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Short 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.

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.

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