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 R(x)

antiderivative

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 revenue x of the total

dR



R

' (

x

)

dx

Total revenue function is the integral with respect to

x

of the marginal revenue function: And, since

R

(

x

)  

R

' (

x

)

dx



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 frequently used to evaluate the constant of integration.

is

R

( 0 )  0

Example 1:

If the marginal revenue function is

R’(x)

= 8 – 6

x

– 2 revenue function.

x

2 determine the total

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

x

If the total cost

y

of producing and marketing units of a commodity is given by the function y = C(x) Then the marginal cost is

C

’(x)

dC dx

=

C’(x)

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

)



fixed cost

Example 3:

Marginal cost as a function of units produced is given by

C’(x)

= 1.064 – 0.005

x

, 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 y o and the corresponding market demand is x o , then the total savings to consumers who are willing to pay more than y product for y 0 o for the product and are still able to buy the is represented by area below the demand curve and above the line

y = y

o and is known as consumer’s surplus (C.S)

price y o

C.S

Figure 1

x

o

p = D(x)

quantity

price

y

o

C.S

p = D(x)

x

o

Figure 2

quantity

price

y

o

C.S

p = D(x)

x

o

Figure 3

quantity

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 y o and the corresponding market supply is x o , the total gain to producers who are willing to supply units at a lower price than y o and are still able to supply units at y o is represented by the area above the supply curve and below the line y = y o is known as producer’s surplus (P.S).

and

price

y

o

P.S

x

o

Figure 5

p = S(x)

quantity

price

y

o

P.S

p = S(x)

x

o

Figure 6

quantity

price

y

o

P.S

x

o

Figure 7

p = S(x)

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 (x

o , y o

),

price

C.S

y

o

P.S

x

o

p = S(x)

E.P(x

o

, y

o

)

p = D(x)

quantity

Figure 9

y

o is the price at which producer is willing to supply and x producer. o is the quantity of the goods purchased by the consumer and supplied by the This equilibrium condition is expressed by the equation :

D(x) = S(x)

Formula to determine Consumers’ surplus

C.S = 0 

x o D

(

x

)

dx



x o y o

Formula to determine Producer’s Surplus

x o

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 – 0.05

x

Example 6

Determine the producers’ surplus region at a price level of RM20 for the price-supply equation:

S(x)

= 2 + 0.0002

producer’s surplus.

x

2 . Then find the

Example 7

Find the equilibrium point, if and

S(x)

= 2 + 0.0002

x

2.

D(x)

= 20 – 0.05

x

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 – x the supply function is y = 2x + 1, find the equilibrium price and determine consumer’s surplus and producer’s surplus area under pure competition.

2 and

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 + 0.0002x

2