The crux of this question involves the pairing of subjects that appear near the beginning of the sentence (in the original example, gasoline and prices). I don't think that any general rule satisfactorily disposes of all such combinations.
Of the poster's two rewritten versions of the original example, "The price of gasoline is going up" sounds natural, but "The gas[oline] price is going up" does not.
On the other hand, consider the subject pair relief and efforts in the sentence "Relief efforts have been successful." Rewriting this statement to match the models provided in the original post, I get "The relief effort has been successful" (which sounds natural) and "The effort for [or toward or in or of] relief has been successful" (which sounds awkward).
Finally, consider the subject pair budget and deficit, in the sentence "Budget deficits are increasing." Here, both "The budget deficit is increasing" and "The deficit in the budget is increasing" sound natural and have virtually identical meanings. But to a U.S. reader, both versions have a more specific meaning than the original plural example does because both implicitly focus on the federal government's budget deficit rather than on budget deficits in general.
To sum up, what sounds right and makes sense in a particular instance depends on so many complicating factors that no usefully predictive general rule governing where and how to use the is possible.