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posts/economics/models-of-production.html
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@ -53,7 +53,8 @@
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>
<div class="fold"><h3>introduction</h3></div> <div class="fold"><h3>introduction</h3></div>
<div> <div>
<p> <p>
@ -70,8 +71,8 @@
<ul> <ul>
<li>\(\bar{A}\): total factor productivity (TFP)</li> <li>\(\bar{A}\): total factor productivity (TFP)</li>
<li> <li>
\(\alpha\): capital&apos;s share of output&mdash;usually \(1/3\) \(\alpha\): capital&apos;s share of output&mdash;usually
based on \(1/3\) based on
<a target="blank" href="https://arxiv.org/pdf/1105.2123" <a target="blank" href="https://arxiv.org/pdf/1105.2123"
>empirical data</a >empirical data</a
> >
@ -127,10 +128,10 @@
</p> </p>
<p> <p>
Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that
\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\). \(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
Utilizing this simplification and its graphical representation \bar{L}^{\frac{2}{3}}\). Utilizing this simplification and its
below, output is clearly characterized by the cube root of graphical representation below, output is clearly characterized
capital: by the cube root of capital:
</p> </p>
<div class="graph"> <div class="graph">
<div id="solow-visualization"></div> <div id="solow-visualization"></div>
@ -202,9 +203,9 @@
</div> </div>
</div> </div>
<p> <p>
When investment is completely disincentivized by depreciation (in When investment is completely disincentivized by depreciation
other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates at a (in other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates
so-called &quot;steady-state&quot; with equilibrium at a so-called &quot;steady-state&quot; with equilibrium
\((K_t,Y_t)=(K_t^*,Y_t^*)\). \((K_t,Y_t)=(K_t^*,Y_t^*)\).
</p> </p>
<p> <p>
@ -234,7 +235,8 @@
</p> </p>
<ul style="list-style: unset"> <ul style="list-style: unset">
<li> <li>
\(\bar{A}\) has a positive relationship with steady-state output \(\bar{A}\) has a positive relationship with steady-state
output
</li> </li>
<li> <li>
Capital is influenced by workforce size, TFP, and savings rate Capital is influenced by workforce size, TFP, and savings rate
@ -247,51 +249,53 @@
</ol> </ol>
</li> </li>
<li> <li>
Large deviations in capital from steady-state \(K^*\) induce net Large deviations in capital from steady-state \(K^*\) induce
investments of larger magnitude, leading to an accelerated net investments of larger magnitude, leading to an accelerated
reversion to the steady-state reversion to the steady-state
</li> </li>
<li> <li>
Economies stagnate at the steady-state \((K^*,Y^*)\)&mdash;this Economies stagnate at the steady-state
model provides no avenues for long-run growth. \((K^*,Y^*)\)&mdash;this model provides no avenues for
long-run growth.
</li> </li>
</ul> </ul>
<p> <p>
Lastly (and perhaps most importantly), exogenous parameters Lastly (and perhaps most importantly), exogenous parameters
\(\bar{s}, \bar{d}\), and \(\bar{A}\) all have immense \(\bar{s}, \bar{d}\), and \(\bar{A}\) all have immense
ramifications on economic status. For example, comparing the ramifications on economic status. For example, comparing the
difference in country \(C_1\)&apos;s output versus \(C_2\)&apos;s difference in country \(C_1\)&apos;s output versus
using the Solow Model, we find that a difference in economic \(C_2\)&apos;s using the Solow Model, we find that a difference
performance can only be explained by these factors: \[ in economic performance can only be explained by these factors:
\[
\frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha} \frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha}
\] \]
</p> </p>
<p> <p>
We see that TFP is more important in explaining the differences in We see that TFP is more important in explaining the differences
per capital output in per capital output
(\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)). (\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)).
<!-- TODO: poor phrasing --> <!-- TODO: poor phrasing -->
Notably, the Solow Model does not give any insights into how to Notably, the Solow Model does not give any insights into how to
alter the most important predictor of output, TFP. alter the most important predictor of output, TFP.
</p> </p>
</div> </div>
<!-- Solow TODO --> </div>
<!-- TODO: dynamics?????? --> <div class="fold"><h2>romer</h2></div>
<!-- TODO: K_0 --> <div>
<h2>romer</h2>
<div class="fold"><h3>introduction</h3></div> <div class="fold"><h3>introduction</h3></div>
<div> <div>
<p>How, then, can we address these shortcomings?</p> <p>How, then, can we address these shortcomings?</p>
<p> <p>
The Romer Model provides an answer by both modeling ideas \(A_t\) The Romer Model provides an answer by both modeling ideas
(analagous to TFP in the Solow model) endogenously and utilizing \(A_t\) (analagous to TFP in the Solow model) endogenously and
them to provide a justification for sustained long-run growth. utilizing them to provide a justification for sustained long-run
growth.
