Q
d
= m P + b
where
m
is the slope, and
b
is the intercept.
(The subscript
d
on
Q
indicates that we are looking at demand)
The
inverse demand equation
describes
price
as a function of
quantity
.
At various points in this course
we will also refer to the inverse demand function as the
marginal benefit function.
This is because the
“price” indicates the maximum amount of money that an individual would pay to obtain an extra unit of the
commodity, or equivalently, the benefit to the consumer of the extra unit of the commodity.
The appropriate equation will have the form:
P = m Q
d
+ b
This is the form that is typically graphed, as it makes intuitive sense.
At high prices, consumers demand
less quantity, while at low prices, consumers demand more quantity.
You can derive either equation first, but it may make more sense to derive the inverse demand equation
first, since this is what we graph.
Recall that the point-slope formula is
(
Y
2
– Y
1
) / (X
2
– X
1
) = m
,
where
m
is the slope
So given any two
(X,Y)
points, you can find the slope of a line.
For inverse demand,
P
corresponds with
Y
, and
Q
d
corresponds with
X
, so the formula can be written as:
(
P
2
– P
1
) / (Q
d 2
– Q
d 1
) = m
(this flips if we are finding the demand equation)
The points we use need to be in
(Q
d
,P)
form.
Using the first two rows of the demand schedule we can take these points:
(1,100, 0.50)
and
(1,050, 1.00)
(remember that this technique will work with
any
two points)
We can calculate the slope as follows:
(1.00 – 0.50) / (1,050 – 1,100) = (0.50 / - 50) = - 0.01 = m
Plugging
m
into the inverse demand equation, we have:
P = -0.01 Q
d
+ b
Now we need to find
b
.
This is done by selecting any point
(Q,P)
, substituting it into the equation, and
solving.
Using the point (1,100, 0.50), we have:
0.50 = (- 0.01 * 1,100) + b
0.50 = (-11) + b