feat(romer-solow): finish
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2 changed files with 424 additions and 109 deletions
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@ -141,7 +141,7 @@
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<ul>
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<li>
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<div class="slider">
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<label for="sliderSA">\(A:\)</label>
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<label for="sliderSA">\(\bar{A}:\)</label>
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<span id="outputSA">1.00</span>
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<input
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type="range"
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@ -155,11 +155,11 @@
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</li>
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<li>
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<div class="slider">
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<label for="sliderSD">\(d:\)</label>
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<span id="outputSD">0.50</span>
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<label for="sliderSd">\(\bar{d}:\)</label>
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<span id="outputSd">0.50</span>
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<input
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type="range"
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id="sliderSD"
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id="sliderSd"
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min="0.01"
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max="0.99"
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step="0.01"
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@ -173,11 +173,11 @@
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<ul start="3">
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<li>
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<div class="slider">
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<label for="sliderSS">\(s:\)</label>
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<span id="outputSS">0.50</span>
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<label for="sliderSs">\(\bar{s}:\)</label>
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<span id="outputSs">0.50</span>
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<input
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type="range"
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id="sliderSS"
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id="sliderSs"
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min="0.01"
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max="0.99"
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step="0.01"
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@ -187,11 +187,11 @@
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</li>
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<li>
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<div class="slider">
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<label for="sliderSAlpha">\(\alpha:\)</label>
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<span id="outputSAlpha">0.33</span>
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<label for="sliderSalpha">\(\alpha:\)</label>
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<span id="outputSalpha">0.33</span>
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<input
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type="range"
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id="sliderSAlpha"
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id="sliderSalpha"
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min="0.01"
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max="0.99"
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step="0.01"
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@ -375,11 +375,11 @@
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<ul>
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<li>
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<div class="slider">
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<label for="sliderRZ">\(\bar{z}:\)</label>
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<span id="outputRZ">0.50</span>
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<label for="sliderRz">\(\bar{z}:\)</label>
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<span id="outputRz">0.50</span>
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<input
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type="range"
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id="sliderRZ"
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id="sliderRz"
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min="0.1"
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max="0.99"
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step="0.01"
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@ -422,14 +422,14 @@
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<li>
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<div class="slider">
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<label for="sliderRA0">\(\bar{A}_0:\)</label>
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<span id="outputRA0">5000</span>
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<span id="outputRA0">500</span>
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<input
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type="range"
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id="sliderRA0"
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min="1"
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max="10000"
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min="0"
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max="1000"
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step="100"
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value="5000"
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value="500"
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/>
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</div>
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</li>
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@ -473,10 +473,6 @@
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output can be solved the production function: \[Y_t=A_t
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L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]
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</p>
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<!-- <p> -->
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<!-- It follows that the intensive form can be written as: -->
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<!-- \[y_t=\frac{Y_t}{\bar{L}}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\]. -->
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<!-- </p> -->
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</div>
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<div class="fold"><h3>analysis</h3></div>
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<div>
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@ -688,11 +684,13 @@
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rate and a negative relationship with the depreciation rate.
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</p>
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<p>
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However, do economics grow <i>faster</i>/<i>slower</i> the
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further <i>below</i>/<i>above</i> they are from their Balanced
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Growth Path, as initially desired? While this can be
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mathematically proven (of course), sometimes a visualization
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helps.
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Using the visualization below, we see a growth pattern similar
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to that of the Romer Model. However, the Romer-Solow economy
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indeed grows at a faster rate than the Romer model—I had
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to cap \(\bar{L}\) at \(400\) and \(\alpha\) at \(0.4\) because
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output would be
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<i> too large </i> for JavaScript to contain in a number (the
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graph would disappear).
