Imported Dollar Multipliers and Local Economies (14jan09 update)

George Rebane

With the worldwide economic downturn that is being amplified by the mavens in Sacramento, there is a lot of interest and attention paid to local economies.  Here we are trying hard to keep existing businesses (by buying locally) and at the same time attract more firms or cash importers to our community.  In this piece I attempt to give a working understanding of the ‘output multiplier’ which summarizes the economic effect on a (local) economy from the arrival of an imported dollar.  (A more general discussion of multipliers is available in this short pdf Download EconMultipliers.)

It’s long been known that every dollar that comes into a local economy produces more than just a dollar’s worth of benefit to the community before it leaks back out.  This benefit is calculated using the so-called ‘Output Multiplier’ or simply ‘multiplier’ which is just a number like 1.83. 

The total benefit of an imported dollar is then equal to the multiplier in dollars – for example $1.00 * 1.83 = $1.83.  It’s as if the multiplied dollar amount was actually making the rounds in the local economy.  The reasoning behind such a multiplier is easy to understand.  Say, a retired person living here gets a check from a pension plan or an investment account.  Depending on her lifestyle, she may spend 60% (the retention factor) of her check in the community and the remaining 40% (the leakage factor) for expenses that are outside the community – for example state and federal taxes, online shopping, a big box store in the next county, and so on.

If we now assume that she’s a pretty typical spender, then we see that the 60% of the cash that stayed in the county will be spent in such a way that 60% (or 0.6 * 0.6 = 0.36 = 36%) of that cash also stays in the community, and so on until all the money leaks back out of the local economy.  The question now is, what is the sum of all the diminishing pieces or percentages of that imported dollar that benefit the local economy – that is the multiplier number that we seek.

Well, it turns out that we can calculate that multiplier from a formula that gives the sum of what we techies call an ‘infinite series’.  Using the above example, what we want is the sum of terms that represent the remaining parts of the original imported dollar which are simply 1 + 0.6 + (0.6*0.6) + (0.6*0.6*0.6) + … and on forever.  Actually, the follow-on terms become pretty small very quickly and we can stop the exercise then and there.

Let’s call the multiplier, which is the sum of the above series, by the letter M.  And let’s call the retention factor r which is a number from zero (0%) to one (100%).  Then from the above example, and remembering a little of your high school algebra, we can write the formula for the multiplier as

Notice that when you continue adding the smaller and smaller terms on forever, the sum of this series actually converges and can be expressed by the very simple formula on the right.  In our example r = 60% = 0.6, therefore the multiplier M is calculated easily as 1/(1-0.6) = 1/0.4 = 2.5.  So we conclude that if every succeeding transaction using the imported dollar keeps 60% of it in the local economy, then the cumulative effect on the economy is as if $2.50 were spent in the economy instead of only the original $1.  So far so good, and we can all see the benefit of such a multiplication process and also the importance of knowing the value the multiplier M for a community.

But at this point some of you may say ‘Wait, there’s a problem.  We don’t know the value of our M, and besides, that’s not the way a dollar will ripple through any local economy.’  And, of course, you would be right.  What actually happens is that the second person, who gets the 60 cents (in our example) from the retired lady, will only spend 40% of it locally.  And the third person, who gets the remaining 24 cents, may actually spend 70% (or about 17 cents) of it locally, and so on.  In reality, r, the retention factor or percentage can vary from person to person as the diminishing amount makes the rounds.

This should make it clear that r is really what techies call a ‘random variable’ that, at best, can only be known probabilistically (i.e. to within a probability distribution).  In a practical manner, the simplest such distribution is expressed as a range of values of r that defines a uniform or ‘boxcar’ distribution.  Don’t roll your eyes yet, specifying a range is just stating two reasonable numbers for the retention factor.  A reasonable answer may be ‘Well, I know that everyone has to ultimately pay taxes on their received monies, and those taxes total about 30% which go out of the community.  So the maximum retention would be around 70% or the maximum r = 0.7.  And we’re a community with lots of different kinds of shops, entertainment, and service outlets (doctors, dentists, auto repair), so I know that at least a third or 33% of every received dollar gets spent locally which gives the minimum r = 0.33.’

