feat(post): solow model

posts/economics/models-of-production.html

@@ -212,7 +212,48 @@
             </p>
           </div>
           <div class="fold"><h3>analysis</h3></div>
-          <p>discuss limitations</p>
+          <p>
+            Using both mathematical intuition and manipulating the visualization
+            above, we find that:
+          </p>
+          <ul style="list-style: unset">
+            <li>
+              \(\bar{A}\) has a positive relationship with steady-state output
+            </li>
+            <li>
+              Capital is influenced by workforce size, TFP, and savings rate
+            </li>
+            <li>
+              Capital output share's \(\alpha\) impact on output is twofold:
+              <ol>
+                <li>Directly through capital quantity</li>
+                <li>Indirectly through TFP</li>
+              </ol>
+            </li>
+            <li>
+              Large deviations in capital from steady-state \(K^*\) induce net
+              investments of larger magnitude, leading to an accelerated
+              reversion to the steady-state
+            </li>
+          </ul>
+          <p>
+            Lastly (and perhaps most importantly), exogenous parameters
+            \(\bar{s}, \bar{d}\), and \(\bar{A}\) all have immense ramifications
+            on economic status. For example, comparing the difference in country
+            \(C_1\)&apos;s output versus \(C_2\)&apos;s using the Solow Model,
+            we find that a difference in economic performance can only be
+            explained by these factors: \[
+            \frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha}
+            \]
+          </p>
+          <p>
+            We see that TFP is more important in explaining the differences in
+            per capital output
+            (\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)).
+            However, the Solow Model does not give any insight in to how to
+            alter what it considers to be the most important predictor of
+            output.
+          </p>
           <h2>romer</h2>
           <h2>romer-solow</h2>
         </article>

scripts/posts/models-of-production.js

@@ -163,6 +163,24 @@ function drawSolowGraph() {
     .attr("stroke", "black")
     .attr("stroke-width", 1)
     .attr("stroke-dasharray", "5,5");
+
+  const y_star = solowOutput(k_star);
+  svg
+    .append("foreignObject")
+    .attr("width", "20em")
+    .attr("height", "2em")
+    .attr("x", x(k_star) - 150)
+    .attr("y", y(y_star) - 50)
+    .append("xhtml:body")
+    .style("font-size", "0.75em")
+    .html(`<div class="solow-visualization-eq"></div>`);
+  katex.render(
+    `(K^*,Y^*)=(${k_star.toFixed(0)},${y_star.toFixed(0)})`,
+    document.querySelector(".solow-visualization-eq"),
+    {
+      throwOnError: false,
+    },
+  );
 }
 
 document.addEventListener("DOMContentLoaded", function () {

styles/post.css

@@ -13,6 +13,14 @@ header {
   justify-content: space-between;
 }
 
+ul {
+  list-style: unset;
+}
+
+li {
+  margin: 5px 0;
+}
+
 .main {
   display: flex;
   justify-content: center;