<p>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:</p>

<p>$$ Y(K(t), L(t)) = K(t)^{\alpha} L(t)^{1-\alpha} \; \; 0&lt; \alpha &lt; 1 $$ </p>

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

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

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

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

<p>$$ \frac{Y(K,L)}{2^{\alpha} L} &lt; Y/L $$</p>

<p>So yes in this sense as you described doubling labour decreased productivity.  </p>

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

CC BY-SA 3.0
2016-03-03T02:50:35.883Z
Does it decrease productivity by exactly half its former value, though?
Kelsey
CC BY-SA 3.0
2016-03-03T02:55:07.667Z
No it decreases by $2^{\alpha}$ which is between $(1,2)$ (not including 1 or 2 by definition of alpha)
Sunhwa
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2016-03-03T03:23:38.167Z
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.
Sunhwa
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2016-03-03T03:28:26.320Z
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.
Kelsey
CC BY-SA 3.0
2016-03-03T02:48:11.523Z
2016-03-03T02:56:02.043Z
2016-03-03T02:56:02.043Z
Sunhwa
<p>I am assuming that the production function for the economy has constant returns to scale, and all other inputs stay the same.</p>

<p>These are the steps I have made to try to answer this question:</p>

<p>1) Y/L = Productivity</p>

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

<p>3) L doubled, so now L=4</p>

<p>4) Now Productivity is equal to 3/4. </p>

<p>5) Productivity has therefore fallen by half of its former value.</p>

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

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

(Asked by: username_0 on 2016-03-03)

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?

**Tags:** macroeconomics, productivity


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## 1 answers

### Answer by username_1 on 2016-03-03. Score: 3 | Accepted answer

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

> **Comments:**
> **username_0**: Does it decrease productivity by exactly half its former value, though?  
> **username_1**: No it decreases by $2^{\alpha}$ which is between $(1,2)$ (not including 1 or 2 by definition of alpha)  
> **username_1**: 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.  
> **username_0**: 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.  

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