<p>Think of your function as</p>

<p>$$Y=[A_1Q^\sigma+ A_2X^\sigma]^{1/\sigma}$$</p>

<p>where $Q=L^\alpha K^\beta$ (I removed unnecessary bits like $\gamma$ and exponent of $A_i$). This is a standard CES, where the optimal use of $Q$ and $X$ depend on $\frac{A_1}{A_2}$ and $\sigma$.</p>

<p>For a relatively low substitution ($\sigma&lt;0$), an increase in $A_1$ (<strong>holding factor prices fixed</strong>) means firms make use of that extra increase in productivity of factor $Q$ by using less of it and more of $X$. Since $A_1$ corresponds to a Hick neutral change regarding $Q$, you will use the bundle $Q$ with the same initial $K/L$ ratio than before the change in $A_1$. Therefore, you will <strong>reduce both $K$ and $L$</strong>. In consequence, an increase in $A_1$ could be thought as labour-saving technical change <em>with respect to $X$</em>, and in absolute value (although notice that $K/L$ remains unchanged).</p>

<p>The reverse is true if $0&lt;\sigma \leq 1$, namely an increase in $A_1$ leads to more use of $Q$ relative to $X$, thereby expanding the use of $L$ and $K$. </p>

<p>Meanwhile, in the inbetween case ($\sigma=0$), the function <a href="https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function">collapses to</a> a Codd-Douglas:</p>

<p>$$ Y = \left(L^\alpha K^\beta\right)^{\phi} X^{1-\phi} $$ </p>

<p>where $\phi = \frac{A_1}{A_1 + A_2}$. Here a change in $A_1$ does not affect the K/L ratio, albeit it does affect the optimal ratio with respect to $X$. But commonly technical change is though as a change in some $A$ rather than in the elasticities of factors, so this case does not nest well with the above. I think this might be because the original function <a href="https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf?9fcd5fddf87498417164716f2bbee9fd" rel="nofollow noreferrer">is not properly normalised</a>. This is, the terms $A_1$ and $A_2$ do not add up to one, which is necessary given your original assumption of constant returns to scale. </p>

CC BY-SA 3.0
2017-09-04T12:17:18.330Z
Once I get the time I will add the mathematical proof.
luchonacho
CC BY-SA 3.0
2017-09-05T06:41:14.650Z
First, the in-between case is $\sigma=1$ not $\sigma=0$. $\sigma=1$ yields directly a linear $Y=A_1Q+A_2X$. In the case of $\sigma=0$ you get of course $Y=Q^\alpha X^{1-\alpha}$ where $\alpha=A_1/(A_1+A_2)$.
Fato
CC BY-SA 3.0
2017-09-05T06:53:53.427Z
@Fato You are right. Confused $\sigma$ with $\rho$. Updated.
luchonacho
CC BY-SA 3.0
2017-09-04T12:06:13.317Z
2017-09-05T07:10:09.043Z
2017-09-05T07:10:09.043Z
luchonacho
<p>Assume, using the following equation
$Y=[((A_1L)^\alpha K^\beta)^\sigma+ (A_2X^\gamma)^\sigma]^{1/\sigma}$, we back out (obtain) the evolution of $A_1$ &amp; $A_2$. Can we interpret $A_1$ as labour-saving technical change index since it augments labour in the model?</p>

CC BY-SA 3.0
2017-09-04T10:50:12.587Z
Say, there is a limited substitutability between the C-D composite and X.
london
<economic-growth><production-function><ces-function>
Cobb-Douglas nested in CES model
london
# Cobb-Douglas nested in CES model

(Asked by: username_0 on 2017-09-04)

Assume, using the following equation
$Y=[((A\_1L)^\alpha K^\beta)^\sigma+ (A\_2X^\gamma)^\sigma]^{1/\sigma}$, we back out (obtain) the evolution of $A\_1$ & $A\_2$. Can we interpret $A\_1$ as labour-saving technical change index since it augments labour in the model?

**Tags:** economic-growth, production-function, ces-function
> **Comments:**
> **username_0**: Say, there is a limited substitutability between the C-D composite and X.  

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## 1 answers

### Answer by username_1 on 2017-09-04. Score: 3 | Accepted answer

Think of your function as

$$Y=[A\_1Q^\sigma+ A\_2X^\sigma]^{1/\sigma}$$

where $Q=L^\alpha K^\beta$ (I removed unnecessary bits like $\gamma$ and exponent of $A\_i$). This is a standard CES, where the optimal use of $Q$ and $X$ depend on $\frac{A\_1}{A\_2}$ and $\sigma$.

For a relatively low substitution ($\sigma<0$), an increase in $A\_1$ (**holding factor prices fixed**) means firms make use of that extra increase in productivity of factor $Q$ by using less of it and more of $X$. Since $A\_1$ corresponds to a Hick neutral change regarding $Q$, you will use the bundle $Q$ with the same initial $K/L$ ratio than before the change in $A\_1$. Therefore, you will **reduce both $K$ and $L$**. In consequence, an increase in $A\_1$ could be thought as labour-saving technical change *with respect to $X$*, and in absolute value (although notice that $K/L$ remains unchanged).

The reverse is true if $0<\sigma \leq 1$, namely an increase in $A\_1$ leads to more use of $Q$ relative to $X$, thereby expanding the use of $L$ and $K$.

Meanwhile, in the inbetween case ($\sigma=0$), the function [collapses to](https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function) a Codd-Douglas:

$$ Y = \left(L^\alpha K^\beta\right)^{\phi} X^{1-\phi} $$

where $\phi = \frac{A\_1}{A\_1 + A\_2}$. Here a change in $A\_1$ does not affect the K/L ratio, albeit it does affect the optimal ratio with respect to $X$. But commonly technical change is though as a change in some $A$ rather than in the elasticities of factors, so this case does not nest well with the above. I think this might be because the original function [is not properly normalised](https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf?9fcd5fddf87498417164716f2bbee9fd). This is, the terms $A\_1$ and $A\_2$ do not add up to one, which is necessary given your original assumption of constant returns to scale.

> **Comments:**
> **username_1**: Once I get the time I will add the mathematical proof.  
> **username_2**: First, the in-between case is $\sigma=1$ not $\sigma=0$. $\sigma=1$ yields directly a linear $Y=A_1Q+A_2X$. In the case of $\sigma=0$ you get of course $Y=Q^\alpha X^{1-\alpha}$ where $\alpha=A_1/(A_1+A_2)$.  
> **username_1**: @Fato You are right. Confused $\sigma$ with $\rho$. Updated.  

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