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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
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