thus henceforth lives a just life. This depends on how much various things contribute to and detract from the happiness of a life.

The truest pleasures are the ones proper to the parts

Socrates argues that the just life is the most pleasurable (and presumably for this reason happier than the unjust life).

Socrates says that there is pleasure for each part of the soul, that the objects of desire approved by reason are most true, and that the life of reason is the "sweetest." A few lines later he provides more detail. He argues that a part of the soul will get the "truest" pleasure when it gets what is "proper" to it, that when the part of the soul with reason dominates, all parts of the soul get their "truest" pleasures because the knowledge in the part with reason directs action, and that when the appetitive or spirited parts dominate, the parts of the soul do not get their "truest" pleasures.

"The three parts of the soul have, it appears to me, three kinds of pleasure, one peculiar to each, and similarly three appetites and controls" (Republic IX.580d).

  "But since the tests are experience and wisdom and reason (ἐμπειρίᾳ καὶ φρονήσει καὶ λόγῳ), what follows?
  Of necessity, that the things approved by the lover of wisdom and the lover of reason (ὁ φιλόσοφός τε καὶ ὁ φιλόλογος) are most true (ἀληθέστατα).
  Then of the three kinds of pleasure, the pleasure of that part of the soul whereby we learn is the sweetest, and the life of the man in whom that part dominates is the most pleasurable (ἡ τούτου τοῦ μέρους τῆς ψυχῆς ᾧ μανθάνομεν ἡδίστη ἂν εἴη, καὶ ἐν ᾧ ἡμῶν τοῦτο ἄρχει, ὁ τούτου βίος ἥδιστος;)?
  How coud it be otherwise" (Republic IX.582e-583a)?

"[T]o be filled with what befits (προσηκόντων) nature is pleasure..." (Republic IX.585d).

"Let us confidently assert that those desires of even the profit-loving and honor-loving parts of the soul, which follow knowledge and reason (ἐπιστήμῃ καὶ λόγῳ) and pursue with their help those pleasures which intelligence (φρόνιμον) prescribes, will attain the truest (ἀληθεστάτας) pleasures possible for them, since they are following the truth. These pleasures are proper (οἰκείας) to them, if that which is best for each thing may be said to be most proper to it. So if the whole soul follows the wisdom-loving part (φιλοσόφῳ) and there is no internal dissension, then each part will be able to fulfill its own task and be just in other respects, and also each will reap its own pleasures, the best and the truest as far as possible. And when one of the other two [parts of the soul] gets the mastery the result for it is that it does not find its own proper pleasure and constrains the others to pursue an alien pleasure and not the true" (Republic IX.586d-587a).

Virtue allows for Contemplation

The life in which the soul has virtue is the good life because virtue frees the part of the soul with reason and thus allows for "contemplation" (θεωρία). As Socrates puts the point in his against Thrasymachus, the just life is "more profitable" (λυσιτελέστερον) than the unjust life (Republic I.354a). That is to say, justice "frees the end" more than injustice.

The just individual is wise because of the part of the soul with reason has knowledge and is not confused about what is good and what is bad, brave because the spirited part holds to the declarations of reason in the face of pleasure and pain, moderate because all three parts are controlled, and just because each part is doing its own job. Because the part with reason is wise, it exercises "forethought in behalf of the entire soul" (Republic IV.441e). In this way, the individual arranges things in his life so that as much as possible he engages in "contemplation" (θεωρία).

"The soul is conceived of as preexisting and as just temporarily joined to the body. It thus has two lives and two sets of concerns. Its own concern is to live a life of contemplation of truth. But, joined to the body, it also has to concern itself with the needs of the body. In doing this it easily forgets itself and its own needs, it easily gets confused so as to make the needs of the body its own. To know how to live well is to know how to live in such a way that the soul is free again to clearly see and mind its own business, namely to contemplate the truth. Thus we have an extremely complex inversion of the relative weight of one's theoretical understanding of reality and one's practical knowledge of how to live. It is one's understanding of reality, and the position of the soul in it, that saves the soul by restoring it to the extent that this is possible in this life to its natural state, in which it contemplates the truth. Hence a good life will crucially involve, as part of the way one lives, contemplation of the truth. Practicing the right way to live will also be a means to enable the soul to free itself from the body, to see the truth, and to engage in the contemplation of truth" (Michael Frede, "The Philosopher," 9).

