Redistribution between generations

I ought to start a series on common macroeconomic misunderstandings. (I do not watch zombie films.) One would be that the central bank’s balance sheet normally matters, although this nice comment on my last post does the job pretty well. Here is one that crops up fairly regularly - that government debt does not involve redistribution between generations. The misunderstanding here is obvious once you see that generations overlap.

Take a really simple example. Suppose the amount of goods produced each period in the economy is always 100. Now if each period was the life of a generation, and generations did not overlap, then obviously each generation gets 100, and there can be no redistribution between them. But in real life generations do overlap.

So instead let each period involve two generations: the old and young. Suppose each produced 50 goods. But in one period, call it period T, the government decides that the young should pay 10 goods into a pension scheme, and the old should get that pension at T, even though they contributed nothing when young. In other words, the young pay the old. A fanciful idea? No, it is called an unfunded pension scheme, and it is how the state pension works in the UK. As a result of the scheme, the old at T get 60 goods, and the young only 40, of the 100 produced in period T. The old at T are clear winners. Who loses? Not the young at T if the scheme continues, because they get 60 when old (and assume for simplicity that people do not care when they get goods). The losers are the generation who are old in the period the scheme stops. Say that is period T+10, when the young get to keep their 50, but the old who only got 40 when young only get 50 when old. So we have a clear redistribution from the old in period T+10 to the old in period T. Yet output in period T and T+10 is unchanged at 100.

That example did not involve any debt, but I started with it because it shows so clearly how you can have redistribution between generations even if output is unchanged. To bring in debt, suppose government taxes both the old and young by 10 each period, and transforms this 20 into public goods. So each generation has a lifetime consumption of 80 of private goods.

Now in period T the government says that the young need pay no taxes, but will instead give 10 goods in exchange for a paper asset - government debt - that can be redeemed next period for 10 goods. In period T nothing changes, except that the young now have this asset. In period T+1 this allows them (the now old) to consume 50 private goods rather than 40: the 40 it produces less tax and the 10 it now gets from the government by selling the debt. Their total consumption of private goods has increased from 80 to 90. How does the government obtain these 10 to give the now old? It says to the young: either you pay 20 rather than 10 in taxes, or you can buy this government debt for 10. As people only care about their total consumption, the young obviously buy the debt. They now consume 30 in private goods in T+1, but 50 in T+2 when they sell their debt, which gets us back to the original 80 in total lifetime consumption.

This process continues until period T+10, say, when the government refuses to give the young the choice of buying debt, and just raises an extra 10 in taxes on the young. So the debt disappears, but the young are worse off, as they only have 30 of private goods to consume this period. Their total lifetime consumption of private goods is 70. We have a clear redistribution of 10 from the young in period T+10 to the young in period T enacted by the government issuing debt in period T.

If you are thinking that these redistributions need not occur if the debt is never repaid or the pension scheme never wound up, then we need to get a bit more realistic and bring in interest rates and growth (and the famous r<>g relationship), which these posts of mine (and these at least as good posts from Nick Rowe) discuss. But the idea with this post is to get across in a very simple way how redistribution between generations can work because generations overlap.

Unfunded pension schemes and intergenerational equity

Government debt and the burden on future generations

The Burden of Government Debt

Nick Rowe

How time travel is possible,

The burden of the (bad monetary policy) on future generations

Retiring macroeconomic theory

Dear Professor Diamond

Thank you for sending your paper ‘National Debt in a Neoclassical Growth Model’ to the American Economic Review. The paper has now been read by two referees, and I’m afraid the news is not good.

Referee A raises a fundamental objection. Your model has a two period structure, where agents work in the first period but do not work in the second. This assumption is simply stated in one paragraph on your page 2, but is not justified in any way. In that sense it appears entirely ad hoc. Furthermore, as referee A stresses, it appears to contradict (is internally inconsistent with) another fundamental part of you model, which is that agents attempt to smooth consumption over time. The referee is quite happy with that assumption, as it clearly comes from standard postulates about the utility of the consumption of goods. Yet why should these postulates not also apply to the consumption of leisure? As the referee points out, if agents tried to smooth leisure in the same way as they smoothed consumption, there would not be any ‘retirement’. As this concern strikes at the heart of your model, it is troubling.

