feat(posts): models of production solow model
This commit is contained in:
parent
cae425da1a
commit
26815a522b
3 changed files with 346 additions and 10 deletions
|
|
@ -6,6 +6,7 @@
|
||||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||||
<link rel="stylesheet" href="/styles/common.css" />
|
<link rel="stylesheet" href="/styles/common.css" />
|
||||||
<link rel="stylesheet" href="/styles/post.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="icon" type="image/webp" href="/public/logo.webp" />
|
||||||
<link rel="stylesheet" href="/public/katex/katex.css" />
|
<link rel="stylesheet" href="/public/katex/katex.css" />
|
||||||
<script defer src="/public/katex/katex.js"></script>
|
<script defer src="/public/katex/katex.js"></script>
|
||||||
|
|
@ -17,7 +18,7 @@
|
||||||
<script defer src="/public/d3.js"></script>
|
<script defer src="/public/d3.js"></script>
|
||||||
<title>Barrett Ruth</title>
|
<title>Barrett Ruth</title>
|
||||||
</head>
|
</head>
|
||||||
<body class="graph">
|
<body class="graph-background">
|
||||||
<header>
|
<header>
|
||||||
<a
|
<a
|
||||||
href="/"
|
href="/"
|
||||||
|
|
@ -74,7 +75,9 @@
|
||||||
</li>
|
</li>
|
||||||
<li>
|
<li>
|
||||||
Capital accumulates according to investment \(I_t\) and
|
Capital accumulates according to investment \(I_t\) and
|
||||||
depreciation \(\bar{d}\), beginning with \(K_0\)
|
depreciation \(\bar{d}\), beginning with \(K_0\) (often called
|
||||||
|
the
|
||||||
|
<u>Law of Capital Motion</u>)
|
||||||
</li>
|
</li>
|
||||||
<li>Labor \(L_t\) is time-independent</li>
|
<li>Labor \(L_t\) is time-independent</li>
|
||||||
<li>
|
<li>
|
||||||
|
|
@ -87,13 +90,13 @@
|
||||||
the Solow Model:
|
the Solow Model:
|
||||||
</p>
|
</p>
|
||||||
<div style="display: flex; justify-content: center">
|
<div style="display: flex; justify-content: center">
|
||||||
<div style="margin: 0 20px">
|
<div style="padding-right: 50px">
|
||||||
<ol>
|
<ol>
|
||||||
<li>\(C_t + I_t = Y_t\)</li>
|
<li>\(C_t + I_t = Y_t\)</li>
|
||||||
<li>\(\Delta K_{t+1} = I_t - \bar{d} K_t\)</li>
|
<li>\(\Delta K_{t+1} = I_t - \bar{d} K_t\)</li>
|
||||||
</ol>
|
</ol>
|
||||||
</div>
|
</div>
|
||||||
<div>
|
<div style="padding-left: 50px">
|
||||||
<ol start="3">
|
<ol start="3">
|
||||||
<li>\(L_t = \bar{L}\)</li>
|
<li>\(L_t = \bar{L}\)</li>
|
||||||
<li>\(I_t = \bar{s} Y_t\)</li>
|
<li>\(I_t = \bar{s} Y_t\)</li>
|
||||||
|
|
@ -111,14 +114,105 @@
|
||||||
</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}}
|
\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\).
|
||||||
\bar{L_t}^{\frac{2}{3}}\). Utilizing this simplification and its
|
Utilizing this simplification and its graphical representation
|
||||||
graphical representation below, output is clearly characterized by
|
below, output is clearly characterized by the cube root of
|
||||||
the cube root of capital:
|
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="1"
|
||||||
|
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="1"
|
||||||
|
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="1"
|
||||||
|
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>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
<div class="fold"><h3>conclusions</h3></div>
|
<div class="fold"><h3>analysis</h3></div>
|
||||||
<p>hello conclusions</p>
|
<p>discuss limitations</p>
|
||||||
<h2>romer</h2>
|
<h2>romer</h2>
|
||||||
<h2>romer-solow</h2>
|
<h2>romer-solow</h2>
|
||||||
</article>
|
</article>
|
||||||
|
|
@ -126,5 +220,6 @@
|
||||||
</main>
|
</main>
|
||||||
<script src="/scripts/common.js"></script>
|
<script src="/scripts/common.js"></script>
|
||||||
<script src="/scripts/post.js"></script>
|
<script src="/scripts/post.js"></script>
|
||||||
|
<script src="/scripts/posts/models-of-production.js"></script>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
|
|
||||||
171
scripts/posts/models-of-production.js
Normal file
171
scripts/posts/models-of-production.js
