I've talked quite a bit recently about overlapping generation models, in particular Samuelson. I thought it would be useful to put Samuelson's model into some simple illustrations, because I believe the process by which it creates a functional relationship between income and wealth is a very important one.
In this model, agents live for three periods. They work and earn for the first two periods, but do not work for the third. They spread their spending however, so that in the first two periods they are spending less than they earn and are therefore saving, which they do by acquiring assets. In the last period, they realise their savings and use the proceeds to fund their spending. The life cycle of each agent is shown below.
Some accounting identities are being used here. Saving is the difference between income and spending. Saving or dis-saving then changes the balance of assets held. The value of assets may also be affected by changes in the price; the older generation may find it can sell the assets for more than it paid for them. More on this later.
After the third period, each agent "dies" but a new generation is born to repeat the process. There is therefore a constant chain of newer generations saving and older generations dis-saving. It is useful in this model to think of savings as being held in the form of some kind of financial instrument, in fixed supply. We therefore have a permanent market where the younger generations (1&2) are buying assets to facilitate their saving, and the older generation (3) is selling them. This is shown below:
We have said nothing about how agents spread their spending. The chart suggests they are spending equally in each period, but this need not be so. They may wish to spend more earlier in life or even to defer spending. This will change the numbers slightly, but the overall pattern will stay the same. We will still get a position whereby each generation accumulates wealth and then runs it down. Furthermore, if the time preferences do not change and aggregate income stays the same then the aggregate balance of assets will stay the same. Each individual's balance rises then falls, but the saving cancels out the dis-saving.
The market for assets consists then of younger generations as buyers and older generations as sellers. This is where Samuelson's results come from. In his simple case, with no growth, the number of buyers and sellers is the same from one period to the next, so the price of assets stays constant. However, if we have a situation where each new generation is larger than the last, the number of buyers is growing and the value of assets must rise. With the quantity of assets in fixed supply, the price must grow in line with the population in order to be able to fulfill the growing demand for savings. This growth in price creates a yield on the asset - Samuelson's rate of interest.
Samuelson's result should be viewed with caution. For the interest rate to be equal to the growth rate requires that there is no outside sectors (government or overseas) and that there is no internal transfer element to the return on assets. However, I do believe that the idea of viewing the demand for assets as a function of a natural life-cycle and a trade between different generations gives us important insights into things we see in the real world.