### Archive

Archive for the ‘Factor Markets’ Category

## The effect of immigration on labour markets

This is a contentious subject and the analysis I am giving here is far from comprehensive, I’m just using one of the general arguments to show how simple factor market analysis is used in this context.

One of the theories surrounding the effect of immigration on labour markets is that you can divide labour into ‘skilled’ and ‘unskilled’ labour, and the immigration of one of the two forms of labour in the absence of sufficient immigration of the other, will lead to negative effects for domestic workers in the type of labour that is receiving immigration, and positive effects for the other type.

This rests on the idea that the two factors are complementary, in the way that capital is complementary to labour – so an increase in unskilled labour will ‘augment’ the labour of skilled labour by making it more productive (presumably skilled labour will be able to achieve more output if it has a wider pool of unskilled labour to work with) and an increase in skilled labour will ‘augment’ the labour of unskilled labour (possibly by designing products and processes that will improve the productivity of unskilled labour – capital could be involved here as well if the skilled labour improves the amount and quality of capital available through its skilled design).

So lets consider the case where there is immigration only of unskilled labour.

First lets see the effects on the labour market for unskilled labour:

At first where there are only home workers, L1 workers are employed and are paid wage w1. The arrival of the immigrants increases the supply of unskilled labour so the total amount of labour hired rises to L2 but the market wage is driven down to w2. Returning to the original supply curve of home workers, you can see that some of the workers that were willing to work at wage w1 are now unwilling to work at wage w2 and so they become voluntarily unemployed.

Now lets see the effects on the labour market for skilled labour:

Remember here that there is no migration of skilled labour, all the migrants are unskilled. So the supply of skilled labour does not change. But as unskilled labour is a complementary factor (ie it augments the productivity of skilled labour) it increases the marginal productivity of skilled labour, so it shifts out the marginal revenue product of skilled labour curve. This results in an increase in the amount of skilled labour being hired, and an increase in the wage of skilled labour.

You could have the same type of graph if you were looking at the effects on the owners of capital. If there is an increase in a complementary factor, and unskilled labour was complementary to capital, then you would have the same graph as above – the marginal revenue product of capital would increase, and so the amount of capital demanded and the return to capital would increase.

So the conclusion from this type of model is that migration of unskilled labour benefits skilled labour and owners of capital and harms domestic unskilled labour.

You can think of this in terms of an “immigration surplus” by looking at the labour market for unskilled labour:

First think of the perspective of “firms”, as these are the consumers in a labour market, they are the buyers of unskilled labour. The “consumer surplus” ie the surplus to firms, is a in the first case where there are only home workers in the market. After the immigrant workers join, then the surplus rises to a + b + c + d so there has been a gain of b + c + d. This is the immigration surplus. This surplus will go to the owners of the complementary factors, ie the owners of capital and skilled workers.

Now think of the perspective of home unskilled workers. At first their “producer surplus” as the producers of labour in this market is area b + e. But after the immigrant labour joins the workforce, the surplus is reduced to e, so home workers have lost b to firms.

The overall “producer surplus” for total workers after the immigrants join is e + f + g + h, however all the areas other than e are below the supply curve for home workers, so areas f + g + h are gained by migrant workers.

So the predictions of this model are that migration of unskilled labour will cause lower wages and more unemployment for home unskilled workers while increasing the returns for owners of capital and increasing the wages of skilled workers. However the empirical literature on this field is mixed, so the model may not be an entirely accurate reflection of reality – as with most economic models it makes a lot of assumptions. But it is a good idea to understand the model at least.

## Adjustments in the market demand for labour

We have seen how in the short run, changes in the market wage can trigger changes in the demand for labour. The simple intuition here is that if the price of labour falls, then because firms are hiring labour at the point where the wage equals the marginal revenue product of labour, a lower wage means that point comes at a lower point on the marginal revenue product of labour curve, ie at a higher amount of labour hired.

