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feat(posts): slider for romer time dynamics

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Barrett Ruth 2024-07-03 15:09:58 -05:00
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172
posts/economics/models-of-production.html
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@ -525,6 +525,24 @@
</li>
</ul>
</div>
<div style="padding-left: 20px">
<ul start="3">
<li>
<div class="slider">
<label for="slidert0">\(t_0:\)</label>
<span id="outputt0">50</span>
<input
type="range"
id="slidert0"
min="1"
max="99"
step="1"
value="50"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
Notably, while both the Romer and Solow Models help to analyze
@ -588,68 +606,98 @@
</div>
</div>
<div class="fold"><h3>solving the model</h3></div>
<p>
Based on the the motivations for creating this model, it is more
useful to first analyze the growth rates of equilibrium long run
output \(Y_t^*\).
</p>
<p>
According to the production function, \[g_Y=g_A+\alpha
g_K+(1-\alpha)g_{L_{y}}\]
</p>
<p>
From previous analysis it was found that
\(g_A=\bar{z}\bar{l}\bar{L}\).
</p>
<p>
Based on the <u>Law of Capital Motion</u>, \[g_K=\frac{\Delta
K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\]
</p>
<p>
Because growth rates are constant on the Balanced Growth Path,
\(g_K\) must be constant as well. Thus, so is
\(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that
\(g_K^*=g_Y^*\).
</p>
<p>
The model assumes population is constant, so
\(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well.
</p>
<p>
Combining these terms, we find:
\[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\]
\[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\]
</p>
<p>
Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\)
must hold on a balanced growth path.
</p>
<p>
Invoking the <u>Law of Capital Motion</u> with magic chants,
\[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow
K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\]
</p>
<p>
Isolating \(Y_t^*\), \[Y_t^*=A_t^*
(\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha
({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow
{Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\]
</p>
<p>
Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a
final expression for the Balanced Growth Path output as a function
of the endogenous parameters and time is obtained: \[
Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\]
</p>
</div>
<div class="fold"><h3>analysis</h3></div>
<div>
<p>
Intuitively, this means that idea-driving factors, as well as an
increased allocation of labor to output, will increase the
Balanced Growth Path (the <i>level</i> of long-run growth),
combining both the Romer and Solow model.
</p>
<div>
<p>
Based on the the motivations for creating this model, it is more
useful to first analyze the growth rates of equilibrium long run
output \(Y_t^*\).
</p>
<p>
According to the production function, \[g_Y=g_A+\alpha
g_K+(1-\alpha)g_{L_y}\]
</p>
<p>
From previous analysis it was found that
\(g_A=\bar{z}\bar{l}\bar{L}\).
</p>
<p>
Based on the <u>Law of Capital Motion</u>, \[g_K=\frac{\Delta
K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\]
</p>
<p>
Because growth rates are constant on the Balanced Growth Path,
\(g_K\) must be constant as well. Thus, so is
\(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that
\(g_K^*=g_Y^*\).
</p>
<p>
The model assumes population is constant, so
\(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well.
</p>
<p>
Combining these terms, we find:
\[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\]
\[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\]
</p>
<p>
Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\)
must hold on a balanced growth path.
</p>
<p>
Invoking the <u>Law of Capital Motion</u> with magic chants,
\[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow
K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\]
</p>
<p>
Isolating \(Y_t^*\), \[Y_t^*=A_t^*
(\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha
({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow
{Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\]
</p>
<p>
Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a
final expression for the Balanced Growth Path output as a
function of the endogenous parameters and time is obtained: \[
Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\]
</p>
</div>
<div class="fold"><h3>analysis</h3></div>
<div>
<p>
First looking at the growth rate of output,
\(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), idea-driving
factors and an increased allocation of labor to output increase
the equilibrium Balanced Growth Path&mdash;the
<i>level</i> of long-run growth. Thus, this model captures the
influences of both capital and ideas on economic growth.
<!-- TODO: t_0 graph break in romer-model and post -->
</p>
<p>
Looking at \(Y_t^*\), ideas have both a direct and indirect
effect on output. Firstly, ideas raise output because they
increase productivity (directly); second, with the introduction
of capital stock, ideas also increase capital, in turn
increasing output further (indirectly). Mathematically, this is
evident in both instances of \(g_A^*\) in the formula for output
\(Y_t^*\)&mdash;note that
\(\frac{1}{1-\alpha},\frac{\alpha}{1-\alpha}>0\) for any
\(\alpha\in(0,1)\), so \(\frac{d}{dg_A^*}Y_t^*>0\).
</p>
<p>
Expectedly, output has a positive relationship with the savings
rate and a negative relationship with the depreciation rate.
</p>
<p>
However, do economics grow <i>faster</i>/<i>slower</i> the
further <i>below</i>/<i>above</i> they are from their Balanced
Growth Path, as initially desired? While this can be
mathematically proven (of course), sometimes a visualization
helps.
</p>
<div class="graph">
<div id="romer-solow-visualization"></div>
</div>
</div>
</div>
</article>
</div>