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.


1 Answer 1


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
    Commented May 15, 2021 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
    Commented May 15, 2021 at 20:51
  • $\begingroup$ Now everything is clear to me. Thanks, Giskard! $\endgroup$
    – PGabriel96
    Commented May 15, 2021 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.