But now think about this in terms of the market rather than just the firm. The firm is responding to the fall in wages by hiring more labour which will mean it is increasing its output. Presumably other firms are doing the same. So if everybody is producing more that means the supply of the good is increasing, which will then push down the market price of the good.

A fall in the market price of the good is equivalent to a reduction in the marginal revenue product of labour, as in a perfectly competitive market this is $P(MP_L)$.

So this represents a downwards shift of the marginal revenue product of labour curve.

Lets see these effects on a graph:

The fall in wages from w1 to w2 encourages firms to increase more output and hire more labour, which reduces the market price of the good. This means that the marginal revenue product of labour falls. The actual market response in terms of the demand for labour is where the new marginal revenue product of labour curve intersects the new market price.

## Long run factor demand: Competitive markets

In the short run a firm’s capital is fixed so the only thing it can vary is labour. If it wants to produce more it has to hire more labour. But in the long run it can vary both the amount of capital and labour.

So we can express the production function in this form: $q=q(K,L)$.

The firm’s revenue will be a function of its output, because the more output it produces the more revenue it will get, the revenue will be of the form $R = R(q(K,L))$.

The firm will face a cost for capital (which is fixed) and a cost for labour (which varies according to the amount hired). So the cost will be of the form $C = wL + rK$ where w is the wage and r is the return to capital (r is usually used as it denotes the ‘rent’ to the owner of capital).

So the profit function, which is revenue minus cost, will be $\pi = R(q(K,L)) - wL - rK$.

If the firm wants to maximise its profits then it has to choose the amount of capital such that $\frac{\partial \pi}{\partial K} = 0$, and the amount of labour such that $\frac{\partial \pi}{\partial L} = 0$. This has to be differentiated using the chain rule:

$\frac{\partial \pi}{\partial K} = \frac{\partial R}{\partial Q}\frac{\partial Q}{\partial K} - w$ so when $\frac{\partial \pi}{\partial K} = 0$, $\frac{\partial R}{\partial Q}\frac{\partial Q}{\partial K} = r$.

$\frac{\partial \pi}{\partial L} = \frac{\partial R}{\partial Q}\frac{\partial Q}{\partial L} - w$ so when $\frac{\partial \pi}{\partial L} = 0$, $\frac{\partial R}{\partial Q}\frac{\partial Q}{\partial L} = w$.

$\frac{\partial R}{\partial Q}$ is the change in revenue when output is increased, ie marginal revenue, $MR$.
$\frac{\partial Q}{\partial K}$ is the change in output when labour is increased, ie marginal productivity of labour, $MP_K$.
$\frac{\partial Q}{\partial L}$ is the change in output when labour is increased, ie marginal productivity of labour, $MP_L$.

So we can express $\frac{\partial R}{\partial Q}\frac{\partial Q}{\partial K} = r$ as $MR(MP_K) = r$ and $\frac{\partial R}{\partial Q}\frac{\partial Q}{\partial L} = w$ as $MR(MP_L) = w$.

This tells us that the profit maximising amount of factors to hire, is the amount of capital at which the marginal revenue multiplied by the marginal productivity of capital, is equal to the return to capital and the amount of labour at which the marginal revenue multiplied by the marginal productivity of labour, is equal to the wage.

Marginal revenue multiplied by marginal productivity of capital is also called the marginal revenue product of capital, $MRP_K$ so here we have $MRP_K = r$.

Marginal revenue multiplied by marginal productivity of labour is also called the marginal revenue product of labour, $MRP_L$ so here we have $MRP_L = w$.

In a competitive market, MR = P, so $P(MP_K) = r$ and $P(MP_L) = w$.

This gives us the firm’s long run labour demand function, the firm will hire capital up to the point where $P(MP_K) = r$ or $P\frac{\partial Q}{\partial K} = r$ and labour up to the point where $P(MP_L) = w$ or $p\frac{\partial Q}{\partial L} = w$.

Now we can see how the short run demand and long run demand for a factor differs. Lets consider the demand for labour. The short run demand is simply the marginal revenue product of labour.

