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In Cities: Skylines, I notice that my income can fluctuate wildly over the course of a few weeks with no inputs from me. I'll be sitting at, say +4000, and a few short weeks later I'm at -2000. I added no new city services and there is no significant population change which would account for the big loss in income. A couple months after that and I could be at +7000. I would expect any changes resulting from population change to be gradual (barring a mass exodus or disaster) and occur over several months, if not longer.

What causes these big changes in the weekly income?

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    I don't have a definitive answer, but I've noticed the day-night cycle is roughly one month each and during the "day" month, my commercial income is 2-2.5x higher than during the "night" month. This is the base game so I don't have the specialized commercial districts from After Dark. – Troyen Sep 28 '18 at 8:34
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This most likely is caused by the day/night cycle. During night, commercial income is reduced (except for leisure specialization from the After Dark DLC). The "day" lasts a few weeks, so there will be noticeable swings in budget. Industrial income should also suffer, because the commercial zones do not buy the goods.

This can also be seen under Statistics => City Budget, where the average budget looks like it is modulated by a sinusoidal.

An additional source of variation could be industry from the Industries DLC, where completed production / import can lead to noticeable changes. These should not be periodic, however.

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This could be because of your industries. They don't always produce the same amount of profit consistently because of the change in supply and demand which causes dips. I've noticed this in my own game-play.

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