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feat: romer model

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Barrett Ruth 2024-07-03 10:41:10 -05:00
parent b6b3e08886
commit 71a959708d
7 changed files with 515 additions and 31 deletions
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1
aggieland.txt Normal file
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the 12th man

1
compile_flags.txt Normal file
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@ -0,0 +1 @@
-std=c++20

5
filename.txt Normal file
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@ -0,0 +1,5 @@
This is an example text file.
This file contains several words, some of which are repeated.
For example, the word "this" appears multiple times in this file.
We can use a shell command to find the most frequent words.
Let's see how many times each word appears in this example text file.

10
map.cc Normal file
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@ -0,0 +1,10 @@
#include <fstream>
#include <iostream>
int main() {
std::ifstream ifs{"aggieland.txt"};
std::string read1;
int read2;
ifs >> read1 >> read2;
std::cout << read1 << read2;
}

254
posts/economics/models-of-production.html
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@ -147,7 +147,7 @@
type="range" type="range"
id="sliderD" id="sliderD"
min="0.01" min="0.01"
max="1" max="0.99"
step="0.01" step="0.01"
value="0.50" value="0.50"
/> />
@ -165,7 +165,7 @@
type="range" type="range"
id="sliderS" id="sliderS"
min="0.01" min="0.01"
max="1" max="0.99"
step="0.01" step="0.01"
value="0.50" value="0.50"
/> />
@ -179,7 +179,7 @@
type="range" type="range"
id="sliderAlpha" id="sliderAlpha"
min="0.01" min="0.01"
max="1" max="0.99"
step="0.01" step="0.01"
value="0.33" value="0.33"
/> />
@ -213,8 +213,6 @@
</div> </div>
<div class="fold"> <div class="fold">
<h3>analysis</h3> <h3>analysis</h3>
</div>
<div>
<p> <p>
Using both mathematical intuition and manipulating the Using both mathematical intuition and manipulating the
visualization above, we find that: visualization above, we find that:
@ -262,10 +260,250 @@
alter the most important predictor of output, TFP. alter the most important predictor of output, TFP.
</p> </p>
</div> </div>
<h2>romer</h2> <!-- Solow TODO -->
<!-- TODO: transition talking about "Romer model does, though" -->
<!-- TODO: say the romer model provides avenue for LR growth -->
<!-- TODO: dynamics?????? --> <!-- 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&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">\(\bar{l}:\)</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> <h2>romer-solow</h2>
</article> </article>
</div> </div>

