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How the World Bank calculates aid requirements

The World Bank use the Harrod-Domar model as a basis for calculating foreign aid requirements for developing countries. This is a simplified version of the model they use to calculate the financing gap which justifies the amount of aid a country gets.

Remember how the key value in the Harrod-Domar model is the capital-output ratio: $\theta = \frac{K}{Y}$. In terms of output, that means $Y = \frac{K}{\theta}$.

In the Harrod-Domar model, the capital-output ratio stays constant, which is a bit of a strong assumption, and one which is relaxed in the Solow growth model which is probably a more realistic model of growth. Because the capital-output ratio stays constant, then $Y = \frac{K}{\theta}$ tells us the only way you can get growth in Y is with proportionate growth in K, ie $\Delta Y = \frac{\Delta K}{\theta}$.

Economic growth is basically the proportionate change in output. If output one year is 100 and the next year is 103, then growth is $\frac{103-100}{100}=1.03=3\%$. So we can represent growth algebraically as $\frac{\Delta Y}{Y}$.

So if we go back to $\Delta Y = \frac{\Delta K}{\theta}$ and divide both sides by Y, we get $\frac {\Delta Y}{Y} = \frac{\Delta K}{\theta Y}$.

Now we can think about the change in capital stock, ${\Delta K}$. We can express an equation for capital accumulation: $K_{t+1}=K_t+I_t-\delta(K_t)$, which can be rearranged to $K_{t+1} - K_t=I_t-\delta(K_t)$ to put the LHS in terms of change in capital stock.

So $\Delta K=I_t-\delta(K_t)$. Let’s drop the time subscripts and substitute this in to the equation we had above: $\frac {\Delta Y}{Y} = \frac{I-\delta K}{\theta Y}$.

The increase in capital stock has come as a result of firms investing in new capital. If we were in a closed economy, ie an economy which does not trade and has no involvement with the rest of the world, then this investment, I, would entirely be financed by domestic savings, ie the proportion of income that households were not spending and were instead putting in banks or other financial intermediaries that the firms go to to borrow funds to invest. The proportion of income that is being saved in the economy is called the saving rate, s, so the amount of savings available is $sY$, the proportion of total income being saved, times total income itself. We would have an equation like this: $\frac {\Delta Y}{Y} = \frac{sY-\delta K}{\theta Y}$

When we have an open economy which does engage with the rest of the world then we allow for some other forms of funding the investment – as well as $sY$ which is the saving from domestic savers, we can borrow from foreign savers, or, if we are a developing country, we can get aid to finance investment. So we have an expression for investment, $I=sY+F+A$ where F means private foreign inflows (funds from foreign lenders) and A means aid. Using that in our Harrod-Domar equation above and we get $\frac {\Delta Y}{Y} = \frac{sY+F+A-\delta K}{\theta Y}$.

We can also rearrange the capital-output ratio, $\theta = \frac{K}{Y}$, to $\theta Y = K$ and substitute this into our equation to get $\frac {\Delta Y}{Y} = \frac{sY+F+A-\delta (\theta Y)}{\theta Y}$.

This means if we know all the other parameters of the economy, the level of GDP (income), the volume of private foreign inflows, the depreciation rate and the capital-output ratio, and we have a target growth rate for the economy, we can find an estimation for the aid requirement, or the ‘financing gap’ that represents funds required for investment that won’t be made up by domestic saving and private foreign inflows alone. Define the target growth rate as $\frac {\Delta Y}{Y} = g$ and rearrange so:

$g \theta Y + \delta (\theta Y) - sY - F = A$

So now we can look at an example. Imagine we had a developing country with a GDP of $10bn, and development economists at the World Bank had determined our target growth rate should be 6%, they modelled the capital-output ratio as being 3.5, the depreciation rate as being 10%, the saving rate as 10% which would mean$1bn of domestic savings available. They also estimated the country would receive $2bn of foreign inflows. How much aid would the country need? $0.06 (3.5) (10) + 0.1 (3.5) (10) - 0.1(10) - 2 = A = 2.6$. So the model would recommend it needed$2.6bn.

Now this World Bank model has its sceptics. William Easterly wrote a piece showing that Zambia has been receiving funding on this basis for many years, and the growth targets; that were used in the formula would have seen Zambia have an equivalent level of GDP to Switzerland now….