Conjecture: innovation kills growth; so does growth

Attention Conservation Notice: conjectures that economic growth is inherently self-limiting, because innovation causes diversification, which limits the possible impact of future innovation; and income growth increases relative demand for low-productivity services.  Somewhat condensed and without discussion of its assumptions.

There are two fundamental reasons why growth has to slow. One is inherent in the process of innovation, and one is inherent in people: diversification, and attention-seeking.

Innovation creates new products and new industries, but most of the time, the old industries affected by an innovation don't die out: they just lose importance in the economy. Even when they do die out altogether, the number of new industries created more than compensates. So innovation increases the number of different industries (and occupations) in the economy. This means that each later innovation, which is likely to affect only one industry, has a smaller potential impact than earlier innovations did.

Let's work through this with an example. Start with an economy in which 60 percent of people aree doing the same thing, growing food as in 1700 England.  A single innovation in food-growing such as Jethro Tull's seed drill could have a huge impact on productivity of the economy as a whole.  And that one did: it increased farm productivity by a fifth, increasing the productivity of the whole economy by over ten percent. But among its consequences were mass unemployment of farm labourers--and a decrease in the relative price of food, which allowed households to spend more on other goods, expanding employment in those other industries.

150 years later, in an economy where about a third of people were either manufacturing things or transporting them, innovations that made those activities more productive could also have a large impact on aggregate production. But each innovation, say the Siemens-Martin process for making steel, had a much smaller impact than agricultural improvements did earlier, simply because the affected industry was a smaller part of the whole economy.  Completely eliminating all workers from steel-making could only have increased total productivity by ten per cent or so; and of course that didn't happen. But the Siemens-Martin process innovation enabled the creation of all sorts of manufacturing industries.

Now? The biggest industry in employment terms is retail, about an eighth of total employment, and itself so diverse that it is hard to call it a single industry. It's unlikely that any single innovation could  have a significant effect on the productivity of all of retail, let alone on that of other industries as well. And there are thousands of industries and occupations that didn't exist when the Siemens-Martin process was revolutionising steel-making. Because there are so many of them, it's really hard for any potential innovation to have a significant impact on every one of those thousands of industries, varying as they do from smart-phone design to dog-walking.
 

Each innovation, then, reduces the potential impact of all future innovations: the process of economic growth through innovation has decay built into it. 

What about the other point?

The other point is about demand: what people will pay for, and how that changes as they get richer. 

A person with nothing wants food, clothing and shelter. It turned out that producing food and clothing is something that can be made almost arbitrarily productive. Three centuries ago, more than 60% of people were occupied in growing things. Now, about 2% of workers in the USA are in agriculture, and this figure is widely expected to halve soon.  (Think of that: 100%+ productivity growth in a few years. A major, major innovation. What effect will it have on the productivity of the economy as a whole? Almost none.)

A person's demand for food is limited. Pretty soon she wants something else instead, if she can afford it.  It's this aspect of people that is the other limitation of economic growth.

As people get higher incomes their demands change. At first, the focus is on material goods and services: water supply, heating, furniture, white goods, mobility. Again, it turned out that producing these things can be done with less and less labour. Even entertainment can be provided with relatively little labour. (It used to be labour intensive things like stage plays and bands in cafes; now it's movies, TV, and the internet.)  There was plenty of potential for productivity growth in providing these goods and services, and the potential was used.

But as people get richer still, they start to demand something else: services. Health care. Education. Security. It appears that producing these can not (yet) be done without using a lot of labour: people buying these services need a lot of attention focused on them.

But it gets worse. For other goods and services that rich people buy, like designer shoes, architect-designed homes, or expensive restaurant meals, value resides in the fact that you are using someone's labour time and excluding other people from having the benefit of it. 

The value of boutique items such as designer clothes and architect-designed homes is that they are not mass-produced, that they take a lot of time and skill to produce. The skill can't be leveraged by the use of machines. An architect may be able to speed up the drawing of her designs, but she can't take the same house design and use it to produce hundreds or thousands of houses, because then the design loses the value of exclusivity. Similarly, she  can't speed up the production of designs too much: clients are unlikely to pay for a house design that is produced in five minutes after a five-second chat, whether or not it is unique. What clients are paying for is the architect's time and attention.

It's the same with other boutique items. Take shoes. A shoe designer can't have her designs mass-produced, because if everyone has the same 'designer' shoes, the 'design' value is gone. They're just shoes, the same as everybody else's.  Mass-produced art is merely wallpaper.

Many services are inherently time-related. Baby-sitting. Dog-walking: a dog walker might be able to double her productivity by walking six dogs at a time instead of three, but she can't multiply it by 60 by walking 180 dogs or by walking dogs for 60 seconds instead of 60 minutes. Psychotherapy, physiotherapy, yoga teaching, garden design, interior design, tutoring, early childhood education: the list of low-productivity-potential occupations like this is growing all the time, while employment in productive industries such as integrated circuit manufacturing steadily decreases in importance.
 

