The Value of Simple Models, with Examples of Economic Dynamics

[This is posted in the new Category Economics, Technical]

[Published 14 March at Real World Economics Review Blog.  Also posted 12 March at Steve Keen’s Debtwatch]

Many people, including many heterodox economists, understand that the neoclassical equilibrium approach to understanding economies is futile and misleading [1], because modern economies are far from equilibrium.  The neoclassical prediction of equilibrium or near equilibrium requires a string of patently absurd assumptions.  However the development of better theories seems to be significantly hindered by a feeling that any superseding theory has to be thoroughly quantified before it can be useful, and a feeling that the neoclassical theory has set a benchmark for sophisticated mathematics that must be matched before another theory can be respectable.  Less fundamentally there seems to be a common perception that empirical insights can only be gained through elaborate statistical treatments of observations.

Here I offer some discussion from my experience as a natural scientist, and some examples regarding the Global Financial Crisis, to counter these hindrances.  Useful and relatively simple models can be constructed that can immediately overcome major neoclassical limitations, for example by permitting non-equilibrium behaviour.  The solution of the mathematics can be done using very standard numerical integration methods that are readily available in commercial packages.  Mathematical machismo is not required.  There are also situations in which the empirical lesson is obvious with no analysis, as will be noted here.

I should be clear that there are certainly many modellers who operate outside neoclassical confines, reported for example in Beinhocker’s excellent survey of “complexity economics” [2].  The lessons offered here will not be news to them.    Also some of them are constructing quite complex models that are nevertheless very instructive, such as models with many interacting adaptive agents.  This article is prompted by my reading of some heterodox blog discussions, and is addressed to anyone who may have some difficulty seeing how to move beyond the neoclassical approach.  Nor are the models here are offered as original investigations, though they may lead to such.

General points on quantification and mathematics

Theories do not necessarily even have to be quantified to provide important insights.  I have argued that the recognition of economies as self-organising systems with many possible states already implies three important conclusions:  that economies can be restored to their appropriate place serving society, that there can be a diversity of economic styles rather than a monoculture, and that economies can be compatible with the living world.

Nor does quantification have to be comprehensive or to involve highly sophisticated mathematics to yield useful insights.  Indeed when a field is new, useful insights can often be gained from rather simple models, even from back-of-the-envelope, order-of-magnitude estimates.  Some striking examples of this difference in outlook between physicists and economists are recounted by Waldrop in his excellent book Complexity [3].

Economics is new in this sense, as the recognition of its complexity (in the technical sense) requires that it be thoroughly re-explored and re-conceived.  In the natural sciences quite rough approximations are frequently used in such situations to, in effect, roughly map out the territory.  More careful quantifications can then be appropriate to clarify and refine early findings.  Indeed, unless guided by clear preliminary concepts and rough estimates, a more elaborate quantification can turn out to be a waste of time, or even misleading, if inappropriate parameter values have been used.

An example from my own field is an estimate that a magma ocean, resulting from the collision of two proto-planets, might solidify within only a few thousand years [4].  Even if this estimate is uncertain by one or two orders of magnitude it still makes clear that the magma ocean will freeze much more quickly than the time it takes for the final planet to aggregate, which is tens of millions of years.  That is an important insight.

More basically, neoclassical economists seem to have a fundamental misconception that if they are doing sophisticated mathematics then they are doing science.  This misconception goes back, via Milton Friedman, to the founding work of Walras among others.  In science, mathematics is a very useful tool, but it is only a tool.  In science, a hypothesis is proposed and its implications are compared to observations of the world.  The objective is to find a hypothesis (or theory or story) that provides a useful guide to how the world is observed to behave.  Mathematics is very useful for deducing the implications of a hypothesis, which can then be compared to observations.  It can also be useful for processing observations (e.g. via statistics).  The crucial distinction between mathematics and science is the comparison with observations, and the subsequent judgement as to whether the hypothesis is proving to be a useful guide.  Neoclassical economics seems to have missed this fundamental distinction at the beginning, well over a century ago, and never to have noticed the oversight.

