Inverse demand function: Difference between revisions

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{{Short description|Mathematical function in economics}}

In [[economics]], an '''inverse demand function''' is the mathematical relationship that expresses price as a [[function (mathematics)|function]] of quantity demanded (it is therefore also known as a '''price function''').<ref>{{Cite book|title=Intermediate microeconomics : with calculus|last=R.|first=Varian, Hal|date = 7 April 2014|isbn=9780393123982|edition= First|___location=New York|pages=115|oclc=884922812}}</ref>

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 [[economics]], an '''inverse demand function''' is a function that maps the quantity of output demanded to the market price (dependent variable) for that output. Quantity demanded, Q, is a function of price; the inverse demand function treats price as a function of quantity demanded, and is also called the price function{{Citation needed|date=September 2009}}.

In mathematical terms, if the [[demand curve|demand function]] is <math>{demand} = f(x{price})</math>, then the inverse demand function is f<supmath>{price} = f^{-1}({demand})</supmath>(x). The Thisvalue is to say thatof the inverse demand function is the [[demandhighest curve|demandprice function]]that withcould be charged and still generate the axesquantity switcheddemanded.<ref>Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London</ref> 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>

The Inverse Demand Function therefore measures what price a good has to be in order for the consumer to choose that level of consumption.<ref>Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London</ref>

To compute the inverse demand function, simply solve for P infrom the [[demand curve|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>.5Q<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.

==Relation to marginal revenue==

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 = PQ. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) x Q = 120Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function or MR = 120 - Q. Note that 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 ablie to quickly calculate MR is that the profit maximizing conditions for firms regardless of market structure is to produce where marginal revenue equals marginal cost. To derive MC you take the first derivative of the total cost function.

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 inverse linear demand function and the marginal revenue 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 intercept.<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.

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.

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 = PQ. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) x 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 ablieable to quickly calculate MR is that the profit -maximizing conditionscondition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC). To derive MC you take 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 economist place the independent variable on the y axis and the dependent variable on the y axis. The slope of the inverse funtion is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for calcularing elasticity is PED = (∆Q/∆P) x (P/Q).

== References ==

{{reflist}}

==See also==

* [[Supply andHicksian demand function]]

* [[Marshallian demand function]]

* [[Excess demand function]]

* [[Supply and demand]]

* [[Demand]]

* [[Law of demand]]

* [[Profit (economics)]]

== References ==

[[Marginal Revenue]]

{{Reflist}}

== Further reading ==

[[Category:Mathematical finance]]

* {{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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[[Category:Mathematical finance]]

[[Category:Demand]]