Income growth increases the demand for high-touch, inherently low-productivity goods and services. (It also increases the demand for status goods: a house in a Good Neighbourhood being the canonical example. The value of these goods also resides in exclusivity, and therefore they can't be mass-produced. More anon.)

So we have two reasons why economic growth must slow down. First, innovations that increase productivity in one industry cause the creation of new industries, lowering the potential scope of impact of future innovations in the original industry, and shifting relative employment out of the productive industry towards lower-productivity activities.  Second, increasing household income by itself changes the structure of demand, increasing the share given to  low-productivity services.

The end result of both processes is that the production of high-productivity goods and services becomes a smaller and smaller part of the total economy, as a bigger and bigger proportion of the workforce is employed meeting the demand for newly invented time-consuming services, services in which productivity cannot grow quickly, if at all (and more and more debt is incurred in a zero-sum race for status goods).

Classifying Innovations: type, breadth, impact, fertility.

Attention Conservation Notice: In the hope that people will stop talking past each other quite so much, discusses the different parts of the innovation elephant, and proposes  that a classification scheme be adopted, based on type, breadth, impact, and fertility.

Last year Tyler Cowen wrote a book called The Great Stagnation, which claimed that the flow of benefits from innovation is drying up. No matter how many scientist there are, or how many patents, or whatever things that others like to counter with, we're not seeing the same rate of return from these inputs as we did from innovations in the nineteenth and early twentieth century. The investment is great, but the rewards are few. The low-hanging fruit has been eaten. (Or: the law of diminishing marginal returns applies to the Solow Residual, too.)

A few months ago Robert Gordon published a paper with the same theme: no new inventions can compare in impact with that of indoor plumbing, the internal combustion engine, and electric motors. (Perhaps the last really world-changing inventions were the contraceptive pill, and high-response, high-yield dwarf grain varieties, a.k.a. the Green Revolution.)

There has been some econoblogospheric debate on the topic of innovation, with  Isabella Kaminska at FT Alphaville being one of the more important nodes in the debate graph.

A couple of days ago Daron Acemoğlu and James Robinson joined the debate with a post on their blog for their marvellous book, Why Nations Fail. Here's an extract, which itself is only supporting material for their theme, but illustrates what I want to say:-

First, much evidence shows that what determines technological innovations isn’t some sort of “exogenous innovation capacity,” but incentives. This point was stated forcefully by the great economist Jacob Schmookler in his Invention and Economic Growth where he argued (p. 206): 

invention is largely an economic activity which, like other economic activities, is pursued for gain…

Schmookler illustrated these ideas vividly with the example of the horseshoe. He documented that there was a very high rate of innovation leading to improvements in the horseshoe throughout the late 19th and early 20th centuries because the increased demand for transport meant increased demand for better and cheaper horseshoes. It didn’t look like there was any sort of limit to the improvements or any evidence of an “exogenous innovation capacity” in this ancient technology, which had been around since 2nd century BC.

Acemoğlu and Robinson reiterate the excellent points that the things that matter most for innovation are institutions and incentives. True, every word of it, but missing the point of what Cowen, Gordon et al. are going on about.

It's clear from the example in passage quoted above, and a from few other passages in their piece that Acemoglu and Robinson are talking about innovations in the broad sense. Cowen, Gordon et al. on the other hand, seem to be pining for more of the kinds of innovation that change the structure of the economy, and sometimes the nature of work itself, by generating a slew of follow-on, derivative innovations. Cowen:-

I think in terms of general purpose technologies and platform-like breakthroughs.  Once you get them, innovation runs wild, otherwise it is tough sledding, with incentives still accounting for some of the variation within a particular place on the innovation curve.

In short, Acemoğlu and Robinson talk about the supply of innovations, without considering their effects either in terms of changes to production processes or in terms of benefits to households, while Cowen and Gordon are talking about their social and economic effects. To be fair to Acemoglu and Robinson, no-one so far in the debate has been very clear about what they are talking about, or proposed much at all in the way of a taxonomy of innovations. So here's a first stab at one,  based on effects.

Looking along one dimension, the effects of innovations could be divided into three categories. An innovation creates something new; it does or creates something better; or it substitutes inputs or eliminates an industry from the money economy.  

"Something" here means a new category of product, or a new industry,  not just another product in an existing category.  So "television" is a type I innovation, whereas "LCD television" is a type II innovation, and within that industry, 3D TV is a minor (perhaps negligible) type II innovation, and things such as Sony's "X-Reality Pro" are really insignificant type II innovations. The spread of schooling and resulting widespread literacy eliminated "letter-writer" as a specialist industry, and so it could be considered a type III innovation, at least in part. 

In the real world things don't divide neatly into categories, of course. Some innovations can be hard to classify into one of the three types, because they have multiple effects.  The spread of literacy had many other effects besides putting scribes out of work. Among them, it increased the demand for printed books and greatly increased social mobility. But the categorisation into new/better/gone still tells us something useful, I think.

