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<title>Expenditure function</title>

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Expenditure function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Microeconomics" title="Microeconomics">microeconomics</a>, the expenditure function represents the minimum amount of expenditure needed to achieve a given level of <a href="Utility" title="Utility">utility</a>, given a <a href="Utility_function" class="mw-redirect" title="Utility function">utility function</a> and the prices of goods.

Formally, if there is a utility function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> that describes <a href="Preference_(economics)" title="Preference (economics)">preferences</a> over n goods, the expenditure function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p,u^{*})}">

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<annotation encoding="application/x-tex">{\displaystyle e(p,u^{*})}</annotation>

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</math><img src="./7dc7882839829917cf4d4c8d03c9ab4b6ac82f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.48ex; height:2.843ex;" alt="{\displaystyle e(p,u^{*})}" loading="lazy"> is defined as:

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<annotation encoding="application/x-tex">{\displaystyle e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x}</annotation>

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</math><img src="./ff25b112d0bbd7ca0e4852db959e938bc558212b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.51ex; height:4.509ex;" alt="{\displaystyle e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is the price vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle u^{*}}</annotation>

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</math><img src="./cfeddccb8efb954b55c85480a6fe7d4e33b60ab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle u^{*}}" loading="lazy"> is the desired utility level, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}</annotation>

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</math><img src="./70881d7137ac5db3e11106ecb6844a514388840d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.642ex; height:3.009ex;" alt="{\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}" loading="lazy"> is the set of providing at least utility <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle u^{*}}</annotation>

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Expressed equivalently, the individual minimizes expenditure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}p_{1}+\dots +x_{n}p_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1}p_{1}+\dots +x_{n}p_{n}}</annotation>

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</math><img src="./53bc5a25a8146eaadca5d5bbfe226bff02c25e0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.948ex; height:2.343ex;" alt="{\displaystyle x_{1}p_{1}+\dots +x_{n}p_{n}}" loading="lazy"> subject to the minimal utility constraint that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},\dots ,x_{n})\geq u^{*},}">

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},\dots ,x_{n})\geq u^{*},}</annotation>

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</math><img src="./4d0fe2bddfb309e922bda28369aa2abf136da55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.378ex; height:2.843ex;" alt="{\displaystyle u(x_{1},\dots ,x_{n})\geq u^{*},}" loading="lazy"> giving optimal quantities to consume of the various goods as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{*},\dots x_{n}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1}^{*},\dots x_{n}^{*}}</annotation>

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</math><img src="./dfec8e2d804c0a2fd11715c2f152a4aeaa49d9b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.076ex; height:2.843ex;" alt="{\displaystyle x_{1}^{*},\dots x_{n}^{*}}" loading="lazy"> as function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle u^{*}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.}</annotation>

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</math><img src="./ef07e7b88db2567559ab4516362a8088b3f84501.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.793ex; height:3.009ex;" alt="{\displaystyle e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is a continuous utility function representing a <a href="Local_nonsatiation" title="Local nonsatiation">locally non-satiated</a> preference relation on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {R}}_{+}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {R}}_{+}^{n}}</annotation>

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</math><img src="./6fb5276f0a8b8fd22e9088a073888a56349d9f26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:3.009ex;" alt="{\displaystyle {\textbf {R}}_{+}^{n}}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p,u^{*})}">

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<annotation encoding="application/x-tex">{\displaystyle e(p,u^{*})}</annotation>

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<ol><li><a href="Homogeneous_function" title="Homogeneous function">Homogeneous</a> of degree one in p: for all and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation>

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</math><img src="./eea25afc0351140f919cf791c49c1964b8b081de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(\lambda p,u)=\lambda e(p,u);}">

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<annotation encoding="application/x-tex">{\displaystyle e(\lambda p,u)=\lambda e(p,u);}</annotation>

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</math><img src="./34765f5995cb6dfb8c441777886e8f93619d207b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.307ex; height:2.843ex;" alt="{\displaystyle e(\lambda p,u)=\lambda e(p,u);}" loading="lazy"></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u;}">

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<annotation encoding="application/x-tex">{\displaystyle u;}</annotation>

