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Lewis and MacDonald (2002) and a host of others have demonstrated that long-run aggregate demand for labour in Australia is well-approximated by the first order condition for labour derived by solving a standard profit maximisation problem of a representative firm subject to a constant elasticity of substitution (CES) production function. The representative firm's profit maximisation problem can be presented as follows:

(1)

subject to a constant returns to scale and CES production function:

(2)

where: *∏ _{t}* is profit at time

*t*;

*p*is output prices;

_{t}*Y*is output (volume);

_{t}*w*is nominal wages; is the payroll tax rate;

_{t}*N*is employment;

_{t}*q*is the rental rate of capital;

_{t}*K*is capital;

_{t}*A*is Hicks-neutral technical change; 0 < θ < 1 is a weighting parameter;

_{t}*X*is labour-augmenting technical change (Harrod-neutral); and σ > 0 is the elasticity of substitution between capital and labour.

_{t}For ease of exposition and without loss of generality, we assume that the firm's production function is always binding, which allows us to solve the following problem:

(3)

The resulting necessary first order condition for labour implies:

(4)

Substituting (2) into (4):

(5)

and solving for labour (*N _{t}*) implies:

(6)

Taking logarithms gives the following labour demand relationship, conditional on the level of output, labour-augmenting technical change and real wages:

(7)

This expression can be simplified further by noting that there are no cyclical fluctuations over the long-run, so in the long-run *A _{t}* will be at its average level

*A*, which yields the following long-run labour demand equation:

(8)

where:

Equation (8) gives the logarithm of long-run labour demand as equal to three terms: a constant, the logarithm of real output adjusted for labour-augmenting technical change, and the elasticity of substitution multiplied by the logarithm of the real wage adjusted for labour-augmenting technical change (that is, the logarithm of long-run real unit labour costs). The effect of an increase in the level of labour-augmenting technical change on labour demand depends on the size of the elasticity of substitution between capital and labour: other things equal, labour demand decreases if σ < 1 and increases if σ > 1.