When inequalities are minimized in any situation, the group of individuals is more likely to behave in a cooperative manner. Generally speaking, people that feel equal to one another in an office environment or even on a task force that requires manual labor, are more likely to freely share ideas, address concerns, and alert the rest of the group to issues that are present or could be faced in the future. While leadership is a must for any group, when all members of a party feel equal and inequality is shrunk to nearly nothing, there are few, if any, barriers to progress of thought, action, pro-action, and progress which is sure to strengthen an economy already struggling with a wage and income gap as egregious as our own.
Yes, history and current events tells us that as inequality shrinks an economy experiences growth. One example of this cause and effect can be seen in the progress of LGBT Rights within the United States. In numerous studies, such as one conducted by the Williams Institute, the benefits of same-sex marriage would be more than substantial for local, state and national economies. Within the coming years, hundreds of millions of dollars will be spend on flowers, gifts, food, apparel and the many other products and services associated with same-sex marriage ceremonies.
It's very unlikely that shrinking inequality will help grow the economy. The simple fact is that there has to be a firm source for the funding that will be used to shrink the fiscal gap. This will most likely come from an increase in taxes on the upper and upper/middle classes, so right off the bat you will damage the economy because those that are currently stimulating it will be unable to continue to spend at their current rate. In addition, there is no guarantee that those who are currently low income will suddenly start stimulating the economy with meaningful purchases just because they now have more money. In fact, there would most likely be an initial urge to spend frivolously due to their new-found wealth, which would ultimately land them in a worse situation that they were at before trying to force equality.
No, shrinking inequality will not help to grow the economy, because people have to work for the things they have. If people are guaranteed the same income whether or not they work, they will not work as hard as they could. When people know they keep what they make, they have more incentive to work hard.
Reducing the gap between rich and poor at first thought seems like a very good idea. The reality is, shrinking or taking away incentives might be the worst thing to happen. Think about those who start companies or work extremely hard for the life that they want. The possible future of being in the 1% is a strong driving factor among those who want to succeed. Conversely, those who do not work as hard would still not be too far behind the top economic performers, which may allow them a comfort zone where they believe they do not have to do much but still can remain above the poverty line.