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Archive for the ‘Harrod-Domar Model’ Category

## 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….

Categories: Harrod-Domar Model, Macro

## The per-capita Harrod-Domar model

We can extend the basic Harrod-Domar model by taking into account the population size. This is important because it is per capita income that gives us a relative idea of living standards in each country. We might hear that China has become the world’s second largest economy, and its overall GDP (national income) is higher than most developed countries, but when you divide the total national income by the population you find that China has a much lower per capita income.

To take population into consideration (and the fact that population will grow over time) we can go back to the equation $\theta Y_{t+1}=(1-\delta)\theta Y_t +sY_t$.

Define population as $P_t$ and divide that equation throughout by it, $\frac {\theta Y_{t+1}}{P_t}=\frac{(1-\delta)\theta Y_t}{P_t} +\frac{sY_t}{P_t}$.

Now we have population in there can start thinking in terms of per capita income rather than absolute income. The way we will do this is to use small letters, ie define $y_t = \frac{Y_t}{P_t}$ so the equation above becomes $\frac {\theta Y_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

This is tidier but the left hand side (LHS) is a bit of a nuisance because we have Y in terms of year t+1 and P in terms of t, so we can’t just use the small letter to express it in per capita terms. So there is a trick to deal with this LHS, multiply top and bottom by the same thing, ie multiply it by $\frac{P_{t+1}}{P_{t+1}}$.

This means you get $\frac {\theta Y_{t+1}}{P_t+1} \frac {P_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

Now we are in per capita terms for year t+1, so we have $\theta y_{t+1} \frac {P_{t+1}}{P_t}=(1-\delta)\theta y_t +sy_t$.

Now divide both sides by $\theta y_t$ to get $\frac{y_{t+1}}{y_t} \frac {P_{t+1}}{P_t}=(1-\delta) +\frac{s}{\theta}$.

Look carefully at those terms on the LHS. $\frac{y_{t+1}}{y_t}$ is basically 1 + the rate of per capita income growth. $\frac {P_{t+1}}{P_t}$ is the rate of population growth.

We can define per capita income growth as $\frac{y_{t+1}}{y_t} = (1+g*)$ and population growth as $\frac {P_{t+1}}{P_t} = (1+n)$.

This makes our equation above $(1+g*)(1+n)=(1-\delta) +\frac{s}{\theta}$ so we can say
$(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$.

This is the Harrod-Domar equation with population growth.

Growth depends on the ability to save and invest, the ability to convert capital into output (which depends inversely on $\theta$), the depreciation rate and the rate of population growth.

Lets try an example of this.
Suppose an economy has a capital-output ratio of 2.2, a depreciation rate of 8%, a saving rate of 25% and a population growth rate of 2%, what would you predict the per capita growth rate of income to be?

Substitute the values into $(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$ to get $(1+g*)=\frac{(1-0.08)}{(1.02)} +\frac{0.25}{(2.2)(1.02)} = 1.0134$.

So if (1+g*)=1.0134 then g*=0.0134 which is equivalent to a 1.34% per capita growth rate.

Categories: Harrod-Domar Model, Macro

## The basic Harrod-Domar model

The Harrod-Domar model is the easiest model to start learning about growth and the long-run. We start by using the general concepts of income, saving and consumption, and capital accumulation, to give us a few equations that will form a framework of thinking about growth:

Income, saving and consumption:
$Y_t = C_t + S_t$.
$S_t = I_t$ so $Y_t = C_t + I_t$.
Also $S = sY$ so $I_t = sY_t$. Given that all income will either be saved or consumed, $S = sY \Rightarrow C=(1-s)Y$.

Capital accumulation:
$K_{t+1}=I_t+K_t(1-\delta)$ so $K_{t+1}=sY_t+K_t(1-\delta)$

The Harrod-Domar model adds the concept of a capital-output ratio. This is basically the efficiency of production for an economy, measured in terms of capital. If the capital-output ratio is low, then the economy can produce a lot of output from a little capital. If the capital-output ratio is high then it needs a lot of capital for production, and it will not get as much value of output for the same amount of capital. The capital-output ratio can take into account things like the ‘quality’ of capital, if a country has high quality capital that is very productive then it will have a low capital-output ratio. The capital-output ratio is denoted as $\theta$ where $\theta = \frac{K_t}{Y_t}$.

