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

Housing affordability: An honest debate

I spoke on Sunday at an event in Sydney, hosted by Sustainable Australia, called Housing Affordability: An honest debate. A video should be on the Facebook page and is below, as are the notes for my talk.



Imagine explaining 2017 to someone from 1987
I would like to start today by imagining a conversation. It’s the conversation I would have with my grandfather if I could go back in time 30 years to 1987. It might go a bit like this.

Me: 
Grandad, how are you? I’m so glad I could come back from the future to tell you all about the amazing progress we have made.

Grandad:
Oh, look at how you’ve grown. What amazing things can you tell me all about it.

Me:
Well, remember how you were the first person out of everyone you knew to fly in an aeroplane? Well, in 2017, people fly all around the world all the time. Not only that, you know how you saw a computer once? Now everyone has a computer in their pocket. It’s like every encyclopaedia in the world has been shrunk down. And we can video call anyone, anywhere in the world, anytime. We have clean solar energy now, and China, South Korea, and many countries in Asia have become wealthy and brought their people out of poverty.

Grandad:
That does sound rather fantastical. And how are you?

Me:
Well, I’m 34 now. I got a university degree, and a PhD, so I’m the first in the family for that. And I have a loverly wife and two young boys who are 6 and 9.

Grandad:
Oh how lovely. You must have a beautiful house and garden then with all those grand things the future holds.

Me:
Well. Actually no Grandad. We live in a house smaller that the one Mum and Dad moved into when they were first married. It’s very similar in fact – I don’t think it has been renovated or improved in 30 years. And our garden? I’ve tried, but I don’t want to put too much money and effort into it. Because I rent Grandad. Although I have a university degree, and so does my wife, we can’t buy a house. They cost $1 million in and around Sydney. Even in Brisbane, homes are around half a million. Less if you go out of town, but it’s hard to find work outside of the city. And we have no chance of saving more than one hundred thousand dollars for a deposit when our rent takes up 40% of our income.

And what’s worse, the price of the average house in Australian capital cities went up 11% last year. In Sydney and Melbourne, the average home went up in price in one year by more than the average wage! So saving up for a deposit is like running up the down escalator. You never make progress!

So we are in a pickle. I can’t have a nice garden, because if we do spend the money and time to spruce it up, the real estate agent will see it and realise that they can rent the house for more, then ask us to pay for it by putting the rent up at the end of our lease. And every year, when the lease is up, we don’t know if we can stay anyway. We’ve moved 4 times in the last 10 years. I don’t really think we will ever have a house and nice garden. Probably not even a nice apartment.

Grandad:
What an absurd situation. You have all this technology – a computer in your pocket, world travel – and you can’t have a place with a nice garden to call your own.

How can we change my story?
Despite being absurd, this conversation contains many truths about modern Australia. And it doesn’t have to be this way. The past 30 years could have taken another path.

By many metrics Australia is one of the wealthiest handful of countries in the world. So why are we having an event today talking about the basic challenge of making housing secure and affordable?

If anyone can do it, we should. Many of our wealthy peer countries have established home ownership is a genuine option, and they’ve made law that make renting a secure alternative.

You see, the heart of this story not about housing, but about economic diversity. When wealthy countries ensure that economic investment is made across many different sectors, they establish demand for skilled workers, demand for cheap housing also comes from the companies who need to attract those workers. Mobility, in terms of households moving cities for work, has declined, mostly because of the challenge of finding affordable accomodation.

We could be in a different place today. We could be talking instead about Australia cementing its place as a global manufacturer of solar and wind energy, one of a our natural advantages. We could be talking about how a world class satellite industry and growing remote sensing industry has revolutionised agriculture, and how we’re exporting this expertise to the world. I could have regaled my grandfather with these wonders instead.

The reality however is that Australia has dropped in the world rankings of economic productivity and diversity. Harvard University researchers called Australia a ‘laggard’ of the Asian region, along with Mongolia and Papua New Guinea. We’ve dropped in their rankings from 32 in 1994 to 54 in 2008, and are still declining. Our exports are now dirt. Mostly coal and iron ore. Canada still beats us, ranked 29, but having declined from being ranked 19 just 20 years ago. Canada too has similar housing affordability problems to us.

As a side note, Canada is second place in the developed world in terms of immigration rates. If Australia just decreased the immigration intake to match the rate of Canada over the past ten years, there would be 1.3 million fewer people. That’s three Canberras!

