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@ -53,477 +53,542 @@
in Intermediate Macroeconomics (ECON 3020) during the Spring in Intermediate Macroeconomics (ECON 3020) during the Spring
semester of 2024 at the University of Virginia. semester of 2024 at the University of Virginia.
</p> </p>
<h2>solow</h2> <div class="fold"><h2>solow</h2></div>
<div class="fold"><h3>introduction</h3></div>
<div> <div>
<p> <div class="fold"><h3>introduction</h3></div>
The Solow Model is an economic model of production that <div>
incorporates the incorporates the idea of capital accumulation. <p>
Based on the The Solow Model is an economic model of production that
<a incorporates the incorporates the idea of capital accumulation.
target="blank" Based on the
href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function"
>Cobb-Douglas production function</a
>, the Solow Model describes production as follows:
\[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
</p>
<ul>
<li>\(\bar{A}\): total factor productivity (TFP)</li>
<li>
\(\alpha\): capital&apos;s share of output&mdash;usually \(1/3\)
based on
<a target="blank" href="https://arxiv.org/pdf/1105.2123"
>empirical data</a
>
</li>
</ul>
<p>
In this simple model, the following statements describe the
economy:
</p>
<ol>
<li>
Output is either saved or consumed; in other words, savings
equals investment
</li>
<li>
Capital accumulates according to investment \(I_t\) and
depreciation \(\bar{d}\), beginning with \(K_0\) (often called
the
<u>Law of Capital Motion</u>)
</li>
<li>Labor \(L_t\) is time-independent</li>
<li>
A savings rate \(\bar{s}\) describes the invested portion of
total output
</li>
</ol>
<p>
Including the production function, these four ideas encapsulate
the Solow Model:
</p>
<div style="display: flex; justify-content: center">
<div style="padding-right: 50px">
<ol>
<li>\(C_t + I_t = Y_t\)</li>
<li>\(\Delta K_{t+1} = I_t - \bar{d} K_t\)</li>
</ol>
</div>
<div style="padding-left: 50px">
<ol start="3">
<li>\(L_t = \bar{L}\)</li>
<li>\(I_t = \bar{s} Y_t\)</li>
</ol>
</div>
</div>
</div>
<div class="fold">
<h3>solving the model</h3>
</div>
<div>
<p>
Visualizing the model, namely output as a function of capital,
provides helpful intuition before solving it.
</p>
<p>
Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that
\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\).
Utilizing this simplification and its graphical representation
below, output is clearly characterized by the cube root of
capital:
</p>
<div class="graph">
<div id="solow-visualization"></div>
</div>
<div class="sliders">
<div style="padding-right: 20px">
<ul>
<li>
<div class="slider">
<label for="sliderA">\(A:\)</label>
<span id="outputA">1.00</span>
<input
type="range"
id="sliderA"
min="0.1"
max="2"
step="0.01"
value="1"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderD">\(d:\)</label>
<span id="outputD">0.50</span>
<input
type="range"
id="sliderD"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
</ul>
</div>
<div style="padding-left: 20px">
<ul start="3">
<li>
<div class="slider">
<label for="sliderS">\(s:\)</label>
<span id="outputS">0.50</span>
<input
type="range"
id="sliderS"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderAlpha">\(\alpha:\)</label>
<span id="outputAlpha">0.33</span>
<input
type="range"
id="sliderAlpha"
min="0.01"
max="0.99"
step="0.01"
value="0.33"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
When investment is completely disincentivized by depreciation (in
other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates at a
so-called &quot;steady-state&quot; with equilibrium
\((K_t,Y_t)=(K_t^*,Y_t^*)\).
</p>
<p>
Using this equilibrium condition, it follows that:
\[Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \rightarrow
\bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}\]
\[\rightarrow
K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}\]
\[\rightarrow
Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}\]
</p>
<p>
Thus, the equilibrium intensive form (output per worker) of both
capital and output are summarized as follows:
\[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}})
=((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha},
\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\]
</p>
</div>
<div class="fold">
<h3>analysis</h3>
</div>
<div>
<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>
<li>
Economies stagnate at the steady-state \((K^*,Y^*)\)&mdash;this
model provides no avenues for long-run growth.
</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
(\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)).
<!-- TODO: poor phrasing -->
Notably, the Solow Model does not give any insights into how to
alter the most important predictor of output, TFP.
</p>
</div>
<!-- Solow TODO -->
<!-- TODO: dynamics?????? -->
<!-- TODO: K_0 -->
<h2>romer</h2>
<div class="fold"><h3>introduction</h3></div>
<div>
<p>How, then, can we address these shortcomings?</p>
<p>
The Romer Model provides an answer by both modeling ideas \(A_t\)
(analagous to TFP in the Solow model) endogenously and utilizing
them to provide a justification for sustained long-run growth.
