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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.

> 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.

Where isn't there one? Offhand, I can think of several:

*) Retirement agreements and regulations are all based around the idea of a "retirement age," after which a worker becomes eligible for benefits. Retirement decisions are also generally difficult or impossible to revoke once made, such as converting an RRSP into an RRIF.

*) Businesses have both minimal work units and fixed costs per worker. Two half-time workers can't effectively share a single desk with the same comfort and luxury that a single full-time worker has with an individual desk. In some respects, this is the same as the "household effect," of why not everyone wants to rent out spare rooms or why communities want control over immigration.

*) Jobs have fixed commitments per employee. Few people would argue that working two half-time jobs is equivalent to working one full-time job, because each workplace has its own distinct social situation, workplace rules, commute, and paperwork burden.

*) The marginal utility of leisure slopes upwards, at least to a point. This is why people try to schedule vacation time as a block, including arranging for three-day weekends over taking Wednesdays off. At the extreme, this is why people don't try to spend their vacation time on two-hour lunches to take one-onehundredth of a canoeing trip.

*) Career advancement has threshold effects, both in effort required and in gains received. Gaining tenure isn't something that can be done by a part-time professor over 25 years. Likewise, the discrete nature of job titles means that one can't have a half-increment of salary through being half-promoted.

*) Leisure time and wages have risks in the opposite direction. Everyone is free to quit their job and enjoy as much leisure as they wish, but they are not always free to scale up their labour output and wage income. Any period of voluntary unemployment risks being turned into involuntary unemployment.

*) People do not have free access to credit. Generally speaking, unsecured credit (which would finance leisure) is only advanced upon a proven source of income; this means that a grad student (for example) cannot freely spend the present value of the eventual income they would have as a tenured professor. This happens because consumer credit is really a sort of options contract, since bankruptcy discharges the debt for obvious public policy reasons. (Even if slavery were permitted, we'd still be bitten by the prior point: there's no guarantee that the assumed future income would be available.)

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.

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