Applied Calculus for Business Worksheets
By O. Pauline Chow
HACC
1
Math 110 Applied Calculus for Business
Lecture Notes on Chapter 7.2
(audio dated 10/08/07 at 9:30 am, 19 minutes long)
Applications in Business and Economics
Consumers’ Surplus
Producers’ Surplus
Applications in Business and Economics
One application is the consumers’ and producers’ surplus in business.
Consumers’ Surplus
( )
Let p = D(x) be the price-demand function and x, p be a point on the curve. Consumers’ surplus is the area between p = D(x) and
p = p.
x
Consumers’ surplus = CS =
∫ ( D( x) − p ) dx
0
Ex 1 The price-demand equation is p = D(x) = 200 – 0.02x. Find the consumers’ surplus at a price level of p = $120 .
You need to find the corresponding x-value, x .
120 = 200 − 0.02 x
0.02 x = 80; x = 4000
x
(
)
CS = ∫ D( x) − p dx
0
CS =
4000
4000
0
0
4000
2
∫ ( 200 − 0.02 x − 120 ) dx = ∫ (80 − 0.02 x ) dx = 80 x − 0.01x  0
Producers’ Surplus
= $160, 000
( )
Let p = S(x) be the price-supply function and x, p be a point on the curve. Producers’ surplus is the area between p = S(x) and p
= p.
Calculus for Business, Economics, Life Sciences, and Social Sciences, by Barnett, Ziegler, Byleen, 11th edition, Prentice Hall
Applied Calculus for Business Worksheets
By O. Pauline Chow
HACC
2
x
Producers’ surplus = PS =
∫ ( p − S ( x) ) dx
0
Ex 2 The price-supply equation is p = S(x) = 15 + 0.1x + 0.003x2. Find the producers’ surplus at a price level of p = $55 .
You need to find the corresponding x-value, x .
55 = 15 + 0.1x + 0.003 x
2
2
0.003x + 0.1x − 40 = 0
−0.1 ± 0.49
= 100 or − 133.3
0.006
x = 100
x=
x
(
)
PS = ∫ p − S ( x) dx
0
100
PS =
100
∫ ( 55 − 15 − 0.1x − 0.003x ) dx = ∫ ( 40 − 0.1x − 0.003x ) dx =  40 x − 0.05x
2
2
0
2
100
− 0.001x 3 
0
= $2500
0
Ex 3 Given: p = D(x) = 25 – 0.004x2, p = S(x) = 5 + 0.004x2. Find the consumers’ and producers’ surpluses.
First you need to find the point(s) of intersection which is the same as the equilibrium point.
25 – 0.004x2 = 5 + 0.004x2
20 = 0.008x2
x2 = 2500, x = 50
p = 25 – 0.004(50)2 = 15
( x, p ) = (50,15)
50
CS =
∫ ( 25 − 0.004 x
2
− 15 ) dx =
0
0.004

∫ (10 − 0.004 x ) dx = 10 x − 3 x
2
3
0
50
PS =
50
50
50
0.004 

∫ (15 − 5 − 0.004 x ) dx = ∫ (10 − 0.004 x ) dx = 10 x − 3 x 
2
0
2
0

 = 333.3 ≈ 333.33
0
50
= 333.3 ≈ 333.33
3
0
Calculus for Business, Economics, Life Sciences, and Social Sciences, by Barnett, Ziegler, Byleen, 11th edition, Prentice Hall
Applied Calculus for Business Worksheets
By O. Pauline Chow
HACC
3
Ex 4 Given: p = D(x) = 185e –0.005x, p = S(x) = 25e 0.005x . Find the consumers’ and producers’ surpluses.
First you need to find the point(s) of intersection which is the same as the equilibrium point.
185e −0.005 x = 25e0.005 x
185 e0.005 x
=
= e0.005 x + 0.005 x
25 e −0.005 x
7.4 = e0.01x
ln 7.4 = 0.01x
ln 7.4
= 200.148 ≈ 200
0.01
p = 185e −0.005(200.148) = 68
x=
( x, p ) = (200, 68)
200
200
CS =
∫ (185e
0
−0.005 x
 185 −0.005 x

e
− 68 ) dx = 
− 68 x 
 −0.005
0
=  −37000e −0.005 x − 68 x 
200
200
0
= −37000e −1 − 13600 + 37000 ≈ 9788.46
200
25 0.005 x 

0.005 x
0.005 x 200
∫0 ( 68 − 25e ) dx = 68x − 0.005 e  0 = 68 x − 5000e  0
= 13600 − 5000e + 5000 ≈ 5008.59
PS =
Calculus for Business, Economics, Life Sciences, and Social Sciences, by Barnett, Ziegler, Byleen, 11th edition, Prentice Hall