My last post was about the implications for the natural rate
of interest of different time horizons of households, comparing overlapping
generations (OLG) with infinitely lived agents.

I think I overstated the case a bit in that post, in
claiming that the OLG version implied no natural rate of interest, so I
thought I'd better expand on the point.

I showed in the post how different interest rates in an OLG model
would change the consumption pattern of individual agents, but would always
result in constant aggregate consumption over time (in a steady-state, nil
growth economy). What I didn't make
clear was that, for any given level of aggregate non-interest income, the interest rate will affect what the actual level
of consumption is. In other words, it
will not change the slope of the aggregate consumption line, but it will change
the level, as in the chart below.

The reason for this can be seen from a simple accounting
identity. As we are talking about a
steady state, the aggregate level of wealth is constant, which means there is
no net saving, which means that aggregate consumer spending equals disposable
income. Disposable income is national
output less taxes plus interest on net wealth.
We can write this as:

c = y - t
+ r.v

where c is consumer spending, y is income, t is taxes (other
than taxes on interest income), r is the average after-tax return on net wealth
and v is net wealth. We can immediately
see that the steady state level of consumption depends amongst other things on the flow of income from wealth. This is the interest income effect discussed
in my post before last.

To ensure no output gap in our closed economy, we need to
make sure that consumer spending is equal to the difference between full employment
output y* and government spending, g:

y* - g = c = y* -
t + r.v

or rearranged:

r = ( t -
g ) / v

This then means that we do have a sort of natural rate of
interest, an interest rate at which consumer spending is at the right level to
achieve a nil output gap. However, this
rate is driven entirely by an income effect, rather than the inter-temporal
substitution effect of the infinite horizon models. Its role is purely to neutralise the impact
of a primary budget surplus or deficit.

This result is consistent with Samuelson's consumption loan model, the classic
paper on OLG models. That paper looks at
a purely private sector economy. As g - t is
zero, the natural rate of interest (in a nil growth economy) must also be zero.

Samuelson then finds that in a growing economy, the interest rate
needs to be equal to the growth rate. The
intuition here is simple. If the economy
is growing smoothly, the value of wealth must be growing at the same rate. In Samuelson's economy, there is no external
sector so there is no net acquisition of financial assets. The rate of growth of wealth must therefore be
equal to the rate of return on wealth.

The household budget constraint requires that consumer
spending plus the change in assets is equal to income plus the return on assets:

c + Δv
= y + r.v

Since c = y, this implies that:

r = Δv
/ v = ω

i.e. the rate of return equals the growth rate, ω.

As noted, this is the result where there is no external
sector so there is no flow of new assets to households. We can modify this by substituting y - g for
c on the left hand side of the budget constraint and subtracting tax from the
right hand side.

y - g +
Δv = y - t + r.v

which gives:

r = ω +
( t - g ) / v

This is the growth version of our previous result.

Those of you familiar with the stock-flow models of Godley and Lavoie may notice some strong parallels with their results. In particular, the main role of the interest
rate here is in relation to its impact on the flow of funds between the public
and private sectors. For any target
level of output, a given level of government expenditure and taxation requires
a unique interest rate. See, for example, section 4.5.2 of Godley and Lavoie.

This similarity should be no surprise. OLG models naturally produce steady state
ratios between wealth and income. The
spending pattern that drops out of such models are closely approximated by the private
expenditure functions used by Godley and Lavoie.

After writing my previous post, I noticed comments on other
forums about Samuelson's result being that the natural rate of interest should
be equal to the growth rate. Given that I
had been writing on OLG models, I really should have looked at Samuelson first. Had I done so, I might have expressed my
point differently. Unfortunately, I've always been rather lazy
about reading other people's work, even the greats, preferring to try and work
things out for myself.

So, I think I was incorrect to say that there is no natural
rate of interest in an OLG model.
However, I hope I've shown at least that it is a very different animal than that used
in an infinite horizon model.