Why goods prices are normalized in each period in growth models? As it is known, in simple economic growth models such as the Solow or the Ramsey model, all goods prices are set equal to 1 in each period. 

Also, as we known from general equilibrium theory, homogeneity of degree zero of demand implies that only relative prices matter. Therefore, we can always set one price equal to 1 (or do other types of "normalization"). 

However, in economic growth models (which are general equilibrium models) we have an infinite number of good prices and all are set equal to 1. Why do we always do that? Doesn't it affect the analysis of the model?

I have search for an explanation, but I've been unable to find one. I have a hypothesis, though. Maybe we can normalize goods prices in each period because we assume the economy is in equilibrium at each instant. Then, by the same argument we use in static models, we can normalize goods price in each period. Is this explanation true? Or there is some other reason? And, if it is true, how do models that feature inflation are made?
In fact, in growth models, people usually assume that there exists a one generic good for which the price is equal to one and not an infinite number of goods. Then, it is possible to set the price equal to one. 

You can look at Acemoglu's Introduction to Modern Economic Growth textbook. In this book, you can see in a decentralized economy (where there are different sectors, final good producer, intermediate good producer etc.) you can see that the price of the final good is normalized to one and there exists a price $p$ for the intermediate good. 

Therefore, to sum up, in simple canonical solow model with only one sector, there exists a one generic good and for more complicated growth models (endogenous growth models), where there are different sectors, there are different goods, then this implies different prices.

Hope that is a clear answer for you !