Chapter Two: Point Elasticity, Consumer and Producer Surplus
Section 1: Instantaneous Percent Change
We know that percent change is computed by
changein the amount
original amount
Example 1: Suppose the price of cabbage increased from $1.50 per pound to $1.70 per pound. What is
the percent increase?
Solution:
Determine the price increase: $1.70 - $1.50 = $0.20
Then divide by the original amount: $0.20 / $1.50 = 0.13333
Convert to a percentage: 13.333%
Answer: The price increased by about 13.3%
When we want to know an instantaneous percent change, use the same formula, but use derivatives to
compute the change in the amount:
If f(x) determines the amount of something, then
f ' (x)
Instantaneous percent change = f ( x )
Example 2: The number of ants in an ant colony can be determined by the function p(x) = 23t2 + 4t +
9, where t is the number of days since March 31, 2014. Find the instantaneous percent growth on April
20.
Solution: First find p’
p’ = 26t + 4
On April 20, t = 20, the instantaneous rate of change is p’(20) = 26(20) + 4 = 524
Next, find p
When t = 20, p(20) = 23(20)2 + 4(20) + 9 = 9289
Finally, divide: 524 / 9289 ≈ 0.0564
Answer: On April 20, the ant colony is growing at about 5.64%
Section 2: Point Elasticity
So far, we have seen how to compute elasticity between two points by computing the percent change
from point to point. However, there is another kind of elasticity, called point elasticity, that uses
derivatives to find the instantaneous percentage change at just one point.
To do this, the elasticity formula, E = (%Δq)/(%Δp) must be revised in terms of derivatives. We know
that the instantaneous percent change can be computed by:
f ' (x)
f (x)
So if q is the quantity and p is the price, then
E
% Q Q  / Q

% P P  / P
Example 3: Find the point elasticity of the demand function p = -0.04q2 + 3000 when q = 150
Solution: First, find p' when q = 150:
Find the derivative: p' = -0.08q
when q = 150, then the instantaneous rate of change is p' = (-0.08)(150) = -12
Next, find p when q = 150:
p = -0.04q2 + 3000 = -0.04(150)2 + 3000 = 2100
Next, find p’/p: (-12)/(2100) = -0.0057143
Therefore, when q = 150, the price is decreasing about 0.57%
How to find the derivative of q? It's actually really easy: use the equation q = q.
then, q' = 1
then q’/q = 1/q = 1/150 = 0.006667
Therefore, when q = 150, quantity is increasing by about 0.67%
Finally, the elasticity is (percent change in q)/(percent change in p) = 0.006667 / -0.0057143
= -1.16667
Answer: the point elasticity is -1.6667
Section 3: Consumer and Producer Surplus
Below is an illustration of consumer and producer surplus:
Source:
http://www.economicsonline.co.uk/Competitive_markets/Consumer_and_producer_surplus.html
Consumer Surplus
Consumer surplus measures the consumers’ gain from trade. It is the total amount of
savings from consumers who were willing to pay more than the actual price. On the
diagram below, the shaded area represents the consumer surplus:
Declining consumer surplus
Consumer surplus generally declines with consumption. One explanation for this is the law
of diminishing marginal utility, which suggests that the first unit of a good or service consumed
generates much greater utility than the second, which generates greater utility than the third
and subsequent units. A very thirsty consumer will be prepared to pay a relatively high
price for their first soft drink, but, as they drink more, less utility is derived and the price
they would be prepared to pay falls. Therefore, in the above diagram, as consumption rises
from zero, at C, to Q, marginal utility falls. As utility falls, the price that consumers are
prepared to pay declines, causing the demand curve to slope down from A to B.
Some firms can capture this consumer surplus by charging the highest price that consumers
would be prepared to pay, rather than charge price P for all units consumed.
Producer surplus
On the other hand, Producer Surplus measures the producers’ total gain from trade,
resulting from producers selling a product for a price that is higher than they were willing to
sell.
Producer surplus is the additional private benefit to producers, in terms of profit, gained
when the price they receive in the market is more than the minimum they would be
prepared to supply for. In other words they received a reward that more than covers
their costs of production.
The producer surplus derived by all firms in the market is the area from the supply curve to
the price line, EPB.
Economic welfare
Economic welfare is the total benefit available to society from an economic transaction or
situation.
Economic welfare is also called community surplus. Welfare is represented by the area ABE
in the diagram below, which is made up of the area for consumer surplus, plus the area for
producer surplus.
In market analysis economic welfare at equilibrium can be calculated by adding consumer
and producer surplus.
Welfare analysis considers whether economic decisions by individuals, organisations, and the
government increase or decrease economic welfare.
Section 4: Calculating consumer and producer surplus
We already have seen the formula for consumer surplus:
ConsumerSu rplus 

