Why loose lending and foreign buying can have large effects on property prices

Key Points
In asset trading markets prices are set through buyer competition
It takes only one extra buyer to bid up the price at an auction, whether they end up winning it or not

Price changes snowball
This month, extra buyers bid up the price. Then next month, they bid prices up from the new higher price that was established last month

Impact on asset trading markets
These two phenomena are why prices in all asset trading markets (where people buy and sell, like most houses and shares), change so much and so often compared to prices of consumer good (that consumers only buy, like most groceries and household items)

Conclusion
Any policy that brings more buyers into the housing market can have large effects on prices (and the same in reverse if you take those buyers away)

Extra buyer effects
In NSW last year, 11% of home buyers were foreigners. But is 11% a big number? What sort of effect on price could that much foreign buying have?

What we know for sure is that 11% of buyers does not mean that the presence of foreign buyers has made prices 11% higher. The price effect could be lower, or far higher. I suspect higher.

One trick to understanding the price effect of additional buyers in the housing market is to understand that potential buyers can affect prices without ever buying a home. It doesn’t matter if the extra buyers are foreigners or investors funded by loose lending. In all cases, not only do the extra buyers who end up buying a home affect prices but so to do other new buyers who didn’t end up buying.

Consider a home auction scenario. The highest bidder wins by exceeding the second highest bid by a tiny amount. But it may well be the case that this one person was willing to bid much higher to buy the property, but didn't have to.

Let's say for simplicity that the winning bidder was willing to pay $1.2 million (it is a Sydney house after all), and the second highest bidder (the under-bidder) was willing to pay $1 million. In this case, the winning bidder need only bid a little over $1 million to win the auction and set the price.

What happens if another buyer shows up at the auction and is willing to pay $1.1 million. They will take up the bidding after the previous under-bidder stops. Pushing the price to $1.1 million by bidding against the person willing to pay $1.2 million. The eventual result will be the same person wins the auction and buys the home, but the bidding process with the extra potential buyer sets the price at $1.1 million, or 10% higher.

What this small example demonstrates is that in a market like housing, additional buyers can influence the price even if they never actually buy anything!

My reasoning, therefore, suggests the price effect of the presence of extra buyers at the margin can have a large effect on prices relative to how many homes they actually buy. This is actually likely to be exacerbated in an asset market like property, as small rises in prices ‘reset’ expectations for future buyers about what the price should be next week or next month. So any small price effect at each auction with an extra buyer in attendance, setting a slightly higher price, is cumulative across the market and over time. These effects are why asset markets can be so volatile and cyclical.

One implication of this is that a sudden reduction in the presence of investors or foreign buyers in the Australia residential property market is likely to have a large negative effect on prices.

Demonstration with auction simulation
To get a feel of the potential size of the price effects from a new group of buyers such as foreign investors, who end up buying 11% of properties, I do the following auction simulations. In these simulations, the new buyers have exactly the same distribution of willingness to pay for homes as local do. The price effect comes from both additional under-bidding and addition winning of bids.

In the ‘before foreign buyers’ case, I draw 89 people out of a statistical distribution of willingness to pay. I use 89 people for the auction so that in the ‘after foreign buyers’ case I use 100, and the new people win the bid 11% of the time on average. The bidders are drawn from a normal distribution with a mean of $1 million and standard deviation of $150,000 to represent the likely willingness to pay in the Sydney housing market.

I then play an auction with the 89 people, where the price paid is the second-highest bid based on the slightly different willingness to pay of each person. The mean winning bid is $1.305 million. It is higher than the mean willingness to pay because the mean potential buyer almost never wins, as they are outbid by the people higher in the distribution of willingness to pay.

The ‘after foreign buyers’ case simply adds 11 extra people to the auction, so that there are 100 people, all drawn from the same distribution of willingness to pay. Here, the mean winning bid is $1.313 million.

That’s 0.6% higher.

That’s not much. In fact, that’s somewhat in keeping with analysis on the price effect of foreign buyers by Treasury. Their analysis looked at price difference between suburbs with high levels of foreign buyers and low levels, to conclude that the price effect of their presence is small. Others have argued similarly.

The cumulative effects
But this is not the end of the story.

There is a problem with my method, and with the method used by the Treasury. Treasury’s analysis assumes that the price effect caused by additional buyers in one area is fully independent of the way prices are set in neighbouring areas. This is unlikely to be true. In my analysis, I assume that the price effect at one auction has no bearing on the willingness to pay of all potential buyers at future auctions. Again, probably not true.

