feat: post refactor + fix title
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
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async
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||||
></script>
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||||
<title>Barrett Ruth</title>
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||||
<title>competitive programming log</title>
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||||
</head>
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||||
<body class="graph-background">
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||||
<site-header></site-header>
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||||
<main class="main">
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||||
<div class="post-container">
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||||
<header class="post-header">
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||||
<h1 class="post-title">Competitive Programming Log</h1>
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<h1 class="post-title">competitive programming log</h1>
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||||
</header>
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||||
<article class="post-article">
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<h2>
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||||
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@ -14,7 +14,7 @@
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
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||||
async
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||||
></script>
|
||||
<title>Barrett Ruth</title>
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||||
<title>extreme circular buffer</title>
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||||
</head>
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||||
<body class="graph-background">
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||||
<site-header></site-header>
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||||
|
|
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|||
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@ -14,14 +14,14 @@
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|||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
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||||
async
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||||
></script>
|
||||
<title>Barrett Ruth</title>
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||||
<title>leetcode daily</title>
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||||
</head>
|
||||
<body class="graph-background">
|
||||
<site-header></site-header>
|
||||
<main class="main">
|
||||
<div class="post-container">
|
||||
<header class="post-header">
|
||||
<h1 class="post-title">Leetcode Daily</h1>
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||||
<h1 class="post-title">leetcode daily</h1>
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||||
</header>
|
||||
<article class="post-article">
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||||
<h2>
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913
posts/algorithms/models-of-production.html
Normal file
913
posts/algorithms/models-of-production.html
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<!doctype html>
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<html lang="en">
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<head>
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||||
<meta charset="UTF-8" />
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<meta http-equiv="X-UA-Compatible" content="IE=edge" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
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||||
<link rel="stylesheet" href="/styles/common.css" />
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||||
<link rel="stylesheet" href="/styles/post.css" />
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||||
<link rel="stylesheet" href="/styles/graph.css" />
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||||
<link rel="icon" type="image/webp" href="/public/logo.webp" />
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<script
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||||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
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async
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||||
></script>
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<script src="/public/d3.js"></script>
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<title>models of production</title>
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</head>
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<body class="graph-background">
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<site-header></site-header>
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<main class="main">
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<div class="post-container">
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<header class="post-header">
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<h1 class="post-title">models of production</h1>
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<p class="post-meta">
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<time datetime="2024-06-22">22/06/2024</time>
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</p>
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</header>
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<article class="post-article">
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<p>
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This post offers a basic introduction to the Solow, Romer, and
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Romer-Solow economic models, as taught by
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<a target="blank" href="https://www.vladimirsmirnyagin.com/"
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>Vladimir Smirnyagin</a
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>
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and assisted by
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<a target="blank" href="https://www.donghyunsuh.com/"
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>Donghyun Suh</a
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>
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in Intermediate Macroeconomics (ECON 3020) during the Spring
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semester of 2024 at the University of Virginia.
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</p>
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<h2>solow</h2>
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<div>
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<h3>introduction</h3>
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<div>
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<p>
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The Solow Model is an economic model of production that
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incorporates the incorporates the idea of capital accumulation.
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Based on the
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<a
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target="blank"
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href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function"
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>Cobb-Douglas production function</a
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>, the Solow Model describes production as follows:
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\[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
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</p>
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<ul style="list-style: unset">
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<li>\(\bar{A}\): total factor productivity (TFP)</li>
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<li>
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\(\alpha\): capital's share of output—usually
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\(1/3\) based on
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<a target="blank" href="https://arxiv.org/pdf/1105.2123"
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>empirical data</a
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>
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</li>
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</ul>
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<p>
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In this simple model, the following statements describe the
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economy:
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</p>
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<ol>
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<li>
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Output is either saved or consumed; in other words, savings
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equals investment
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</li>
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<li>
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Capital accumulates according to investment \(I_t\) and
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depreciation \(\bar{d}\), beginning with \(K_0\) (often called
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the
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<u>Law of Capital Motion</u>)
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</li>
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<li>Labor \(L_t\) is time-independent</li>
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<li>
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A savings rate \(\bar{s}\) describes the invested portion of
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total output
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</li>
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</ol>
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<p>
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Including the production function, these four ideas encapsulate
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the Solow Model:
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</p>
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<div style="display: flex; justify-content: center">
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<div style="padding-right: 50px">
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<ol>
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<li>\(C_t + I_t = Y_t\)</li>
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<li>\(\Delta K_{t+1} = I_t - \bar{d} K_t\)</li>
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</ol>
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</div>
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<div style="padding-left: 50px">
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<ol start="3">
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<li>\(L_t = \bar{L}\)</li>
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<li>\(I_t = \bar{s} Y_t\)</li>
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</ol>
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</div>
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</div>
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</div>
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<h3>solving the model</h3>
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<div>
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<p>
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Visualizing the model, namely output as a function of capital,
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provides helpful intuition before solving it.
