Consumer, Producer, & Total Surplus

Consumer, Producer, & Total Surplus#

Consumer & Producer Surplus#

Definition

Consumer surplus is the difference between what customers are willing to pay for a product and what they actually pay.

For a given demand function, $D (x)$ , if $\bar{x}$ is the quantity demanded and $\bar{p}$ is the corresponding unit market price, then consumer surplus, $C S$ , is given by:

C S = \int_{0}^{\bar{x}} D (x) - \bar{p} d x = \int_{0}^{\bar{x}} D (x) d x - \bar{p} \bar{x} .

Geometric Interpretation of Consumer Surplus

Consumer surplus can also be interpreted as the area of the region between the demand curve and the horizontal line $p = \bar{p}$ on the interval $[0, \bar{x}]$ .

../_images/pic_applicationintegrals_consumer_producer_total_1.png — Fig. 13 Consumer surplus#

Definition

Producer surplus is the difference between what sellers receive for their product and what they are willing to receive.

For a given supply function, $S (x)$ , if $\bar{x}$ is the quantity demanded and $\bar{p}$ is the corresponding unit market price, then producer surplus, $P S$ , is given by:

P S = \int_{0}^{\bar{x}} \bar{p} - S (x) d x = \bar{p} \bar{x} - \int_{0}^{\bar{x}} S (x) d x .

Geometric Interpretation of Producer Surplus

Producer surplus can also be interpreted as the area of the region between the supply curve and the horizontal line $p = \bar{p}$ on the interval $[0, \bar{x}]$ .

../_images/pic_applicationintegrals_consumer_producer_total_2.png — Fig. 14 Producer surplus#

Example 1#

Determining surplus at market equilibrium using geometry

The demand function for Penn State Learning’s Calculus on Demand video series is $p = D (x) = - 3 x + 44$ and the corresponding supply function is $p = S (x) = 4 x + 2$ . Determine the consumer and producer surplus at the market equilibrium values by finding the area of the corresponding region between two curves.

Total Surplus#

Definition

Total surplus is the sum of the consumer and the producer surpluses.

For given demand and supply functions, $D (x)$ and $S (x)$ , if $\bar{x}$ is the quantity demanded, then total surplus, $T S$ , is given by:

T S = C S + P S = \int_{0}^{\bar{x}} D (x) - S (x) d x

Geometric Interpretation of Total Surplus

Total surplus can also be interpreted as the area of the region between the demand curve and the supply curve on the interval $[0, \bar{x}]$ .

Total surplus is maximized at the equilibrium quantity.

../_images/pic_applicationintegrals_consumer_producer_total_5.png — Fig. 15 Total surplus#

Example 2#

Determining total surplus

Continuing with Example 1, determine the total surplus at the equilibrium quantity by finding the area of the corresponding region between two curves.

Example 3#

Determining surplus using integration

The quantity demanded for computer chips is given by

p = 225 - x^{2}

where $x$ is the quantity demanded (in hundreds of chips per week), and $p$ is the price of a single chip (in dollars). The manufacturer will make $x$ units available to the market each week if

p = 100 + \frac{x^{2}}{4}

Determine the consumer and producer surplus at the market equilibrium values.

Example 4#

Determining surplus using integration

The supplier of a custom pen will make $x$ units of pens available to the market when the wholesale unit price is $p = \sqrt{9 + 2 x}$ where $p$ is in dollars. Determine the producer surplus when the market unit price is set to $5 a unit.

Consumer, Producer, & Total Surplus

Contents

Consumer, Producer, & Total Surplus#

Consumer & Producer Surplus#

Example 1#

Total Surplus#

Example 2#

Example 3#

Example 4#