The lovers of wisdom rule as part of a plan

In a just city, the lovers of wisdom are the rulers. How can this be? Wouldn't the rulers be better off abandoning their posts so that they can spend more of their time in the love of wisdom, since "contemplation" is the activity in which happiness consists most of all?

Socrates says that the rulers rule out of "necessity."

"Each ruler will spend much of his time in the love of wisdom (φιλοσοφίᾳ), but, when his turn comes, he labors in politics and for city's sake, not as if he were doing something fine (καλόν), but as a necessity (ἀναγκαῖον)" (Republic VII.540b).

Why do the lovers of wisdom rule? The answer not completely clear, but maybe part of the idea is that ruling is part of a plan to maximize happiness.

The lovers of wisdom realize that a just city requires rulers and that only they are qualified to rule, but maybe the lovers of wisdom also know that each taking his turn in ruling will maximize the time each spends in "contemplation" and hence maximize the satisfaction each takes in his life. In this case, each thinks it is not rational for him to shirk his duty. If he were to not take his turn in ruling, he would increase the probability of spending less time in "contemplation" and hence increase his probability of having a less good life.

"And so the man proposes the penalty of death. Well, then, what shall I propose as an alternative? Clearly that which I deserve, shall I not? And what do I deserve to suffer or to pay, because in my life I did not keep quiet, but neglecting what most men care for--money-making and property, and military offices, and public speaking, and the various offices and plots and parties that come up in the state--and thinking that I was really too honorable to engage in those activities and live, refrained from those things by which I should have been of no use to you or to myself, and devoted myself to conferring upon each citizen individually what I regard as the greatest benefit? For I tried to persuade each of you to care for himself and his own perfection in goodness and wisdom (βέλτιστος καὶ φρονιμώτατος) rather than for any of his belongings, and for the state itself rather than for its interests, and to follow the same method in his care for other things. What, then, does such a man as I deserve? Some good thing, men of Athens, if I must propose something truly in accordance with my deserts; and the good thing should be such as is fitting for me. Now what is fitting for a poor man who is your benefactor, and who needs leisure to exhort you? There is nothing, men of Athens, so fitting as that such a man be given his meals in the prytaneum. That is much more appropriate for me than for any of you who has won a race at the Olympic games with a pair of horses or a four-in-hand. For he makes you seem to be happy (εὐδαίμονας), whereas I make you happy in reality; and he is not at all in need of sustenance, but I am needy. So if I must propose a penalty in accordance with my deserts, I propose maintenance in the prytaneum" (Apology 36b-37a).

Contemplation is the Reward

It is paradoxical to think that the life of "contemplation" (θεωρία) is the good life.

Plato no doubt knew this is paradoxical and thought that the more common view of the good life rests on a confusion about what human beings are and about what is in their interest. The common view more or less identifies human beings with their bodies, but Plato seems to have thought that a human being is a soul and reason that is temporarily in a body.

"[W]hereas Socrates had thought that there was no need to gain theoretical knowledge about the world or reality and that perhaps it was even impossible to do so, since it was not the function of reason to gain such knowledge, both Plato and Aristotle disagreed. They thought that it was crucial not only for a good life, but also for an understanding of how to live well, to have an adequate general understanding for the world. Moreover, though they granted that it was a function of reason to determine the way we live, they, each in their own way, did not think that this was the sole function of reason. Plato rather seems to have thought that guiding us through our embodied life is a function which reason takes on, but that it, left to itself, is concerned to theoretically understand things quite generally" (Michael Frede, "Introduction" in Rationality in Greek Thought, 13).

Perseus Digital Library:
Plato's, Republic
Henry George Liddell, Robert Scott, A Greek-English Lexicon:
λυσιτελής, (λύω + τέλος), lysitelēs, "profitable"