Now we can use these equations to find an expression for growth.

First, rearrange the capital-output ratio equation: $\theta Y_t = K_t$.

Now substitute this into $K_{t+1}=sY_t+K_t(1-\delta)$ to get $\theta Y_{t+1}=sY_t+ \theta Y_t (1-\delta)$.

This can be expanded to $\theta Y_{t+1}=sY_t+ \theta Y_t -\delta \theta Y_t \Rightarrow \theta Y_{t+1} - \theta Y_t =sY_t -\delta \theta Y_t$.

Now divide both sides by $\theta$ to get $Y_{t+1} - Y_t =\frac{sY_t}{\theta} -\delta Y_t$.

Now divide both sides by $Y_t$ to get $\frac{Y_{t+1} - Y_t}{Y_t} =\frac{s}{\theta} -\delta$.

Notice what the left-hand side of that equation is: $\frac{Y_{t+1} - Y_t}{Y_t}$. This is basically the proportionate increase of output compared to this year’s output, which is the rate of output growth. If we define this as g, then we get a simple Harrod-Domar equation for growth:

$g =\frac{s}{\theta} -\delta$.

The Harrod-Domar model is nice and simple but it does have some weaknesses. It is based around two concepts, the saving rate and the capital-output ratio. It assumes that there are constant returns to factor, ie the capital-output ratio stays constant and the more of the factor (capital) that you add, the more growth you will get, so if you add more and more capital then your growth rate will go up and up. The weakness here is that physical capital needs another factor, labour, to operate. We are not in a situation yet where capital can operate itself and is fully automated. In reality if you just keep adding more and more capital, and the population and labour force doesn’t grow quickly enough to keep up, there won’t be enough workers to use the new capital effectively, so adding more capital will give you diminishing returns to factor. In the Harrod-Domar model this would mean the capital-output ratio dropped as you accumulated more capital. The Solow model takes care of this concept of diminishing returns to factor, which is why the Solow model is probably more realistic.

Categories: Harrod-Domar Model, Macro

## Capital stock

The next key idea on which growth models are based is that the capacity of a country to produce goods depends on the amount of capital stock in the economy. A poor country with only basic agricultural equipment and little electricity supply or infrastructure is obviously not going to be able to produce very much, because it has a low capital stock. A richer country with hi tech equipment, fast railways, high speed internet connections, large scale production plants and so on, is going to be able to produce more because it has a high capital stock. To get more growth you need to increase your productive capacity, that means increasing your capital stock.

Every year you add to your capital stock through investment, firms getting more machinery, plants etc. But because capital stock depreciates, every year you lose some of last year’s stock through depreciation. So you can express this as $K_{t+1}=K_t+I_t-\delta(K_t)$. This is saying that the capital stock you will have next year is equal to the capital stock this year plus investment (the new capital you add) minus the depreciation rate, $\delta$ multiplied by this year’s capital stock. The last part of that equation gives you the amount of capital that has been depreciated. You can rewrite the equation as $K_{t+1}=I_t+K_t(1-\delta)$, which basically says capital next year is equal to investment this year plus the amount of your existing capital stock that is left after depreciation. Usually the depreciation rate is constant.

Categories: Harrod-Domar Model, Macro

## Saving, consumption and growth

In simplest terms, economic growth is the result of deferring consumption now, so that you can get more production later.

An economy produces a variety of goods, the act of production generates income. That income is used to buy goods. We can roughly divide these into two types: consumption goods and capital goods.