The importance of dynamics, and the role of money and debt

A financial market bubble and subsequent crash is an intrinsically dynamical event.  In other words the market is driven by internal forces or feedbacks that move it through and beyond any perceived “true” value or equilibrium state.  In October 1987, financial market values changed by thirty to forty percent in a single day, though there was no  corresponding external event affecting the real economy.  This demonstrated, starkly and with no analysis required, that financial markets are strongly affected by internal forces, and that they must have been far from equilibrium before, after or both.  Such events clearly cannot be modelled by the equilibrium or quasi-equilibrium concepts that are the foundation of neoclassical economics.  Indeed it must be doubted if the highly volatile financial markets could ever be so modelled.

Money must play a pivotal role in the internal dynamics of economies;  more specifically this applies to token money, which is money without its own intrinsic worth.  This is because token money links the present to the future, so in dynamical terms it operates on the time-derivatives of economic variables.  This can be explained as follows.  If I receive a ten-dollar note I am, in effect, receiving a promise from the community that it can be exchanged, in the future, for ten dollars’ worth of real goods or services.  Thus the token (the ten dollar note) amounts to an implicit social contract between me and my community.  That contract links the present (when I receive the note) to the future (when I spend it, i.e. when I exchange it for real goods or services).

The implicit contract involves a debt (the community owes me) and a corresponding credit (I am owed by the community).  Because token money, like all debt, links to the future, it involves our expectations of the future, which fluctuate within the uncertainty of whether the future will actually deliver.  Before I spend the note, it is merely a token of potential wealth.  If the future does not turn out as we expect, my potential wealth may not be realised.  This is also true of other forms of token money, such as entries in books and bits in computers.  It is also true of other forms of debt, and in this context there is no difference between money and debt:  both involve contracts, implicit or explicit, for delayed payments.  Therefore both money and debt must be included in modelling of the dynamics of an economy, meaning its development in time.

Yet according to Keen [5, 6], the role of money in macroeconomics has been seriously neglected.  Indeed much economic theory ignores both money and debt and treats exchange (the fundamental event of economics) as barter.  Notable exceptions have been Keynes’ qualitative discussion of a revolving fund of finance [7], and the circuit theories of money  initiated by Graziani [8].  Keen has been, according to his claim, the first to show quantitatively how Keynes’ revolving fund of finance works, and the first to show, building on circuit theory, quantitatively that a manufacturer can borrow money and still make a profit [9].

Simple dynamical models of economies

The models by Keen [5], and extensions of them by Davies [10] illustrate the intrinsically dynamic nature of economies, and the role of money in those dynamics.  They also illustrate the value of relatively simple models, and contrast with the opaque complexity and irrelevance of neoclassical equilibrium models.

Keen [5] starts with a simple economy that develops by its own internal workings into a steady state.  This steady state demonstrates how Keynes’ “revolving fund of finance” can work, and also serves as a reference state from which to explore other things.  Keen  goes on to demonstrate how a credit squeeze causes a drop in the bank deposits of business and employees and a rise in unemployment.  He extends this model to demonstrate that a government stimulus directed to households is much more effective in quickly restoring employment and circulating money than is a stimulus directed to boosting bank reserves.  These are potentially important findings from rather simple models.

Davies [10] extends Keen’s model to include property, as well as goods and services, so an asset price bubble might be simulated.  The following examples come from this work.  The methods are explained in detail in the references.

Keen’s economy comprises banks, firms and employees (“workers”).  Firms borrow money from banks and use the funds to manufacture goods.  Those goods are sold to employees and bank personnel.  Employees work for the firms in return for wages, and bank personnel work for the banks, deriving income from the interest charged on loans.  Starting with the firms having no money, this little economy quickly approaches a steady state.  Davies’ version includes housing property and employee mortgages and it comes to an analogous steady state, which is illustrated in Figure 1.

Figure 1.