We can actually see part of the cause of our narrowing of our economic base in the pattern of bank lending. Fundamentally, the pattern of new lending determines the patterns of real investment in the economy. It determines where we invest in new manufacturing and production capabilities, or instead take leveraged gambles by reselling homes to each other. That’s why the fast growing Asian countries use State-owned banks to direct investment towards new production facilities that expand their capabilities.

In the early 1990s, over 50% of Australia new bank lending was for business investment. Only 4% was for investor housing. 25% OO housing.

Now? It’s down to 30% for business. Investor housing has jumped from 4% to 20% of lending, and OO lending too has leaped from 25% to nearly 40%.

But it’s worse. 95% if that massively increase investor housing lending is for established homes, up from 50% in the late 1980s. Only 5% is for newly constructed homes, down from 50% a quarter of a century ago. We are directing our powerful monetary system, our bank lending, away from new investment in new housing, towards trading the same homes with each other. Home ownership rates have steadily declined for twenty years.

Why is this?

In many ways it was a conscious political choice. While the decline of our economic diversity, has failed the average worker, it has been a boon for the landlord class. Those who already own land and housing benefit at the expense of those who want access to housing for their own household security.  Those who own the banks benefit too. And we have seen the enormous lengths to which government will go to support the way things are. Every “affordable housing” policy, from FHBG to super access, is designed not to let housing prices fall, and housing become genuinely more affordable.

What can work?
While the main challenge here is political, one thing we can do is have a shortlist of effective policies proposed, analysed, and scrutinised, that are ready to be adopted when the political timing is right.

Let’s start by debunking a furphy. Anyone who says that we need less regulation and freer markets to solve housing affordability is a fool. They have not though this through. They have ignored all the evidence from extensive regulations in wealthy countries that do promote affordable housing.  Australia’s problem come in many way from “market forces”.

I’m an economist, I love markets. But I’m not an idiot.

In terms of the larger economic diversity problems, much of this stems from lobbying pressure to sell the assets of the nation to well-connected interests on the cheap, with these new monopolies doing the rational market thing of constraining output and increases prices. No longer do we have a public sector able to use these institutions to take risks, or invest in new technologies that have obvious public benefits.

Public investment has always laid a platform upon which the private sector has built, innovated, and thrived. We know that the space race of the 1960s had huge economic benefits for the economy broadly. We know the many waves of public investment in our city infrastructure also supported private sector producers in terms of transport, communications, water, energy and health. The same is true today. We actually need to defy market forces if we want to make progress. We need to gamble with public investment for major capital projects – yes, like the NBN – for the future.

For example. it’s not too late to do economic CPR on the manufacturing sector if we want to, and that sector will help us invest in the clean energy and information technology future.

Moving on to banking. We should see new lending and money creation, as a social function. It’s a public good. And we have let this money creation gift that we provide our banks to be used to fund land and housing speculation, rather than new businesses and real investment. This is the natural market outcome in the absence of regulations that direct our monetary system is be used for broad economic goals.

We need to massively tighten lending rules for banks to the housing market, and force them to look elsewhere and take risks on genuine productive investment. We could go beyond LVR rules, to loan-to-income rules, that restrict investor lending based on the rental income of the property. We could completely ban investor housing lending except for newly built homes, which do not expand our productive capacity. While it seems extreme, being a wealthy nation where citizens cant afford housing IS AN EXTREME problem!

We should be taxing unproductive activities, and un-taxing productive ones. That means charging for our land and mineral rights. The ACT is six years into a 20 year transition to land value taxation. Loopholes in the PRRT can be closed, and the system expanded onshore as we once tried with the MRRT. As you might now be realising that one of the problems with getting back our economic diversity is the concentration of political power by a few narrow interests. Many came out against the attempt to retain control of mineral right with the MRRT. Expect nothing less next time effective reforms are proposed.

In terms of housing more generally, we could ban foreign purchases altogether. I see absolutely no reason not to. It is clearly a myth that we need the money – after all our domestic lending for housing investment is off the charts! There was around $120 billion if new lending to investors last year. If this was direct to new housing only, it could have funded over 300,000 new homes, far more than one each for our new population over the period. Removing foreign buyers would take a large chunk of the wealthiest potential buyers out of the price-bidding contest for homes. 11% of home buyers in NSW last year alone.

And on the same note, the 50% CGT discount merely serves to increase the payoff from speculation rather than investment and production. It was a mistake. We can admit it. Let’s move on.

Renters
I want renting to be a valid and equal alternative to ownership. To that end my general view is that balance of bargaining power needs to shift towards renters, which would decrease the price-competition that keeps rents high.

There are a number of ways to do this.

