Why don't they put soft material in the sleeves of clothes?

There are 2 reasons. 1 its for looks, if the fur is really nice looking which makes it more expensive for the manufacturer the manufacturer will cut back on how much they use of the expensive fur (to save money) so when your walking around town and showing off your jacket to your friends they will be looking on the interior not the inner sleeves. 2 its for movement, you could have the nice fur around the inner coat because it doesn't move around as much but if you had the puffy maybe heavy fur around your arms it would be harder to move your arms around and you. (but my main though is that the manufacturing companies are trying to just cut back on costs of making the jacket)