Wednesday, 27 November 2013

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.


  1. Another great, in depth, post. Thanks!

    Great post offer the opportunity for deeper thought. Toward a goal of deeper understanding, I offer this comment for you and readers to consider.

    I am going to use your first three equations and logic but replace the definition of "r.v" with the unknown X. My term X will represent the change in money supply. A change in money supply during the measuring period could occur or an undetected counterfeiter could be at work. With this substitution of terms, we have:

    r.v = X

    (I will use your words but substitute where appropriate with words in CAPITALS.)

    "Disposable income is national output less taxes plus NEW MONEY. We can write this as:

    c = y - t + X

    where c is consumer spending, y is income, t is taxes (other than taxes on interest income), AND X IS NEW MONEY. We can immediately see that the steady state level of consumption depends amongst other things on the flow of income from NEW MONEY.

    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 + X

    or rearranged:

    (STRIKE r = ( t - g ) / v)

    X = t - g

    (This ends my shameless use of your words and logic. Please forgive me if my technique offends you.)

    It is obvious from inspection that the unknown change in money supply has come from an imbalance between government spending and receipts.

    Having presented that a change in money supply has the same derivation as "natural interest rate times wealth", I come to the question of why we would want to equate:

    X = r.v

    It seems to me to be a very arbitrary association. In other words, why would we want to associate two unknowns, natural interest rate and wealth, with a difference between government expenses and income?

    We probably should notice that we could substitute any logical sector association with the term r.v and come to the identical final result.

    1. Roger,

      That's interesting, but I'm not sure how closely it relates to what I'm saying. The equation "c = y - t + r.v" was specifically supposed to be in a steady state situation where wealth is not changing. Maybe it was my fault for the way I set it out, but the bit you quote doesn't make sense if you don't include the line before. In general, consumption is not equal to income. That equation was only supposed to apply to the situation where all income is being spent, so there is no saving - no accumulation of new assets. In that situation, there should be no new money (except where there is switching of assets - say more money in return for less bonds).

      If you are including changes in money, then normally you would expect it to be on the other side of the equation, i.e. income equals consumption plus increase in money holdings. All income that is not spend represents an increase in money holdings.

  2. I am not sure of the connection of Samuelson with Godley/Lavoie.

    In this paper the interest rate is sort of different from the growth rate and can even be higher.

    1. I could probably write a whole paper on this, if not a book. Maybe, I'll do a post or two on it.

      On the face of it, they are very different approaches, but I was thinking of this: Samuelson models household behaviour as a series of overlapping generations, each working for only part of their life, but maximising utility over a lifetime. Given a fixed growth rate, this behaviour generates a steady-state ratio of aggregate wealth to aggregate household income. Each generation's wealth is continually changing, but the ratio of the aggregates is constant. The ratio depends on the assumptions about income pattern, interest rates and utility function.

      If you shock this, by changing disposable income (perhaps imposing higher taxes), it changes the steady state wealth. However, the adjustment takes time. Because of the OLG structure, there is an incremental adjustment. Older generations, who no longer earn, are less affected by the change so their spending doesn't change much. That of younger generations changes more. Wealth slowly adjusts to the new equilibrium level.

      If you choose the parameters appropriately, this pattern is very similar to what you get in G&L models which use a private expenditure function px = f(r).yd + b.v(-1) where px is expenditure, yd is disposable income, v(-1) is lagged wealth, f(r) is a function of the interest rate and b is a parameter. It's not exact, but it's close.

      In a sense therefore, an OLG model like Samuelson's provides a sort of micro-foundation for the G&L private expenditure function. It explains the behaviour in terms of utility-maximising agents - the only difference with standard NK models being the time horizon. (There is the additional point that G&L are using current income, rather than expected lifetime income, but that doesn't affect what I'm saying here and would take me way too much space to go into.) This is not to say that G&L needs a microfoundation to justify itself - I think the more interesting point is that microfoundations with different assumptions about preferences can produce very different results.

      So that's the connection. In his paper, Samuelson doesn't have an external sectors, so he doesn't get into the sorts of issues that G&L do. He just has households. With a growing economy, he needs financial assets to grow at the same rate, but with no external sectors there can be no net acquisition of such assets. The assets therefore have to grow organically by yielding a return equal to the growth rate.

      If you introduce a public sector, with spending and taxation, then there is now a potential net flow of financial assets to or from households. This means that the equilibrium return on assets is no longer equal to the growth rate, but has to be modified for the steady state primary budget deficit. This is entirely consistent with Samuelson's methodology, but is now starting to look more like the things going on in G&L. Of course, Samuelson is starting from equilibrium output and deducing the interest rate. G&L take the interest rate as given and deduce the resultant level of output. But the mechanics are similar.

      Last point. Samuelson is working with real quantities and real returns, which is necessary for the utility-maximising analysis. In the simple G&L models, I'm not sure if it's specified whether these are real or nominal, but they are probably best understood as nominal. However, they can easily be understood in real terms with appropriate inflation accounting.

      I think I'll have to do some posts on this.

    2. Nick,

      Maybe I am talking past but my interest was because I have seen a lot of people connecting the "r versus g" and OLG and throw something in the air.

      I linked the G/L paper to actually show that there isn't a connection - at least there is no connection in the way people go about it.

      As you say there are differences but my point was that these differences are too big.

