What societal problem does the poem "Richard Cory" depict?

I think there are two major social criticisms contained within the poem "Richard Cory." First, there is the poem's depiction of class inequality, by which Richard Cory's wealth and upper-class refinement is contrasted against the lower-class narrator, who wishes to switch places with Richard Cory himself. The contrast Robinson makes between the wealth and status of Cory and the poverty that surrounds him is one of the critical components that shapes the poem.

At the same time, however, the poem ends with the startling image of Richard Cory's suicide. Ultimately, though many of the town's lower-class residents might have envied Richard Cory and wished to trade places with him, this ending reveals the degree to which Richard Cory had his own personal demons and struggles he had to contend with. In this sense, these lower-class residents never really knew Richard Cory to begin with.

Thus, the poem can be viewed as a sobering reflection on the degree to which public personas and reputations rarely match up to the private realities of people's lives, even as it also depicts the degree to which other people will make judgments and assumptions based on those same public personas. His lower-class contemporaries, on account of Richard Cory's own wealth and status, assume that Richard Cory must live a blessed and happy life, even though reality proves much to the contrary.

How does the poem "Richard Cory" exemplify the idea of social class in America?

This poem, "Richard Cory," was written by Edward Arlington Robinson after the 1893 Depression when the average person in America could not afford meat and had a diet consisting mainly of bread. Thus, during this period and during the period of the 1920's Great Depression, there was a very sharp divide between the "haves" and the "have nots." 

When the wealthy Richard Cory goes downtown, although the people admire him, they do not speak to him, for he is in a place apart from them, an enviable place because in the United States there was once  the "American Dream" that promised anyone he/she could rise out of poverty.  With the depression, this dream now seems unattainable and Richard Cory appears distantly "imperial" and a gentleman from "sole to crown" to the disenfranchised of the 1893 American depression who do not understand that one who has food and comfort can be desperate enough to commit suicide. 

How does the poem "Richard Cory" exemplify the idea of social class in America?

I think the basic content of this poem, as it relates to social class, is very American in outlook.  The poem shows that one may have wealth and social class yet not be satisfied with one's life, or strong enough to survive it's turmoils, while the workers had the strength to survive and could find enough satisfaction in their life to continue despite their complaints. The workers are presented as stronger individuals than Richard Cory.  Further, the fact that Richard Cory "fluttered pulses" when he walked and inspired envy seems more American in nature. In Europe where class divisions were much more strictly adhered to than in the U.S., there would less likely be any jealousy or "fluttered pulses," because there was more acceptance of one's place in society because there was no hope to advance. In America, workers at least had the hope that by hard work or by marrying someone with money, that one could advance one's social status.  Thus when they saw Richard Cory, they could aspire to become like him or, for women, even to marry him or someone like him.