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.

The Great Blog Debate about Debt: A Reading List

I applaud everyone who has weighed in on the Great Blog debate about debt (Simon,  Bob,  me, and others too numerous to link. All of the issues that have been raised on Nick's blog were the topic of frontier research in economics journals in the 1950s -- 1970s.  Nick has links to earlier posts here.

The paper that started all of this (at least in the English speaking world) was by Paul Samuelson. "An exact consumption-loan model of interest with or without the social contrivance of money", Journal of Political Economy 1958, Vol 66 No. 6. The French lay claim to an earlier version by Maurice Allais, but that's another story. 

Samuelson's paper was a revelation to economists because it provided an example where markets don't work. In Samuelson's example, there is an equilibrium, (people optimize taking prices as given and all markets clear) that can be improved upon by a government institution. Samuelson's paper is a good starting point for those who would like to read more about this.

Samuelson provided a model of pure exchange, like the examples Nick has developed. In a pure exchange model there is no production. In 1965, Peter Diamond introduced capital to this model and he discussed the role of government debt in "crowding out" private capital. His paper was published in the American Economic Review, Vo. 55, no 5 under the title "National Debt in a Neoclassical Growth Model". Peter uses a mathematical tool called a 'difference equation'; and if you are sticking with my reading program, you will need to know a little bit about difference equations. There are many good undergraduate books on the topic; I like "Fundamental Methods of Mathematical Economics" by Chiang, but that probably dates me.

The next paper I would recommend in this literature is by a mathematician, David Gale, "Pure Exchange Equilibria of Dynamic Economic Models" Journal of Economic Theory 6 (1973). I include David's paper on the reading list of my first year Ph.D. class. In it, David distinguishes what he calls a "Samuelson economy' from a 'classical economy' and he shows that every overlapping generations model has at least two steady state equilibria; one in which the interest rate equals the population growth rate and one in which the aggregate saving by the young is zero. This divide is the key to understanding when government debt is a burden in the sense we have been discussing.

Throughout the 1960s and 1970s there was a very muddled discussion in the journals, trying to understand why markets can sometimes fail to be optimal. Some people thought that it was because not everybody can meet, due to the one way flow of time. That issue was cleared up by Karl Shell in 1971, "Notes on the Economics of Infinity", Journal of Political Economy, Vol. 79. Karl attributed the problem to what he called the 'double infinity' of people and goods. This is the paper to cite at parties if you want to appear knowledgeable about the topic. It probably won't enlighten you much unless you're enrolled in an economics Ph.D. program.

Any question that you have has, almost surely, been answered already in the literature. How do the conclusions of the model depend on the assumption of no bequests? What happens if some people live forever? What happens if there are multiple goods in each period? Many of these questions are answered in my book "The Macroeconomics of Self-Fulfiling Prophecies".

I'm sorry if the answers are not always obvious, or the papers I have cited seem impenetrable to you. But realize that mathematics is a language and often it is the best language for answering questions of logic. "Everything should be made as simple as possible, but no simpler".

If you think that we are debating esoteric issues that are unrelated to the real world; you are entitled to that opinion. An economic model is only useful if helps us to understand the world. I happen to think that the overlapping generations model contains a great deal of useful insight. If you read, and understand, all of the papers I have cited, you will never again utter the phrase: "debt is money that we owe to ourselves".

Sam and Janet Learn about Debt

In a recent post on the (non)-importance of debt buildup worldwide, Antonio Fatas makes the point that debt is not necessarily a problem. While I agree with that statement: a great deal hinges on the qualification “not necessarily”.  

Paul Krugman goes further than Antonio. According to Paul debt is “money that we owe to ourselves”. That is at best misleading and at worst;  false. Money is money we owe to ourselves. Debt is money that some of us owe to others. 

In the real world, debt matters; and it matters a lot. Why? Because different generations are not all connected by operative chains of bequests. Government debt is the liability not only of current generations, but also of future generations. An increase in government debt always places a burden on future generations. The right question is: do the benefits of increased government debt outweigh the cost?

Paul, in a separate post, invites us to think of a world where some people (spendthrift Sams) spend more than others (Judicious Janets). I like that model; but lets develop it a little bit further and suppose that Sams and Janets reproduce (asexually since this is, after all, a brave new world). That allows us to think about the real world in which different generations coexist.

