More on the Natural Rate of Interest in OLG Models

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.

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.

New Keynesians and Interest Rate Effects

John Cochrane's recent blog post on the differences between New Keynesians and Old Keynesians prompted some good responses.  I particularly liked that of Simon Wren-Lewis.

There are two assumptions in the simple NK model that I find particularly troublesome.  The first is that agents have an infinite time horizon.  The second is that agents are homogenous, or that capital markets are perfect, so that everyone can borrow at the risk-free rate.  There are certainly versions of the model where these assumptions are relaxed, but I'm not convinced how well  the whole NK architecture copes with this.

I was thinking about this, whilst reading some recent posts on Brian Romanchuk's excellent blog, where he talks about interest rate effects (especially here).

The NK assumptions described above are quite important to the impact of interest rates in that world.  Taking a very simple model, where GDP (y) is made up of only consumer spending (c) and government spending (g), we have:

y = c + g

all in real terms.

In the long run, there is assumed to be no output gap, so GDP is at its natural level (y*), so we can determine equilibrium consumption (c*) as:

c* = y* - g*

The assumption of consumers maximising utility over an infinite time period, then implies that the relationship between consumption today and consumption tomorrow is a function of the interest rate.  The exact form depends on the assumed utility function, but might be something like this:

c_t = c_t+1 - α . log [(1+r)/(1+β)]

where r is the real interest rate and α and β are preference parameters.  Current consumption is then given by working backwards from the long run equilibrium level using expected future interest rates.  Higher interest rates will imply less consumption today.

In this model, therefore, interest rates have a substitution effect but no income effect.  Changes in interest rates change how households allocate spending between periods, but don't make households feel any better off overall.  This is because households look at long run income and long run income is simply determined by equilibrium output less what the government spends. 

This conclusion on interest rates however stands in sharp contrast with that of many Post Keynesians, who see an important income effect from rate changes .  In PK models, this arises as a result of the additional interest income of households due to higher interest paid on government debt.  For most, PK economists, this effect tends to override any substitution effect.

In the basic NK model, it's basically Ricardian Equivalence that is eliminating this income effect.  Interest paid on government debt is just a form of transfer income, comparable to a negative tax, so an increase has no effect because agents know they will just pick up the cost later.

And this is really where the assumption about time horizons matters.  Agents maximising over infinite lives might indeed be Ricardian (perhaps given other conditions as well).  But this is a strong assumption, and not in my opinion a very realistic one.  Once we assume agents with a shorter time perspective, the dynamics become somewhat different.

Rather than consider a world with infinitely lived agents, a good way of looking at this is to think of a world with agents who live and die, with agents at all different stages of the lifecycle.  Now, a rise in interest rates redistributes future purchasing power from those without assets to those that do.  To a significant extent, this actually means a redistribution away from agents that have not yet been born, or at least are not yet active consumers.  The potential negative impact on their spending does not therefore materialise currently. 

In effect, we are inducing today's consumers to spend more by promising them a bigger share of tomorrow's resources at the expense of people who have no impact on current demand.

The impact of interest rate changes is a complicated one.  There are different effects working in different directions.  To my mind, the simple NK model assumes away some of the more interesting aspects of what might be going on.

Keen on Krugman Again

Steve Keen comments again on Paul Krugman and endogenous money.

Keen has done much to highlight the important role of debt in driving demand in recent years.  However, in doing so, he appeals to the role of money as a key element.  I think this is a mistake.

Keen's analysis can be illustrated by the tables below showing changes in balance sheets (a +ve indicates an increase in an asset or a decrease in a liability; a -ve indicates an increase in a liability or a decrease in an asset).

Following Krugman, he has patient entities (lenders) and impatient entities (borrowers), but he also has banks.  He distinguishes two cases.  In the first, patient lends directly to impatient.  This increases direct loans, decreases patient's deposits and increases impatient's deposits.  In the second, a bank lends to impatient  This increases bank loans and increases impatient's deposits.

First scenario

	Patient	Impatient	Bank
Bank Loans
Direct Loans	+Ld	-Ld
Deposits	-Dp	+Di

 Second scenario

	Patient	Impatient	Bank
Bank Loans		-Lb	+Lb
Direct Loans
Deposits		+Di	-Di

Keen believes these two scenarios will have a different result, because of the position of patient.  In the first scenario, patient's overall asset remain the same, but he has reduced his claims on the bank and increased his claims on impatient by the same amount.  We can presume that he did so willingly - the terms of his loan to impatient must have been attractive enough - so he probably does not feel any less well off as a result.

Nevertheless, Keen's position (and in fact his entire result) rests on the assumption that patient will respond to this loan by cutting his own spending by an amount equivalent to that loaned.  The reasoning behind this seems to based on the importance of money, i.e. bank debt.  As patient's holding of bank debt has decreased, he will want to save more, to make it up again.

I find this highly implausible.  That is not to say that I don't think there are important differences between bank lending and non-bank lending, but suggesting that spending is based on holding of bank debt only, with everything else is ignored, just doesn't ring true.  I think the problem here is that a focus on the mechanics of money creation has distracted from the real relationship between debt and spending.

Interestingly, one of the papers Keen references is this one.  In my view, Krugman and Eggertsson's approach in this paper actually picks up quite well one of the important features of intermediated lending, even though it doesn't actually mention banks.

The paper uses a simple New Keynesian style model.  Its particular feature is the two distinct groups of entities: patient and impatient.  However, the amount of debt in this economy is not determined as an amount that patient consumers wish to save.  Instead, there is a separate exogenous debt level (described as a limit, but functioning as an actual amount).  This level is not explained in the paper, but could easily be interpreted as a bank lending limit.

This model does not then function as a loanable funds model.  If the patient consumer suddenly becomes even more patient and wishes to save more, this does not enable the impatient consumer to spend more, because this is determined by the debt limit.  All that happens is that overall spending drops until the patient consumer's expected consumption path meets with his new super-patient preferences.  In the end, the patient consumer can only save an amount equal to the debt limit, since the two amounts are equal by accounting necessity.  If he tries to save a greater share of his income, it is the absolute amount of income that has to give.

On the other hand, if we suddenly raised the debt limit this would lead to increased overall spending without any change in consumer preferences.  In fact, the scenario considered in the paper is a deflation caused by a sudden reduction in the debt limit.

This distinction exactly captures the point about intermediated lending.  The issue is whether the decision by a household to consume less today and save more automatically implies a decision to provide more loans to credit constrained borrowers.  If it does not, then the lending decision will drive saving and not visa versa.  And this is much more likely with intermediated lending, particularly with banks but also with other types of intermediary.  It has nothing to do with whether the intermediary's liabilities count as money.