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Barrett Ruth 2024-07-30 10:00:34 -05:00
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289
posts/economics/models-of-production.html
View file

@ -68,7 +68,7 @@
>, the Solow Model describes production as follows: >, the Solow Model describes production as follows:
\[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With: \[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
</p> </p>
<ul> <ul style="list-style: unset">
<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 \(\alpha\): capital&apos;s share of output&mdash;usually
@ -141,11 +141,11 @@
<ul> <ul>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderA">\(A:\)</label> <label for="sliderSA">\(\bar{A}:\)</label>
<span id="outputA">1.00</span> <span id="outputSA">1.00</span>
<input <input
type="range" type="range"
id="sliderA" id="sliderSA"
min="0.1" min="0.1"
max="2" max="2"
step="0.01" step="0.01"
@ -155,11 +155,11 @@
</li> </li>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderD">\(d:\)</label> <label for="sliderSd">\(\bar{d}:\)</label>
<span id="outputD">0.50</span> <span id="outputSd">0.50</span>
<input <input
type="range" type="range"
id="sliderD" id="sliderSd"
min="0.01" min="0.01"
max="0.99" max="0.99"
step="0.01" step="0.01"
@ -173,11 +173,11 @@
<ul start="3"> <ul start="3">
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderS">\(s:\)</label> <label for="sliderSs">\(\bar{s}:\)</label>
<span id="outputS">0.50</span> <span id="outputSs">0.50</span>
<input <input
type="range" type="range"
id="sliderS" id="sliderSs"
min="0.01" min="0.01"
max="0.99" max="0.99"
step="0.01" step="0.01"
@ -187,11 +187,11 @@
</li> </li>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderAlpha">\(\alpha:\)</label> <label for="sliderSalpha">\(\alpha:\)</label>
<span id="outputAlpha">0.33</span> <span id="outputSalpha">0.33</span>
<input <input
type="range" type="range"
id="sliderAlpha" id="sliderSalpha"
min="0.01" min="0.01"
max="0.99" max="0.99"
step="0.01" step="0.01"
@ -316,7 +316,7 @@
The Romer Models&apos; production function can be modelled as: The Romer Models&apos; production function can be modelled as:
\[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With: \[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:
</p> </p>
<ul> <ul style="list-style: unset">
<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li> <li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
<li> <li>
\(L_{yt}\): the population working on production-facing \(L_{yt}\): the population working on production-facing
@ -375,11 +375,11 @@
<ul> <ul>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderZ">\(\bar{z}:\)</label> <label for="sliderRz">\(\bar{z}:\)</label>
<span id="outputZ">0.50</span> <span id="outputRz">0.50</span>
<input <input
type="range" type="range"
id="sliderZ" id="sliderRz"
min="0.1" min="0.1"
max="0.99" max="0.99"
step="0.01" step="0.01"
@ -389,11 +389,11 @@
</li> </li>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderL">\(\bar{L}:\)</label> <label for="sliderRL">\(\bar{L}:\)</label>
<span id="outputL">505</span> <span id="outputRL">505</span>
<input <input
type="range" type="range"
id="sliderL" id="sliderRL"
min="10" min="10"
max="1000" max="1000"
step="19" step="19"
@ -407,11 +407,11 @@
<ul start="3"> <ul start="3">
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderl">\(\bar{l}:\)</label> <label for="sliderRl">\(\bar{l}:\)</label>
<span id="outputl">0.50</span> <span id="outputRl">0.50</span>
<input <input
type="range" type="range"
id="sliderl" id="sliderRl"
min="0.01" min="0.01"
max="0.99" max="0.99"
step="0.01" step="0.01"
@ -421,15 +421,15 @@
</li> </li>
<li> <li>
<div class="slider"> <div class="slider">
<label for="sliderA0">\(\bar{A}_0:\)</label> <label for="sliderRA0">\(\bar{A}_0:\)</label>
<span id="outputA0">5000</span> <span id="outputRA0">500</span>
<input <input
type="range" type="range"
id="sliderA0" id="sliderRA0"
min="1" min="0"
max="10000" max="1000"
step="100" step="100"
value="5000" value="500"
/> />
</div> </div>
</li> </li>
@ -473,10 +473,6 @@
output can be solved the production function: \[Y_t=A_t output can be solved the production function: \[Y_t=A_t
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\] L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]
</p> </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>
<div class="fold"><h3>analysis</h3></div> <div class="fold"><h3>analysis</h3></div>
<div> <div>
@ -687,6 +683,134 @@
Expectedly, output has a positive relationship with the savings Expectedly, output has a positive relationship with the savings
rate and a negative relationship with the depreciation rate. rate and a negative relationship with the depreciation rate.
