Money holds the power to shape the flow of games - from single player games to MMOs. With every game we make, we are designing currency. Sometimes the currency is simply points (or points in a more colorful guise). Other times it is a means of drawing the player towards challenges - collect x widgets and you can continue. In a third case the player collects money to gain power directly or indirectly: Direct mechanisms being things like Mario's 100 coins for extra lives or experience points to earn level-ups, and indirect ones being shopping and bartering.
One thing unifies those examples: The game has absolute control over the money supply. And in many games, it tends towards an initial scarcity that later collapses into an abundance of wealth; the player starts off weak, and has to pick opportunities carefuly, but inevitably progresses in skill or power, or finds loopholes in the system, and legitimately or not, they collect hundreds of extra lives, store thousands of pieces of equipment, boost their character's abilities to god-like ratings.
If we look at this in economic terms, the player starts off with a low productivity, and in games that allow productivity to increase by simply incrementing numbers(the case with experience points and equipment), productivity goes up very quickly, much faster than in the real world, and much faster than a game that is strictly based on player skill. Those games make the ability to become more productive a major part of the mechanic, and relate it to the player's preexisting wealth, rather than isolating it with the player's skill. For those games, unless there are strict caps set, eventually the productivity is so high that the old challenges of the game cease to exist: the game becomes a sandbox of abundant wealth.
For single-player games, this is not really a problem. Everyone knows the addicitiveness of this artificial rise to greatness, and it makes for a fine capstone to typical game story progressions, albeit in many cases the story's tone will get wildly out-of-sync with the gameplay. "Great hero becomes stronger; faster; wiser" is hard to reconcile with typical dramatic devices.
Short-term multiplayer games like Halo or Mario Kart typically set caps on productivity via preexisting wealth: you can collect a few items, and they provide situational advantages, but you can't hoard them. The goal is fairness and balance, and depending the game style this may mean a model of "scarcity for everyone" with very few advantages given monetarily, or "abundance for everyone," where everything is similarly overpowered, but it's rarely a case of "wealth inequality." Inequality works if your game seeks to be as bitter and dystopian as reality.
MMOs face complex issues with money because a large and unknown number of players engage in the game over a long period of time. In the short-term game it can be assumed that everyone's working to the fullest of their abilities for the duration. But in the long-term game people appear and disappear "at random," and work at vastly different rates. Thus efforts of the game to manipulate money in the world and give all players a consistent experience are constantly foiled because of player actions, direct or indirect - for example, too many people camping too few monster spawns. Or item markets where pricing is destroyed because people at the top of the productivity chain can give away their goods for next-to-nothing.
For the most part, the biggest problems of money have been resolved in existing MMOs with an array of checks, caps and balances; they can provide an "every player the same" experience with high consistency. However, it would be far more interesting to discuss ways to use money to drive other gameplay, online or offline, rather than to preserve "every player the same," which reeks of artificiality. In particular, I want to point to gift economies.
The economics of the natural world are mostly oriented towards abundance: anyone or anything can take whatever it needs to the best of their ability, and the environment alone limits you. Just like what a game does. And for the humans of past eras that were able to operate like this, it had a major impact on society: It made people, not goods, be the thing that mattered economically. So gifts and reputation held a higher status. This is obviously not the case in our modern society, but in small ways, we have the power to turn back the clock.
What I like about this is that games, as pointed out earlier, tend to converge on abundance and extremely wealthy, productive players; a game with too much scarcity is obviously broken, and it takes conscious design to balance scarcity for the purpose of challenge. But it's fairly easy to make a game with "lots of everything." So the most natural thing to do to a game design would be a focus on the gift economy in some way, shape, or form. Essentially: Making it so that the world acts as building blocks for people's skill and creativity. This has been done in games many times: sometimes the entire system is built for creative purposes(e.g. art and music programs), sometimes it is a small portion of a game with other currencies(character customization), sometimes we make the world a meta-world(level editing tools). And sometimes we simply inspire reams of fan creations because of a powerful setting, characters or story - we turn literary elements into their own kind of money.
The most exciting aspect of an game (for me) is that it can have effects that change the real world. If not that, it can at least allow a temporary transcendence of reality - that thing we call immersion. And I think that it is in the money system that the most is done to cause those effects.