Now with this kind of approach you’re cooking.  You have just said in effect that the retention factor will range from 33% to 70% every time an exchange takes place, and it can be anywhere in between – you just don’t know any more than that.  This describes a useful (also ‘uniform’) probability distribution that works in the above formula.  All we have to do now is take the average of the specified range, that is the average r = (0.33 + 0.70)/2 = 0.515 and use it in M = 1/(1-r) = 1/(1-0.515) = 2.06.

How do we know that this multiplier value actually comes out of hundreds or thousands of such transactions in a community when each transaction retains a random percentage drawn from the above distribution – i.e. here between 33% and 70%?  Well, you can verify it by performing what is known as a Monte Carlo experiment on a computer.  I did a little noodling on the problem trying to get my arms around a practical way of how a community could find its multiplier. So I derived a couple of Monte Carlo models and wrote some code to demonstrate that, indeed, the above simple method does give you the expected value of the multiplier which is the average of thousands of transactions each exercised with various retention factors or percentages.

One can say a few more words about setting such limits for r.  The upper end will almost always be determined by the minimum ‘tax leakage’, and the lower end will be determined mostly by the size of the community or economy.  The larger the community that has more types of places for its residents to spend their money locally, the higher is the minimum value of r.  For example, a large metropolitan economy like Los Angeles may let us argue that about 90% of every spendable dollar is spent ‘locally’ - a remarkable concept given the sheer size of that ‘community’.  If so, then 90% of the remaining 70% (remember the 30% tax rate from the above example) comes to 63% as the minimum value of r.  The average of this tight range is approximately 67%, and putting 0.67 into our formula for M gives a large metropolitan area a multiplier of about 3.  This demonstrates why big cities thrive when they provide full services and lots stores; every imported dollar has the impact of $3 before it finally leaks out of such a large economy.

From these arguments, one can also see how lowering tax rates will increase the economic stimulus of every new dollar into an economy.  Just work out the same problem with the maximum value of r = 0.85 that represents a minimum total tax rate of 15%.  The multiplier goes up to 5.84; every new dollar in gives the economic benefit of $5.84.

Finally, consider possibly the smallest economy – a family.  Except for paying some allowances, and the kids paying each other for chores, the retention factor of what the wage earner(s) bring in is close to zero since everything has to be bought from outside the family unit.  Putting r = 0 into our formula gives the multiplier M = 1, so there is no economic benefit beyond the original dollar that arrives in the paycheck.  This argument can now be extended to very small communities, and here’s the rub.

When an economy loses the ability to import a dollar from the outside, it doesn’t just lose the benefit of that dollar.  Instead, it loses the benefit of that dollar multiplied by its current value of M.  The economic leverage of the multiplier is a two-edged sword.  In a small community such as western Nevada County, it is pretty safe to say that every time another dollar is not imported, the economic impact is as if about two dollars is taken out of our local economy.

Determining the multiplier for any given community is usually done by hiring a consultant who will go through an economic analysis based on lots of dubious data that gets crunched with complex but equally questionable methods.  It is not clear what incremental value such efforts deliver beyond what can reasonably be developed in the community with readily available ballpark data and local knowledge.  But having a believable value for a community’s output multiplier in hand does inject elements of reason into an otherwise economic development planning process based entirely on brown numbers, or no numbers at all.

[14jan09 update.  Here is a simple spreadsheet for estimating M as described above. Download MultiplierEstimation ]

[For the technical reader.  I was interested to see whether the random process that generates the aggregate multiplier is ergodic in the sense that the expected value of its ‘spatial distribution’ equaled its ‘temporal distribution’.  To test this I wrote two models and embedded them into one (Matlab) program.  The first computed the mean of the above function 1/(1-r) of a random variable the approximate formula for which involves the first and third terms of its Taylor series expansion that pulls in the dispersion (variance) of the underlying distribution for r.  The simulation runs entailed sampling the uniform distribution once and computing M.  The ensemble average compared well with the approximation formula.  However, as noted above, these samples don’t represent the realistic process of each transaction having a different retention factor known only to within a probability distribution.

