Review Consumers Producers Surplus

8/2/2019 Review Consumers Producers Surplus

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Calculus

Review: Consumers' and Producers' Surplus

We have already worked with price-demand and price-supply equations but let's refresh our

memory. p = D(x) is the price-demand equation for a product, where x is the number of units

of the product that consumers will purchase at a price of $p per unit and p = S(x) is the price-supply equation for a product where x is the number of units of the product that producers will

supply at a price of $p per unit. Let's look at some examples.

Example 1: Let p = D(x) = 20 - 0.05x. (a) Find the price per item if the demand for the item

is 100 units and (b) find the number of units consumers will purchase if the price is $8.

Solution: (a) p = D(100) = 20 - 0.05(100) = 20 - 5 = 15. So if the demand is 100 units the

price will be $15.

(b) We wish to find x so that p = D(x) = 20 - 0.05x = 8

20 - 0.05x = 8

- 0.05x = 8 - 20

-0.05 x = -12

x = 240

So, if the price is $8, 240 consumers will purchase the item.

Example 2: Let p = S(x) = 2 + 0.0002x2. (a) Find the price at which the producers would be

willing to provide 550 units and (b) find the number of units the producers will supply if the

price is $6.50.

Solution: (a) p = S(550) = 2 + 0.0002H550L2 = 62.5

So, the producers would be willing to provide 550 units if the price were $62.50 per unit.

(b) We wish to find x so that p = S(x) = 2 + 0.0002x2 = 6.50

0.0002x2 = 4.50

x2 = 22500

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x = 150

So, at a price of $6.50, the producers would be willing to provide 150 units.

Enough review. Let's get on to the new stuff!

Look at the graph of the price-demand equation below. p represents the current price and x

represents the number of units that can be sold at that price.

x

x

p

p

p = DHxL

Now look at the same graph below. Notice that if the price were higher than p, the demand is

less than x

, but some consumers are still willing to pay the higher price.

x

xx

p

p

p

p = DHxL

This means that consumers who were willing to pay more for the item ($p) but were able to

buy the product for less ($p) have saved money. Let's put this into perspective. You have

saved your money to buy a new CD player. You walk into the store and find out it's on sale

for half off! You were perfectly willing to pay the full price but didn't have to. So you saved



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money. The amount consumers save on an item when they are able to pay less than they are

willing to pay is called consumers' surplus. We can use definite integrals to find the total

amount all of the consumers saved who were willing to pay more for an item. The consumers'

surplus is represented by the area between the price-demand equation and the price-level equa-

tion. Look at the definition below.

Consumers' Surplus

If (x, p) is a point on the graph of the price-demand equation p = D(x) for a particular prod-

uct, then the consumers' surplus, CS, at a price level p is

CS = 0x

@DHxL - pD x

which is the area between p = p and p = D(x) from x = 0 to x = x, as shown below.

The consumers' surplus represents the total savings to consumers who are willing to pay

more than p for the product but are still able to buy the product for p.

x

x

p

p

p = DHxL

CS

x

x

p

p

Similarly, if p = S(x) is the price-supply equation for a product , p is the current price, and

x is the current supply, some suppliers are still willing to supply some units at a lower price

than p. The additional money that these suppliers gain from the higher price is called produc-



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ers' surplus and can be expressed in terms of a definite integral. See the definition below.

Producers' Surplus

If (x, p) is a point on the graph of the price-supply equation p = S(x), then the producers'

surplus, PS, at a price level p is

PS =0x@p - SHxLD x

which is the area between p = p and p = S(x) from x = 0 to x = x, as shown below.

The producers' surplus represents the total gain to producers who are willing to supply

units at a lower price than p but are still able to supply units at p.

x

x

p

p

PS

p = SHxL

x

x

p

p

In a free competitive market, the price of a product is determined by the relationship

between supply and demand. If p = D(x) and p = S(x) are the price-demand and price-supply

equations, respectively, for a product and if (x, p) is the point of intersection of these equa-

tions, p is called the equilibrium price and x is called the equilibrium quantity. This is the

price level that often determines both the consumers' surplus and the producers' surplus.

Look at the example below.

Example 3: Find the equilibrium price and then find the consumers'surplus and the producers'

surplus at the equilibrium price level if



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p = D(x) = 20 - 0.05x and p = S(x) = 2 + 0.0002x2

Solution: First find the equilibrium point by setting the price-demand and price-supply equa-

tions equal to each other. Enter both equations into your calculator as Y1and Y2. Set the Xmin

and Xmax to 0 and 400 respectively (notice that the price-demand equation is 0 at x = 400).

Now ZoomFit the curve. Use your calculator to find the point of intersection. You should

find the intersection to be (200, 10). The x-value in your calculator is the equilibrium quantity

and the y-value is the equilibrium price. Look at the graph below.

100 200 300 400x

5

10

15

20

p

CS

PS

p = DHxL= 20 - 0.05x

p = SHxL= 2 + 0.0002x2

Hx, pL

100 200 300 400x

5

10

15

20

p

The consumers' surplus is the area between the graph of p = D(x) = 20 - 0.05x and the graph of

p = 10 from x = 0 to x = 200. This results in the following definite integral

CS = 0200H20 - 0.05x - 10L x

Evaluating on our calculator gives us 1,000. So the consumers' surplus is $1,000.

The producers' surplus is the area between the graph of p = 10 and the graph of p = S(x) = 2 +

0.0002x2 from x = 0 to x = 200.

PS = 0200@10 - H2 + 0.0002x2LD x = $1,067 Rounded to the nearest dollar.


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Review Consumers Producers Surplus