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Indices and production functions

Make sure you’re familiar with the rules of indices and logs because you use them quite a lot, especially with production functions.

Basically a production function is something which tells you how much output you get, depending on the amount of inputs you use. Usually the inputs are capital (K) and labour (L). We think in terms of ‘units’ of capital and labour. If you have a production function $Q=2K^{0.4} L^{0.6}$ then this allows us to predict how much output (Q) we will get depending on the amounts of capital and labour we have. Eg if we use 100 units of capital and 200 units of labour, we will get $Q=2(100^{0.4}) (200^{0.6})=303.143$

Production functions are often written in the form $Q=AK^\alpha L^\beta$. This is called the Cobb-Douglas production function. An important economic concept is the idea of returns to scale, ie if you were to for instance double your inputs (use twice as much capital and labour), will you double your output? If you do, you are said to have constant returns to scale. If you more than double your output you have increasing returns to scale, and if you get less than double the output you have decreasing returns to scale.

As an example, go back to the production function we used before, $Q=2K^{0.4} L^{0.6}$. When we had 100 units of capital and 200 units of labour we had output of 303.143. Now if we double both of our inputs, we have $Q=2(200^{0.4}) (400^{0.6})=606.286$. We have also doubled our output. This production function is a constant returns to scale function.

You can tell that it’s a constant returns to scale function because capital was raised to the power 0.4 and labour raised to the power 0.6, and when you add 0.4 and 0.6 together they sum to 1.

Think of the general case $Q=AK^\alpha L^\beta$. What happens if you double inputs? Lets call the new output you get Q’ so $Q'=A(2K)^\alpha (2L)^\beta$. You use the rules of logs to manipulate this a bit and it becomes $Q'=A(2^\alpha )K^\alpha (2 ^\beta) L^\beta = (2^{\alpha + \beta})AK^\alpha L^\beta$ so $Q'=2^{\alpha + \beta}Q$

This gives you an important result – if $\alpha + \beta = 1$ then you will have constant returns to scale, because then doubling your inputs will double your output, you will have $Q'=2^1 Q$

If $\alpha + \beta > 1$ you will have increasing returns to scale and if $\alpha + \beta < 1$ you will have decreasing returns to scale.

A lot of the time in economics you use constant returns to scale functions. Production functions which are concerned with the economy as a whole, rather than just a firm (you can tell these because instead of output being Q, it is Y, which is usually used for national income or the output of the economy), are usually modelled as being constant returns to scale functions. The idea is that if you increase just one of the factors (eg population growth means labour force grows and you keep capital constant) you will get diminishing returns to factor…your national output still goes up, but by a smaller amount each time you add another unit of labour. But if you could ‘clone’ the economy, ie just double all the people and capital they are using, it would be reasonable to expect that the ‘clones’ would produce the same as the original people did, so you would double output. So national income Cobb-Douglas production functions usually have $\alpha + \beta = 1$ and so are written like this: $Y=AK^\alpha L^{1-\alpha}$

Lets use an example here to illustrate what I was talking about with ‘diminishing returns to factor’. Returns to scale are what you get when you increase both factors (capital and labour) by the same multiple, returns to factor are what you get when you increase one factor while holding the other constant.

So lets assume an economy with the production function $Y=3K^{0.3} L^{0.7}$. This is a constant returns to scale function because 0.3 + 0.7 add to 1. I haven’t just chosen those numbers randomly, usually the share of capital in output is modelled as being around a third and the share of labour in output being around two thirds, so I chose 0.3 and 0.7 to be along realistic lines.

Lets say originally we have 1000 units of capital and 3000 units of labour. Output will be $Y=3(1000)^{0.3} (3000)^{0.7}=6473.008$. Now lets assume the labour force stays constant and we add a unit of capital so we have 1001 units of capital. The new output is $Y=3(1001)^{0.3} (3000)^{0.7}=6474.949$. The increase of 1 unit of capital has yielded an increase in output of 1.941. This is the marginal productivity of capital, the amount of extra output that you have got by increasing capital by one unit, whilst holding the other input constant. Now let’s add another unit of capital. $Y=3(1002)^{0.3} (3000)^{0.7}=6476.889$ This time adding another unit of capital has yielded an increase of 1.940. Notice that the marginal productivity of capital has dropped slightly compared to when we added the unit of capital before that. As long as we keep L fixed, each time we add another unit of K, we will get a smaller and smaller gain each time, eventually it will reach a point where the gain by adding another unit of labour tends to zero. This is the principle of diminishing returns to factor.

You can think of this in terms of people using capital (eg machines) to produce output. If you keep the number of workers the same, then at first adding more capital means each worker has more capital to work with, so each worker will become more productive. But over time these productivity gains start to diminish, there won’t be enough workers to use the additional capital effectively. So after a while, even though adding more capital does give you a bit of output gain, the benefit you are getting from every additional unit of capital becomes lower and lower and eventually approaches zero. Unless of course, the labour force is growing as well!

In reality you rarely get a situation in an economy where capital or labour is fixed, but you will often have a situation where one of the factors is growing faster than the other, and so diminishing returns to factor kick in. This is a concept which comes up in a lot of fields, particularly growth models.