515 lines
20 KiB
HTML
515 lines
20 KiB
HTML
<!doctype html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="UTF-8" />
|
|
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
|
<link rel="stylesheet" href="/styles/common.css" />
|
|
<link rel="stylesheet" href="/styles/post.css" />
|
|
<link rel="stylesheet" href="/styles/graph.css" />
|
|
<link rel="icon" type="image/webp" href="/public/logo.webp" />
|
|
<link rel="stylesheet" href="/public/katex/katex.css" />
|
|
<script defer src="/public/katex/katex.js"></script>
|
|
<script
|
|
defer
|
|
src="/public/katex/katex-render.js"
|
|
onload="renderMathInElement(document.body);"
|
|
></script>
|
|
<script defer src="/public/d3.js"></script>
|
|
<title>Barrett Ruth</title>
|
|
</head>
|
|
<body class="graph-background">
|
|
<header>
|
|
<a
|
|
href="/"
|
|
style="text-decoration: none; color: inherit"
|
|
onclick="goHome(event)"
|
|
>
|
|
<div class="terminal-container">
|
|
<span class="terminal-prompt">barrett@ruth:~$ /economics</span>
|
|
<span class="terminal-cursor"></span>
|
|
</div>
|
|
</a>
|
|
</header>
|
|
<main class="main">
|
|
<div class="post-container">
|
|
<header class="post-header">
|
|
<h1 class="post-title">Models of Production</h1>
|
|
<p class="post-meta">
|
|
<time datetime="2024-06-22">22/06/2024</time>
|
|
</p>
|
|
</header>
|
|
<article class="post-article">
|
|
<h2>solow</h2>
|
|
<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/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's share of output—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 "steady-state" 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>
|
|
<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 \((K^*,Y^*)\)—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 \(C_1\)'s output versus \(C_2\)'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
|
|
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' 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 "speed of ideas". 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—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—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 "higher" 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">\(\bar{l}:\)</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>
|
|
</div>
|
|
<h2>romer-solow</h2>
|
|
</article>
|
|
</div>
|
|
</main>
|
|
<script src="/scripts/common.js"></script>
|
|
<script src="/scripts/post.js"></script>
|
|
<script src="/scripts/posts/models-of-production.js"></script>
|
|
</body>
|
|
</html>
|