Sunday, February 15, 2015

Sam and Janet go to College

My reading list on the overlapping generations model has already generated some questions. Rather than respond in the comment section to each question individually, I will answer these questions in a new post. Here goes.

In a comment on my previous blog Brian Romanchuck has a “good grounding in mathematics” and he “understands the [overlapping generations] models.” He is my ideal reader. Brian raises a number of points that may be shared by others with a similar background. If you also have a good grounding in mathematics and you think you understand the models: this post is for you.

Let's start with a little background about the overlapping generations (OLG) model. When Samuelson introduced the model in 1958 it revolutionized the way that economists think about the interest rate. Economists have long wondered why the interest rate is positive. The dominant view, before Samuelson’s article, was that most people prefer to consume early in life rather than later in life: a bird in the hand is worth two in the bush. The interest rate is compensation for waiting and it is governed by what we call, the ‘rate of time preference’.

Samuelson pointed out that, in equilibrium models with overlapping generations, there is another possibility. The interest rate may be equal to the rate of population growth, even when everyone has a positive rate of time preference. The ‘biological theory of the rate of interest’ was born.

Does the overlapping generations model explain money?

Most people earn very little when they are young, and very little when they are old. The bulk of earnings arise in middle age. The balance of earnings over a persons life is called his or her income profile. For income profiles that are tilted towards youth, there is an equilibrium in the OLG model in which the interest rate is less than the growth rate. If the size of the population is constant, this is a negative number.

Samuelson pointed out that, an equilibrium where the interest rate is negative, is inefficient. This is true in every dynamic equilibrium model and it is referred to as dynamic inefficiency
In the simplest case, everyone lives for two periods and has one apple when young and none when old. The equilibrium, in the absence of government, is that everyone eats their apple when young and starves when old.

Samuelson thought that this model can explain why money has value. He pointed out that the initial old generation could invent what he called the ‘social contrivance’ of money. This is a worthless piece of paper that the old pass to the young in exchange for one half of their apple. This contrivance supports a new equilibrium in which the interest rate is equal to the population growth rate and everyone is better off forever.

Some economists took this idea seriously as a model of money. See for example, the conference volume edited by Kareken and Wallace. However, other economists pointed out that the object that is passed from old to young does not have to be money. Anything that is valued and in fixed supply will have the same effect; for example, Rembrandt paintings.

The consensus now is that the OLG model is not a model of money. But it IS the best way of thinking about the determination of the interest rate in the world in which we live. That is a world where we have finite lives and not all of us care enough about our children to leave them bequests.

The issue of dynamic efficiency is important because it has implications for the impact of government debt on the welfare of different generations. Do we live in a world that is dynamically inefficient? Abel and co-authors say no.

Can the OLG model explain inflation?
I do not personally believe that Samuelson’s  contrivance of money is a good description of why we use money. But the OLG framework CAN be used to understand money and inflation.

There is a group of purists who insist that we model money by explaining the frictions that cause us to use money in exchange. The new monetarists, Randy Wright and Steve Willamson are in this camp. My own view is that it is acceptable to assume that the real value of money yields utility, as first suggested by Don Patinkin in his seminal book Money Interest and Prices

If you accept this point of view, explaining money in an overlapping generations model is no different from explaining money in any other inter-temporal general equilibrium model. Money is an asset that is held because it is useful in exchange and it is different from government debt because as Robert Clower famously stated:

Money buys goods and goods buy money but in a monetary economy goods do not buy goods. 
General equilibrium theorists have used a variety of devices to capture this idea varying from cash-in-advance, to money in the production function, money in the utility function or money that reduces transactions costs.  Here is a link to Olivier Blanchard's MIT lecture notes on this topic,

Does the OLG model offer useful policy insights?
Of course it does. It is one of two widely used frameworks to think about intertemporal macroeconomics: the other is the infinite horizon Ramsey/Cass/Koopmans model. The OLG model is indispensable for asking, and answering questions related to the design of pension schemes and for all issues relating to intergenerational allocation of resources.

For examples of practical questions that can only be addressed by overlapping generations models, see the book by Auerbach and Kotlikoff or the set of generation accounts prepared by Auerbach Gokhale and Kotlikoff.