</p>
<p>The Model divides the world into two parts:</p>
<ul style="list-style: unset">
<li>
<u>Objects</u>: finite resources, like capital and labor in the
Solow Model
</li>
<li>
<u>Ideas</u>: infinite,
<a <a
target="blank" target="blank"
href="https://en.wikipedia.org/wiki/Rivalry_(economics)" href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function"
>non-rivalrous</a >Cobb-Douglas production function</a
> >, the Solow Model describes production as follows:
items leveraged in production (note that ideas may be \[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
<a href="blank" href="https://www.wikiwand.com/en/Excludability" </p>
>excludable</a <ul>
>, though) <li>\(\bar{A}\): total factor productivity (TFP)</li>
</li> <li>
</ul> \(\alpha\): capital&apos;s share of output&mdash;usually
<p> \(1/3\) based on
The Romer Models&apos; production function can be modelled as: <a target="blank" href="https://arxiv.org/pdf/1105.2123"
\[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With: >empirical data</a
</p> >
<ul> </li>
<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li> </ul>
<li> <p>
\(L_{yt}\): the population working on production-facing In this simple model, the following statements describe the
(output-driving) tasks economy:
</li> </p>
</ul> <ol>
<p> <li>
Assuming \(L_t=\bar{L}\) people work in the economy, a proportion Output is either saved or consumed; in other words, savings
\(\bar{l}\) of the population focuses on making ideas: equals investment
\(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\). </li>
</p> <li>
<!-- TODO: footnotes - dynamic JS? --> Capital accumulates according to investment \(I_t\) and
<p> depreciation \(\bar{d}\), beginning with \(K_0\) (often called
Further, this economy garners ideas with time at rate the
<u>\(\bar{z}\)</u>: the &quot;speed of ideas&quot;. Now, we can <u>Law of Capital Motion</u>)
describe the quantity of ideas tomorrow as function of those of </li>
today: the <u>Law of Ideal Motion</u> (I made that up). <li>Labor \(L_t\) is time-independent</li>
\[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta <li>
A_{t+1}=\bar{z}A_tL_{at}\] A savings rate \(\bar{s}\) describes the invested portion of
</p> total output
<p> </li>
Analagously to capital in the solow model, ideas begin in the </ol>
economy with some \(\bar{A}_0\gt0\) and grow at an <p>
<i>exponential</i> rate. At its core, this is because ideas are Including the production function, these four ideas encapsulate
non-rivalrous; more ideas bring about more ideas. the Solow Model:
</p> </p>
<p>Finally, we have a model:</p> <div style="display: flex; justify-content: center">
<div style="display: flex; justify-content: center"> <div style="padding-right: 50px">
<div style="padding-right: 50px"> <ol>
<ol> <li>\(C_t + I_t = Y_t\)</li>
<li>\(Y_t=A_tL_{yt}\)</li> <li>\(\Delta K_{t+1} = I_t - \bar{d} K_t\)</li>
<li>\(\Delta A_{t+1} = \bar{z}A_tL_{at}\)</li> </ol>
</ol> </div>
</div> <div style="padding-left: 50px">
<div style="padding-left: 50px"> <ol start="3">
<ol start="3"> <li>\(L_t = \bar{L}\)</li>
<li>\(L_{yt}+L_{at}=\bar{L}\)</li> <li>\(I_t = \bar{s} Y_t\)</li>
<li>\(L_{at}=\bar{l}\bar{L}\)</li> </ol>
</ol> </div>
</div> </div>
</div> </div>
<p> <div class="fold">
A visualization of the Romer Model shows that the economy grows <h3>solving the model</h3>
exponentially&mdash;production knows no bounds (<a
target="blank"
href="https://en.wikipedia.org/wiki/Ceteris_paribus"
><i>ceteris parbibus</i></a
>, of course). A graph of \(log_{10}(Y_t)\) can be seen below:
</p>
<div class="graph">
<div id="romer-visualization"></div>
</div> </div>
<div class="sliders"> <div>
<div style="padding-right: 20px"> <p>
<ul> Visualizing the model, namely output as a function of capital,
<li> provides helpful intuition before solving it.