The cellphone (a type I innovation, spawning several new industries) enabled many other industries to work at a faster tempo, so it has type II aspects. It has also caused the virtual elimination of pay phones, and it seems likely to eventually eliminate traditional landline phones, so it has some type III-ness, too. The iPhone, a type II improvement over preëxisting smartphones, has enabled the creation of new industries besides affecting the structure of the mobile phone industry, and changing its economic importance.

Considering innovations like these points us toward three other dimensions of innovation: breadth, the number of industries affected by the innovation; impact: how much effect the innovation has on methods of production or how great the benefits to households are; and fertility: how many industries or products and services arise that depend on the innovation.

Type, breadth, impact, fertility. Let's look at some examples from above.

Schmookler's horseshoes are clearly of type II: each was an improvement on the preceding one, in terms of the services provided to horse users. The scope or breadth of effect of these innovations was on the fair to middling fraction of industries and households that employed horses on a daily basis in the late nineteenth and early twentieth centuries.  In terms of impact on productivity, on changes to economic importance of the industries concerned, and on benefits to households, the impact of each new kind of horseshoe was more or less unnoticeable. Fewer horses lamed, incremental increases in load size, slightly cheaper tickets and fewer delays due to horse trouble. All nice for farm workers, travellers, and transport firms, but not the stuff of letters to mother.  In terms of fertility, each new horseshoe may have inspired its successors, but that's the best that can be said. There was no impact on manufacturing or services other than transport.

In the end, a type I invention provided overwhelmingly greater benefits than even the best horseshoe, even in its scope of impact, as Acemoğlu and Robinson admit:

Then suddenly, innovations came to an end, but this had nothing to do with running out of low hanging fruit. Instead, as Schmookler put it (p. 93), it was because the incentives to innovate in this technology disappeared because

the steam traction engine and, later, internal combustion engine began to displace the horse… 

So despite A and R's claim that "[i]t didn't look like there was any sort of limit to the improvements" to be made to the horseshoe, in the end they ceased to matter. 

More importantly for our purposes, the structure of production in those years would not have differed noticeably if none of those improvements had been made, so they didn't matter to start with, either. So: type II, breadth minor, impact almost nil, fertility almost nil. 

The steam traction engine, on the other hand, had great impact in agriculture, transport, and  construction, which at the time were three large industries in terms of employment, and three important suppliers of services to households. It reduced employment in agriculture, and increased the size of the agricultural  contracting industry, and it transformed and increased the importance of maintenance engineering (blacksmithing, as it used to be). The traction engine was also quite fertile, causing the spread of new kinds of  earthmoving equipment such as the bulldozer and the steam excavator, and inspiring various machines used in forestry such as 'portable' sawmills, besides stimulating the development of new, larger combine harvesters and ploughs.

The classification for the traction engine is therefore: type I, breadth moderate, impact major, and fertility moderate.

The steam traction engine was itself a type I child of a much more fruitful type II innovation: Watt's improvements to the steam engine. This was a type II innovation which had enormous breadth and strength of impact, and was moreover  fertile. (Once Watt's patents for his invention had expired, the number of different applications for steam engines took off, and the resulting innovations swept through nearly all of the major industries of the time, the main exceptions being retail trade and domestic service.) 

The external condenser steam engine is classified as type II despite its fertility. One of its more visible effects was to enable the creation of the railway industry (type I). But that's all it did: enable. The actual innovation of rail was a separate process with the steam engine as one of its major parts.

(It has been asserted that if rail had not been invented, economic progress would not have been greatly delayed or stunted. People and goods would have been moved via canals and rivers instead, with no great loss of benefits. Nevertheless, the rail industry did become a significant part of the structure of production.)

The proposed classification scheme enables convenient definition of what we are talking about, when we talk about innovation.  Cowen and Gordon are focused on highly fertile, wide breadth, high impact type I and type II innovations: transformational innovations, we could call them, which are the tiny minority. Acemoğlu and Robinson, by contrast, have a broader view, taking in the whole range of innovations from the most common and trivial up to the transformational.

Anyway, there it is: type, breadth, impact, and fertility. Enjoy!

Jørgen Randers: 2052: A Global Forecast for the Next Forty Years

Attention conservation notice: a summary of the book, which is worth reading. Or at least, the better summary linked below is worth reading.

Website: 2052.info. Has link to a spreadsheet with the model, which spreadsheet seems to be password protected. The password isn't in the Kindle ebook version of the book.

