Following on from the happiness maths and the associated notes about the value of toy models, here is a toy economic model and some notes about what it might mean for regeneration of local economies (also known as ‘are you sure you want to knock down those shops and build a supermarket?’). Comments on both the economics and the epistomology very welcome…

Imagine that a local economy consists of players (businesses, individuals) which spend a certain amount of money. Some of that money they spend within the local economy, so it becomes part of the income of other players (ie it multiplies). Some of it leaks getting spent elsewhere (saved in banks, put under the bed, frittered away on Amazon – it doesn’t matter, at a first approximation that money has disappeared from the local economy). The amount each player has to spend is defined by their total income, and the total income is made up of the money they get from other players plus an amount which is independent of the local economy. This part, lets call it the external income is made up of the money paid from outside the local economy to that player, plus anything they generate themselves (i.e. something reflecting their productivity); for the model it doesn’t really matter how this money is generated – just that it is independent of what else is going on in the local economy.

Given an economy like this, it is possible to use a model to show that the following can happen: one of the players can be replaced with a more productive one (it generates higher external income), but overall there comes to be less money in the local economy because that player spends less of their income locally.

The conclusion I’m going to argue for is that you can’t economically justify replacing units in the local economy by virtue of increased income alone (a static value), but instead need to take into account the multiplier effect. First i’ll go through the model, and then i’ll cover the logic of coming to this conclusion (so if you’re already convinced of the existence of such a scenario, but skeptical, of the logic you may want to skip to there).

a toy model of a local economy

There are three players. At each point in time, t, each player has a total income which is the sum of external income and internal income. External income is a constant value for each player. Internal income for each player is defined by the total income of the other players at the previous time-step, t-1, and a set of fractions which determine how much of each player’s total income is spent with each other player (the multipliers).

The precise numbers used for the constants don’t matter, and you can see how i’ve implemented the equations here.

A couple of observations on this basic system:

• while the sum of multipliers for each player is less than 1 (ie they don’t spent more money locally than they make) the income of each player reaches an equilibrium level.
• the equilibrium levels are defined entirely by the external incomes of each player and the local multipliers each posseses (more on this below). In other words they are independent of the starting income you provide the model with.
• the equilibrium values are stable under perterbation – i.e. adding zero-mean noise means incomes just fluctuate around the equilibrium values that would exist without noise.

You can see these points illustrated here:

Now the incomes for the three players in the above graphs are different, because I chose different random external incomes and multipliers – but the same principles apply if you make all the players identical, so that they have the same external income and the same multiplier fractions. If you do this then you can vary the external incomes and multipliers as single parameters, and look at the effect on total income.

Total income varies linearly with the amount of external income:

However, total income is a non-linear (exponential) function of the multiplier:

This implies that total income is more responsive to changes in the multiplier than to changes in the external income (although it is hard to directly compare the two precisely because external income is an absolute value and the multiplier is a fraction – is there any meaningful way in which you can say that, for example, it is easier to double external income than it is to double the multiplier?)

Because of this, it is possible to replace one of the players with a player which generates more external income, but because they spend less of it locally (ie their multipliers go down) overall the total income of the local economy suffers:

So this, say, is something like building a supermarket over some local shops because it will bring in more money. It does bring in more money, but because it spends a lower proportion locally the local economy suffers. Notice how the income of the replaced player goes up initially, then drops. The increase is because it brings in more money and the other players still have a high income, part of which goes to the replaced player. But as the replaced player spends less with the other players the total income of the economy drops and the replaced player’s income drops in line.

It doesn’t have to happen like this. Depending on how you adjust the numbers you can get the whole economy benefitting, or even just the replacement player benefitting and the rest suffering a drop in income:

This scenario is generated by taking the same initial values as all the other scenarios, but when replacing one of the players increasing the external income by the same proportion as the multipliers are decreased by. Specifically the external income for player 3 is increased from 5 to 40 and the multipliers dropped from 0.43 and 0.46 to 0.05 and 0.06 (so a change of a factor of 8 for both variables, but in different directions). In this scenario the income of the replacement player increases, but the income of the economy as a whole drops.

the logic

This post is motivated by the sentiment of economist Paul Krugman, We just don’t see what we can’t formalize, and by the very real possibility that a development masterplan in Sheffield will knock down some of the best bits of Burngreave and build a supermarket (link). The motivation for this is – or at least is claimed to be – explicitly regeneration based. Supermarkets, of course, generate lots of income, income is good for the local economy, is regeneration, right?

The comeback is that supermarkets force local shops to close down, increase car use, damage community, etc, etc. But aside from this, and specifically, it’d like to make a point about assuming that more income implies a richer local economy.

The purpose of the discussion of the model above is to show, in detail, how including the concept of money circulation in your thinking can affect your conclusions. Specifically, total income in a local economy can decrease when individual income and spending increases, if enough of that additional income is not spent locally. You don’t need the model to prove this – it’s just true – but, remembering Krugman, the model might help get some grips on the angles.

So does the model show? How can we use it? What does it, fundamentally, mean?

At a minimum, only this: systems defined in this way (discussed above) can behave in these ways (discussed above). This is about as exciting as saying that adding 1 to 1 gives you 2. The outcome is defined completely by what you start with.

Perhaps a little further to this we can generalise our model and say: in a toy economy with external inputs and local multipliers the total equilibrium income is defined by both factors (and decreases in one can offset increases in the other).

Our toy model shows that such systems can exist, and can behave in the way discussed. It is another step again to work out if our toy bears any resemblance to a real economy. Scientifically, this is the important thing, but, i’m going to argue, perhaps for the present purposes not essential. Perhaps it is enough to show that such a result could plausibly occur, not that it must occur. Doesn’t that put the onus of proof on those advocating building the supermarket? It certainly, to my mind, suggests that using a single static statistic (the amount of extra money that will be spent in a supermarket if it is built) is an inadequate way of conceptualising the economic benefits of building a supermarket.

All the model proves – all any model can ever prove – is the equivalent of ‘1 + 1 = 2’. Sometimes, with complex models, that’s a result in itself – if it wasn’t obvious to everybody that ‘1 + 1’ would lead to ‘2’.

Beyond that, though, the model doesn’t do any other scientific work. There’s extra research in the real world needed to show that we’ve actually got ‘1 + 1’. If we can show that, then we can combine it with an existence proof, from the model, that this might add up to ‘2’ (but we still can’t prove it definitely will add up to ‘2’, because the model doesn’t show that ‘1 + 1’ is all we have, we might have an extra ‘+ 3’ as well, so to speak).

If we can prove that we’ve got ‘2’, then the model offers us a sufficiency proof that having ‘1 + 1’ is enough to get it (but it doesn’t show that the ‘2’ was definitely made by ‘1 + 1’, it could have been made something else, maybe ‘7 – 6’).

In terms of proof then, this toy model is utterly banal – it just shows that this certain kind of result can exist. Wider than this, however, there is the other value of models – lets call it their suggestive power – and hopefully this toy model also has some minimal worth here. Are you sure that a supermarket that brings in lots of income is good for the local economy? Hopefully less sure now.

And remember, kids, All models are false, some are useful