<div class="slider"> </p>
<label for="sliderZ">\(\bar{z}:\)</label> <p>
<span id="outputZ">0.50</span> Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that
<input \(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
type="range" \bar{L}^{\frac{2}{3}}\). Utilizing this simplification and its
id="sliderZ" graphical representation below, output is clearly characterized
min="0.1" by the cube root of capital:
max="0.99" </p>
step="0.01" <div class="graph">
value="0.50" <div id="solow-visualization"></div>
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderL">\(\bar{L}:\)</label>
<span id="outputL">505</span>
<input
type="range"
id="sliderL"
min="10"
max="1000"
step="19"
value="505"
/>
</div>
</li>
</ul>
</div> </div>
<div style="padding-left: 20px"> <div class="sliders">
<ul start="3"> <div style="padding-right: 20px">
<li> <ul>
<div class="slider"> <li>
<label for="sliderl">\(\bar{l}:\)</label> <div class="slider">
<span id="outputl">0.50</span> <label for="sliderA">\(A:\)</label>
<input <span id="outputA">1.00</span>
type="range" <input
id="sliderl" type="range"
min="0.01" id="sliderA"
max="0.99" min="0.1"
step="0.01" max="2"
value="0.50" step="0.01"
/> value="1"
</div> />
</li> </div>
<li> </li>
<div class="slider"> <li>
<label for="sliderA0">\(\bar{A}_0:\)</label> <div class="slider">
<span id="outputA0">5000</span> <label for="sliderD">\(d:\)</label>
<input <span id="outputD">0.50</span>
type="range" <input
id="sliderA0" type="range"
min="1" id="sliderD"
max="10000" min="0.01"
step="100" max="0.99"
value="5000" step="0.01"
/> value="0.50"
</div> />
</li> </div>
</ul> </li>
</ul>
</div>
<div style="padding-left: 20px">
<ul start="3">
<li>
<div class="slider">
<label for="sliderS">\(s:\)</label>
<span id="outputS">0.50</span>
<input
type="range"
id="sliderS"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderAlpha">\(\alpha:\)</label>
<span id="outputAlpha">0.33</span>
<input
type="range"
id="sliderAlpha"
min="0.01"
max="0.99"
step="0.01"
value="0.33"
/>
</div>
</li>
</ul>
</div>
</div> </div>
<p>
When investment is completely disincentivized by depreciation
(in other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates
at a so-called &quot;steady-state&quot; with equilibrium
\((K_t,Y_t)=(K_t^*,Y_t^*)\).
</p>
<p>
Using this equilibrium condition, it follows that:
\[Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \rightarrow
\bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}\]
\[\rightarrow
K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}\]
\[\rightarrow
Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}\]
</p>
<p>
Thus, the equilibrium intensive form (output per worker) of both
capital and output are summarized as follows:
\[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}})
=((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha},
\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\]
</p>
</div> </div>
<p> <div class="fold">
Playing with the sliders, this graph may seem underwhelming in <h3>analysis</h3>
comparison to the Solow Model. However, on a logarithmic scale, </div>
small changes in the parameters lead to massive changes in the <div>
growth rate of ideas and economices: <p>
</p> Using both mathematical intuition and manipulating the
<div class="romer-table-container"> visualization above, we find that:
<table id="romer-table"> </p>
<thead> <ul style="list-style: unset">
<tr id="romer-table-header"></tr> <li>
</thead> \(\bar{A}\) has a positive relationship with steady-state
<tbody> output
<tr id="row-A_t"></tr> </li>
<tr id="row-Y_t"></tr> <li>
</tbody> Capital is influenced by workforce size, TFP, and savings rate
</table> </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>
<li>
Economies stagnate at the steady-state
\((K^*,Y^*)\)&mdash;this model provides no avenues for
long-run growth.
</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
(\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)).
<!-- TODO: poor phrasing -->
Notably, the Solow Model does not give any insights into how to
alter the most important predictor of output, TFP.
</p>
</div> </div>
</div> </div>
<div class="fold"><h3>solving the model</h3></div> <div class="fold"><h2>romer</h2></div>
<div> <div>
<p> <div class="fold"><h3>introduction</h3></div>
To find the output in terms of exogenous parameters, first note <div>
that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\] <p>How, then, can we address these shortcomings?</p>
</p> <p>
<p> The Romer Model provides an answer by both modeling ideas
Now, all that remains is to find ideas \(A_t\). It is assumed that \(A_t\) (analagous to TFP in the Solow model) endogenously and
ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\] utilizing them to provide a justification for sustained long-run
</p> growth.