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</math><img src="./4f511ca1cadf3d2f1be744433d72cab2a82f1830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle u;}" loading="lazy"></li>

<li><a href="Monotonic_function" title="Monotonic function">Nondecreasing</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and strictly increasing in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\gg 0;}">

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<annotation encoding="application/x-tex">{\displaystyle p\gg 0;}</annotation>

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</math><img src="./d7add2f30695a34aff14d59f140ecb491e398f92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle p\gg 0;}" loading="lazy"></li>

<li><a href="Concave_function" title="Concave function">Concave</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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<li>If the utility function is strictly quasi-concave, there is <a href="Shephard's_lemma" title="Shephard's lemma">Shephard's lemma</a></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Proofs">Proofs</h3></div>

(1) As in the above proposition, note that

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<annotation encoding="application/x-tex">{\displaystyle e(\lambda p,u)=\min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}</annotation>

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</math><img src="./c34aad56f7404d6d233728f624ec41f8a9b24dd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.141ex; height:4.676ex;" alt="{\displaystyle e(\lambda p,u)=\min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda p\cdot x=\lambda \min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda p\cdot x=\lambda \min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}</annotation>

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</math><img src="./21395cd0e6cd20682eac914a766ff84358b65075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.635ex; height:4.509ex;" alt="{\displaystyle \lambda p\cdot x=\lambda \min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot x=\lambda e(p,u)}">

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<annotation encoding="application/x-tex">{\displaystyle p\cdot x=\lambda e(p,u)}</annotation>

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</math><img src="./83874e26061ee24cca858513ce88adcfbfd01bc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.147ex; height:2.843ex;" alt="{\displaystyle p\cdot x=\lambda e(p,u)}" loading="lazy">

(2) Continue on the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {R}}_{++}^{N}*{\textbf {R}}\rightarrow {\textbf {R}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {R}}_{++}^{N}*{\textbf {R}}\rightarrow {\textbf {R}}}</annotation>

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</math><img src="./a33087765129e414c3600c9db9c482734b3dda12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.608ex; height:3.343ex;" alt="{\displaystyle {\textbf {R}}_{++}^{N}*{\textbf {R}}\rightarrow {\textbf {R}}}" loading="lazy">

(3) Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\prime }>p}">

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<annotation encoding="application/x-tex">{\displaystyle p^{\prime }>p}</annotation>

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</math><img src="./878658f581bb5e03d193471c8596de0f8cda0777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.211ex; height:2.843ex;" alt="{\displaystyle p^{\prime }>p}" loading="lazy"> and suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in h(p^{\prime },u)}">

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<annotation encoding="application/x-tex">{\displaystyle x\in h(p^{\prime },u)}</annotation>

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</math><img src="./d71c67eebc2637b8db6fbff2e81b29ee42fef87d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.536ex; height:3.009ex;" alt="{\displaystyle x\in h(p^{\prime },u)}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(h)\geq u}">

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<annotation encoding="application/x-tex">{\displaystyle u(h)\geq u}</annotation>

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</math><img src="./a7ff0cec8ebf8ddbe5cd5634f278a92d702b8b7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.906ex; height:2.843ex;" alt="{\displaystyle u(h)\geq u}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p^{\prime },u)=p^{\prime }\cdot x\geq p\cdot x}">

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<annotation encoding="application/x-tex">{\displaystyle e(p^{\prime },u)=p^{\prime }\cdot x\geq p\cdot x}</annotation>

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</math><img src="./fedfbcf3960e611dc65b88463ba2ced0191badd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.349ex; height:3.009ex;" alt="{\displaystyle e(p^{\prime },u)=p^{\prime }\cdot x\geq p\cdot x}" loading="lazy"> . It follows immediately that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p,u)\leq e(p^{\prime },u)}">

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<annotation encoding="application/x-tex">{\displaystyle e(p,u)\leq e(p^{\prime },u)}</annotation>

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</math><img src="./aa08f7a2f438664310ae26f5933ebb77a41594eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.635ex; height:3.009ex;" alt="{\displaystyle e(p,u)\leq e(p^{\prime },u)}" loading="lazy">.

