feat(posts): models of production solow model
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3 changed files with 346 additions and 10 deletions
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@ -6,6 +6,7 @@
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<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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<link rel="stylesheet" href="/public/katex/katex.css" />
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<script defer src="/public/katex/katex.js"></script>
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@ -17,7 +18,7 @@
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<script defer src="/public/d3.js"></script>
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<title>Barrett Ruth</title>
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</head>
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<body class="graph">
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<body class="graph-background">
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<header>
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<a
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href="/"
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@ -74,7 +75,9 @@
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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 \(K_0\)
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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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@ -87,13 +90,13 @@
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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="margin: 0 20px">
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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>
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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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@ -111,14 +114,105 @@
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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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\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
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\bar{L_t}^{\frac{2}{3}}\). Utilizing this simplification and its
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graphical representation below, output is clearly characterized by
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the cube root of capital:
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\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\).
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Utilizing this simplification and its graphical representation
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below, output is clearly characterized by the cube root of
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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="sliderA">\(A:\)</label>
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<span id="outputA">1.00</span>
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<input
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type="range"
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id="sliderA"
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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="sliderD">\(d:\)</label>
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<span id="outputD">0.50</span>
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<input
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type="range"
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id="sliderD"
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min="0.01"
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max="1"
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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="sliderS">\(s:\)</label>
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<span id="outputS">0.50</span>
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<input
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type="range"
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id="sliderS"
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min="0.01"
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max="1"
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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="sliderAlpha">\(\alpha:\)</label>
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<span id="outputAlpha">0.33</span>
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<input
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type="range"
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id="sliderAlpha"
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min="0.01"
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max="1"
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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 (in
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other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates at a
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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>
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<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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<div class="fold"><h3>conclusions</h3></div>
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<p>hello conclusions</p>
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<div class="fold"><h3>analysis</h3></div>
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<p>discuss limitations</p>
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<h2>romer</h2>
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<h2>romer-solow</h2>
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</article>
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@ -126,5 +220,6 @@
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</main>
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<script src="/scripts/common.js"></script>
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<script src="/scripts/post.js"></script>
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<script src="/scripts/posts/models-of-production.js"></script>
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</body>
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</html>
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