A monopolist faces the following demand curve:<\/p>\n\n\n\n
Q = 144\/P2 where Q is the quantity demanded and P is price.<\/p>\n\n\n\n
Its average variable cost is AVC = Q1\/2 and its fixed cost is 5.<\/p>\n\n\n\n
What are its profit-maximizing price and quantity? What is the resulting profit?<\/p>\n\n\n\n
Suppose the government regulates the price to be no greater than $4 per unit. How much will the monopolist produce, and what will its profit be?<\/p>\n\n\n\n
Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will do this?<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n Given:<\/p>\n\n\n\n We can express the price ( P ) as a function of quantity ( Q ) by rearranging the demand curve:<\/p>\n\n\n\n [ Total Revenue is the price times the quantity, ( TR = P \\times Q ). Substituting the inverse demand function:<\/p>\n\n\n\n [ Total Cost is the sum of fixed cost and variable cost. The variable cost is derived from the Average Variable Cost (AVC):<\/p>\n\n\n\n [ Thus, the Total Cost (TC) is:<\/p>\n\n\n\n [ Profit (( \\pi )) is Total Revenue minus Total Cost:<\/p>\n\n\n\n [ To find the profit-maximizing quantity, we take the first derivative of the profit function with respect to ( Q ) and set it equal to zero:<\/p>\n\n\n\n [ [ Set the derivative equal to zero:<\/p>\n\n\n\n [ Solve for ( Q ):<\/p>\n\n\n\n [ Substitute ( Q = 4 ) into the inverse demand function to find the price:<\/p>\n\n\n\n [ Thus, the monopolist’s profit-maximizing price is ( P = 6 ), and the quantity produced is ( Q = 4 ).<\/p>\n\n\n\n To find the profit, we calculate Total Revenue and Total Cost at ( Q = 4 ).<\/p>\n\n\n\n Profit:<\/p>\n\n\n\n [ So, the monopolist\u2019s profit-maximizing quantity is 4, the price is 6, and the profit is 11.<\/p>\n\n\n\n Suppose the government regulates the price to no greater than ( P = 4 ).<\/p>\n\n\n\n At ( P = 4 ), we can find the quantity demanded using the demand curve:<\/p>\n\n\n\n [ So, the monopolist will produce 9 units.<\/p>\n\n\n\n Now, we calculate the monopolist’s profit at ( Q = 9 ) and ( P = 4 ).<\/p>\n\n\n\n Profit:<\/p>\n\n\n\n [ So, under the price regulation, the monopolist will produce 9 units and make a profit of 4.<\/p>\n\n\n\n To induce the monopolist to produce the largest possible output, the government needs to set a price that causes the monopolist to produce the highest quantity.<\/p>\n\n\n\n The monopolist maximizes output when price is equal to marginal cost (MC), since production will occur as long as MC is less than or equal to the price.<\/p>\n\n\n\n Marginal Cost is the derivative of Total Cost with respect to ( Q ):<\/p>\n\n\n\n [ To find the price that leads to the largest quantity, set price equal to marginal cost:<\/p>\n\n\n\n [ From the inverse demand function:<\/p>\n\n\n\n [ Set the two expressions for price equal:<\/p>\n\n\n\n [ Squaring both sides:<\/p>\n\n\n\n [ Solve for ( Q ):<\/p>\n\n\n\n [ At ( Q = 8 ), the price is:<\/p>\n\n\n\n [ Thus, the price that induces the monopolist to produce the largest possible output is approximately ( P = 4.24 ).<\/p>\n\n\n\n A monopolist faces the following demand curve: Q = 144\/P2 where Q is the quantity demanded and P is price. Its average variable cost is AVC = Q1\/2 and its fixed cost is 5. What are its profit-maximizing price and quantity? What is the resulting profit? Suppose the government regulates the price to be no […]<\/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-204666","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204666","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=204666"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204666\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=204666"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=204666"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=204666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Profit Maximization for the Monopolist<\/h3>\n\n\n\n
\n
1.1: Inverse Demand Function<\/h4>\n\n\n\n
Q = \\frac{144}{P^2} \\quad \\Rightarrow \\quad P = \\sqrt{\\frac{144}{Q}}
]<\/p>\n\n\n\n1.2: Total Revenue (TR)<\/h4>\n\n\n\n
TR = \\sqrt{\\frac{144}{Q}} \\times Q = 12 \\sqrt{Q}
]<\/p>\n\n\n\n1.3: Total Cost (TC)<\/h4>\n\n\n\n
VC = AVC \\times Q = Q^{1\/2} \\times Q = Q^{3\/2}
]<\/p>\n\n\n\n
TC = FC + VC = 5 + Q^{3\/2}
]<\/p>\n\n\n\n1.4: Profit Function<\/h4>\n\n\n\n
\\pi = TR – TC = 12\\sqrt{Q} – (5 + Q^{3\/2})
]<\/p>\n\n\n\n1.5: Maximizing Profit<\/h4>\n\n\n\n
\\frac{d\\pi}{dQ} = \\frac{d}{dQ}\\left( 12\\sqrt{Q} – 5 – Q^{3\/2} \\right)
]<\/p>\n\n\n\n
\\frac{d\\pi}{dQ} = \\frac{6}{\\sqrt{Q}} – \\frac{3}{2} Q^{1\/2}
]<\/p>\n\n\n\n
\\frac{6}{\\sqrt{Q}} = \\frac{3}{2} Q^{1\/2}
]<\/p>\n\n\n\n
12 = 3 Q \\quad \\Rightarrow \\quad Q = 4
]<\/p>\n\n\n\n1.6: Finding the Profit-Maximizing Price<\/h4>\n\n\n\n
P = \\sqrt{\\frac{144}{Q}} = \\sqrt{\\frac{144}{4}} = \\sqrt{36} = 6
]<\/p>\n\n\n\n1.7: Profit<\/h4>\n\n\n\n
\n
\\pi = TR – TC = 24 – 13 = 11
]<\/p>\n\n\n\nStep 2: Government Price Regulation<\/h3>\n\n\n\n
Q = \\frac{144}{P^2} = \\frac{144}{4^2} = \\frac{144}{16} = 9
]<\/p>\n\n\n\n2.1: Profit under Price Regulation<\/h4>\n\n\n\n
\n
\\pi = TR – TC = 36 – 32 = 4
]<\/p>\n\n\n\nStep 3: Ceiling Price for Maximum Output<\/h3>\n\n\n\n
3.1: Marginal Cost (MC)<\/h4>\n\n\n\n
MC = \\frac{d}{dQ} (5 + Q^{3\/2}) = \\frac{3}{2} Q^{1\/2}
]<\/p>\n\n\n\n
P = MC = \\frac{3}{2} Q^{1\/2}
]<\/p>\n\n\n\n
P = \\sqrt{\\frac{144}{Q}}
]<\/p>\n\n\n\n
\\sqrt{\\frac{144}{Q}} = \\frac{3}{2} Q^{1\/2}
]<\/p>\n\n\n\n
\\frac{144}{Q} = \\frac{9}{4} Q
]<\/p>\n\n\n\n
144 = \\frac{9}{4} Q^2 \\quad \\Rightarrow \\quad Q^2 = \\frac{576}{9} = 64 \\quad \\Rightarrow \\quad Q = 8
]<\/p>\n\n\n\n
P = \\sqrt{\\frac{144}{8}} = \\sqrt{18} \\approx 4.24
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
\n