The second simulation treated each transaction with a sampled value of r from which the random sum gave the sample value of M.  Happily, the mean of the aggregate sample values did indeed agree with the 1/(1-r) when computed with the mean of r from its underlying distribution.  Therefore, the output multiplier process is not ergodic, but does have a simple and practical solution.]   
       

 

Comments

[Nolan Love]

A regular reader of RR, this is my first comment, posted with the caveat that I have no background in economic theory other than a casual layman's interest and the occasional Wikipedia article. However, something about the logic of this post set off some warning bells, and I wanted to ask for some clarification and to posit my own theory of the potential benefit of an imported dollar.

To begin with, a challenge to the thesis of the output multiplier via reductio ad absurdum: Imagine a dollar imported to a micro-economy of baseball collectors via the sale of a baseball card. Then picture the same dollar being spent for an indefinite period of time among collectors to buy $1 cards from each other (for the sake of perpetual variety in each collection, let's say). Using the above logic, the multiplier approaches infinity (1/(1-r)) as r approaches 1.0, meaning the economic benefit to the community is immeasurable. But the dollar entered the economy at the same time as a one-dollar card left the economy, so there was no change in value of the overall system, only a conversion of one baseball card into currency, and a lot of transaction. For this reason, I don't think the metric is sophisticated enough on its own to measure economic benefit.

It seems then that the multiplier is a measure of the number of dollar-valued transactions a given dollar is likely to engage in during its tenure in an economy. This, however, is not a guarantee of economic benefit because it is not a guarantee of value creation. The true metric that makes the multiplier valuable is the degree to which spending locally stimulates value-creating activity within the local economy. Buying baseball cards from other members of the economy is economic activity, but it does not make the overall system more valuable. If however, a member of the community carves a walking stick and sells it for the aforementioned dollar, the overall system has increased in value by one walking stick.

The other question that needs to be addressed is what value left the system when the dollar was imported. Each economy has value in the form of property (e.g., land) and potential (e.g. the ability to create net value or wealth from raw materials or by performing services). If an economy imports dollars by exporting/selling non-renewable resources such as oil, or real-estate, or even personal property such as diamond necklaces, the net value of the economy does not change (given a stable currency). If, however, the citizens of the economy trade value that they themselves have created for dollars (e.g. by teaching kickboxing or by sculpting clay), they increase the value of the overall economy through their own voluntary industry which is itself renewable.

The next piece of my somewhat simplified puzzle (which does not address appreciation and depreciation among other things) is the question of how an exported dollar is spent. This dollar can be spent in one of two ways: either it increases the net value of the economy through the purchase of something that increases the overall property or potential (e.g. a tractor or a math lesson), or it decreases the net value because the dollar has not economically edified the community -- e.g. the purchase of entertainment.

Finally, getting back to the original subject of the output multiplier, we have to ask this key question: How much is the spending of each dollar within an economy likely to spur wealth creation therein? The simple, almost tautological answer seems to be "only as much as each citizen demands via his purchasing power something that requires creation anew within the system". This means that buying baseball cards and cars within Nevada County, even if done with great frequency (http://en.wikipedia.org/wiki/Velocity_of_money) will not increase the value to the overall economy. What is needed is the purchase of kickboxing lessons and locally made clay pots while there, the selling of locally created goods and services to those from outside, and the purchase of "economically edifying" property and skills from those outside of the system.

In conclusion, I believe that the output multiplier must be itself multiplied by the percentage of local transactions which are "value-creating" in order to get a sense of the economic benefit of an imported dollar.

[George Rebane]

Good points Nolan. The presumed use of these multipliers in promoting the building of economies is that, indeed, the succession of trades involves something more than the same type of baseball cards. However, your critique is equally valid to how GDP is computed by the government where each such transaction, no matter what kind of 'value' is produced or not, adds to the quoted total.

[Steve Frisch]

Nolan Love just hit on the reason why "Buy Local" and "Think Local" are not the same thing.

Buying local only creates wealth if value is created: if the item purchased is imported, if its local and the external costs are higher than the multiplier, or the dollar immediately leaves the community, the new wealth created is insignificant.