In reality, the prices that are set this week, or month, inform how much every buyer will be willing to pay next week, or next month. After all, where does the willingness to pay come from if not informed by previous prices and how they are changing?

So to get an understanding the total cumulative impact of this larger buyer pool we can take the 0.6% price effect at each auction and compound it to reflect the higher prices becoming incorporated in the willingness to pay of all buyers. There is no clear and correct way to do this, but two options that jump out are to compound weekly (people update their willingness to pay after last week’s auctions), or monthly (the update based on new price information once a month).

If we compound weekly, we get a cumulative price effect of 34.9% over a year. If we compound monthly, it is 7.1%.

What we see is that small effects at the margin matter if they are cumulative, and certainly the effect of more buyers in the property market will have such a cumulative feedback effect on prices.

It is important to note however, that these numbers just demonstrate what could be happening. They are not true of correct, unless by chance my simulation is a perfect reflection of reality. They simply demonstrate the mechanism by which a new pool of buyers who buy 11% of properties can effect prices.

What is definitely not happening is that 11% foreign buying means prices are 11% higher. They probably are higher, but we have no idea by how much. This simulation just shows the sort of range of price effects if 11% of buyers were foreign and they were willing to pay exactly the same as local buyers.

There is also a case where foreign buyers have a different distribution of willingness to pay. Because some foreign buyers may receive benefits from purchasing that are external to the property, like in some cases permanent residency, they may on average be willing to pay more than each local buyer.

If I extend the same simulation account for foreign buyers being willing to pay just 1% more, then the cumulative price effects could be in the range of 14% to 75%. Obviously the higher the difference in the willingness to pay, the much larger effect on prices!

So what?
Unfortunately we can’t say a lot about the real price effect from additional buyers in the housing market, be they foreign buyers or investors. But what we can say is that

• The share of foreign buying doesn’t really help understand the price effects very much 
• Additional buyers will increase the price of properties they do not buy through under-bidding 
• Small price effects from additional buyers are cumulative if all buyers incorporate the new market price into their future willingness to pay 
• If foreign buyers have a higher willingness to pay for other reasons, the price effect will be much larger 

Do economists even know what firms do?

Standard economic theory says that firms maximise profits, or revenue minus costs. I had always taken this to be a self-evident truth given how entrenched in economic thought it is.

But when we unpick what it really means we encounter problems and contradictions that are rarely discussed, and that undermine the core of the theory itself.

Consider the problem of comparing projects of different sizes. It is well known that the net present value method is not a good way to compare projects with different total costs. Spending $10 to make $10 in net present value terms is far superior to spending $100 to make $11 in NPV terms.

Yet the same logic applies to all the additional costs necessary to vary output in a firm. Each level of output requires different total input costs. Therefore, making a comparison between two output levels must consider alternative uses of any additional costs. Maximising profits without any consideration of the size of the costs incurred to obtain them makes no sense. But this logic still forms the core economic theory of what firms do when they choose how much to produce.

The only way to appropriately compare between projects of different sizes is to fully account for the opportunity cost of any extra costs if they were instead invested elsewhere. In this case, the optimal choice for a project is to produce at a level that maximises the rate of return on all costs, or profits divided by costs.

This is quite similar in many ways to choosing projects based on their profitability index, which is the NPV of future positive cash flows divided by the current costs.

I call this decision process return-seeking, and it is the focus of a working paper I have written with my colleague Brendan Markey-Towler (now published here). What is striking is how different the expected behaviour of a return-seeking firm is compared to a profit-maximising one, despite the relatively minor change in their objective. In general, this helps bridge some of the long-existing gap between the economic model of firm production behaviour, and the empirical realities of firm behaviour.

An example
To further explain this hidden contradiction within profit-maximisation by way of example, consider following the choice of output for a firm.

A firm faces fixed costs of $4, variable costs of $4, and revenues of $10 by producing 10 units of output. Profit is $2, and the rate of return on all costs is 25% (calculated by 10/(4+4) -1). This is shown in Panel A below.

The firm can increase output to 20 units and receive $20 revenue, but variable costs will rise to $13. In this case, profit is $3, and the rate of return on all costs is 17.6% (calculated by 20/(4+13) -1). This is shown in Panel B below. 