For the second statement, suppose to the contrary that for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\prime }>u}">

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<annotation encoding="application/x-tex">{\displaystyle u^{\prime }>u}</annotation>

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</math><img src="./e2e440a6da6cbd82b3b8474ebf7289bb8a3f950b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle u^{\prime }>u}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p,u^{\prime })\leq e(p,u)}">

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<annotation encoding="application/x-tex">{\displaystyle e(p,u^{\prime })\leq e(p,u)}</annotation>

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</math><img src="./88d7fbcb4e462e70c1ce519382930ecaf1fd6b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.635ex; height:3.009ex;" alt="{\displaystyle e(p,u^{\prime })\leq e(p,u)}" loading="lazy"> Than, for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in h(p,u)}">

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<annotation encoding="application/x-tex">{\displaystyle x\in h(p,u)}</annotation>

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</math><img src="./7a10daa52eeb95509f995f8a3679b861de1ba396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.852ex; height:2.843ex;" alt="{\displaystyle x\in h(p,u)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=u^{\prime }>u}">

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<annotation encoding="application/x-tex">{\displaystyle u(x)=u^{\prime }>u}</annotation>

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</math><img src="./6000a178d07b4860770ab579208b6bab1c5d7b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.01ex; height:3.009ex;" alt="{\displaystyle u(x)=u^{\prime }>u}" loading="lazy">, which contradicts the "no excess utility" conclusion of the previous proposition

(4) Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in (0,1)}">

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<annotation encoding="application/x-tex">{\displaystyle t\in (0,1)}</annotation>

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</math><img src="./47e93bf1b33e520ce1c54ecd00d977a91cc29075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.848ex; height:2.843ex;" alt="{\displaystyle t\in (0,1)}" loading="lazy"> and suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in h(tp+(1-t)p^{\prime })}">

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</math><img src="./9de83dbfa7a1fbd1dc60de259ec8e25d4e9bdbdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.674ex; height:3.009ex;" alt="{\displaystyle x\in h(tp+(1-t)p^{\prime })}" loading="lazy">. Then, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot x\geq e(p,u)}">

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<annotation encoding="application/x-tex">{\displaystyle p\cdot x\geq e(p,u)}</annotation>

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</math><img src="./f41c79b6de0c7aa3c19592215cb1bf9cf1bfa45f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.792ex; height:2.843ex;" alt="{\displaystyle p\cdot x\geq e(p,u)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\prime }\cdot x\geq e(p^{\prime },u)}">

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<annotation encoding="application/x-tex">{\displaystyle p^{\prime }\cdot x\geq e(p^{\prime },u)}</annotation>

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</math><img src="./7796de914b329bb4fbe4393c1600e3fbc72fa988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.161ex; height:3.009ex;" alt="{\displaystyle p^{\prime }\cdot x\geq e(p^{\prime },u)}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(tp+(1-t)p^{\prime },u)=(tp+(1-t)p^{\prime })\cdot x\geq }">

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<annotation encoding="application/x-tex">{\displaystyle e(tp+(1-t)p^{\prime },u)=(tp+(1-t)p^{\prime })\cdot x\geq }</annotation>

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</math><img src="./c8d0fde374c58ea58e9e7aa41c687a35a90256e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.337ex; height:3.009ex;" alt="{\displaystyle e(tp+(1-t)p^{\prime },u)=(tp+(1-t)p^{\prime })\cdot x\geq }" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle te(p,u)+(1-t)e(p^{\prime },u)}">

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<annotation encoding="application/x-tex">{\displaystyle te(p,u)+(1-t)e(p^{\prime },u)}</annotation>

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</math><img src="./5dfd2cf036a472a26575e2b6798d5b3b5a3c571f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.868ex; height:3.009ex;" alt="{\displaystyle te(p,u)+(1-t)e(p^{\prime },u)}" loading="lazy">.

