<blockquote>
<p>Why write an implicit function instead of simply solving for <span class="math-container">$w_{u,t}$</span>?</p>
</blockquote>
<p>Because solving for it explicitly is time consuming and would not yield any new information.</p>
<p>If someone writes <span class="math-container">$y=x+10$</span> that is exactly the same information as <span class="math-container">$x=y-10$</span>. If you have complex function it could take long time to solve it explicitly for some variable. Time is a scarce resource.</p>
<p>If you work with general functions it might even be impossible to explicitly solve for one of the variables although that does not seem to be an issue in your case.</p>
<blockquote>
<p>Finally, why the following derivative has a negative sign and has been written as a ratio?</p>
</blockquote>
<p>Because that’s the rule for implicit derivatives along level curve. Generally speaking slope of a level curve can be found realizing that:</p>
<p><span class="math-container">$$F(x,y)=c \implies y’ = - \frac{F’_1(x,y)}{F’_2(x,y)}, F’_2 \neq 0$$</span></p>

CC BY-SA 4.0
2022-09-01T10:37:22.017Z
2022-09-01T10:37:22.017Z
1muflon1
<p>I'm studying an economic model, and I'd like to understand why the authors of the book wrote as follows:</p>
<p>Consider the following equation:</p>
<p><span class="math-container">$L_u = \phi(q_{L,t}-Q_t)l_{u,t}$</span> <span class="math-container">$~~~~~~[1]$</span></p>
<p>then, since <span class="math-container">$q_{L,t} = \frac{\eta}{w_{u,t}}$</span> and <span class="math-container">$l_{u,t}= {\frac{w_{u,t}}{\alpha \epsilon^2 L_s^{1- \epsilon}}}^{\frac{1}{\alpha \epsilon -1}}$</span></p>
<p>You simply plug these values in <span class="math-container">$[1]$</span></p>
<p>Then, why the book says: <strong>we obtain the wage rate of low-skilled workers, <span class="math-container">$w_{u,t}$</span>, as implicitly given by</strong>:</p>
<p><a href="https://i.sstatic.net/wrYWF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wrYWF.png" alt="enter image description here" /></a></p>
<p>Why write an implicit function instead of simply solving for <span class="math-container">$w_{u,t}$</span>?</p>
<p>Finally, why the following derivative has a negative sign and has been written as a ratio?:</p>
<p><a href="https://i.sstatic.net/EhrIM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EhrIM.png" alt="enter image description here" /></a></p>
<p>I'd expected <span class="math-container">$\displaystyle \frac{\partial G}{\partial Q_t} \displaystyle \frac{\partial w}{\partial Q_t}$</span> instead of <span class="math-container">$-\displaystyle \frac{\partial G}{\partial Q_t}/ \displaystyle \frac{\partial G}{\partial w_{u,t}}$</span></p>

<macroeconomics>
Implict function in economic models
John M.
# Implict function in economic models

(Asked by: username_0 on 2022-09-01)

I'm studying an economic model, and I'd like to understand why the authors of the book wrote as follows:

Consider the following equation:

$L\_u = \phi(q\_{L,t}-Q\_t)l\_{u,t}$ $~~~~~~[1]$

then, since $q\_{L,t} = \frac{\eta}{w\_{u,t}}$ and $l\_{u,t}= {\frac{w\_{u,t}}{\alpha \epsilon^2 L\_s^{1- \epsilon}}}^{\frac{1}{\alpha \epsilon -1}}$

You simply plug these values in $[1]$

Then, why the book says: **we obtain the wage rate of low-skilled workers, $w\_{u,t}$, as implicitly given by**:

[![enter image description here](https://i.sstatic.net/wrYWF.png)](https://i.sstatic.net/wrYWF.png)

Why write an implicit function instead of simply solving for $w\_{u,t}$?

Finally, why the following derivative has a negative sign and has been written as a ratio?:

[![enter image description here](https://i.sstatic.net/EhrIM.png)](https://i.sstatic.net/EhrIM.png)

I'd expected $\displaystyle \frac{\partial G}{\partial Q\_t} \displaystyle \frac{\partial w}{\partial Q\_t}$ instead of $-\displaystyle \frac{\partial G}{\partial Q\_t}/ \displaystyle \frac{\partial G}{\partial w\_{u,t}}$

**Tags:** macroeconomics

---

## 1 answers

### Answer by username_1 on 2022-09-01. Score: 2 | Accepted answer

> Why write an implicit function instead of simply solving for $w\_{u,t}$?

Because solving for it explicitly is time consuming and would not yield any new information.

If someone writes $y=x+10$ that is exactly the same information as $x=y-10$. If you have complex function it could take long time to solve it explicitly for some variable. Time is a scarce resource.

If you work with general functions it might even be impossible to explicitly solve for one of the variables although that does not seem to be an issue in your case.

> Finally, why the following derivative has a negative sign and has been written as a ratio?

Because that’s the rule for implicit derivatives along level curve. Generally speaking slope of a level curve can be found realizing that:

$$F(x,y)=c \implies y’ = - \frac{F’\_1(x,y)}{F’\_2(x,y)}, F’\_2 \neq 0$$


---

