5.1 copyright © 2014 pearson education, inc. an economics applications: consumer surplus and...
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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.
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CONSUMER SURPLUS
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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
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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
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PRODUCER SURPLUS
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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