</p> </p>
<p>The Model divides the world into two parts:</p> <p>The Model divides the world into two parts:</p>
<ul style="list-style: unset"> <ul style="list-style: unset">
<li> <li>
<u>Objects</u>: finite resources, like capital and labor in the <u>Objects</u>: finite resources, like capital and labor in
Solow Model the Solow Model
</li> </li>
<li> <li>
<u>Ideas</u>: infinite, <u>Ideas</u>: infinite,
@ -301,7 +305,9 @@
>non-rivalrous</a >non-rivalrous</a
> >
items leveraged in production (note that ideas may be 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 >excludable</a
>, though) >, though)
</li> </li>
@ -318,9 +324,10 @@
</li> </li>
</ul> </ul>
<p> <p>
Assuming \(L_t=\bar{L}\) people work in the economy, a proportion Assuming \(L_t=\bar{L}\) people work in the economy, a
\(\bar{l}\) of the population focuses on making ideas: proportion \(\bar{l}\) of the population focuses on making
\(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\). ideas: \(L_{at}=\bar{l}\bar{L}\rightarrow
L_{yt}=(1-\bar{l})\bar{L}\).
</p> </p>
<!-- TODO: footnotes - dynamic JS? --> <!-- TODO: footnotes - dynamic JS? -->
<p> <p>
@ -454,8 +461,8 @@
that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\] that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\]
</p> </p>
<p> <p>
Now, all that remains is to find ideas \(A_t\). It is assumed that Now, all that remains is to find ideas \(A_t\). It is assumed
ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\] that ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\]
</p> </p>
<p> <p>
Using the growth rate formula, we find: \[g_A=\frac{\Delta Using the growth rate formula, we find: \[g_A=\frac{\Delta
@ -474,26 +481,27 @@
<div class="fold"><h3>analysis</h3></div> <div class="fold"><h3>analysis</h3></div>
<div> <div>
<p> <p>
We see the Romer model exhibits long-run growth because ideas have We see the Romer model exhibits long-run growth because ideas
non-diminishing returns due to their nonrival nature. In this have non-diminishing returns due to their nonrival nature. In
model, capital and income eventually slow but ideas continue to this model, capital and income eventually slow but ideas
yield increasing, unrestricted returns. continue to yield increasing, unrestricted returns.
</p> </p>
<p> <p>
Further, all economy continually and perpetually grow along a Further, all economy continually and perpetually grow along a
"Balanced Growth Path" as previously defined by \(Y_t\) as a "Balanced Growth Path" as previously defined by \(Y_t\) as a
function of the endogenous variables. This directly contrasts the function of the endogenous variables. This directly contrasts
Solow model, in which an economy converges to a steady-state with the Solow model, in which an economy converges to a steady-state
transition dynamics. with transition dynamics.
</p> </p>
<p> <p>
Changes in the growth rate of ideas, then, alter the growth rate Changes in the growth rate of ideas, then, alter the growth rate
of output itself&mdash;in this case, parameters \(\bar{l}, of output itself&mdash;in this case, parameters \(\bar{l},
\bar{z}\), and \(\bar{L}\). This is best exemplified by comparing \bar{z}\), and \(\bar{L}\). This is best exemplified by
the growth rate before and and after a parameter changes. In the comparing the growth rate before and and after a parameter
below example, a larger \(\bar{l}\) initially drops output due to changes. In the below example, a larger \(\bar{l}\) initially
less workers being allocated to production. Soon after, though, drops output due to less workers being allocated to production.
output recovers along a &quot;higher&quot; Balanced Growth Path. Soon after, though, output recovers along a &quot;higher&quot;
Balanced Growth Path.
</p> </p>
<div class="graph"> <div class="graph">
<div id="romer-lchange-visualization"></div> <div id="romer-lchange-visualization"></div>
@ -518,12 +526,69 @@
</ul> </ul>
</div> </div>
</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>
<div class="fold"><h2>romer-solow</h2></div>
<div>
<div class="fold"><h3>introduction</h3></div> <div class="fold"><h3>introduction</h3></div>
<div> <div>
<p>hi</p> <p>
<p>hello</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>

57
scripts/post.js
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@ -3,6 +3,38 @@ document.documentElement.style.setProperty(
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); );
const tagToHeader = new Map([
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["H3", "##"],
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const makeFold = (h, i) => {
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toggle.style.fontStyle = "normal";
toggle.textContent = "v";
// only unfold first algorithm problem
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mdHeading.textContent = `${header} `;
mdHeading.style.color = getTopicColor(urlToTopic());
h.prepend(mdHeading);
h.prepend(toggle);
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const mdHeading = document.createElement("span"); const mdHeading = document.createElement("span");
@ -11,27 +43,6 @@ document.addEventListener("DOMContentLoaded", () => {
h2.prepend(mdHeading); h2.prepend(mdHeading);
}); });
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