255
scripts/posts/models-of-production.js
View file

@ -11,10 +11,10 @@ function drawSolowGraph() {
}; };
}); });
const A = document.getElementById("outputA").textContent, const A = parseFloat(document.getElementById("outputA").textContent),
D = document.getElementById("outputD").textContent, D = parseFloat(document.getElementById("outputD").textContent),
S = document.getElementById("outputS").textContent, S = parseFloat(document.getElementById("outputS").textContent),
alpha = document.getElementById("outputAlpha").textContent; alpha = parseFloat(document.getElementById("outputAlpha").textContent);
const solowOutput = (K) => A * Math.pow(K, alpha) * Math.pow(L, 1 - alpha); const solowOutput = (K) => A * Math.pow(K, alpha) * Math.pow(L, 1 - alpha);
const solowDepreciation = (K) => D * K; const solowDepreciation = (K) => D * K;
const solowInvestment = (Y) => S * Y; const solowInvestment = (Y) => S * Y;
@ -34,12 +34,10 @@ function drawSolowGraph() {
.attr("transform", `translate(${margin.left}, ${margin.top})`); .attr("transform", `translate(${margin.left}, ${margin.top})`);
const x = d3.scaleLinear().domain([0, K_MAX]).range([0, width]); const x = d3.scaleLinear().domain([0, K_MAX]).range([0, width]);
const xAxis = svg svg
.append("g") .append("g")
.attr("transform", `translate(0, ${height})`) .attr("transform", `translate(0, ${height})`)
.call(d3.axisBottom(x)); .call(d3.axisBottom(x))
xAxis
.append("text") .append("text")
.attr("fill", "#000") .attr("fill", "#000")
.attr("x", width + 10) .attr("x", width + 10)
@ -50,9 +48,9 @@ function drawSolowGraph() {
const Y_MAX = solowOutput(K_MAX) + K_MAX / 10; const Y_MAX = solowOutput(K_MAX) + K_MAX / 10;
const y = d3.scaleLinear().domain([0, Y_MAX]).range([height, 0]); const y = d3.scaleLinear().domain([0, Y_MAX]).range([height, 0]);
const yAxis = svg.append("g").call(d3.axisLeft(y)); svg
.append("g")
yAxis .call(d3.axisLeft(y))
.append("text") .append("text")
.attr("fill", "#000") .attr("fill", "#000")
.attr("x", 0) .attr("x", 0)
@ -87,9 +85,7 @@ function drawSolowGraph() {
.append("xhtml:body") .append("xhtml:body")
.style("font-size", "0.75em") .style("font-size", "0.75em")
.html(`<div class="solow-visualization-y"></div>`); .html(`<div class="solow-visualization-y"></div>`);
katex.render("Y", document.querySelector(".solow-visualization-y"), { katex.render("Y", document.querySelector(".solow-visualization-y"));
throwOnError: false,
});
const depreciationData = Array.from({ length: K_MAX }, (_, k) => ({ const depreciationData = Array.from({ length: K_MAX }, (_, k) => ({
K: k, K: k,
@ -118,9 +114,7 @@ function drawSolowGraph() {
.append("xhtml:body") .append("xhtml:body")
.style("font-size", "0.75em") .style("font-size", "0.75em")
.html(`<div class="solow-visualization-d"></div>`); .html(`<div class="solow-visualization-d"></div>`);
katex.render("\\bar{d}K", document.querySelector(".solow-visualization-d"), { katex.render("\\bar{d}K", document.querySelector(".solow-visualization-d"));
throwOnError: false,
});
const investmentData = outputData.map((d) => ({ const investmentData = outputData.map((d) => ({
K: d.K, K: d.K,
@ -149,9 +143,7 @@ function drawSolowGraph() {
.append("xhtml:body") .append("xhtml:body")
.style("font-size", "0.75em") .style("font-size", "0.75em")
.html(`<div class="solow-visualization-i"></div>`); .html(`<div class="solow-visualization-i"></div>`);
katex.render("I", document.querySelector(".solow-visualization-i"), { katex.render("I", document.querySelector(".solow-visualization-i"));
throwOnError: false,
});
const k_star = L * Math.pow((S * A) / D, 1 / (1 - alpha)); const k_star = L * Math.pow((S * A) / D, 1 / (1 - alpha));
svg svg
@ -177,13 +169,230 @@ function drawSolowGraph() {
katex.render( katex.render(
`(K^*,Y^*)=(${k_star.toFixed(0)},${y_star.toFixed(0)})`, `(K^*,Y^*)=(${k_star.toFixed(0)},${y_star.toFixed(0)})`,
document.querySelector(".solow-visualization-eq"), document.querySelector(".solow-visualization-eq"),
{
throwOnError: false,
},
); );
} }
const formatNumber = (num) => {
return `~${num.toExponential(0)}`;
};
const normalFont = `style="font-weight: normal"`;
const updateRomerTable = (romerData) => {
const tableHeader = document.getElementById("romer-table-header");
const rowA_t = document.getElementById("row-A_t");
const rowY_t = document.getElementById("row-Y_t");
tableHeader.innerHTML = `<th ${normalFont}><div class="romer-table-time"></th>`;
katex.render(`t`, document.querySelector(".romer-table-time"));
rowA_t.innerHTML = `<td class="romer-table-at"></td>`;
rowY_t.innerHTML = `<td class="romer-table-yt"></td>`;
katex.render("A_t", document.querySelector(".romer-table-at"));
katex.render("Y_t", document.querySelector(".romer-table-yt"));
romerData.forEach((d) => {
if (d.year % 20 === 0 || d.year === 1) {
tableHeader.innerHTML += `<th ${normalFont}>${d.year}</th>`;
rowA_t.innerHTML += `<td>${formatNumber(d.A)}</td>`;
rowY_t.innerHTML += `<td>${formatNumber(d.Y)}</td>`;
}
});
};
function drawRomerGraph() {
const T_MAX = 100;