To create new wealth by "Thinking Local", (which includes encouraging diversity in local products and services, local manufacture, renewable resource management, local investment, local philanthropy and building local intellectual or physical capacity) is to leave lasting value in the community, which will then multiply.

[George Rebane]

Appreciably, the notions introduced by Nolan Love and Steve Frisch go beyond the mere definition of output multiplier which is fairly straightforward as seen in my explication. We all understand that each money transaction serves at least one of two major functions – creation of wealth (capital) and supporting consumption. The claimed benefit from the imported dollar, before it leaks out of the economy, is made up of both parts, but most certainly the dollar must support consumption with or without increasing local capital.

Before going on, I want to restrict my discussion to only sane types of economic behavior which will exclude most kinds of repeated exchanges of, say, baseball cards for money. In such exchanges the imported dollar will quickly evaporate from the community, and its departure as such will be captured in the value of the multiplier (M).

Both Nolan and Steve are correct in identifying that some imported money will (should?) be used in local transactions that build retained capital (recall, capital is anything that can be used to create wealth – cash, labor, machinery, land, intellectual property, …). But it is easy to see that there exist plenty of communities, mostly with low M values, that use the imported dollar only for consumption and little or no capital creation, save through the relative scarcity or increase in demand (an exogenous factor) for their very location and/or its resources.

When we consider the entire ‘buy/think locally’ issue, we are really talking about the attempt to make a community, a piece of land more self-sufficient economically. In modern times there are great economic forces arrayed against such insularity. Given facile means of communications and transportation, and the absence of government diktats – directly by checkpoints and barriers, and indirectly by taxes and tariffs – the modern world has on the whole increased our quality of life (QoL) by promoting the exact opposite of such aims and behavior.

Economics and human propensities have long made specialization and the economies of scale the basis for developing and delivering excellence in goods and services. The downside (if any) of such national and international progress has been the requirement of stable and mutually supportive civilizations which understand and teach the benefits of such behavior. For increasing the overall QoL around the world, the arguments for continuing in this direction are legion.

Multiple times on these pages (here) and elsewhere (here) I am on record for supporting more than less specialization in communities and countries. I am still not sure where the boundaries of self-sufficiency and insularity lie – they most certainly depend on the perceived goodwill of one’s partners and peers in such efforts as globalization – but I am fairly certain that expending the resources of a small economy, such as western Nevada County, to achieve a ‘balanced community’ in modern America is a counter-productive fool’s errand. Of course, as always, I invite reasoned argument to counter these beliefs.

[stan]

Ricardo showed that trade created more real wealth because of "comparative advantage". Having people engage in economic behavior which is comparatively disadvantageous simply because of a misplaced reliance on the multiplier effect is foolish.

[George Rebane]

Excellent point Stan and agreed. My piece is an explication of the multiplier itself and not an endorsement of one kind of trade policy vs another in order to maximize the creation of wealth in a given economy. To the extent that most small communities have little or no comparative advantage to offer, their ability to sustain themselves as a distinct community (for objectives other than maximizing wealth) mostly depends on the multiplier effect. In our own little community of western Nevada County, the only comparative advantage that we have is a desirable living environment for the retired (cash importers) and tourism drawn by our natural environment and cultural events. Nevertheless, this is not a widely held view, and therefore we continue to promote local policies that are designed to force/maintain a 'balanced community'.

[Cy Englert]

I appreciate all the information here and I am still struggling to find 'ball park' multipliers for different size communities. Does anyone know of a source of such estimations? For example, I'm wondering if multipliers are larger for remote, rural communitities since they must be by design more self-sufficient. Anyway, I'm building an ROI model incorporating benefits of microfinancing. The output multiplier is an important benefit in estimating the benefit to local communities.

[George Rebane]

Couple of points Cy -
1. The multiplier would be higher for larger remote communities that provide a large variety of goods/services. Small remote communities would suffer higher leakage especially as abetted by ecommerce.
2. The spreadsheet should let you develop usable bounds for the multiplier.

I think RR readers would be interested in your ROI work (especially incorporating micro-financing), if you care to share it.