In a model that I developed here, there is a stationary population that consists of a bunch of Sams and Janets of different ages.  The Sams spend more than their income when young, and less when old, by borrowing from the Janets. Janets do the opposite. They spend less than their incomes when young and more when old. Sams and Janets are selfish, and although they love their children, they don’t love them all equally. 

Several centuries of social evolution has created a society composed of equal sized populations of Sams and Janets. In each population, there are ancient dynasties and newly created dynasties all of whom are borrowing and lending to each other in amounts that depend on their ages and their types.

Let's expand this idea a little further. Although the incomes earned by Sams and Janets are equal every year (the Scandinavian model) the amount that they each earn fluctuates from one year to the next. In most years, income is high; but occasionally there is a recession and income is low. Since Sams and Janets are risk averse, they insure themselves against these fluctuations by trading in the financial markets. 

Sams and Janets trade two financial assets. One asset, debt, is a promise to pay one dollar next period whether income is high or low. The other asset, equity, is a claim to the future stream of income. Since there are only two events in each period, income my be high or low. The financial markets are complete and Sam and Janet are perfectly insured against fluctuations. But their unloved children are not.

So far so good. Along comes Government Gus. Government Gus issues a boatload of debt that it sells to existing generations of Sams and Janets. Government Gus is in a heated political race with the opposition, and seeking political popularity, he spends the proceeds of government debt creation on pork belly projects that favor existing dynasties of Sams and Janets. Since interest rates are currently low, Government Gus borrows $10 trillion in the form of 1% consols; these are promises to pay $10 billion in coupon payments to the bearers of the securities every year forever. To finance these interest payments, Government Gus introduces an income tax.

Now we can evaluate Paul’s statement that “debt is money that we owe to ourselves”. Not quite. The benefits of  the pork barrel spending have all accrued to existing generations of Sams and Janets. But debt finance has created a perpetual stream of tax obligations that fall not only on current generations of Sams and Janets; but also on all future generations. That is the sense in which Paul’s claim is misleading. ‘We’: are not a homogenous group. Debt benefits some groups at the expense of others.

What about Antonio’s qualification that debt is “not always bad”. That depends on what causes recessions. John Cochrane points out that, if recessions are all supply side events, those pork belly projects are an unambiguous theft from our children. But if most recessions are caused by deficient “animal spirits” as I believe they are and have modeled here, the additional pork-belly spending may increase income (and employment) by shifting the economy into a higher, more efficient, equilibrium. Spending on roads, bridges and Warp drive technology that benefits future generations, as well as current ones, would be even better.

Lets recap; if aggregate income and employment is determined by aggregate supply, government borrowing to finance current expenditure is an unambiguous transfer from future to current generations.

If aggregate income and employment is determined by aggregate demand, government borrowing to finance current expenditure can potentially increase income and employment and make everyone better off. 

Irregardless of whether recessions are induced by supply or demand failures: government debt creation has distributional consequences. A better way of financing expenditure in the midst of a deep recession, (Europe take note), is to create lots of little pieces of colored paper and distribute them to all of the hungry Sams and Janets out there who don't understand why Gus is standing idly by while they go hungry.

Lessons from the Great Galactic Depression

A long time ago, in a galaxy far far away, there were two planets orbiting a star, not unlike our own sun.  The inhabitants of these planets share a common ancestry but, over the years, they have developed somewhat different temperaments. 

The names of these planets are difficult to pronounce in English, but we will call them Nordus and Sudus. The name of their star is Sol. Nordus, being further away from Sol, has a colder climate than Sudus and its inhabitants are known to be frugal and patient. The Sudusians, in contrast, live for the moment.  Using the language of economics, earth people would say that the Sudusians have a higher rate of time preference. 

For many centuries, the Nordusians and the Sudusians were often at war. But in recent years, the development of a new and dangerous weapon, the Death Star, has made war infinitely more risky. Some fifty Sols ago, the political elites of the Solarian system, realizing the danger of mutual destruction, entered into an economic and monetary union.  At the inception of the union, each planet abandoned its own monetary unit and a new Solarian Central Bank (SCB) was created; its President is a wise Solarian by the name of Obi-Mar Draghobi. Under Obi-Mar's guidance, the SCB issued a new currency, the Solo, which takes the form of little pieces of colored plastic called Solarian credits, or screds.