</p> </p>
<p>
Using the visualization below, we see a growth pattern similar
to that of the Romer Model. However, the Romer-Solow economy
indeed grows at a faster rate than the Romer model&mdash;I had
to cap \(\bar{L}\) at \(400\) and \(\alpha\) at \(0.4\) because
output would be
<i> too large </i> for JavaScript to contain in a number (the
graph would disappear).
</p>
<div class="graph">
<div id="romer-solow-visualization"></div>
</div>
<div class="sliders">
<div style="padding-right: 20px">
<ul>
<li>
<div class="slider">
<label for="sliderRSz">\(\bar{z}:\)</label>
<span id="outputRSz">0.50</span>
<input
type="range"
id="sliderRSz"
min="0.1"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSA0">\(A_0:\)</label>
<span id="outputRSA0">500</span>
<input
type="range"
id="sliderRSA0"
min="0"
max="1000"
step="10"
value="500"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSd">\(\bar{d}:\)</label>
<span id="outputRSd">0.50</span>
<input
type="range"
id="sliderRSd"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSs">\(\bar{s}:\)</label>
<span id="outputRSs">0.50</span>
<input
type="range"
id="sliderRSs"
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="sliderRSalpha">\(\alpha:\)</label>
<span id="outputRSalpha">0.33</span>
<input
type="range"
id="sliderRSalpha"
min="0.01"
max="0.40"
step="0.01"
value="0.33"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSl">\(\bar{l}:\)</label>
<span id="outputRSl">0.50</span>
<input
type="range"
id="sliderRSl"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSL">\(\bar{L}:\)</label>
<span id="outputRSL">200</span>
<input
type="range"
id="sliderRSL"
min="0"
max="400"
step="10"
value="200"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
Playing with the parameters, the previous mathematical findings
are validated. For example, because
\(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), only changes
in parameters \(\alpha,\bar{z},\bar{l}\), and \(\bar{L}\) affect
the growth rate of output, manifesting as the y-axis scaling
up/down on a ratio scale.
</p>
<p> <p>
However, do economics grow <i>faster</i>/<i>slower</i> the However, do economics grow <i>faster</i>/<i>slower</i> the
further <i>below</i>/<i>above</i> they are from their Balanced further <i>below</i>/<i>above</i> they are from their Balanced
@ -694,9 +818,106 @@
mathematically proven (of course), sometimes a visualization mathematically proven (of course), sometimes a visualization
helps. helps.
</p> </p>
<p>
The graph below illustrates the transition dynamics of
Romer-Solow Model. Namely, \((\bar{z}, \bar{l}, \bar{L},
\alpha)=(0.5, 0.5, 100, 0.33)\forall t&lt;t_0\), then update to
the slider values when \(t>t_0\).
</p>
<div class="graph"> <div class="graph">
<div id="romer-solow-visualization"></div> <div id="romer-solow-change-visualization"></div>
</div> </div>
<div class="sliders">
<div style="padding-right: 20px">
<ul>
<li>
<div class="slider">
<label for="sliderRSCz0">\(\bar{z}_0:\)</label>
<span id="outputRSCz0">0.50</span>
<input
type="range"
id="sliderRSCz0"
min="0.1"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSCalpha0">\(\alpha_0:\)</label>
<span id="outputRSCalpha0">0.33</span>
<input
type="range"
id="sliderRSCalpha0"
min="0.01"
max="0.54"
step="0.01"
value="0.33"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSCL0">\(\bar{L}_0:\)</label>
<span id="outputRSCL0">100</span>
<input
type="range"
id="sliderRSCL0"
min="0"
max="200"
step="10"
value="100"
/>
</div>
</li>
</ul>
</div>
<div style="padding-left: 20px">
<ul start="3">
<li>
<div class="slider">
<label for="sliderRSCl0">\(\bar{l}_0:\)</label>
<span id="outputRSCl0">0.50</span>
<input
type="range"
id="sliderRSCl0"
min="0.01"
max="0.99"
step="0.01"
value="0.50"
/>
</div>
</li>
<li>
<div class="slider">
<label for="sliderRSCt0">\(t_0:\)</label>
<span id="outputRSCt0">50</span>
<input
type="range"
id="sliderRSCt0"
min="0"
max="100"
step="1"
value="50"
/>
</div>
</li>
</ul>
</div>
</div>
<p>
Finally, it is clear that economies converge to their Balanced
Growth Path as desired&mdash;something slightly more convoluted
to prove from the complex expression for \(Y^*\) derived
earlier. For example, with an increase in \(\alpha_0\), output
grows at an increasing rate after the change, then increases at
a decreasing rate as it converges to the new higher Balanced
Growth Path. Increasing parameters \(\bar{z},\bar{l},\bar{L}\)
yield similar results, although the changes are visually less
obvious.
