Time Horizons and the Natural Rate of Interest

In my last post, I looked at how our assumptions about the time horizon of households affect our conclusions about a possible income effect of interest rate changes.  I want to continue that theme in this post by looking at the implications of time horizons for the concept of the natural rate of interest.

The standard microfoundations of New Keynesian models have households maximising utility across time.  This requires comparing utility in different periods and taking account of a budget constraint. With a positive real interest rate, the budget constraint implies that foregoing an amount of real goods today allows consumption of a greater amount tomorrow.  Although it would be possible to construct utility functions that implied otherwise, the usual conclusion is that the higher the interest rate, the more consumers will defer consumption to later periods.

Let's take a simple New Keynesian model with infinitely lived households who expect a constant (non-interest) income into the indefinite future.  If the interest rate is higher than a certain level, this implies that households will want to defer consumption, so that each period's consumption will be a little higher than the previous.  Initially consumption may be below current income, but it will slowly increase and will go on increasing indefinitely.  The initial period of low consumption is required to accumulate some wealth and, as time goes on, households are increasingly living off interest on savings, rather than earned income.  This is shown in the chart below.

If all households are doing this, in a closed economy, there is obviously going to be a problem since consumption is constrained to be equal to national output less government spending (ignoring private investment).  (Theoretically, a small open economy might be able to operate like this, increasingly becoming a rentier living off the rest of the world.)  Hence, the natural rate of interest.  This is the interest rate at which consumption growth matches the rate of achievable output growth less government spending.  In terms of our graph, we need to reduce the real interest rate until consumption lines up with income.

One of the things that I find rather odd about this analysis is that the level of household wealth isn't really determined by the preference function.  If the real interest rate is too high, we get ever increasing wealth.  If it is too low, wealth declines and we eventually get ever increasing debt.  When, we set the real interest rate at its natural level, wealth remains constant but the level is just whatever it happens to be.  It plays a role in determining household income (because of interest income), but that appears to be all.

Looking now at households with finite time horizons, we can see that a rather different pattern emerges.  For this, I am going to assume consumers that live for a fixed amount of time, with new ones being continually born.  We therefore have a whole series of overlapping generations.  We can still use the same assumptions about utility functions and income expectations (I don't necessarily buy into these assumptions, but I'm going to run with them here).  This will mean for each consumer that an increase in the real rate of interest will lead to them deferring consumption, spending less today and more tomorrow.  A plot of their individual consumption looks rather like that of the infinitely lived households, with the difference being that it doesn't go on forever.  This is illustrated in the chart below.

When we come to aggregate, this makes a big difference.  Now, nobody's consumption is trending off to infinity.  If we add up the spending of the overlapping generations, we get a constant level of consumption, whatever the real rate of interest.  However steep the individual lines, when we add them up they still give a flat line.  We don't need a natural rate of interest to get level consumption.

Now we also have a structure that clearly determines an aggregate level of wealth.  The shape of each consumer's consumption line implies an accumulation and erosion of wealth during that consumer's life.  Although an individual's wealth is continually changing, aggregate wealth is constant.  And the aggregate level of wealth depends on the pattern of lifetime expenditure, which depend on the real interest rate.

I should make clear that I am talking about a steady state position.  If we unexpectedly change the real rate of interest, there will be a process of adjustment.  Young generations can quickly change to the new pattern; older ones cannot do so, as the wealth they actually hold will be different from that implied by the new pattern.  When the interest rate changes, it takes time for accumulated wealth to adjust to the new equilibrium level.

The main point about all this is that a model with overlapping generations with finite horizons does not imply a natural rate of interest, even with rational maximising agents.  And we now have structure in which net wealth positions are playing an integral role in behaviour.  For me this is important.  I believe that many of the most important ways in which economies evolve come down to the structure of claims between economic agents - the pattern that is reflected in the national financial balance sheet.  I think that models that marginalise this aspect of behaviour can risk missing important insights.