Wealth in a non-monetary economy
What does wealth look like in a non-reciprocal gifting, non-monetary economy? Well, I think it looks a little simpler than wealth in an exchange economy. The major reason is that in an exchange economy wealth is tied to exchange-value, and in a non-reciprocal gifting economy this doesn't exist.
Because of this, the concepts of wealth, and the motivations attached to it, are radically different: people are not as incentivised to accrue assets, and businesses are not incentivised to accrue profits, which results in a fundamentally - and often more ideal - motivation for businesses.
Wealth and possession
In an exchange econoomy, a person or a business' possessions have a use-value, a cost-value, and an exchange-value. The cost-value could be the cost of obtaining the possession, or the cost of storing, maintaining or securing the possession. So, a person might own a car, which cost the person some amount of money or labour to get, and there may be a cost to storing the car (for example, it takes up space in their garage) and maintaining the car (even if they are not using it, they may desire to keep it clean in case they want to use it soon, or so that they can sell it later). The car also has a use-value - it helps them get around. And it has an exchange-value: if necessary, the person could trade in the car in order to buy something else.
If the person isn't using the car, and they evaluate the exchange-value as higher than the cost-value, then the person is likely to keep the car so that they can sell it at some point (or they could sell it immediately). But it is for these reasons, as I explored earlier, that a business might hold onto resources, thinking that they will have more exchange-value later, or withhold resources so that the exchange-value of the resources goes up.
In a non-reciprocal gifting, non-monetary economy, the circumstances are, of course, different. There is no exchange-value. The consequence of this is that decisions will be makde by people and businesses on the use-value and the cost-value alone. So if something is very useful and does not cost that much to store and maintain, a person or a business will keep it. And if something is not very useful and costs something to store or maintain, then a person or business is likely to want to pass it on.
If the person isn't going to want to use the car, then they probably don't want it taking up space in their garage. They will instead want to pass it on. This is especially the case if the car doesn't have sentimental value or isn't a particularly rare car, because, in a non-reciprocal gifitng economy, it wouldn't cost them exchange capacity to go and get another one. Similarly, a business will have no clear motivation to sit on resources that they don't find useful, because those resources are not valuable to them to store.
No abstract wealth
A lot of measures of wealth in an exchange economy are of things that only have exchange-value and no use-value, such as money, stock, bonds, and other financial instruments. These things are functional in an exchange economy because they help facilitate exchanges and store exchange capacity, but they don't have use-value beyond that, like a house or food does. These forms of wealth make up an enormous amount of the wealth measured in the modern liberal market economy.
With no exchange-value, these financial instruments are unnecessary and would be absent from the economy. By various measures, the disappearance of these financial instruments would mean that a lot of wealth would now be missing from the world. But, in fact, all the things with use-value would still be present. If financial instruments were suddenly removed, there would still be the same amount of wood, oil, people, food, water, and so on. So what is the difference? The major difference wold simply be who had the power to direct where those resources went. That is what financial instrument are. So the removal of these financial instruments doesn't actually change the resources in the world or what can be done with them, but it does change whose interests these resources are directed to satisfy.
So what would be quite different is wealth inequality. The people who are in very wealthy positions today would not be able to direct resources around using their abstract wealth. I'll come to how resources are directed in a later section, but one primary difference is that social consideration - that is, convincing other people that the resources are going to be useful - would have a higher priority. The likelihood is that wealth inequality would be nowhere so severe, and that the ability to use wealth for personal projects or political persuasion would be fundamentally diminished.
Think of a person who owns a mansion and a mega-yacht. Let's imagine that, all of a sudden, the laws are changed from an exchange economy to a non-reciprocal gifting economy (let's not fuss ourselves with how at the moment). In the exchange economy, the mansion and the mega-yacht are assets. They can, if necessary, be exchanged for other things. There is also a lot of use-value that the person experiences from them, because they have a wonderful time relaxing in them.
But once the economy is transformed into a non-reciprocal gifting economy, things change. First, the mansion and the mega-yacht can't be sold, so they have no exchange-value. Our wealthy person is immediately less wealthy. Moreover, the mega-yacht and the mansion probably lose some use-value and gain some cost-value. Previously, a fleet of staff cleaned the mansion, did the gardening, performed the maintenance, piloted the yacht, and maybe cooked meals and did administrative tasks. The staff did this for money. However, if the staff could get food and shelter and other items through gifting, they no longer need to work in the mansion, and they might choose not to do so unless they think that there is some good benefit to them or to society. Now the mansion has a host of cleaning and maintenance issues that the owner either needs to convince people to attend to as volunteer labour, or which they have to attend to themselves. In many ways, the mansion has become a bit of a burden, and the yacht has become useless.
In an exchange economy, the abstract wealth with which they paid the staff, and the exchange-value of their assets made them wealthy and able to direct resources to their interests. In a non-reciprocal gifting economy, this wealth doesn't really exist, and they need to use the quality of the ideas or purpose of the work to convince people to do it. This applies not just to cleaning a mansion, but also political projects, where money has funded political parties to sway or control their behaviour.