Consumption goods are just things that are produced for the purpose of satisfying human desires in the here and now. Capital goods are things that are produced for the purpose of producing other things. The mechanism by which income ends up being spent on capital goods, is through saving. When households get their income, they will spend a portion of it on consumption goods and they save the rest of it. This saving becomes ‘loanable funds’ which firms use to borrow for investment. If you think about it, saving is really just the opposite of borrowing, when somebody takes out a loan, they can only do that because somebody else is saving. The reason you get paid interest on your savings in a savings account, is because when you put your savings in a bank, the bank is lending them out to other people and charging them interest.

So the higher the proportion of household income that is used for current consumption, the less is being saved, and when there is not as much being saved, there are not as many funds available for firms to borrow to invest in capital goods. So what you want is a higher ‘saving rate’ in the economy, because if people can hold back a bit on current consumption, and save a bit more, then firms have more loanable funds available with which to invest in capital goods. These capital goods are what is going to drive increased levels of production in the future. In models like the Harrod-Domar and Solow model, economic growth comes from accumulating more capital goods, often described as increasing the capital stock.

Households spend income on consumption goods and save the rest, so of the overall income in the economy you get an equation $Y_t = C_t + S_t$, or national income in year t is equal to the amount of consumption in year t plus the amount of savings in year t. The savings make available a pool of loanable funds that firms use to buy capital goods (investment).

From this you get the first idea which is used in growth models: the idea that the amount of saving in the economy determines the amount of investment. The more saving is going on, the more loanable funds are available to allow firms to invest. It’s not an exact one for one relationship but it is a useful assumption on which the simple forms of growth models are based: if you assume that savings are equal to investment then you get $S_t = I_t$ so $Y_t = C_t + I_t$.

Remember that the total amount of saving is a proportion of national income, so we can use the concept of a saving rate. If say national income is 1000 and the saving rate is 20% or 0.2, then total saving is 200. If we define small letter s as the saving rate then $S = sY$. This means that $I_t = sY_t$.

Categories: Harrod-Domar Model, Macro

## First principles of long-run growth

When you first start learning Macro you spend a lot of time looking at short-run fluctuations around a medium-run trend, eg all of the stuff around being above or below a natural level of output, or natural rate of unemployment.

When you step outside the concept of these fluctuations and look at the concept of economic growth you can start to think about the long-run, which tells you how the medium-run concept of the natural level of output will grow over time. Looking at growth models gives us some idea of answers to questions like why do some countries grow faster than others, why have some not grown and stayed poor. Long run growth is key to raising living standards, reducing poverty and promoting economic development.

From the end of the Roman Empire to the year 1500 there was essentially no growth of output per person in Europe. Most workers were employed in agriculture, there was little technological progress and because agriculture’s share of output was so large, inventions with applications outside agriculture did not make much difference to total production and output. Output growth was roughly proportionate to increase in population, keeping output per person constant. This was the Malthusian era. Malthus argued that any increase in output would lead to a decrease in mortality, which would mean population grew until output per person was back to its initial level. Eventually Europe escaped the Malthusian trap, growth rates were positive after 1500 (around 0.1% per year until 1700, then about 0.2% per year until 1820. Starting with the Industrial Revolution growth rates increased but even there in the US output averaged only 1.5% per year between 1820 and 1950. It is in the past 50-60 years that we have had the big explosion in growth rates.

Since 1950 rich countries have seen a large increase in output per person. There has been a convergence of output per person amongst the richer countries (eg in the OECD, countries like Japan catching up the US). There has also been some convergence from the four tigers (Singapore, Hong Kong, Taiwan, South Korea), and more recently from China, who started a long way behind but is now growing at a very high rate. Economies with high growth rates but still have low output per person are called emerging economies.

However convergence is not uniform. Uruguay, Argentina and Venezuela were roughly at the same level of output per person as France in 1950, by 2004 they were between 0.25 and 0.5 of the level of France. Many African countries were very poor in 1960 and some have had negative growth of output per person since then, eg Madagascar has been falling back 1.1% per year, Niger is about 60% of its 1960 level.

So neither growth nor convergence is inevitable.

Categories: Harrod-Domar Model, Macro