{"id":219562,"date":"2025-05-26T10:43:45","date_gmt":"2025-05-26T10:43:45","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219562"},"modified":"2025-05-26T10:43:48","modified_gmt":"2025-05-26T10:43:48","slug":"a-firm-has-two-factories-for-which-costs-are-given-by","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/a-firm-has-two-factories-for-which-costs-are-given-by\/","title":{"rendered":"A firm has two factories, for which costs are given by"},"content":{"rendered":"\n<p>A firm has two factories, for which costs are given by:<\/p>\n\n\n\n<p>Factory #I: C1(Q1) = 10Q12<\/p>\n\n\n\n<p>Factory #2: C2(Q2) = 20Q22<\/p>\n\n\n\n<p>The firm faces the following demand curve:<\/p>\n\n\n\n<p>P = 700 &#8211; 5Q<\/p>\n\n\n\n<p>where Q is total output, i.e., Q = Q1 + Q2.<\/p>\n\n\n\n<p>On a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal cost of producing Q = Q1 + Q2). Indicate the profit-maximizing output for each factory, total output, and price.<br>Calculate the values of Q1, Q2, Q, and P that maximize profit.<br>Suppose labor costs increase in Factory #I but not in Factory #2. How should the firm adjust (i.e., raise, lower, or leave unchanged): Output in Factory #1? Output in Factory #2? Total output? Price?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Solution:<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 1: Given<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Factory 1 Cost<\/strong>: C1(Q1)=10Q12C_1(Q_1) = 10Q_1^2 \u2192 MC1=dC1dQ1=20Q1MC_1 = \\frac{dC_1}{dQ_1} = 20Q_1<\/li>\n\n\n\n<li><strong>Factory 2 Cost<\/strong>: C2(Q2)=20Q22C_2(Q_2) = 20Q_2^2 \u2192 MC2=dC2dQ2=40Q2MC_2 = \\frac{dC_2}{dQ_2} = 40Q_2<\/li>\n\n\n\n<li><strong>Demand<\/strong>: P=700\u22125QP = 700 &#8211; 5Q, where Q=Q1+Q2Q = Q_1 + Q_2<\/li>\n\n\n\n<li><strong>Total Revenue<\/strong>: TR=P\u22c5Q=(700\u22125Q)Q=700Q\u22125Q2TR = P \\cdot Q = (700 &#8211; 5Q)Q = 700Q &#8211; 5Q^2<\/li>\n\n\n\n<li><strong>Marginal Revenue (MR)<\/strong>: MR=dTRdQ=700\u221210QMR = \\frac{dTR}{dQ} = 700 &#8211; 10Q<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Profit Maximization Condition<\/strong><\/h3>\n\n\n\n<p>The firm maximizes profit when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>MR=MC1=20Q1MR = MC_1 = 20Q_1<\/li>\n\n\n\n<li>MR=MC2=40Q2MR = MC_2 = 40Q_2<\/li>\n<\/ul>\n\n\n\n<p>So: 20Q1=700\u221210Q(1)40Q2=700\u221210Q(2)20Q_1 = 700 &#8211; 10Q \\quad \\text{(1)} \\\\ 40Q_2 = 700 &#8211; 10Q \\quad \\text{(2)}<\/p>\n\n\n\n<p>Since Q=Q1+Q2Q = Q_1 + Q_2, substitute this into both:<\/p>\n\n\n\n<p>From (1):<br>Q1=700\u221210(Q1+Q2)20Q_1 = \\frac{700 &#8211; 10(Q_1 + Q_2)}{20}<\/p>\n\n\n\n<p>From (2):<br>Q2=700\u221210(Q1+Q2)40Q_2 = \\frac{700 &#8211; 10(Q_1 + Q_2)}{40}<\/p>\n\n\n\n<p>Now solve these equations simultaneously.<\/p>\n\n\n\n<p>From (1):<br>20Q1=700\u221210(Q1+Q2)20Q_1 = 700 &#8211; 10(Q_1 + Q_2)<br>\u2192 20Q1+10Q1+10Q2=70020Q_1 + 10Q_1 + 10Q_2 = 700<br>\u2192 30Q1+10Q2=70030Q_1 + 10Q_2 = 700<br>\u2192 (a) 3Q1+Q2=703Q_1 + Q_2 = 70<\/p>\n\n\n\n<p>From (2):<br>40Q2=700\u221210(Q1+Q2)40Q_2 = 700 &#8211; 10(Q_1 + Q_2)<br>\u2192 40Q2+10Q1+10Q2=70040Q_2 + 10Q_1 + 10Q_2 = 700<br>\u2192 10Q1+50Q2=70010Q_1 + 50Q_2 = 700<br>\u2192 (b) Q1+5Q2=70Q_1 + 5Q_2 = 70<\/p>\n\n\n\n<p>Solving (a) and (b):<\/p>\n\n\n\n<p>Multiply (a) by 5:<br>\u2192 15Q1+5Q2=35015Q_1 + 5Q_2 = 350<br>Now subtract (b):<br>15Q1+5Q2\u2212(Q1+5Q2)=350\u22127015Q_1 + 5Q_2 &#8211; (Q_1 + 5Q_2) = 350 &#8211; 70<br>\u2192 14Q1=28014Q_1 = 280<br>\u2192 Q1=20Q_1 = 20<\/p>\n\n\n\n<p>Substitute into (a):<br>3(20)+Q2=703(20) + Q_2 = 70<br>\u2192 Q2=10Q_2 = 10<\/p>\n\n\n\n<p>Then:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Total Output<\/strong>: Q=20+10=30Q = 20 + 10 = 30<\/li>\n\n\n\n<li><strong>Price<\/strong>: P=700\u22125(30)=550P = 700 &#8211; 5(30) = 550<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer Summary:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Q\u2081<\/strong> = 20<\/li>\n\n\n\n<li><strong>Q\u2082<\/strong> = 10<\/li>\n\n\n\n<li><strong>Q<\/strong> = 30<\/li>\n\n\n\n<li><strong>P<\/strong> = 550<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Adjustment to Labor Cost Increase in Factory 1:<\/strong><\/h3>\n\n\n\n<p>If labor costs increase in <strong>Factory 1<\/strong>, its <strong>marginal cost curve shifts upward<\/strong>. Thus, for any given price, <strong>Factory 1 should produce less<\/strong>, and <strong>Factory 2 should produce more<\/strong>, since its cost structure remains unchanged. However, <strong>total output should fall<\/strong>, and <strong>price should rise<\/strong> slightly, because the overall marginal cost of production has increased.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation <\/strong><\/h3>\n\n\n\n<p>To determine the firm&#8217;s optimal output and pricing strategy, we analyze marginal revenue (MR) and marginal costs (MC) for both factories. Each factory\u2019s cost structure is quadratic, so their MCs are linear: MC1=20Q1MC_1 = 20Q_1, MC2=40Q2MC_2 = 40Q_2. Given a market demand curve P=700\u22125QP = 700 &#8211; 5Q, the firm\u2019s total revenue is TR=P\u22c5Q=700Q\u22125Q2TR = P \\cdot Q = 700Q &#8211; 5Q^2, making MR = 700\u221210Q700 &#8211; 10Q.<\/p>\n\n\n\n<p>For profit maximization, the firm allocates output so that MR equals the marginal cost of each factory. Setting MR=MC1=20Q1MR = MC_1 = 20Q_1 and MR=MC2=40Q2MR = MC_2 = 40Q_2 leads to a system of equations that, when solved, yields Q1=20Q_1 = 20, Q2=10Q_2 = 10, and total output Q=30Q = 30. Plugging into the demand curve, the corresponding price is P=550P = 550.<\/p>\n\n\n\n<p>If labor costs increase in Factory 1, the cost and thus the marginal cost of producing each unit in Factory 1 rises. This shifts MC1MC_1 upward. To maintain profit-maximization (where MR = MC), Factory 1 should reduce output. The firm will compensate by increasing production in Factory 2, up to the point where MR=MC2MR = MC_2 again. However, since Factory 2 has a steeper MC curve, total output will decrease overall. Lower total supply leads to a higher market-clearing price.<\/p>\n\n\n\n<p>In summary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Output in Factory 1<\/strong>: Decrease<\/li>\n\n\n\n<li><strong>Output in Factory 2<\/strong>: Increase<\/li>\n\n\n\n<li><strong>Total Output<\/strong>: Decrease<\/li>\n\n\n\n<li><strong>Price<\/strong>: Increase<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner5-24.jpeg\" alt=\"\" class=\"wp-image-219563\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A firm has two factories, for which costs are given by: Factory #I: C1(Q1) = 10Q12 Factory #2: C2(Q2) = 20Q22 The firm faces the following demand curve: P = 700 &#8211; 5Q where Q is total output, i.e., Q = Q1 + Q2. On a diagram, draw the marginal cost curves for the two [&hellip;]<\/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-219562","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219562","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=219562"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219562\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}