When we model production functions in macroeconomics, the broad ingredients we have for output growth are labour and capital – our factors of production. Both of these factors need to be remunerated, which raises the question of what share of income goes to each.
This is the question of factor income shares.
An income share for a broad category like “labour” or “capital” will depend on two things – what happens with the quantity of the factor and what happens with the price of the factor. If we have more labour for the same amount of capital it will imply that labour’s income share must rise. However, as the marginal product of labour will decline in that case it is likely that labour’s price – the wage – will decline.
As a result, it is unclear whether a higher quantity of labour (or capital) leads to a higher share of income for that factor.
In this context, we can start of by thinking about a constant elasticity of substitution (CES) framework – where production function has a constant change in factor shares (e.g. capital and labour). In this case, if the amount of capital rises then the relative return to capital falls (with an increase in wages or reduction in the rate of return to capital) such that the relative income share of labour and capital changes by less than the increase in the quantity shares.
This makes a lot of sense. More capital makes labour more productive, by providing more tools to do the work. However, capital can also be used to do some tasks that labour could also perform. For a given technology, the only way for it to make sense to change that mix would be if capital was to become relatively cheaper.
Following Piketty’s Capital it has been made clear that the nature of these changes matters. Both because the income shares themselves matter because of the difference in the distribution of the factors among individuals. Also because the nature of the change (do rates of return decline, or do wages rate) matter in terms of the livelihoods of people.
In this post I want to briefly describe three scenarios and what do they mean from economic perspective. The scenarios are
- When CES is equal to 1;
- CES is less than 1 (complementarity);
- CES is greater than 1 (substitutability).
What happens when CES is equal to 1
In this case if the amount of capital rises, then the relative return to capital falls (with an increase in wages or reduction in the rate of return to capital) such that the relative income share of labour and capital remains unchanged.
The use of these functions stem from the Kaldor Facts – stylised facts that were used to describe macroeconomic statistics in the past.
However, we know that this CES form doesn’t have to hold in reality – and so want to think about other forms.
What happens when CES is less than 1
This is the case that has, globally and historically, been most likely to hold in aggregate.
Here we state that labour and capital are considered to complement each other – at an extreme this can be though of in terms of a Leontief production function.
In this case increase in the capital stock will increase the relative share of income that is earned by labour. The simplest way to explain this result would be to think about how the change influences each price – the increase in capital leads to a large decline in the marginal product of capital as labour is fixed (in the extreme Leontiff case, the MPk = 0 past the current labour input), while the increase in capital lifts the marginal product of labour.
The vast majority of literature suggests that labour-capital income substitution is below 1 and the assumptions that capital-augmenting technology would raise the labour share is drawn around that assumption.
What happens when the substitution is above 1?
This is the case when labour and capital instead shift towards becoming perfect substitutes. Here an increase in the quantity of capital does less to increase the marginal product of labour and/or has less of an impact on reducing the marginal product of capital for a fixed amount of labour. As a result, there is less downward pressure on rates of return and less upward pressure on wages. Overall, an increase in the quantity of capital will lead to an increased income share for capital.
Thomas Piketty in his book Capital suggests that elasticity of substitution for some nations is above 1, and in the case of France and Spain this appears to be the case. However, estimates for the US and a number of other countries have not shown this form.
This all leads to the question – how does New Zealand stack up?
What is the New Zealand evidence?
However, both papers also give us different results for the aggregate elasticity – with the Stats NZ paper suggesting a figure that is less than 1 while the RBNZ paper indicates a figure that is greater than 1.
Understanding what the true elasticity of substitution is in the New Zealand context – and how technological change may influence it – seems like an important question to get a definitive answer on!