Referee B raised rather different issues. They pointed out that the model implies a constant interest rate that is only a function of the population growth rate. The model therefore makes a clear prediction, but as the referee points out interest rates have fallen in this country over the last two decades, without any matching declines in the population growth rate. So the model has been clearly falsified by events, and therefore cannot be the basis of any meaningful discussion of the impact of national debt. The referee is also concerned that you failed to locate your analysis within an ontological discussion of the open rather than closed nature of the social realm, which makes your deductivist and formalist reasoning about socially constructed variables problematic, to say the least.

I am therefore very sorry to inform you that we will be unable to publish your paper. Referee A did make a number of helpful suggestions about how ‘retirement’ could be microfounded, and I am sure you will find the extensive reading list referee B provided on economic methodology helpful in any future work. 

My apologies to Nick Rowe, whose post gave me the idea. I actually think asking the question why we have retirement is revealing, but writing the above was easier than attempting an answer. (And I also think economic methodology is important!)  

Secular Stagnation and Three Period OLG

For macroeconomists. This post is a kind of introduction to the new paper on secular stagnation by Eggertsson and Mehrotra. As usual, any misinterpretations are my fault.

A basic idea behind secular stagnation is that the natural real rate of interest might become negative for a prolonged period of time. A simple way to model this would be to allow the steady state real interest rate to become negative. That cannot happen in basic representative agent models, where the steady state real interest rate (absent growth) is given by

1+r = 1/b

where b<1 is the utility discount factor. With population growth (at rate = n) this becomes

1+r = n +1/b

Note that a fall in n will reduce the real interest rate, which is a useful result if we want to relate secular stagnation to falling population growth, but rates cannot fall below the rate of time preference.

In a standard two period OLG model we have more flexibility. If agents only work in the first period, then they need to save in that period to be able to smooth consumption between their working lives and retirement. If we allow them to do that through investing in capital, and if α is the exponent on capital in a Cobb Douglas production function, then with log utility the real interest rate in steady state is given by

r = k + kn        where              k = α(1+b)/b(1- α)

If one period is about 25 years, then b could be 0.5 (annual b = 0.973), and with α = 0.4 then k=2. So now the impact of a fall on population growth on the real interest rate is magnified, but the steady state real interest rate is also likely to be above the representative agent case. (If n=0 and b = 0.5, then we have r=1 and r=2 respectively. For a 25 year period this would correspond to annual interest rates of around 2.8% and 4.5%.)

In a three period OLG setup, we can have saving without capital. The middle aged work (receiving income Y), and they lend to the young, and in retirement get paid back by the now middle aged. Suppose, however, that because of some credit friction the amount the young can borrow gross of interest payments is fixed at D, and let d=D/Y<1. The middle aged would like to lend them enough to smooth consumption, so the supply of loans in steady state is (given log utility)

b (Y-D)/ (1+b)

where Y-D is middle age income net of repaying loans taken out when young. The demand for loans is

D (1+n)/(1+r)

The borrowing limit is gross of interest, so with no population growth actual borrowing is D/(1+r). With population growth there are more of the young than middle aged, so we need to scale up loan demand accordingly. The real interest rate equates demand and supply, which implies

1+r = j + jn       where              j = (1+b)d/b(1-d)

Now if d is small, j could be less than one, which reduces the sensitivity of interest rates to population growth, although a fall in population growth still reduces rates. However this also means that the gross interest rate (1+r) could be less than one, so the steady state real interest rate could be negative.

The middle aged need to save for retirement, but the only way they can do this is by lending to the young. The higher the real interest rate, the less the young can borrow because of the credit friction. In that situation, the real interest rate could easily be negative, because only then will the young be able to borrow enough to allow the middle age to consumption smooth when they retire.

The key result that Eggertsson and Mehrotra explore is that a credit crunch - a fall in D - could lower real interest rates into negative territory, and could therefore generate secular stagnation. They consider how inequality could be incorporated into the model, and then embed the model in a nominal framework. Nominal wage rigidity is added (using a similar mechanism to that in the Schmitt-Grohe and Uribe paper I discussed here), and the implications for monetary and fiscal policy explored. So I have only touched on the paper here, but as this three period OLG set-up is not standard I thought this post might be useful.