(5) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta (p^{0},u^{0})}{\delta p_{i}}}=x_{i}^{h}(p^{0},u^{0})}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta (p^{0},u^{0})}{\delta p_{i}}}=x_{i}^{h}(p^{0},u^{0})}</annotation>

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</math><img src="./262be4b8152f80e952d66981ddba8800c66c504c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.394ex; height:6.343ex;" alt="{\displaystyle {\frac {\delta (p^{0},u^{0})}{\delta p_{i}}}=x_{i}^{h}(p^{0},u^{0})}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Expenditure_and_indirect_utility">Expenditure and indirect utility</h2></div>

The expenditure function is the inverse of the <a href="Indirect_utility" class="mw-redirect" title="Indirect utility">indirect utility</a> function when the prices are kept constant. I.e, for every price vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and income level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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<annotation encoding="application/x-tex">{\displaystyle e(p,v(p,I))\equiv I}</annotation>

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</math><img src="./620147caa55d5e19f8acefb3b2d4ada0cab94637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.678ex; height:2.843ex;" alt="{\displaystyle e(p,v(p,I))\equiv I}" loading="lazy"></dd></dl>

There is a duality relationship between the expenditure function and the utility function. If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function. In addition to the property that prices are once homogeneous and utility is monotonically increasing, the expenditure function usually assumes

<ol><li>Is a non-negative function, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P\cdot u)>O;}">

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</math><img src="./2124d2386eb54ae37281ff510c8ba5eee6de6c7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.858ex; height:2.843ex;" alt="{\displaystyle E(P\cdot u)>O;}" loading="lazy"></li>

<li>For P, it is non-decreasing, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(p^{1}u)>E(p^{2}u),u>Op^{l}>p^{2}>O_{N}}">

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<annotation encoding="application/x-tex">{\displaystyle E(p^{1}u)>E(p^{2}u),u>Op^{l}>p^{2}>O_{N}}</annotation>

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</math><img src="./464062afef31062c50466f8756f4202e45afcd20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.388ex; height:3.176ex;" alt="{\displaystyle E(p^{1}u)>E(p^{2}u),u>Op^{l}>p^{2}>O_{N}}" loading="lazy">;</li>

<li>E(Pu) is a concave function. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(np^{l}+(1-n)p^{2})u)>\lambda E(p^{1}u)(1-n)E(p^{2}u)y>0}">

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<annotation encoding="application/x-tex">{\displaystyle e(np^{l}+(1-n)p^{2})u)>\lambda E(p^{1}u)(1-n)E(p^{2}u)y>0}</annotation>

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</math><img src="./7e271b0513a57ec1794ee02e1a5db1897fafdbb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.038ex; height:3.176ex;" alt="{\displaystyle e(np^{l}+(1-n)p^{2})u)>\lambda E(p^{1}u)(1-n)E(p^{2}u)y>0}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O<\lambda <1p^{l}\geq O_{N}p^{2}\geq O_{N}}">

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<annotation encoding="application/x-tex">{\displaystyle O<\lambda <1p^{l}\geq O_{N}p^{2}\geq O_{N}}</annotation>

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</math><img src="./22c4cfef17cb314f93c501be1c110035f19c284d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.73ex; height:3.009ex;" alt="{\displaystyle O<\lambda <1p^{l}\geq O_{N}p^{2}\geq O_{N}}" loading="lazy"></li></ol>

Expenditure function is an important theoretical method to study consumer behavior. Expenditure function is very similar to cost function in production theory. Dual to the utility maximization problem is the cost minimization problem <a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

Suppose the utility function is the <a href="Cobb-Douglas_function" class="mw-redirect" title="Cobb-Douglas function">Cobb-Douglas function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},x_{2})=x_{1}^{.6}x_{2}^{.4},}">

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=x_{1}^{.6}x_{2}^{.4},}</annotation>

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</math><img src="./14d467cba2dec9623ffd6ffdaa99eccd004fcf77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.369ex; height:3.176ex;" alt="{\displaystyle u(x_{1},x_{2})=x_{1}^{.6}x_{2}^{.4},}" loading="lazy"> which generates the demand functions<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle x_{1}(p_{1},p_{2},I)={\frac {.6I}{p_{1}}}\;\;\;\;{\rm {and}}\;\;\;x_{2}(p_{1},p_{2},I)={\frac {.4I}{p_{2}}},}</annotation>

