If responding to an inner need of fulfilling your tasks -- which derive from your very way of being- means happiness for each and every person living from the city, then it is easy to understand how personal happiness and justice contribute to social justice. Justice is the realization of the potential of all the individuals. This realization is at its turn a manifestation of nature which is a reflection of the idea of good (here we must underline the importance of the "ideas" in Plato's Republic).

The city can very well be interpreted as a metaphor for the individual. If for the individual happiness means the harmony between reason and feeling and the manifestation of his own nature, then it is easy to see the analogy with the just people living in the just city obeying a principle of justice because that principle is their very own nature....

In both cases reason plays a fundamental role and Plato actually underlines that it is better for people to have their actions guided by this particular part of the soul.
Bibliography:

Anderson, Doug. The difficulties of nature: the good city in Plato's Republic. Retrieved October 21, 2009 from http://www.geocities.com/SunsetStrip/Alley/7028/plato1.htm

Bloom, Allan. The Republic of Plato. Basic Books.

Haslanger, Sally. Plato on happiness: the Republic's answer to Thrasymachus. Retrieved October 21, 2009 from http://ocw.mit.edu/NR/rdonlyres/Linguistics-and-Philosophy/24-200Fall-2004/E9798F70-1826-4469-87A8-8C42AABB6A36/0/repsum.pdf

LaVange, Don. Plato's treatment of the arts and the artist in the Just society. 2003. Retrieved October 21, 2009 from http://donlavange.livejournal.com/1863.html