</p>
</div> </div>
</div> </div>
</article> </article>

326
scripts/posts/models-of-production.js
View file

@ -1,23 +1,31 @@
function setUpParameters(render, parameters, modelPrefix) {
parameters.forEach((param) => {
const slider = document.getElementById(`slider${modelPrefix}${param}`);
slider.oninput = function () {
slider.previousElementSibling.innerText = this.value;
render();
};
});
return parameters.map((param) => {
return parseFloat(
document.getElementById(`output${modelPrefix}${param}`).textContent,
);
});
}
function drawSolowGraph() { function drawSolowGraph() {
const L = 150, const L = 150,
K_MAX = 500, K_MAX = 500,
margin = { top: 20, right: 30, bottom: 20, left: 50 }; margin = { top: 20, right: 30, bottom: 20, left: 50 };
["A", "D", "S", "Alpha"].forEach((param) => { const [A, d, s, alpha] = setUpParameters(
const slider = document.getElementById(`slider${param}`); drawSolowGraph,
slider.oninput = function () { ["A", "d", "s", "alpha"],
slider.previousElementSibling.innerText = this.value; "S",
drawSolowGraph(); );
};
});
const A = parseFloat(document.getElementById("outputA").textContent),
D = parseFloat(document.getElementById("outputD").textContent),
S = parseFloat(document.getElementById("outputS").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;
const container = document.getElementById("solow-visualization"); const container = document.getElementById("solow-visualization");
const width = container.clientWidth - margin.left - margin.right; const width = container.clientWidth - margin.left - margin.right;
@ -145,11 +153,11 @@ function drawSolowGraph() {
.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"));
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
.append("line") .append("line")
.attr("x1", x(k_star)) .attr("x1", x(k_star))
.attr("y1", y((D * k_star) / S)) .attr("y1", y((d * k_star) / s))
.attr("x2", x(k_star)) .attr("x2", x(k_star))
.attr("y2", y(0)) .attr("y2", y(0))
.attr("stroke", "black") .attr("stroke", "black")
@ -200,21 +208,14 @@ const updateRomerTable = (romerData) => {
}; };
function drawRomerGraph() { function drawRomerGraph() {
const T_MAX = 100; const T_MAX = 100,
margin = { top: 20, right: 100, bottom: 20, left: 50 }; margin = { top: 20, right: 100, bottom: 20, left: 50 };
["Z", "L", "l", "A0"].forEach((param) => { const [z, L, l, A0] = setUpParameters(
const slider = document.getElementById(`slider${param}`); drawRomerGraph,
slider.oninput = function () { ["z", "L", "l", "A0"],
slider.previousElementSibling.innerText = this.value; "R",
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 container = document.getElementById("romer-visualization");
const width = container.clientWidth - margin.left - margin.right; const width = container.clientWidth - margin.left - margin.right;
@ -300,19 +301,10 @@ function drawRomerlGraph() {
const T_MAX = 100, const T_MAX = 100,
z = 0.01, z = 0.01,
L = 50, L = 50,
A0 = 50; A0 = 50,
margin = { top: 20, right: 100, bottom: 20, left: 50 }; margin = { top: 20, right: 100, bottom: 20, left: 50 };
["lChange", "t0"].forEach((param) => { const [l, t0] = setUpParameters(drawRomerlGraph, ["lChange", "t0"], "");
const slider = document.getElementById(`slider${param}`);
slider.oninput = function () {
slider.previousElementSibling.innerText = this.value;
drawRomerlGraph();
};
});
const l = parseFloat(document.getElementById("outputlChange").textContent),
t0 = parseFloat(document.getElementById("outputt0").textContent);
const container = document.getElementById("romer-lchange-visualization"); const container = document.getElementById("romer-lchange-visualization");
const width = container.clientWidth - margin.left - margin.right; const width = container.clientWidth - margin.left - margin.right;
@ -388,7 +380,6 @@ function drawRomerlGraph() {
.y((d) => y(d.Y)), .y((d) => y(d.Y)),
); );
console.log(t0)
svg svg
.append("line") .append("line")
.attr("x1", x(t0)) .attr("x1", x(t0))
@ -439,6 +430,249 @@ function drawRomerlGraph() {
katex.render("log_{10}Y", document.querySelector(".romer-changel-y")); katex.render("log_{10}Y", document.querySelector(".romer-changel-y"));
} }
function calculateRomerSolowData(
T_MAX,
L,
l,
A0,
alpha,
s,
d,
z,
t0 = Infinity,
L0,
l0,
alpha0,
z0,
) {
let A = A0,
K_t = 1,
romerSolowData = [];
