{"id":2921,"date":"2022-01-03T23:38:21","date_gmt":"2022-01-03T15:38:21","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=2921"},"modified":"2022-03-02T16:51:26","modified_gmt":"2022-03-02T08:51:26","slug":"mrs-and-mrts","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=2921","title":{"rendered":"MRS and MRTS"},"content":{"rendered":"\n<h4 class=\"wp-block-heading\">Derivations<\/h4>\n\n\n\n<p>We here derive why \\(MRS_{x,y}=\\frac{MU_x}{MU_y}\\).<\/p>\n\n\n\n<p>Let \\(U(x,y)=f(x,y)\\), and we know, by definition, MRS measures how many units of x is needed to trade y holding utility constant. Thus, we keep the utility function unchanged, \\(U(x,y)=C\\), and take differentiation and find \\(-dy\/dx\\).<\/p>\n\n\n\n<p>$$f(x,y) dx=C dx$$<\/p>\n\n\n\n<p>$$ \\frac{\\partial f(x,y)}{x}+\\frac{\\partial f(x,y)}{\\partial y}\\frac{\\partial y}{\\partial x}=0 $$<\/p>\n\n\n\n<p>$$\\frac{\\partial y}{\\partial x}=-\\frac{\\frac{\\partial f(x,y)}{\\partial x}}{\\frac{\\partial f(x,y)}{\\partial y}}=\\frac{MU_x}{MU_y}$$<\/p>\n\n\n\n<p>Therefore,<\/p>\n\n\n\n<p> <span class=\"katex math multi-line\">MRS_{x,y}=-\\frac{dy}{dx}=\\frac{MU_x}{MU_y}<\/span><\/p>\n\n\n\n<p>$$|MRS_{x,y}|=-\\frac{dy}{dx}=\\frac{MU_x}{MU_y} $$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Example 1<\/h4>\n\n\n\n<p>$$U=x^2+y^2$$<\/p>\n\n\n\n<p>$$MRS_{x,y}=\\frac{MU_x}{MU_y} =\\frac{x}{y}$$<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-2.png\" alt=\"\" class=\"wp-image-2925\" width=\"426\" height=\"306\" srcset=\"http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-2.png 850w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-2-300x216.png 300w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-2-768x554.png 768w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Example 2<\/h4>\n\n\n\n<p>$$U=x\\cdot y$$<\/p>\n\n\n\n<p>, which is similar as the Cobb-Douglas form but has exponenets zero.<\/p>\n\n\n\n<p>$$MRS_{x,y}=\\frac{MU_x}{MU_y} =\\frac{y}{x}$$<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-3.png\" alt=\"\" class=\"wp-image-2927\" width=\"496\" height=\"352\" srcset=\"http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-3.png 870w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-3-300x213.png 300w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-3-768x545.png 768w\" sizes=\"(max-width: 496px) 100vw, 496px\" \/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Example 3<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"439\" height=\"392\" src=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-4.png\" alt=\"\" class=\"wp-image-2928\" srcset=\"http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-4.png 439w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-4-300x268.png 300w\" sizes=\"(max-width: 439px) 100vw, 439px\" \/><figcaption>Perfect Substitution: MRS constant<\/figcaption><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"407\" height=\"385\" src=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-6.png\" alt=\"\" class=\"wp-image-2930\" srcset=\"http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-6.png 407w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/01\/image-6-300x284.png 300w\" sizes=\"(max-width: 407px) 100vw, 407px\" \/><figcaption>Perfect Complement<\/figcaption><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">MRTS<\/h4>\n\n\n\n<p>Marginal Rate of Technical Substitution (MRTS) measures the amount of cost which a specific input can be replaced for another resource of production while maintaining a constant output.<\/p>\n\n\n\n<p>$$MRTS_{K,L}=-\\frac{\\Delta K}{\\Delta L}=-\\frac{d K}{d L}=\\frac{MP_L}{MP_K}$$<\/p>\n\n\n\n<p>How to derive that?<\/p>\n\n\n\n<p>Recall the Isoquant that is equivalent to the contour line of the output function. MRTS is like the slope of the isoquant line. We let,<\/p>\n\n\n\n<p>$$Q=L^a K^b$$<\/p>\n\n\n\n<p>Then,<\/p>\n\n\n\n<p>$$MP_K=\\frac{\\partial Q}{\\partial K}=b L^A K^{b-1}$$<\/p>\n\n\n\n<p>$$MP_L=\\frac{\\partial Q}{\\partial L}=a L^{a-1}K^b$$<\/p>\n\n\n\n<p>$$MRTS=\\frac{ b L^A K^{b-1} }{  a L^{a-1}K^b  }=\\frac{aK}{bL}$$<\/p>\n\n\n\n<p>In short, MRTS is a similar concept to MRS, but in the output aspect.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derivations We here derive why \\(MRS_{x,y}=\\frac{MU_x}{MU_y}\\). Let \\(U(x,y)=f(x,y)\\), and we know, by definition, MRS measures how many units of x is needed to trade y holding utility constant. Thus, we keep the utility function unchanged, \\(U(x,y)=C\\), and take differentiation and find \\(-dy\/dx\\). $$f(x,y) dx=C dx$$ $$ \\frac{\\partial f(x,y)}{x}+\\frac{\\partial f(x,y)}{\\partial y}\\frac{\\partial y}{\\partial x}=0 $$ $$\\frac{\\partial y}{\\partial &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=2921\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">MRS and MRTS<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,6,18],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2921"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2921"}],"version-history":[{"count":31,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2921\/revisions"}],"predecessor-version":[{"id":3028,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2921\/revisions\/3028"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2921"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2921"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}