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class="vector-toc-numb">2</span> <span>Map and properties</span> </div> </a> <ul id="toc-Map_and_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assumptions_of_consumer_preference_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Assumptions_of_consumer_preference_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Assumptions of consumer preference theory</span> </div> </a> <button aria-controls="toc-Assumptions_of_consumer_preference_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Assumptions of consumer preference theory subsection</span> </button> <ul id="toc-Assumptions_of_consumer_preference_theory-sublist" class="vector-toc-list"> <li id="toc-Application" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Application</span> </div> </a> <ul id="toc-Application-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_indifference_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_indifference_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Examples of indifference curves</span> </div> </a> <ul id="toc-Examples_of_indifference_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Preference_relations_and_utility" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Preference_relations_and_utility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Preference relations and utility</span> </div> </a> <button aria-controls="toc-Preference_relations_and_utility-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Preference relations and utility subsection</span> </button> <ul id="toc-Preference_relations_and_utility-sublist" class="vector-toc-list"> <li id="toc-Preference_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Preference_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Preference relations</span> </div> </a> <ul id="toc-Preference_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_link_to_utility_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_link_to_utility_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Formal link to utility theory</span> </div> </a> <ul id="toc-Formal_link_to_utility_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Linear_utility" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Linear_utility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Linear utility</span> </div> </a> <ul id="toc-Linear_utility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cobb–Douglas_utility" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cobb–Douglas_utility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Cobb–Douglas utility</span> </div> </a> <ul id="toc-Cobb–Douglas_utility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CES_utility" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#CES_utility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.3</span> <span>CES utility</span> </div> </a> <ul id="toc-CES_utility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.4</span> <span>Biology</span> </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Criticisms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Criticisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Criticisms</span> </div> </a> <ul id="toc-Criticisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Indifference curve</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" 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Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%AD%D9%86%D9%89_%D8%A7%D9%84%D8%B3%D9%88%D8%A7%D8%A1" title="منحنى السواء – Arabic" lang="ar" hreflang="ar" data-title="منحنى السواء" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/F%C9%99rqsizlik_%C9%99yrisi" title="Fərqsizlik əyrisi – Azerbaijani" lang="az" hreflang="az" data-title="Fərqsizlik əyrisi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%B0%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7_%E0%A6%B0%E0%A7%87%E0%A6%96%E0%A6%BE_(%E0%A6%85%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%A8%E0%A7%80%E0%A6%A4%E0%A6%BF)" title="নিরপেক্ষ রেখা (অর্থনীতি) – Bangla" lang="bn" hreflang="bn" data-title="নিরপেক্ষ রেখা (অর্থনীতি)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Corba_d%27indifer%C3%A8ncia" title="Corba d'indiferència – Catalan" lang="ca" hreflang="ca" data-title="Corba d'indiferència" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Indiferen%C4%8Dn%C3%AD_k%C5%99ivka" title="Indiferenční křivka – Czech" lang="cs" hreflang="cs" data-title="Indiferenční křivka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Indifferenzkurve" title="Indifferenzkurve – German" lang="de" hreflang="de" data-title="Indifferenzkurve" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Curva_de_indiferencia" title="Curva de indiferencia – Spanish" lang="es" hreflang="es" data-title="Curva de indiferencia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Indiferentzia-kurba" title="Indiferentzia-kurba – Basque" lang="eu" hreflang="eu" data-title="Indiferentzia-kurba" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%D8%AD%D9%86%DB%8C_%D8%A8%DB%8C%E2%80%8C%D8%AA%D9%81%D8%A7%D9%88%D8%AA%DB%8C" title="منحنی بی‌تفاوتی – Persian" lang="fa" hreflang="fa" data-title="منحنی بی‌تفاوتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Courbe_d%27indiff%C3%A9rence" title="Courbe d'indifférence – French" lang="fr" hreflang="fr" data-title="Courbe d'indifférence" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AC%B4%EC%B0%A8%EB%B3%84_%EA%B3%A1%EC%84%A0" title="무차별 곡선 – Korean" lang="ko" hreflang="ko" data-title="무차별 곡선" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%BF%D5%A1%D6%80%D5%A2%D5%A5%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%AF%D5%B8%D6%80" title="Անտարբերության կոր – Armenian" lang="hy" hreflang="hy" data-title="Անտարբերության կոր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%89%E0%A4%A6%E0%A4%BE%E0%A4%B8%E0%A5%80%E0%A4%A8%E0%A4%A4%E0%A4%BE_%E0%A4%B5%E0%A4%95%E0%A5%8D%E0%A4%B0" title="उदासीनता वक्र – Hindi" lang="hi" hreflang="hi" data-title="उदासीनता वक्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Krivulja_indiferencije" title="Krivulja indiferencije – Croatian" lang="hr" hreflang="hr" data-title="Krivulja indiferencije" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kurva_indiferensi" title="Kurva indiferensi – Indonesian" lang="id" hreflang="id" data-title="Kurva indiferensi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Curva_di_indifferenza" title="Curva di indifferenza – Italian" lang="it" hreflang="it" data-title="Curva di indifferenza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%95%D7%9E%D7%AA_%D7%90%D7%93%D7%99%D7%A9%D7%95%D7%AA" title="עקומת אדישות – Hebrew" lang="he" hreflang="he" data-title="עקומת אדישות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%AA%E0%BA%B1%E0%BB%89%E0%BA%99%E0%BB%82%E0%BA%84%E0%BB%89%E0%BA%87%E0%BA%9A%E0%BB%8D%E0%BB%88%E0%BA%88%E0%BA%B3%E0%BB%81%E0%BA%99%E0%BA%81" title="ເສັ້ນໂຄ້ງບໍ່ຈຳແນກ – Lao" lang="lo" hreflang="lo" data-title="ເສັ້ນໂຄ້ງບໍ່ຈຳແນກ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6z%C3%B6mb%C3%B6ss%C3%A9gi_g%C3%B6rbe" title="Közömbösségi görbe – Hungarian" lang="hu" hreflang="hu" data-title="Közömbösségi görbe" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0_%D0%BD%D0%B0_%D1%80%D0%B0%D0%BC%D0%BD%D0%BE%D0%B4%D1%83%D1%88%D0%BD%D0%BE%D1%81%D1%82" title="Крива на рамнодушност – Macedonian" lang="mk" hreflang="mk" data-title="Крива на рамнодушност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Keluk_puas_sama" title="Keluk puas sama – Malay" lang="ms" hreflang="ms" data-title="Keluk puas sama" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indifferentiecurve" title="Indifferentiecurve – Dutch" lang="nl" hreflang="nl" data-title="Indifferentiecurve" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%84%A1%E5%B7%AE%E5%88%A5%E6%9B%B2%E7%B7%9A" title="無差別曲線 – Japanese" lang="ja" hreflang="ja" data-title="無差別曲線" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Krzywa_oboj%C4%99tno%C5%9Bci" title="Krzywa obojętności – Polish" lang="pl" hreflang="pl" data-title="Krzywa obojętności" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Curva_de_indiferen%C3%A7a" title="Curva de indiferença – Portuguese" lang="pt" hreflang="pt" data-title="Curva de indiferença" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Curba_de_indiferen%C8%9B%C4%83" title="Curba de indiferență – Romanian" lang="ro" hreflang="ro" data-title="Curba de indiferență" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0%D1%8F_%D0%B1%D0%B5%D0%B7%D1%80%D0%B0%D0%B7%D0%BB%D0%B8%D1%87%D0%B8%D1%8F" title="Кривая безразличия – Russian" lang="ru" hreflang="ru" data-title="Кривая безразличия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Indifferenssik%C3%A4yr%C3%A4" title="Indifferenssikäyrä – Finnish" lang="fi" hreflang="fi" data-title="Indifferenssikäyrä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Indifferenskurva" title="Indifferenskurva – Swedish" lang="sv" hreflang="sv" data-title="Indifferenskurva" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%A3%E0%AF%88%E0%AE%AA%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AF%80" title="இணைபயன் வளையீ – Tamil" lang="ta" hreflang="ta" data-title="இணைபயன் வளையீ" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kay%C4%B1ts%C4%B1zl%C4%B1k_e%C4%9Frileri" title="Kayıtsızlık eğrileri – Turkish" lang="tr" hreflang="tr" data-title="Kayıtsızlık eğrileri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0_%D0%B1%D0%B0%D0%B9%D0%B4%D1%83%D0%B6%D0%BE%D1%81%D1%82%D1%96" title="Крива байдужості – Ukrainian" lang="uk" hreflang="uk" data-title="Крива байдужості" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/B%C3%A0ng_quan_(kinh_t%E1%BA%BF_h%E1%BB%8Dc)" title="Bàng quan (kinh tế học) – Vietnamese" lang="vi" hreflang="vi" data-title="Bàng quan (kinh tế học)" data-language-autonym="Tiếng 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src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/240px-Simple-indifference-curves.