The Solarian social structure is composed of dynastic families who accumulate wealth and pass it down to their children. This system is fairly stable but dynasties do not last forever. Sometimes, a Solarian family produces a "bad wookie"  who squanders the family fortune. And often children quarrel with their parents and are disinherited. As a consequence, there is constant regeneration of dynasties and a dynastic wealth distribution that depends on dynastic age and dynastic time preference. The governments of Nordus and Sudus sometimes engage in infrastructure projects that benefit not only current generations of Solarian citizens, but also generations yet to come. A good example is the investment by the Nordusians in Warp drive technology to enhance trade with neighboring star systems. 

Solarian economists are an advanced lot who agree about most things; the Solarians call them “dentists". When the Nordusian government embarked upon the Warp program, it consulted with the Solarian Association of Dentists (SAD) who recommended that Warp investment be funded not just from current taxes, but also from Nordusian government bonds. Since the benefits of the Warp drive are very long-lived, the initial investment was paid for by issuing 3% consols. These are scred-denominated claims to a flow of 3% of the principal in perpetuity. The Sudusian government, like its Nordusian neighbors, also issues consols. But the profligacy of the Sudusians is legendary and the citizens of Solaria, both Nordusians and Sudusians, are unwilling to buy Sudusian consols at 3%. Instead, they require a 2% default premium to reflect the considerable possibility that a future Sudusian government may decide to default on its debt. Since much of the debt has been used to finance transfers to Sudusian pensioners, and, some say, star cruisers for the Sudusian political classes, the probability of default is only too real.

On  setting up the Solarian Central Bank, SAD dentists were once again consulted. They pointed out that screds are costless to produce and they have value solely because of their use in exchange. Because scred creation is costless, the SCB, as the monopoly provider of screds, can create wealth. But how is that wealth to be distributed? The founders of the Solarian union were far sighted and wise. They created a rule that distributes the benefits of scred creation back to the citizens of Nordus and Sudus in proportion to the number of citizens of each planet. Since there are approximately equal numbers of people on each planet, in practice, the Nordusian and Sudusian peoples receive equal shares of seigniorage revenues created by the SCB.

These revenues are of three kinds.  First, there was the creation of sc1 trillion at the inception of the union. Second, there is the ongoing flow from the creation of new screds to meet the need for liquidity as the Solarian economy grows. And third, there is the flow of revenues generated by the interest payments on the asset holdings of the SCB. At its inception, the SCB needed a way to introduce screds into circulation. Following a recommendation by the Solarian Dental Council, the SCB purchased sc1 trillion of consols split equally between Nordusian and Sudusian bonds. And every twelfthsol, the SCB purchases a further basket of consols, equally weighted, between Nordus and Sudus. The exact amount purchased depends on economic conditions in the Galactic Empire; but there has not been a single twelfthsol since the inception of the SCB, when the Bank did not purchase at least some consols in the financial markets.

The Nordusian government pays 3% interest every Sol to the SCB and, under its charter of incorporation, the SCB turns around and pays those revenues right back to the Nordusian Treasury. Similarly, the Sudusian government pays 5% (3% plus the default premium) every Sol to the SCB which the SCB returns to the Sudusian Treasury.  The situation I have described worked like a charm for more than fifty Sols, from the inception of the union in Sol MMCX through to the onset of the Great Solarian Depression in Sol MMCLX. The history of this episode is well known so I will be brief. 

During the Protracted Expansion that lasted for almost twenty five Sols, the Nordusian banks invested heavily in foreign bonds issued to fund res-pod construction at the Galactic Center. As pod prices escalated there was an unsustainable wave of new pod development that ended in disaster with the bankruptcy of the Jabba Brothers bank in Sol MMCLII. As we all know, the collapse of Jabba Brothers caused the entire galactic economy to spiral into a deep depression. The Galactic Depression had profound effects for the Solarian economy and the Sudusian government defaulted on its debt. Much of this debt had been held by Nordusian banks. Although the default caused consternation for Nordusian shareholders, the Solarian Association of Dentists pointed out, correctly in my view, that the Nordusians should not have been too surprised. They had, after all, been earning a premium of 2% on their investments in Sudusian bonds for more than ten Sols. 

