The point that T-Rex raises is that money might affect happiness in varying degrees depending on the levels of income. For example, he postulates that awarding enough money to lower-income individuals such that they could purchase food or or clothes for their children, where they could not previously, would increase their happiness levels. However, he also notes that billionaires are not uniformly happy and that affording more money to such billionaires might not increase their happiness levels by as large a magnitude as it would for the lower-income populations.
Within the framework of economics, this phenomenon is known as diminishing marginal utility (the marginal utility decreases as money increases) and implies that the utility function of money is both nonlinear and concave as the following graph shows:
Of course, this curve need not always be the case. In fact, the notion of risk plays quite a significant role in determining the value of money towards satisfaction or utility. The above graph implies that an individual is risk-averse. However, an individual that is risk-seeking would actually derive more marginal happiness from increases in money. In other words, the individual is more willing to gamble for the possibility of large payoffs than receiving a lesser amount of money for certain. Unlike the above graph, a risk-seeking individual would have a convex utility function.
Indeed, this is all theoretical and in reality happiness can take a meaning that is quite distinct from what economists have identified as utility. Moreover, the factors influencing happiness are still relatively uncertain and consequently happiness is pretty difficult to measure. For someone like T-Rex, it's fairly simple: stomping on houses and X-box equals happiness. But for Utahraptor, it might be something else entirely. Like negating T-Rex's arguments.
So, what then are the empirical results of the relationship between money and happiness? According to one study from 2001 in the UK, money can apparently buy happiness:
We find that, as theory predicts, a windfall of money in year t is followed by lower mental stress and higher reported happiness. As a conservative estimate, a windfall of 50,000 pounds (75,000 US dollars) improves mental wellbeing between 0.1 and 0.3 standard deviations.This study seems interesting in that it uses mental stress and psychological health as measures of happiness. It also follows individuals longitudinally, thereby gauging their happiness levels over a period of time. Although, I believe the study is only limited to two years after the sample, possibly suggesting that people might adjust to some baseline level of happiness after receiving monetary windfalls.
Here is something more recent by Justin Wolfers and Betsey Stevenson at the University of Pennsylvania. Here is a NYTimes summary of the research and here is the Freakonomics Blog. In this paper, it is argued that self-reported measures of happiness rise with income not only within a society, but on a more macro level between poorer and richer countries (rejecting the notable Easterlin Paradox, whichargues that there is no link between a society's economic development and well-being).
I am not too familiar with the major research on money and happiness, so I would love if the readers could point me to some of the bigger studies. It is a fascinating subject. But as of my quick reading, T-Rex's (and Notorious BIG's) assertion that Mo' Money == Mo' Problems seems unlikely overall (assuming problems lead to unhappiness).