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/360px-Simple-indifference-curves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/480px-Simple-indifference-curves.svg.png 2x" data-file-width="217" data-file-height="217" /></a><figcaption>An example of an indifference map with three indifference curves represented</figcaption></figure> <p>In <a href="/wiki/Economics" title="Economics">economics</a>, an <b>indifference curve</b> connects points on a graph representing different quantities of two goods, points between which a consumer is <i>indifferent</i>. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no <a href="/wiki/Preference_(economics)" title="Preference (economics)">preference</a> for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of <a href="/wiki/Utility" title="Utility">utility</a> (satisfaction) for the consumer. In other words, an indifference curve is the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent <a href="/wiki/Preference" title="Preference">preferences</a> rather than something from which preferences come.<sup id="cite_ref-Geanakoplis_(1987),_p._117_1-0" class="reference"><a href="#cite_note-Geanakoplis_(1987),_p._117-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The main use of indifference curves is in the <a href="/wiki/Mathematical_problem" title="Mathematical problem">representation</a> of potentially observable <a href="/wiki/Demand" title="Demand">demand</a> patterns for individual consumers over commodity bundles.<sup id="cite_ref-Böhm_and_Haller_(1987),_p._785_2-0" class="reference"><a href="#cite_note-Böhm_and_Haller_(1987),_p._785-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Indifference curve analysis is a purely technological model which cannot be used to model consumer behaviour. Every point on any given indifference curve must be satisfied by the same budget (unless the consumer can be indifferent to different budgets). As a consequence, every budget line for a given budget and any two products is tangent to the same indifference curve and this means that every budget line is tangent to, at most, one indifference curve (and so every consumer makes the same choices). </p><p>There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an <b>indifference map</b>. The <a href="/wiki/Slope" title="Slope">slope</a> of an indifference curve is called the MRS (marginal rate of substitution), and it indicates how much of good y must be sacrificed to keep the utility constant if good x is increased by one unit. Given a utility function u(x,y), to calculate the MRS, one takes the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of the function <i>u</i> with respect to good <i>x</i> and divide it by the partial derivative of the function <i>u</i> with respect to good <i>y</i>. If the marginal rate of substitution is diminishing along an indifference curve, that is the magnitude of the slope is decreasing or becoming less steep, then the preference is convex. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theory of indifference curves was developed by <a href="/wiki/Francis_Ysidro_Edgeworth" title="Francis Ysidro Edgeworth">Francis Ysidro Edgeworth</a>, who explained in his 1881 book the mathematics needed for their drawing;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> later on, <a href="/wiki/Vilfredo_Pareto" title="Vilfredo Pareto">Vilfredo Pareto</a> was the first author to actually draw these curves, in his 1906 book.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The theory can be derived from <a href="/wiki/William_Stanley_Jevons" title="William Stanley Jevons">William Stanley Jevons</a>' <a href="/wiki/Ordinal_utility" title="Ordinal utility">ordinal utility</a> theory, which posits that individuals can always rank any consumption bundles by order of preference.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Map_and_properties">Map and properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=2" title="Edit section: Map and properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Indifference_curve_example.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Indifference_curve_example.png/240px-Indifference_curve_example.png" decoding="async" width="240" height="642" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Indifference_curve_example.png/360px-Indifference_curve_example.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Indifference_curve_example.png/480px-Indifference_curve_example.png 2x" data-file-width="966" data-file-height="2584" /></a><figcaption>An example of how indifference curves are obtained as the <a href="/wiki/Level_curves" class="mw-redirect" title="Level curves">level curves</a> of a utility function</figcaption></figure> <p>A graph of indifference curves for several utility levels of an individual consumer is called an <b>indifference map</b>. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical graph. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive <a href="/wiki/Marginal_utility" title="Marginal utility">marginal utility</a> for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "<a href="/wiki/Bliss_point_(economics)" title="Bliss point (economics)">bliss point</a>," a consumption bundle that is preferred to all others. </p><p>Indifference curves are typically<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness"><span title="Give a few citations of authors who share these requirements; give some notes and citations on authors with deviating requirements. (December 2015)">vague</span></a></i>]</sup> represented<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Should probably be 'required'. (December 2015)">clarification needed</span></a></i>]</sup> to be: </p> <ol><li>Defined only in the non-negative <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">quadrant</a> of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).</li> <li>Negatively sloped. That is, as the consumption of one good increases, to maintain constant utility, a lesser quantity of the other good just be consumed. This is equivalent to assuming <a href="/wiki/Local_nonsatiation" title="Local nonsatiation">Local non-satiation</a> (an increase in the consumption of either good increases, rather than decreases, total utility). The counterfactual to this assumption is assuming a <a href="/wiki/Bliss_point_(economics)" title="Bliss point (economics)">bliss point</a>. If utility <i>U = f(x, y)</i>, <i>U</i>, in the third dimension, does not have a <a href="/wiki/Maxima_and_minima#Functions_of_more_than_one_variable" class="mw-redirect" title="Maxima and minima">local maximum</a> for any <i>x</i> and <i>y</i> values.) The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); an upward sloping indifference curve would imply that a consumer is indifferent between a bundle A and another bundle B because they lie on the same indifference curve, even in the case in which the quantity of both goods in bundle B is higher. Because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, and lie on a different indifference curve at a higher utility level. The negative slope of the indifference curve implies that the <a href="/wiki/Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a> is always positive;</li> <li><a href="/wiki/Total_order" title="Total order">Complete</a>, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated since the point(s) of intersection would have equal utility).</li> <li><a href="/wiki/Transitive_relation#Examples" title="Transitive relation">Transitive</a> with respect to points on distinct indifference curves. That is, if each point on <i>I<sub>2</sub></i> is (strictly) preferred to each point on <i>I<sub>1</sub></i>, and each point on <i>I<sub>3</sub></i> is preferred to each point on <i>I<sub>2</sub></i>, each point on <i>I<sub>3</sub></i> is preferred to each point on <i>I<sub>1</sub></i>. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.