<p>You did nothing wrong. If <span class="math-container">$A\ne0$</span>, <span class="math-container">$y^*\ne0$</span> and <span class="math-container">$\alpha\ne1$</span>, then
<span class="math-container">$$\left(\,\frac{y^*}{A}\right)^\frac{1}{\alpha-1}=\left(\left(\,\frac{A}{y^*}\right)^{-1}\right)^\frac{1}{\alpha-1}=\left(\,\frac{A}{y^*}\right)^{-\frac{1}{\alpha-1}}=\left(\,\frac{A}{y^*}\right)^{\frac{1}{1-\alpha}}$$</span></p>

CC BY-SA 4.0
2024-01-11T12:54:25.540Z
2024-01-11T12:54:25.540Z
VARulle
<p>I'm working with growth models in a course, and keep stumbling into this when trying to isolate variables of interest.
I have a formula <span class="math-container">$y^*=AL^{\alpha-1}$</span> and I want to isolate <span class="math-container">$L$</span> in this expression. When I do this, I get the answer that
<span class="math-container">$$L=\left(\,\frac{y^*}{A}\right)\,^\frac{1}{\alpha-1}$$</span>
According to my lecture slides, the fraction should be flipped and the exponent should now read <span class="math-container">$1-\alpha$</span> instead of <span class="math-container">$\alpha-1$</span>, like so:
<span class="math-container">$$L=\left(\,\frac{A}{y^*}\right)\,^\frac{1}{1-\alpha}$$</span>
I cannot find any hints online as to what I'm doing wrong, hope some of you can explain the reason for this.</p>

CC BY-SA 4.0
2024-01-10T13:02:29.387Z
two hints: (i) $1-\alpha = -(\alpha -1)$, (ii) for $x\ne 0,\, x^{-1}=\frac{1}{x}$.
VARulle
CC BY-SA 4.0
2024-01-10T14:19:04.600Z
@VARulle I think you can just post that as the answer.
Giskard
CC BY-SA 4.0
2024-01-11T12:54:56.137Z
@Giskard, O.k., done.
VARulle
CC BY-SA 4.0
2024-01-11T15:08:19.533Z
So, the point is the following: input elasticities of output (namely, $\alpha$ in your model) are generally between 0 and 1 (most of the Cobb-Douglas production functions exhibit decreasing returns to scale). Hence, you generally flip everything, so that you have a clearer representation.
Matteo Bulgarelli
<economic-growth>
Question regarding math rules for fractions and exponents when isolating variables of interest
Daniel
# Question regarding math rules for fractions and exponents when isolating variables of interest

(Asked by: username_0 on 2024-01-10)

I'm working with growth models in a course, and keep stumbling into this when trying to isolate variables of interest.
I have a formula $y^\*=AL^{\alpha-1}$ and I want to isolate $L$ in this expression. When I do this, I get the answer that
$$L=\left(\,\frac{y^\*}{A}\right)\,^\frac{1}{\alpha-1}$$
According to my lecture slides, the fraction should be flipped and the exponent should now read $1-\alpha$ instead of $\alpha-1$, like so:
$$L=\left(\,\frac{A}{y^\*}\right)\,^\frac{1}{1-\alpha}$$
I cannot find any hints online as to what I'm doing wrong, hope some of you can explain the reason for this.

**Tags:** economic-growth

> **Comments:**
> **username_1**: two hints: (i) $1-\alpha = -(\alpha -1)$, (ii) for $x\ne 0,\, x^{-1}=\frac{1}{x}$.  
> **username_2**: @username_1 I think you can just post that as the answer.  
> **username_1**: @username_2, O.k., done.  
> **username_3**: So, the point is the following: input elasticities of output (namely, $\alpha$ in your model) are generally between 0 and 1 (most of the Cobb-Douglas production functions exhibit decreasing returns to scale). Hence, you generally flip everything, so that you have a clearer representation.  

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## 1 answers

### Answer by username_1 on 2024-01-11. Score: 5

You did nothing wrong. If $A\ne0$, $y^\*\ne0$ and $\alpha\ne1$, then
$$\left(\,\frac{y^\*}{A}\right)^\frac{1}{\alpha-1}=\left(\left(\,\frac{A}{y^\*}\right)^{-1}\right)^\frac{1}{\alpha-1}=\left(\,\frac{A}{y^\*}\right)^{-\frac{1}{\alpha-1}}=\left(\,\frac{A}{y^\*}\right)^{\frac{1}{1-\alpha}}$$


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