Almost a quarter of a century ago, Robert Lucas posed the simple question: “Why does capital flow from poor to rich countries?” (Lucas 1990). It remains as relevant today given that the poorer countries of the world tend to run current-account surpluses (thus exporting capital) and the richer ones (most notably the US) tend to run current-account deficits (thus importing capital).^{1} Table 1 below shows that the global financial crisis has not really changed this pattern.

**Table 1**.

*Source*: IMF, WEO database. The set of poorer countries corresponds to the IMF classification ‘emerging and developed countries’.

Here I argue that the direction of capital flows makes economic sense given savings behaviour. But the real puzzle is why savings rates are high in poor countries and low in rich ones.

The starting point of the analysis is the simple growth model that that all economists are familiar with since Robert Solow’s work in the 1950s. Each national economy is conceptually a single production relation whereby capital and labour are combined to produce ‘GDP’ – with this linked mediated by technical productivity, i.e. ‘total factor productivity’.

As is well known, the capital-to-output ratio is a sufficient statistic for the return to capital (interested readers will find the algebra in the annex below). This simple result has a strong implication:

- The output-to-capital ratio is a sufficient statistic for the relative returns to capital between nations that have the same functional relationship between capital, labour and output.
- The important point here is that in a cross country comparison the return to capital can be related not the capital-to-labour ratio, but the capital-to-output ratio.

A country with a lower output-to-capital ratio will have a lower rate of return to capital – and the reverse holds as well. Such ratios are readily calculated, so it is straightforward to infer relative returns to capital from widely available data.

# Why poor to rich?

The question posed by Robert Lucas in 1990, “Why Doesn’t Capital Flow from Rich to Poor Countries?”, has thus a simple answer. Capital should flow from countries with a high capital-to-output ratio to countries with a low capital-to-output ratio and the latter are not necessarily the poor ones.

A natural comparison using this metric is China versus the US. China still has a much lower capital-to-labour ratio (according to many sources by a factor of 20). But from this it does not follow necessarily that the return to capital in China will be higher because total factor productivity in China could be much lower.

It follows that if the capital-to-output (as opposed to the capital-to-labour) ratio should be used as an indicator of the marginal productivity of capital. Hence one should focus on the dynamics of this capital-to-output ratio over time and across countries if one wants to predict capital flows – not income per capita or the capital-to-labour ratio.

# What determines the capital-to-output ratio?

The evolution of the capital-to-output ratio depends essentially on the rate of investment (investment as a percent of GDP) to the growth rate and the rate of depreciation (see annex for details). A higher investment rate raises the capital-to-output ratio, but a higher growth rate (or a higher rate of depreciation) increases the denominator and thus lowers the capital-to-output ratio.

Returning to the question posed by Lucas one should thus look at the combination of growth rates and investment rates to look for an answer of why capital flows ‘uphill’, i.e. from emerging economies to advanced economies.

Somewhat surprisingly one finds that the capital-to-output ratios of these two broad groups are not too distant and that the ratio for the emerging market economies is poised to overtake that of the advanced economies if these groups maintain present investment and growth rates. This result can be easily verified by using the available data for investment rates and trend GDP growth rates for these two groups from the WEO database of the IMF.

For advanced economies (as defined by the IMF) the investment rate is at present approximately 20% of GDP (and is expected to remain at that level) and real trend growth about 2%. This implies a steady state capital-to-output ratio of around 2.5 if, following the literature, it is assumed that the depreciation rate is 6%; the steady state calculation is that 2.5= 0.2/(0.02+0.06).

For emerging economies (emerging and developed economies in the IMF classification) the investment rate is at present about 30% and the trend growth rate about 6% (average to 2018). This also leads to a steady state capital-to-output ratio of 2.5 since 2.5=0.30/(0.06+0.06).

**Table 2**.

*Source*: IMF, WEO database. Note: The set of poorer countries corresponds to the IMF classification ‘emerging and developed countries’.

Table 2 uses the projections of the IMF to calculate the steady-state capital output ratios. The key point is that by 2018, according to the IMF, investment rate of the emerging economies should rise to almost a third of GDP. This implies a slightly higher capital-to-output ratio for the poorer nations than the richer ones. Moreover today’s capital-to-output ratios are already very close (albeit with important differences within the two groups).

- The fact that both regions have similar capital-to-output ratios does not mean that their GDP per capita should be similar.
- However, this equality in capital-to-output ratios has one important implication, i.e. the return to capital should be equalised as well.