</li> <li>(Strictly) <a href="/wiki/Convex_function" title="Convex function">convex</a>. <a href="/wiki/Convex_preferences" title="Convex preferences">Convex preferences</a> imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the <a href="/wiki/Composite_good" title="Composite good">other good</a> are required to keep satisfaction unchanged. Convex preferences are assumed in concordance with the principle of <a href="/wiki/Declining_marginal_utility" class="mw-redirect" title="Declining marginal utility">declining marginal utility</a>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Assumptions_of_consumer_preference_theory">Assumptions of consumer preference theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=3" title="Edit section: Assumptions of consumer preference theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Preferences are <b>complete</b>. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.</li></ul> <dl><dd>Assume that there are two consumption bundles <i>A</i> and <i>B</i> each containing two commodities <i>x</i> and <i>y</i>. A consumer can unambiguously determine that one and only one of the following is the case: <ul><li><i>A</i> is preferred to <i>B</i>, formally written as <i>A</i> <sup>p</sup> <i>B</i><sup id="cite_ref-Binger_7-0" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><i>B</i> is preferred to <i>A</i>, formally written as <i>B</i> <sup>p</sup> <i>A</i><sup id="cite_ref-Binger_7-1" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><i>A</i> is indifferent to <i>B</i>, formally written as <i>A</i> <sup>I</sup> <i>B</i><sup id="cite_ref-Binger_7-2" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul></dd> <dd>This axiom precludes the possibility that the consumer cannot decide,<sup id="cite_ref-Perloff_2008._p._62_8-0" class="reference"><a href="#cite_note-Perloff_2008._p._62-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.<sup id="cite_ref-Binger_7-3" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <ul><li>Preferences are <b>reflexive</b></li></ul> <dl><dd>This means that if <i>A</i> and <i>B</i> are identical in all respects the consumer will recognize this fact and be indifferent in comparing <i>A</i> and <i>B</i> <ul><li><i>A</i> = <i>B</i> ⇒ <i>A</i> <sup>I</sup> <i>B</i><sup id="cite_ref-Binger_7-4" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <ul><li>Preferences are <b>transitive</b><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><ul><li>If <i>A</i> <sup>p</sup> <i>B</i> and <i>B</i> <sup>p</sup> <i>C</i>, then <i>A</i> <sup>p</sup> <i>C</i>.<sup id="cite_ref-Binger_7-5" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>Also if <i>A</i> <sup>I</sup> <i>B</i> and <i>B</i> <sup>I</sup> <i>C</i>, then <i>A</i> <sup>I</sup> <i>C</i>.<sup id="cite_ref-Binger_7-6" class="reference"><a href="#cite_note-Binger-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul></dd> <dd>This is a consistency assumption.</dd></dl> <ul><li>Preferences are <b>continuous</b></li></ul> <dl><dd><ul><li>If <i>A</i> is preferred to <i>B</i> and <i>C</i> is sufficiently close to <i>B</i> then <i>A</i> is preferred to <i>C</i>.</li> <li><i>A</i> <sup>p</sup> <i>B</i> and <i>C</i> → <i>B</i> ⇒ <i>A</i> <sup>p</sup> <i>C</i>.</li></ul></dd> <dd>"Continuous" means infinitely divisible - just like there are infinitely many numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.</dd></dl> <ul><li>Preferences exhibit <b>strong monotonicity</b></li></ul> <dl><dd><ul><li>If <i>A</i> has more of both <i>x</i> and <i>y</i> than <i>B</i>, then <i>A</i> is preferred to <i>B</i>.</li></ul></dd> <dd>This assumption is commonly called the "more is better" assumption.</dd> <dd>An alternative version of this assumption requires that if <i>A</i> and <i>B</i> have the same quantity of one good, but <i>A</i> has more of the other, then <i>A</i> is preferred to <i>B</i>.</dd></dl> <p>It also implies that the commodities are <b>good</b> rather than <b>bad</b>. Examples of <b>bad</b> commodities can be disease, pollution etc. because we always desire less of such things. </p> <ul><li>Indifference curves exhibit <b>diminishing marginal rates of substitution</b></li></ul> <dl><dd><ul><li>The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="After having explained 'marginal rate of substitution', explain when they are called 'diminishing. (December 2015)">clarification needed</span></a></i>]</sup></li> <li>This assumption assures that indifference curves are smooth and convex to the origin.</li> <li>This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive.</li> <li>Another name for this assumption is the <b>substitution assumption</b>. It is the most critical assumption of <a href="/wiki/Consumer_theory" class="mw-redirect" title="Consumer theory">consumer theory</a>: Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations"<sup id="cite_ref-Silberberg_10-0" class="reference"><a href="#cite_note-Silberberg-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.<sup id="cite_ref-Silberberg_10-1" class="reference"><a href="#cite_note-Silberberg-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Application">Application</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=4" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Indifference_curves_showing_budget_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/220px-Indifference_curves_showing_budget_line.svg.png" decoding="async" width="220" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/330px-Indifference_curves_showing_budget_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/440px-Indifference_curves_showing_budget_line.svg.png 2x" data-file-width="230" data-file-height="217" /></a><figcaption>To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.</figcaption></figure> <p><a href="/wiki/Consumer_theory" class="mw-redirect" title="Consumer theory">Consumer theory</a> uses indifference curves and <a href="/wiki/Budget_constraint" title="Budget constraint">budget constraints</a> to generate <a href="/wiki/Supply_and_demand" title="Supply and demand">consumer demand curves</a>. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g., carrots, and let the other be a composite of all other goods. <a href="/wiki/Budget_constraint" title="Budget constraint">Budget constraints</a> give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than the household, the household will sell it; if the market values a good less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal.<sup id="cite_ref-Lipsey_1975._pp._182-186_11-0" class="reference"><a href="#cite_note-Lipsey_1975._pp._182-186-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve.<sup id="cite_ref-Lipsey_1975._pp._182-186_11-1" class="reference"><a href="#cite_note-Lipsey_1975._pp._182-186-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Stated precisely, a set of indifference curve for representative of different price ratios between two goods are used to generate the <a href="/wiki/Price-consumption_curve" title="Price-consumption curve">Price-consumption curve</a> in good-good vector space, which is equivalent to the <a href="/wiki/Demand_curve" title="Demand curve">demand curve</a> in good-price vector space. The line connecting all points of tangency between the indifference curve and the <a href="/wiki/Budget_constraint" title="Budget constraint">budget constraint</a> as the budget constraint changes is called the <a href="/wiki/Expansion_path" title="Expansion path">expansion path</a>,<sup id="cite_ref-Salvatore_12-0" class="reference"><a href="#cite_note-Salvatore-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and correlates to shifts in demand. The line connecting all points of tangency between the indifference curve and budget constraint as the price of either good changes is the price-consumption curve, and correlates to movement along the demand curve. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Examples_of_indifference_curves">Examples of indifference curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=5" title="Edit section: Examples of indifference curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Simple-indifference-curves.svg" class="mw-file-description" title="Figure 1: An example of an indifference map with three indifference curves represented"><img alt="Figure 1: An example of an indifference map with three indifference curves represented" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/120px-Simple-indifference-curves.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/180px-Simple-indifference-curves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Simple-indifference-curves.svg/240px-Simple-indifference-curves.svg.png 2x" data-file-width="217" data-file-height="217" /></a></span></div> <div class="gallerytext">Figure 1: An example of an indifference map with three indifference curves represented</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Indifference-curves-perfect-substitutes.svg" class="mw-file-description" title="Figure 2: Three indifference curves where Goods X and Y are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel."><img alt="Figure 2: Three indifference curves where Goods X and Y are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Indifference-curves-perfect-substitutes.svg/120px-Indifference-curves-perfect-substitutes.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Indifference-curves-perfect-substitutes.svg/180px-Indifference-curves-perfect-substitutes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Indifference-curves-perfect-substitutes.svg/240px-Indifference-curves-perfect-substitutes.svg.png 2x" data-file-width="217" data-file-height="145" /></a></span></div> <div class="gallerytext">Figure 2: Three indifference curves where Goods <i>X</i> and <i>Y</i> are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel. </div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Indifference-curves-perfect-complements.svg" class="mw-file-description" title="Figure 3: Indifference curves for perfect complements X and Y. The elbows of the curves are collinear. The grey line shows the Income–consumption curve (the consumer theory equivalent to the Expansion path) of a series of Leontief utility curves."><img alt="Figure 3: Indifference curves for perfect complements X and Y. The elbows of the curves are collinear. The grey line shows the Income–consumption curve (the consumer theory equivalent to the Expansion path) of a series of Leontief utility curves." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Indifference-curves-perfect-complements.svg/120px-Indifference-curves-perfect-complements.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Indifference-curves-perfect-complements.svg/180px-Indifference-curves-perfect-complements.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Indifference-curves-perfect-complements.svg/240px-Indifference-curves-perfect-complements.svg.png 2x" data-file-width="217" data-file-height="145" /></a></span></div> <div class="gallerytext">Figure 3: Indifference curves for perfect complements <i>X</i> and <i>Y</i>. The elbows of the curves are <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a>. The grey line shows the <a href="/wiki/Income%E2%80%93consumption_curve" title="Income–consumption curve">Income–consumption curve</a> (the consumer theory equivalent to the <a href="/wiki/Expansion_path" title="Expansion path">Expansion path</a>) of a series of <a href="/wiki/Leontief_utilities" title="Leontief utilities">Leontief utility curves</a>.</div> </li> </ul> <p>In Figure 1, the consumer would rather be on <i>I<sub>3</sub></i> than <i>I<sub>2</sub></i>, and would rather be on <i>I<sub>2</sub></i> than <i>I<sub>1</sub></i>, but does not care where he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), known by economists as the <a href="/wiki/Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a>, shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For <i>most</i> goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin, describing the negative <a href="/wiki/Substitution_effect" title="Substitution effect">substitution effect</a>. As price rises for a fixed money income, the consumer seeks the less expensive substitute at a lower indifference curve. The substitution effect is reinforced through the <a href="/wiki/Income_effect" class="mw-redirect" title="Income effect">income effect</a> of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb–Douglas function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35c447fe95f9a449eb5016339591f0406787969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.246ex; height:2.343ex;" alt="{\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1}"></span>. The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.<sup id="cite_ref-Silberberg_10-2" class="reference"><a href="#cite_note-Silberberg-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>If two goods are <a href="/wiki/Substitute_good" title="Substitute good">perfect substitutes</a> then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> <mi>y</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c00836104a9a206bf554cb5a9b50d8eca15644ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.063ex; height:2.176ex;" alt="{\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y}"></span>. </p><p>If two goods are <a href="/wiki/Complement_good" class="mw-redirect" title="Complement good">perfect complements</a> then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: the consumer is no better off having several right shoes if she has only one left shoe - additional right shoes have zero marginal utility without more left shoes, so bundles of goods differing only in the number of right shoes they include - however many - are equally preferred. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is the Leontief function: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mi>x</mi> <mo>,</mo> <mi>β</mi> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facfdc47b69fed439d16d9ea80680114d7ea9676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.626ex; height:2.176ex;" alt="{\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}}"></span>. </p><p>The different shapes of the curves imply different responses to a change in price as shown from demand analysis in <a href="/wiki/Consumer_theory" class="mw-redirect" title="Consumer theory">consumer theory</a>. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium on the same indifference curve: </p> <dl><dd>in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.</dd> <dd>in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the <a href="/wiki/Budget_constraint" title="Budget constraint">budget constraint</a>) or would change quantity demanded from one end of the <a href="/wiki/Budget_constraint" title="Budget constraint">budget constraint</a> to the other.</dd> <dd>in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Preference_relations_and_utility">Preference relations and utility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=6" title="Edit section: Preference relations and utility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Choice theory formally represents consumers by a <b>preference relation</b>, and use this representation to derive indifference curves showing combinations of equal preference to the consumer. </p> <div class="mw-heading mw-heading3"><h3 id="Preference_relations">Preference relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=7" title="Edit section: Preference relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> be a set of mutually exclusive alternatives among which a consumer can choose.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span> be generic elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span>.</dd></dl> <p>In the language of the example above, the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> is made of combinations of apples and bananas. The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is one such combination, such as 1 apple and 4 bananas and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span> is another combination such as 2 apples and 2 bananas. </p><p>A preference relation, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \succeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⪰</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \succeq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a76e2e7e4851d6699b00468080d1e25720f854d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \succeq }"></span>, is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> define on the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span>. </p><p>The statement </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\succeq b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⪰</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\succeq b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31d3fb826bac0f1b73190c324c9e8a90c0fc503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.971ex; height:2.343ex;" alt="{\displaystyle a\succeq b\;}"></span></dd></dl> <p>is described as '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is weakly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span>.' That is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is at least as good as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span> (in preference satisfaction). </p><p>The statement </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\sim b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∼</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\sim b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4076ef5896b650efb33617495310ede09959e9d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle a\sim b\;}"></span></dd></dl> <p>is described as '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is weakly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span> is weakly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span>.' That is, one is <i>indifferent</i> to the choice