A useful summary: cpsl.cam.ac.uk/Resources/SoSL/SoSL_2012 (PDF)

The 2052 Model

The model is based on population. Population trends partly determine economic trends and those in turn determine energy use trends. Randers believes that some of the large trends operating today will continue. He puts them together and examines the consequences.  In summary:

• In advanced and developing countries, the birth rate will continue to decline, falling below the death rate around 2040. Total fertility falls well below replacement before 2030.  Therefore the population of these countries peaks around 6 billion.  The working-age population peaks in the 2030s. (These predictions are slightly below the UN's 2010 medium estimates.) Urbanization continues.
• The least developed countries (mainly in Africa) will not develop significantly, because they lack appropriate governance. (In this view, he matches Acemoglu and Robinson's view expressed in Why Nations Fail--and my own view.) Citizens of these nations won't be relevant economically but will be victims, especially of closed borders. And they will devastate the ecosystems around them. Total world population will peak at around 8 billion.
• The key assumption: Short-termism will continue to dominate politics in the West, as it's built into both capitalism and democracy. (Neither does anything about problems until they're already hurting.) As a result of using these institutions, we'll be unable to act in time to prevent serious harm.
• Thanks to short-termism, we'll continue to use fossil fuels until climatic instability gets really bad, which will mainly be after 2502. Since gas-powered power stations are cheapest and quickest to build, we'll mainly use gas to produce new electricity, because we'll just react to brown-outs.
  
  Renewables will continue to rise, contributing 40% of total energy by 2052 - the largest "single" contribution; nuclear will continue to decline in relative terms. CO2 emissions will peak in 2030.
• Energy efficiency will continue to improve at recent rates, so we'll need about twice as much electricity in 2052 as we do now (allowing for limited GDP growth and some movement from other forms of energy to electricity).
• The workforce will continue to grow till about 2035, but employment will continue to move into low-productivity-growth services such as elder care and community care in general. So GDP growth will be slow.
• As the bulk of GDP growth will take place in developing countries, incomes in western countries will feel static at best. As well, holiday spots will be crowded with the new middle class from the newly developed countries.  However, something (minimal) will be done about increasing inequality, eventually.
• Investment: Climate change will cause increasing damage to structures and infrastructure, and require extensive new infrastructure (e.g. massive irrigation schemes).  This implies that an increasing proportion of GDP will have to be used to fix things and supply services that used to be free.
  
  In addition, it will cost more (take more materials and work) to get the raw materials we need (fossil fuels, minerals).
  
  As an increasing fraction of GDP has to be set aside for investment, less will be available for consumption. Household disposable incomes will stagnate; in the west, they will decline in real terms (especially when including disamenity from worse food, expensive water, bad weather (storms and heat waves), and crowded holiday spots).
• Climate change: the effects will be increasingly felt, and repairs and strengthening costs will absorb an increasing fraction of GDP. Cities will be buttressed, and eventually they may be moved as well, when buttressing fails. But the full effects of climate change won't be felt until the second half of the century, when he expects them to overwhelm remedial investment.
• Technology: Randers expects AI by 2052, but doesn't think that it will have a significant impact on people's lives. Some new services may be developed, and some old ones may be improved or made cheaper.  As above, the main effect of technology will be to continue pushing people into low-productivity industries, ironically reducing economic growth. (This is where he may go the wrongest. But it's not very wrong.)
• Food will be expensive but there won't be famines, except in the least developed countries, which already have famines thanks to their poor governance. To westerners, the diet will seem to get worse: less red meat and more vegetables. To Asians, the reverse.
• Medical technology will be more advanced, but much of the new stuff will be outside the price range of most people. Lifespans are unlikely to increase much, but people may be working for longer, since work is increasingly less hard on the body.
• Water will be priced, and people will discover that agriculture can use water much more efficiently than it does today.
• In international politics, there will be a race to be the last to lose, i.e. competition for resources, especially arable land in least-developed countries.
  
  China's development (if it doesn't run out of steam about now) will be seen by other Asian countries as a model, i.e. the idea that democracy is necessary for development will be discarded. (Would-be copycats won't be as successful as China, IMO: they mostly lack China's cultural legacy of bureaucratic service to the state.)  The USA will lose its status as undisputed leader and policeman of the world.
• Natural ecosystems will be all gone. Such "wildlife" and "wild" ecosystems as remain will be carefully managed.
• The zeitgeist: life in 2052 will be lived by grannies in cities who are afraid of the sky, live on soy and lentils, and watch historical nature documentaries on TV, reminiscing about the good old days. For once, the reminiscences won't be entirely wrong about the "good" bit.

My Thoughts

Broadly in agreement, with a few quibbles.

As he usually does, Randers ignores the possibility of catastrophes: global epidemics, continental-scale wars, famines, drawn-out international disputes and closing of borders, etc. So growth could be even lower than his already pessimistic forecast. It definitely will be if the effects of climate change come on sooner than he estimates.

Randers seems to think that expectations won't play too much of a role in productivity growth. IMO, investment in the past has been bolstered by underlying population growth: with 2% per year population growth, a business can expect to double in size in 35 years, if it lasts that long. So the end of population growth will add a risk premium to the required rate of return on investments in enterprises and innovations.  Declining disposable income may exacerbate this effect.

The dementia epidemic almost certainly will catch us by surprise. (At the age of 80, about 20% of people have dementia, and this figure rises by a few percent for every year of age thereafter, according to Charlie Stross. Update: a more accurate figures is about 8% at age 80; or: 1%  in the 65 - 69 age group, with the proportion doubling every 5.5 years: about 4% of 70 - 79 year olds, about 17% of over-80s. Source: UK Alzheimer's Society ) A world with lots of demented codgers is going to need a lot of nursing labour. Or a lot of fairly smart robots--my next point.

