<p>In the Cobb Douglas production function technological progress can be thought of as either labor or capital augmenting, it doesn't matter. </p>

<p>Under Cobb Douglas:</p>

<p>$Y_t = F(A_t, K_t, L_t) = A_t \cdot K_t^\alpha \cdot L_t^{1-\alpha}$</p>

<p>Which can be written as Labor augmenting:</p>

<p>$Y_t = K_t^\alpha \cdot (A_t^{1/(1-\alpha)} \cdot L_t)^{1-\alpha} = F(K_t,\hat A_t L_t) $</p>

<p>Where $\hat{A}_t = A_t^{1/(1-\alpha)} $</p>

<p>But which can also be written as capital augmenting:</p>

<p>$Y_t = ( A_t^{1/\alpha} \cdot K_t)^\alpha \cdot L_t^{1-\alpha} = G(\check{A}_tK_t, L_t)$</p>

<p>Where $\check{A}_t = A_t^{1/\alpha} $</p>

<p>I believe that there is a larger class of production functions for which this is true. If I recall correctly, this is the homothetic production functions with factor augmenting technologies. </p>

CC BY-SA 3.0
2015-12-08T16:52:05.710Z
Doesn't it already break down for the natural extension of Cobb-Douglas, CES?
FooBar
CC BY-SA 3.0
2015-12-08T18:20:42.877Z
How about asking this as a separate question? I believe I can answer it.
BKay
CC BY-SA 3.0
2015-01-08T13:16:42.727Z
2015-01-08T13:21:53.677Z
2015-01-08T13:21:53.677Z
BKay
<p>The mathematical reason, is that this happens in order for the model to have a steady-state in terms of growth rates: variables like Consumption, Capital, Income, grow at the steady-state, but grow at the same rate, so their ratios remain constant (and it is in this sense that this situation represents a "steady"-state). If they were to grow at different rates, their ratios would tend to either zero or infinity which is not very realistic, since it would imply that the economy tends towards one or the other "corner" situation.  </p>

<p>The mathematical proof can be found in <a href="http://down.cenet.org.cn/upfile/8/200751171644184.pdf">Barro &amp; Sala-i-Martin book (2nd ed)</a> , section 1.5.3, pp 78-80. Relevant and useful is also the discussion in section 1.2.12, pp 51-53.</p>

<p>For functional forms like (generalized, even) Cobb-Douglas, it is really indistinguishable (not separately identifiable), especially since we predominantly use the exponential function:</p>

<p>$$Y_t = A\cdot \left (K_te^{zt}\right)^\alpha \left (L_te^{vt}\right)^\beta = A\cdot K_t^{\alpha}\left (L_te^{(v+\frac {\alpha}{\beta}z)t}\right)^\beta = A\cdot K_t^{\alpha}\left (L_te^{wt}\right)^\beta$$</p>

<p>So strictly speaking in such a functional setup we can say that technology is also capital augmenting.</p>

<p>But since for other functional forms, the above does not hold, and so we must explicitly assume that technology is "labor-augmenting" for the reason stated previously, authors settled in labeling it as such in order to cover all cases, and when they want to keep the functional form unspecified.  </p>

<p>Regarding the conceptual issue the OP poses, which is insightful, a conceptual way out is to think of "Technology" more like "Knowledge". So "Knowledge" that goes into the machines, is part of the Investment that augments capital, while the <em>other</em> knowledge turns raw labor $L$ into human capital: essentially a production function with "exogenous labor-augmenting technology", is equivalent to a formulation that includes Human Capital instead of labor but where the investment in Human Capital is not subject to optimizing behavior but "automatic" (which points to Arrow's "Learning-by-Doing" concept of human capital accumulation).</p>

CC BY-SA 4.0
2022-08-06T14:42:54.150Z
@user253751 Notice that the "corner" I am referring to is in relation to Capital/Income or consumption ratio, not Capital/Labor. And moreover, it depends on the functional forms used. We need a different functional form to allow for what you describe.
Alecos Papadopoulos
CC BY-SA 3.0
2015-01-09T01:27:04.377Z
Thank you so much for the reference. As stated, then, it is an assumption needed for a certain kind of steady state. I also agree with your argument that we could conceive capital technology as part of investment. The consequences of it, though, are severe. Romer spends most of its early chapters showing how capital accumulation can't matter for growth because it would require enormous investments to numerically explain it. But if we start thinking of all technology as capital investment, then well capital accumulation sounds again like a good explanation.
CarrKnight
CC BY-SA 3.0
2015-01-09T01:51:05.240Z
@CarrKnight A rather neglected aspect of the issue is investment in non-human intangible assets (software and intellectual property rights being the two most prominent). As you can see, both are directly linked to "technology".
Alecos Papadopoulos
CC BY-SA 3.0
2015-01-08T14:49:46.877Z
2015-01-08T18:46:28.127Z
2015-01-08T18:46:28.127Z
Alecos Papadopoulos
<p>Take Romer's advanced macro book as reference. In it the Solow model,the Ramsey model and the Diamond OLG all contain the fundamental $A_t$  variable representing technological progress.<br>
In all these models, technology affects only labor, that is:<br>
$Y_t = F(K_t,A_t L_t)$</p>

<p>Now my question is why is such assumption so prevalent in these models. It seems to me that when we imagine technology as affecting output we think of the Northrop loom, the Bessemer steel, the container, the railroad. You know, stuff. All these seem to me to be mostly capital-augmenting technologies.<br>
So why do we tend to assume labor-augmenting technology instead? </p>

CC BY-SA 3.0
2015-01-13T14:13:22.183Z
As a quick reference note, I believe I recall King and Rebelo's (1999) "[Resuscitating real business cycles](http://www.sciencedirect.com/science/article/pii/S1574004899100223)" paper from the Handbook of Macro having a nice discussion of this in their appendix. At least, that was one of the first places it "clicked" for me. The references provided in the answers, of course, are also very good (but textbooks always cost something...)
CompEcon
<macroeconomics><technology>
Why in most macro models technology is labor-augmenting?
CarrKnight
# Why in most macro models technology is labor-augmenting?