For the economically-trained it is clear that such a choice is the logical result of a situation where this project has rising marginal (and average) costs beyond 10 units of output, and where the marginal cost curve meets the demand curve (red) at q=20, and where at that point the average cost is $0.85.

It is the standard textbook treatment. And in this standard profit-maximisation view the extra dollar of profit available in Panel B should be sought even though it costs an extra $9 to get it. The extra costs required to earn the extra profits are ignored.

This is a problem.

We are now comparing two projects with different total costs. The correct way to judge whether the choice in Panel B is better is to consider not just Panel A as the opportunity cost, but Panel A plus $9 of investment elsewhere, so that the true opportunity cost of $17 of investment can be assessed.

So let us now take a step back and look how to evaluate whether spending $17 to make more profit by choosing Panel B is worthwhile.

In the panels below I show the project choices with an output of 10 units and of 20 units, in addition to two other project choices, C and D, that are potential alternative investment opportunities in the marketplace. If I now compare the true opportunity cost of the $17 necessary to take option B, I can see that if I had those resources to spend, I could invest in Panel A, C and D together, and make $4.30 profit on my $17.

Also note that Panel C is simply a duplication of the project in Panel A, which is there to make the point that to get double the output without increasing average costs, one can always duplicate their existing capital project.

The point here is that each dollar used to cover variable costs to increase the output of a project could be used to purchase new capital goods instead or invest in alternative projects, which could get a higher rate of return than it does at the margin by expanding the output of the single project. 

Where there are alternative opportunities for investment of these inputs costs, the logical thing to do is to make decisions about the output quantities of each capital project based on maximising the rate of return on all costs of each. This way, each scarce dollar is directed most effectively to maximising both its own profit and its overall rate of return. That is, the profit of each dollar of input is maximised.

Think of it this way

An intuitive story that clarifies the logic of the return-seeking model starts with imagining that every dollar of cost necessary for a firm to spend on production comes from a different investor. To expand output in the face of increasing cost you need to add more investors, who share profits in proportion to their contribution. You do this only if the overall rate of return on the total costs is increasing. Once you hit that maximum rate of return, adding additional investors to cover greater costs reduces both profits and returns for existing investors.

Overall, return-seeking is actually not much different to the standard view. All it does is make explicit the implicit denominator of fixed capital that exists the short-run profit maximisation model, and say that all costs (not just capital costs) have the option to be capital in alternative projects.

Of course, the standard view can be rescued by assuming that there are no alternative options to spend those additional input costs on. In that case, it is consistent with return-maximisation and is equivalent to having absolutely zero alternative ways to spend those additional input costs.

Making sense of economies of scale

What makes the return-seeking model interesting is that it precludes projects that do not have any economies of scale. If costs are always rising with output, the rate of return on costs can be increased by decreasing the output of the project.

Below I change our example to show a modified project that does not exhibit economies of scale. If output is one, as per the new Panel A, the unit cost is lowest and the rate of return is highest, at 33%. Here, the choice confronting the firm if it wishes to expand output is whether the extra $7.25 in cost necessary to get to Panel B is better spent on investing in nine new extra firms like Panel A, producing one unit each at a cost of $0.75. Clearly producing 10 units of output with ten projects like Panel A is superior, as it makes more profit ($2.50) on less cost ($7.50), and maintains the high 33% rate of return of all costs incurred.

Overall, the view of firms as return-seekers gives a new way to look at how output choices might be made. Indeed, one of the great problems of the standard profit maximisation view is that it rarely matches the empirical record, which includes evidence from surveys of firm managers about their output and pricing decisions. Return-seeking does, in fact, match many of these empirical results more closely, and could offer clues about how to bridge the reality gap between the economic model and real economic decisions.

Read more about return-seeking in our paper, and see some previous blog discussions of this idea here, here, and here.

Economics of empty homes

Prosper Australia has for years been conducting research into how many of Australia’s 9.8 million homes are left vacant. Their major finding is that of the 1.7 million homes in greater Melbourne alone, around 82,000 are vacant, or 4.8%. Their research has been cited by a recent United Nations study on the pernicious effects of the financialisation of the housing sector, and has likely been a key reason for the adoption of a vacant housing tax in Victoria, and probably in Canada as well.

It is timely, therefore, to consider some of the economic and practical realities of vacant housing.

Why keep property vacant?
What gets lost in the hype is this important question. Very few people understand the economic rationale behind leaving homes vacant, as the commonsense view is that a vacant home is always costly since it is forgoing rental income for its owner.

