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<title>Inverse demand function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Inverse demand function</span></span>

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<p>In <a href="Economics" title="Economics">economics</a>, an <b>inverse demand function</b> is the mathematical relationship that expresses price as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of quantity demanded (it is therefore also known as a <b>price function</b>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Historically, the economists first expressed the price of a good as a function of demand (holding the other economic variables, like income, constant), and plotted the price-demand relationship with demand on the x (horizontal) axis (the <a href="Demand_curve" title="Demand curve">demand curve</a>). Later the additional variables, like prices of other goods, came into analysis, and it became more convenient to express the demand as a <a href="Multivariate_function" class="mw-redirect" title="Multivariate function">multivariate function</a> (the <b>demand function</b>):

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<annotation encoding="application/x-tex">{\displaystyle {demand}=f({price},{income},...)}</annotation>

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</math></span><img src="./01af305be1aaba98c77c674c9c387f4156f8264e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.103ex; height:2.843ex;" alt="{\displaystyle {demand}=f({price},{income},...)}" loading="lazy"></span>, so the original demand curve now depicts the <i>inverse</i> demand function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {price}=f^{-1}({demand})}">

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<annotation encoding="application/x-tex">{\displaystyle {price}=f^{-1}({demand})}</annotation>

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

<p>In mathematical terms, if the demand function is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {demand}=f({price})}">

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<annotation encoding="application/x-tex">{\displaystyle {demand}=f({price})}</annotation>

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</math></span><img src="./e3f7dc8799f41d4674c6e7a3d77b3a7d5cb49adb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.477ex; height:2.843ex;" alt="{\displaystyle {demand}=f({price})}" loading="lazy"></span>, then the inverse demand function is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {price}=f^{-1}({demand})}">

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<annotation encoding="application/x-tex">{\displaystyle {price}=f^{-1}({demand})}</annotation>

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</math></span><img src="./9d0f8ef240fb33b0f280302e984e1fc2d06b6a7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.941ex; height:3.176ex;" alt="{\displaystyle {price}=f^{-1}({demand})}" loading="lazy"></span>. The value of the inverse demand function is the highest price that could be charged and still generate the quantity demanded.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This is useful because economists typically place price (P) on the vertical axis and quantity (demand, Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see.

</p><p>The inverse demand function is the same as the average revenue function, since P = AR.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>To compute the inverse demand function, simply solve for P from the demand function. For example, if the demand function has the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=240-2P}">

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<annotation encoding="application/x-tex">{\displaystyle Q=240-2P}</annotation>

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</math></span><img src="./79ade75bfa212f90579ab6e3f20c53617b30d626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.172ex; height:2.509ex;" alt="{\displaystyle Q=240-2P}" loading="lazy"></span> then the inverse demand function would be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=120-{\frac {1}{2}}Q}">

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<annotation encoding="application/x-tex">{\displaystyle P=120-{\frac {1}{2}}Q}</annotation>

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</math></span><img src="./b2e489b7d9437936e0072fa56c01b0a54f420e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.009ex; height:5.176ex;" alt="{\displaystyle P=120-{\frac {1}{2}}Q}" loading="lazy"></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse.

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_marginal_revenue">Relation to marginal revenue</h2></div>

<p>There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function. For any linear demand function with an inverse demand equation of the form P = a - bQ, the marginal revenue function has the form MR = a - 2bQ.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The inverse linear demand function and the marginal revenue function derived from it have the following characteristics:

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<ul><li>Both functions are linear.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>The marginal revenue function and inverse demand function have the same y intercept.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li>

<li>The x intercept of the marginal revenue function is one-half the x intercept of the inverse demand function.</li>

<li>The marginal revenue function has twice the slope of the inverse demand function.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li>

<li>The marginal revenue function is below the inverse demand function at every positive quantity.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul>

<p>The inverse demand function can be used to derive the total and marginal revenue functions. Total revenue equals price, P, times quantity, Q, or TR = P×Q.

Multiply the inverse demand function by Q to derive the total revenue function: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TR=(120-{\frac {1}{2}}Q)\cdot Q=120Q-{\frac {1}{2}}Q^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle TR=(120-{\frac {1}{2}}Q)\cdot Q=120Q-{\frac {1}{2}}Q^{2}}</annotation>

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</math></span><img src="./41af9a02a052046f8c549e309c0a813922bc9462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.146ex; height:5.176ex;" alt="{\displaystyle TR=(120-{\frac {1}{2}}Q)\cdot Q=120Q-{\frac {1}{2}}Q^{2}}" loading="lazy"></span>.

The marginal revenue function is the first derivative of the total revenue function or MR = 120 - Q. Note that in this linear example the MR function has the same y-intercept as the inverse demand function, the x-intercept of the MR function is one-half the value of the demand function, and the slope of the MR function is twice that of the inverse demand function. This relationship holds true for all linear demand equations. The importance of being able to quickly calculate MR is that the profit-maximizing condition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC). To derive MC the first derivative of the total cost function is taken.

</p><p>For example, assume cost, C, equals 420 + 60Q + Q<sup>2</sup>. then MC = 60 + 2Q.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Equating MR to MC and solving for Q gives Q = 20. So 20 is the profit-maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P.

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

<li><a href="Marshallian_demand_function" title="Marshallian demand function">Marshallian demand function</a></li>

<li><a href="Excess_demand_function" title="Excess demand function">Excess demand function</a></li>

<li><a href="Supply_and_demand" title="Supply and demand">Supply and demand</a></li>

<li><a href="Demand" title="Demand">Demand</a></li>

<li><a href="Law_of_demand" title="Law of demand">Law of demand</a></li>

<li><a href="Profit_(economics)" title="Profit (economics)">Profit (economics)</a></li></ul>

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

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</style><cite id="CITEREFR.2014" class="citation book cs1">R., Varian, Hal (7 April 2014). <i>Intermediate microeconomics : with calculus</i> (First ed.). New York. p. 115. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780393123982</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/884922812">884922812</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link)</span></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFKaraivanov" class="citation web cs1">Karaivanov, Alexander. <a rel="nofollow" class="external text" href="https://www.sfu.ca/~akaraiva/demfun.pdf">"The demand function and the demand curve"</a> <span class="cs1-format">(PDF)</span>. <i>sfu.ca</i>. <a href="Simon_Fraser_University" title="Simon Fraser University">Simon Fraser University</a><span class="reference-accessdate">. Retrieved <span class="nowrap">29 August</span> 2023</span>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London</span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Chiang & Wainwright, Fundamental Methods of Mathematical Economics 4th ed. Page 172. McGraw-Hill 2005</span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003)</span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Perloff, J: Microeconomics Theory & Applications with Calculus page 363. Pearson 2008.</span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Perloff, J: Microeconomics Theory & Applications with Calculus page 362. Pearson 2008.</span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.<a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-321-27794-5</bdi></span>

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

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

<ul><li><cite id="CITEREFRyanPearce1977" class="citation book cs1">Ryan, W. J. L.; Pearce, D. W. (1977). "Demand Functions". <i>Price Theory</i>. London: Macmillan Education UK. pp. <span class="nowrap">31–</span>69. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-349-17334-1_2">10.1007/978-1-349-17334-1_2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-333-17913-0</bdi>.</cite></li></ul>

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