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## Total factor productivity

Using a production function like $Y=K^{\alpha} L^{1-\alpha}$ is a nice and convenient way of modelling an economy, the problem is if you do that you are almost always going to find the model doesn’t work.

As an example, lets say you have a model economy with 1000 people (all workers) and 5000 units of capital, and a production function $Y=K^{0.3} L^{0.7}$. That means you will have output of $Y=5000^{0.3} 1000^{0.7}=1620.657$.

Now say over the next year the population grows by 4% and capital grows by 2%, so you have 1040 people and 5100 units of capital. The model would predict that you would have output of $Y=5100^{0.3} 1040^{0.7}=1675.693$. But the odds are when you actually measure output, you are going to get a higher figure than the model predicted. Say for example you find output is actually 1750. How can we explain this?

It is quite logical given the technological progress story. If there has been some form of technological progress which has augmented labour and made it more productive than labour was last year, so the model doesn’t hold any more.

We could take care of this by changing the model to this form: $Y=AK^{\alpha} L^{1-\alpha}$.

If you totally differentiate this to get an expression for the change in Y, you get $\Delta Y = \Delta A \frac{\partial Y}{\partial A} + \Delta K (\alpha) \frac{\partial Y}{\partial K} + \Delta L (1-\alpha) \frac{\partial Y}{\partial L}$.

Now we can sort out these partials:

If $Y=AK^{\alpha} L^{1-\alpha}$ then $\frac{\partial Y}{\partial A} = K^{\alpha} L^{1-\alpha}$. But this is just the same as $\frac{Y}{A}$.

Similarly $\frac{\partial Y}{\partial K} = \alpha A K^{\alpha - 1} L^{1-\alpha}$. This is the same as $\alpha \frac{Y}{K}$.

Finally $\frac{\partial Y}{\partial L} = (1-\alpha) A K^{\alpha} L^{-\alpha}$. This is the same as $(1-\alpha) \frac{Y}{L}$.

So this gives us a nice expression: $\Delta Y = \Delta A \frac{Y}{A}+ \Delta K \alpha \frac{Y}{K} + \Delta L (1-\alpha) \frac{Y}{ L}$.

Divide the whole thing through by Y and we get $\frac{\Delta Y}{Y} = \frac{\Delta A }{A}+ \alpha \frac{\Delta K }{K} + (1-\alpha) \frac{\Delta L }{ L}$.

This is now in terms of ‘proportionate change’ in A, K and L. We can think of it in terms of ‘growth of A’ and ‘growth of K’ and ‘growth of L’ by defining notation like this:

$g_A = \frac{\Delta A}{A}$, $g_K = \frac{\Delta K}{K}$, $g_L = \frac{\Delta L}{L}$.

So $g_Y = g_A + \alpha g_K + (1-\alpha) g_L$.

This tells us the growth in output is made up of three things. It is the growth in A, plus the share of capital in output times the growth in capital, plus the share of labour in output times the growth in labour. The growth in A then gives us a tool for capturing the growth that has developed not through increases in inputs (capital and labour) but through something else. That something else is technological progress.

A is usually described as Total Factor Productivity and the growth in A is called the Solow Residual.

You can relate the growth in A back to the concept of labour augmenting technological progress, because the Solow Residual is equivalent to (proof not shown here) $g_A = (1- \alpha)g_\pi$. The growth in TFP is the share of labour in output multiplied by the growth in labour augmenting technological progress.

The Solow Residual is a useful tool for growth accounting, in that it gives us a way of estimating how much technology is progressing. If we know how much capital and labour has increased, and we know how much output has increased, then we can work out the Solow Residual and get an estimate of technological progress from that.

Returning to the example from before, where capital had gone up from 5000 to 5100, labour had gone up from 1000 to 1040 and output had gone up from 1675.693 to 1750.

$g_Y = \frac{1750-1675.693}{1675.693}=0.0443$, $g_K = \frac{5100-5000}{5000}=0.02$, $g_L = \frac{1040-1000}{1000}=0.04$,

So $0.0443 = g_A + 0.7 (0.02) + (0.3) (0.04) \Rightarrow g_A = 0.0183$. This means we have had TFP growth of 1.83%.

And from this we can say that ) $0.0183 = 0.7 g_\pi \Rightarrow g_\pi = 0.0261$ so we have had labour augmenting technological progress of 2.61%.

Of course the Solow Residual is just a way of estimating technological progress, it is very difficult to be totally accurate in practice, but it’s not a bad framework for thinking about it.