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14.2. CONSUMER AND PRODUCER SURPLUS 1

14.2 Consumer and Producer Surplus

Consumer and Producer Surplus

Let us direct our attention towards applying some of the concepts that we learned in regards to the area of

region bounded by two curves. In this section we will use the following notation.

D(q) will denote the price-demand equation of product.

S(q) will denote the price-surplus equation of a product.

The variableq will denote the quantity of the product that is in question. In a free-market economy, the

price of an product is determined by the relationship between its supply and demand. A positive surplus

occurs when the price is above a certain price-level; whereas, a negative surplus (i.e. shortage) occurs the

price is below this price-level. This price-level is known as the equilibrium price. It is the price in which the

effects of demand is in balance with a products supply.

Definition 14.1: Equilibrium-Point for Supply and Demand

The equilibrium-point between forD(q)and S(q)is the point(q0,p0)such thatp0 =D(q0) = S(q0).

Thep0 is referred to as the equilibrium price, andq0 is known as the equilibrium quantity.

The equilibrium-pricep0is the price per unit that the consumers are willing to purchase, and the price that

producers are willing to sell.

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Example1.2 Equilibrium-point

Find the equilibrium-point if the D(q) = 25 + 0.2q and S(q) = 1 + 0.01q2

Solution

The equilibrium-point between D(q)and S(q)can be found by solving the equation D(q) = S(q).

D(q) = S(q)

25 0.2q = 1 + 0.01q2

(q 60)(q + 40) = 0

Since it is impractical for q = 40, it must be the case that q = 60; in other words, the equilibrium-quantity is q0 =60,

which in turns yields the equilibrium-price as S(60) = D(60) = 37. Therefore, the equilibrium-point point is(60, 37).

Sometimes it necessary to determine the difference between the maximum price and the actual price that a

consumer is willing to pay for a product. This quantity is know as the consumer surplus.

Definition 14.2: Consumer Surplus

The consumer surplus at the price-level of p = D(q)is defined as

CS =

q

0

(D(q) p)dq.

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14.2. CONSUMER AND PRODUCER SURPLUS 3

Example2.2 Consumers Surplus

Determine the consumer surplus at a price-level of of$5for the price-demand equation D(q) = 41 0.01q2.

Solution

The question that first needs to be addressed is what is the value ofq when p = 5.

p = D(q)

5 = 41 0.01q2

3, 600 = q2

Since quantity is a nonnegative measure, then it follows that q = 6; therefore, the consumer surplus, at the given

price-level, is found by determining the area bounded by the functions D(q)and p = 5on the interval[0, 6].

60

(41 0.01q2 5)dq = 36q q3

3000

60=215.28

This value$215.28represents the total savings to consumers who are willing to pay a higher price than $5in order to

obtain the product.

The producer surplus measures the difference between the price in which the producers are willing to

supply to the consumers and the actual price that they receive from the consumers.

Definition 14.3: Producer Surplus

The producer surplus at the price-level of p = S(q)is defined as

CS =

q0

( p S(q))dq.

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Example3.2 Producers Surplus

Find the producers surplus at a price level of$30if the price-supply equation is S(q) = 5 + 0.01q2.

Solution

In this situation, we are given p = 30and we need to determine q. We can do this solving the equation p0 =S(q0).

30 = 5 + 0.01q2

2500 = q2

Since quantity is a nonnegative measure, then it follows that q =

50; therefore, the producer surplus, at the givenprice-level, is found by determining the area bounded by the functions S(q)and p = 30on the interval[0, 50].

500

(30 (5 0.01q2))dq =q3

3000+ 25q

500=

5000

3

This value$ 50003 $1666.67represents the total amount gained by the producers who are willing to sell the product

to the consumers at a lower price than$8.

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14.2. CONSUMER AND PRODUCER SURPLUS 5

Consumer Surplus Producer Surplus

The total savings to the consumers who

are willing to pay more than the given price

ofD(q) = pfor a product.

The total gain to the producers who are

willing to supply a product at a price lower

than a given priceS(q) = p.

Ifpis the equilibrium-price forD(q) andS(q), then there is stability between supply and demand.