<p> </p>
Using the growth rate formula, we find: \[g_A=\frac{\Delta <p>The Model divides the world into two parts:</p>
A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}\] <ul style="list-style: unset">
</p> <li>
<p> <u>Objects</u>: finite resources, like capital and labor in
Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally, the Solow Model
output can be solved the production function: \[Y_t=A_t </li>
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\] <li>
</p> <u>Ideas</u>: infinite,
<!-- <p> --> <a
<!-- It follows that the intensive form can be written as: --> target="blank"
<!-- \[y_t=\frac{Y_t}{\bar{L}}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\]. --> href="https://en.wikipedia.org/wiki/Rivalry_(economics)"
<!-- </p> --> >non-rivalrous</a
</div> >
<div class="fold"><h3>analysis</h3></div> items leveraged in production (note that ideas may be
<div> <a
<p> href="blank"
We see the Romer model exhibits long-run growth because ideas have href="https://www.wikiwand.com/en/Excludability"
non-diminishing returns due to their nonrival nature. In this >excludable</a
model, capital and income eventually slow but ideas continue to >, though)
yield increasing, unrestricted returns. </li>
</p> </ul>
<p> <p>
Further, all economy continually and perpetually grow along a The Romer Models&apos; production function can be modelled as:
"Balanced Growth Path" as previously defined by \(Y_t\) as a \[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:
function of the endogenous variables. This directly contrasts the </p>
Solow model, in which an economy converges to a steady-state with <ul>
transition dynamics. <li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
</p> <li>
<p> \(L_{yt}\): the population working on production-facing
Changes in the growth rate of ideas, then, alter the growth rate (output-driving) tasks
of output itself&mdash;in this case, parameters \(\bar{l}, </li>
\bar{z}\), and \(\bar{L}\). This is best exemplified by comparing </ul>
the growth rate before and and after a parameter changes. In the <p>
below example, a larger \(\bar{l}\) initially drops output due to Assuming \(L_t=\bar{L}\) people work in the economy, a
less workers being allocated to production. Soon after, though, proportion \(\bar{l}\) of the population focuses on making
output recovers along a &quot;higher&quot; Balanced Growth Path. ideas: \(L_{at}=\bar{l}\bar{L}\rightarrow
</p> L_{yt}=(1-\bar{l})\bar{L}\).
<div class="graph"> </p>
<div id="romer-lchange-visualization"></div> <!-- TODO: footnotes - dynamic JS? -->
</div> <p>
<div class="sliders"> Further, this economy garners ideas with time at rate
<div style="padding-right: 20px"> <u>\(\bar{z}\)</u>: the &quot;speed of ideas&quot;. Now, we can
<ul> describe the quantity of ideas tomorrow as function of those of
<li> today: the <u>Law of Ideal Motion</u> (I made that up).
<div class="slider"> \[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta
<label for="sliderlChange">\(\bar{l}_1:\)</label> A_{t+1}=\bar{z}A_tL_{at}\]
<span id="outputlChange">0.50</span> </p>
<input <p>
type="range" Analagously to capital in the solow model, ideas begin in the
id="sliderlChange" economy with some \(\bar{A}_0\gt0\) and grow at an
min="0.1" <i>exponential</i> rate. At its core, this is because ideas are
max="0.99" non-rivalrous; more ideas bring about more ideas.
step="0.01" </p>
value="0.50" <p>Finally, we have a model:</p>
/> <div style="display: flex; justify-content: center">
</div> <div style="padding-right: 50px">
</li> <ol>
</ul> <li>\(Y_t=A_tL_{yt}\)</li>
<li>\(\Delta A_{t+1} = \bar{z}A_tL_{at}\)</li>
</ol>
</div>
<div style="padding-left: 50px">
<ol start="3">
<li>\(L_{yt}+L_{at}=\bar{L}\)</li>
<li>\(L_{at}=\bar{l}\bar{L}\)</li>
</ol>
</div>
</div>
<p>
A visualization of the Romer Model shows that the economy grows
exponentially&mdash;production knows no bounds (<a
target="blank"
href="https://en.wikipedia.org/wiki/Ceteris_paribus"
><i>ceteris parbibus</i></a
>, of course). A graph of \(log_{10}(Y_t)\) can be seen below:
</p>
<div class="graph">
<div id="romer-visualization"></div>
</div>
<div class="sliders">
<div style="padding-right: 20px">
<ul>
<li>
<div class="slider">
<label for="sliderZ">\(\bar{z}:\)</label>
<span id="outputZ">0.50</span>
<input
type="range"
id="sliderZ"
min="0.1"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderL">\(\bar{L}:\)</label>
<span id="outputL">505</span>
<input
type="range"
id="sliderL"
min="10"
max="1000"
step="19"
value="505"
/>
</div>
</li>
</ul>
</div>
<div style="padding-left: 20px">
<ul start="3">
<li>
<div class="slider">
<label for="sliderl">\(\bar{l}:\)</label>
<span id="outputl">0.50</span>
