225 lines
8.3 KiB
HTML
225 lines
8.3 KiB
HTML
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<title>Barrett Ruth</title>
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href="/"
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onclick="goHome(event)"
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>
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<span class="terminal-prompt">barrett@ruth:~$ /economics</span>
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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 \(K_0\) (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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<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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\(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\).
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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>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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</div>
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