first post here. I'm an economist and I'm a bit rusty with my math. Would someone help out? I've done my homework and searched but couldn't find answers.

Question: what is this equation for? $$\frac{\Delta y_{t}}{y_{t}}=s_{t}\frac{\Delta k_{t}}{k_{t}}+\Delta s_{t}\left(\log k_{t}+\frac{\Delta B(s_{t})}{B(s_{t})}\right)$$

  • sorry for my notations. $\Delta x/x$ is a growth rate over time, $s_{t}\in[0,1]$ is some parameter that varies over time and $B(s_{t})$ is some polynomial in $s_{t}$.
  • clearly the first part relates the growth rate of $y_{t}$ to the growth rate of $k_{t}$ in a linear way with (time-varying) slope $s_{t}$.
  • But I don't want to dismiss the rest of the equation as noise.
  • I suspect some kind of circular movement or oscillations. But I may be wrong.
  • What would a plot look like? Any help is appreciated!!!
  • $\begingroup$ Where did you encounter this equation? Was there any information at all about what the variables represent? $\endgroup$ Jan 5 '18 at 23:17
  • $\begingroup$ It's something I've found. the y's are income per capita, the k's are capital per capita and s is the capital share of income $\endgroup$
    – frencho
    Jan 6 '18 at 3:04
  • $\begingroup$ I just don't even know if this is a differential equation or what. $\endgroup$
    – frencho
    Jan 6 '18 at 3:05

This is not a differential equation. It is close to being convertable to a differential equation, but if you do, the contribution of the final term is lost.

$\Delta x$ means the change in $x$, presumably over some time interval in this case. So $\frac {\Delta x}{x}$ is the relative change in $x$. That is, the change in $x$ as a fraction of its size. Usually, this is measured in percentages, but for various reasons, mathematicians prefer to leave it a ratio out of $1$. It only becomes the rate of change if you divide it by the length of the time interval it is measured over. More on that later.

Based on the construction here, I would say that $y$ is not straight "income per capita", but is "income from capital per capita", because there does not appear to be any contribution that is not related to capital. Understanding that everything is per capita, the equation is saying that relative change in capital income is the product of the capital share of income times the relative change in capital plus the change in capital share of income times (the logarithm of capital plus the relative change in the $B$ polynomial). Whether this equation is derived from some principles or is a choice for modeling the behavior of these variables, I can't tell (I am not an economist).

We can try to convert it to a differential equation by dividing by the change in time $\Delta t$ (I'm dropping the $t$ subscript for simplicity - just remember that everything here depends on $t$: $$\frac{\Delta y}{y\Delta t}=s\frac{\Delta k}{k\Delta t}+\frac{\Delta s}{\Delta t}\left(\log k +\frac{\Delta B(s)}{B(s)}\right)$$

If we let $\Delta t$ decrease, each of the $\Delta$ expressions will also go to $0$, but where there is a fraction of two of them, we get a derivative. But the final $\frac{\Delta B(s)}{B(s)}$ does not have a $\Delta$ expression in the denominator. The numerator goes to $0$ while the denominator stays at a fixed size. As a result, the term is lost and we get the following differential equation: $$\frac 1y \frac {dy}{dt} = \frac sk \frac {dk}{dt} + \frac{ds}{dt}\log k$$ This is easily solved. It just means that $$y = Pk^s$$ where $P$ is some constant value.

That is, capital income has sublinear growth (since $s < 1$) with respect to the amount of capital invested, but exponential growth with respect to the ratio capital income to total income.

This is the basic behavior behind your equation, but remember that the contribution of $B(s)$ was lost. The author of this equation is saying that the differential version is not quite accurate. There is a correction factor of $\frac {\Delta B(s)}{B(s)}$ that must also be considered. If $B(s)$ is increasing with $s$, then $y$ will be somewhat higher than the formula $y = Pk^s$ predicts. if $B(s)$ is decreasing as $s$ increases, then $y$ will be somewhat lower than predicted by $y = Ak^s$.

Beyond that I cannot say without knowing more about the context.

  • $\begingroup$ Thank you Paul! Lots of insights here. Allow me to be a bit more specific. I already seen the result that $y_{t}=Pk_{t}^{s_{t}}$ and you are right, the contribution of the $B(s_{t})$ parameter should be taken into account and when you do, you actually get $y_{t}=B(s_{t})k_{t}^{s_{t}}$. But I realized my question was ill-posed. I am not so much interested in the equation per se, as I am interested in the right-hand side only, specifically: $s_{t}\frac{\Delta k}{k}+\Delta s_{t}.\log k_{t}$ $\endgroup$
    – frencho
    Jan 6 '18 at 13:17
  • $\begingroup$ I just don't know what to do with that expression, beyond saying that the variable I'm trying to explain, $y_{t}$, is linear in $\frac{\Delta k}{k}$ with (time-varying) slope $s_{t}$, "plus the other term". But I feel that something else is going on here. I don't recognize this expression, I don't know how to factor it out, or even how to plot it to gain further insights. I'm stuck :/ $\endgroup$
    – frencho
    Jan 6 '18 at 13:31
  • $\begingroup$ $y_t = B(s_t)k_t^{s_t}$ would give you $$\frac{\Delta y_{t}}{y_{t}}=s_{t}\frac{\Delta k_{t}}{k_{t}}+\Delta s_{t}\log k_{t}+\frac{\Delta B(s_{t})}{B(s_{t})}$$I.e., the $\Delta s_t$ doesn't multiply $\Delta B / B$. Your equation does something different. As for $s_t\frac{\Delta k}{k}+\Delta s_t.\log k_t$, it is equal to $$\Delta(s_t\cdot \log k_t)$$. $\endgroup$ Jan 6 '18 at 15:12
  • $\begingroup$ Thank you again Paul. You had provided some much needed insights on a previous post of mine, which led me to some much-needed rethinking of my economic and math! I have posted a follow-up question here, if you are interested. $\endgroup$
    – frencho
    Apr 5 '18 at 14:23
  • $\begingroup$ Ross Millikan appears to have anything I could say on that question covered. $\endgroup$ Apr 5 '18 at 18:52

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