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).

  • $\begingroup$ 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. $\endgroup$ – PGabriel96 May 15 at 20:12
  • $\begingroup$ @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. $\endgroup$ – Giskard May 15 at 20:51
  • $\begingroup$ Now everything is clear to me. Thanks, Giskard! $\endgroup$ – PGabriel96 May 15 at 20:53

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