The answer is ‘options’. What I mean here are real options. That is, keeping the home vacant keeps open the valuable option of selling the property vacant and earning a higher price.

Let me explain by way of example.

Say you have a property that could sell for $500,000 if there is a sitting tenant in place, or $520,000 if it is vacant (a 4% price boost from vacancy). This means that the option to sell vacant is worth $20,000.

If you are considering selling in the near future because you want to time your exit from the market, then you may want to forgo rent in order to keep your option of selling vacant open. If the annual rent is $19,000, it might be worthwhile to forgo a whole year’s rent because it is less than the value of the option of selling vacant.

Quite clearly, if you make this decision, you aren’t in the housing market to be a long term supplier of rental housing, but to time your exit and cashing your capital gains.

This is why the financialisation of housing, which encourages speculative buying and selling (getting most of your return from capital growth), rather than long-term investing (get most of your return from rental income), makes housing markets fail in their primary social function of supplying secure housing.

The same logic is at play with vacant land. Given that there is always a positive return to be had from developing land, the very existence of vacant land should be a puzzle. But it is again a real options problem.

If I have a vacant site that I can economically build a 5 storey building on today, going ahead with the build removes my option of making even larger profits from building a 10 storey building in a few year’s time when higher prices make a larger building more profitable. This happens in the absence of any zoning controls, since as prices rise, a larger scale of development becomes more profitable.

So vacant land is only vacant because the landowner is waiting for their development options to increase in value.

But this can be stopped. In my example, if there was a zoning limit of five storeys, there would be no future option of building a 10 storey building, only a 5 storey one at a later date. This makes the value of waiting less, and encourages faster supply.

This is not just my opinion. Here’s an excerpt from a 1985 article by Sheridan Titman who asked this exact question and published his results in a little journal called the American Economic Review, in an article entitled Land Prices under Uncertainty.

It is shown that the initiation of height restrictions, perhaps for the purpose of limiting growth in an area, may lead to an increase in building activity in the area because of the consequent decrease in uncertainty regarding the optimal height of the buildings, and thus has the immediate affect of increase in the number of building units in an area.

A ballpark estimate
The next question is to ask how many homes may be vacant primarily because of this speculative motive. Prosper uses water meter data to asses whether a property has been vacant. By looking at properties that have used no water over a 12 month period (25,000 dwellings), and those that used less than 50L per day over a 12 month period (82,000 dwellings), they make a judgment that these extremely-low-water-use homes are vacant.

To answer how many vacant dwellings there are nationwide we can make a ballpark estimate by scaling up the results of Prosper’s research to other capital cities based purely on the relative size of the dwelling stocks. This method relies on the assumption that if it is logical for owners in Melbourne to keep that share of dwellings vacant, it is equally logical for owners in other states.

The reason to do this, rather than simply recreate the research using water meter data, is that in Queensland and New South Wales, apartments are not all individually metered for water, but metered only once of the whole apartment building. So their approach fails to catch vacant apartments when used in other areas. Electricity usage data, or data from other utilities such as internet and gas, can also be troublesome for this purpose, both in obtaining reliable data from utility companies, and making judgements about what constitutes a vacancy.

When scaling up Prosper’s results from Melbourne we can be conservative, and instead of taking the 4.8% number as the share of vacant dwellings, take a clean 4%. We can also make some other (somewhat) justifiable downwards adjustments for other states where the value of the “vacancy option” is lower because prices have been more stable. The table below shows this calculation.

	Total dwellings ('000)	Potentially vacant ('000)	Adjusted	Adjusted reason
Vic	2,507	100	100	None
NSW	3,026	121	121	None
QLD	1,956	78	59	Stable prices means less value from vacancy option (x0.75)
SA	765	31	23	Stable prices means less value from vacancy option (x0.75)
WA	1,058	42	21	Falling prices means vacancy options less valuable when cashflow a priority. (x0.5)
Tas	242	10	7	Stable prices means less value from vacancy option (x0.75)
ACT	164	7	5	Stable prices means less value from vacancy option (x0.75)
NT	84	3	2	Stable prices means less value from vacancy option (x0.6)
TOTAL		392	338

As a ballpark, around 300,000 out of 9.8 million dwellings are likely to be sitting vacant each year, or about 3% of them. For the last five year,s the country has built about 153,000 net new dwellings each year, so these vacant homes represent about two years of new supply at our recent historically high rates of dwelling construction.

