2.2
Application of Integration
in
Economics and Business
OBJECTIVE
:
At the end of the lesson, students should
be able to
• find the revenue and cost functions from
the marginal revenue and marginal cost
functions
• define consumer’s surplus and
producer’s surplus regions
• find the market equilibrium point
REVENUE FUNCTION
MARGINAL
REVENUE
R’(x)
TOTAL REVENUE
FUNCTION
antiderivative
R(x)
For any demand function : y = f(x)
y is the price per unit and
x is the number of units (demand)
TOTAL REVENUE, R(x)
and
MARGINAL REVENUE, R’(x)
R is the product of x and y
R(x) = xy = x.f(x)
Marginal revenue with respect to demand is
the derivative with respect to x of the total
revenue
dR
 R' ( x)
dx
Total revenue function is the integral with
respect to x of the marginal revenue function:
R ( x)   R ' ( x)dx
And, since
R
'
(
x
)
dx

R
(
x
)

k

An initial condition must be specified to obtain
a unique total revenue function.
The initial condition :
x = 0, R(x) = 0
Revenue is zero if demand is zero is
frequently used to evaluate the constant
of integration.
R(0)  0
Example 1:
If the marginal revenue function is
R’(x) = 8 – 6x – 2x 2 determine the total
revenue function.
Example 2:
The marginal revenue function for a company’s
product is R’(x) = 50 000 – x, where x equals
the number of units produced and sold.
If total revenue equals 0 when no units is sold,
determine the total revenue function for the
product.
COST FUNCTION
If the total cost y of producing and marketing
x units of a commodity is given by the function
y = C(x)
Then the marginal cost is C ’(x)
dC
= C’(x)
dx
MARGINAL COST
Marginal cost is the derivative with
respect to x, C’(x), of the total cost
function y = C(x). Thus total cost is
the integral with respect to x of the
marginal cost function that is,
y   C ' ( x ) dx = C(x) + k
an initial condition must be specified to
obtain a unique total cost function
Frequently this specification is in terms
of a fixed cost or initial overhead that
is, the cost when x = 0.
x  0, C ( x)  fixedcost
Example 3:
Marginal cost as a function of units produced
is given by C’(x) = 1.064 – 0.005x, find the
total cost function if fixed cost is 16.3.
Example 4:
The marginal cost of producing a
product is C’(x) = x + 100 where x
equals the number of units produced.
It is also known that total cost equals
RM40 000 when x = 100.
Determine the total cost function.
CONSUMER’S SURPLUS REGION
Suppose that p is a price that consumer
willing to pay for a quantity x of a
particular goods. The demand curve can
be written as follows:
p = D(x)
In general, the demand curve is
a decreasing function.
If the market price is yo and the
corresponding market demand is xo,
then the total savings to consumers who
are willing to pay more than yo for the
product and are still able to buy the
product for y0
is represented by area below the
demand curve and above the line
y = yo and is known as
consumer’s surplus (C.S)
price
yo
C.S
p = D(x)
xo
Figure 1
quantity
price
C.S
yo
p = D(x)
xo
quantity
Figure 2
price
C.S
yo
p = D(x)
xo
quantity
Figure 3
Producer’s Surplus Region
Suppose the price p that a producer is
willing to charge for a quantity x of
particular goods is governed by the
supply curve.
p = S(x)
In general, the supply function S is an
increasing function.
If the market price is yo and the
corresponding market supply is xo ,
the total gain to producers who are
willing to supply units at a lower price
than yo and are still able to supply units
at yo
is represented by the area above the
supply curve and below the line y = yo and
is known as producer’s surplus (P.S).
price
p = S(x)
yo
P.S
xo
Figure 5
quantity
price
p = S(x)
yo
P.S
xo
quantity
Figure 6
price
p = S(x)
yo
P.S
xo
Figure 7
quantity
MARKET EQUILIBRIUM POINT
The point of intersection of the demand
curve and the supply curve is called the
equilibrium point of a free market,
which indicated by point E.P.
The coordinate of E.P is (xo, yo),
price
p = S(x)
C.S
yo
E.P(xo, yo)
P.S
p = D(x)
xo
Figure 9
quantity
yo is the price at which producer is
willing to supply and xo is the
quantity of the goods purchased by
the consumer and supplied by the
producer.
This equilibrium condition is expressed
by the equation :
D(x) = S(x)
Formula to determine Consumers’
surplus
xo
C.S =
 D( x)dx  x
o
yo
0
Formula to determine Producer’s
Surplus
xo
P.S

= x o y o  S ( x)dx
0
Example 5:
Determine the consumers’ surplus region at a
price level of RM8 for the price-demand equation
D(x) = 20 – 0.05x
Example 6
Determine the producers’ surplus region at a
price level of RM20 for the price-supply
equation: S(x) = 2 + 0.0002x 2. Then find the
producer’s surplus.
Example 7
Find the equilibrium point, if D(x) = 20 – 0.05x
and S(x) = 2 + 0.0002x 2.
Example 8
Find the producers’ surplus at a price level of
RM 7 for the price-supply equation,
S(x) = 2x + 1
Example 9
Find the consumers’ surplus at a price level of
RM8 for the price-demand equation,
D(x) = 20 - 0.05x
Example 10
Find the consumers’ surplus at a price level of RM4
for the price-demand equation, D(x) = 7 – x
Example 11
If the demand function is y = 16 – x2 and
the supply function is y = 2x + 1, find
the equilibrium price and determine
consumer’s surplus and producer’s
surplus area under pure competition.
Exercise
Find the equilibrium price and then find the
consumers surplus and producers surplus at the
equilibrium price level if
D(x) = 20 – 0.05x and S(x) = 2 + 0.0002x2