5.1

Copyright © 2014 Pearson Education, Inc.

An Economics Applications:Consumer Surplus and Producer Surplus

OBJECTIVE• Given demand and supply functions, find the consumer surplus and the producer surplus at the equilibrium point.

Slide 5- 2Copyright © 2014 Pearson Education, Inc.

CONSUMER SURPLUS

Slide 5- 3Copyright © 2014 Pearson Education, Inc.

DEFINITION:

Suppose that p = D(x) describes the demand function for a commodity. Then, the consumer surplus is defined for the point (Q, P) as

D(x) dx QP0

Q

.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

0( ) .

QD x dx QP

Slide 5- 4Copyright © 2014 Pearson Education, Inc.

Example 1: Find the consumer surplus for the demand function given by

When x = 3, we have Then,

D(x) (x 5)2 when x 3.

D(3) (3 5)2 4.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

3 2

0( 10 25) 12x x dx

3 2

0( 5) 3 4x dx

Consumer

Surplus 0

( )ExD x dx Q P

Slide 5- 5Copyright © 2014 Pearson Education, Inc.

Example 1 (concluded):

5.1 An Economics Application:Consumer Surplus and Producer Surplus

3 2

0( 10 25) 12x x dx

332

0

5 25 123

xx x

3 32 23 0

5 3 25 3 5 0 25 0 123 3

9 45 75 0 12

$27.00

Slide 5- 6Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 1

Find the consumer surplus for the demand function given by 2 6 16 when 1.D x x x x

Finding when 1, we get:D x x

21 1 6 1 16 1 6 16 11.D

Then, consumer surplus is:

0( ) .

QCP D x dx QP

Slide 5- 7Copyright © 2014 Pearson Education, Inc.

PRODUCER SURPLUS

Slide 5- 8Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 1 Concluded

1

2

0

6 16 1 11CP x x dx 1

3 2

0

13 16 11

3CP x x x

3 2 2 21 11 3 1 16 1 0 3 0 16 0 11

3 3CP

1 13 16 11 2 or $2.33

3 3CP

Slide 5- 9Copyright © 2014 Pearson Education, Inc.

DEFINITION:

Suppose that p = S(x) is the supply function for a commodity. Then, the producer surplus is defined for the point (Q, P) as

QP S(x) dx0

Q

.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Slide 5- 10Copyright © 2014 Pearson Education, Inc.

Example 2: Find the producer surplus for

When x = 3, Then,

.3 when 3)( 2 xxxxS

.15333)3( 2 S

5.1 An Economics Application:Consumer Surplus and Producer Surplus

3 2

03 15 ( 3)x x dx

Producer

Surplus 0

( )Ex

Q P S x dx

Slide 5- 11Copyright © 2014 Pearson Education, Inc.

Example 2 (continued):

5.1 An Economics Application:Consumer Surplus and Producer Surplus

$22.50

945 9 9 0

2

3 2 3 23 3 0 045 3 3 3 0

3 2 3 2

33 2

0

45 33 2

x xx

3 2

03 15 ( 3)x x dx

Slide 5- 12Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 2

Find the producer surplus for 21 44 when 1.

3 3S x x x x

When Then, 21 4 21, 1 1 1 4 5 .

3 3 3x S

0

Producer Surplus .Q

QP S x dx 1

2

0

2 1 4 1 5 4

3 3 3x x dx

Slide 5- 13Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 2 Concluded

13 2

0

2 1 25 4

3 9 3x x x

3 2 3 22 1 2 1 25 1 1 4 1 0 0 4 0

3 9 3 9 3

2 1 2 2 7 85 4 5 4

3 9 3 3 9 9

$0.89

12

0

2 1 4 1 5 4

3 3 3x x dx

Slide 5- 14Copyright © 2014 Pearson Education, Inc.

DEFINITION:

The equilibrium point, (xE, pE), is the point at which the supply and demand curves intersect. It is that point at which sellers and buyers come together and purchases and sales actually occur.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Slide 5- 15Copyright © 2014 Pearson Education, Inc.

