Labour Saving vs. Labour Augmenting - what is the difference? It seems reasonable to say that technology generally increases output per unit of labour.  Therefore in any factory which has installed new technology one can reasonably expect that for the same output one is likely to need less employees than before (or perhaps the firm will retain the same number of employees but output will increase).  Therefore, in either case, the output per employee has increased.  Therefore one could say that labour has been saved by the new technology.  One could also say that labour has been augmented by the new technology as each worker is supported by the new technology and is now more productive.

My question is: do we have clear definitions of these two terms and what is the difference?
Consider a Leontief production function:

$$ Y = \text{min}(aL,bK) $$

Optimal capital-labour ratio is:

$$ \frac{K^*}{L^*}=\frac{a}{b} $$ 

Labour-saving technical change is such that $a$ increases. The optimal capital-labour ratio increases, this is, firms use less labour per unit of capital. This is also called Capital-augmenting technical change (see reference [2]).

Labour-augmenting technical change is such that $b$ increases. The optimal capital-labour ratio falls, this is, firms use more labour per unit of capital.

Notice that how output per worker changes depend on how $Y$ changes. For example, assume that the above adjustment is such that $Y$ is unchanged after the technical change. Then, output per worker increases in the labour-saving case but falls in the labour-augmenting change. A complete analysis would require a general equilibrium approach, where factor prices are endogenous. All I am showing here is that it is not necessarily the case that either type of technical change increases output per worker.

Some references: [1], [2].

PS: I use the Leontief for simplicity. In a Cobb-Douglas, any technical change is Hicks-neutral, as it does not affect the K/L ratio (assuming fixed factor prices). In a CES, the nature of technical change depends on the elasticity of substitution.