So what?
What’s the big deal then? Two things. First, if you think that the supply side of the housing market is a major determinant of prices, having two year’s of new supply already built but sitting vacant is bad. Second, even if you don’t think this much supply has any significant effect on prices or rents (which I don’t, probably around 1-2% at most) then the main rationale for concern is on economic efficiency grounds. These vacancies are a symptom of bad housing policy.

When the housing market turns downwards, much of this vacancy will solve itself as owners look to buckle down to ride out the downturn and generate the rental incomes instead of capital gains.

The historical data from Prosper’s Speculative Vacancies Report confirm this pattern. After the financial crisis their speculative vacancy measure fell from 7% to 4.4% in the following four years, but since 2013 has begun to rise again as property prices started once again began to increase rapidly.

When the next downturn comes, those who are unwilling to accept the price they can get from their option to sell, will probably take the option of renting instead, bringing a massive dose of new supply into rental markets.

In this context, a tax on vacant housing can act as a dampener on speculation, as it makes more costly the speculative option of keeping property vacant. In practice, Canada’s vacant home tax will rely on declarations by owners and spot checks to ensure compliance. This is really the only way. Relying on water data would simply encourage owners to leave taps on to avoid the tax.

But in many ways, speculative vacancies are a symptom of a poorly regulated housing market that is attracting speculative buying (undesirable) rather than long term investing (desirable). We can some of the underlying causes in a much broader way, such as by restricting speculative lending into the housing market in the first place, like by banning interest only loans. With stable prices, only investor buyers looking to earn an income from renting will be likely to invest.

Revisiting the mathematics of economic expectations

The below presentation by Dr Ole Peters opened my mind. If there was one thing I believed was reasonable about economics, it was the assumption that expectation values upon which agents base their decisions are the “ensemble mean” of a large number of draws from a distribution.

Surely there is nothing about this simple method that could undermine the main conclusions of rational expectations? Surely this is a logical benchmark, regardless of whether actual human behaviour deviates from it.

But now I’m not so sure. Below is a video of Dr Peters making the case that non-ergodicity of many economic processes means that taking the ensemble mean as an expectation for an individual is probably not a good, or rational, expectation upon which to base your decisions.

I encourage you to watch it all.

Let me first be very clear about the terminology he is using. He uses the term ergodic to describe a process where the average across the time dimension is the same as the average across another dimension.

Rolling a dice is a good example. The expected distribution of outcomes from rolling a single dice in a 10,000 roll sequence is the same as the expected distribution of rolling 10,000 dice once each. That process is ergodic [1].

But many processes are not like this. You cannot just keep making the same gamble over time and expect to converge to the mean of the result that you get if you made that gamble independently many times.

An example

Peter’s example is this. You start with a $100 balance. You flip a coin. Heads means you win 50% of your current balance. Tails means you lose 40%. Then repeat.

Taking the ensemble mean entails reasoning by way of imagining a large number coin flips at each time period and taking the mean of these fictitious flips. That means the expectation value based on the ensemble mean of the first coin toss is (0.5x$50 + 0.5*$-40) = $5, or a 5% gain. Using this reasoning, the expectation for the second sequential coin toss is (0.5*52.5 + 0.5 * $-42) = $5.25, another 5% gain.

The ensemble expectation is that this process will generate a 5% compound growth rate over time.

But if I start this process and keep playing long enough over time, I will never converge to that 5% expectation. The process is non-ergodic.

In the left graph below I show in blue the ensemble mean at each period of a simulation of 40,000 runs of this process for 100 time periods (on a log scale). It looks just like our 5% compound growth rate (as it should).

The dashed orange lines are 10 sample runs of the simulation. Notably, the distribution of those runs is heavily biased towards low final balances, with a median payoff after 100 rounds of $0.52 Recall that the starting balance was $100, so this is a 99.5% loss of your original balance.

In fact, out of the 40,000 runs in my simulation, 34,000 lost money over the 100 time periods, having a final balance less than their $100 starting balance (or 85% of runs). Even more starkly, more than half the runs had less than $1 after 100 time periods. 

The right-hand graph shows the final round balances of the 40,000 simulations on a log scale. You can read more about the mathematics here.

If almost everybody losses from this process, how can the ensemble mean of 5% compound growth be a reasonable expectation value? It cannot. For someone who is only going to experience a single path through a non-ergodic process, who has a finite budget to play with, basing your behaviour on an expectation using the ensemble mean probably won’t be an effective way to navigate economic processes that are non-ergodic [2].

