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@@ -475,7 +487,7 @@ <h1><span class="section-number">37. </span>Irrelevance of Capital Structures wi
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<sectionid="introduction">
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<h2><spanclass="section-number">37.1. </span>Introduction<aclass="headerlink" href="#introduction" title="Permalink to this heading">#</a></h2>
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<p>This is a prolegomenon to another lecture <aclass="reference internal" href="BCG_incomplete_mkts.html"><spanclass="doc">Equilibrium Capital Structures with Incomplete Markets</span></a> about a model with
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incomplete markets 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>.</p>
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incomplete markets authored by Bisin, Clementi, and Gottardi <spanid="id1">[<aclass="reference internal" href="zreferences.html#id30" 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>.</p>
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<p>We adopt specifications of preferences and technologies very close to
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Bisin, Clemente, and Gottardi’s but unlike them assume that there are complete
<li><p>indeterminacy of consumers’ portfolio choices</p></li>
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<li><p>indeterminacy of firms’ financial structures that underlies a
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Modigliani-Miller theorem <spanid="id2">[<aclass="reference internal" href="zreferences.html#id30" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span></p></li>
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Modigliani-Miller theorem <spanid="id2">[<aclass="reference internal" href="zreferences.html#id31" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span></p></li>
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</ul>
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</li>
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<li><p>introducing <codeclass="docutils literal notranslate"><spanclass="pre">Big</span><spanclass="pre">K,</span><spanclass="pre">little</span><spanclass="pre">k</span></code> issues in a simple context that will
<p>which is the same expression that we obtained above when we assumed that
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the firm issued only equity.</p>
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<p>We thus obtain a version of the celebrated Modigliani-Miller theorem <spanid="id3">[<aclass="reference internal" href="zreferences.html#id30" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span>
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<p>We thus obtain a version of the celebrated Modigliani-Miller theorem <spanid="id3">[<aclass="reference internal" href="zreferences.html#id31" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span>
@@ -410,7 +405,7 @@ <h1><span class="section-number">38. </span>Equilibrium Capital Structures with
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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>
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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#id30" 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>
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in which financial markets are <strong>incomplete</strong>.</p>
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<p>Instead of being able to trade equities and a full set of one-period
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Arrow securities as they can in <aclass="reference internal" href="BCG_complete_mkts.html"><spanclass="doc">Irrelevance of Capital Structure with Complete Markets</span></a>, here consumers and firms trade only equity and a bond.</p>
<li><p>there is a unique stochastic discount factor that prices all assets</p></li>
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<li><p>consumers’ portfolio choices are indeterminate</p></li>
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<li><p>firms’ financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem <spanid="id2">[<aclass="reference internal" href="zreferences.html#id30" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span></p></li>
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<li><p>firms’ financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem <spanid="id2">[<aclass="reference internal" href="zreferences.html#id31" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span></p></li>
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<li><p>the aggregate of all firms’ financial structures are indeterminate, a consequence of there being redundant assets</p></li>
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</ul>
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<p>In the incomplete markets economy studied here</p>
<li><p>different stochastic discount factors price different assets</p></li>
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<li><p>consumers’ portfolio choices are determinate</p></li>
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<li><p>while <strong>individual</strong> firms’ financial structures are indeterminate, thus conforming to part of a Modigliani-Miller theorem,
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<spanid="id3">[<aclass="reference internal" href="zreferences.html#id30" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span>, the <strong>aggregate</strong> of all firms’ financial structures <strong>is</strong> determinate.</p></li>
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<spanid="id3">[<aclass="reference internal" href="zreferences.html#id31" title="Franco Modigliani and Merton H. Miller. Corporation finance and the theory of investment. American Economic Review, XLVIII(3):261-297, 1958.">Modigliani and Miller, 1958</a>]</span>, the <strong>aggregate</strong> of all firms’ financial structures <strong>is</strong> determinate.</p></li>
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</ul>
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<p>A <codeclass="docutils literal notranslate"><spanclass="pre">Big</span><spanclass="pre">K,</span><spanclass="pre">little</span><spanclass="pre">k</span></code> analysis played an important role in the previous lecture <aclass="reference internal" href="BCG_complete_mkts.html"><spanclass="doc">Irrelevance of Capital Structure with Complete Markets</span></a>.</p>
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<p>A more subtle version of a <codeclass="docutils literal notranslate"><spanclass="pre">Big</span><spanclass="pre">K,</span><spanclass="pre">little</span><spanclass="pre">k</span></code> features in the BCG incomplete markets environment here.</p>
<h3><spanclass="section-number">38.1.1. </span>Setup<aclass="headerlink" href="#setup" title="Permalink to this heading">#</a></h3>
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<p>We adopt specifications of preferences and technologies used by Bisin,
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Clemente, and Gottardi (2018) <spanid="id4">[<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> and in our earlier lecture on a complete markets
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Clemente, and Gottardi (2018) <spanid="id4">[<aclass="reference internal" href="zreferences.html#id30" 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> and in our earlier lecture on a complete markets
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version of their model.</p>
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<p>The economy lasts for two periods, <spanclass="math notranslate nohighlight">\(t=0, 1\)</span>.</p>
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<p>There are two types of consumers named <spanclass="math notranslate nohighlight">\(i=1,2\)</span>.</p>
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