Normal file
|
|
@ -0,0 +1,171 @@
|
||||||
|
function drawSolowGraph() {
|
||||||
|
const L = 150,
|
||||||
|
K_MAX = 500,
|
||||||
|
margin = { top: 20, right: 30, bottom: 20, left: 50 };
|
||||||
|
|
||||||
|
["A", "D", "S", "Alpha"].forEach((param) => {
|
||||||
|
const slider = document.getElementById(`slider${param}`);
|
||||||
|
slider.oninput = function () {
|
||||||
|
slider.previousElementSibling.innerText = this.value;
|
||||||
|
drawSolowGraph();
|
||||||
|
};
|
||||||
|
});
|
||||||
|
|
||||||
|
const A = document.getElementById("outputA").textContent,
|
||||||
|
D = document.getElementById("outputD").textContent,
|
||||||
|
S = document.getElementById("outputS").textContent,
|
||||||
|
alpha = document.getElementById("outputAlpha").textContent;
|
||||||
|
const solowOutput = (K) => A * Math.pow(K, alpha) * Math.pow(L, 1 - alpha);
|
||||||
|
const solowDepreciation = (K) => D * K;
|
||||||
|
const solowInvestment = (Y) => S * Y;
|
||||||
|
|
||||||
|
const container = document.getElementById("solow-visualization");
|
||||||
|
const width = container.clientWidth - margin.left - margin.right;
|
||||||
|
const height = container.clientHeight - margin.top - margin.bottom;
|
||||||
|
|
||||||
|
container.innerHTML = "";
|
||||||
|
|
||||||
|
const svg = d3
|
||||||
|
.select("#solow-visualization")
|
||||||
|
.append("svg")
|
||||||
|
.attr("width", width + margin.left + margin.right)
|
||||||
|
.attr("height", height + margin.top + margin.bottom)
|
||||||
|
.append("g")
|
||||||
|
.attr("transform", `translate(${margin.left}, ${margin.top})`);
|
||||||
|
|
||||||
|
const x = d3.scaleLinear().domain([0, K_MAX]).range([0, width]);
|
||||||
|
const xAxis = svg
|
||||||
|
.append("g")
|
||||||
|
.attr("transform", `translate(0, ${height})`)
|
||||||
|
.call(d3.axisBottom(x));
|
||||||
|
|
||||||
|
xAxis
|
||||||
|
.append("text")
|
||||||
|
.attr("fill", "#000")
|
||||||
|
.attr("x", width + 10)
|
||||||
|
.attr("y", -10)
|
||||||
|
.style("text-anchor", "end")
|
||||||
|
.style("font-size", "1.5em")
|
||||||
|
.text("K");
|
||||||
|
|
||||||
|
const Y_MAX = solowOutput(K_MAX) + K_MAX / 10;
|
||||||
|
const y = d3.scaleLinear().domain([0, Y_MAX]).range([height, 0]);
|
||||||
|
const yAxis = svg.append("g").call(d3.axisLeft(y));
|
||||||
|
|
||||||
|
yAxis
|
||||||
|
.append("text")
|
||||||
|
.attr("fill", "#000")
|
||||||
|
.attr("x", 0)
|
||||||
|
.attr("y", -10)
|
||||||
|
.style("text-anchor", "start")
|
||||||
|
.style("font-size", "1.5em")
|
||||||
|
.text("Y");
|
||||||
|
|
||||||
|
const outputData = Array.from({ length: K_MAX }, (_, k) => ({
|
||||||
|
K: k,
|
||||||
|
Y: solowOutput(k),
|
||||||
|
}));
|
||||||
|
svg
|
||||||
|
.append("path")
|
||||||
|
.datum(outputData)
|
||||||
|
.attr("fill", "none")
|
||||||
|
.attr("stroke", getTopicColor(urlToTopic()))
|
||||||
|
.attr("stroke-width", 2)
|
||||||
|
.attr(
|
||||||
|
"d",
|
||||||
|
d3
|
||||||
|
.line()
|
||||||
|
.x((d) => x(d.K))
|
||||||
|
.y((d) => y(d.Y)),
|
||||||
|
);
|
||||||
|
svg
|
||||||
|
.append("foreignObject")
|
||||||
|
.attr("width", "2em")
|
||||||
|
.attr("height", "2em")
|
||||||
|
.attr("x", x(K_MAX))
|
||||||
|
.attr("y", y(outputData[K_MAX - 1].Y))
|
||||||
|
.append("xhtml:body")
|
||||||
|
.style("font-size", "0.75em")
|
||||||
|
.html(`<div class="solow-visualization-y"></div>`);
|
||||||
|
katex.render("Y", document.querySelector(".solow-visualization-y"), {
|
||||||
|
throwOnError: false,
|
||||||
|
});
|
||||||
|
|
||||||
|
const depreciationData = Array.from({ length: K_MAX }, (_, k) => ({
|
||||||
|
K: k,
|
||||||
|
Y: solowDepreciation(k),
|
||||||
|
}));
|
||||||
|
svg
|
||||||
|
.append("path")
|
||||||
|
.datum(depreciationData)
|
||||||
|
.attr("fill", "none")
|
||||||
|
.attr("stroke", "red")
|
||||||
|
.attr("stroke-width", 2)
|
||||||
|
.attr(
|
||||||
|
"d",
|
||||||
|
d3
|
||||||
|
.line()
|
||||||
|
.x((d) => x(d.K))
|
||||||
|
.y((d) => y(d.Y)),
|
||||||
|
);
|
