Inverse Demand Function

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In economics, an 'inverse demand function', P = f^{-1}(Q), 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.^{[1]} Note that the inverse demand function is not the reciprocal of the demand function--the word "inverse" refers to the mathematical concept of an inverse function.

In mathematical terms, if the 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.^{[2]} This is to say that the inverse demand function is the 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.

The inverse demand function is the same as the average revenue function, since P = AR.^{[3]}

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 - 0.5Q.^{[4]}

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 + Q^{2}. then MC = 60 + 2Q.^{[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.

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).

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.^{[6]} The marginal revenue function and inverse linear demand function have the following characteristics:

- Both functions are linear.
^{[7]} - The marginal revenue function and inverse demand function have the same y interecept.
^{[8]} - 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.
^{[9]} - The marginal revenue function is below the inverse demand function at every positive quantity.
^{[10]}

**^**Samuelson, W and Marks, S Managerial Economics 4th ed. page 35. Wiley 2003.**^**Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London**^**Chiang & Wainwright, Fundamental Methods of Mathematical Economics 4th ed. Page 172. McGraw-Hill 2005**^**Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003)**^**Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.ISBN 0-321-27794-5**^**Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.**^**Perloff, J: Microeconomics Theory & Applications with Calculus page 363. Pearson 2008.**^**Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.**^**Samuelson, W & Marks, S Managerial Economics 4th ed. Page 47. Wiley 2003.**^**Perloff, J: Microeconomics Theory & Applications with Calculus page 362. Pearson 2008.

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