We don't normally publish our failures. But perhaps we should. Because we can (sometimes) learn as much from our failures as from our successes.
Academic economists spend a lot of time trying to model things. One reason we do this, and a good reason for doing this, is as a check on our imperfect intuitions. That's what I was doing here. My intuition didn't check out.
We are richer than we used to be, and live longer than we used to, and spend more years retired than we used to. So we want a bigger stock of savings to finance our retirement. An increase in desired savings would cause a lower equilibrium rate of interest. It sounded plausible to me that this could be a reason for falling real interest rates. So I tried to build a model to show that this was not just plausible, but correct under reasonable assumptions.
But my model started to tell me that it wasn't correct.
There is no obvious relationship between higher wages and retiring earlier. (I should have suspected this, because income and substitution effects go in opposite directions.)
Longer lives mean people retire for longer, but also work for longer. It has no obvious effect on the ratio of working years to retired years. (Frances mentioned this, when I was telling her about my little model outside Tim Horton's.) And since firms' desired stock of capital would increase in proportion to working years (labour supply), there is no obvious effect of increased longevity on the equilibrium rate of interest.
But I still like my (incomplete) little model, even though it was disobedient and didn't tell me what I had wanted it to tell me. It is a neat and simple way to make retirement endogenous, and to make the value of the desired pension plan endogenous. I may come back to it, some other time.
Here's my (unfinished) original post:
I'm trying to build a simple macro model where the retirement age is endogenous. I want to examine the effect of longer lives and higher incomes on the desired stock of savings and on the equilibrium rate of interest. The effect isn't obvious, because if people live one extra year they might decide to retire one year later. It's not even obvious why people retire at all. Why do they smooth their consumption of goods, but consume a big lump of leisure at the end of their lives? We need a non-convexity somewhere, either in technology or in preferences.
Suppose you had diminishing marginal utility of consumption, but constant marginal utility of leisure. You would smooth your consumption over time, but you would not smooth your leisure over time. You would work full-time and save, and when your stock of savings got big enough, you would stop working completely and live off your savings.
Which is roughly what we observe. So let's run with that assumption.
U = log(C) – L, where C > 0 is consumption and 1 >= L >= 0 is labour.
Let W be the (real) wage (in terms of the consumption good). At the date t* where the person is indifferent at the margin between retiring and working, we know that the marginal utility of leisure (which = 1), divided by the marginal utility of consumption (which = (1/C)), must equal the wage. So
C(t*) = W
So people are consuming all their wages on the day they retire. They choose to retire when their stock of savings is big enough that they no longer save any of their wages. (This exact equality between consumption and wages is an artefact of the assumed logarithmic form of the utility function; more generally, consumption will be some fraction of wages at retirement.) They choose to retire when their actual stock of savings hits their desired target stock of savings.
People discount future utility for two reasons: because they are imperfectly patient; and because they might die. Let 0 < B < 1 be the degree of patience, and let 0 < S(t) < 1 be the survival rate, so 0 < BS(t) < 1 is the total subjective discount factor.
Assume that people save by buying life annuities. If R(t) is the safe gross interest factor, so that R= 1/(1+r) where r is the (real) interest rate, then the gross interest factor on life annuities will be R(t)S(t). Optimal consumption smoothing implies:
BS(t)C(t)/C(t+1) = R(t)S(t)
So we can delete S(t) from both sides, and find the gross growth rate of a person's consumption over the lifetime as:
C(t+1)/C(t) = B/R(t)
Notice that the survival rate has no effect on the growth rate of consumption over a person's lifetime.
The lifetime budget constraint is:
Present Value [C(t)] = Present Value [W(t)L(t)]
where future consumption and future wage income are discounted by R(t)S(t).
A doubling of wages W would cause a doubling of consumption C at all dates over a person's lifetime, but would have no effect on the retirement date. Again, this result is an artefact of the logarithmic (or CRRA) utility function, where the income and substitution effects of wages on the demand for leisure exactly cancel out.