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</p>
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<p>
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Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that
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\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
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\bar{L}^{\frac{2}{3}}\). Utilizing this simplification and its
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graphical representation below, output is clearly characterized
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by the cube root of capital:
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</p>
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<div class="graph">
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<div id="solow-visualization"></div>
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</div>
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<div class="sliders">
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<div style="padding-right: 20px">
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<ul>
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<li>
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<div class="slider">
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<label for="sliderSA">\(\bar{A}:\)</label>
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<span id="outputSA">1.00</span>
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<input
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type="range"
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id="sliderSA"
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min="0.1"
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max="2"
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step="0.01"
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value="1"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderSd">\(\bar{d}:\)</label>
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<span id="outputSd">0.50</span>
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<input
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type="range"
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id="sliderSd"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.50"
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/>
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</div>
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</li>
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</ul>
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</div>
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<div style="padding-left: 20px">
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<ul start="3">
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<li>
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<div class="slider">
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<label for="sliderSs">\(\bar{s}:\)</label>
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<span id="outputSs">0.50</span>
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<input
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type="range"
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id="sliderSs"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.50"
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/>
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</div>
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</li>
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<li>
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<div class="slider">
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<label for="sliderSalpha">\(\alpha:\)</label>
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<span id="outputSalpha">0.33</span>
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<input
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type="range"
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id="sliderSalpha"
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min="0.01"
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max="0.99"
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step="0.01"
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value="0.33"
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/>
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</div>
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</li>
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</ul>
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</div>
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</div>
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<p>
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When investment is completely disincentivized by depreciation
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(in other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates
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at a so-called "steady-state" with equilibrium
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\((K_t,Y_t)=(K_t^*,Y_t^*)\).
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</p>
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<p>
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Using this equilibrium condition, it follows that:
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\[Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \rightarrow
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\bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}\]
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\[\rightarrow
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K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}\]
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\[\rightarrow
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Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}\]
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||||
</p>
|
||||
<p>
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Thus, the equilibrium intensive form (output per worker) of both
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capital and output are summarized as follows:
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\[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}})
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=((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha},
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\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\]
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</p>
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</div>
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<h3>analysis</h3>
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<div>
|
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<p>
|
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Using both mathematical intuition and manipulating the
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visualization above, we find that:
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</p>
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<ul style="list-style: unset">
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<li>
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\(\bar{A}\) has a positive relationship with steady-state
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output
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</li>
|
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<li>
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Capital is influenced by workforce size, TFP, and savings rate
|
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</li>
|
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<li>
|
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Capital output share's \(\alpha\) impact on output is twofold:
|
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<ol>
|
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<li>Directly through capital quantity</li>
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<li>Indirectly through TFP</li>
|
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</ol>
|
||||
</li>
|
||||
<li>
|
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Large deviations in capital from steady-state \(K^*\) induce
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net investments of larger magnitude, leading to an accelerated
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reversion to the steady-state
|
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</li>
|
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<li>
|
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Economies stagnate at the steady-state
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\((K^*,Y^*)\)—this model provides no avenues for
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long-run growth.
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</li>
|
||||
</ul>
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<p>
|
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Lastly (and perhaps most importantly), exogenous parameters
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\(\bar{s}, \bar{d}\), and \(\bar{A}\) all have immense
|
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ramifications on economic status. For example, comparing the
|
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difference in country \(C_1\)'s output versus
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\(C_2\)'s using the Solow Model, we find that a difference
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in economic performance can only be explained by these factors:
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\[
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\frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha}
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\]
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</p>
|
||||
<p>
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We see that TFP is more important in explaining the differences
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in per capital output
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(\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)).
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<!-- TODO: poor phrasing -->
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Notably, the Solow Model does not give any insights into how to
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alter the most important predictor of output, TFP.
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</p>
|
||||
</div>
|
||||
</div>
|
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<h2>romer</h2>
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<div>
|
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<h3>introduction</h3>
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<div>
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<p>How, then, can we address these shortcomings?</p>
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<p>
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The Romer Model provides an answer by both modeling ideas
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\(A_t\) (analagous to TFP in the Solow model) endogenously and
|
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utilizing them to provide a justification for sustained long-run
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growth.
