Wednesday, 20 November 2013

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.


  1. The standard microfoundations of New Keynesian models have households maximising utility across time.

    This has always struck me as an absurdly one dimensional starting point for an analysis. Apart from the fact that the word household itself implies more than one generation, it denies that maximization (assuming it takes place at all) has more dimensions than just for individuals of a homogenous age group over time. The much more plausible dimension, at least when considered in real (i.e. non. monetary) terms, is between individuals of different age groups in each instant of time. And from there one can go on and slice society into segments other than age, such as class, corporate sectors, gender, you name it. But hey, that would imply maximazition beyond the individual. And who other than smelly socialists would want to go there?

    1. Yes, the word household does imply more than one generation, which is why these models are described in terms of households rather than consumers - after all, it would be a very strange to base on model on individuals who lived forever. Empirical plausibility has never been a major concern of this approach - but that would be stretching it too far.

      It is only a starting point. There are versions which try to modify this assumption. But you shouldn't have to get complicated to get to more realistic results. If the simple version of the model is giving results that don't fit with the real world, then it's not a good starting point.

    2. Maybe I should have put it this way: it seems logically wrong to me to think of maximisation over an infinite time period. It's either personal maximisation, in which case the relevant time period is a lifetime (as in your example). Or it's intergenerational maximisation, in which case the relevant time period is 0? Or am I the one who's confused?

    3. No, if it's inter-generational, the idea is that consumers leave bequests to their offspring, so they are maximising between what they spend during their lives and what they leave behind. This is then treated as if the consumer expected to live for ever.

      I have no doubt that a desire to leave bequests does factor into people's spending choices (but I've ignored that in this post). However, translating that to an infinite time horizon requires further assumptions about time preferences and that's where I'm not convinced.

  2. I guess people are so different though. Some people do seem to act as though they were immortal. They want to gather a fortune as a challenge -as an an end in itself or to endow a foundation. If it is such people who gather the most wealth, then they will be setting the nature of the economy even if that isn't the economy that most people want.
    Steve R Waldman pointed out this study:
    Why Do the Rich Save So Much?
    Christopher D. Carroll

    "The paper concludes that the simplest model that
    explains the relevant facts is one in which either consumers regard the accumulation
    of wealth as an end in itself, or unspent wealth yields a flow of services (such as power
    or social status) which have the same practical effect on behavior as if wealth were
    intrinsically desirable."

    1. An excellent point, and it did occur to me whilst writing this. I do think that the finite horizon model is a better representation of how most people behave. It should ideally take account of a desire to leave bequests, but that doesn't automatically turn it into an infinite horizon model. However, I think you are right that amongst the very rich, there is this accumulation behaviour that is clearly not maximising over a lifetime. And, because of the skewed influence of such savers, this has a big impact on aggregate behaviour. I had been meaning to do a post on it because it has major implications for the sort of stock-flow models I tend to use, which generally assume steady state wealth to income ratios.

      It's worth noting however that taking account such behaviour doesn't really get you to the standard NK household either. Thanks for the reference to the Carroll paper - I'll look at that.

    2. My impression is that there is a sort of ecological succession over time. After an economy has been rebooted (eg after WWII or by a massive colonization such as of North America or whatever), wealth is somewhat evenly distributed so the "average Joe's" behavior of maximizing lifetime utility prevails. There are always a handful of full on wealth gathers but initially they won't yet have gathered so much as to skew the economy. Overtime, they end up being in a position that dominates the economy and we end up as we were in the 1930s and once again are today. The resulting low output economic state can persist for a very long time. Haiti has been like that for two hundred years I guess.

    3. I also think that it is much much harder to sensibly make use of a very large fortune. If someone gains a large fortune simply by doing a good job -eg creating an innovative company or whatever- that doesn't mean that they will have a way to spend it. It is very easy to spend £10k but well nigh impossible to spend £1B.
      Bill Gates basically has had to make a tough full time job of trying to give his money away and even then he gets criticized for not being careful enough with how it gets spent. The default -go with the flow- result once someone has a great fortune is for that money to gather more -pretty much by itself.

    4. I suspect that a realistic individual consumption should have consumption based on income (or expected lifetime income), but with a declining propensity to consume out of relative income, i.e. income relative to national or local averages. I certainly think that learned behaviour plays an important part - we base our spending on what we see other people doing. Aggregate consumption would then depend on income and wealth distribution.

  3. Reading the comments, I think this might interest you:

    See Nick Rowe's comment:

  4. I appreciate your point that real world conditions can be a series of incremental money accumulations, each of which disperse. This dispersion prevents the infinite accumulation intentionally predicted by the NK model with any positive natural rate of interest.

    The NK model has at least two additional subtle but unstated aspects:

    Has anyone noticed that the concept of accumulating-fiat-money-for-future-consumption has a natural-conversion-to-gold aspect? That is, by intentionally saving fiat money for future consumption, there certainly would be the expectation that the future value of money would continue to exist, exactly like gold.

    Another subtle aspect: The increased flow of interest over time requires more money to flow. How could this money increase be accommodated? Considering one lender and one borrower, and assuming constant money supply conditions, interest could only be paid in incremental amounts, requiring an exchange interaction between borrower and lender on an ongoing basis. At contract completion, lender would hold the original amount of money (remember this is constant money supply conditions) but lender would have received more of the exchanged product or service (such as labor).

    For other than stable money supply conditions, we need to consider the source of the money. Presumably, borrower and lender can not make their own money supply. .Any money supply increase in perpetuity would require government participation.

    Is the NK model then simply MMT in mathematical expression?

    1. Roger,

      When we solve the models, we don't get the continual accumulation, precisely because the natural rate of interest prevents it. So once everything has adjusted, the actual rate is assumed to be at the natural rate, which will mean all income is consumed and there is no accumulation (in a non-growth version). These issues are therefore avoided.