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feat(models): romer-solow solution

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Barrett Ruth 2024-07-03 14:37:47 -05:00
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posts/economics/models-of-production.html
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@ -471,7 +471,7 @@
<p>
Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally,
output can be solved the production function: \[Y_t=A_t
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\]
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]
</p>
<!-- <p> -->
<!-- It follows that the intensive form can be written as: -->
@ -488,10 +488,10 @@
</p>
<p>
Further, all economy continually and perpetually grow along a
"Balanced Growth Path" as previously defined by \(Y_t\) as a
function of the endogenous variables. This directly contrasts
the Solow model, in which an economy converges to a steady-state
with transition dynamics.
constant "Balanced Growth Path" as previously defined by \(Y_t\)
as a function of the endogenous variables. This directly
contrasts the Solow model, in which an economy converges to a
steady-state via transition dynamics.
</p>
<p>
Changes in the growth rate of ideas, then, alter the growth rate
@ -569,12 +569,12 @@
Intuitively, incorporating capital into output via the Solow
Model&apos;s production function, as well as including the
<u>Law of Capital Motion</u> seems like one way to legitimately
create the so-called &apos;Romer-Solow&apos; model:
create this so-called &quot;Romer-Solow&quot; model:
</p>
<div style="display: flex; justify-content: center">
<div style="padding-right: 50px">
<ol>
<li>\(Y_t=K_t^\alpha L_{yt}^{1-\alpha}\)</li>
<li>\(Y_t=A_t K_t^\alpha L_{yt}^{1-\alpha}\)</li>
<li>\(\Delta K_{t+1}=\bar{s}Y_t-\bar{d}K_t\)</li>
<li>\(\Delta A_{t+1} = \bar{z}A_tL_{at}\)</li>
</ol>
@ -588,7 +588,68 @@
</div>
</div>
<div class="fold"><h3>solving the model</h3></div>
<div>content</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>
</article>
</div>