||||||
|
|
||||||
|
svg
|
||||||
|
.append("foreignObject")
|
||||||
|
.attr("width", "2em")
|
||||||
|
.attr("height", "2em")
|
||||||
|
.attr("x", x(K_MAX))
|
||||||
|
.attr("y", y(depreciationData[K_MAX - 1].Y))
|
||||||
|
.append("xhtml:body")
|
||||||
|
.style("font-size", "0.75em")
|
||||||
|
.html(`<div class="solow-visualization-d"></div>`);
|
||||||
|
katex.render("\\bar{d}K", document.querySelector(".solow-visualization-d"), {
|
||||||
|
throwOnError: false,
|
||||||
|
});
|
||||||
|
|
||||||
|
const investmentData = outputData.map((d) => ({
|
||||||
|
K: d.K,
|
||||||
|
Y: solowInvestment(d.Y),
|
||||||
|
}));
|
||||||
|
svg
|
||||||
|
.append("path")
|
||||||
|
.datum(investmentData)
|
||||||
|
.attr("fill", "none")
|
||||||
|
.attr("stroke", "purple")
|
||||||
|
.attr("stroke-width", 2)
|
||||||
|
.attr(
|
||||||
|
"d",
|
||||||
|
d3
|
||||||
|
.line()
|
||||||
|
.x((d) => x(d.K))
|
||||||
|
.y((d) => y(d.Y)),
|
||||||
|
);
|
||||||
|
|
||||||
|
svg
|
||||||
|
.append("foreignObject")
|
||||||
|
.attr("width", "1em")
|
||||||
|
.attr("height", "2em")
|
||||||
|
.attr("x", x(K_MAX))
|
||||||
|
.attr("y", y(investmentData[K_MAX - 1].Y))
|
||||||
|
.append("xhtml:body")
|
||||||
|
.style("font-size", "0.75em")
|
||||||
|
.html(`<div class="solow-visualization-i"></div>`);
|
||||||
|
katex.render("I", document.querySelector(".solow-visualization-i"), {
|
||||||
|
throwOnError: false,
|
||||||
|
});
|
||||||
|
|
||||||
|
const k_star = L * Math.pow((S * A) / D, 1 / (1 - alpha));
|
||||||
|
svg
|
||||||
|
.append("line")
|
||||||
|
.attr("x1", x(k_star))
|
||||||
|
.attr("y1", y((D * k_star) / S))
|
||||||
|
.attr("x2", x(k_star))
|
||||||
|
.attr("y2", y(0))
|
||||||
|
.attr("stroke", "black")
|
||||||
|
.attr("stroke-width", 1)
|
||||||
|
.attr("stroke-dasharray", "5,5");
|
||||||
|
}
|
||||||
|
|
||||||
|
document.addEventListener("DOMContentLoaded", function () {
|
||||||
|
drawSolowGraph();
|
||||||
|
window.onresize = drawSolowGraph;
|
||||||
|
});
|
||||||
70
styles/graph.css
Normal file
70
styles/graph.css
Normal file
|
|
@ -0,0 +1,70 @@
|
||||||
|
.graph {
|
||||||
|
height: 50vh;
|
||||||
|
width: 100%;
|
||||||
|
display: flex;
|
||||||
|
justify-content: center;
|
||||||
|
margin: 50px 0;
|
||||||
|
}
|
||||||
|
|
||||||
|
.graph div {
|
||||||
|
height: 100%;
|
||||||
|
width: 60%;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider {
|
||||||
|
display: flex;
|
||||||
|
align-items: center;
|
||||||
|
justify-content: space-between;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider label {
|
||||||
|
margin-right: 10px;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider input {
|
||||||
|
margin-left: 20px;
|
||||||
|
-webkit-appearance: none;
|
||||||
|
appearance: none;
|
||||||
|
width: 100%;
|
||||||
|
height: 2px;
|
||||||
|
background: black;
|
||||||
|
cursor: pointer;
|
||||||
|
outline: none;
|
||||||
|
transform: translateY(-50%);
|
||||||
|
width: 150px;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider input::-webkit-slider-thumb {
|
||||||
|
-webkit-appearance: none;
|
||||||
|
width: 2px;
|
||||||
|
height: 15px;
|
||||||
|
background: black;
|
||||||
|
cursor: col-resize;
|
||||||
|
position: relative;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider input::-moz-range-thumb {
|
||||||
|
width: 2px;
|
||||||
|
height: 15px;
|
||||||
|
background: black;
|
||||||
|
cursor: col-resize;
|
||||||
|
position: relative;
|
||||||
|
}
|
||||||
|
|
||||||
|
.slider input::-webkit-slider-runnable-track,
|
||||||
|
.slider input::-moz-range-track {
|
||||||
|
width: 100%;
|
||||||
|
height: 2px;
|
||||||
|
background: black;
|
||||||
|
border: none;
|
||||||
|
}
|
||||||
|
|
||||||
|
.sliders {
|
||||||
|
display: flex;
|
||||||
|
justify-content: center;
|
||||||
|
}
|
||||||
|
|
||||||
|
ul {
|
||||||
|
list-style: none;
|
||||||
|
margin: 0;
|
||||||
|
}
|
||||||
Loading…
Add table
Add a link
Reference in a new issue