A popular model which is often used to teach economic growth (and explain why some countries are richer than others) is the Exogenous Growth Model, attributable to the work of Trevow Swan (ANU) and Rober Solow (MIT) in the 1950s. This model says that for most countries, economic growth occurs due to the relentless ability of firms to work out ways of doing the same work with fewer inputs. This contrasted with earlier models which said that all countries needed to do was save more in order to buy machines.
While simple, the model captures some of the interesting relations in an economy, and can help non-economists with high-school mathematics understand why some countries are rich and others poor.
At the core of most contemporary macroeconomic thinking lies some notion of “exogenous growth”. This post describes this important concept as intuitively as possible, for economics students, or for people who may be interested in why some countries are richer than other countries.
Before launching into the details of the model, though, it’s important to recognise why economists like it so much. When building a toy model of the economy, you must keep in mind a set of things which you want your model to replicate. The rules Swan and Solow wanted to replicate were Nicholas (later Lord) Kaldor’s Facts:
Lord Kaldor, a British economist, noticed a few rules which tended to hold across countries and time. These are:
1. There is no long-run trend in the return to capital. This is contrary to the Marxian conclusion that capitalists earn more and more on the same investment. While there are periods when capital earns more, and less, there is little time-trend to these.
2. There is no long-run trend in the capital share or labour share of income. This is a bit different to #1. In #1, we were interested in the return to invested capital. In #2 we are interested in how income is split between wages and retained earnings/dividend payments. #2 says that the amount of income that goes to capital (which in turn may be owned by anyone) has no time trend.
This is not to say that it is impossible for such a thing to occur (it may have in the past while in the US), just that it’s not such a dominant force as to include it in a toy model of the economy.
3. Income per capita seems to show fairly constant growth over time.
4. The ratio of national income to labour-force exhibits a (positive) time trend. That is to say, I could make a fairly safe bet that the GDP per person will be higher in 2030 than now.
5. The ratio of invested capital to income, or income to invested capital, exhibits no such time trend. That means that on average, the stock of invested capital grows as quickly as the economy.
6. We see capital deepening; more capital per worker over time. This simply follows from #4 and #5. If you’re studying a course on growth or macroeconomics, you will definitely want to memorise these facts.
While they are a crass generalisation (and may not describe economies like Australia’s very well) they do describe most economies most of the time, and also underpin this exceptionally powerful little model. In particular the related concept of a Balanced Growth Path, which is something you want a growth model to replicate, is one that complies with these facts.
There are a few key ingredients in the model. They are:
1. The savings rate of the economy: s. This is the proportion of production that is not consumed;
2. The rate at which the average unit of capital (building, computer, whatever) depreciates each year: d;
3. The rate at which Labour Augmenting Technological Change occurs: g (this is described in more detail below);
4. The rate at which the population grows: n;
There are also a few assumptions needed for the model. All are unrealistic, and most growth modelling is focussed on relaxing the assumptions:
1. There is no trade, and no international capital flows;
2. All people work, and the 1 product they produce is one they consume and invest (think about corn: you can eat it, plant it, and harvest it);
3. All the people who work live in the same house and live forever;
4. Contracts (and insurance) exist, and always work, and so there is no economic risk;
5. People do not get to decide their savings rate, I just tell them what it is;
6. The technology which grows doesn’t require that you need to invest in research to get it. This is a bit like technological growth outside the US; in Australia, we don’t need to invent things, just buy the snazzy capital equipment from Seimens, Toshiba, or Westinghouse;
7. Finally, the amount of stuff produced in the economy is determined by the amount of capital, the amount of people, and the amount of technology.
A short note on Labour Augmenting Technical Change
Labour augmenting technology is technology which allows us to use less people to do the same amount of stuff. In Mexico, where I am at the moment, they don’t seem to have caught on to the idea of automatically-wringing mop-buckets. They wring their mops out by hand, taking twice as long and getting bleach all over their skin.