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</p>
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<div class="graph">
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<div id="romer-solow-visualization"></div>
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@ -702,25 +700,53 @@
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<ul>
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<li>
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<div class="slider">
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<label for="sliderRSA">\(A:\)</label>
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<span id="outputRSA">1.00</span>
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<label for="sliderRSz">\(\bar{z}:\)</label>
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<span id="outputRSz">0.50</span>
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<input
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type="range"
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id="sliderRSA"
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id="sliderRSz"
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min="0.1"
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max="2"
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max="0.99"
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step="0.01"
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value="1"
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value="0.50"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSD">\(d:\)</label>
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<span id="outputRSD">0.50</span>
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<label for="sliderRSA0">\(A_0:\)</label>
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<span id="outputRSA0">500</span>
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<input
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type="range"
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id="sliderRSD"
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id="sliderRSA0"
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min="0"
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max="1000"
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step="10"
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value="500"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSd">\(\bar{d}:\)</label>
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<span id="outputRSd">0.50</span>
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<input
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type="range"
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id="sliderRSd"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.50"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSs">\(\bar{s}:\)</label>
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<span id="outputRSs">0.50</span>
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<input
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type="range"
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id="sliderRSs"
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min="0.01"
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max="0.99"
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step="0.01"
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@ -734,11 +760,25 @@
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<ul start="3">
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<li>
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<div class="slider">
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<label for="sliderRSS">\(s:\)</label>
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<span id="outputRSS">0.50</span>
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<label for="sliderRSalpha">\(\alpha:\)</label>
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<span id="outputRSalpha">0.33</span>
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<input
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type="range"
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id="sliderRSS"
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id="sliderRSalpha"
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min="0.01"
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max="0.40"
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step="0.01"
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value="0.33"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSl">\(\bar{l}:\)</label>
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<span id="outputRSl">0.50</span>
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<input
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type="range"
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id="sliderRSl"
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min="0.01"
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max="0.99"
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step="0.01"
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@ -748,21 +788,136 @@
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSAlpha">\(\alpha:\)</label>
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<span id="outputRSAlpha">0.33</span>
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<label for="sliderRSL">\(\bar{L}:\)</label>
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<span id="outputRSL">200</span>
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<input
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type="range"
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id="sliderRSAlpha"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.33"
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id="sliderRSL"
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min="0"
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max="400"
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step="10"
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value="200"
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/>
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</div>
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</li>
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</ul>
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</div>
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</div>
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<p>
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Playing with the parameters, the previous mathematical findings
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are validated. For example, because
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\(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), only changes
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in parameters \(\alpha,\bar{z},\bar{l}\), and \(\bar{L}\) affect
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the growth rate of output, manifesting as the y-axis scaling
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up/down on a ratio scale.
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</p>
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<p>
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However, do economics grow <i>faster</i>/<i>slower</i> the
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further <i>below</i>/<i>above</i> they are from their Balanced
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Growth Path, as initially desired? While this can be
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mathematically proven (of course), sometimes a visualization
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helps.
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</p>
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<p>
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The graph below illustrates the transition dynamics of
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Romer-Solow Model. Namely, \((\bar{z}, \bar{l}, \bar{L},
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\alpha)=(0.5, 0.5, 100, 0.33)\forall t<t_0\), then update to
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the slider values when \(t>t_0\).
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</p>
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<div class="graph">
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<div id="romer-solow-change-visualization"></div>
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</div>
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<div class="sliders">
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<div style="padding-right: 20px">
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<ul>
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<li>
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<div class="slider">
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<label for="sliderRSCz0">\(\bar{z}_0:\)</label>
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<span id="outputRSCz0">0.50</span>
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<input
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type="range"
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id="sliderRSCz0"
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min="0.1"
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max="0.99"
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step="0.01"
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value="0.50"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSCalpha0">\(\alpha_0:\)</label>
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<span id="outputRSCalpha0">0.33</span>
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<input
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type="range"
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id="sliderRSCalpha0"
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min="0.01"
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max="0.54"
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step="0.01"
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value="0.33"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSCL0">\(\bar{L}_0:\)</label>
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<span id="outputRSCL0">100</span>
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<input
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type="range"
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id="sliderRSCL0"
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min="0"
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max="200"
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step="10"
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value="100"
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/>
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</div>
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</li>
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</ul>
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</div>
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<div style="padding-left: 20px">
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<ul start="3">
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<li>
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<div class="slider">
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<label for="sliderRSCl0">\(\bar{l}_0:\)</label>
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<span id="outputRSCl0">0.50</span>
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<input
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type="range"
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id="sliderRSCl0"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.50"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderRSCt0">\(t_0:\)</label>
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<span id="outputRSCt0">50</span>
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<input
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type="range"
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id="sliderRSCt0"
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min="0"
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max="100"
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step="1"
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value="50"
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/>
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</div>
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</li>
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</ul>
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</div>
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</div>
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<p>
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Finally, it is clear that economies converge to their Balanced
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Growth Path as desired—something slightly more convoluted
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to prove from the complex expression for \(Y^*\) derived
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earlier. For example, with an increase in \(\alpha_0\), output
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grows at an increasing rate after the change, then increases at
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a decreasing rate as it converges to the new higher Balanced
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Growth Path. Increasing parameters \(\bar{z},\bar{l},\bar{L}\)
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yield similar results, although the changes are visually less
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obvious.
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</p>
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</div>
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</div>
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</article>
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