In terms of upside risks, he also may underestimate the (admittedly still potential) acceleration of automation that is in the offing, enabled by the machine learning-internet-cellphone-cheap sensors-robotics revolution. Autonomous water-pipe-laying and -repairing machines may bring water to millions of new urban residents, and analogous construction machines may make or repair the buildings they live in. Autonomous "waste mining" machines may make recycling more effective.  And so on, and on. This "revolution" will, of course, take time to play out, and it has downside risks of its own. (The British agricultural and steam revolutions caused massive unemployment, as did the internal combustion-electric motor revolution.) 

Of course the big growth sectors are education, health, and personal and community services. Some effects will be felt here, but education in particular is a tough nut to automate.

Growth, 1960 - 2011: Summary.

In the top 11 advanced economies, per capita real GDP growth has been linear over the last 50 years:

GDP Per Person for the top 11 countries combined.  Data from FRED.

I first looked at it in proportional terms, charting growth as a percentage of  the current level of GDP per capita for each of the top 10 advanced countries:

GDP: is 50 years long enough for a trend?

Rounding out the top 10

After that, I combined the data for the top 11 advanced countries (dropping Mexico in favour of Australia and the Netherlands), in order to help reduce the variance of the data, and found that GDP per capita has grown linearly for the last 50 years: GDP Growth is Ruled by the Clock.

Per capita GDP growth in dollars per year. The trend is essentially flat.

How do we interpret the data?

The Solow model of growth implies that per-capita growth should depend on growth in "technology", other things being equal. If that is correct, technology grows at a fairly fixed rate ($565 per person per year), irrespective of absolute population size or GDP.

If we want to dispute that but retain the Solow framework, we need complicated models that explain the linearity. Perhaps exponentially increasing effort in innovation is matched by logarithmic returns. Perhaps productivity growth has been counterbalanced by movements of workers from high-productivity industries to low-productivity industries, and also explain why increasing and then reducing the participation rate didn't seem to have much effect. Perhaps the depreciation rate has smoothly increased and/or the savings rate decreased, these changes nearly cancelling changes in technology, which is growing exponentially as required. Or both. Or something else.

In other words, we need a lot of coincidences.  

Much better to stick with the simplest model that fits the data: per person, GDP grows at a fixed rate of $565 per year.

GDP Growth is Ruled by the Clock.

Attention conservation notice: It turns out that in advanced countries, GDP per capita seems to increase by about the same amount every year. In 2011 US dollar terms, about $565 per person per year.

 The first two posts in this series looked at the last fifty years of growth in GDP per capita for each of the top ten advanced countries: the biggest countries in GDP terms, excluding China, Russia, Brazil, and India.  For those countries FRED doesn't seem to have good long-term data--or maybe I haven't looked in the right places.

In this post I'm looking at the same data, for the "big advanced countries" as a unit. Adding countries together will, I hope, smooth out some of the year-to-year variation in real GDP. (When one country is having a banner year, another might be having a poor one, and the effects partly cancel.)

I have also dropped Mexico. Mexico was not considered an advanced country at the start of the period, so including it might introduce some "winner bias" in the results. (South Korea, by contrast, had adopted some modern institutions and a policy of industrialisation.) In Mexico's place, I wasn't sure whether to include Australia or the Netherlands...so I chose both.

These charts present combined data for the top 11 advanced countries, for which FRED has real GDP and population series going back to 1960: the USA, Japan, Germany, France, Great Britain, Italy, Spain, South Korea, Canada, Australia, and the Netherlands.

The year-over-year percent change for the group (above) shows a pretty clear trend. Per-person GDP growth was high in the 1960s, and the trend has steadily declined.

The charts in the last two posts were built on the assumption that growth in GDP depends on its current size. The underlying idea is that a country produces stuff, and a part of what it produces (machinery, buildings, roads) can be used to produce more stuff the next year. This implies that the more a country produces in one year, the more new machinery it has for the next year's production.

With this perspective it made sense to focus on growth in terms of percentages. But as the chart shows, the trend in the percentage is continuously declining.  That suggest that maybe the underlying assumption is wrong, or countries are continuously reducing their savings rates (the fraction of output that is machinery and offices).

So let's look at the raw data.

Looks pretty close to a straight line.  What does a chart of the absolute increments look like--the number of dollars per year that each person's GDP goes up? Like this:

There's a slight downward trend, but it's not significant. Given that there were a couple of large-ish recessions in the 1950s, if we started the chart a few years earlier the trend would be flat, or even point up-hill.

So the simplest interpretation that fits the data is that GDP per person in these countries increases the same amount every year (plus some random variation).  That amount comes out to be about $565, in 2011 US dollars.

The Solow Model II: the Inada Condition

Attention Conservation Notice: Another step in the exegesis of the Solow model with technology, Hicks-neutral version. Growth submits to diminishing returns, but production is proportional.

