{"id":185940,"date":"2025-01-23T18:33:36","date_gmt":"2025-01-23T18:33:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185940"},"modified":"2025-01-23T18:33:38","modified_gmt":"2025-01-23T18:33:38","slug":"b-suppose-a-consumer-has-900000-shs-to-be-allocated-between-commodities-x-and-y","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/23\/b-suppose-a-consumer-has-900000-shs-to-be-allocated-between-commodities-x-and-y\/","title":{"rendered":"(b). Suppose a consumer has 900,000 shs. To be allocated between commodities x and y"},"content":{"rendered":"\n<p>(b). Suppose a consumer has 900,000 shs. To be allocated between commodities x and y. The price of commodity y is 2000.find the demand function for x and y if the utility function is<br><\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the demand function for commodities xx and yy given a consumer&#8217;s utility function and budget, we proceed as follows:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Problem Setup<\/h3>\n\n\n\n<p>Let:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>U(x,y)U(x, y): Utility function.<\/li>\n\n\n\n<li>PxP_x: Price of commodity xx.<\/li>\n\n\n\n<li>Py=2000P_y = 2000: Price of commodity yy.<\/li>\n\n\n\n<li>M=900,000M = 900,000: Total income or budget.<\/li>\n<\/ul>\n\n\n\n<p>Suppose the utility function is of the form U(x,y)=xaybU(x, y) = x^a y^b, where aa and bb are positive constants that define the relative preference for xx and yy. Without a specific utility function provided, we assume this general Cobb-Douglas utility function.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Demand Function Derivation<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Budget Constraint:<\/strong> The total expenditure must equal the budget: Pxx+Pyy=MP_x x + P_y y = M<\/li>\n\n\n\n<li><strong>Optimization Problem:<\/strong> The consumer maximizes utility U(x,y)=xaybU(x, y) = x^a y^b subject to the budget constraint Pxx+Pyy=MP_x x + P_y y = M.<\/li>\n\n\n\n<li><strong>Lagrangian Method:<\/strong> Define the Lagrangian function: L=xayb+\u03bb(M\u2212Pxx\u2212Pyy)\\mathcal{L} = x^a y^b + \\lambda (M &#8211; P_x x &#8211; P_y y) Take partial derivatives with respect to xx, yy, and \u03bb\\lambda, and set them equal to zero: \u2202L\u2202x=axa\u22121yb\u2212\u03bbPx=0\\frac{\\partial \\mathcal{L}}{\\partial x} = a x^{a-1} y^b &#8211; \\lambda P_x = 0 \u2202L\u2202y=bxayb\u22121\u2212\u03bbPy=0\\frac{\\partial \\mathcal{L}}{\\partial y} = b x^a y^{b-1} &#8211; \\lambda P_y = 0 \u2202L\u2202\u03bb=M\u2212Pxx\u2212Pyy=0\\frac{\\partial \\mathcal{L}}{\\partial \\lambda} = M &#8211; P_x x &#8211; P_y y = 0<\/li>\n\n\n\n<li><strong>Solving for Demand Functions:<\/strong> Divide the first two equations to eliminate \u03bb\\lambda: axa\u22121ybbxayb\u22121=PxPy\\frac{a x^{a-1} y^b}{b x^a y^{b-1}} = \\frac{P_x}{P_y} Simplify: ab\u22c5yx=PxPy\\frac{a}{b} \\cdot \\frac{y}{x} = \\frac{P_x}{P_y} Rearrange: yx=bPxaPy\\frac{y}{x} = \\frac{b P_x}{a P_y} Solve for yy in terms of xx: y=bPxaPyxy = \\frac{b P_x}{a P_y} x Substitute yy into the budget constraint: Pxx+Py(bPxaPyx)=MP_x x + P_y \\left(\\frac{b P_x}{a P_y} x\\right) = M Simplify: Pxx(1+ba)=MP_x x \\left(1 + \\frac{b}{a}\\right) = M x=MPx(1+ba)x = \\frac{M}{P_x \\left(1 + \\frac{b}{a}\\right)} Similarly, find yy by substituting xx back: y=bPxaPy\u22c5MPx(1+ba)y = \\frac{b P_x}{a P_y} \\cdot \\frac{M}{P_x \\left(1 + \\frac{b}{a}\\right)} y=bMPy(a+b)y = \\frac{b M}{P_y (a + b)}<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Final Demand Functions:<\/h3>\n\n\n\n<p>The demand functions for xx and yy are: x=aMPx(a+b)x = \\frac{a M}{P_x (a + b)} y=bMPy(a+b)y = \\frac{b M}{P_y (a + b)}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>The demand functions for xx and yy are derived based on the consumer&#8217;s objective to maximize utility given a fixed income and prices. The utility function U(x,y)=xaybU(x, y) = x^a y^b reflects the consumer&#8217;s preferences, with aa and bb indicating the relative importance of each commodity. The budget constraint ensures the consumer does not spend more than their income, M=900,000M = 900,000.<\/p>\n\n\n\n<p>The Lagrangian optimization method is used to solve this constrained maximization problem. By setting up and solving the first-order conditions, we derive relationships between the quantities of xx and yy and their prices. The ratio y\/x=bPxaPyy\/x = \\frac{b P_x}{a P_y} indicates that the consumer&#8217;s allocation between xx and yy depends on their relative prices and preferences.<\/p>\n\n\n\n<p>The demand function for xx shows that the quantity demanded decreases as the price of xx increases (law of demand). Similarly, the demand for yy decreases with higher PyP_y. The income MM is allocated proportionally based on the weights aa and bb, ensuring optimal satisfaction.<\/p>\n\n\n\n<p>These functions provide a foundational tool in consumer theory, helping to predict behavior under different price and income scenarios.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(b). Suppose a consumer has 900,000 shs. To be allocated between commodities x and y. The price of commodity y is 2000.find the demand function for x and y if the utility function is The correct answer and explanation is: To find the demand function for commodities xx and yy given a consumer&#8217;s utility function [&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-185940","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185940","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=185940"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185940\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185940"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185940"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}