<input
type="range"
id="sliderl"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderA0">\(\bar{A}_0:\)</label>
<span id="outputA0">5000</span>
<input
type="range"
id="sliderA0"
min="1"
max="10000"
step="100"
value="5000"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
Playing with the sliders, this graph may seem underwhelming in
comparison to the Solow Model. However, on a logarithmic scale,
small changes in the parameters lead to massive changes in the
growth rate of ideas and economices:
</p>
<div class="romer-table-container">
<table id="romer-table">
<thead>
<tr id="romer-table-header"></tr>
</thead>
<tbody>
<tr id="row-A_t"></tr>
<tr id="row-Y_t"></tr>
</tbody>
</table>
</div> </div>
</div> </div>
<div class="fold"><h3>solving the model</h3></div>
<div>
<p>
To find the output in terms of exogenous parameters, first note
that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\]
</p>
<p>
Now, all that remains is to find ideas \(A_t\). It is assumed
that ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\]
</p>
<p>
Using the growth rate formula, we find: \[g_A=\frac{\Delta
A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}\]
</p>
<p>
Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally,
output can be solved the production function: \[Y_t=A_t
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\]
</p>
<!-- <p> -->
<!-- It follows that the intensive form can be written as: -->
<!-- \[y_t=\frac{Y_t}{\bar{L}}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\]. -->
<!-- </p> -->
</div>
<div class="fold"><h3>analysis</h3></div>
<div>
<p>
We see the Romer model exhibits long-run growth because ideas
have non-diminishing returns due to their nonrival nature. In
this model, capital and income eventually slow but ideas
continue to yield increasing, unrestricted returns.
</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.
</p>
<p>
Changes in the growth rate of ideas, then, alter the growth rate
of output itself&mdash;in this case, parameters \(\bar{l},
\bar{z}\), and \(\bar{L}\). This is best exemplified by
comparing the growth rate before and and after a parameter
changes. In the below example, a larger \(\bar{l}\) initially
drops output due to less workers being allocated to production.
Soon after, though, output recovers along a &quot;higher&quot;
Balanced Growth Path.
</p>
<div class="graph">
<div id="romer-lchange-visualization"></div>
</div>
<div class="sliders">
<div style="padding-right: 20px">
<ul>
<li>
<div class="slider">
<label for="sliderlChange">\(\bar{l}_1:\)</label>
<span id="outputlChange">0.50</span>
<input
type="range"
id="sliderlChange"
min="0.1"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
Notably, while both the Romer and Solow Models help to analyze
growth across countries, they both are unable to resolve one
question: why can and do investment rates and TFP differ across
contries? This is a more fundamental economic question involving
culture, institutions, and social dynamics&mdash;one day I hope
we&apos;ll have an answer.
</p>
</div>
</div> </div>
<h2>romer-solow</h2> <div class="fold"><h2>romer-solow</h2></div>
<div class="fold"><h3>introduction</h3></div>
<div> <div>
<p>hi</p> <div class="fold"><h3>introduction</h3></div>
<p>hello</p> <div>
<p>
While the Romer Model provides an avenue for long-run economic
growth, it is anything but realistic&mdash;surely economies due
not grow at an ever-increasing blistering rate into perpetuity.
A model in which:
</p>
<ul style="list-style: unset">
<li>
Economies grow <i>faster</i> the further <i>below</i> they are
from their balanced growth path
</li>
<li>
Economies grow <i>slower</i> the further <i>above</i> they are
from their balanced growth path
</li>
</ul>
<p>
would certainly be more pragmatic. The Solow Model&apos;s
capital dynamics do, in some sense, mirror part of this behavior
with respect to the steady-state (output converges
quicker/slower to the steady state the further/closer it is from
equilibrium).
</p>
<p>
Combining the dynamics of the Romer Model&apos;s ideas and the
Solow Model&apos;s capital stock could yield the desired result.
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:
</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>\(\Delta K_{t+1}=\bar{s}Y_t-\bar{d}K_t\)</li>
<li>\(\Delta A_{t+1} = \bar{z}A_tL_{at}\)</li>
</ol>
</div>
<div style="padding-left: 50px">
<ol start="4">
<li>\(L_{yt}+L_{at}=\bar{L}\)</li>
<li>\(L_{at}=\bar{l}\bar{L}\)</li>
</ol>
</div>
</div>
</div>
<div class="fold"><h3>solving the model</h3></div>
<div>content</div>
</div> </div>
</article> </article>
</div> </div>

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