For a simple example, imagine a relatively inefficient combined cycle is competing with a coal generator. At $4/MMBtu, the natural gas combined cycle would have a dispatch cost of around $40/MWh. Say the coal generator has a dispatch cost of $43/MWh. In order to decrease natural gas demand, natural gas prices would have to increase until the dispatch cost for the coal generator is lower. In this case, natural gas prices would have to increase more than $0.30/MMBtu to drive dispatch costs higher than $43/MWh.
If natural gas demand needs to increase (due to relative short term oversupply), the opposite happens, with natural gas prices going down until natural gas replaces coal through power market competition. Note that coal prices will also change as power sector coal demand decreases or increases; this is the major reason why coal prices approximately track natural gas prices.
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