Home > Factor Markets, Micro concepts > Long run factor demand: Competitive markets

## Long run factor demand: Competitive markets

January 17, 2012

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.