<p>You have most of this; it seems like you might have a miscalculation when differentiating the fraction:
<span class="math-container">$$
\begin{align*}
\frac{dk_t}{dt} = \frac{d\frac{K_t}{N_t}}{dt} &amp; = \frac{\frac{dK_t}{dt}N_t - \frac{dN_t}{dt}K_t}{N_t^2} \\
&amp; = \frac{\frac{dK_t}{dt}}{N_t} - nk_t
\end{align*}
$$</span>
thus
<span class="math-container">$$
\frac{\frac{dK_t}{dt}}{N_t} = \frac{dk_t}{dt} + nk_t,
$$</span>
and you can plug this into your equation (4).</p>

CC BY-SA 4.0
2021-05-15T20:12:28.140Z
Just to clarify it a little bit more: is it mathematically wrong to do $\frac{\frac{d K_t}{dt}}{N_t} = \frac{d K_t}{dt} * \frac{1}{N_t} = \frac{d K_t}{dt N_t} = \frac{d k_t}{dt}$? Because that was the way I was instinctively seeing the $\frac{\frac{d K_t}{dt}}{N_t}$ term and I think it was the main source of my wrong approach.
PGabriel96
CC BY-SA 4.0
2021-05-15T20:51:01.350Z
@PGabriel96 The final equation is indeed wrong, it implies $ \frac{1}{N_t} = \frac{d\frac{t}{N_t}}{dt} $, which does not hold here. You can see this in more detail in the first equation of my answer.
Giskard
CC BY-SA 4.0
2021-05-15T20:53:05.000Z
Now everything is clear to me. Thanks, Giskard!
PGabriel96
CC BY-SA 4.0
2021-05-15T19:20:27.530Z
2021-05-15T19:20:27.530Z
Giskard
<p>I started studying Olivier Blanchard and Stanley Fischer's Lectures on Macroeconomics, but I couldn't follow the reasoning behind a differentiation step (at least I think it is something involving differentiation) in the chapter 2 (Consumption and Investment: Basic Infinite Horizon Models), page 38.</p>
<p>There, it is exposed by a simple production function the idea that output is either consumed or invested:</p>
<p>(1) <span class="math-container">$Y_t = F(K_t, N_t) = C_t + \frac{K_t}{dt}$</span></p>
<p>Then, as one of this production function's properties is homogeneity of degree one, everything is divided by <span class="math-container">$N_t$</span> for us to get the per capita form of this production function:</p>
<p>(2) <span class="math-container">$f(k_t) = c_t + \frac{dk_t}{dt} + nk_t$</span></p>
<p>And that is my question: I suppose that the <span class="math-container">$nk_t$</span> term is originated by the <span class="math-container">$N_t$</span> division of equation 2, since <span class="math-container">$\frac{d\frac{K_t}{N_t}}{dt}$</span> can be determinated as a multivariate chain rule, since both <span class="math-container">$K_t$</span> and <span class="math-container">$N_t$</span> are functions of <span class="math-container">$t$</span>. But when I proceed like that, I always end up with:</p>
<p>(3) <span class="math-container">$f(k_t) = c_t + \frac{dk_t}{dt} - nk_t$</span> instead of a positive <span class="math-container">$nk_t$</span>.</p>
<p>After some hours of trying to figure it out, I realized that my approach might not even be appropriated, since when we divide equation 1 by <span class="math-container">$N_t$</span>, it makes more sense that equation 1 gets a form:</p>
<p>(4) <span class="math-container">$\frac{Y_t}{N_t} = \frac{C_t}{N_t} + \frac{\frac{dK_t}{d_t}}{N_t}$</span></p>
<p>Instead of what I originally thought:</p>
<p>(5) <span class="math-container">$\frac{Y_t}{N_t} = \frac{C_t}{N_t} + \frac{d\frac{K_t}{N_t}}{dt}$</span></p>
<p>And then I am kinda stuck. It is a basic question, but if someone could shed some light on it, I would be really, really grateful.</p>

<macroeconomics><mathematical-economics>
A differentiation step in a economic growth model
PGabriel96
# A differentiation step in a economic growth model

(Asked by: username_0 on 2021-05-15)

I started studying Olivier Blanchard and Stanley Fischer's Lectures on Macroeconomics, but I couldn't follow the reasoning behind a differentiation step (at least I think it is something involving differentiation) in the chapter 2 (Consumption and Investment: Basic Infinite Horizon Models), page 38.

There, it is exposed by a simple production function the idea that output is either consumed or invested:

(1) $Y\_t = F(K\_t, N\_t) = C\_t + \frac{K\_t}{dt}$

Then, as one of this production function's properties is homogeneity of degree one, everything is divided by $N\_t$ for us to get the per capita form of this production function:

(2) $f(k\_t) = c\_t + \frac{dk\_t}{dt} + nk\_t$

And that is my question: I suppose that the $nk\_t$ term is originated by the $N\_t$ division of equation 2, since $\frac{d\frac{K\_t}{N\_t}}{dt}$ can be determinated as a multivariate chain rule, since both $K\_t$ and $N\_t$ are functions of $t$. But when I proceed like that, I always end up with:

(3) $f(k\_t) = c\_t + \frac{dk\_t}{dt} - nk\_t$ instead of a positive $nk\_t$.

After some hours of trying to figure it out, I realized that my approach might not even be appropriated, since when we divide equation 1 by $N\_t$, it makes more sense that equation 1 gets a form:

(4) $\frac{Y\_t}{N\_t} = \frac{C\_t}{N\_t} + \frac{\frac{dK\_t}{d\_t}}{N\_t}$

Instead of what I originally thought:

(5) $\frac{Y\_t}{N\_t} = \frac{C\_t}{N\_t} + \frac{d\frac{K\_t}{N\_t}}{dt}$

And then I am kinda stuck. It is a basic question, but if someone could shed some light on it, I would be really, really grateful.

**Tags:** macroeconomics, mathematical-economics

---

## 1 answers

### Answer by username_1 on 2021-05-15. Score: 3 | Accepted answer

You have most of this; it seems like you might have a miscalculation when differentiating the fraction:
$$
\begin{align\*}
\frac{dk\_t}{dt} = \frac{d\frac{K\_t}{N\_t}}{dt} & = \frac{\frac{dK\_t}{dt}N\_t - \frac{dN\_t}{dt}K\_t}{N\_t^2} \\
& = \frac{\frac{dK\_t}{dt}}{N\_t} - nk\_t
\end{align\*}
$$
thus
$$
\frac{\frac{dK\_t}{dt}}{N\_t} = \frac{dk\_t}{dt} + nk\_t,
$$
and you can plug this into your equation (4).

> **Comments:**
> **username_0**: Just to clarify it a little bit more: is it mathematically wrong to do $\frac{\frac{d K_t}{dt}}{N_t} = \frac{d K_t}{dt} * \frac{1}{N_t} = \frac{d K_t}{dt N_t} = \frac{d k_t}{dt}$? Because that was the way I was instinctively seeing the $\frac{\frac{d K_t}{dt}}{N_t}$ term and I think it was the main source of my wrong approach.  
> **username_1**: @PGabriel96 The final equation is indeed wrong, it implies $ \frac{1}{N_t} = \frac{d\frac{t}{N_t}}{dt} $, which does not hold here. You can see this in more detail in the first equation of my answer.  
> **username_0**: Now everything is clear to me. Thanks, Giskard!  

---