Example 3: Given

find each of the following:a) The equilibrium point.b) The consumer surplus at the equilibrium point.c) The producer surplus at the equilibrium point.

D(x) (x 5)2 and S(x) x2 x 3,

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Slide 5- 16Copyright © 2014 Pearson Education, Inc.

Example 3 (continued):a) To find the equilibrium point, set D(x) = S(x) and solve.

Thus, xE = 2. To find pE, substitute xE into either D(x)

or S(x) and solve.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

x2

2( 5)x 2 3x x

2 10 25x x 2 3x x

10 25x 3x

22 11x

Slide 5- 17Copyright © 2014 Pearson Education, Inc.

Example 3 (continued):If we choose D(x), we have

Thus, the equilibrium point is (2, $9).

5.1 An Economics Application:Consumer Surplus and Producer Surplus

E Ep D x 2D

22 5

23

$9

Slide 5- 18Copyright © 2014 Pearson Education, Inc.

Example 3 (continued):b) The consumer surplus at the equilibrium point is

5.1 An Economics Application:Consumer Surplus and Producer Surplus

$14.67

27 125 44

183 3 3

3 3(2 5) (0 5)

183 3

2 2

0( 5) 2 9x dx

23

0

( 5)18

3

x

Slide 5- 19Copyright © 2014 Pearson Education, Inc.

Example 3 (concluded):c) The producer surplus at the equilibrium point is

5.1 An Economics Application:Consumer Surplus and Producer Surplus

23 22 2

00

2 9 ( 3) 18 33 2

x xx x dx x

$7.33

3 2 3 2(2) (2) (0) (0)18 3 2 3 0

3 2 3 2

8 4 2218 6 0

3 3 3

Slide 5- 20Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 3

Given find each of the following. Assume

a.) The equilibrium point

b.) The consumer surplus at the equilibrium point

c.) The producer surplus at the equilibrium point

2 21 46 16 and 4,

3 3D x x x S x x x

5.x

Slide 5- 21Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

a.) To find the equilibrium point, set and solve. D x S x

2 21 46 16 4

3 3x x x x

22 17 12 0

3 3x x

2 and 9.x x Through the quadratic formula, we see that Since we assume that , we know that To find substitute into either or and solve.

5x 2.Ex ,Ep

Ex D x S x

Quick Check 3 Continued

Slide 5- 22Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 3 Continued

If we choose we have ,D x

2Ep D x D

22 6 2 16

4 12 16

8

So the equilibrium point is 2,$8 .

Slide 5- 23Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 3 Continued

b.) The consumer surplus at the equilibrium point is 0

.Ex

E ED x dx x p

22

0

6 16 2 8x x dx 2

3 2

0

13 16 16

3x x x

3 2 3 21 12 3 2 16 2 0 3 0 16 0 16

3 3

8 212 32 16 22 16

3 3

$6.67

Slide 5- 24Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Quick Check 3 Concluded

c.) The producer surplus at the equilibrium point is 0

.Ex

E Ex p S x dx 2

2

0

1 42 8 4

3 3x x dx

23 2

0

1 216 4

9 3x x x

3 2 3 21 2 1 216 2 2 4 2 0 0 4 0

9 3 9 3

8 8 516 8 16 11

9 3 9

$4.44

Slide 5- 25Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Section Summary

• A demand curve is the graph of a function which represents the unit price a consumer is willing to pay for items. It is usually a decreasing function.

• A supply curve is the graph of a function which represents the unit price a producer is willing to accept for items. It is usually an increasing function.

,p D xp x

,p S xp x

Slide 5- 26Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Section Summary Continued

• Consumer surplus at point is defined as

•Producer surplus at point is defined as ,Q P

0

.Q

QP S x dx

,Q P

0

.Q

D x dx QP

Slide 5- 27Copyright © 2014 Pearson Education, Inc.

5.1 An Economics Application:Consumer Surplus and Producer Surplus

Section Summary Concluded

• The equilibrium point is the point at which the supply and demand curves intersect. The consumer surplus at the equilibrium point is

The producer surplus at the equilibrium point is

,E Ex p

0

.Ex

E ED x dx x p

0

.Ex

E Ex p S x dx