Figure 1:  Approach to steady state in a simple economy.  Quantities shown are accounting balances:  Firm loans, Firm deposits, Bank transaction account, Employee deposits, Bank “vault” account, total money supply and total employee mortgages.  From Davies [10].

In this example the population is steady and the total amount of money is constant.  The money starts off in the bank’s reserve account.  To avoid common confusions, Keen suggests it is useful to think of paper money, and to regard the reserve money as being kept in the bank’s vault.  The model is started with some of this money already in the possession of employees, through their mortgages (otherwise it takes generations for the mortgages to come to steady state).  However the firms start with no money, and both the Firm Loan and Firm Deposit accounts rise from zero.  As firms take loans the bank vault is further depleted, and as firms conduct their business money flows to employees’ deposit accounts.  The bank’s transaction account also rises from zero as interest charges are added.  After a few years a constant amount of money circulates through the various accounts.

This example illustrates fundamentally non-neoclassical behaviour.  During the initial transient phase, lasting 3-5 years, quantities change rapidly – they are dynamic.  The subsequent steady state is not the same as a neoclassical model equilibrium, because the model is not constrained a priori to reach or approach a steady state, it does so through its internal interactions.  During the initial transient the economy is far from a steady state, and a neoclassical model is not capable of representing this phase.  Furthermore the model can deviate far from equilibrium under the action of internal forces, as two more examples will illustrate.

In a variation on the steady model, the price of property is assumed to rise exponentially, simulating a speculative bubble.  If the money supply is also taken to rise exponentially, and some of the bank’s profit is re-invested in its reserve fund, then an economy with perpetual inflation results, as shown in Figure 2 (note the logarithmic vertical scale, in which exponentials become linear).  This “growth” behaviour is readily induced in the model.  It is true that the growing property prices and money supply are imposed from the outside, but this type of model accommodates these influences just as readily as it accommodates the steady state of Figure 1.

Figure 2. An economy with perpetual inflation, driven by inflating property prices, with the money supply increased to match.

It is easy to experiment with this model.  One can, for example, keep the money supply constant as property prices increase.  In that case, not surprisingly, the bank vault is soon depleted, but the response of the other variables is also instructive:  firm and employee deposits and the bank transaction account continue to increase, but the firm loans peak and decline.  A recession would soon ensue.

Another experiment was to couple the property price to employee indebtedness.  Although the total of employee deposits was increasing in the model just mentioned, some deposits went into overdraft (not shown here:  more detail is in the reference).  Therefore the rate of increase of the property price was reduced in proportion to the overdrafts.  The result was that property prices peaked and crashed, taking the rest of the economy with them.  The primary variables are shown in Figure 3.

Figure 3. Primary variables for the case of a property crash.


Prices and wages are compared in Figure 4 for the three cases.  Prices are nominal prices for goods, whereas property prices are prescribed as already described.  Wages and unemployment (below) are calculated from a Phillips curve.  The exponential increase in land prices in the inflationary case (Figure 4b) can be contrasted with the peak and decline in the crash case (Figure 4c).  Wages and goods prices both decline in the latter two cases, though in proportion so that real spending power is roughly maintained.  However many employees are heavily in deficit.

Figure 4. Nominal prices for goods, wages and land prices for the three cases considered.


Unemployment is dramatically different in the three cases (Figure 5).  It is near 6% in the steady case, only about 2.5% in the inflationary case, but high early and then skyrocketing in the crash case.  It should be borne in mind that the unemployment rate is calculated from an empirical Phillips curve that merely characterises the way economies have behaved over the past few decades.  Further details of these models can be seen in the reference [10].

Figure 5. Unemployment for the three cases considered.


The point being made here does not so much concern the details of the models, nor their potential veracity, but rather the relative simplicity of the models and their ability to yield instructive detail on the interactions among the variables characterising the model economy.  They are also readily amenable to experimenting with various assumptions, as these examples illustrate.

In an example of instructive detail, Figure 3 shows that during the crash the bank vault reserves increase as firm deposits and employee deposits crash.  In other words money is withdrawn from circulation, and this will slow the productive economy.  Keen also found that bank reserves increase during his simulated credit crunch.  The important policy implication of this is that it does little good to boost bank reserves, as was done in the United States.  It does more good to boost consumer spending, as was done in Australia.

