What problem is laid out in the first section of "A Modest Proposal?"

The problem laid out in the first section of "A Modest Proposal" is the severe poverty in Ireland. Swift describes seeing

...the roads and cabbin-doors crowded with beggars of the female sex, followed by three, four, or six children, all in rags, and importuning every passenger for an alms. 

Swift focuses especially on the plight of the Irish children, who are a burden on their families in their youth, and, he says, a burden on...

The problem laid out in the first section of "A Modest Proposal" is the severe poverty in Ireland. Swift describes seeing

...the roads and cabbin-doors crowded with beggars of the female sex, followed by three, four, or six children, all in rags, and importuning every passenger for an alms. 

Swift focuses especially on the plight of the Irish children, who are a burden on their families in their youth, and, he says, a burden on Ireland as a whole when they grow up. Many wind up serving as mercenaries or as indentured servants on sugar plantations in Barbadoes, and those who stay behind are doomed to lives of desperate poverty. Ireland, it should be understood, was part of the British Empire in the early eighteenth century, and most of the land in Ireland was owned not by the Irish people but by landlords, many of whom were absentee Englishmen (Swift himself was the son of an English landowner in Ireland). So Irish poverty is the problem, and Swift frames his essay as a "proposal" for a solution. Nothing in the introduction prepares the reader for the shock of learning what this solution actually is.