Linear Demand, Elasticity, and Total Revenue

To find out how to work the applet controls, see the instructions below.

This applet is designed to help you visualize the relationship between a linear demand curve, elasticity, and the total revenue of a monopolist.  The total revenue may also be thought of as the total expenditure by consumers.


Linear Demand

The Demand Equation

The first demand equation beginning economics students usually see is that of a linear, or straight-line demand curve.  This is the simplest type of equation to work with.  Economists write the equation for demand in the following form:

Q = a + bP
(demand equation)

Q is on the left-hand side because we usually think of the quantity demanded as depending on price and not the other way around.  So Q is the dependent variable.  That is, we consider Q to be a function of P, so the demand equation may also be written as

Q(P)= a + bP
(Q written as a function of P)

which emphasizes the dependence of Q on P.

The parameters a and b are called the intercept and slope, respectively.  The value of a tells us what the value of Q will be when P is equal to zero.  The value of b, tells us the amount of change in Q that will occur due to a unit change in P.  The value of b will always be negative, meaning that P and Q move in opposite directions, because the law of demand tells us that when the price goes up, the quantity demanded goes down.

The Inverse Demand Equation

According to tradition, when economists draw a demand curve in a two dimensional diagram, they put the price on the vertical axis, and the quantity on the horizontal axis, like the diagram in the upper right corner of the applet.  Mathematical tradition dictates that the dependent variable be drawn on the vertical axis, and the independent variable be drawn on the horizontal.  Thus, we must also be able to think of P as function that is dependent on Q.   The function P(Q), which is also linear, is called the inverse of the function Q(P).  The corresponding inverse demand equation is written like this:

P = c + dQ
(inverse demand equation)

If you start with the demand equation Q(P), and do a little algebra to isolate P on the left-hand side, you can verify that c = -a/b and d = 1/b.  Since, as we determined above,  the sign of a is positive and the sign of b is negative, we know from these equalities that c must be positive and d must be negative.

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Elasticity is a measure of the responsiveness of one variable to changes in some other variable.  We are interested in the price elasticity of demand, which measures the response of the quantity demanded to a change in price.  Formally, the price elasticity of demand, which we will call E, is the ratio of the percent change in quantity over the percent change in price.

For small changes in P and Q, a reasonable approximation of the percent change in P is the difference between the initial price P1  and an adjacent price P2 divided by the initial price.  The percent change in price is thus approximately (P2 - P1)/P1.  The approximation of the percent change in Q is obtained in the same way, where  Q1 = Q(P1) and Q2=Q(P2).  Using these definitions for percent change, an approximation for elasticity is

(price elasticity approximation)
where the vertical bars surrounding the ratio on the right side of the equation indicates absolute value.  The smaller the change in P, the closer the approximation.

It is possible to calculate an exact value of the elasticity for a small change in P, call it P, using the demand equation Q(P) = a + bP and a little algebra.  Let P1 = P and P2= P + P so that P = P2 - P1.   Then let Q = Q(P2) - Q(P1).  Substituting the demand equation into the right side of this equation we get

Q = a + bP2 - (a + bP1) = b(P2 - P1) = bP.

When we substitute P and Q into the exact version of the definition of elasticity, we get the following:

(exact price elasticity)

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The equation for demand, Q = a + bP, and the inverse demand equation, P = c + dQ, are displayed in the upper left of the applet. 

1. The parameters a and b can be changed to any values that define a downward-sloping curve.  Therefore, a must be a positive number and b must be negative.  The parameters of the inverse demand equation, c and d, will change automatically when you change a or b.  You will probably notice that entering different values for a and b do not change the appearance of the demand curve.  This is because the applet adjusts the scale of each axis based on where the demand curve crosses it.

2.  P and Q can be changed directly or by dragging either of the sliders.  When you change the value of P or Q, the corresponding value of the other is changed to maintain the equality of the equations.

NOTE:  When changing any of the 4 values directly, you must press your Enter or Return key immediately after editing, while the cursor is still in the box, otherwise your changes will not take effect. 

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Copyright 1997,1998, Geoffrey Gerdes
Last Update: November 24, 1998