The mathematics

The mathematics behind these models comprises a coupled set of ordinary, first-order differential equations.  An example is the equation for the balance, ED, in Employees’ deposit accounts.

where t is time FD is the balance in Firms’ deposit accounts, and the other factors are rate constants representing the rate of pay received from Firms, the rate of interest received from the bank, and the rate of consumption expenditure.

Such a coupled set of equations may not easily yield analytical solutions, but it is readily integrated numerically.  Commercial packages such as Matlab, MathCad or Mathematica will do this routinely on a desktop computer.  Analytical solutions can be valuable if they can be obtained without undue simplification, because they reveal the internal interactions of variables explicitly.  However these equations are still simple enough (even though the set of them is quite large in Davies’ models) that the behaviour resulting from numerical integration can be understood fairly readily with careful examination.  Therefore these models can lead to useful insights.

Contrast with a neoclassical approach

Keen [6] has drawn attention to a draft paper by prominent economists Gauti B. Eggertsson (NY Fed) and Paul Krugman (Princeton, NY Times columnist, Nobel Laureate) that attempts to apply equilibrium modelling to the Global Financial Crisis [11].  Their paper illustrates fundamental problems with the neoclassical approach.  Keen gives a detailed critique, and only a few main points will be made here.

Most basically, equilibrium models cannot follow the system through a bubble and crash like that illustrated in Figures 3, 4(c) and 5(c).  They have to make do with before-and-after models.  However the “before” condition was not at equilibrium (otherwise it would not have crashed) so they cannot properly represent it.  Neither is there any assurance that the “after” condition is at any equilibrium.  Indeed, as there has been no fundamental reform, the global financial system is probably moving into another boom and bust sequence, which is intrinsically out of equilibrium.

Next, the models do not include money, or debt of any form, astounding as this seems to an outsider to the field.  The authors reveal a fundamental misconception by stating “Ignoring the foreign component, or looking at the world as a whole, the overall level of debt makes no difference to aggregate net worth – one person’s liability is another person’s asset.”  In the real world, when banks issue money by creating it out of nothing and “loaning” it, the “borrowers” can spend it, even though there is the formality of a book-keeping entry treating the borrower’s debt as a bank asset.  People can also fail to pay back the “loan”, thus creating a problem for the bank.  This is central to the dynamic of a boom and bust.  Keen notes that the authors also reveal their ignorance of Minsky and Schumpeter.

Lacking money, the authors contrive obligations between “impatient” agents and “patient” agents, where what is borrowed is not money, but “risk-free bonds denominated in the consumption good” (whatever that might mean).   They assume there is a ceiling on the amount that impatient agents can borrow and they contrive a crisis by lowering that limit for the second of their two equilibrium models.  Such contrivances are not necessarily a bad thing, if the model is carefully posed to reasonably represent an observed aspect of the world.  They might still be instructive in principle, if carefully interpreted, but in this case the models are so unrealistic that little useful is likely to be learned.

The neoclassical situation can be contrasted with the simple non-equilibrium models presented above.  Analogous contrivances have been used in the cases illustrated in Figures 2 and 3.  In Figure 2, the price of property is assumed to rise exponentially with time.  In Figure 3 the price is assumed to respond negatively to the level of overdraft of employees.  However these models were conceived as steps towards a more satisfactory kind of model.  Although they are already instructive in some respects, they have the potential to do much better.  The land price and the money supply can be made mutually dependent, which creates the potential for an internally-driven instability, and the model will then follow the dynamics that result, however far from any notional equilibrium it may stray.  Neoclassical models can never do that.


1. Davies, G.F., Economia: New Economic Systems to Empower People and Support the Living World. 2004, Sydney: ABC Books.  Pdf available here.

