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?


1 Answer 1


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

  • $\begingroup$ Does it decrease productivity by exactly half its former value, though? $\endgroup$
    – Kelsey
    Commented Mar 3, 2016 at 2:50
  • $\begingroup$ No it decreases by $2^{\alpha}$ which is between $(1,2)$ (not including 1 or 2 by definition of alpha) $\endgroup$
    – Sunhwa
    Commented Mar 3, 2016 at 2:55
  • $\begingroup$ 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. $\endgroup$
    – Sunhwa
    Commented Mar 3, 2016 at 3:23
  • $\begingroup$ 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. $\endgroup$
    – Kelsey
    Commented Mar 3, 2016 at 3:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.