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</p>
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<p>The Model divides the world into two parts:</p>
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<ul style="list-style: unset">
|
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<li>
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<u>Objects</u>: finite resources, like capital and labor in
|
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the Solow Model
|
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</li>
|
||||
<li>
|
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<u>Ideas</u>: infinite,
|
||||
<a
|
||||
target="blank"
|
||||
href="https://en.wikipedia.org/wiki/Rivalry_(economics)"
|
||||
>non-rivalrous</a
|
||||
>
|
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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 style="list-style: unset">
|
||||
<li>\(A_t\): the amount of ideas \(A\) in period \(t\)</li>
|
||||
<li>
|
||||
\(L_{yt}\): the population working on production-facing
|
||||
(output-driving) tasks
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||||
</li>
|
||||
</ul>
|
||||
<p>
|
||||
Assuming \(L_t=\bar{L}\) people work in the economy, a
|
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proportion \(\bar{l}\) of the population focuses on making
|
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ideas: \(L_{at}=\bar{l}\bar{L}\rightarrow
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L_{yt}=(1-\bar{l})\bar{L}\).
|
||||
</p>
|
||||
<!-- TODO: footnotes - dynamic JS? -->
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<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="sliderRz">\(\bar{z}:\)</label>
|
||||
<span id="outputRz">0.50</span>
|
||||
<input
|
||||
type="range"
|
||||
id="sliderRz"
|
||||
min="0.1"
|
||||
max="0.99"
|
||||
step="0.01"
|
||||
value="0.50"
|
||||
/>
|
||||
</div>
|
||||
</li>
|
||||
<li>
|
||||
<div class="slider">
|
||||
<label for="sliderRL">\(\bar{L}:\)</label>
|
||||
<span id="outputRL">505</span>
|
||||
<input
|
||||
type="range"
|
||||
id="sliderRL"
|
||||
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="sliderRl">\(\bar{l}:\)</label>
|
||||
<span id="outputRl">0.50</span>
|
||||
<input
|
||||
type="range"
|
||||
id="sliderRl"
|
||||
min="0.01"
|
||||
max="0.99"
|
||||
step="0.01"
|
||||
value="0.50"
|
||||
/>
|
||||
</div>
|
||||
</li>
|
||||
<li>
|
||||
<div class="slider">
|
||||
<label for="sliderRA0">\(\bar{A}_0:\)</label>
|
||||
<span id="outputRA0">500</span>
|
||||
<input
|
||||
type="range"
|
||||
id="sliderRA0"
|
||||
min="0"
|
||||
max="1000"
|
||||
step="100"
|
||||
value="500"
|
||||
/>
|
||||
</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>
|
||||
<h3>solving the model</h3>
|
||||
<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: \[Y_t=A_t
|
||||
L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]
|
||||
</p>
|
||||
</div>
|
||||
<h3>analysis</h3>
|
||||
<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
|
||||
constant "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 via 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">\(\bar{l}_1:\)</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 style="padding-left: 20px">
|
||||
<ul start="3">
|
||||
<li>
|
||||
<div class="slider">
|
||||
<label for="slidert0">\(t_0:\)</label>
|
||||
<span id="outputt0">50</span>
|
||||
<input
|
||||
type="range"
|
||||
id="slidert0"
|
||||
min="1"
|
||||
max="99"
|
||||
step="1"
|
||||
value="50"
|
||||
/>
|
||||
</div>
|
||||
</li>
|
||||
</ul>
|
||||
</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—one day I hope
|
||||
we'll have an answer.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<h2>romer-solow</h2>
|
||||
<div>
|
||||
<h3>introduction</h3>
|
||||
<div>
|
||||
<p>
|
||||
While the Romer Model provides an avenue for long-run economic
|
||||
growth, it is anything but realistic—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'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's ideas and the
|
||||
Solow Model's capital stock could yield the desired result.
|
||||
Intuitively, incorporating capital into output via the Solow
|
||||
Model's production function, as well as including the
|
||||
<u>Law of Capital Motion</u> seems like one way to legitimately
|
||||
create this so-called "Romer-Solow" model:
|
||||
</p>
|
||||
<div style="display: flex; justify-content: center">
|
||||
<div style="padding-right: 50px">
|
||||
<ol>
|
||||
<li>\(Y_t=A_t K_t^\alpha L_{yt}^{1-\alpha}\)</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>
|
||||
<h3>solving the model</h3>
|
||||
<div>
|
||||
<p>
|
||||
Based on the the motivations for creating this model, it is more
|
||||
useful to first analyze the growth rates of equilibrium long run
|
||||
output \(Y_t^*\).
|
||||
</p>
|
||||
<p>
|
||||
According to the production function, \[g_Y=g_A+\alpha
|
||||
g_K+(1-\alpha)g_{L_y}\]
|
||||
</p>
|
||||
<p>
|
||||
From previous analysis it was found that
|
||||
\(g_A=\bar{z}\bar{l}\bar{L}\).