An increase in non-wage income (for example, a tax on wages that was redistributed as lump-sum transfer payments) would have no effect on consumption but would all be "spent" on extra leisure, through earlier retirement. That's because consumption has diminishing marginal utility, while leisure has constant marginal utility.
An increase in R (a reduction in the real interest rate) would cause people to retire later. To see this, remember that consumption grows at the rate of interest, and so a worker who chooses to retire at date t* would choose a pension plan that pays C(t*)=W(t*) initially, and annual benefits to survivors that grow at the rate of interest, and then consumes those benefits. The present value of such a pension plan would be independent of the rate of interest. It depends only on the current and future survival rates S(t). A reduction in the rate of interest also means a slower growth rate of consumption over the working years, and higher initial consumption for a new worker with no assets. He will save less, and earn less interest on what he has saved, so it will take longer for his stock of savings to reach an amount sufficient to finance his optimal pension plan. So he will retire later. (At the limit, with R=1 (a zero real interest rate) people will be indifferent about when they consume leisure.)
An increase in survival rates S (a fall in the death rate) causes people to retire later, but they would also spend more years retired. Proof is by contradiction. If they retire at the same age, the increased survival rate would increase the present value of the optimal pension plan, but have no effect on the growth rate of consumption for survivors, so it would have take longer for workers to save enough to buy that optimal pension plan. Which is a contradiction. If they spend the same number of years retired, the present value of the optimal pension plan would be unchanged, but they would spend more years working and save too much to buy that optimal pension plan. Which is a contradiction.
An increase in survival rates S(t) (a fall in the death rate) causes people to have a higher average stock of savings over their lifetimes. "Proof" is by eyeball geometry. Draw a curve that begins at zero, slopes up to a point, then slopes down towards zero again. The height of that curve represents the total assets of a cohort at a given age. The average height represents the average stock of assets over their lifetime. The height at the point represents the present value of the pension plans at their retirement date. An increase in S(t) shifts the whole curve to the right (which doesn't affect the average height) but also increases the height at the point (which increases the average height). (OK, that's the best "proof" I can do, but someone else could do better.)
An increase in R (a reduction in the rate of interest) would have (I think) an ambiguous effect on the desired average stock of savings over the lifetime. Because I can't prove it either way.
All the above was partial equilibrium analysis. Now let's do macro.
Everyone is identical, but they are all born on different dates. If we take a snapshot of this economy we will observe young people working and saving part of their wages, and old people who have retired and are living off their pensions.
The zero-profit pension funds own all the land and capital in the economy, and firms produce consumption and investment goods by hiring labour from young workers and hiring land and capital from the pension funds. Factor markets are competitive, so factors are paid their marginal products. So the prices of land and capital goods are equal to the present values of their marginal products. Investment is a positive function of the ratio of the price of capital goods to consumption goods. Usual stuff.
Worthwhile Canadian Initiative 
I see problems with some of your assumptions:
Young people might be working (if they are lucky enough to get a job that pays money, and not an unpaid internship) but not saving, only paying off their student loans. There is no surplus money going to young people that they could be saving, at the typical salary and cost of living.
Old people, many of whom didn’t save enough, probably aren’t retiring when you expect them to, because they didn’t save enough. So they keep working (which mean fewer jobs for those young people, see above).
Markets are not competitive. In successful industries you have a monopoly or duopoly.
Mike R: there are always “problems” with the assumptions in any model, because a model is always a simplification of reality. But they key is to find those assumptions that matter for the question at hand.
It would be interesting to see how demographics affect these and are affected by them which is the most likely affect on the real interest rate.
Lord: I think a decline in the birthrate would mean there are fewer workers as a ratio of the desired stock of savings, which would push down the rate of interest.
Mike R. In the more-or-less standard way of thinking about saving, paying down debt is, in fact, counted as saving. Simply because the effect of paying down debt is to increase net worth.
