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.
Contents
Linear Demand
The Demand Equation
The first demand equation beginning economics students usually see is
that of a linear, or straightline 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 lefthand 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 lefthand 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.
Go back to the applet.
Elasticity
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
P_{1} and an adjacent price P_{2}
divided by the initial price. The percent change in price is
thus approximately (P_{2 }
P_{1})/P_{1}. The approximation of the
percent change in Q is obtained in the same way, where
Q_{1} = Q(P_{1}) and
Q_{2}=Q(P_{2}). 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 P_{1 }= P and P_{2}= P
+ P so that P = P_{2} 
P_{1}. Then let Q = Q(P_{2}) 
Q(P_{1}). Substituting the demand equation into the
right side of this equation we get
Q = a + bP_{2 } (a
+ bP_{1}) = b(P_{2}  P_{1}) = bP.
When we substitute P
and Q
into the exact version of the definition of elasticity, we get the
following:

(exact price elasticity)

Go back to the applet.
HOW TO WORK THE APPLET CONTROLS
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 downwardsloping 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.
Go back to the applet. 
Copyright 1997,1998, Geoffrey
Gerdes
Last Update: November 24, 1998
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