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</math><img src="./9d4d56a468e309b987aea07fb51797206082af24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.502ex; height:5.676ex;" alt="{\displaystyle x_{1}(p_{1},p_{2},I)={\frac {.6I}{p_{1}}}\;\;\;\;{\rm {and}}\;\;\;x_{2}(p_{1},p_{2},I)={\frac {.4I}{p_{2}}},}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> is the consumer's income. One way to find the expenditure function is to first find the <a href="Indirect_utility_function" title="Indirect utility function">indirect utility function</a> and then invert it. The indirect utility function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(p_{1},p_{2},I)}">

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<annotation encoding="application/x-tex">{\displaystyle v(p_{1},p_{2},I)}</annotation>

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</math><img src="./61ad43bcfebf011768f0eac8bd0f83d3fdeb2146.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.624ex; height:2.843ex;" alt="{\displaystyle v(p_{1},p_{2},I)}" loading="lazy"> is found by replacing the quantities in the utility function with the demand functions thus:

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<annotation encoding="application/x-tex">{\displaystyle v(p_{1},p_{2},I)=u(x_{1}^{*},x_{2}^{*})=(x_{1}^{*})^{.6}(x_{2}^{*})^{.4}=\left({\frac {.6I}{p_{1}}}\right)^{.6}\left({\frac {.4I}{p_{2}}}\right)^{.4}=(.6^{.6}\times .4^{.4})I^{.6+.4}p_{1}^{-.6}p_{2}^{-.4}=Kp_{1}^{-.6}p_{2}^{-.4}I,}</annotation>

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</math><img src="./b7afdbde211d2339d0147a95721a8eacd8e41cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.267ex; height:6.676ex;" alt="{\displaystyle v(p_{1},p_{2},I)=u(x_{1}^{*},x_{2}^{*})=(x_{1}^{*})^{.6}(x_{2}^{*})^{.4}=\left({\frac {.6I}{p_{1}}}\right)^{.6}\left({\frac {.4I}{p_{2}}}\right)^{.4}=(.6^{.6}\times .4^{.4})I^{.6+.4}p_{1}^{-.6}p_{2}^{-.4}=Kp_{1}^{-.6}p_{2}^{-.4}I,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle K=(.6^{.6}\times .4^{.4}).}</annotation>

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</math><img src="./226c324240f62aa00b5f70ea594aad038956dd0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.103ex; height:3.176ex;" alt="{\displaystyle K=(.6^{.6}\times .4^{.4}).}" loading="lazy"> Then since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(p_{1},p_{2},u)=e(p_{1},p_{2},v(p_{1},p_{2},I))=I}">

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<annotation encoding="application/x-tex">{\displaystyle e(p_{1},p_{2},u)=e(p_{1},p_{2},v(p_{1},p_{2},I))=I}</annotation>

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</math><img src="./105398263f9c242114fa7a34b71df29c9b5a335f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.138ex; height:2.843ex;" alt="{\displaystyle e(p_{1},p_{2},u)=e(p_{1},p_{2},v(p_{1},p_{2},I))=I}" loading="lazy"> when the consumer optimizes, we can invert the indirect utility function to find the expenditure function:

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<annotation encoding="application/x-tex">{\displaystyle e(p_{1},p_{2},u)=(1/K)p_{1}^{.6}p_{2}^{.4}u,}</annotation>

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</math><img src="./53670857d39523a9a0511176eb1df172d7525bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.375ex; height:3.176ex;" alt="{\displaystyle e(p_{1},p_{2},u)=(1/K)p_{1}^{.6}p_{2}^{.4}u,}" loading="lazy"></dd></dl>

Alternatively, the expenditure function can be found by solving the problem of minimizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{1}x_{1}+p_{2}x_{2})}">

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</math><img src="./6ae1ed52d8140c08a5574c3db837323faf36975c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.865ex; height:2.843ex;" alt="{\displaystyle (p_{1}x_{1}+p_{2}x_{2})}" loading="lazy"> subject to the constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},x_{2})\geq u^{*}.}">