2. Beinhocker, E.D., The Origin of Wealth. 2006, Boston: Harvard Business School Press.

3. Waldrop, M.M., Complexity. 1992, New York: Touchstone. 380.

4. Davies, G.F., Heat and mass transport in the early earth, in Origin of the Earth, H.E. Newsome and J.H. Jones, Editors. 1990, Oxford University Press. p. 175-194.

5. Keen, S. (2010) Solving the paradox of monetary profits. Economics E-Journal 4,  DOI: 10.5018/economics-ejournal.ja.2010-31.

6. Keen, S., “Like a Dog Walking on its Hind Legs”: Krugman’s Minsky Model, 4 March 2011, in Steve Keen’s Debtwatch.  Also in Real World Economics Review Blog.

7. Keynes, J.M., Alternative theories of the rate of interest. Economic Journal, 1937. 47: p. 241-252.

8. Graziani, A., The theory of the monetary circuit. Economies et Societes, 1990. 24(6): p. 7-36.

9. Bruun, C. and C. Heyn-Johnsen, The Paradox of Monetary Profits: An Obstacle to Understanding Financial and Economic Crisis?, in Economics Discussion Papers, No 2009-52. 2009.

10. Davies, G.F., Exploring the dynamics of asset bubbles. 2011. 14 pp.  AssetBubbleDynamics (pdf 885 kb).

11. Eggertsson, G.B. and P. Krugman, Debt, Deleveraging, and the Liquidity Trap:  A Fisher-Minsky-Koo approach. 2010, 33 pages, here.

6 thoughts on “The Value of Simple Models, with Examples of Economic Dynamics

  1. Kevin Cox

    What you say is spot on. However, I think the system is even simpler than you imagine and it is easier to control and it easy stabilise the monetary system so that we do get a system in equilibrium – at which time the classical analysis will provide useful insights.

    Here is another way of expressing how the monetary system works and how to fix it.

    Each time a bank makes a loan, receives a payment, transfers money to another bank, or receives a transfer from another bank it can increase or decrease the amount of money in the system by deciding whether or not to increase its reserves. This is a random variable because on each transaction the banks have no idea whether to increase or decrease money.

    What happens is that they tend to try to increase the number of loans they make because they think they will make more profit from making a loan (providing it gets paid back) than not making the loan. Because of this tendency banks will tend to try to increase the number of money tokens in the system if there are not enough available. Now the way we increase the number of money tokens in the system is only limited by the reserves that banks have and by the loans they can make. In practise a bank can always get its reserves increased if it can make what appears to be a sound loan. Thus there is no limit on the number of money tokens that the system will create. In the current system these money tokens attract interest the moment they are created and hence they are expensive.

    However, if we put in place another way of creating extra money tokens that did not attract interest and that would increase the money supply with funds that were guaranteed to be paid back and guaranteed to create more value than the value of the newly created funds, then banks would tend not to create extra funds but would try to increase their reserves with these new money tokens. Once we have this in place then the money supply will become limited by the banks because they have a limit of having 100% reserves. In other words the system will rapidly move to banks not increasing the money supply. This will stabilise the system because the money supply increase will be limited by the number of interest free loans than can be profitably created. My guess is that in a stable system this will be of the order of 10% of all loans.

    The interest rate of money for regular loans will tell us if we are creating too much or too little money with interest free loans. We can address bubbles in the economy by putting interest free loans to work in the areas of the economy where there are price increases due to shortages.

    This system will stabilise the amount of money in the system and lead to a system that is in equilibrium – which will mean the classic economic models may well work as expected.


  2. Geoff Davies Post author

    Kevin – Not sure I follow your proposal yet, I’ll look at it some more. However I can say that it will take more than a better monetary system to stabilise the economy, and even if it can be made more stable the neoclassical assumptions will still not be relevant. The economy would be a less unstable nonlinear dynamical system (in a local equilibrium), rather than the stable linear dynamical system (in a global equilibrium) of the neoclassical fantasy.


  3. Kevin Cox

    The thing is often forgotten is that value is created and destroyed. When we invest successfully then we create more value than existed before. This is forgotten when people simply look at numbers. The financial system is really about dividing up the value of the system and not about the creation of value.