If you had a very large area to mop, and only one mop bucket, you would need fewer people to mop it if your mop bucket had an automatic-wringer. And so the automatic wringer is labour-augmenting. If Mexico adopted automatically-wringing mop buckets, this would be labour-augmenting technical change.
Note that not all labour-augmenting technologies need to be encapsulted in a capital good, like a mop bucket. Methods used in businesses can determine how many people you need to do a particular thing. Last January, I wanted to climb a mountain right next to Solo (Surakarta) in Central Java. We figured it would be cold up top, and went to buy some jumpers from the local equivalent of Myer or Sears. Instead of there being a lot of employees, as in Myer or Sears, there were thousands of them. The process to buy a jumper included dealing with a sales assistant, then giving the chosen jumper it to another person, and another person giving you an identifying ticket. When you were done collecting tickets you took them to the cash register. The cashier then gave your tickets to another person, who collected your goods while you paid. That meant there were five people doing the job one or two would do in Australia. The difference is in method, not in capital, yet is still labour-augmenting technology.
A formal statement of the model:
We have a production function Y(t) = F(K(t),L(t)E(t)), where Y is output, L is the number of people, and E is labour-augmenting technology. This is continuous, upward sloping, and concave. These assumptions say that if I have a pizza shop and employ more people, I can make more pizzas, but as I put on more and more staff, the increase in pizza-making is hampered by crowding. Likewise, we cannot indefinitely add pizza ovens and expect that we can make more and more pizzas without increasing the number of workers.
The t subscripts indicate that the value of the variable is for perid t; t-1 was the value for last period, and t+1 is for the next period.
A capital accumulation function
K(t+1) = K(t)(1-d) + I(t), where I is investment, and capital takes one period to come online. This equation simply states that next period’s capital is all the capital which existed today that did not depreciate, plus whatever we invested in new capital today.
Because we have a closed economy, savings must equal investment, and so
I(t) = sY
Because n and g are growth rates:
L(t+1) = L(t)(1+n)
E(t+1) = E(t)(1+g)
Remember growth facts 4 and 5? They state that while capital per person K/L grows over time and income per person Y/L grows over time, income per unit of capital Y/K does not grow over time. This means we can define a growth rate of E, g, such that y=Y/LE and k=K/LE do not grow over time. This gives the capital accumulation equation (expressed in terms of effective units of labour, LE):
K(t+1)/L(t)E(t) = K(t)(1-d)/L(t)E(t) + sY(t)/L(t)E(t)
= K(t+1)/L(t+1)E(t+1)*L(t+1)E(t+1)/L(t)E(t)= k(t)(1-d) +sy(t)
= k(t+1)*L(t)E(t)(1+n)(1+g)/L(t)E(t)= k(t)(1-d) +sy(t)
= k(t+1)*(1+n+g+ng) = k(t)(1-d) +sy(t)
Now recall under Balanced Growth, when the Kaldor rules hold, both k and y have no time trend. This implies k(t+1) = k(t) = k, and y(t+1) = y(t) = y, and so: k(1+n+g+ng) = k(1-d) +sy which gives:
sy = k(n+g+d+ng), or because both n and g are small,
sy = k(n+g+d) under balanced growth.
We should really take some time to discuss this result. The term (n+g+d)k is the amount you need to invest to simply not go backwards (in terms of capital per effective unit of labour). To invest this much, your savings need to be sufficient. So what happens if your savings decrease? Then your capital will depreciate, and your population will grow, and your technology will progress, until there is less capital per effective labour unit.
How about what happens when you save more? Exactly the opposite happens– you acquire more capital, which will allow you to produce a bit more, but not as much as the existing capital helped you produce. You can see this in the following chart.
The top curve is the amount you can produce at all the given levels of capital per effective unit of labour. The curve below (that has a similar shape) is the amount of production saved given some savings rate s, for all the different possible levels of capital. The straight line is the “necessary replacement” line in order to keep capital growing as quickly as effective labour units (as it does in the Kaldor Fact). The intersection of the straight line and the savings line determines the amount of capital per effective unit of labour, which in turn determines output.
