This is a long post. It is partly in response to some questions from first-year grad students, and so it can be considered as notes for a lecture that might potentially be given to them at some point in the near future. For all that, it isn’t pitched at a high mathematical level. I think you could understand this without having ever done any kind of dynamic optimization work before.
Do savings matter for growth? This is an ill-formed question for several reasons. First, implicitly hiding in that question is the assumption that savings = investment. That doesn’t necessarily have to be true, and the events of the financial crisis in 2008 provide some evidence of this. So let us be more careful and say “Do investment rates matter for growth?”.
What do we mean by growth? You could really be referring to several things. “Do investment rates matter for…
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Utopia, you are standing in it! 
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