Retirement, saving, and the rate of interest

I want to sketch out a very simple model of the effect of retirement on saving and on the rate of interest. Because I think that saving for retirement is the biggest motive for saving, and that the increase in the length of time people are retired is having a big effect on the rate of interest. And I want a model to try to make my intuition a little bit clearer.

To keep it very simple, I want a model where nothing changes over time. A stationary state. Employment, population, the stocks of capital and land, technology, and output, are all constant over time.

Here is a picture of the model I have in mind:

We normally put the desired flows of saving and investment on the horizontal axis. But in my diagram I have the stock of assets on the horizontal axis. And I measure that stock not in physical units but in value units (with the consumption good as numeraire).

Let me first explain the red curve.

There are two assets: land N and capital K, which together with labour L can be used to produce consumption goods C and new capital goods I (gross investment). Capital depreciates at a fixed rate d, so in steady state, where K is constant over time, I=dK.

There is a production function F(C,I,L,K,N)=0

If there were no capital, and a fixed amount of land were the only asset, the red curve would be a simple rectangular hyperola. The annual rent on an acre of land would be the marginal product of land, and the price of an acre of land would be the annual rent divided by the rate of interest. So the value of the total stock of land (in terms of consumption goods) would be inversely proportional to the rate of interest.

If there were no investment or depreciation, so the total stock of capital were fixed, the value of the total stock of capital goods (in terms of consumption goods) would also be inversely proportional to the rate of interest. Just like land.

But capital goods can be produced. The price of capital goods will be Pk=MPK/(r+d). Gross investment I will be an increasing function of Pk. MPK is a decreasing function of the stock of capital K. So the steady state value of the stock of capital, Pk.K, will be a decreasing function of the rate of interest. But (I think) we cannot say whether a rise in r will cause PkK to fall by a more than or less than proportional amount.

Now let me explain the blue curve.

It's a simple lifecycle model of saving.

Assume people work for L periods, retire for R periods, then die. The ratio R/(L+R) is very important in determining the average stock of savings people will choose to hold. For example, if the rate of interest and rate of time preference are both zero, people will want a constant level of consumption over time. Assume they produce y per period when working (y is annual output per worker net of depreciation of capital), so have a lifetime output of yL units of the consumption good. And if they consume C per period, their lifetime budget constraint is C(L+R)=yL. If their average stock of savings over their working years is A, we know that 2A=CR (because the stock of saving rises linearly to their retirement date, and must be sufficient to finance their consumption in retirement). Simple algebra gives us:

A = yL[R/(L+R)]/2 or A/yL = [R/(L+R)]/2

That gives us the position of the blue curve where r=0%.

In this simple example, where consumption is constant over time, and the rate of interest is zero, the desired asset/lifetime income ratio is determined by R/(L+R). And if R is small, a doubling of R will nearly double the desired asset/lifteime income ratio.

If the rate of interest increases, that will have two effects on the desired stock of savings. The substitution effect leads people to want a growing level of consumption over time, which means an increased desired stock of savings. The income effect leads to a decreased desired stock of savings, because the interest income will pay for part of their consumption when retired. The net effect could go either way, depending on the intertemporal elasticity of substitution. So the blue curve could have either a positive or a negative slope.

Putting the two curves together.

The stationary state equilibrium is where the red and blue curves intersect.

If people live longer after retirement (if R increases): the blue curve shifts to the right; the value of the stock of assets increases (partly through the physical quantity of capital increasing, and partly from the price of capital goods and land increasing); and the rate of interest falls.

How big will this effect be? That depends. If the blue curve is vertical (it could be), and if the red curve is a rectangular hyperbola (it could be), then if people double the percentage of their lives they spend retired, the rate of interest will halve.

That's a big effect. And that is not an implausible thought-experiment.

[Doing the math to check if what I said is right is left as a fun little exercise for the reader. Remember that Pk=Fk/Fc, MPK=Fk/Fc, and MPN=Fn/Fc. Assume constant returns to scale so that MPK.K+MPN.N+MPL.L=C+Pk.I. I think I got that right.]

Comments

[Miami Vice]

http://www.statcan.gc.ca/pub/75-001-x/2011004/charts-graphiques/11578/cg00l-eng.jpg

http://www.statcan.gc.ca/pub/75-001-x/2011004/charts-graphiques/11578/cg00m-eng.jpg

http://www.statcan.gc.ca/pub/75-001-x/2011004/article/11578-eng.htm#a7

Interesting thought experiment.

[The Market Fiscalist]

I judge your posts by the number of times I have to read them before I have the feeling (or illusion) of comprehension. This was a 2-read one.

My synopsis is:

The longer people expect to be retired the more they plan to save, and the more they plan to save the lower the rate of interest will be. In the extreme case (where people's savings is unrelated to the rate of interest?) there will be a direct and inverse correlation between r and R. Is that correct ?

[Nick Rowe]

Miami: good find. But interest rates are (mostly) set in world markets. It's world retirement lengths that will (mostly) matter.

TMF: Nearly. Your last line should be: in the average/simple(?) case, where the stock of saving is unrelated to the rate of interest, and the value of the stock of assets firms produce is inversely proportional to the rate of interest, in steady state, there will be an inverse proportional relationship between r and R/(R+L). If R/(R+L) doubles, r halves.

[The Market Fiscalist]

Got it, its R as a proportion of total years lived, not R in itself that drives the change in r.

[Lord]

If a substantial portion of retirement is provided through paygo taxes, much of the savings would be in the form of taxes but would not lower interest rates much, so raising taxes and benefits would raise interest rates.

[Nick Rowe]

Lord: correct. Not in the model, but could be added.

[anon]

Retirement financed through PAYGO taxes is a lot like forced saving + government debt. It does raise the rate of interest, but that may be desirable if (r < g) and dynamic inefficiency applies.

[Nick Rowe]

anon: yep. But in my model, g=0 by assumption, and r > 0 because land exists, so r > g. (More generally, land will prevent the r < g dynamic inefficiency.)

[Richard]

Stylized facts, at least as I know them:
--Until 1950, reductions in mortality were largely reductions in infant mortality. After 1950, reductions in mortality were increases in longevity.
--Actuarial tables have historically increased longevity every decade they are produced, i.e., UP84 life expectancy at 65 is about 1 year larger than UP75.
--My guess is that life expectancy for adults hasn't increased enough to seriously impact interest rates. I do wonder if the social *expectation* of retiring early, especially by high wage earners, might have that effect.

[Nick Rowe]

Richard: thanks. Is that Canadian data? My guess is that the big change is coming from China, and similar places. When people are poor, they work until they die. When people are richer, they plan to stop working before they die, so they need to save. (There's also the effect of not being looked after by one's kids, and the effect of smaller families, but that's not formally in my "model".)

[Majromax]

> I do wonder if the social *expectation* of retiring early, especially by high wage earners, might have that effect.

I'd start looking at senior poverty levels. In Canada, prior to OAS and full CPP eligibility, seniors were the most-poor age group. That indicates that while they may have *wished* to maintain a level-ish consumption path into retirement, they were unable to do so.

The modern phenomenon of not-poor seniors means that collectively they are now saving significantly more. Some of this savings trend may be from private savings (RRSP), but a substantial amount is "savings" from public funds. It looks something like the above-mentioned paygo pensions.