Two types of investment
One where people invest to earn a return…have more wealth than they started
The second is investment without expectation of return at least directly …and many times no return even indirectly
Both are needed and without the second type of investment economies are not viable and will collapse which
Consumption (Sales) = Income plus Investment
That is C = Y + I
That is in order for the investor to invest, he has to get a return on his investment which equals all of that which he invested plus income (total income (Y) of everyone involved in the project, workers and other factors of production, entrepreneurs and investors)
But this cannot happen for the whole economy
Because for the whole economy we have
Y = I + C
So
C = Y – I meaning consumption is less than income
So the only way to get the extra income is either increase the propensity to consume all the way to 100% of income being spent which does not happen
Or to have extra investment add to the system from preexisting or newly created wealth.
This is true because Y=kI so
C = (k-1)I then to increase consumption we need to increase investment
So consumption is always less than total income and that’s why we need investment of preexisting wealth to make the economy viable
(does this mean we call k the investment income multiplier…..and {k-1} the consumption investment multiplier??)
All this is derivable from the general theory by Keynes
Secular stagnation in a society that is not supply constrained.. is generally caused by inadequate investment of the second type
And longevity and retirement are involved in how society defines full employment
Sorry
C = Y + I is the first type of investment
Y = C + I. Shows we need the second type of investment to make the economy viable
djb: C=Y+I and Y=C+I don’t make sense.
income equals consumption(sales) plus investment Y = C + I is right out of the general theory
and for a business who wants to make profit, consumption ie sales must equal the total income of the venture plus that which is invested in it
that is they expect the sales to pay for all the income and returns plus the that which they invested
thats C = Y plus I
but for the whole economy this cannot work
since Y must equal C plus I
so the part of the economy where investors invest for profit is only part of the economy
this must be balanced with investment of preexisting wealth or newly printed money to make the economy viable
is used to think that both C = Y + I and Y = C + I could both be true at the same time, but in reality i never let myself think about it,
so i was believing in an escher waterfall type of situation
for the whole economy they cannot both be true
Keynes was Pretty clear about the Y = C + I part of it.
Nick,
have you considered modelling the impact of health in old age? I don’t know what the data would show, but perhaps people are not able to increase their working years to the same extent as life expectancy has increased due to physical and or mental afflictions. In essence, medical advancements may have prolonged life, but not the number of fully productive working years that we have. There might be some sort of insurance market friction whereby people cannot insure themselves against the risk of dementia, arthritis, or other conditions that are not necessarily lethal, but could prevent a person from working much into old age. This might be modeled by simply assuming that there is some fixed maximum number of years that people can work. I suspect an OLG framework might be the easiest approach for this. I haven’t tried to work out the algebra on this but I suspect that increased lifespans would then lead to a larger desired stock of savings because there would be more years where consumption would have to be financed purely from savings. This working years hypothesis also explains why people might consume leisure in a big lump in retirement – it’s not that this is their ideal path of leisure consumption, but that they may be forced to plan for it this way due insurance market frictions. Some people might get lucky and find that they reach retirement age with both health and a large stock of savings, while others may be forced to stop working and depend on their savings due to health issues.
Patrick: the puzzle is why do we bunch leisure? Falling health and productivity as we age is one reason why we might postpone leisure. But falling health also means we might enjoy leisure less as we age too. Which deteriorates faster: my ability to do my job or my ability to paddle my canoe?
And even if productivity at work declines faster than productivity at leisure, we still need to explain why most people suddenly switch from working full-time to working zero time. Why doesn’t we go smoothly into retirement? There must be a non-convexity somewhere.
Majro: OK, a non-convexity in : preferences; technology; or regulations. I’ve assumed it’s preferences. But it could be in one of the other two.
“A doubling of wages W would cause a doubling of consumption C at all dates over a person’s lifetime, but would have no effect on the retirement date.”
Mr Money Mustache and other fans of financial independence or early retirement would shake their heads at the stupidity of people who behave as they are assumed to do in your model. I don’t know what percentage of the real population are that stupid. I would imagine that it is quite a large one. But I doubt that it is 100%.
Derek: nothing is 100%.
Our hourly (real) wages are (say) 10 times higher than our ancestors’. We could work one tenth the time they did, and consume like they did. But we don’t.
If I changed my model slightly, and assumed that wages were taxed, and the taxes used to finance lump-sum transfers, then an increase in wages would cause both increased consumption and earlier retirement.