feat: romer model
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type="range"
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id="sliderD"
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min="0.01"
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type="range"
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id="sliderS"
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min="0.01"
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max="1"
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max="0.99"
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/>
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type="range"
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id="sliderAlpha"
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min="0.01"
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max="1"
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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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/>
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@ -213,8 +213,6 @@
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</div>
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<div class="fold">
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<h3>analysis</h3>
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</div>
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<div>
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<p>
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Using both mathematical intuition and manipulating the
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visualization above, we find that:
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@ -262,10 +260,250 @@
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alter the most important predictor of output, TFP.
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</p>
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</div>
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<h2>romer</h2>
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<!-- TODO: transition talking about "Romer model does, though" -->
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<!-- TODO: say the romer model provides avenue for LR growth -->
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<!-- Solow TODO -->
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<!-- TODO: dynamics?????? -->
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<!-- TODO: K_0 -->
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<h2>romer</h2>
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<div class="fold"><h3>introduction</h3></div>
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<div>
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<p>How, then, can we address these shortcomings?</p>
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<p>
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The Romer Model provides an answer by both modeling ideas \(A_t\)
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(analagous to TFP in the Solow model) endogenously and utilizing
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them to provide a justification for sustained long-run growth.
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</p>
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<p>The Model divides the world into two parts:</p>
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<ul style="list-style: unset">
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<li>
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<u>Objects</u>: finite resources, like capital and labor in the
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Solow Model
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</li>
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<li>
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<u>Ideas</u>: infinite,
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<a
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target="blank"
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href="https://en.wikipedia.org/wiki/Rivalry_(economics)"
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>non-rivalrous</a
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>
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items leveraged in production (note that ideas may be
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<a href="blank" href="https://www.wikiwand.com/en/Excludability"
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>excludable</a
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>, though)
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</li>
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</ul>
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<p>
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The Romer Models' production function can be modelled as:
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\[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:
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</p>
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<ul>
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<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
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<li>
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\(L_{yt}\): the population working on production-facing
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(output-driving) tasks
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</li>
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</ul>
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<p>
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Assuming \(L_t=\bar{L}\) people work in the economy, a proportion
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\(\bar{l}\) of the population focuses on making ideas:
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\(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\).
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</p>
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<!-- TODO: footnotes - dynamic JS? -->
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<p>
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Further, this economy garners ideas with time at rate
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<u>\(\bar{z}\)</u>: the "speed of ideas". Now, we can
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describe the quantity of ideas tomorrow as function of those of
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today: the <u>Law of Ideal Motion</u> (I made that up).
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\[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta
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A_{t+1}=\bar{z}A_tL_{at}\]
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</p>
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<p>
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Analagously to capital in the solow model, ideas begin in the
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economy with some \(\bar{A}_0\gt0\) and grow at an
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<i>exponential</i> rate. At its core, this is because ideas are
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non-rivalrous; more ideas bring about more ideas.
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</p>
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<p>Finally, we have a model:</p>
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<div style="display: flex; justify-content: center">
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<div style="padding-right: 50px">
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<ol>
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<li>\(Y_t=A_tL_{yt}\)</li>
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<li>\(\Delta A_{t+1} = \bar{z}A_tL_{at}\)</li>
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</ol>
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</div>
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<div style="padding-left: 50px">
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<ol start="3">
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<li>\(L_{yt}+L_{at}=\bar{L}\)</li>
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<li>\(L_{at}=\bar{l}\bar{L}\)</li>
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</ol>
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</div>
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</div>
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<p>
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A visualization of the Romer Model shows that the economy grows
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exponentially—production knows no bounds (<a
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target="blank"
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href="https://en.wikipedia.org/wiki/Ceteris_paribus"
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><i>ceteris parbibus</i></a
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>, of course). A graph of \(log_{10}(Y_t)\) can be seen below:
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</p>
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<div class="graph">
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<div id="romer-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="sliderZ">\(\bar{z}:\)</label>
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<span id="outputZ">0.50</span>
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<input
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type="range"
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id="sliderZ"
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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="sliderL">\(\bar{L}:\)</label>
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<span id="outputL">505</span>
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<input
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type="range"
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id="sliderL"
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min="10"
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max="1000"
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step="19"
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value="505"
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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="sliderl">\(\bar{l}:\)</label>
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<span id="outputl">0.50</span>
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<input
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type="range"
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id="sliderl"
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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="sliderA0">\(\bar{A}_0:\)</label>
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<span id="outputA0">5000</span>
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<input
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type="range"
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id="sliderA0"
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min="1"
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max="10000"
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step="100"
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value="5000"
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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 sliders, this graph may seem underwhelming in
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comparison to the Solow Model. However, on a logarithmic scale,
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small changes in the parameters lead to massive changes in the
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growth rate of ideas and economices:
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</p>
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<div class="romer-table-container">
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<table id="romer-table">
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<thead>
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<tr id="romer-table-header"></tr>
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</thead>
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<tbody>
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<tr id="row-A_t"></tr>
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<tr id="row-Y_t"></tr>
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</tbody>
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</table>
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</div>
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</div>
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<div class="fold"><h3>solving the model</h3></div>
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<div>
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<p>
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To find the output in terms of exogenous parameters, first note
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that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\]
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</p>
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<p>
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Now, all that remains is to find ideas \(A_t\). It is assumed that
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ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\]
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</p>
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<p>
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Using the growth rate formula, we find: \[g_A=\frac{\Delta
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A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}\]
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</p>
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<p>
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Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally,
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output can be solved the production function: \[Y_t=A_t
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L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\]
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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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<p>
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We see the Romer model exhibits long-run growth because ideas have
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non-diminishing returns due to their nonrival nature. In this
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model, capital and income eventually slow but ideas continue to
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yield increasing, unrestricted returns.
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</p>
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<p>
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Further, all economy continually and perpetually grow along a
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"Balanced Growth Path" as previously defined by \(Y_t\) as a
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function of the endogenous variables. This directly contrasts the
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Solow model, in which an economy converges to a steady-state with
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transition dynamics.
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</p>
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<p>
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Changes in the growth rate of ideas, then, alter the growth rate
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of output itself—in this case, parameters \(\bar{l},
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\bar{z}\), and \(\bar{L}\). This is best exemplified by comparing
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the growth rate before and and after a parameter changes. In the
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below example, a larger \(\bar{l}\) initially drops output due to
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less workers being allocated to production. Soon after, though,
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output recovers along a "higher" Balanced Growth Path.
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</p>
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<div class="graph">
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<div id="romer-lchange-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="sliderlChange">\(\bar{l}:\)</label>
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<span id="outputlChange">0.50</span>
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<input
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type="range"
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id="sliderlChange"
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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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</ul>
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</div>
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</div>
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</div>
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<h2>romer-solow</h2>
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</article>
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</div>
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