of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span>, meaning not that they are unwanted but that they are equally good in satisfying preferences. </p><p>The statement </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\succ b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≻</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\succ b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5f884bb0adcea530479753ed77b6c1717c0e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle a\succ b\;}"></span></dd></dl> <p>is described as '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is weakly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span>, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span> is not weakly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span>.' One says that '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> is strictly preferred to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dcf8d0b14c23e87a35c672d3939936dab5671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.643ex; height:2.176ex;" alt="{\displaystyle b\;}"></span>.' </p><p>The preference relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \succeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⪰</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \succeq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a76e2e7e4851d6699b00468080d1e25720f854d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \succeq }"></span> is <b>complete</b> if all pairs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08691c28f11d140ba29f6f2c72cc0d93026ba973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.906ex; height:2.509ex;" alt="{\displaystyle a,b\;}"></span> can be ranked. The relation is a <a href="/wiki/Transitive_relation" title="Transitive relation">transitive relation</a> if whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\succeq b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⪰</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\succeq b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31d3fb826bac0f1b73190c324c9e8a90c0fc503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.971ex; height:2.343ex;" alt="{\displaystyle a\succeq b\;}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\succeq c,\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>⪰</mo> <mi>c</mi> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\succeq c,\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d29b34bd89d68e88ce7116ce6ff72bacde32fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.395ex; height:2.509ex;" alt="{\displaystyle b\succeq c,\;}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\succeq c\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⪰</mo> <mi>c</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\succeq c\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1f169890961c3f9d8a90a021ddbe3000e985d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.98ex; height:2.176ex;" alt="{\displaystyle a\succeq c\;}"></span>. </p><p>For any element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44cceef289772ea3b38027d4a957cccdfa1bd1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.459ex; height:2.176ex;" alt="{\displaystyle a\in A\;}"></span>, the corresponding indifference curve, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acee3447e65eb0a8aaecdf10ae823614ff702d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.327ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{a}}"></span> is made up of all elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> which are indifferent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Formally, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mi>b</mi> <mo>∼</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee723210a468985ba0df339bf6bfb3171e4a878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.595ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Formal_link_to_utility_theory">Formal link to utility theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=8" title="Edit section: Formal link to utility theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the example above, an element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span> of the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> is made of two numbers: The number of apples, call it <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709f697fecb2ad226cc2bdb0f0fca25d264c8620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.622ex; height:2.009ex;" alt="{\displaystyle x,\;}"></span> and the number of bananas, call it <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y.\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>.</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432608df474c7edcb2945d345222635c74edfb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.447ex; height:2.009ex;" alt="{\displaystyle y.\;}"></span> </p><p>In <a href="/wiki/Utility" title="Utility">utility</a> theory, the <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a> of an <a href="/wiki/Agent_(economics)" title="Agent (economics)">agent</a> is a function that ranks <i>all</i> pairs of consumption bundles by order of preference (<i>completeness</i>) such that any set of three or more bundles forms a <a href="/wiki/Transitive_relation" title="Transitive relation">transitive relation</a>. This means that for each bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88bc79d6340964e46aff9d1e003f8aecbaf3949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)}"></span> there is a unique relation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4afcdf1b4790ec62489113d1539a169f3625b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.498ex; height:2.843ex;" alt="{\displaystyle U\left(x,y\right)}"></span>, representing the <a href="/wiki/Utility" title="Utility">utility</a> (satisfaction) relation associated with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88bc79d6340964e46aff9d1e003f8aecbaf3949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)}"></span>. The relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)\to U\left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)\to U\left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f3414e59d7ecd0351c39083371a10392927a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.441ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)\to U\left(x,y\right)}"></span> is called the <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a>. The <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> of the function is a set of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x,y)\geq U(x',y')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x,y)\geq U(x',y')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a3592c7be6fad798faf37f97789b9079f0a021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.695ex; height:3.009ex;" alt="{\displaystyle U(x,y)\geq U(x',y')}"></span>, then the bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88bc79d6340964e46aff9d1e003f8aecbaf3949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)}"></span> is described as at least as good as the bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x',y'\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x',y'\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb6a0c24edc64b1884b8d120e2d0fe05c30524a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.703ex; height:3.009ex;" alt="{\displaystyle \left(x',y'\right)}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)>U\left(x',y'\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>></mo> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)>U\left(x',y'\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccca90458e9dbd3609527f7a0a418f4f028733da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.469ex; height:3.009ex;" alt="{\displaystyle U\left(x,y\right)>U\left(x',y'\right)}"></span>, the bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88bc79d6340964e46aff9d1e003f8aecbaf3949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)}"></span> is described as strictly preferred to the bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x',y'\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x',y'\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb6a0c24edc64b1884b8d120e2d0fe05c30524a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.703ex; height:3.009ex;" alt="{\displaystyle \left(x',y'\right)}"></span>. </p><p>Consider a particular bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{0},y_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{0},y_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384022e4cd04beca44d5927eb34661ae1d1e793b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle \left(x_{0},y_{0}\right)}"></span> and take the <a href="/wiki/Total_derivative" title="Total derivative">total derivative</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4afcdf1b4790ec62489113d1539a169f3625b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.498ex; height:2.843ex;" alt="{\displaystyle