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975
posts/economics/models-of-production.html
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@ -53,477 +53,542 @@
in Intermediate Macroeconomics (ECON 3020) during the Spring
semester of 2024 at the University of Virginia.
</p>
<h2>solow</h2>
<div class="fold"><h3>introduction</h3></div>
<div class="fold"><h2>solow</h2></div>
<div>
<p>
The Solow Model is an economic model of production that
incorporates the incorporates the idea of capital accumulation.
Based on the
<a
target="blank"
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,
<div class="fold"><h3>introduction</h3></div>
<div>
<p>
The Solow Model is an economic model of production that
incorporates the incorporates the idea of capital accumulation.
Based on the
<a
target="blank"
href="https://en.wikipedia.org/wiki/Rivalry_(economics)"
>non-rivalrous</a
>
items leveraged in production (note that ideas may be
<a href="blank" href="https://www.wikiwand.com/en/Excludability"
>excludable</a
>, though)
</li>
</ul>
<p>
The Romer Models&apos; production function can be modelled as:
\[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:
</p>
<ul>
<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
<li>
\(L_{yt}\): the population working on production-facing
(output-driving) tasks
</li>
</ul>
<p>
Assuming \(L_t=\bar{L}\) people work in the economy, a proportion
\(\bar{l}\) of the population focuses on making ideas:
\(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\).
</p>
<!-- TODO: footnotes - dynamic JS? -->
<p>
Further, this economy garners ideas with time at rate
<u>\(\bar{z}\)</u>: the &quot;speed of ideas&quot;. Now, we can
describe the quantity of ideas tomorrow as function of those of
today: the <u>Law of Ideal Motion</u> (I made that up).
\[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta
A_{t+1}=\bar{z}A_tL_{at}\]
</p>
<p>
Analagously to capital in the solow model, ideas begin in the
economy with some \(\bar{A}_0\gt0\) and grow at an
<i>exponential</i> rate. At its core, this is because ideas are
non-rivalrous; more ideas bring about more ideas.
</p>
<p>Finally, we have a model:</p>
<div style="display: flex; justify-content: center">
<div style="padding-right: 50px">
<ol>
<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>
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>
<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 class="fold">
<h3>solving the model</h3>
</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>
<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 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 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>
<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 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>
</div>
<div class="fold"><h3>solving the model</h3></div>
<div class="fold"><h2>romer</h2></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 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
target="blank"
href="https://en.wikipedia.org/wiki/Rivalry_(economics)"
>non-rivalrous</a
>
items leveraged in production (note that ideas may be
<a
href="blank"
href="https://www.wikiwand.com/en/Excludability"
>excludable</a
>, though)
</li>
</ul>
<p>
The Romer Models&apos; production function can be modelled as:
\[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:
</p>
<ul>
<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
<li>
\(L_{yt}\): the population working on production-facing
(output-driving) tasks
</li>
</ul>
<p>
Assuming \(L_t=\bar{L}\) people work in the economy, a
proportion \(\bar{l}\) of the population focuses on making
ideas: \(L_{at}=\bar{l}\bar{L}\rightarrow
L_{yt}=(1-\bar{l})\bar{L}\).
</p>
<!-- TODO: footnotes - dynamic JS? -->
<p>
Further, this economy garners ideas with time at rate
<u>\(\bar{z}\)</u>: the &quot;speed of ideas&quot;. Now, we can
describe the quantity of ideas tomorrow as function of those of
today: the <u>Law of Ideal Motion</u> (I made that up).
\[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta
A_{t+1}=\bar{z}A_tL_{at}\]
</p>
<p>
Analagously to capital in the solow model, ideas begin in the
economy with some \(\bar{A}_0\gt0\) and grow at an
<i>exponential</i> rate. At its core, this is because ideas are
non-rivalrous; more ideas bring about more ideas.
</p>
<p>Finally, we have a model:</p>
<div style="display: flex; justify-content: center">
<div style="padding-right: 50px">
<ol>
<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 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>
<h2>romer-solow</h2>
<div class="fold"><h3>introduction</h3></div>
<div class="fold"><h2>romer-solow</h2></div>
<div>
<p>hi</p>
<p>hello</p>
<div class="fold"><h3>introduction</h3></div>
<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>
</article>
</div>

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