Inverse demand function: Difference between revisions

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{{Short description|Mathematical function in economics}}

{{citation style|date=May 2012}}

In [[economics]], an '''inverse demand function', P = f⁻¹(Q),'' is a function that maps the quantitymathematical ofrelationship outputthat demanded to the marketexpresses price (dependent variable) for that output. Quantity demanded, Q, isas a [[function of price; the inverse demand function treats price as a (mathematics)|function]] of quantity demanded, and(it is therefore also calledknown theas a '''price function''').<ref>Samuelson,{{Cite Wbook|title=Intermediate andmicroeconomics : Markswith calculus|last=R.|first=Varian, SHal|date = Managerial7 EconomicsApril 4th2014|isbn=9780393123982|edition= ed.First|___location=New page 35. Wiley 2003.York|pages=115|oclc=884922812}}</ref> Note that the inverse demand function is not the reciprocal of the demand function&mdash;the word "inverse" refers to the mathematical concept of an [[inverse function]].

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 [[demand curve]]). Later the additional variables, like prices of other goods, came into analysis, and it became more convenient to express the demand as a [[multivariate function]] (the '''demand function'''):

<math>{demand} = f({price}, {income}, ...)</math>, so the original demand curve now depicts the ''inverse'' demand function <math>{price} = f^{-1}({demand})</math> with extra variables fixed.<ref>{{cite web |last1=Karaivanov |first1=Alexander |title=The demand function and the demand curve |url=https://www.sfu.ca/~akaraiva/demfun.pdf |website=sfu.ca |publisher=[[Simon Fraser University]] |access-date=29 August 2023}}</ref>

==Definition==

In mathematical terms, if the [[demand curve|demand function]] is f(P), then the inverse demand function is f⁻¹(Q), whose value is the highest price that could be charged and still generate the quantity demanded Q.<ref>Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London</ref> This is to say that the inverse demand function is the [[demand curve|demand function]] with the axes switched. This is useful because economists typically place price ('''P''') on the vertical axis and quantity ('''Q''') on the horizontal axis.

In mathematical terms, if the [[demand curve|demand function]] is <math>{demand} = f(P{price})</math>, then the inverse demand function is f<supmath>−1{price} = f^{-1}({demand})</supmath>(Q),. whoseThe value of the inverse demand function is the highest price that could be charged and still generate the quantity demanded Q.<ref>Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London</ref> This is to say that the inverse demand function is the [[demand curve|demand function]] with the axes switched. 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.

The inverse demand function is the same as the average revenue function, since P = AR.<ref>Chiang & Wainwright, Fundamental Methods of Mathematical Economics 4th ed. Page 172. McGraw-Hill 2005</ref>

To compute theThe inverse demand function, simplyis solvethe for Psame fromas the [[demandaverage curve|demandrevenue function]]. For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would besince P = 120 - 0.5QAR.<ref>SamuelsonChiang & MarksWainwright, ManagerialFundamental Methods of Mathematical Economics 4th ed. (WileyPage 172. McGraw-Hill 2003)2005</ref>

To compute the inverse demand function, simply solve for P from the demand function. For example, if the demand function has the form <math>Q = 240 - 2P</math> then the inverse demand function would be <math>P = 120 - \frac{1}{2}Q</math>.<ref>Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003)</ref> 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.

==Applications==

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: TR = (120 - .5Q) × Q = 120Q - 0.5Q². 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.

For example, assume cost, C, equals 420 + 60Q + Q². then MC = 60 + 2Q.<ref>Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.{{ISBN|0-321-27794-5}}</ref> 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.

The inverse demand function is the form of the demand function that appears in the famous [[Marshallian Scissors]] diagram. The function appears in this form because economists place the independent variable on the y-axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for elasticity is (∆Q/∆P) × (P/Q).

==Relation to marginal revenue==

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.<ref>Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</ref> The marginalinverse revenuelinear demand function and inversethe linearmarginal demandrevenue function derived from it have the following characteristics:

*Both functions are linear.<ref>Perloff, J: Microeconomics Theory & Applications with Calculus page 363. Pearson 2008.</ref>

*The marginal revenue function and inverse demand function have the same y intereceptintercept.<ref>Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</ref>

*The x intercept of the marginal revenue function is one-half the x intercept of the inverse demand function.

* The marginal revenue function has twice the slope of the inverse demand function.<ref>Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.</ref>

* The marginal revenue function is below the inverse demand function at every positive quantity.<ref>Perloff, J: Microeconomics Theory & Applications with Calculus page 362. Pearson 2008.</ref>

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: <math>TR = (120 - \frac{1}{2}Q)\cdot Q = 120 Q - \frac{1}{2}Q^2 </math>.

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: TR = (120 - .5Q) × Q = 120Q - 0.5Q². 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.

For example, assume cost, C, equals 420 + 60Q + Q². then MC = 60 + 2Q.<ref>Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.{{ISBN|0-321-27794-5}}</ref> 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.

==See also==

* [[Supply andHicksian demand function]]

* [[Marshallian demand function]]

*[[Demand]]

* [[DemandExcess demand Curvefunction]]

* [[LawSupply ofand demand]]

* [[Profit (economics)Demand]]

* [[Law of demand]]

* [[Profit (economics)]]

== References ==

{{Reflist}}

== Further reading ==

* {{cite book | last1=Ryan | first1=W. J. L. | last2=Pearce | first2=D. W. | title=Price Theory | chapter=Demand Functions | publisher=Macmillan Education UK | publication-place=London | year=1977 | isbn=978-0-333-17913-0 | doi=10.1007/978-1-349-17334-1_2 | pages=31–69}}

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== References ==

{{reflist}}

{{DEFAULTSORT:Inverse Demand Function}}