Peters says the logical thing to do is maximise the average expected rate of growth of wealth over time, rather than the average outcome across many alternatives. In this case, the average rate of growth over time of all runs in the simulation is actually negative 5.03%, meaning it is not a good bet to partake in despite the traditional assessment of expected returns being 5%.

Lessons for Economics

I see two areas of economics where we may have been misled by thinking of the ensemble mean as a reasonable expectation.

First is a very micro level concern: behavioural biases. The whole idea of endowment effects and loss-aversion make sense in a world dominated by non-ergodic processes. We hate losing what we have because it decreases our ability to make future gains. Mathematics tells us we should avoid being on one of the many losing trajectories in a non-ergodic process.

The second is a macro level concern: insurance and retirement. Insurance pools resources at a given point in time across individuals in the insurance scheme in order that those who are lucky enough to be winners at that point in time are able to make transfers to those who are losers. By doing this, risk is shared amongst the pool of insurance scheme participants [3].

Retirement and disability support schemes are social insurance schemes. They pool the resources of those lucky enough to be able to earn money at each point in time and transfer it to those that are unable to.

But there has been a big trend towards self-insurance for retirement. In the US they are 401k plans, and in Australia there are superannuation schemes. The idea of these schemes is that rather than pooling with others at each point in time (as in public pension systems), why not pool with your past and future self to smooth out your income?

You can immediately see the problem here. If the process of earning and saving is non-ergodic and similar in character to the example above, such a system won’t be able to replace public pensions at all. Many earning and saving paths of individuals will never recover during their working life to support their retirement. Unless you want the poor elderly living on the street, some public retirement insurance will be necessary.

Undoubtedly there are many more areas of economics where this subtle shift in thinking can help improve out understanding of the world. I’m thinking especially about Gigerenzer’s idea of a heuristics approach as a generally effective way for humans to navigate non-ergodic processes.

I will leave the last word to Robert Solow, who has had similar misgivings (for over 30 years!) about our assumptions of ergodicity (a stationary stochastic process) which undermine rational expectations.

I ask myself what I could legitimately assume a person to have rational expectations about, the technical answer would be, I think, about the realization of a stationary stochastic process, such as the outcome of the toss of a coin or anything that can be modeled as the outcome of a random process that is stationary. If I don’t think that the economic implications of the outbreak of World war II were regarded by most people as the realization of a stationary stochastic process. In that case, the concept of rational expectations does not make any sense. Similarly, the major innovations cannot be thought of as the outcome of a random process. In that case the probability calculus does not apply.

Footnote [1]. He does not use the term, as it is often used in economics, to describe what is called the term Lucas critique, or in sociology is called performativity. Basically, it is the idea that the introducing a model of the world creates a reaction to that model. Take a sports example. As a basketball coach, I look at the past data and see that three-point shots should be taken more because they aren’t defended well. I then create plays (models) that capitalise on this. But because my opponents respond to the model, the success of the model is fleeting.

Footnote [2]. In theory, if you start with an infinitely small gamble, or have infinite wealth, you could ‘double down’ after a loss in such a process to regain the ensemble mean outcome.

Footnote [3]. Peters himself has a paper on The Insurance Puzzle. The puzzle is that if it is profitable to offer insurance, it is not profitable to get insurance. The typical solution invokes non-linear utility to solve it. Peters offers an alternative. My take is on the economic implications of this is that if people can individually smooth consumption through time for retirement than there is no logic to social insurance.

This is an update of a post from June 2016.

UPDATE (26/03/2017 9.10pm): On Twitter, it has been mentioned that I have simply restated the logic of the Kelly Criterion. This is true. The logic here, and there, is at odds with the naive way in which odds are translated into rational expected payoffs in economics. In fact, adopting the Kelly Criterion when playing the betting game in the above example generates an expected rate of growth of wealth over time of 8.65%, instead of negative 5.03%, and a far higher and narrower distribution of final period wealth outcomes. The paths of the simulation for betting under this condition, and the distribution of final period wealth, are shown in the below graphs. Notice that this strategy is highly effective at both changing the distribution of outcomes AND increasing the overall rate of growth of wealth. Regardless, the very fact that such a strategy is needed tells us that there is a problem with what a rational expected outcome should be for non-ergodic processes