diff --git a/posts/economics/models-of-production.html b/posts/economics/models-of-production.html index 1648df6..00c6623 100644 --- a/posts/economics/models-of-production.html +++ b/posts/economics/models-of-production.html @@ -68,7 +68,7 @@ >, the Solow Model describes production as follows: \[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
-+ Using the visualization below, we see a growth pattern similar + to that of the Romer Model. However, the Romer-Solow economy + indeed grows at a faster rate than the Romer model—I had + to cap \(\bar{L}\) at \(400\) and \(\alpha\) at \(0.4\) because + output would be + too large for JavaScript to contain in a number (the + graph would disappear). +
++ Playing with the parameters, the previous mathematical findings + are validated. For example, because + \(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), only changes + in parameters \(\alpha,\bar{z},\bar{l}\), and \(\bar{L}\) affect + the growth rate of output, manifesting as the y-axis scaling + up/down on a ratio scale. +
However, do economics grow faster/slower the further below/above they are from their Balanced @@ -694,9 +818,106 @@ mathematically proven (of course), sometimes a visualization helps.
++ The graph below illustrates the transition dynamics of + Romer-Solow Model. Namely, \((\bar{z}, \bar{l}, \bar{L}, + \alpha)=(0.5, 0.5, 100, 0.33)\forall t<t_0\), then update to + the slider values when \(t>t_0\). +
+ Finally, it is clear that economies converge to their Balanced + Growth Path as desired—something slightly more convoluted + to prove from the complex expression for \(Y^*\) derived + earlier. For example, with an increase in \(\alpha_0\), output + grows at an increasing rate after the change, then increases at + a decreasing rate as it converges to the new higher Balanced + Growth Path. Increasing parameters \(\bar{z},\bar{l},\bar{L}\) + yield similar results, although the changes are visually less + obvious. +