    3. For example I am not sure why Nick Rowe keeps bringing it.

    4. I don't know exactly what people are saying about "r versus g" - I know Nick Rowe's written about it quite a bit, but I haven''t read it. I think my point is that Samuelson's result that r = g is a very special case as it assumes no net acquisition of financial assets. It is a slightly odd feature of Samuelson's model, that it has some form of outside money, but no flows involving any outside sector. I suppose you could come up with some sort of story to explain it, but the important point is that as soon as you get these external flows, r no longer equals g.

      The model in the G&L paper you linked is actually quite a good one to demonstrate my point, so I'll maybe use that as the basis for a post.

    5. Ramanan,

      I just had a look at the numbers in that G&L paper and I think they're consistent with what I'm saying. Of course, G&L are assuming the interest rate is given and are then asking what level of real government expenditure is needed; I was assuming the level of real government spending is given and was solving for the interest rate. Not because I think that's how it works necessarily, but because the context of the discussion was the supposed natural rate of interest.

      If you take the numbers for each of the three scenarios in their table 2, you should find that they all satisfy my equation r = ω + ( t - g ) / v , to within about 0.01%, which I assume is just rounding. You have to make a few adjustments, because I defined my terms slightly differently (see above). My t doesn't include tax on interest income and my r is the after-tax real interest rate. Also, you need to take account of the lagging - they (correctly) base current interest on previous period wealth; I didn't specify that above as I was keeping it simple.

    6. Yes of course I know you'd get the G&L stuff easily.

      What I am saying is that the Samuelson stuff and its various offshoots is used in public discussions about notions of sustainability, social security etc. I think this was one of the reasons Tobin was opposed to using it because the users tried to use it to say things such as rising burden on society etc.

      My impression is that offshoots of Samuelson use it to say for r<g there is a bubble economy and the other case burden on society keeps rising and things such as that.

    7. I wasn't aware of that. To be honest, I wasn't even of aware of the Samuelson until Nick Rowe drew my attention to it after reading my post on the meaning of wealth and national balance sheets

      It's a pity because I think this approach has something very important to say (although I don't think it's anywhere near a complete description). It gives a lot more meaning to social accounting than you get from infinite horizon models, where everything gets waved away by Ricardian equivalence.

  3. Thanks for considering my previous comment. The goal of my previous comment was to point out the arbitrary nature of the relationship between r.v and g-t. I don't think I got my point across.

    In a nutshell, I believe that the relationship between r.v and g-t is true only at the zero point where two unrelated but intersecting functions cross zero.

    We begin your derivation with

    c = y - t + r.v

    where r.v is assumed to be NOT zero.

    Then we condition y expense = y income and g expense = t income. This would result in the balanced equation:

    ye + ge = yi + ti.

    Next we redefine y to be y* but leave ge and ti unchanged. ge continues to equal ti.

    For the next step, we equate the expense side of the two sector GDP model with the right side of the two sector GDP model while adding the r.v term with the resulting equation:

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

    We simplify by subtracting y* from each side leaving:

    0 - g = 0 - t + r.v.

    We can simplify further by subtracting g = t from each side, leaving

    0 - 0 = 0 - 0 + r.v.

    Obviously, r.v = zero.

    Now we specified that g = t which also allows us to say that g - t = 0. It is thus correct to say that:

    0 = 0 or r.v = g - t

    are both ways of mathematically writing that zero equals zero.

    The relationship only holds at the one point where both terms equal zero. I think that to then relax the preconditions and equate the two terms results in a very arbitrary association. (which was the point of my previous comment.)

    I believe the zero equivalent problem continues into your dynamic analysis, making that also an arbitrary association.

    Just to reinforce this conclusion by examining the problem from an entirely different approach, consider how GDP is determined. GDP is the sum of transactions. Financial transactions are specifically excluded. This financial exclusion does not mean that funds from financial transactions are not used for GDP counted transactions;

    undetected counterfeit money would increase ye, ti and yi but not ge;
    government borrowing would increase ye,ge,ti and yi;
    saving by an individual would not change measured GDP (but would change potential GDP);
    individuals spending savings would increase ye, ti and yi but not ge, (identical to counterfeit money).

    There is nothing here to suggest that an interest rate, especially a "natural interest rate", can be deduced or teased from this data. No one would deny that an arbitrary association could be claimed and defended.

    We can approach the issue from a third approach: There are other macroeconomic associations for the "g - t" term. MMT holds that the deficit is the principal way that government injects money supply into the economy. I made a similar assertion in my post Government Provided Money Supply found at Any such association is purely arbitrary; the appropriateness of the association depends completely upon the logic applied.

    You end your post by saying "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."

    You can see from these comments that I am not yet convinced that there is more than an arbitrary association between "natural rate of interest" and an OLG model. Any relationships between the two should be expected to describe two intersecting paths, interest following one one formula and the OLG model following a second formula.

    1. Roger,

      Again, my c = y - t + r.v is specifically in the case where there is no change in net financial assets (what I'm calling wealth). Since the government sector is the only source of such assets, this requires that the government has a balanced budget, so g - t + r.v = 0. So if r.v is not zero, then g cannot be equal to t.

      I maybe haven't helped by not setting out here a full balance sheet and precise description of my assumptions. It's one of the problems, I'm afraid, of trying to keep the posts relatively short.