Is the OLG model unrealistic?
Some might think that because the original model has only three periods of life that it is an unrealistic description of the world we live in. Although many of the insights of the model have been developed in a two or three period framework, the model is easily extended to multiple periods and the same insights remain. 

Economists often construct seventy period models that they solve and simulate on a computer to answer important questions about the incidence of taxes and transfers on welfare.

Olivier Blanchard, the research director of the IMF, developed an elegant version of the OLG model in which people have long lives, but they die each period with some probability. Olivier’s model is set up in continuous time, but adapting it to a discrete model with a period of a year or a quarter to match real world data is a simple task.

Do these models ignore intragenerational trade?
Absolutely not. The model accommodates lives of arbitrary length, many people of different types and many goods within each generation. The best source for the general overlapping generations model with many people and many goods is the Econometrica paper by Tim Kehoe and David Levine.

If you are interested in how all of these pieces fit together and like Brian, you have a good grounding in mathematics, try reading my book, The Macroeconomics of Self-Fulfilling Prophecies.

Economics is a fascinating subject that combines economic history and the history of thought with mathematics and mathematical statistics to shed light on important issues of public policy. All of the questions that commentators have left on my blog were answered in the literature more then forty years ago but understanding these answers, sometimes, takes a little effort.


  1. Hello,

    Thanks for the lengthy response to my earlier comment. I will have to take a look when I have more time. I hope to be able to give a longer response where I explain my comments and/or ask questions in more detail. Since I do not want to put a 1000 word comment here, I will probably post it on my blog.

    But to give an example of a problem that I hinted at, imagine that the time step within a 3-period lifetime model corresponds to 20 years. It is reasonable to for individuals to treat the average price level in the current quarter or month as being a well defined quantity. But how do I mentally relate the average price level of a 20-year interval (with the current time being at some arbitrary offset within that interval), to the average price in the following 20-year interval? But if I cannot do this, the accounting constructs within the models break down.

    This may or may not be dealt with by finer time scales (e.g. annual), but from what I have seen of those models, those finer time scales come at the cost of losing the ability to simulate other components of the economy.

  2. Brian
    The time scale is not the issue. All of the quantitative work that uses the OLG model is based either on models with seventy period lives or with the Blanchard model linked in my blog that is in continuous time.

  3. Hi Roger,

    Here is a concise example showing the the burden on the future generation of tax payers.

    The government borrows £1 Billion from the central bank and pays that to its employees . 30 years later the government repays that debt by taxing the then current generation of tax payers. The result is the heirs of the government spending through inheritance or commerce have more money and the tax payers have less money.

    1. This assumes that the heirs of government spending through inheritance or commerce are not taxpayers. That's an excellent argument for higher taxes on commerce and a high inheritance tax. Saving money is less of a civic virtue when it is seen for what it is, taking money from those less fortunate who have to pay taxes.

    2. It doesn't require that assumption, because in my example all the individuals pay tax and still some have more money because it came from the legacy of the debt spending.

      However the taxation and debt repayment in my example there, is actually an unnecessary complexity to the example , a more concise illustration is this -

      The government borrows £1 Billion from the central bank and pays that to its employees. 30 years later, even if the debt is not repaid there is still an effect on how money is distributed amongst the individual members of the population as some people have more money due to receiving the debt spending money by the process of inheritance or commerce, and others do not.

  4. Hello Professor Farmer,

    I read some of the articles that looked interesting relative to the questions that I have.

    Unfortunately, they do not address my concerns. I explain what criteria I am using, and why the papers listed do not meet them, in

    I recognise that I am unaware of the full breadth of the OLG literature. Unfortunately, I do not have access to an academic library, so I have had to work with a few textbooks (not yours, at least not yet) as well as articles in the public domain.

    A very brief summary:

    - It is my opinion that we can draw insights from models with long time increments, as they do not correspond to the actual policy choices faced in the real world. (I recognise that academic economists would disagree.)

    - I am aware of frameworks such as Kotlikoff, with a finer-grained time step. In my opinion, those are accounting frameworks, and all of the interesting economic variables are exogenous.

    I do not want to post too long a comment, so I will let anyone who wants a better explanation read my article.

    Once again, thanks for the response, and the references.


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