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Really, neither will run appreciably slower than the other.
In the strictest sense, the clock in the upstairs bedroom is moving a few centimeters per year faster than the clock downstairs in the kitchen, because it is a few feet further from the Earth's center than the clock in the kitchen; therefore, according to the theory of relativity, time will pass more slowly in its location than in the kitchen. However, this difference in times is less than one nanosecond, this is as testable as the question "If a tree falls in the forest, and no one is there to hear it, does it still make a sound?"
There are also reasons for the clock in the Kitchen to run slower than the one in the upstairs bedroom. Grease smoke might condense on its parts, interfering with its function (this happened to a cuckoo clock in my grandparent's house). And as I said, the difference due to relativity (if it exists) is immeasurable. I've even tested it myself. Two identical alarm clocks, one in an upstairs bedroom, one in a basement bedroom, after 5 years the clocks were still in perfect synch.
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