# Labour Productivity Growth and Reallocation of Labour

Using a shift-share methodology, labour productivity growth can be arise from productvity growth within a sector (within effect) or reallocation of labour to sectors with higher productivity (reallocation effect).

### 1)

However we know that Productivity = Real Value-added / Employment.

So won't an increase in employment in a sector result in a fall in the sector's productivity and hence overall productivity?

employment intensity is the direct inverse of labour productivity

So does this mean the government should not employ policies to reallocate labour to higher productivity sectors?

It will be good to refer me to some paper that address these questions

You make a hand waiving claim. In these scenarios making concise mathematic statements helps improving your arguments.

Denote employment as $L$, define real value added as $F(L) - wL$, and productivity as real value / employment:

$$A = \frac{F(L) - w(L)}{L}$$

The labor share should be some constant $\beta$ of total value added. Without going deep into mathematics, straight away assume $w(L) = \beta\alpha F(L)$. Hence, we have that

$$A = \frac{F(L) (1 - \beta\alpha)}{L}$$

Now, what happens if $L$ increases? Let $L_2 > L_1$

$$A_2 - A_1 = (1-\alpha\beta)(\frac{F(L_2)}{L_2} - \frac{F(L_1)}{L_1})$$

Assume $F$ to be concave. Then we have that the change of productivity, $A_2 - A_1$ is negative - as you say.

However, $A$ here is average productivity. Total productivity is $AL$.

$$A_2L_2 - A_1L_1 = (1-\alpha\beta)(F(L_2) - F(L_1))$$.

Independent of the curvature of $F$, for any strictly increasing production function, this will be positive.

tl;dr

• Don't confuse average productivity with total productivity.
• Whenever you find something weird, use algebra to make your statements precise

Again, you are not being precise enough. What is the government's objective? If government is only about maximizing average productivity, the solution is to let $L\to 0$, as then $F'(L) \to \infty$ (and for Cobb-Douglas, $F'(L)$ is proportional to average productivity).

Maximizing total output implies allocating workers to the sectors with highest productivity. I do not understand how the functional relationship between employment intensity and labor productivity affects that premise. Furthermore, unless there is a market distortion, the role of any government intervention is not clear at all.

• Sorry for Ad1) why is it that if F is concave, A2-A1 is negative. And why if F is a strictly increasing function, A2 L2- A1 L1 is positive. Thanks May 11 '15 at 1:13
• Sorry for Ad2), Government objective is to maximise labour productivity growth rate. So should Government reallocate labour to sectors with higher productivity level or growth rate? Thanks May 11 '15 at 1:16