(Asked by: username_0 on 2015-01-08)

Take Romer's advanced macro book as reference. In it the Solow model,the Ramsey model and the Diamond OLG all contain the fundamental $A\_t$ variable representing technological progress.  
In all these models, technology affects only labor, that is:  
$Y\_t = F(K\_t,A\_t L\_t)$

Now my question is why is such assumption so prevalent in these models. It seems to me that when we imagine technology as affecting output we think of the Northrop loom, the Bessemer steel, the container, the railroad. You know, stuff. All these seem to me to be mostly capital-augmenting technologies.  
So why do we tend to assume labor-augmenting technology instead?

**Tags:** macroeconomics, technology
> **Comments:**
> **username_1**: As a quick reference note, I believe I recall King and Rebelo's (1999) "[Resuscitating real business cycles](http://www.sciencedirect.com/science/article/pii/S1574004899100223)" paper from the Handbook of Macro having a nice discussion of this in their appendix. At least, that was one of the first places it "clicked" for me. The references provided in the answers, of course, are also very good (but textbooks always cost something...)  

---

## 2 answers

### Answer by username_4 on 2015-01-08. Score: 12 | Accepted answer

The mathematical reason, is that this happens in order for the model to have a steady-state in terms of growth rates: variables like Consumption, Capital, Income, grow at the steady-state, but grow at the same rate, so their ratios remain constant (and it is in this sense that this situation represents a "steady"-state). If they were to grow at different rates, their ratios would tend to either zero or infinity which is not very realistic, since it would imply that the economy tends towards one or the other "corner" situation.

The mathematical proof can be found in [Barro & Sala-i-Martin book (2nd ed)](http://down.cenet.org.cn/upfile/8/200751171644184.pdf) , section 1.5.3, pp 78-80. Relevant and useful is also the discussion in section 1.2.12, pp 51-53.

For functional forms like (generalized, even) Cobb-Douglas, it is really indistinguishable (not separately identifiable), especially since we predominantly use the exponential function:

$$Y\_t = A\cdot \left (K\_te^{zt}\right)^\alpha \left (L\_te^{vt}\right)^\beta = A\cdot K\_t^{\alpha}\left (L\_te^{(v+\frac {\alpha}{\beta}z)t}\right)^\beta = A\cdot K\_t^{\alpha}\left (L\_te^{wt}\right)^\beta$$

So strictly speaking in such a functional setup we can say that technology is also capital augmenting.

But since for other functional forms, the above does not hold, and so we must explicitly assume that technology is "labor-augmenting" for the reason stated previously, authors settled in labeling it as such in order to cover all cases, and when they want to keep the functional form unspecified.

Regarding the conceptual issue the OP poses, which is insightful, a conceptual way out is to think of "Technology" more like "Knowledge". So "Knowledge" that goes into the machines, is part of the Investment that augments capital, while the *other* knowledge turns raw labor $L$ into human capital: essentially a production function with "exogenous labor-augmenting technology", is equivalent to a formulation that includes Human Capital instead of labor but where the investment in Human Capital is not subject to optimizing behavior but "automatic" (which points to Arrow's "Learning-by-Doing" concept of human capital accumulation).

> **Comments:**
> **username_0**: Thank you so much for the reference. As stated, then, it is an assumption needed for a certain kind of steady state. I also agree with your argument that we could conceive capital technology as part of investment. The consequences of it, though, are severe. Romer spends most of its early chapters showing how capital accumulation can't matter for growth because it would require enormous investments to numerically explain it. But if we start thinking of all technology as capital investment, then well capital accumulation sounds again like a good explanation.  
> **username_4**: @CarrKnight A rather neglected aspect of the issue is investment in non-human intangible assets (software and intellectual property rights being the two most prominent). As you can see, both are directly linked to "technology".  
> **username_4**: @user253751 Notice that the "corner" I am referring to is in relation to Capital/Income or consumption ratio, not Capital/Labor. And moreover, it depends on the functional forms used. We need a different functional form to allow for what you describe.  

---

### Answer by username_2 on 2015-01-08. Score: 7

In the Cobb Douglas production function technological progress can be thought of as either labor or capital augmenting, it doesn't matter.

Under Cobb Douglas:

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

Which can be written as Labor augmenting:

$Y\_t = K\_t^\alpha \cdot (A\_t^{1/(1-\alpha)} \cdot L\_t)^{1-\alpha} = F(K\_t,\hat A\_t L\_t) $

Where $\hat{A}\_t = A\_t^{1/(1-\alpha)} $

But which can also be written as capital augmenting:

$Y\_t = ( A\_t^{1/\alpha} \cdot K\_t)^\alpha \cdot L\_t^{1-\alpha} = G(\check{A}\_tK\_t, L\_t)$

Where $\check{A}\_t = A\_t^{1/\alpha} $

I believe that there is a larger class of production functions for which this is true. If I recall correctly, this is the homothetic production functions with factor augmenting technologies.

> **Comments:**
> **username_3**: Doesn't it already break down for the natural extension of Cobb-Douglas, CES?  
> **username_2**: How about asking this as a separate question? I believe I can answer it.  

---

