Such things as cost of living are pretty hard, if not impossible, to precisely quantify for the ancient world. Likewise, the Roman world knew no such concept as a steady, fixed wage, which makes it difficult to talk about the cost to an average worker. We have very little information about such things, so we have to rely on spotty evidence and generalizations. The "average" wage, whatever that means, for the inhabitant of the city of Rome is usually said to be around three denarii a day, on the basis of a passage in Cicero where he mentions that even an unskilled slave could earn that much in a day as a day-worker. That estimate is probably too high (one denarius even is probably too much), but it works as a decent-enough foundation, particularly if we consider that in the city most people were not working steadily. The vast majority of the city's population had no steady employment or source of income at all. The urban population was composed almost entirely of unskilled day-workers who hired themselves out at the monumental building projects, or perhaps at the dockyards, or as temporary extra fieldhands during the suburban harvests (something that was actually still done until recently). If these people were lucky enough to find work that day they were paid, at variable rates. If not, they went without. Skilled workers like shopkeepers or tradesmen were lucky enough to have a stable source of income, even if it wasn't all that much, but most of the plebs urbana lived on a razor's edge, making up what Harrison calls a "fluid" or "floating" population, which is to say that the urban plebs were essentially semi-homeless. With rents often being paid daily workers and their families probably moved frequently to chase work, with the relatively well-off "plebs media," the skilled tradesmen who could afford permanent accommodations, no matter how spartan, making up the majority of the "settled" population. Between migration, immigration, and disease the demographics of any individual neighborhood must have been changing nearly constantly.

While precise costs are nearly impossible to determine, we can pretty confidently say that almost the entirety of an individual's pay was going towards food, clothing, and shelter, not necessarily in that order. Grain for bread, the staple of the Roman diet, was not cheap. It's hard to understand this in a post-Green Revolution world, but famine was a constant concern in antiquity. The city of Rome, and indeed most large cities, maintained an artificially inflated population far larger than the ability of the surrounding hinterland to support and feed. This was only made possible by the ability to import provincial grain, but even then before the Green Revolution a single poor harvest might have devastating effects for the rest of the season. Combined with manipulation of the market through private stockpiling and trading in grain futures, both common abuses of the late Republic, and the price of grain might be radically different from one day to the next, causing artificial scarcity through vastly inflated prices when there was in fact plenty. Legislation under the Principate largely solved the issue by instituting fixed price margins and establishing greater incentives (and punishments) for traders who maintained a steady supply, but even then the situation was spotty sometimes. In the late Republic it was a total disaster, and attempts by C. Gracchus, Saturninus (although I disagree with Badian that this law was ever actually passed), and later P. Clodius to instituted leges frumentariae were often only marginally useful. While state-subsidized grain was available for free to every citizen by the time of Clodius' lex frumentaria (leading to abuses, in which some slave masters freed their slaves en masse in order to collect part of their grain) the actual amount of the state-subsidized grain was relatively small--a single modius, or alternately 5 modii per month, was enough to feed an individual, but not a family (as my thesis director puts it, the lex Clodia "was nice, but it wasn't going to feed your family"). Additional grain had to be purchased on the market, at market prices, which even in the Principate could be rather high. By the Principate the recipients of state-subsidized grain had shrunk to a minority, and the establishment of inalienable tickets, which could be inherited and sold, essentially turned the segment of the population receiving subsidized grain into a privileged minority that did not at all represent the actual poorest citizens. Besides this, rent was at a premium in the city, and with daily rents a good chunk of one's pay--Harrison (on the basis of the estimates of others) suggests nearly the entirety of an individual's three denarii, and suggests that even if we double that there's not much left over. The situation for a family, which even with child labor would have had mouths to feed that weren't contributing much (and families, given the high mortality rate to disease, probably not entirely offset by immigration into the city, were almost certainly widespread), would have been even more economically dire. What little money was left over after the purchase of grain and payment of rent probably mostly went towards additional food (the urban plebs could not cook for themselves in the crowded firetraps that were their insulae, as most rooms had no hearths) or payment for baking bread (this could also be done in communal ovens, but I'm not sure how much evidence we have as to whether there was a cost to use these), or otherwise for clothing and other minor amenities. On days when a member of the family could not find work there was probably nothing left over at all.

The high cost of living at Rome and the abysmal pay of most day-workers mostly has significance in the study of politics, since it leads to the argument of how involved the urban plebs actually were in state affairs (a hot topic these days, since constitutionally nearly all electoral and legislative power technically rested with the citizen body, but we don't see it being exercised much for various reasons). But it has implications, naturally, for society as a whole. The concept of the city's population as the "idle mob," so popular in discussions even as late as the 70s, is considered thoroughly disproven--the Roman urban dweller had to work to survive, and work quite hard at that, and nearly constantly. Most entertainments in the city were at the public expense, or only had nominal fees, but how much actual time could the plebs urbana spend at them? What about at contiones, the comitia, and other public political business? Nothing legally prevented them, but were they actually able to find the time, other than being out of work for the day? Some interesting work is being done on the idea of political activity as essentially a job in itself--for example, Morstein-Marx has proposed, quite convincingly, that politicians maintained paid entourages (which then raises the questions of how much clients could rely on their patrons--in my opinion clientage was not in fact particularly important among the semi-migratory lowest class and did not in fact contribute very much to a steady worker at all) to support them at the vote or in contiones. Likewise we hear from Tacitus, etc. of duces theatralium operarum, leaders of theater gangs (and other gangs--Sallust mentions in passing a class of political gang-leaders whom he seems to think could be hired out at will by any party) who seem to have worked as if leading these cliques was a permanent job--from whom were they receiving their pay? The topic of wages and costs is a very difficult one, and one to which attention has been increasingly paid recently--just a few years ago Andreas Bendlin pretty thoroughly (it's already being considered the orthodox opinion) dismantled the idea of collegia tenuiorum, voluntary associations of the poorest individuals for social and funerary purposes, partly on the basis that the only evidence we have for their registration wages would have been impossible even for many tradesmen (also because there's like literally no evidence for them, Mommsen kinda made them up)