Growth and the natural rate:

In std models, r* increases with trend growth, but the reason for that is not as simple as it seems: it actually comes from the demand side.

Here is a short explanation (without
referring to the Euler equation)

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A higher growth rate means you need more investment to keep the capital stock on its growth trend.

That’s more demand ex ante, so the real interest rate must increase to offset it.

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The offset comes from a lower capital intensity of production (& maybe also a higher saving rate, but that depends on the model).

Yes, that’s right: higher trend growth means lower capital intensity of production.

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In other words, investment becomes more extensive, less intensive.

So it’s not higher productivity gains or innovations per se that increase the natural rate, it’s their effect on investment demand.

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