|
||||
</p>
|
||||
<p>
|
||||
Based on the <u>Law of Capital Motion</u>, \[g_K=\frac{\Delta
|
||||
K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\]
|
||||
</p>
|
||||
<p>
|
||||
Because growth rates are constant on the Balanced Growth Path,
|
||||
\(g_K\) must be constant as well. Thus, so is
|
||||
\(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that
|
||||
\(g_K^*=g_Y^*\).
|
||||
</p>
|
||||
<p>
|
||||
The model assumes population is constant, so
|
||||
\(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well.
|
||||
</p>
|
||||
<p>
|
||||
Combining these terms, we find:
|
||||
\[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\]
|
||||
\[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\]
|
||||
</p>
|
||||
<p>
|
||||
Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\)
|
||||
must hold on a balanced growth path.
|
||||
</p>
|
||||
<p>
|
||||
Invoking the <u>Law of Capital Motion</u> with magic chants,
|
||||
\[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow
|
||||
K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\]
|
||||
</p>
|
||||
<p>
|
||||
Isolating \(Y_t^*\), \[Y_t^*=A_t^*
|
||||
(\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha
|
||||
({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow
|
||||
{Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\]
|
||||
</p>
|
||||
<p>
|
||||
Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a
|
||||
final expression for the Balanced Growth Path output as a
|
||||
function of the endogenous parameters and time is obtained: \[
|
||||
Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\]
|
||||
</p>
|
||||
</div>
|
||||
<h3>analysis</h3>
|
||||
<div>
|
||||
<p>
|
||||
First looking at the growth rate of output,
|
||||
\(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), idea-driving
|
||||
factors and an increased allocation of labor to output increase
|
||||
the equilibrium Balanced Growth Path—the
|
||||
<i>level</i> of long-run growth. Thus, this model captures the
|
||||
influences of both capital and ideas on economic growth.
|
||||
<!-- TODO: t_0 graph break in romer-model and post -->
|
||||
</p>
|
||||
<p>
|
||||
Looking at \(Y_t^*\), ideas have both a direct and indirect
|
||||
effect on output. Firstly, ideas raise output because they
|
||||
increase productivity (directly); second, with the introduction
|
||||
of capital stock, ideas also increase capital, in turn
|
||||
increasing output further (indirectly). Mathematically, this is
|
||||
evident in both instances of \(g_A^*\) in the formula for output
|
||||
\(Y_t^*\)—note that
|
||||
\(\frac{1}{1-\alpha},\frac{\alpha}{1-\alpha}>0\) for any
|
||||
\(\alpha\in(0,1)\), so \(\frac{d}{dg_A^*}Y_t^*>0\).
|
||||
</p>
|
||||
<p>
|
||||
Expectedly, output has a positive relationship with the savings
|
||||
rate and a negative relationship with the depreciation rate.
|
||||
</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—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>
|
||||
However, do economics grow <i>faster</i>/<i>slower</i> the
|
||||
further <i>below</i>/<i>above</i> they are from their Balanced
|
||||
Growth Path, as initially desired? While this can be
|
||||
mathematically proven (of course), sometimes a visualization
|
||||
helps.
|
||||
</p>
|
||||
<p>
|
||||
The graph below illustrates the transition dynamics of
|
||||
Romer-Solow Model. Namely, \((\bar{z}, \bar{l}, \bar{L},
|
||||
\alpha)=(0.5, 0.5, 100, 0.33)\forall t<t_0\), then update to
|
||||
the slider values when \(t>t_0\).
|
||||
</p>
|
||||
<div class="graph">
|
||||
<div id="romer-solow-change-visualization"></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—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>
|
||||
</article>
|
||||
</div>
|
||||
</main>
|
||||
<site-footer></site-footer>
|
||||
<script src="/scripts/common.js"></script>
|
||||
<script src="/scripts/post.js"></script>
|
||||
<script src="/scripts/posts/models-of-production.js"></script>
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -14,14 +14,14 @@
|
|||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
|
||||
async
|
||||
></script>
|
||||
<title>Barrett Ruth</title>
|
||||
<title>practice makes perfect</title>
|
||||
</head>
|
||||
<body class="graph-background">
|
||||
<site-header></site-header>
|
||||
<main class="main">
|
||||
<div class="post-container">
|
||||
<header class="post-header">
|
||||
<h1 class="post-title">Practice Makes Perfect</h1>
|
||||
<h1 class="post-title">practice makes perfect</h1>
|
||||
<p class="post-meta">
|
||||
<time datetime="2024-06-22">05/07/2024</time>
|
||||
</p>
|
||||
|
|
|
|||
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