The Story So Far

• The model is of the rate of production of a single good (service) produced by using the services, called capital and labour, of the stocks of capital goods and workers available at the time. (This single good can be called "everything", I guess.)
• The rate of production depends only on the available services of capital and labour: nothing else is scarce. The stocks of capital and workers are assumed (initially) to be fully used, to provide the maximum flow of services that can be got from combining them in a production process.
• Over time, people think of better ways to combine capital and labour (and perhaps make use of non-scarce things more effectively, too), so the same flows of each produce more. This is added to the model in the form of a term $B(t)$, the production method at time $t$. 
• For convenience $B$ is associated with $L$, and has its name changes to $A$ as a result.
• In maths: the production function:   $Y(t) = f(K(t), A(t)L(t))$  , where:

• Explicit assumption: changes in $L$ and $A$ are unaffected by $Y$: they are exogenous.
• A few implicit assumptions:  $Y$, $K$, $A$, and $L$, are all greater than or equal to $0$ for all times $t$, and at time $t=0$ they are all strictly greater than zero - in the beginning, there are  some things, some people, and some ways of making things .

...and for capital accumulation,

• $K$ is decreased by depreciation of the capital stock at a constant rate $\delta$
• $K$ is increased by using a fraction $s$ of the output flow $Y$ to create new capital:

$$\frac{dK}{dt} = sY - \delta K = sf(K, AL) - \delta K .$$ 

There is an equilibrium point ($dK/dt=0$, i.e., $K$ does not change) when $sY = \delta K$.

So: a two-equation model of a single-good economy using two factors of production.

More on the Production Function

What forms can $f(K, AL)$ take?  Quadratic? exponential?

In economic terms we require  a few things.  When either $K$ or $L$ is $0$, $Y$ has to be $0$ too. When $K$ and $L$ are both positive, $Y = f(K, AL)$ has to be positive too, and  bigger amounts of either or both of them should increase $Y$. So far, so obvious.

Diminishing Returns

For economic plausibility, we need diminishing returns to each factor of production. Holding $AL$ constant, when there is an  increase from a small amount of $K$ to a slightly larger amount, this should  have a large positive effect on how much is produced. When there's a lot of $K$, increasing it by the same increment should have a much smaller effect on production.

DeLong New California Economy example: If 100 baristas have one espresso machine for them all, and then you give them another one (or replace the existing one with one that works faster), we expect a large increase in the number of lattes produced per hour.  Similarly with yoga instructors and yoga mats.

At the other extreme, if 100 baristas have 1000 espresso machines and 100,000 cup sets and all the other facilities in excess, adding another espresso machine will have almost no effect on production.

Putting these into mathematical terms:$$\lim_{K \to 0} \frac{\partial f} {\partial K} = \infty  ,  \lim_{K \to \infty} \frac{\partial f}{\partial K} = 0 , $$ and we require $\partial f /\partial K$ to be differentiable throughout, and the second partial derivative of each factor to be negative throughout.

Similarly for $L$, and $AL$: When not much of these services are around, increasing them a bit produces a large increase in output, and when there's a lot (with no change in the capital they can use), the same increase has almost no effect.

So:$f$ is defined, continuous, and $>=0$ for $K >=0$ and $AL >=0$, $f(0,0) = 0$ , $f$ is monotonically increasing, and the partial derivatives of $f$ are concave down (have derivatives that are $< 0$).  Collectively these are called the Inada Conditions.

 These conditions add up to requiring that in terms of each factor of production, $f$ is a "concave down" function passing through $0$.

.... but Constant Returns

Also for economic plausibility, we require proportionality overall:  double the amounts of both capital and labour (by putting the economy into an EconoClone^™ duplicating machine), and you should get twice as much output. 

In maths:  $f(cK, cAL) = cf(K, AL)$ for $c >=0$.

What can satisfy all these conditions? Stay tuned for the next thrilling instalment.

The Solow Model

The Solow-Swan model of economic growth is the basic framework used by economists to model economic growth.  Here are some notes about it.

The Basic Ideas

Flows, not stocks: the Solow model is a very simple flow model, a model of the rate of production of something, specified as the quantity produced in one time period. Stocks enter the model only indirectly and implicitly. The model uses two equations, one for production, and one describing how one of the two free variables of the production function, "capital", changes. It also uses several rather ad hoc and unrealistic auxiliary assumptions. (But the point of a model is to be simple, and just realistic enough to be useful, so these aren't necessarily problematic.)

(To jump ahead a bit, although the Solow model appears to talk about stocks of capital and workers, what it actually talks about or assumes is the flow of services that these stocks provide when used--generally called "capital" and "labour". It is assumed that both stocks are fully used at all times, so the flow of services can be measured by measuring the stocks, but sometimes it pays to be clear about the difference between capital-the-stock and capital-the-flow.)

The rate of production, usually given as $Y$, is often referred as "the quantity produced", with "...per unit time" usually omitted.

$Y$ depends on $K$ and $L$, the rate of flow of services from the stock of capital, and the rate of flow of labour from the stock of workers.