U\left(x,y\right)}"></span> about this point: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4073fabc25eddc4fca0af0ccc527b3ddbf1fe663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.335ex; height:2.843ex;" alt="{\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}"></span></dd></dl> <p>or, without loss of generality, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mn>.1</mn> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075ac70b602d4868ec22cd06a7a72115d32bee82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.897ex; height:5.843ex;" alt="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}}"></span> <b>(Eq. 1)</b></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}\left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}\left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9365d98f088c2f98445598ce519237847c758724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.357ex; height:2.843ex;" alt="{\displaystyle U_{1}\left(x,y\right)}"></span> is the partial derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4afcdf1b4790ec62489113d1539a169f3625b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.498ex; height:2.843ex;" alt="{\displaystyle U\left(x,y\right)}"></span> with respect to its first argument, evaluated at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88bc79d6340964e46aff9d1e003f8aecbaf3949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \left(x,y\right)}"></span>. (Likewise for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}\left(x,y\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}\left(x,y\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5462a6be4dd84813292f430c49a130f8fda1138f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.391ex; height:2.843ex;" alt="{\displaystyle U_{2}\left(x,y\right).}"></span>) </p><p>The indifference curve through <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{0},y_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{0},y_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384022e4cd04beca44d5927eb34661ae1d1e793b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle \left(x_{0},y_{0}\right)}"></span> must deliver at each bundle on the curve the same utility level as bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{0},y_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{0},y_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384022e4cd04beca44d5927eb34661ae1d1e793b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle \left(x_{0},y_{0}\right)}"></span>. That is, when preferences are represented by a utility function, the indifference curves are the <a href="/wiki/Level_curve" class="mw-redirect" title="Level curve">level curves</a> of the utility function. Therefore, if one is to change the quantity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40983a43a92917b6921e62ddf582bc0a4e6e5015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.933ex; height:2.176ex;" alt="{\displaystyle dx\,}"></span>, without moving off the indifference curve, one must also change the quantity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}"></span> by an amount <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcf06c2e3008204b5b9e7a340df779f411216c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.758ex; height:2.509ex;" alt="{\displaystyle dy\,}"></span> such that, in the end, there is no change in <i>U</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c6f00ca7eed5c7de68708ed2b15322494a8b6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.903ex; height:5.843ex;" alt="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0}"></span>, or, substituting <i>0</i> into <i>(Eq. 1)</i> above to solve for <i>dy/dx</i>:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d157ca467c2561381b859322cff6ac48430f2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.704ex; height:6.509ex;" alt="{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}}"></span>.</dd></dl> <p>Thus, the ratio of marginal utilities gives the absolute value of the <a href="/wiki/Slope" title="Slope">slope</a> of the indifference curve at point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{0},y_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{0},y_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384022e4cd04beca44d5927eb34661ae1d1e793b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle \left(x_{0},y_{0}\right)}"></span>. This ratio is called the <a href="/wiki/Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Linear_utility">Linear utility</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=10" title="Edit section: Linear utility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the utility function is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)=\alpha x+\beta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)=\alpha x+\beta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c6af7027ede09541ab011e1b15634354cd4cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.742ex; height:2.843ex;" alt="{\displaystyle U\left(x,y\right)=\alpha x+\beta y}"></span> then the marginal utility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}\left(x,y\right)=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}\left(x,y\right)=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286b329bdf40fd33eb52df79665ed430e1968e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.943ex; height:2.843ex;" alt="{\displaystyle U_{1}\left(x,y\right)=\alpha }"></span> and the marginal utility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}\left(x,y\right)=\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}\left(x,y\right)=\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b76af64bea6d32e9589c9d8e833096ea0530422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.788ex; height:2.843ex;" alt="{\displaystyle U_{2}\left(x,y\right)=\beta }"></span>. The slope of the indifference curve is, therefore, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dy}}=-{\frac {\beta }{\alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>α</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dy}}=-{\frac {\beta }{\alpha }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253e49126b9f36a851f7cdf4985862154c418746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.259ex; height:5.843ex;" alt="{\displaystyle {\frac {dx}{dy}}=-{\frac {\beta }{\alpha }}.}"></span></dd></dl> <p>Observe that the slope does not depend on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}"></span>: the indifference curves are straight lines. </p> <div class="mw-heading mw-heading4"><h4 id="Cobb–Douglas_utility"><span id="Cobb.E2.80.93Douglas_utility"></span>Cobb–Douglas utility</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=11" title="Edit section: Cobb–Douglas utility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons: </p><p>1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety. </p><p>2. They are very flexible and can be adjusted to fit real-world data very easily. If the utility function is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629a73ef0f54c19a3a92b57648e2037865a5f840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.756ex; height:3.176ex;" alt="{\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }}"></span> the marginal utility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a11abbb672ba8099e61639a185d3f766442a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.785ex; height:3.343ex;" alt="{\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}}"></span> and the marginal utility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2be111904715687b36987cfc3b8d07b525e71b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.497ex; height:3.009ex;" alt="{\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }}"></span>.Where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4769a8dab3c8a2045bc128b9000da5d661f7dab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <1}"></span>. The <a href="/wiki/Slope" title="Slope">slope</a> of the indifference curve, and therefore the negative of the <a href="/wiki/Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a>, is then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>α</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723b6a2e6d9ee1a165ea0dd93eb84f13ee521cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.623ex; height:6.176ex;" alt="{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="CES_utility">CES utility</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=12" title="Edit section: CES utility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general CES (<a href="/wiki/Constant_Elasticity_of_Substitution" class="mw-redirect" title="Constant Elasticity of Substitution">Constant Elasticity of Substitution</a>) form is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x,y)=\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{1/\rho }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x,y)=\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{1/\rho }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4eef40cf0333b8b991656be5996cc00b8436a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.027ex; height:3.509ex;" alt="{\displaystyle U(x,y)=\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{1/\rho }}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7272729c5ddc8692703612118df631b03035d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho \leq 1}"></span>. (The <a href="/wiki/Cobb%E2%80%93Douglas" class="mw-redirect" title="Cobb–Douglas">Cobb–Douglas</a> is a special case of the CES utility, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \rightarrow 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \rightarrow 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea12eb37823ea04c895814dcf0b91ae2f08bb137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.366ex; height:2.676ex;" alt="{\displaystyle \rho \rightarrow 0\,}"></span>.) The marginal utilities are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271402529a5c9be6b2fd2c35e3f55e19914a8aaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.266ex; height:3.509ex;" alt="{\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7843adb77196b9493e16cbe17fcb510d5cf3d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.556ex; height:3.509ex;" alt="{\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}"></span></dd></dl> <p>Therefore, along an indifference curve, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right)^{1-\rho }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>α</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right)^{1-\rho }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878e0492b9a43b345130e5ab2ca65f31a8c5a88e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.031ex; height:6.509ex;" alt="{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right)^{1-\rho }.}"></span></dd></dl> <p>These examples might be useful for <a href="/wiki/Model_(economics)" class="mw-redirect" title="Model (economics)">modelling</a> individual or aggregate demand. </p> <div class="mw-heading mw-heading4"><h4 id="Biology">Biology</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=13" title="Edit section: Biology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As used in <a href="/wiki/Biology" title="Biology">biology</a>, the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability. </p> <div class="mw-heading mw-heading2"><h2 id="Criticisms">Criticisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=14" title="Edit section: Criticisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Indifference curves inherit the <a href="/wiki/Utility#Discussion_and_criticism" title="Utility">criticisms directed at utility</a> more generally. </p><p><a href="/wiki/Herbert_Hovenkamp" title="Herbert Hovenkamp">Herbert Hovenkamp</a> (1991)<sup id="cite_ref-Hovenkamp_15-0" class="reference"><a href="#cite_note-Hovenkamp-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> has argued that the presence of an <a href="/wiki/Endowment_effect" title="Endowment effect">endowment effect</a> has significant implications for <a href="/wiki/Law" title="Law">law</a> and <a href="/wiki/Economics" title="Economics">economics</a>, particularly in regard to <a href="/wiki/Welfare_economics" title="Welfare economics">welfare economics</a>. He argues that the presence of an endowment effect indicates that a person has no indifference curve (see however Hanemann, 1991<sup id="cite_ref-Hanemann_16-0" class="reference"><a href="#cite_note-Hanemann-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>) rendering the neoclassical tools of welfare analysis useless, concluding that courts should instead use <a href="/wiki/Willingness_to_accept" title="Willingness to accept">WTA</a> as a measure of value. Fischel (1995)<sup id="cite_ref-Fischel_17-0" class="reference"><a href="#cite_note-Fischel-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> however, raises the counterpoint that using WTA as a measure of value would deter the development of a nation's infrastructure and <a href="/wiki/Economic_growth" title="Economic growth">economic growth</a>. </p><p>Austrian economist <a href="/wiki/Murray_Rothbard" title="Murray Rothbard">Murray Rothbard</a> criticised the indifference curve as "never by definition exhibited in action, in actual exchanges, and is therefore unknowable and objectively meaningless."<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Budget_constraint" title="Budget constraint">Budget constraint</a></li> <li><a href="/wiki/Community_indifference_curve" title="Community indifference curve">Community indifference curve</a></li> <li><a href="/wiki/Consumer_theory" class="mw-redirect" title="Consumer theory">Consumer theory</a></li> <li><a href="/wiki/Convex_preferences" title="Convex preferences">Convex preferences</a></li> <li><a href="/wiki/Endowment_effect" title="Endowment effect">Endowment effect</a></li> <li><a href="/wiki/Level_curve" class="mw-redirect" title="Level curve">Level curve</a></li> <li><a href="/wiki/Microeconomics" title="Microeconomics">Microeconomics</a></li> <li><a href="/wiki/Rationality" title="Rationality">Rationality</a></li> <li><a href="/wiki/Substitute_good" title="Substitute good">Substitute good</a></li> <li><a href="/wiki/Utility%E2%80%93possibility_frontier" title="Utility–possibility frontier">Utility–possibility frontier</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">The transitivity of weak preferences is sufficient for most indifference-curve analyses: If <i>A</i> is weakly preferred to <i>B</i>, meaning that the consumer likes <i>A</i> <i>at least as much</i> as <i>B</i>, and <i>B</i> is weakly preferred to <i>C</i>, then <i>A</i> is weakly preferred to <i>C</i>.<sup id="cite_ref-Perloff_2008._p._62_8-1" class="reference"><a href="#cite_note-Perloff_2008._p._62-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Indifference curves can be used to derive the individual demand curve. However, the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Geanakoplis_(1987),_p._117-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Geanakoplis_(1987),_p._117_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGeanakoplos1987" class="citation book cs1">Geanakoplos, John (1987). 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Piccola Biblioteca Scientifica. Vol. 13. Milano: Societa Editrice Libraria.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manuale+di+Economia+Politica+%26mdash%3B+con+una+Introduzione+alla+Scienza+Sociale&rft.place=Milano&rft.series=Piccola+Biblioteca+Scientifica&rft.pub=Societa+Editrice+Libraria&rft.date=1919&rft.au=Vilfredo+Pareto&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmanualedieconomi00pareuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://policonomics.com/indifference-curves/">"Indifference curves | Policonomics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Indifference+curves+%7C+Policonomics&rft_id=https%3A%2F%2Fpoliconomics.com%2Findifference-curves%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.policonomics.com/william-stanley-jevons/">"William Stanley Jevons - Policonomics"</a>. <i>www.policonomics.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">23 March</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.policonomics.com&rft.atitle=William+Stanley+Jevons+-+Policonomics&rft_id=http%3A%2F%2Fwww.policonomics.com%2Fwilliam-stanley-jevons%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Binger-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Binger_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Binger_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Binger_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Binger_7-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Binger_7-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Binger_7-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Binger_7-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBingerHoffman1998" class="citation book cs1">Binger; Hoffman (1998). <i>Microeconomics with Calculus</i> (2nd ed.). Reading: Addison-Wesley. pp. 109–117. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-321-01225-9" title="Special:BookSources/0-321-01225-9"><bdi>0-321-01225-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microeconomics+with+Calculus&rft.place=Reading&rft.pages=109-117&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1998&rft.isbn=0-321-01225-9&rft.au=Binger&rft.au=Hoffman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Perloff_2008._p._62-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Perloff_2008._p._62_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Perloff_2008._p._62_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerloff2008" class="citation book cs1"><a href="/wiki/Jeffrey_M._Perloff" title="Jeffrey M. Perloff">Perloff, Jeffrey M.