How we derive satisfaction is crucial for wellbeing
When you hear people say 'money can't buy happiness', it's perfectly legit to roll your eyes. Such sayings are most likely to come from people who have enough money to know that getting a bit more isn't likely to make a big difference to their lives. So take it with a pinch of salt.
Nevertheless, many of the little things that offer a great deal of pleasure are easily accessible - think music, sleep-ins or cups of tea. When we think about how consumers derive satisfaction, 'money can't buy happiness' has a ring of truth.
Music doesn't get better with money
Economics as it applies to consumers is interested in the concept of utility - the amount of satisfaction derived from the consumption of a good or service. The deliciousness when you take a bite of chocolate. The warmth and aesthetic from a fire.
Economic theory suggests that rational consumers maximise their satisfaction within the budget available. However, that doesn't imply that a consumer with a larger budget derives more satisfaction. In theory, a consumer may derive more satisfaction from spending $100 than spending $200. It depends on where the money is spent and the satisfaction derived from that consumption.
Of course, consumers aren't necessarily rational. We make decisions that aren't in our best interests. I remember once going into the supermarket with the intention of buying chicken and veggies for dinner, and walking out with three bags of chips. Shit happens. Nevertheless, utility is a useful way to think about the real value of consumption.
The satisfaction from consumption of the same product isn't necessarily any better for the 'haves' over the 'have nots'. This is interesting to think about from a wellbeing and inequality perspective. Satisfaction from consumption isn't meaningfully considered when assessing inequality. Incomes and wealth metrics might point to big insurmountable disparities, however, these are just surrogate measures for wellbeing.
When Bill Gates listens to 'Under Pressure' by Queen and David Bowie, there is no reason to believe he enjoys it any more than anyone else. The music isn't any better with money. Although, he is probably listening to it through better speakers on a comfier couch.
Satisfaction diminishes as we consume more
Granted, Bill Gates can buy a hell of a lot more stuff from which to derive utility, but that doesn't always mean much.
Eat one slice of cake and it's delicious. Eat the whole cake and you feel shame and guilt. If anything, eating the whole cake offers less utility than eating one slice. It's intuition, but also one of the underlying concepts in consumer economics - diminishing marginal utility.

Diminishing marginal utility theory suggests a couple of things:
As consumption increases, individuals derive less utility from each additional unit.
As income increases, individuals gain a smaller increase in utility from each additional unit.
Past a certain level of income/consumption, people aren't really much happier if they make/consume more. Though obviously enough wealth or income is needed to buy the basics with which to live a fulfilling life.
Money increases life satisfaction when it serves to meet basic needs, but at some point, it makes more sense to focus on relationships and personal wellbeing over careers. This is a potential argument for universal basic income and high marginal tax rates past certain thresholds, depending on how you spin it.
Perhaps in some sense 'money can't buy happiness' is real, though better rephrased as 'if one has enough money, more money doesn't buy more happiness' - but that's a mouthful.
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