    We use money as the mechanism for dividing up the value. Money is not only something we use to exchange goods and services but it is also the mechanism that we use to exchange capital assets or means of creating value.

    The amount of money in the system should never be greater than the capital value of assets now or in the near future which means the time it takes to build an asset.

    Our current system permits the creation of more money than there are assets and more money than is likely to be created in the near future. It should not happen because loans (which are meant to create money) are only supposed to be created if there is an existing asset to back the loan. However, we can create extra money to build new assets by giving a loan backed by another loan. That is how the money supply increases. Assets are used twice or even more times to create extra money. Banks give each other loans and use their existing loans as collateral. The money that is created this way is interest bearing. This money may or may not be used to increase value.

    If however, we had another way of creating money which was guaranteed to create new assets which will be of greater value than the money used to create the assets then there will be less pressure on banks to create interest bearing money.

    This is what interest free loans used to invest in the creation of new value will achieve. Once this is set in place then we will find that the debt pyramid will unwind and that the amount of money in society will be less than the total value of assets because there will be little incentive to create interest bearing money, which is more expensive, than interest free money.

    This will come about by all the small decisions that banks make each day on whether to increase or decrease the ration of reserves to deposits. Banks will know that they are likely to get more deposits from interest free loans that are guaranteed to be repaid. That is a lot less expensive than getting more deposits from interest bearing loans from another bank.

    It is not necessary to do any modelling to see that is what will happen. What decision would you make as a bank? Would you borrow some money from another bank at interest or would you find a way of creating some interest free loans that are certain to be repaid?


  4. Neil Wilson


    “In practise a bank can always get its reserves increased if it can make what appears to be a sound loan”

    Banks are not limited by reserves, but by their capital and the regulation surrounding that.

    The limits of a banks lending in the current system is surely the lower of the number of people wanting loans at the price quoted and the leverage allowed against the banks core capital.

    Which is why you have banks whining about core capital and attempting to invent new wheezes that appear to increase it.


  5. Kevin Cox

    Neil there are regulations around both capital and reserves. Another constraint is that banks tend to only lend if they can get their money back immediately if required. That is, they can seize the collateral and sell it and hence the loans they make are supposed to be less risky.

    The evidence is overwhelming that these constraints do not work and the money supply is out of control. The method used to try control the money supply is the government fixing the rate at which they will increase in the money supply. In other markets we call this price control and we deplore it yet for some strange reason in money markets this is meant to be the way we control the supply of the commodity.

    With money this approach is particularly stupid. When we see there is too much money we increase the interest rate. This means that it becomes even more attractive for people who can create new money to create more of it – and so they find ways around the regulations to do it. In Australia the preferred method is to cause house price bubbles. This does nothing to increase the value of assets but everything for profits in the financial sector and in government revenues.

    When there is a shortage of money – interest rates go up – then we should create more money. We should not try to create more money by dropping interest rates because it makes everything cheaper and most of the demand is for consumption not investment.

    What we want to do is to increase the supply of money but make sure that the increase is used to create assets that will be greater than the increase in the money supply. In other words when we increase the supply of money we want it to be used to create more wealth not used for consumption.

    That is what I am proposing. Do not use the price of money to regulate the money supply. Use the price of money to tell us when we need to increase the supply and increase the supply by creating money that will result in more money than is created. It turns out the best way to do this is to build new assets that use less resources for the same output. This turns out to be what we need to do to give us a sustainable society. Do more with less while at the same time becoming wealthier.

    How do we create money that is guaranteed to create more value than is used? One way is to invest in ANY renewable technology. Do the sums and you will find that all forms of renewable energy (like wind farms, geothermal, solar thermal, tidal power, hydro schemes, solar voltaics), are profitable at today’s energy price – if they are financed with interest free loans that are repaid over the life of the asset.

    It really is that simple. Increase the money supply and use the money to create more value than the amount invested.


  6. Pingback: The Value of Simple Economic Models | Steve Keen's Debtwatch

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