In the basic model, only the magnitudes of the available (services of) capital ($K$) and of labour ($L$)  constrain $Y$. Nothing else does. In particular, the basic model ignores the role of mechanical and chemical work done by energy sources, and scarcity of raw materials. Various extensions attempt to incorporate thermodynamic work and resource scarcity.  For now, we'll stick to the basics.

The Production Function, part I

 $Y$ is a function of $K$ and $L$, which are themselves considered as varying over time:  Capital is a function $K(t)$, and Labour $L(t)$. So we have $$Y(t) = f(K(t), L(t))$$ for some function $f$. ($Y$ is probably from yield, once a synonym of harvest.) $Y$ is thus a function of two functions of time: the factors of production, capital and labour.

 Correction Factor $B$

To allow for the fact that in olden times, a given quantity of both capital and workers produced less than the same quantity of both produces now, a third function is introduced, normally written $B(t)$ or $A(t)$, and called "the level of technology at time $t$", although what it actually is, is problematic. In the equation, $B$ is dimensionless, just a multiplier.

So  we have $$Y(t) = f(K(t), L(t), B(t)).  For no very discernible reason (TODO: consult the early literature), economists like to bind the multiplier to the $L$ term, asserting that increasing "the level of technology" increases the effective rate of labour supplied by a given stock of workers. In this form the $B$ is changed to an $A$. (Actually, this assumption makes the maths easier, which is much less of a concern in these days of Maxima, R and Octave than it was in 1957, and it's not a great reason anyway.)

Summary so far

• Production equation $Y(t) = f(K(t), A(t)L(t))$

• $Y(t)$:  The rate of flow of services produced at time $t$.  It depends only on $K$ and $L$.
• everything else is assumed not to be scarce -- i.e., to be available in whatever quantity is desired.
• $K(t)$: The flow  of services from capital goods at time $t$.
• $L(t)$: The flow of services from the stock of workers at time $t$.
• $B(t)$ or $A(t)$: a correction function to make things come out right.
• Assumption that "the level of technology" increases the flow of labour services.

From now on we'll drop the $(t)$ notation where it can be inferred.

Change in Capital

In the Solow model capital is increased or replaced by saving a proportion of Y, the flow of services produced, and using that to increase the stock of capital and therefore the flow of capital services used. Meanwhile, the flow of capital services is reduced by depreciation of the stock of capital.  The fraction of output saved, $s$, and the rate at which capital depreciates, $\delta$, are both assumed to be fixed constants, mainly for mathematical convenience.  The stock of capital, and hence the flow of services from it, is increased by saving, and decreased by depreciation. In one time period the change in capital $\Delta K$ is thus  $$\Delta K = sY - \delta K .$$ Two further points here. $K$ is assumed to change continuously so in the limit we have $$dK/dt = sY - \delta K ,$$ and $Y$ depends on $K$:$$dK/dt = sf(K, AL) - \delta K .$$

To be continued

OK.  Later posts will have some discussion about $f$; the assumptions, the economic model behind the equations; and exposition, discussion of the key result of the model; and discussion of extensions.

Rounding out the top ten

Attention conservation notice: presents FRED-based charts of GDP growth for four more countries, to give us ten, taken with the six in the last post.

In the last post we observed a common trend in GDP per capita in five of the top six ‘advanced’ countries (and the exception appears to be joining the pack). Here are the four next largest advanced economies, giving us a “top ten”: South Korea, Spain, Mexico, Canada. Spoiler alert: again, we see the same pattern.

No doubt we could continue: Australia, the Netherlands, Switzerland, Sweden, Austria, Portugal, Finland, Denmark, Belgium, Norway, Ireland,... This is left as an exercise for the interested reader. Call back!

In case you are wondering, the top ten list is taken from Wikipedia's “List of Countries by GDP (PPP)”. It's sort of a mixture between the first two columns there: the IMF's and the World Bank's rankings. 

I've omitted the BRICs: China, India, the Russian Federation, Brazil; and several other large developing economies, partly because the data series don't go back far enough (Russian Federation), and partly because most of these countries are starting from a long way behind the advanced countries in terms of their production processes, so they are able to increase their per-capita production by copying the production techniques of the more advanced countries, importing the necessary machinery and advanced materials as they do so. More importantly, as they start the copying process, they get better at it, so they catch up at a faster and faster pace, for a while. In short, their rate of growth grows. (There's a bit more to say about that, but as this topic depends on some economic concepts and assumptions, I'll leave it there for the time being. If you like, you can claim I cherry-picked my sample, and create your own charts for the missing countries...if you can find good fifty-year data series for them.)

South Korea is a game of two halves. The whole fifty-year period shows a rising, then falling trend:-

But the thirty-seven years from 1974 show a secular declining trend:

Spain has had some marked deviations from trend, but the trend asserted itself nonetheless:

Mexico isn't yet thought of as an advanced country by many non-Mexicans, although it has been a member of the OECD for a few years now. We don't have the same data series as for the other countries, being forced to use something called the “PPP converted GDP per capita (chained)” instead of the “Real GDP per capita” we've used up till now. This series goes a little further back, to 1951.

