How does Swift establish himself as an authority on the subject in "A Modest Proposal"?

One must first, and this is important, distinguish between Swift, who wrote the essay, and the clueless narrator he invents who comes up with the "modest" proposal of killing and eating the offspring of the poor in Ireland. Swift wants the reader to understand that his narrator is ridiculous—and dangerous—because he lacks a moral compass. The narrator treats the Irish poor not as people but as things.
Nevertheless, Swift's narrator establishes (or attempts to establish) his authority on the basis of how thoroughly he has researched his subject. For his instance, he points out that he has consulted expert opinion:

I have been assured by a very knowing American of my acquaintance in London, that a young healthy child well nursed, is, at a year old, a most delicious nourishing and wholesome food, whether stewed, roasted, baked, or boiled . . .

He also has all the facts and figures at his fingertips. He has already worked out the cost of feeding a child for year and the probable price he would fetch on the market. He spells out the figures in detail:

I have already computed the charge of nursing a beggar's child (in which list I reckon all cottagers, laborers, and four-fifths of the farmers) to be about two shillings per annum, rags included; and I believe no gentleman would repine to give ten shillings for the carcass of a good fat child, which, as I have said, will make four dishes of excellent nutritive meat . . .

The narrator tries to persuade his audience by showing he has already thought his proposal through and done his homework on its profitability and so-called benefits.