margin = { top: 20, right: 100, bottom: 20, left: 50 };
["Z", "L", "l", "A0"].forEach((param) => {
const slider = document.getElementById(`slider${param}`);
slider.oninput = function () {
slider.previousElementSibling.innerText = this.value;
drawRomerGraph();
};
});
const z = parseFloat(document.getElementById("outputZ").textContent),
L = parseFloat(document.getElementById("outputL").textContent),
l = parseFloat(document.getElementById("outputl").textContent),
A0 = parseFloat(document.getElementById("outputA0").textContent);
const container = document.getElementById("romer-visualization");
const width = container.clientWidth - margin.left - margin.right;
const height = container.clientHeight - margin.top - margin.bottom;
container.innerHTML = "";
const svg = d3
.select("#romer-visualization")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform", `translate(${margin.left}, ${margin.top})`);
let A = A0;
const romerData = [];
for (let t = 1; t <= T_MAX; ++t) {
const A_t = A * (1 + z * l * L);
const Y_t = A_t * (1 - l) * L;
romerData.push({ year: t, A: A_t, Y: Math.log10(Y_t) });
A = A_t;
}
const x = d3.scaleLinear().domain([1, T_MAX]).range([0, width]);
svg
.append("g")
.attr("transform", `translate(0, ${height})`)
.call(d3.axisBottom(x))
.append("text")
.attr("fill", "#000")
.attr("x", width + 10)
.attr("y", -10)
.style("text-anchor", "end")
.style("font-size", "1.5em")
.text("t");
const y = d3
.scaleLinear()
.domain([0, romerData[romerData.length - 1].Y])
.range([height, 0]);
svg
.append("g")
.call(d3.axisLeft(y).ticks(10, d3.format(".1s")))
.append("text")
.attr("fill", "#000")
.attr("x", 0)
.attr("y", -10)
.style("text-anchor", "start")
.style("font-size", "1.5em")
.text("log(Y)");
svg
.append("path")
.datum(romerData)
.attr("fill", "none")
.attr("stroke", getTopicColor(urlToTopic()))
.attr("stroke-width", 2)
.attr(
"d",
d3
.line()
.x((d) => x(d.year))
.y((d) => y(d.Y)),
);
svg
.append("foreignObject")
.attr("width", "4em")
.attr("height", "2em")
.attr("x", x(T_MAX))
.attr("y", y(romerData[T_MAX - 1].Y))
.append("xhtml:body")
.style("font-size", "0.75em")
.html(`<div class="romer-visualization-y"></div>`);
katex.render("log_{10}Y", document.querySelector(".romer-visualization-y"));
updateRomerTable(romerData);
}
function drawRomerlGraph() {
const T_MAX = 100,
z = 0.01,
L = 50,
A0 = 50;
margin = { top: 20, right: 100, bottom: 20, left: 50 };
const slider = document.getElementById(`sliderlChange`);
slider.oninput = function () {
slider.previousElementSibling.innerText = this.value;
drawRomerlGraph();
};
const l = parseFloat(document.getElementById("outputlChange").textContent);
const container = document.getElementById("romer-lchange-visualization");
const width = container.clientWidth - margin.left - margin.right;
const height = container.clientHeight - margin.top - margin.bottom;
container.innerHTML = "";
const svg = d3
.select("#romer-lchange-visualization")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform", `translate(${margin.left}, ${margin.top})`);
let A = A0,
l_ = 0.1;
const romerData = [];
for (let t = 1; t <= Math.floor(T_MAX / 2) - 1; ++t) {
const A_t = A * (1 + z * l_ * L);
const Y_t = A_t * (1 - l_) * L;
romerData.push({ year: t, A: A_t, Y: Math.log10(Y_t) });
A = A_t;
}
for (let t = Math.floor(T_MAX / 2); t <= T_MAX; ++t) {
const A_t = A * (1 + z * l * L);
const Y_t = A_t * (1 - l) * L;
romerData.push({ year: t, A: A_t, Y: Math.log10(Y_t) });
A = A_t;
}
const x = d3.scaleLinear().domain([1, T_MAX]).range([0, width]);
svg
.append("g")
.attr("transform", `translate(0, ${height})`)
.call(d3.axisBottom(x))
.append("text")
.attr("fill", "#000")
.attr("x", width + 10)
.attr("y", -10)
.style("text-anchor", "end")
.style("font-size", "1.5em")
.text("t");
const y = d3
.scaleLinear()
.domain([0, romerData[romerData.length - 1].Y])
.range([height, 0]);
svg
.append("g")
.call(d3.axisLeft(y).ticks(10, d3.format(".1s")))
.append("text")
.attr("fill", "#000")
.attr("x", 0)
.attr("y", -10)
.style("text-anchor", "start")
.style("font-size", "1.5em")
.text("log(Y)");
svg
.append("path")
.datum(romerData)
.attr("fill", "none")
.attr("stroke", getTopicColor(urlToTopic()))
.attr("stroke-width", 2)
.attr(
"d",
d3
.line()
.x((d) => x(d.year))
.y((d) => y(d.Y)),
);
}
document.addEventListener("DOMContentLoaded", function () { document.addEventListener("DOMContentLoaded", function () {
drawSolowGraph(); drawSolowGraph();
window.onresize = drawSolowGraph; window.onresize = drawSolowGraph;
drawRomerGraph();
window.onresize = drawRomerGraph;
drawRomerlGraph();
window.onresize = drawRomerlGraph;
}); });

20
styles/graph.css
View file

@ -68,3 +68,23 @@ ul {
list-style: none; list-style: none;
margin: 0; margin: 0;
} }
.romer-table-container {
display: flex;
justify-content: center;
margin: 20px 0;
}
#romer-table {
text-align: center;
margin-top: 20px;
margin: 0;
font-size: 0.8em;
border-collapse: collapse;
}
#romer-table th,
#romer-table td {
border: 1px solid black;
text-align: center;
padding: 5px;
}