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feat(post): fold h2s as well

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165
posts/economics/models-of-production.html
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@ -53,7 +53,8 @@
in Intermediate Macroeconomics (ECON 3020) during the Spring
semester of 2024 at the University of Virginia.
</p>
<h2>solow</h2>
<div class="fold"><h2>solow</h2></div>
<div>
<div class="fold"><h3>introduction</h3></div>
<div>
<p>
@ -70,8 +71,8 @@
<ul>
<li>\(\bar{A}\): total factor productivity (TFP)</li>
<li>
\(\alpha\): capital&apos;s share of output&mdash;usually \(1/3\)
based on
\(\alpha\): 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
>
@ -127,10 +128,10 @@
</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:
\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
\bar{L}^{\frac{2}{3}}\). 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>
@ -202,9 +203,9 @@
</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
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>
@ -234,7 +235,8 @@
</p>
<ul style="list-style: unset">
<li>
\(\bar{A}\) has a positive relationship with steady-state output
\(\bar{A}\) has a positive relationship with steady-state
output
</li>
<li>
Capital is influenced by workforce size, TFP, and savings rate
@ -247,51 +249,53 @@
</ol>
</li>
<li>
Large deviations in capital from steady-state \(K^*\) induce net
investments of larger magnitude, leading to an accelerated
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.
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: \[
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
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>
<div class="fold"><h2>romer</h2></div>
<div>
<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.
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
<u>Objects</u>: finite resources, like capital and labor in
the Solow Model
</li>
<li>
<u>Ideas</u>: infinite,
@ -301,7 +305,9 @@
>non-rivalrous</a
>
items leveraged in production (note that ideas may be
<a href="blank" href="https://www.wikiwand.com/en/Excludability"
<a
href="blank"
href="https://www.wikiwand.com/en/Excludability"
>excludable</a
>, though)
</li>
@ -318,9 +324,10 @@
</li>
</ul>
<p>
Assuming \(L_t=\bar{L}\) people work in the economy, a proportion
\(\bar{l}\) of the population focuses on making ideas:
\(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\).
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>
@ -454,8 +461,8 @@
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 \(g_A\): \[A_t=A_0(1+g_A)^t\]
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
@ -474,26 +481,27 @@
<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.
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.
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.
\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>
@ -518,12 +526,69 @@
</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>
<h2>romer-solow</h2>
</div>
<div class="fold"><h2>romer-solow</h2></div>
<div>
<div class="fold"><h3>introduction</h3></div>
<div>
<p>hi</p>
<p>hello</p>
<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>

57
scripts/post.js
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@ -3,6 +3,38 @@ document.documentElement.style.setProperty(
getTopicColor(urlToTopic()),
);
const tagToHeader = new Map([
["H2", "#"],
["H3", "##"],
]);
const makeFold = (h, i) => {
const toggle = document.createElement("span");
toggle.style.fontStyle = "normal";
toggle.textContent = "v";
// only unfold first algorithm problem
if (urlToTopic() === "algorithms" && i === 0) toggle.textContent = "v";
h.parentElement.nextElementSibling.style.display =
toggle.textContent === ">" ? "none" : "block";
h.parentE;
toggle.classList.add("fold-toggle");
toggle.addEventListener("click", () => {
const content = h.parentElement.nextElementSibling;
toggle.textContent = toggle.textContent === ">" ? "v" : ">";
content.style.display = toggle.textContent === ">" ? "none" : "block";
});
const mdHeading = document.createElement("span");
const header = tagToHeader.has(h.tagName) ? tagToHeader.get(h.tagName) : "";
mdHeading.textContent = `${header} `;
mdHeading.style.color = getTopicColor(urlToTopic());
h.prepend(mdHeading);
h.prepend(toggle);
};
document.addEventListener("DOMContentLoaded", () => {
document.querySelectorAll("article h2").forEach((h2) => {
const mdHeading = document.createElement("span");
@ -11,27 +43,6 @@ document.addEventListener("DOMContentLoaded", () => {
h2.prepend(mdHeading);
});
document.querySelectorAll(".fold h3").forEach((h3, i) => {
const toggle = document.createElement("span");
toggle.textContent = "v";
// only unfold first algorithm problem
if (urlToTopic() === "algorithms" && i === 0) toggle.textContent = "v";
h3.parentElement.nextElementSibling.style.display =
toggle.textContent === ">" ? "none" : "block";
toggle.classList.add("fold-toggle");
toggle.addEventListener("click", () => {
const content = h3.parentElement.nextElementSibling;
toggle.textContent = toggle.textContent === ">" ? "v" : ">";
content.style.display = toggle.textContent === ">" ? "none" : "block";
});
const mdHeading = document.createElement("span");
mdHeading.textContent = "## ";
mdHeading.style.color = getTopicColor(urlToTopic());
h3.prepend(mdHeading);
h3.prepend(toggle);
});
document.querySelectorAll(".fold h2").forEach(makeFold);
document.querySelectorAll(".fold h3").forEach(makeFold);
});