The second point implies that there should be no need for large flows of capital between the two groups.

- Moreover, given today’s growth rate and investment rates the capital-to-GDP ratio should be lower in advance nations, so capital should continue to flow from emerging economies to developed ones.

# Within group differences

While the differences between the two large groups are small, there are important differences in terms of the capital-to-output ratios within the two groups. These correspond to the large capital flows one observes.

- The most important ‘outlier’ among the emerging markets is China.

China has a high capital-to-output ratio (close to 3) despite being poor. It is thus not surprising that China has become a capital exporter.

- Other poor countries, for example India, have much lower capital-to-output ratios than China and even most developed countries.

It thus makes sense that India tends to import capital (it is running current-account deficits most of the time).

Similar, but smaller differences exist among the group of developed economies:

- Japan and Germany have an above average capital-to-output ratio and the one for the US is below average.

This fits well with the continuing current-account deficits of the US and the continuing surpluses of the other two. All in all it seems that global capital markets seem to be ‘efficient’ in the sense that capital goes where the return is highest.

# The real puzzle

The real puzzle is thus not where the investment goes, but the savings rates which are much higher in emerging economies, allowing them to finance their own development out of their own resources.

When Lucas wrote his seminal paper in 1990 the investment rate in emerging economies was much lower than today and they were running consistent current-account deficits – i.e. their investment rates exceeded their savings rates.

At the time puzzle was why there was not more investment in the capital-poor countries. Today the investment rate is more than 10 percentage points of GDP higher in emerging economies than in advanced economies. If their savings rate had remained unchanged emerging countries would be running very large current-account deficits and would thus be importing a lot of capital. However, their savings rates have increased even more than their investment rates and the real puzzle has become: “Why do poor countries save so much?”

# References

Lucas, Robert (1990). "Why doesn't Capital Flow from Rich to Poor Countries?", *The American Economic Review* 80 (2): 92–96.

Alfaro, Laura, Sebnem Kalemli-Ozcan and Vadym Volosovych (2003) “Why doesn’t Capital Flow from Rich to Poor Countries? An Empirical Investigation” University of Houston, December.

# Annex

**A.1 The AK growth model and the return to capital**

The standard ‘AK’ model can be written as:

(1)

Where y is output, and given that the labour force is normalised to one y represents also output per capita. A stands for the overall productivity level of the country called total factor productivity (TFP), which can vary across countries (and over time), k is the capital stock per capita or the capital labour ratio. Αlpha (α<1) represents the share of capital in production (usually assumed to be in the range 0.3-.04). The (inverse of the) capital output ratio is then given by:

(2)

From (1) it follows that the marginal product of capital in country i, denoted by r, is given by:

(3)

The last two equations are identical except for the constant term alpha. This implies that in a comparison of two countries with the same weight of capital in the production function, but different total factor productivities the output-to-capital ratio is a sufficient statistic for the relative returns to capital.

Taking two countries denoted by i and j, the ratio of the returns to capital is thus equal to the ratio of the capital output ratios:

(1)

However, the last equation also implies that the relative capital-to-output ratios can differ from the relative capital-to-labour ratios (denoted by k) as becomes obvious through a slight rewriting of his relationship:

(1)

One might now think of i=China and j=US. This equation has one important implication: even if China has a much lower capital labour ratio (kChn <<kUS by a factor of 20) it does not follow necessarily that the return to capital in China will be higher because total factor productivity in China could be much lower (AChn <<AUS). This difference in the A factor encompasses of course all the differences in institutions and other variables used in the empirical literature.

**A.2 The dynamics of the capital-to-output ratio**

The dynamics of the capital-to-output ratio is described by its low of motion:

(1)

Where I denotes gross investment, g the growth rate of GDP (Y) and, in the last part of the equation, it is assumed that capital depreciates at a rate δ.

The change of the capital-to-output ratio can thus be written as a function of the investment rate I/Y and the present capital output ratio:

(2)

This implies that one can compute the capital-output ratio at the steady state as a simple ratio of the investment rate to the sum of the trend growth rate and the deprecation rate:

(3)

1 The vast empirical literature on the ‘Lucas paradox’ uses mainly the sum of direct and portfolio equity investment from the balance of payments as a measure of capital flows (e.g. Alfaro et al. 2003). But this does not seem to be appropriate since capital flows can take many different forms and the current account which represents the difference between domestic investment and savings is the ultimate measure of how much foreign capital an economy needs to finance its investment.