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</math><img src="./6e32dc785d42893c7cb5c4714de8c915f5562e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.07ex; height:2.843ex;" alt="{\displaystyle u(x_{1},x_{2})\geq u^{*}.}" loading="lazy"> This yields conditional demand functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{*}(p_{1},p_{2},u^{*})}">

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</math><img src="./8bc6173eae1ec0c5d321d8d0bfe6dfa47bb95fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.092ex; height:3.009ex;" alt="{\displaystyle x_{1}^{*}(p_{1},p_{2},u^{*})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}^{*}(p_{1},p_{2},u^{*})}">

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</math><img src="./b34fa105af1875fc7a1a6da89f2b0bf5444a069b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.092ex; height:3.009ex;" alt="{\displaystyle x_{2}^{*}(p_{1},p_{2},u^{*})}" loading="lazy"> and the expenditure function is then

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<annotation encoding="application/x-tex">{\displaystyle e(p_{1},p_{2},u^{*})=p_{1}x_{1}^{*}+p_{2}x_{2}^{*}}</annotation>

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</math><img src="./4a143afebe2cdf1f4cd7834fb38035a216d4a949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.946ex; height:3.009ex;" alt="{\displaystyle e(p_{1},p_{2},u^{*})=p_{1}x_{1}^{*}+p_{2}x_{2}^{*}}" loading="lazy"></dd></dl>

<ul><li><a href="Expenditure_minimization_problem" title="Expenditure minimization problem">Expenditure minimization problem</a></li>

<li><a href="Hicksian_demand_function" title="Hicksian demand function">Hicksian demand function</a></li>

<li><a href="Slutsky_equation" title="Slutsky equation">Slutsky equation</a></li>

<li><a href="Utility_maximization_problem" title="Utility maximization problem">Utility maximization problem</a></li>

<li><a href="Budget_constraint" title="Budget constraint">Budget constraint</a></li>

<li><a href="Consumption_set" class="mw-redirect" title="Consumption set">Consumption set</a></li>

<li><a href="Shephard's_lemma" title="Shephard's lemma">Shephard's lemma</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFVarian1992" class="citation book cs1"><a href="Hal_Varian" title="Hal Varian">Varian, Hal</a> (1992). Microeconomic Analysis (Third ed.). New York: Norton. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-95735-7</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation book cs1">Jing ji xue da ci dian. Xiaomin Liang, 梁小民. (Di 1 ban ed.). Beijing Shi: Tuan jie chu ban she. 1994. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>7-80061-954-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/34287945">34287945</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: others (link)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www2.econ.iastate.edu/classes/econ501/Hallam/documents/DualConsumer.pdf">"CONSUMER CHOICE AND DUALITY"</a> (PDF). 23 February 2024.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFVarian1992" class="citation book cs1">Varian, H. (1992). <a rel="nofollow" class="external text" href="https://archive.org/details/microeconomicana00vari_0">Microeconomic Analysis</a> (3rd ed.). New York: W. W. Norton.</cite>, pp. 111, has the general formula.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFMas-ColellWhinstonGreen2007" class="citation book cs1"><a href="Andreu_Mas-Colell" title="Andreu Mas-Colell">Mas-Colell, Andreu</a>; Whinston, Michael D.; Green, Jerry R. (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780198089537/page/59">Microeconomic Theory</a>. Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780198089537/page/59">59–60</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-510268-0</bdi>.</cite></li>

<li><cite id="CITEREFMathisKoscianski2002" class="citation book cs1">Mathis, Stephen A.; Koscianski, Janet (2002). Microeconomic Theory: An Integrated Approach. Upper Saddle River: Prentice Hall. pp. 132–133. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-011418-9</bdi>.</cite></li>

<li><cite id="CITEREFVarian1984" class="citation book cs1"><a href="Hal_Varian" title="Hal Varian">Varian, Hal R.</a> (1984). Microeconomic Analysis (Second ed.). New York: W. W. Norton. pp. 121–123. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-95282-7</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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