inverse demand function

In mathematical terms, if the demand function is $\{demand\}\; =\; f(\{price\})$, then the inverse demand function is $\{price\}\; =\; f^\{-1\}(\{demand\})$. The value of the inverse demand function is the highest price that could be charged and still generate the quantity demanded.Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London 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.Chiang & Wainwright, Fundamental Methods of Mathematical Economics 4th ed. Page 172. McGraw-Hill 2005

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

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.Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003. The inverse linear demand function and the marginal revenue function derived from it have the following characteristics:

• Both functions are linear.Perloff, J: Microeconomics Theory & Applications with Calculus page 363. Pearson 2008.
• The marginal revenue function and inverse demand function have the same y intercept.Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.
• 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.Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.
• The marginal revenue function is below the inverse demand function at every positive quantity.Perloff, J: Microeconomics Theory & Applications with Calculus page 362. Pearson 2008.

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: $TR\; =\; (120\; -\; \backslash frac\{1\}\{2\}Q)\backslash cdot\; Q\; =\; 120\; Q\; -\; \backslash frac\{1\}\{2\}Q^2$.

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.Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.{{ISBN|0-321-27794-5}} 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.

• Hicksian demand function
• Marshallian demand function
• Excess demand function
• Supply and demand
• Demand
• Law of demand
• Profit (economics)

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• {{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