130 lines
4.6 KiB
HTML
130 lines
4.6 KiB
HTML
<!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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defer
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></script>
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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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<header>
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<a
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href="/"
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style="text-decoration: none; color: inherit"
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onclick="goHome(event)"
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>
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<div class="terminal-container">
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<span class="terminal-prompt">barrett@ruth:~$ /economics</span>
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<span class="terminal-cursor"></span>
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</div>
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</a>
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</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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<h2>solow</h2>
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<div class="fold"><h3>introduction</h3></div>
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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>
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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 \(1/3\)
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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 \(K_0\)
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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="margin: 0 20px">
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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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<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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<div class="fold">
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<h3>solving the model</h3>
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</div>
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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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\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}}
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\bar{L_t}^{\frac{2}{3}}\). 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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</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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<h2>romer</h2>
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<h2>romer-solow</h2>
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</article>
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</div>
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</main>
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<script src="/scripts/common.js"></script>
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</body>
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</html>
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