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<h1><aclass="toc-backref" href="#id5"><spanclass="section-number">38. </span>Equilibrium Capital Structures with Incomplete Markets</a><aclass="headerlink" href="#equilibrium-capital-structures-with-incomplete-markets" title="Permalink to this heading">#</a></h1>
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<divclass="contents topic" id="contents">
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<pclass="topic-title">Contents</p>
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<ulclass="simple">
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<li><p><aclass="reference internal" href="#equilibrium-capital-structures-with-incomplete-markets" id="id5">Equilibrium Capital Structures with Incomplete Markets</a></p>
<li><p><aclass="reference internal" href="#a-picture-worth-a-thousand-words" id="id12">A picture worth a thousand words</a></p></li>
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</ul>
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</li>
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</ul>
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<h1><spanclass="section-number">38. </span>Equilibrium Capital Structures with Incomplete Markets<aclass="headerlink" href="#equilibrium-capital-structures-with-incomplete-markets" title="Permalink to this heading">#</a></h1>
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<p>In addition to what’s in Anaconda, this lecture will need the following libraries:</p>
<h2><aclass="toc-backref" href="#id6"><spanclass="section-number">38.1. </span>Introduction</a><aclass="headerlink" href="#introduction" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.1. </span>Introduction<aclass="headerlink" href="#introduction" title="Permalink to this heading">#</a></h2>
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<p>This is an extension of an earlier lecture <aclass="reference internal" href="BCG_complete_mkts.html"><spanclass="doc">Irrelevance of Capital Structure with Complete Markets</span></a> about a <strong>complete markets</strong>
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model.</p>
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<p>In contrast to that lecture, this one describes an instance of a model authored by Bisin, Clementi, and Gottardi <spanid="id1">[<aclass="reference internal" href="zreferences.html#id29" title="Alberto Bisin, Gian Luca Clementi, and Piero Gottardi. Capital and hedging demand with incomplete markets. Technical Report, NYU and EUI, 2018.">Bisin <em>et al.</em>, 2018</a>]</span>
<h2><aclass="toc-backref" href="#id7"><spanclass="section-number">38.2. </span>Asset Markets</a><aclass="headerlink" href="#asset-markets" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.2. </span>Asset Markets<aclass="headerlink" href="#asset-markets" title="Permalink to this heading">#</a></h2>
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<p>Markets are incomplete: <em>ex cathedra</em> we the model builders declare that only equities and bonds issued by representative
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firms can be traded.</p>
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<p>Let <spanclass="math notranslate nohighlight">\(\theta^i\)</span> and <spanclass="math notranslate nohighlight">\(\xi^i\)</span> be a consumer of type
<h2><aclass="toc-backref" href="#id8"><spanclass="section-number">38.3. </span>Equilibrium verification</a><aclass="headerlink" href="#equilibrium-verification" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.3. </span>Equilibrium verification<aclass="headerlink" href="#equilibrium-verification" title="Permalink to this heading">#</a></h2>
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<p>On page 5 of BCG (2018), the authors say</p>
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<p><em>If the price conjectures corresponding to the plan chosen by firms in
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equilibrium are correct, that is equal to the market prices</em><spanclass="math notranslate nohighlight">\(\check q\)</span><em>and</em><spanclass="math notranslate nohighlight">\(\check p\)</span>, <em>it is immediate to verify that
<h2><aclass="toc-backref" href="#id9"><spanclass="section-number">38.4. </span>Pseudo Code</a><aclass="headerlink" href="#pseudo-code" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.4. </span>Pseudo Code<aclass="headerlink" href="#pseudo-code" title="Permalink to this heading">#</a></h2>
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<p>Before displaying our Python code for computing a BCG incomplete markets equilibrium,
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we’ll sketch some pseudo code that describes its logical flow.</p>
<h2><aclass="toc-backref" href="#id10"><spanclass="section-number">38.5. </span>Code</a><aclass="headerlink" href="#code" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.5. </span>Code<aclass="headerlink" href="#code" title="Permalink to this heading">#</a></h2>
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<p>We create a Python class <codeclass="docutils literal notranslate"><spanclass="pre">BCG_incomplete_markets</span></code> to compute the
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equilibrium allocations of the incomplete market BCG model, given a set
<h2><aclass="toc-backref" href="#id11"><spanclass="section-number">38.6. </span>Examples</a><aclass="headerlink" href="#examples" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.6. </span>Examples<aclass="headerlink" href="#examples" title="Permalink to this heading">#</a></h2>
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<p>Below we show some examples computed with the class <codeclass="docutils literal notranslate"><spanclass="pre">BCG_incomplete</span><spanclass="pre">markets</span></code>.</p>
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<sectionid="first-example">
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<h3><spanclass="section-number">38.6.1. </span>First example<aclass="headerlink" href="#first-example" title="Permalink to this heading">#</a></h3>
@@ -2763,7 +2726,7 @@ <h3><span class="section-number">38.6.3. </span>Another example economy<a class=
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<sectionid="a-picture-worth-a-thousand-words">
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<h2><aclass="toc-backref" href="#id12"><spanclass="section-number">38.7. </span>A picture worth a thousand words</a><aclass="headerlink" href="#a-picture-worth-a-thousand-words" title="Permalink to this heading">#</a></h2>
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<h2><spanclass="section-number">38.7. </span>A picture worth a thousand words<aclass="headerlink" href="#a-picture-worth-a-thousand-words" title="Permalink to this heading">#</a></h2>
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<p>Please stare at the above panels.</p>
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<p>They describe how equilibrium prices and quantities respond to
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alterations in the structure of society’s <em>hedging desires</em> across
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