# If the Number of Workers in an Economy Doubled, Would Productivity Fall to Half its Former Value?

I am assuming that the production function for the economy has constant returns to scale, and all other inputs stay the same.

These are the steps I have made to try to answer this question:

1) Y/L = Productivity

2) Therefore, if Y=3 and L=2 (There were 2 workers in the economy), so Productivity = 3/2

3) L doubled, so now L=4

4) Now Productivity is equal to 3/4.

5) Productivity has therefore fallen by half of its former value.

Am I right in stating this? If not, can you please explain to me where I am going wrong?

Since you say holding other variables as they were and double $L(t)$ I assume you have a production function that may include capital $(K(t))$, including this and assuming a Cobb-Douglas production function:

$$Y(K(t), L(t)) = K(t)^{\alpha} L(t)^{1-\alpha} \; \; 0< \alpha < 1$$

Doubling $L(t)$ gives, (and omitting $t$ from the writing for clarity):

$$Y(K, 2L) = K^{\alpha} (2 L)^{1-\alpha}$$

$$= 2^{1-\alpha} Y(K,L)$$

$$\Rightarrow \frac{Y(K, 2L)}{2L} = \frac{2^{1-\alpha} Y(K,L)}{2L}$$

$$\frac{Y(K,L)}{2^{\alpha} L} < Y/L$$

So yes in this sense as you described doubling labour decreased productivity.

You mention $Y$ having constant returns to scale, this implies if you double both $K$ and $L$ then $Y$ will double as well, then productivity will stay the same

• Does it decrease productivity by exactly half its former value, though? Commented Mar 3, 2016 at 2:50
• No it decreases by $2^{\alpha}$ which is between $(1,2)$ (not including 1 or 2 by definition of alpha) Commented Mar 3, 2016 at 2:55
• Sorry, I will rephrase, for a Cobb-Douglas production function, doubling $L$ has decreased productivity by $2^{\alpha}$ but for some other production function it may be different ... e.g. I believe $Y= L$ would be a trivial case where doubling $L$ equally doubles $Y$ hence there is no change in productivity. Commented Mar 3, 2016 at 3:23
• Okay, I was assuming using the production function $Y = A F(L, K, H, N)$, which would become $Y/L = A F(1, K/L, H/L, N/L)$ when measuring productivity ($Y/L$ is output per worker, which is a measure of productivity), but thank you for your interesting contribution. Commented Mar 3, 2016 at 3:28