Two basic extensions:
1. The Golden Rule of savings.
The Golden Rule savings rate is the rate that maximises consumption over time. So what would consumption be? Clearly, it is the difference between income and savings. We can graph Balanced Growth Rate of consumption (per effective unit of labour; if you want the per-person measure, multiply this by the amount of technology) as being the difference between y and sy. But in balanced growth, the rate of investment, sy = k(n+g+d).
This has an intuitive appeal. To be investing the amount required to maintain the same capital stock per effective unit of labour, you must invest to keep up with population growth, technology growth, and the depreciation of capital. If you look at the graph of y-sy (below) you see that it has a maximum point.
For this point, there is an associated savings rate which will maintain this level of consumption.
y – sy = f(k) – s f(k) = f(k) – k(n+g+d)
We want the maximum point, so we take the derivative of this function with respect to capital per augmented labour unit and set to zero:
f'(k) – (n+g+d) = 0
And so the savings rate which gets us the most consumption is that which matches the marginal product of effective units of capital (ie, the additional output you get by having one additional unit of capital for each technology-improved worker) to the sum of population growth, technological improvement, and depreciation.
This is found by substituting: recall that sf(k) = (n + g + d)k Then the golden rate of savings s* is s*f(k) = f'(k)k Which, intuitively, is when the total amount saved (LHS) is equal to the total payments to capital plus the total amount used to replace depreciated capital (RHS), all in effective units of labour terms.
2. The International Flow of Capital, or why do the Chinese and Germans send it to Australia and the US?:
Imagine if you had two countries without trade or capital-market relationships. One country has distinctly better legal institutions, rule of law, and a fairly developed pension system (discouraging savings). The other is poorly developed political or legal institutions, and no pension system (encouraging savings). Basic economic theory may tell us that poor countries have more to exploit (so to speak) and so should attract capital from rich countries.
A quick view of these two countries through the lense of the Solow model will tell you that capital will flow not from the rich to the poor country, as you may believe, but from the poor to the rich country. This is because there is less risk in a country with well developed legal institutions. Even if returns on SOME projects in China are very high, returns on all projects are not high enough to compensate for the considerable difficulty of being an inward investor in China (relative to being an inward investor in Australia or the US).
Compounding this is the higher domestic savings rates in China. The investment of savings tends to have a home-bias (that is, even if rates of return on foreign investments are higher, people still invest at home), owing to imperfect global capital markets an information problems across borders. This means that we should otherwise expect returns in China to be LOWER than in perfect capital markets (as more will be invested, decreasing the returns to capital).
IMPLICATIONS OF THE MODEL:
So here’s the juicy bit: what does the model predict?
1) While increased savings will lead you to acquire more capital in the short run, it does not allow your growth rate to permanently increase—this only occurs due to technological progress.
2) A decrease in population growth, likewise, will improve growth, as the amount of investment that needs to cover population growth decreases. However, this increase in growth is temporary, and in the end, only technological growth will help.
3) Countries who face very high capital costs or have crappy neighbours, and so can’t get world-market prices for their products, need to pay a lot more for additional capital, and so need a higher savings rate to afford the capital. However if they have production technologies which are away from the frontier but affordable, this may mean that they get stuck in a poverty trap.
Consider a farmer in Burundi who grows a crop but uses a donkey to plough. If he could save enough, he could afford a tractor from China. However, Burundi is landlocked and has relatively unstable neighbours with poor infrastructure. This makes the Chinese tractors more expensive in Burundi. Also, this lack of market access means the cropper can only sell his crops to others in Burundi, or maybe people in Rawanda, DRC, or Tanzania, who are unable to offer the world price for the crop.
The cropper is stuck in a POVERTY TRAP, whereby saving enough for the tractor—which would make him better off—would involve starving himself.
This concludes my description of a basic growth model. Coming up: how does productivity relate to a growth model like this?