Discounting Ethics in Macroeconomics

For macroeconomists (and perhaps philosophers)

In the Ramsey model (aka the representative agent model, the infinite life model, or what Romer calls the Ramsey-Cass-Koopmans model), agents care about their children’s utility as if it was their own. However, because they are impatient, they discount their children’s utility as they do their own. They therefore act as if they live forever. When teaching this [1] we say that the decentralized equilibrium is identical to the allocation that would be chosen by a benevolent social planner, who maximizes the utility of the representative agent.

When we teach the OLG model, we normally describe this model as involving agents who do not care about their children.[2] We note that the decentralized equilibrium would only be equal to the optimal allocation by chance. This is often done by showing that it differs from the golden rule allocation (the allocation that maximises steady state consumption), but some texts (e.g. Blanchard and Fischer) note that it would also differ from an allocation where the social planner showed some impatience over the utility of the unborn. In either case it is assumed that a benevolent social planner would put some value on the utility of the unborn.

It strikes me that the treatment of the two models is inconsistent. The claim about the Ramsey allocation involves an ethical assumption, which is that we allow the current generation to value the utility of the unborn. The benevolent social planner takes that valuation. Putting it another way, the benevolent social planner only maximizes the utility of the current generation, and makes no independent judgment about the utility of the unborn. Textbooks do not usually put it that way, but it seems to me this has to be what is being assumed.[3]

If we applied the same ethical judgment in the OLG model, where agents were entirely selfish, then the benevolent social planner should aim for an allocation which attempted to exploit the unborn as much as possible for the benefit of current generations. They should not be using a social welfare function which gives any weight to the utility of the unborn, and certainly not be thinking about the golden rule allocation. Instead, they should reflect the preferences of the living generations.

No one as far as I know tries to do this, presumably because it appears morally abhorrent. We want to overrule the selfish preferences of OLG agents. However, why is this acceptable in an OLG context, but not acceptable for agents in the Ramsey model?  If Ramsey agents had impatience (a rate of time preference) of 5% pa, then they are giving the utility of their children a weight of between 0.35 and 0.2 compared to their utility today. That is not so different from a weight of zero.

This point is clearer still if we look at the Blanchard/Yaari Model of Perpetual Youth, where agents do not care about their children but face a constant probability of death. A social planner that maximized the utility of the current living generations would discount at the same rate individuals do: impatience plus the probability of death. However I have not seen any papers that do this. Calvo and Obstfeld take a utilitarian perspective, and explicitly note that there is no necessary connection between the rate (if any) that the social planner uses to discount generations and the personal rate of time preference (with or without the probability of death). Once again, why is this distinction made in the context of this particular OLG model, but not when we look at the Ramsey set up where agents do care about their children?

It seems to me that if macroeconomists want to be consistent[4] they need to do one of two things. If they want to continue to insist that a benevolent social planner should use the personal rate of time preference of the current generation to discount future generations, then they should also make the social planner ignore the utility of the unborn in OLG models where agents are assumed not to care about future generations. They should also be transparent about the ethical assumptions they are making in the Ramsey case. (The potential double meaning in my title was deliberate). Alternatively, if they do not want to adopt this ethical position, they need to allow the rate at which the social planner discounts future generations (if any) to differ from individuals impatience in the Ramsey set-up as well as OLG models.[5] I have my own view on which is the better choice, but the point of this post is to suggest that at the moment macroeconomists are collectively just being inconsistent.

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[1] This post reflects the masters teaching I have just completed. I used to follow Romer in teaching the Ramsey model first, and then the OLG model. This year I have experimented with the reverse order (in the spirit of Obstfeld and Rogoff), which has helped highlight the issue I discuss here.

[2]This is crucial. If we described OLG as involving agents who would like to give Barro bequests but for some reason could not do, then my inconsistency argument does not apply.

[3] Future generations would only be happy with this if they gave the utility of their parents a much higher weight than their own. Somehow I do not think this is very realistic.

[4] The only grounds to be inconsistent would be that agents who give a weight to their children of zero should be treated differently than those that give it a non-zero weight. I cannot see what philosophical argument could be used to justify this, but I am not a philosopher.

[5] This is what the Stern Review on climate change does.