I am going to assume that when women want to marry they only care about how much money a man makes. This may or may not be true.
Some people say polygamy can be good for women because just about all women want to marry a man with lots of money and have babies with him. In a society where there is a huge gap between rich and poor, e.g. Australia under John Howard, this puts a lot of pressure in females to marry the rich men. The females go to huge lengths to compete with other females to marry the rich men before it's too late. If they don't get the rich man by the time they're 30, chances are they will be too old and will have to settle for the poorer guy, the Aussie battler. Why would any woman want to marry a man who both votes for John Howard and complains about him at the same time?
If polygamy were legalized, then rich men would naturally want hundreds of wives each. Women then have a good chance of fulfilling their dream of marrying rich men. The losers from polygamy would be poor men.
Monogamy then can be seen as protectionism by low-income males.
One could argue however that a female would not like to be with a man if she has to share him with hundreds of other females. Suppose a rich man is willing and able to spend $1000 on his wives per week while a poor man is willing and able to spend only $100 on wife per week. If the rich man already has 10 wives and a female is thinking about being his 11th wife, then assuming the $1000 is shared among all wives equally, each wife would get $90.90 (1000/11). That female would have been better off getting the $100 from the poor guy.
Clearly then the degree of polygamy in society will re-equilibrate itself to changes in social inequality. As inequality increases, the female demand for polygamy will increase. As inequality decreases, the female demand for polygamy will decrease.
Suppose a country only allowed monogamy. If all of a sudden there were an inequality shock and all the wealth in the country were put into the hands of one man, then one woman is lucky while every other woman is worse off because she wants to marry that rich man but cannot. We can see monogamy then as a welfare loss for society and women during times of high inequality.
What I would like to do is to collect data on countries around the world where polygamy is legalized and then, for each country, get data on the average number of wives each man has. Then I will see whether there is a statistical relationship between the average number of wives men in that country have and the Gini coefficients.
CRITICISMS OF POLYGAMY
One of the criticisms of polygamy is that poor or ugly men don't get any women to marry because all the alpha males have taken away the women. As I said before, monogamy then is a protectionist policy that protects poor males. If they were not protected then they may go crazy and do silly things. Tyler Cowen in The Economics of Polygamy talks about the problem of "sexually frustrated males who are led to revolt, thus destroying social order."
Economists argue that the wretchedness of losers is no excuse for protectionism. Instead, they recommend that it's better to allow free trade and then set up a redistributive system to transfer wealth from winners to losers. For example, some people believe that capitalism as opposed to communism is evil because not everyone is talented enough to make money. Some people are going to be poor because they are not skilled enough. Under communism it is each according to his needs, each according to his abilities, etc. However, economists recommend there be free trade so that wealth can be generated in the first place. Once the wealth is generated, then taxes take away money from the rich and gives it to the poor in the form of welfare. This is arguably better than the communist system because the communist system usually doesn't work and no wealth is generated, meaning the poor can't eat anyway.
Applying the same concept to the marriage market, what should be done then is that money is taken away from rich and given to poor. Then women will not be so willing to marry the richest men as much.
Even if each man has two wives, it is clear that many men will lose out. But this is not all horrible because if there is equality for males and females then not only will men be able to have multiple wives but women will be able to have multiple husbands. If we assume that women want money then it follows that women will want to collect multiple husbands as well.