Tenancy laws: Remove options for not-fault termination of lease, and limits rental increases.
Vacant homes: Introduce a vacant home tax that would punish owners of multiple dwellings, many of which are held for asset speculation, and kept vacant to time their exit from the market and maximise the sale price by selling it vacant.
Social housing: Broaden the number of state-owned social housing projects, ensuring they are widely integrated into the community. As well as funding them directly, make them part of development conditions for large new subdivisions. This provides an outside option to renters who are no longer being funnelled into the private rental market.

These laws are absolutely standard across the our peers of successful, wealthy, economically diverse countries, like France, Germany, Denmark. If I were in these countries I could have told my grandfather about my beautiful garden, and I could have told him how we have not had to move 4 times.

Before I finish I should comment briefly on supply. Many people see that as the answer, despite the fact that Australia has the most homes per person that any time in history, and has just had the biggest housing construction boom in history. There certainly has been a small effect on rental prices, in the order of a percent or two less growth here or there. But it has done nothing for prices. Sydney prices are up 75% in 5 years.

Nor does rezoning increase supply. It makes no sense to tell a landowner who has for decades decided not to build houses that they can built more densely. It is a pure gift. There are examples of sites rezoned because of the urgency of supply, only for the developer to turn around and tell investors that they want to take 35 years or more to develop. Sure rezone, but if you are doing it for housing supply, make it a condition that homes are built in a short time period. I have seen developers lobby for rezoning and be granted it due to concerns over housing supply, only to then turn around and tell their investors that the housing development will take 35 years and they will sell as slow as necessary to keep prices high.

Additionally, rezoning rights can be sold rather than given away. In the ACT, rezoning rights are taxed at 75% of their market value. In Sau Paulo Brazil, rezoning rights are auctioned months, and they have raised $10 bill USD in 10 years from that.

I will wrap up now.

I would love to have gone back in time and had a different conversation with my grandad from an alternative 2017. One where there was more to life that speculating on housing, and we could laugh at the idea of housing reality television on our screens. We could instead talk about gardening.

But we can learn from these past decades, and if my grandchildren do the same exercise in 30 years, they can tell me about their family, their work and their garden, and their beautiful home.

Controlling futile rental-price competition

In a recent post I explained how home rents are mostly wasted spending arising from futile price competition amongst potential tenants. I likened this to Richard Dawkins’ analogy of energy used to create tree trunks being waste from the perspective of the tree, because the only purpose of the trunk is to win the futile competition for sunlight through investment in height amongst a group of uncooperative forest trees.

I explained how tenants could instead cooperate, by unionising, and exert their bargaining power on landlords to cooperatively reduce rents. This post will look at some examples of how the rental canopy is lowered in practice, and how price competition amongst tenants can be limited by well-written regulations.

The best place to start is wartime. Is such times economic efficiency is a priority, and laws and regulations that cut off wasteful competition are more easily passed.

Almost universally, strict rent controls were put in place across Europe, the US, and Australia, during the major 20th-century wars. These had the effect of containing nominal rental prices despite inflationary pressures from the wartime economic build up. In terms of the primary objective of lowering the ‘rental canopy’, they worked very successfully.

In modern times the main approach to pruning the rental canopy is to enact tenancy laws that restrict the pricing power of landlords for existing tenants. Indeed, almost every wealthy country has laws that improve the bargaining position of tenants, with government agencies tasked with ensuring there is full cooperation amongst tenants, and that landlords cannot sidestep these protective laws. They do this because it works. It reduces rental prices. Even economists know this.

“It is possible to design a set of rent regulations that results in an improvement in efficiency over the unrestricted market equilibrium”
Arnott, 1995

In France, tenancies are secured with an automatic rollover of leases and limits on rent increases set by the L’Indice de Référence de Loyers (IRL), which is currently the consumer price index excluding rent and tobacco. In Germany, tenancies are secured in a similar way, with rental increases limited to local conditions, such as to a maximum of 15% over three years in Munich. In Denmark, there are four different rental control systems, all of which ensure tenant security and limit landlord’s power to increase rents. In all cases, tenant security is assured because landlords must justify the reason for asking a tenant to leave, even when the lease is expired, with very few reasons made acceptable by law, such as extensive renovation, or occupation by the landlord (summaries of European rental laws are here).

Such rules avoid the situation where landlords try to severely hike the rent at the lease expiry, and, if the tenants cannot pay, force them to leave. In the absence of legal protections, landlords are able to squeeze out existing tenants and get new ones at market prices, forcing more tenants into futile price competition with each other. Even the mere threat of this power means that existing tenants are often forced to meet the market price each year. The ‘market’ outcome is that the landlord has all the bargaining power.