Some Sudusian dentists reminded the Nordusians that, after all, they had themselves defaulted on bonds issued to fund the last Galactic War and as one charismatic Sudusian Politician put it; "what's sauce for the hoska is sauce for the hosko". And of course, Obi-Mar Draghobi's masterful handling of SCB affairs is by now the stuff of vidtexts. His successful negotiation with the Nordusian Chancellor, and former Princess, Queen Anga-Leia over the funding of future scred creation is widely credited with saving the Solarian union and heading off another Galactic war.  The Nordusians argued that, since the Sudusians had defaulted on their debt, the SCB should be banned from future purchases of Sudusian consols. Obi-Mar Draghobi pointed out, that the financial markets had long since capitalized the default probability of existing bonds and that nobody should be surprised when the inevitable finally happened. Excluding Sudus from the benefits of future scred creation, that was by then running at approximately sc60 billion a twelthsol, would be an unfair penalty to Sudusian taxpayers and could easily kindle a resurgence of the rebel alliance.

Initially, Queen Anga-Leia listened to her finance minister Darth Scheudius, who argued that the Sudusian default had put the Nordusian taxpayer on the hook and the Sudusian's should be made to repay the full amount. Obi-Mar Draghobi pointed out that the purchase of Sudusian consols was a once and for all transfer and that the debt to the SCB was never expected to be repaid.  The rest as they say, is history. The Sudusian Treasury re-entered the capital markets after a short delay and a considerable shakeup of its political elite. It is to be hoped that the new Sudusian finance minister, Hanis Soloufakis, will manage to pull off the considerable reforms that will be required if the union is to survive a potential future Galactic crisis.

Why the ECB Should Take More Risks

Mrs. Merkel and Mr. Schäuble are worried. The ECB is planning to buy the sovereign debt of its member states and Mr. Schäuble doesn't trust his southern European partners. He thinks that Portuguese, Spanish and Italian debt is risky and he knows that Greek debt is.

Bankers are supposed to be boring. And central bankers are supposed to be boring in spades. What would happen if the Fed were to bet the farm, buying shares in an internet start-up that subsequently goes bust? The public purse would be on the hook for the loss. At least, that’s the theory. That theory is wrong.

The central banking business plan is a money-spinner beyond a venture capitalist’s wildest dream. Buy an asset, any asset, and pay for it by issuing little pieces of colored paper. Collect the interest payments and dividends from the assets and use them to pay for your house, your car and a holiday in Spain. If you happen to be the central bank of a sovereign state, pay the interest and dividends to the treasury to help reduce the tax bill of your citizens.

Does it matter which assets you buy? Conventional wisdom says yes. A central bank should buy safe assets, typically promises issued by its own national government that will never fall in value. The Fed buys T-bills on the private market. The Treasury pays the interest and principal to the Fed, and the Fed turns around and pays them straight back to the Treasury. The point of all of this is to keep enough of the little pieces of colored paper passing from one person to another to “oil the wheels of trade”.

What if the Federal Open Market Committee were to put all of the Fed’s assets into the stock market the day before a stock market crash? If the Fed were privately owned, it would be put into receivership. After the crash, it has liabilities of $2,500b. These are the circulating pieces of colored paper. But its asset portfolio has been decimated by the market crash and the market value of those shares has fallen from $2,500b to $1,000b. What a calamity! Surely the taxpayer must rush in and recapitalize the Bank by pumping in another $1,500b. Not so.

Central banks, like the rich, "are different from you and me". Their debts are all in the form of those little pieces of colored paper. And the magic of central banking is that you and I will carry on passing those pieces of paper from one to another without ever trying to cash them in for something else. There was a time when the UK government printed the promise to “pay the bearer on demand, the sum of one pound” on its notes; at the time, a pound note was redeemable in gold. That time is long gone. All you will get at the Bank of England now, if you try to redeem your ten pound note, is two fives.

What does all of this have to do with the ECB? Mrs. Merkel and Mr. Schäuble should sleep easily in their beds. The interest payments on ECB assets are returned to national governments in proportion to a formula that weights each country by its relative size. As Paul DeGrauwe pointed out recently in a VoxEU post, as long as the ECB sticks to this formula when it buys sovereign debt, the only people to suffer from an Italian default will be private holders of Italian bonds. The net worth of the ECB may take a temporary hit, but that shouldn't bother European taxpayers, least of all the Germans.

Extraordinary times require extraordinary measures. There was a time when central banks were privately owned. The Swiss National Bank still is. But the Bank of England, the Federal Reserve and the European Central Bank are supra-national institutions that play by different rules. We should not be concerned if sovereign states are permanently in debt. This has been the case with the UK and the US treasuries for as long as the US and the UK have existed as sovereign nations. What is true for a national treasury, is also true for a central bank.