<p>Given a Cobb Douglas  </p>

<p><span class="math-container">$Y_t = A (K_t^\alpha L_t^{1-\alpha}) $</span> </p>

<p><span class="math-container">$ K_{t+1} = sY_t + (1-\delta) K_t$</span></p>

<p>How do we get the multiplier on productivity to be equal to <span class="math-container">$ \frac{1}{1-\alpha}$</span>? 
I understand that if productivity increases, output increases, thus we get more capital and thereby more output and so on. But I can't reach this multiplier.</p>

<p>My Attempt: </p>

<p>If have x increase in A, 
then 
<span class="math-container">$Y_t=(1+x)A (K_t^\alpha L_t^{1-\alpha})  $</span></p>

<p>That is an x increase in Y. </p>

<p><span class="math-container">$ K_{t+1} = s(1+x)A (K_t^\alpha L_t^{1-\alpha})  + (1-\delta) K_t$</span></p>

<p><span class="math-container">$Y_{t+1}=(1+x)A * K_{t+1}^\alpha *L_{t+1}^{1-\alpha} $</span>
<span class="math-container">$Y_{t+1}=  (1+x)A * (s(1+x)A (K_t^\alpha L_t^{1-\alpha})  + (1-\delta) K_t)^\alpha  * L_{t+1}^{1-\alpha} $</span></p>

<p>I think the increase here should be <span class="math-container">$x * \alpha $</span> but I can't see it.
so that for a unit increase in productivity i.e x=1;
 <span class="math-container">$\Delta Y  =  1 + \alpha + \alpha^2 +... =\frac{1}{1-\alpha} $</span></p>

<p>Also in the steady state we  have, 
 <span class="math-container">$ Y = A^\frac{1}{1-\alpha} * \frac{s}{\delta}^\frac{\alpha}{1-\alpha} * L$</span>
taking logs we have that percentage change in A increases Y by <span class="math-container">$\frac{1}{1-\alpha}$</span>.</p>

CC BY-SA 4.0
2019-01-27T02:54:46.933Z
@KennyLJ  I updated with my attempt
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<macroeconomics><economic-growth><cobb-douglas><solow><growth-accounting>
How to Calculate the productivity multiplier?
Fatima
# How to Calculate the productivity multiplier?

(Asked by: username_0 on 2019-01-27)

Given a Cobb Douglas

$Y\_t = A (K\_t^\alpha L\_t^{1-\alpha}) $

$ K\_{t+1} = sY\_t + (1-\delta) K\_t$

How do we get the multiplier on productivity to be equal to $ \frac{1}{1-\alpha}$?
I understand that if productivity increases, output increases, thus we get more capital and thereby more output and so on. But I can't reach this multiplier.

My Attempt:

If have x increase in A,
then
$Y\_t=(1+x)A (K\_t^\alpha L\_t^{1-\alpha}) $

That is an x increase in Y.

$ K\_{t+1} = s(1+x)A (K\_t^\alpha L\_t^{1-\alpha}) + (1-\delta) K\_t$

$Y\_{t+1}=(1+x)A \* K\_{t+1}^\alpha \*L\_{t+1}^{1-\alpha} $
$Y\_{t+1}= (1+x)A \* (s(1+x)A (K\_t^\alpha L\_t^{1-\alpha}) + (1-\delta) K\_t)^\alpha \* L\_{t+1}^{1-\alpha} $

I think the increase here should be $x \* \alpha $ but I can't see it.
so that for a unit increase in productivity i.e x=1;
$\Delta Y = 1 + \alpha + \alpha^2 +... =\frac{1}{1-\alpha} $

Also in the steady state we have,
$ Y = A^\frac{1}{1-\alpha} \* \frac{s}{\delta}^\frac{\alpha}{1-\alpha} \* L$
taking logs we have that percentage change in A increases Y by $\frac{1}{1-\alpha}$.

**Tags:** macroeconomics, economic-growth, cobb-douglas, solow, growth-accounting

> **Comments:**
> **username_0**: @KennyLJ  I updated with my attempt  

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## 0 answers

