I would suggest that any discussion of how to alleviate world poverty, or for that matter "income inequality," needs to begin by looking at actual countries out there to see what works and doesn't.  Granted, it's not possible to do a controlled laboratory-type experiment in the field of economic policy.  On the other hand, 190 or so countries all trying somewhat different things can generate a lot of data.

So there we have Haiti, sitting just a few hundred miles southeast of Miami, and it's about as dirt poor as you can get.  Here is a Wikipedia site that aggregates per capita GDP data for all countries of the world for 2013 from the IMF, the World Bank, and the CIA.  Note that, particularly for smaller and poorer countries, the numbers can differ substantially from one source to another.  But Haiti is right near the bottom no matter which source you choose:  $1315 according to the IMF; $1703 per the World Bank; and $1300 per the CIA.  (By comparison, the U.S. is at approximately $53,000 in per capita GDP in all three rankings.)  And, what's more, there have been no recent improvements in the Haitian economy, and there are few prospects for improvement in the immediate future.  Surely there is something those Haitians could sell to the United States to make some money!  How is it even possible for them to maintain this level of poverty?

Perhaps, influenced by the terrible post-colonial history of sub-Saharan Africa, you are thinking of a hypothesis that Haiti is poor because all black-dominated countries are poor.  Well, as Einstein famously said (actually, this may be apocryphal, but the quote is widely attributed to him), "a single experiment can prove me wrong."  In the case of the hypothesis that all black-dominated countries are poor, there are multiple counterexamples that prove it wrong.  Number one is Bermuda (55% black), which is at the very top of the income distribution of countries, ranking way ahead of the United States in the CIA ranking (at $86,000) and just behind the U.S. per the World Bank (at $53,030).  At about 85% black population, we have the Bahamas (IMF - $32,036; WB - $23,102; CIA - $32,000) -- this country is right on the tail of places like France, Japan and the UK.  Even among countries that, like Haiti, are well above 90% black in ethnic mix, we have Botswana (IMF - $16,377; World Bank - $15,675; CIA - $16,400) and Barbados ($25,181 - IMF; $15,566 - WB; $25,100 - CIA).  These countries are solidly middle-income, and are considerably wealthier than, for example, Mexico and China.  So no, black-dominated countries are not inherently poor.

As hypothesis number 2, there is the theory that Haiti is poor because it is being exploited by the evil rich capitalist countries.  According to data here from Wikipedia, Haiti's exports in 2012 were $801 million -- around [$80] per person in a country of about 10 million people, which is rather pitiful.  It seems that the rest of the world is just staying away, perhaps because Haiti doesn't have much to sell -- but it's hard to call that "exploitation."  By comparison, close neighbor Barbados, with a population of only about 250,000, had 2012 exports of almost $500 million -- about 25 times Haiti's level on a per capita basis.  Yet it is a middle income country.  Somehow the evil exploiters are failing to keep Barbados down.  My own view would be that trade makes you rich, not the reverse, but I'm just trying to consider all the hypotheses here.

Now, might I suggest hypothesis number 3, which is that endemic corruption and the absence of the rule of law make it nearly impossible to invest in or trade with Haiti.  Let's just look at a few pieces of data:

•   In an International Finance Corporation project in 2012 called Doing Business, Haiti was ranked 187 out of 189 countries surveyed in terms of ease of starting a business.
• How about tourist hotels?  Lots and lots of international capital is available to build them, and most every Caribbean country has used that capital as a source of good jobs for the people.  Trip Advisor here lists some 47 tourist hotels in Haiti.  Does that sound like a lot?  They list 102 in Barbados -- and Barbados has only one-fortieth the population!  So Barbados has more than 80 times the number of tourist hotels per capita.
• According to Atlas Media here, Haiti's main exports, such as they are, are t-shirts, knit sweaters, other garments, and scrap iron.  Notable for all of those is that these are things that do not require an expensive factory to make.  Or to put it another way, nobody but nobody builds an expensive factory in Haiti.  Why not?  The labor is really cheap -- couldn't you make a lot of money?  The simple answer is, you could make a lot of money if the government wouldn't steal it.   QED -- If you make a big investment, nobody trusts the government not to steal it.  The nice term for that is "absence of the rule of law." 
• According to the Center for Global Development here (2012), the so-called "formal sector" of the Haitian economy accounts for only 10% of employment.  That's another amazing statistic, which says that the large majority of all business activity is essentially hiding from the government.
• Transparency International in 2008 ranked Haiti the 4th most corrupt country in the world.

And finally, let us consider foreign aid.  Haiti is perhaps the all-time champ of foreign aid.  If foreign aid worked to raise a country up out of poverty, then Haiti would be a rich country.  The Center for Global Development (no enemies of foreign aid) here has a history of foreign aid in Haiti, including the statistic that by 1970 "foreign assistance was 70% of the Haitian national treasury revenues" and proceeded to increase from there: "aid levels rose to $35.5 million in 1975."  And "by 1991, Haiti received $380 million from abroad."  And so on.  But the aid thing really exploded after the big earthquake in January 2010.  A compilation here has the total of gifts and pledges to Haiti in the aftermath of the earthquake at $9,337.8 million.  That's about the equivalent of a full year's GDP for Haiti!  But the title of this report from Foreign Policy In Focus in January says it all:  Haiti: billions in Aid, Pennies in Progress since Earthquake.  You really need to read the whole thing to see how one government-imposed obstacle after another has kept Haiti from moving forward.  

Barbados?  They basically don't get any foreign aid.

UPDATE August 8:  A reader points out that $800 million of exports for 10 million people is $80 per person, not $800 as in an earlier version.  It has been corrected.