Mormonism and polygamy/Purpose of plural marriage/Possible benefits
Possible benefits of plural marriage
Important introductory material on plural marriage available here
Joseph Smith era:
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Possible benefits of plural marriage
To try (prove) His people
Polygamy stood as an Abrahamic test for the saints. The willingness to obey a commandment that was inherently distasteful to the vast majority of the members of the Church allowed members to draw close to the Lord.
To "raise up" righteous seed
Specifically it allowed a relatively few righteous men to become very prolific in a time when the West was very wild and there were many unrighteous men. Children were raised in more households with a strong gospel commitment.
It served to "set apart" his people as a peculiar people to the world
This social isolation that gave the church space to solidify itself into an identity independent of the many denominations from which the membership was derived. Sociologists have discovered that in order for a religion to successfully grow it has to be demanding and it has to experience a moderate amount of tension with its host society. The RLDS Church rejected plural marriage, and perhaps not coincidentally are now small in number and virtually indistinguishable from Protestants.
Polygamy was part of the "restoration of all things"
This was a way for Mormons to feel connected with prophets like Abraham and Jacob. 19th century Mormons gained a greater appreciation for covenants that these forefathers made with God.
Numerous family ties that were created, building a network of associations that strengthened the Church.
Higher growth rates
Arguably polygamy affected higher natural growth rates. Ironically plural wives had fewer children than their monogamous Mormon counterparts. 
Temple recommend holders
Professor Kathryn M. Daynes makes the point that in nineteenth century Utah, more women arranged to hold temple recommends and receive their endowments. That is, female rates of temple-worthiness (or, at least, being willing to take the time and effort to get a recommend and actually go to the temple) were higher than male rates. And, these rates didn't really change much, regardless of how common plural marriage was (and, so, these higher rates cannot have been caused by plural marriage). Thus, women in Utah were in a difficult situation--more of them were willing and able to have temple sealings/eternal marriage than there were men willing and able to do so. Plural marriage changed this dynamic enormously. One temple-worthy man being married would not take that man out of the "potential married partners pool." This allowed more members to have temple marriages, sealings, and the blessings that came with these ordinances.
Widows and orphans
Out on the frontier in 19th century life expectancy was low and women were not as economically independent as they are today. Therefore there were many widows (and orphans coming of age) that needed to be taken care of. Some women who joined the Church abroad immigrated without their husbands, leaving them without male financial support. Furthermore, Brigham Young instituted the most liberal divorce policy in the country so women (but not men!) could get out of unhappy marriages. Kathryn Daynes estimated that 30% of plural marriages came from married-before women. 
Education for women
Church Historian Elder Jensen observed how Mormon polygamy enabled women more freedom to earn college degrees and join national women's rights organizations at the time. 
Polygamy helped integrate foreign immigrants into Mormon society. With the marriage market operating so efficiently, women were highly sought after, and so Utah men had to sometimes marry outside their preferred cultural boundaries. This provided a great way to redistribute the wealth to the immigrants families coming. 
Plural marriages provided a social support network while the husbands were off on missions.
- [note] David R. Keller, "And We Multiplied Exceedingly," FAIR Blog (last accessed 9 May 2008) off-site
- [note] Kathryn Daynes, "Single Men in a Polygamous Society: Male Marriage Patterns in Manti, Utah," Journal of Mormon History 24/1 (Spring 1998): 110. off-site
- [note] Marlin K. Jensen, "Polygamy Then and Now," in LDS Newsroom, 5 May 2008 off-site
- [note] Kathryn Daynes, "Single Men in a Polygamous Society: Male Marriage Patterns in Manti, Utah," Journal of Mormon History 24/1 (Spring 1998): 97. off-site