Here if we start from a situation where there is a perfectly elastic supply of labour at the market wage w1, the firm will hire L1 workers, and then if the market wage falls to w2, the firm hires more labour, L2, as it simply equates the level at which the value of the marginal product of labour (the marginal revenue product of labour) equals the wage.

But then in the long run, what if the cheaper labour enabled the firm to purchase more capital and increase its level of capital. More capital would increase the marginal productivity of labour, as it would mean each unit of labour has more capital with which to work. So more capital would result in a shifting out of the marginal revenue product of labour curve.

Here the firm equates the level at which the new marginal revenue product of labour curve equates with the wage, so it hires the amount of labour L3. The demand for labour has increased from the short run to the long run, as a result of the change in the wage.

The long run labour demand curve is shallower than the short run labour demand curve. This means that the employment response from a fall in wages will be more elastic in the long run than the short run.

## Short run factor demand: Competitive markets

In the short run a firm’s capital is fixed so the only thing it can vary is labour. If it wants to produce more it has to hire more labour.

So we can express the production function in this form: $q=q(L,\Bar{K})$ or simply as $q=q(L)$.

The firm’s revenue will be a function of its output, because the more output it produces the more revenue it will get, the revenue will be of the form $R = R(q(L))$.

The firm will face a cost for capital (which is fixed) and a cost for labour (which varies according to the amount hired). So the cost will be of the form $C = wL + F$ where F is the fixed cost of the capital.

So the profit function, which is revenue minus cost, will be $\pi = R(q(L)) - wL - F$.

If the firm wants to maximise its profits then it has to choose the amount of labour such that $\frac{d \pi}{d L} = 0$. This has to be differentiated using the chain rule:

$\frac{d \pi}{d L} = \frac{d R}{d Q}\frac{d Q}{d L} - w$ so when $\frac{d \pi}{d L} = 0$, $\frac{d R}{d Q}\frac{d Q}{d L} = w$.

$\frac{d R}{d Q}$ is the change in revenue when output is increased, ie marginal revenue, $MR$.
$\frac{d Q}{d L}$ is the change in output when labour is increased, ie marginal productivity of labour, $MP_L$.

So we can express $\frac{d R}{d Q}\frac{d Q}{d L} = w$ as $MR(MP_L) = w$.

This tells us that the profit maximising amount of labour to hire, is the amount at which the marginal revenue multiplied by the marginal productivity of labour, is equal to the wage.

Marginal revenue multiplied by marginal productivity of labour is also called the marginal revenue product of labour, $MRP_L$ so here we have $MRP_L = w$.

In a competitive market, MR = P, so $P(MP_L) = w$. This is the firm’s short run labour demand function, the firm will hire labour up to the point where the wage is equal to the price multiplied by the marginal productivity of labour.

Usually when capital is fixed the marginal productivity of labour will diminish as more labour is hired, if you only had a fixed amount of machinery and just hired more and more labour, each additional worker would be less useful than the last as there would not be enough machinery to go around.

Lets look at an example on a graph:

Here we have a competitive factor market where there is a perfectly elastic supply of labour willing to work at wage w1. The firm will hire at the point where the wage equals the marginal revenue product of labour (which will here be the price multiplied by the marginal productivity of labour), at quantity of labour L1.

Now what would happen if the market wage changed, for instance if there was a reduction in the supply of labour that drove the market wage up…?

Now the wage is higher, but the marginal revenue product of labour is unchanged, so the firm reacts by hiring less labour.

What would happen if the marginal revenue product of labour changed?
This could happen for two reasons:
– the market price of the good could change (as $MRP_L = MR(MP_L)$ and here in a competitive market $MRP_L = P(MP_L)$, any change in P will change the marginal revenue product of labour)
– the marginal productivity of labour could change, for instance because workers became more skilled at their job

Take the example where the market price of the good increased so the marginal revenue product of labour increased…

This time the firm hires a greater amount of labour.