# A differentiation step in a economic growth model

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.

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).
• 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 May 15 at 20:12
• @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 May 15 at 20:51