Nonetheless, Mexico follows the rule, albeit with more random noise about its trend than most of the earlier, larger countries.

Canada also succumbs to the iron law of declining GDP growth. I expected this one to be an exception, because of Canada's current resources boom which is taking place during the high-priced part of a commodity super-cycle. But no such luck:

Accounting for Economies

What accounts for this pattern? What I have read of standard economic theory is of little help. I'll say a bit about that in the next post, as I prefer to keep the tea-leaves and the readings a little separated. This was a data post.

GDP: Is 50 Years Long Enough For a Trend?

Attention Conservation Notice: presents data showing that the rate of growth of per-capita real GDP in the large advanced economies is steadily declining and will fall below zero within the next two decades.

This blog is about global economic growth--specifically, the arguments (if I can use the term broadly) that people make about its desirability and possibility this century. I had intended to start with a few backgrounder posts, summarising a little of the background knowledge essential for thinking about problems of this type, and about this domain of study. 

But no! Instead, let's jump in and wade through some numbers. (Cue Brad DeLong with Rudy Dornbusch's jeweller-plumber-pig taxonomy...)

What can we say about economic growth if we try to minimise our assumptions? We'll assume ‘real GDP’ is a meaningful index of economic welfare (without worrying at all what it actually is), and we'll use numbers from the St. Louis Federal Reserve's marvellously easy-to-use FRED repository, thus assuming that FRED is a good store of good numbers.  How are the numbers doing? What do they tell us?

OK, more assumptions. First, the big countries matter the most for global growth. Even if Tuvalu suddenly starts growing really fast, it'll be a long time before it is big enough to have a noticeable impact on the global figure. Not this century.

Second, what matters to people are per-person figures. If Nigeria's GDP (whatever that is) grows 5%, but at the same time its population grows 5%, from the point of view of an ordinary person, nothing has changed.

Third, the ‘advanced’ economies (as they proudly style themselves) represent a kind of end-state towards which other countries will tend to move over time. Therefore, what happens in the advanced countries will eventually happen everywhere.

Let's look at year-over-year changes in GDP per person in the big advanced economies. If growth were constant, this number should be the same each year: a chart of year-over-year percentage change in GDP should be a flat line. Of course there will be deviations above and below--there's always weather--but overall, the trend should be flat. So, is it?

Take it away, FRED. First up, the USA:

USA per-person GDP growth year-over-year for the period 1961 to 2011, with linear trend.

Yep: a simple depiction of GDP growth as fluctuations around a secular trend matches the evidence pretty well. 

(Lots of writers on the topic of GDP growth talk about "periods of high growth" and "periods of low growth," and stationary points in the trend caused by technology shocks. In climate science and in finance, people have learned not to do that kind of cherry-picking. Start with a large deviation away from trend, and finish with another, and you can make it look as though the trend is something else...for a while. Random noise, in the temperature signal or in stock prices, is random noise. Perhaps the same is true in the "GDP per person" signal. With 50 data points, we have, perhaps, enough numbers for the trend to start to dominate the random noise.)

The United States' rate of growth in production per person is on a downward trend which, extrapolated, reaches zero in 2032. Meaning that after 2032, each person in the United States starts to become less well off than they were the previous year, at least to the extent that GDP matches personal welfare. Every year from 2032 onwards.

But perhaps the United States is exceptional?  Let's look at the other large advanced economies.

Japan:

Fluctuations about a linear trend again here. A declining trend here too? Oh dear.

Who's next? Germany?  OK.

A declining trend again. It seems reünification caused a large but short-lived spike in production growth, after which, growth returned to its trend.

Next up, either France or the United Kingdom, depending on whose country-size list you trust.  The United Kingdom is interesting, so let's have a look at it:

Overall, the UK shows a flattish trend, but the later part of the chart seems to show the same declining trend as the other countries seen so far. The first part, from 1960 to about 1975, seems to show relatively stable growth. Let's look at the post-Thatcher period:-

Warning: insufficient data error!  But the prototrend is suggestive.

Although there are not enough data points in this interval to say anything with confidence, what data there are seem to fit the pattern. Time will tell.

Onwards! A vous, la France!

Perhaps we should say downwards.

And lastly Italy:

Summary, with some gobbledygook words.

Overall, our know-nothing assumption of a linear trend fits the data. Perhaps surprisingly, the trend is the same everywhere, and it points downwards.

Of the top six advanced market economies, the ones that have been in the growth business the longest and for which we have long sets of numbers, five show a steady decline in per-capita GDP growth--a decline that means that GDP per person is about to start shrinking, if it hasn't already started doing so.  The other country, the UK, seems to have delayed the trend for about thirty years, but it might also have succumbed. Time will tell.

Interpretation

This does not bode well for economic growth in the 21st century. The countries that are already big are about to start shrinking, so any growing that other countries do will be working against this headwind. Do we know what's causing the trend? (Rhetorical question.) Stay tuned for some speculation.