q*
0
D ( q ) dq  revenue
But we haven’t seen the formula for producer surplus. To understand it, let’s take another look at the
producer surplus picture:
The actual revenue is formed by the rectangle (where P and Q meet with B). If we subtract the area
under the supply curve, we are left with the producer surplus. Therefore, the formula for producer
surplus is:
q*
Pr oducer Surplus  revenue   S ( q ) dq
0
Example 4: A demand function is D(q) = -0.04q2 + 35, and a supply curve is S(q) = 2 + 0.2q2
a)
b)
c)
d)
Find the equilibrium point
Find the revenue at the equilibrium point
Find the consumer surplus
Find the producer surplus
Part a: find the equilibrium point
Solution:
To find the equilibrium point,
set the two equations equal to each other: -0.04q2 + 35 = 2 + 0.2q2
move the constants to one side and variables to the other: 33 = 0.24q2
divide by 0.24:
137.5 = q2
take the square root:
11.726 = q
Find the equilibrium price: D(11.726) = -0.04(11.726)2 + 35
= $29.50
Answer to part a: The equilibrium point is (11.726, 29.50)
Part b: find the revenue:
Solution to part b: The revenue is
price * quantity
= (11.726)(29.50)
= $345.92
Answer to part b: The revenue is $345.92
Part c: Find the consumer surplus
Solution: the formula for consumer surplus is
ConsumerSu rplus 

q*
0
D ( q ) dq  revenue
First, we’ll calculate the area under the demand curve:

11.726
0
0.04 q 2  35 dq
0.04 q 3
11.726

 35 q 0
3
 388.91
Then subtract the revenue:
388.91 – 345.92 = 42.99
Answer to part c: The consumer surplus is $42.99
Part d: Find the producer surplus:
Solution:
The formula for the producer surplus is
q*
Pr oducer Surplus  revenue   S ( q ) dq
0
First, let’s compute the area under the supply curve:

11.726
0
2  0.2 q 2 dq
11.726
0.2 q 3 
 2q 

3 0
 130.94
Then, subtract from the revenue:
345.92-130.94 = 214.98
Answer to part d: The producer surplus is $214.98
Chapter 2 Practice Problems
1) The demand curve for a product is D(q) = 90 - 10q. Find the elasticity of demand when q = 4. If the
price increases by 2%, calculate the corresponding change in demand.
2) The demand curve for a product is D(q) = 45 - 8q. Find the elasticity of demand when q = 3. If the
price increases by 4%, calculate the corresponding change in demand.
3) The supply curve for a product is S(q) = 0.3q2 + 0.5q + 6. Find the elasticity of demand when q = 7
4) The supply curve for a product is S(q) = 0.5q2 + 0.4q + 8. Find the elasticity of demand when q = 5
5) A demand function is D(q) = -0.09q2 - 0.4q + 860, and the graph of this function is shown below:
800
600
400
200
10
20
30
40
50
60
70
80
90
a) Compute the point elasticity for q = 40. Indicate this point on the graph. Is the item elastic,
inelastic, or unit elasticity at this point? Compute the price and the revenue at this point.
b) Compute the point elasticity for q = 55. Is the item elastic, inelastic or unit elasticity at this
point? Also, estimate the revenue at this point.
c) Compute the point elasticity for q = 80. Is the item elastic, inelastic or unit elasticity at this
point? Also, estimate the revenue at this point
d) At which point is the revenue the highest?
e) Based on this example, is the following true or false: “maximum revenue occurs at unit
elasticity”
6) The supply curve for a product is S(q) = 34q0.75. Find the elasticity of supply when q = 7. Then find
the elasticity when q = 9.
7) The supply curve for a product is S(q) = 19q0.4. Find the elasticity of supply when q = 7. Then find
the elasticity when q = 9.
8) Suppose that a supply function for charcoal is
S(q) = 2.3q
and the demand function is
D(q) = -0.4q + 30
(a) Graph the supply and demand curves.
(b) Find the point at which supply and demand are in equilibrium
(c) Find the consumer surplus.
(d) Find the producer surplus.
9) Suppose the supply function of a certain item is given by
S(q) = 1.4q
and the demand function is given by
D(q) = -0.6q + 10.
(a) Graph the supply and demand curves.
(b) Find the point at which supply and demand are in equilibrium
(c) Find the consumer surplus.
(d) Find the producer surplus.
10) Suppose the demand function for corn meal is D(q) = 4.5 – 0.008q2, and that demand and supply
are in equilibrium at q = 15. Find the consumer surplus.
11) Suppose the supply function for concrete is given by
S(q) = 100 + 3q3/2 + q5/2
and that supply and demand are in equilibrium at q = 9. Find the producer surplus.
12) A supply function is S(q) = 0.5q2 + 1.55q, and a demand function is D(q) = 135 – 0.7q2
(a) Graph the supply and demand curves.
(b) Find the point at which supply and demand are in equilibrium.
(c) Find the consumer surplus.
(d) Find the producer surplus.
13) A supply function is S(q) = q2 + 2.75q, and a demand function is D(q) = 150 – q2
(a) Graph the supply and demand curves.
(b) Find the point at which supply and demand are in equi librium.
(c) Find the consumer surplus.
(d) Find the producer surplus.
Answers
1) E = -1.25, 2.5% decrease
3) E = 0.7356
5a) E = -2.3, elastic, revenue = $28000
5b) E = -1.0, unit elastic, revenue = $31,116.25
5c) E = -0.2, inelastic, revenue = $20,160
5d) true; highest revenue is at unit elasticity
7) 2.5 for both
9) b) (5,7) c) $7.50 d) $17.50
11) $1999.54
13) b) (8,86) c) $341.33 d) $429.33