</a> (2008). <i>Microeconomics: Theory & Applications with Calculus</i>. Boston: Addison-Wesley. p. 62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-27794-7" title="Special:BookSources/978-0-321-27794-7"><bdi>978-0-321-27794-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microeconomics%3A+Theory+%26+Applications+with+Calculus&rft.place=Boston&rft.pages=62&rft.pub=Addison-Wesley&rft.date=2008&rft.isbn=978-0-321-27794-7&rft.aulast=Perloff&rft.aufirst=Jeffrey+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Silberberg-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Silberberg_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Silberberg_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Silberberg_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilberbergSuen2000" class="citation book cs1">Silberberg; Suen (2000). <i>The Structure of Economics: A Mathematical Analysis</i> (3rd ed.). Boston: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-118136-9" title="Special:BookSources/0-07-118136-9"><bdi>0-07-118136-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Structure+of+Economics%3A+A+Mathematical+Analysis&rft.place=Boston&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=2000&rft.isbn=0-07-118136-9&rft.au=Silberberg&rft.au=Suen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Lipsey_1975._pp._182-186-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lipsey_1975._pp._182-186_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lipsey_1975._pp._182-186_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipsey1975" class="citation book cs1"><a href="/wiki/Richard_Lipsey" title="Richard Lipsey">Lipsey, Richard G.</a> (1975). <i>An Introduction to Positive Economics</i> (Fourth ed.). <a href="/wiki/Weidenfeld_%26_Nicolson" title="Weidenfeld & Nicolson">Weidenfeld & Nicolson</a>. pp. 182–186. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-297-76899-9" title="Special:BookSources/0-297-76899-9"><bdi>0-297-76899-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Positive+Economics&rft.pages=182-186&rft.edition=Fourth&rft.pub=Weidenfeld+%26+Nicolson&rft.date=1975&rft.isbn=0-297-76899-9&rft.aulast=Lipsey&rft.aufirst=Richard+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Salvatore-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Salvatore_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalvatore1989" class="citation book cs1">Salvatore, Dominick (1989). <i>Schaum's Outline of Theory and Problems of Managerial Economics</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-054513-8" title="Special:BookSources/0-07-054513-8"><bdi>0-07-054513-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Theory+and+Problems+of+Managerial+Economics&rft.pub=McGraw-Hill&rft.date=1989&rft.isbn=0-07-054513-8&rft.aulast=Salvatore&rft.aufirst=Dominick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBingerHoffman1998" class="citation book cs1">Binger; Hoffman (1998). <i>Microeconomics with Calculus</i> (2nd ed.). Reading: Addison-Wesley. pp. 141–143. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-321-01225-9" title="Special:BookSources/0-321-01225-9"><bdi>0-321-01225-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microeconomics+with+Calculus&rft.place=Reading&rft.pages=141-143&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1998&rft.isbn=0-321-01225-9&rft.au=Binger&rft.au=Hoffman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Hovenkamp-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hovenkamp_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHovenkamp1991" class="citation journal cs1">Hovenkamp, Herbert (1991). "Legal Policy and the Endowment Effect". <i>The Journal of Legal Studies</i>. <b>20</b> (2): 225. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F467886">10.1086/467886</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:155051169">155051169</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Legal+Studies&rft.atitle=Legal+Policy+and+the+Endowment+Effect&rft.volume=20&rft.issue=2&rft.pages=225&rft.date=1991&rft_id=info%3Adoi%2F10.1086%2F467886&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A155051169%23id-name%3DS2CID&rft.aulast=Hovenkamp&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Hanemann-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hanemann_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanemann1991" class="citation journal cs1">Hanemann, W. Michael (1991). "Willingness To Pay and Willingness To Accept: How Much Can They Differ? Reply". <i>American Economic Review</i>. <b>81</b> (3): 635–647. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1257%2F000282803321455449">10.1257/000282803321455449</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2006525">2006525</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Review&rft.atitle=Willingness+To+Pay+and+Willingness+To+Accept%3A+How+Much+Can+They+Differ%3F+Reply&rft.volume=81&rft.issue=3&rft.pages=635-647&rft.date=1991&rft_id=info%3Adoi%2F10.1257%2F000282803321455449&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2006525%23id-name%3DJSTOR&rft.aulast=Hanemann&rft.aufirst=W.+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-Fischel-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fischel_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFischel1995" class="citation journal cs1">Fischel, William A. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0144-8188%2894%2900005-F">"The offer/ask disparity and just compensation for takings: A constitutional choice perspective"</a>. <i>International Review of Law and Economics</i>. <b>15</b> (2): 187–203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0144-8188%2894%2900005-F">10.1016/0144-8188(94)00005-F</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Review+of+Law+and+Economics&rft.atitle=The+offer%2Fask+disparity+and+just+compensation+for+takings%3A+A+constitutional+choice+perspective&rft.volume=15&rft.issue=2&rft.pages=187-203&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0144-8188%2894%2900005-F&rft.aulast=Fischel&rft.aufirst=William+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0144-8188%252894%252900005-F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRothbard1998" class="citation book cs1">Rothbard, Murray (1998). <i>The Ethics of Liberty</i>. New York University Press. p. 242. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780814775592" title="Special:BookSources/9780814775592"><bdi>9780814775592</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ethics+of+Liberty&rft.pages=242&rft.pub=New+York+University+Press&rft.date=1998&rft.isbn=9780814775592&rft.aulast=Rothbard&rft.aufirst=Murray&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Indifference_curve&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeattieLaFrance2006" class="citation journal cs1">Beattie, Bruce R.; LaFrance, Jeffrey T. (2006). <a rel="nofollow" class="external text" href="http://ageconsearch.umn.edu/record/25013/files/CUDARE%20959R%20LaFrance.pdf">"The Law of Demand versus Diminishing Marginal Utility"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Applied_Economic_Perspectives_and_Policy" title="Applied Economic Perspectives and Policy">Applied Economic Perspectives and Policy</a></i>. <b>28</b> (2): 263–271. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1467-9353.2006.00286.x">10.1111/j.1467-9353.2006.00286.x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154152189">154152189</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Economic+Perspectives+and+Policy&rft.atitle=The+Law+of+Demand+versus+Diminishing+Marginal+Utility&rft.volume=28&rft.issue=2&rft.pages=263-271&rft.date=2006&rft_id=info%3Adoi%2F10.1111%2Fj.1467-9353.2006.00286.x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154152189%23id-name%3DS2CID&rft.aulast=Beattie&rft.aufirst=Bruce+R.&rft.au=LaFrance%2C+Jeffrey+T.&rft_id=http%3A%2F%2Fageconsearch.umn.edu%2Frecord%2F25013%2Ffiles%2FCUDARE%2520959R%2520LaFrance.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKomlos2015" class="citation journal cs1">Komlos, J (2015). <a rel="nofollow" class="external text" href="http://www.uq.edu.au/economics/AJEE/docs/Volume%2012,%20Number%202,%202015/1.%20Behavioural%20Indifference%20Curves%20-%20John%20Komlos.pdf">"Behavioral Indifference Curves"</a> <span class="cs1-format">(PDF)</span>. <i>Australasian Journal of Economics Education</i>. <b>2</b>: 1–11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Australasian+Journal+of+Economics+Education&rft.atitle=Behavioral+Indifference+Curves&rft.volume=2&rft.pages=1-11&rft.date=2015&rft.aulast=Komlos&rft.aufirst=J&rft_id=http%3A%2F%2Fwww.uq.edu.au%2Feconomics%2FAJEE%2Fdocs%2FVolume%252012%2C%2520Number%25202%2C%25202015%2F1.%2520Behavioural%2520Indifference%2520Curves%2520-%2520John%2520Komlos.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndifference+curve" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span 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