In Australia, this situation is the norm. Landlords can increase rents without limit after each contract, typically one year, and ask tenants to leave without any reason at lease expiry.

The graph below gives an idea of how limits on rental increases and a secure rental tenure system can result in tenant paying below market rents. This occurs over time if the rental increase limit is below the growth rate of local market prices. The total value of the benefit from these laws for this example tenant is shaded in grey. For society as a whole, the benefit is the sum for all tenants of the difference in the actual rental price paid to the market rental price.

How big could the gains for Australian renters be if we adopted similar tenancy laws to what is fairly standard n Europe? Let us take the situation of secure tenant rights and coupled with limiting rental price increases to CPI.

In Sydney for example, the housing component of CPI has increased 55% in the past decade, while the CPI itself has only increased 25%. In Melbourne it is 51% and 25%, and in Brisbane it is 50% and 30%. With these figures in mind, a realistic rental price saving of 20% on market prices is possible for long-term tenants in the major capitals.

With $60 billion in total housing rents paid to landlords each year, even small rental savings across a small share of tenants results in big savings. If just one-quarter of tenants made a 20% saving on market rental price, that would amount to $3 billion per year saved. If half of the tenants made those savings, that’s $6 billion.

The security given by these tenant protections also means that tenants are able to invest in improving the property – with curtains, lights, gardens and other beautification projects – without fear of losing their investment because they will soon be forced to leave. Indeed, unlike Australia, tenants are often obliged to bring their own curtains, lights, and sometimes kitchens, when they rent a new home in countries with such well-established tenant protection laws.

Adopting world’s best tenancy rights would undermine futile rental-price competition and be a $3 billion win for Australian tenants.

Bullshit financialisation

That’s what I’m calling a class of nonsense policies designed specifically to not address a social problem, but to rake profits from those facing the problem. They disguise a grab for economic rents as a solution. But when you actually think about them as solutions for a specific problem, you meet irreconcilable logical errors that reveal their true nature.

Retirement savings
We know that in terms of real goods and services, private compulsory retirement accounts are useless. We also know that $20 billion per year is creamed off in fees and charges on superannuation accounts. And yet, this bullshit continues.

Error: Buying shares and bonds with superannuation is good, yet buying property with superannuation saving is bad because it inflates prices.

Why: Both are bad. They funnel money into a set of asset markets that would mostly be used instead to buy goods and services and make new investments. The economy is no larger and no more able to support consumption by retirees.

Housing
If you want more housing, your build it. Instead, governments tweak the funding settings for social housing, tweak rules about town planning, buy equity in homes, and provide cash gifts to homebuyers.

Imagine running the military. You want more helicopters. Do you give out helicopter production permits to manufacturers, provide a government guaranteed line of credit to manufacturers of helicopters, buy equity in helicopter hire companies, and give cash grants to people who are buying existing helicopters from each other? No. Because it's bullshit.

Error: Rezoning land to allow greater density is good. Forcing rezoned landowners to actually build homes is bad.

Why: Because these policies are not about supplying more housing. They are bullshit financialisation.

Healthcare
This one is a little more US centric. Instead of simply providing health care, they are proposing to get the government to match payments into individual healthcare savings accounts. It’s bullshit. It’s not an insurance program because insurance only works when you have a diverse pool of participants – you can’t self-insure against catastrophic events, which is sort of the point.

In Australia, money is funnelled into private health insurance using tax system incentives to the tune of $5 billion a year. For every dollar given by the public to the private hospital system, it provides only 40 cents worth of equivalent care compared to the public hospital system. From a health perspective, it's 60% wasted money. So why do it?

Error: People don’t have enough money to pay for healthcare, so to solve that we get people to pay for their own healthcare. Or, we don't have enough money to boost the public health system, so we instead pay $1 of tax money for 40 cents worth of health care in the private system.

Why: Because these policies are not about healthcare, but scraping exorbitant fees for ‘managing’ the money in savings accounts and health insurance accounts.

Privatisation and public-private partnerships (PPPs)
Privatisation, and partial privatisation in the form of PPPs, is about nothing more than removing both debts and assets from the government books, then lying to the public by only talking about the debt side of the equation.

Error: Government has no money, so it sells its assets to pay off debts, leaving it with no money.

Why: Because these policies are about putting monopoly enterprises in the hands of the private sector so that they can make stupidly high profits.

Summary
This is just a taste of the bullshit financialisation in the economy. It is likely to get worse before it gets better.