What does the narrator say to make himself appear to be an authority on this subject?
Swift's narrator portrays himself not necessarily as an expert (indeed, this is why he calls it a "modest proposal,") but as a rational, concerned reformer who has consulted a variety of experts in seeking a solution to the social problems he describes early in the essay. For instance, he is informed by a "knowing American of my acquaintance in London" that a "young healthy child" is a "delicious and wholesome food." He considers one person's "refinement" to his scheme (eating teenage boys as a replacement for venison) but rejects it on the grounds that the young men's flesh is said to be too tough.
This said, he claims that he is "wearied out for many years with offering vain, idle, visionary thoughts, and at length utterly despairing of success." He has sought for years, he implies, to find a rational solution to the problem of poverty among the lower classes in Ireland, and has thus arrived at his "proposal" after much consideration. While he does not claim to be an expert, however, he does try to show, albeit with tongue firmly in cheek, that the solution makes mathematical (i.e. rational) sense. He calculates the amount of money that would be saved by individuals as well as added to the coffers of Great Britain, and suggests that it would spark a thriving trade. So he is describing himself as a rational reformer with a common-sense solution to a serious social problem.