for (let t = 1; t <= T_MAX; ++t) {
if (t > t0) {
alpha = alpha0;
z = z0;
l = l0;
L = L0;
}
const Y_t = A * Math.pow(K_t, alpha) * Math.pow((1 - l) * L, 1 - alpha);
const A_t = A * (1 + z * l * L);
K_t = K_t + s * Y_t - d * K_t;
romerSolowData.push({ year: t, A: A_t, K: K_t, Y: Math.log10(Y_t) });
A = A_t;
}
return romerSolowData;
}
function drawRomerSolowGraph() {
const T_MAX = 100,
margin = { top: 20, right: 100, bottom: 20, left: 50 };
const [z, l, L, A0, s, d, alpha] = setUpParameters(
drawRomerSolowGraph,
["z", "l", "L", "A0", "s", "d", "alpha"],
"RS",
);
const container = document.getElementById("romer-solow-visualization");
const width = container.clientWidth - margin.left - margin.right;
const height = container.clientHeight - margin.top - margin.bottom;
container.innerHTML = "";
const svg = d3
.select("#romer-solow-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})`);
const romerSolowData = calculateRomerSolowData(
T_MAX,
L,
l,
A0,
alpha,
s,
d,
z,
);
const x = d3.scaleLinear().domain([1, T_MAX]).range([0, width]);
svg
.append("g")
.attr("transform", `translate(0, ${height})`)
.call(d3.axisBottom(x))
.append("text")
.attr("fill", "#000")
.attr("x", width + 10)
.attr("y", -10)
.style("text-anchor", "end")
.style("font-size", "1.5em")
.text("t");
const y = d3
.scaleLinear()
.domain([0, romerSolowData[romerSolowData.length - 1].Y])
.range([height, 0]);
svg
.append("g")
.call(d3.axisLeft(y).ticks(10, d3.format(".1s")))
.append("text")
.attr("fill", "#000")
.attr("x", 0)
.attr("y", -10)
.style("text-anchor", "start")
.style("font-size", "1.5em")
.text("log(Y)");
svg
.append("path")
.datum(romerSolowData)
.attr("fill", "none")
.attr("stroke", getTopicColor(urlToTopic()))
.attr("stroke-width", 2)
.attr(
"d",
d3
.line()
.x((d) => x(d.year))
.y((d) => y(d.Y)),
);
svg
.append("foreignObject")
.attr("width", "4em")
.attr("height", "2em")
.attr("x", x(T_MAX))
.attr("y", y(romerSolowData[T_MAX - 1].Y))
.append("xhtml:body")
.style("font-size", "0.75em")
.html(`<div class="romer-solow-visualization-y"></div>`);
katex.render(
"log_{10}Y",
document.querySelector(".romer-solow-visualization-y"),
);
}
function drawRomerSolowChangeGraph() {
const T_MAX = 100,
margin = { top: 20, right: 100, bottom: 20, left: 50 },
s = 0.2,
d = 0.2,
A0 = 50,
alpha = 0.33,
l = 0.5,
L = 100,
z = 0.5;
const [z0, l0, L0, alpha0, t0] = setUpParameters(
drawRomerSolowChangeGraph,
["z0", "l0", "L0", "alpha0", "t0"],
"RSC",
);
const container = document.getElementById("romer-solow-change-visualization");
const width = container.clientWidth - margin.left - margin.right;
const height = container.clientHeight - margin.top - margin.bottom;
container.innerHTML = "";
const svg = d3
.select("#romer-solow-change-visualization")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform", `translate(${margin.left}, ${margin.top})`);
const romerSolowData = calculateRomerSolowData(
T_MAX,
L,
l,
A0,
alpha,
s,
d,
z,
t0,
L0,
l0,
alpha0,
z0,
);
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, romerSolowData[romerSolowData.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(romerSolowData)
.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("line")
.attr("x1", x(t0))
.attr("y1", y(romerSolowData[T_MAX - 1].Y))
.attr("x2", x(t0))
.attr("y2", height)
.attr("stroke", "black")
.attr("stroke-width", 1)
.attr("stroke-dasharray", "4");
svg
.append("foreignObject")
.attr("width", "4em")
.attr("height", "2em")
.attr("x", x(T_MAX))
.attr("y", y(romerSolowData[T_MAX - 1].Y))
.append("xhtml:body")
.style("font-size", "0.75em")
.html(`<div class="romer-solow-change-visualization-y"></div>`);
katex.render(
"log_{10}Y",
document.querySelector(".romer-solow-change-visualization-y"),
);
}
document.addEventListener("DOMContentLoaded", function () { document.addEventListener("DOMContentLoaded", function () {
drawSolowGraph(); drawSolowGraph();
window.onresize = drawSolowGraph; window.onresize = drawSolowGraph;
@ -448,4 +682,10 @@ document.addEventListener("DOMContentLoaded", function () {
drawRomerlGraph(); drawRomerlGraph();
window.onresize = drawRomerlGraph; window.onresize = drawRomerlGraph;
drawRomerSolowGraph();
window.onresize = drawRomerSolowGraph;
drawRomerSolowChangeGraph();
window.onresize = drawRomerSolowChangeGraph();
}); });