{"id":250703,"date":"2025-07-10T12:01:17","date_gmt":"2025-07-10T12:01:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=250703"},"modified":"2025-07-10T12:01:24","modified_gmt":"2025-07-10T12:01:24","slug":"marginal-revenue-function-find-the-marginal-revenue-mr","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/marginal-revenue-function-find-the-marginal-revenue-mr\/","title":{"rendered":"[Marginal Revenue Function] Find the marginal revenue (MR)"},"content":{"rendered":"\n

[Marginal Revenue Function] Find the marginal revenue (MR) functions for each of the following demand functions and evaluate them at Q = 4 and Q = 10. (a) Demand Function Q = 36 – 2P. (b) Demand Function 44 – 4P – Q = 0<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find the marginal revenue (MR)<\/strong> function, we need to first express the revenue function (R)<\/strong> in terms of quantity (Q)<\/strong>, and then differentiate it with respect to Q. Here’s how we can proceed for each case:<\/p>\n\n\n\n

(a) Demand Function: Q = 36 – 2P<\/strong><\/h3>\n\n\n\n
    \n
  1. Step 1<\/strong>: Express the demand function in terms of price (P). Rearranging the equation, we get: Q=36\u22122P\u2005\u200a\u27f9\u2005\u200aP=36\u2212Q2Q = 36 – 2P \\implies P = \\frac{36 – Q}{2}Q=36\u22122P\u27f9P=236\u2212Q\u200b<\/li>\n\n\n\n
  2. Step 2<\/strong>: Find the revenue function, RRR. Revenue is given by: R=P\u00d7QR = P \\times QR=P\u00d7Q Substituting the expression for PPP into the revenue formula: R=(36\u2212Q2)\u00d7Q=(36\u2212Q)Q2=36Q\u2212Q22R = \\left( \\frac{36 – Q}{2} \\right) \\times Q = \\frac{(36 – Q)Q}{2} = \\frac{36Q – Q^2}{2}R=(236\u2212Q\u200b)\u00d7Q=2(36\u2212Q)Q\u200b=236Q\u2212Q2\u200b<\/li>\n\n\n\n
  3. Step 3<\/strong>: Differentiate the revenue function with respect to QQQ to find the marginal revenue function: MR=dRdQ=ddQ(36Q\u2212Q22)MR = \\frac{dR}{dQ} = \\frac{d}{dQ} \\left( \\frac{36Q – Q^2}{2} \\right)MR=dQdR\u200b=dQd\u200b(236Q\u2212Q2\u200b) Using basic differentiation: MR=12(36\u22122Q)=18\u2212QMR = \\frac{1}{2} \\left( 36 – 2Q \\right) = 18 – QMR=21\u200b(36\u22122Q)=18\u2212Q<\/li>\n\n\n\n
  4. Step 4<\/strong>: Evaluate the MR function at Q=4Q = 4Q=4 and Q=10Q = 10Q=10:\n
      \n
    • When Q=4Q = 4Q=4: MR(4)=18\u22124=14MR(4) = 18 – 4 = 14MR(4)=18\u22124=14<\/li>\n\n\n\n
    • When Q=10Q = 10Q=10: MR(10)=18\u221210=8MR(10) = 18 – 10 = 8MR(10)=18\u221210=8<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

      Thus, the marginal revenue function is MR=18\u2212QMR = 18 – QMR=18\u2212Q, and at Q=4Q = 4Q=4, MR=14MR = 14MR=14, and at Q=10Q = 10Q=10, MR=8MR = 8MR=8.<\/p>\n\n\n\n

      \n\n\n\n

      (b) Demand Function: 44 – 4P – Q = 0<\/strong><\/h3>\n\n\n\n
        \n
      1. Step 1<\/strong>: Express the demand function in terms of price (P). Rearranging the equation: 44\u22124P\u2212Q=0\u2005\u200a\u27f9\u2005\u200a4P=44\u2212Q\u2005\u200a\u27f9\u2005\u200aP=44\u2212Q444 – 4P – Q = 0 \\implies 4P = 44 – Q \\implies P = \\frac{44 – Q}{4}44\u22124P\u2212Q=0\u27f94P=44\u2212Q\u27f9P=444\u2212Q\u200b<\/li>\n\n\n\n
      2. Step 2<\/strong>: Find the revenue function, RRR. Revenue is: R=P\u00d7QR = P \\times QR=P\u00d7Q Substituting for PPP: R=(44\u2212Q4)\u00d7Q=(44\u2212Q)Q4=44Q\u2212Q24R = \\left( \\frac{44 – Q}{4} \\right) \\times Q = \\frac{(44 – Q)Q}{4} = \\frac{44Q – Q^2}{4}R=(444\u2212Q\u200b)\u00d7Q=4(44\u2212Q)Q\u200b=444Q\u2212Q2\u200b<\/li>\n\n\n\n
      3. Step 3<\/strong>: Differentiate the revenue function with respect to QQQ to find the marginal revenue: MR=dRdQ=ddQ(44Q\u2212Q24)MR = \\frac{dR}{dQ} = \\frac{d}{dQ} \\left( \\frac{44Q – Q^2}{4} \\right)MR=dQdR\u200b=dQd\u200b(444Q\u2212Q2\u200b) Using basic differentiation: MR=14(44\u22122Q)=11\u2212Q2MR = \\frac{1}{4} \\left( 44 – 2Q \\right) = 11 – \\frac{Q}{2}MR=41\u200b(44\u22122Q)=11\u22122Q\u200b<\/li>\n\n\n\n
      4. Step 4<\/strong>: Evaluate the MR function at Q=4Q = 4Q=4 and Q=10Q = 10Q=10:\n
          \n
        • When Q=4Q = 4Q=4: MR(4)=11\u221242=11\u22122=9MR(4) = 11 – \\frac{4}{2} = 11 – 2 = 9MR(4)=11\u221224\u200b=11\u22122=9<\/li>\n\n\n\n
        • When Q=10Q = 10Q=10: MR(10)=11\u2212102=11\u22125=6MR(10) = 11 – \\frac{10}{2} = 11 – 5 = 6MR(10)=11\u2212210\u200b=11\u22125=6<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

          Thus, the marginal revenue function is MR=11\u2212Q2MR = 11 – \\frac{Q}{2}MR=11\u22122Q\u200b, and at Q=4Q = 4Q=4, MR=9MR = 9MR=9, and at Q=10Q = 10Q=10, MR=6MR = 6MR=6.<\/p>\n\n\n\n

          \n\n\n\n

          Summary:<\/h3>\n\n\n\n
            \n
          • For demand function Q=36\u22122PQ = 36 – 2PQ=36\u22122P, the MR function is MR=18\u2212QMR = 18 – QMR=18\u2212Q, with values at Q=4Q = 4Q=4 and Q=10Q = 10Q=10 as 14 and 8, respectively.<\/li>\n\n\n\n
          • For demand function 44\u22124P\u2212Q=044 – 4P – Q = 044\u22124P\u2212Q=0, the MR function is MR=11\u2212Q2MR = 11 – \\frac{Q}{2}MR=11\u22122Q\u200b, with values at Q=4Q = 4Q=4 and Q=10Q = 10Q=10 as 9 and 6, respectively.<\/li>\n<\/ul>\n\n\n\n
            \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

            [Marginal Revenue Function] Find the marginal revenue (MR) functions for each of the following demand functions and evaluate them at Q = 4 and Q = 10. (a) Demand Function Q = 36 – 2P. (b) Demand Function 44 – 4P – Q = 0 The Correct Answer and Explanation is: To find the marginal […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-250703","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=250703"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250703\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=250703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=250703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=250703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}