Class 2- Average Value+Producer andConsumer Surplus

Terry Mullen

September 2, 2019

1 Area between two curves

The area between two curves f(x) and g(x) is

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Example

Find the area between y = x + 5 and y = 2x + 1 from x = 0 and x = 2.

(A) -6

(B) 10

(C) 6

(D) 0

2 The Fundamental Theorem of Calculus

If F ′(x) is continuous for a ≤ x ≤ b, then∫ b

a

F ′(x)dx =

The definite integral of the derivative of a function gives the total change inthe function.

Example

Suppose ice is forming on a pond at a rate of dydt =

√t2 inches per hour. If y

represents the thickness of the ice in inches at time t in hours since the icestarted forming. Estimate the thickness of the ice after 5 hours.

(A)∫ 5

0

√t2 dt

(B)∫ √

t2 dt

(C)∫ 5

0

√t2

(D)√t2 × 5

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3 Marginal Cost and Change in Total Cost

C(q) represents the cost of producing q items, then C ′(q) (the derivative) isthe marginal cost.

Therefore....

Why????

Fixed Cost:

Total variable cost:

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Example

A marginal cost curve is given in Figure 5.63. If the fixed cost is $1000,estimate the total cost of producing 250 items.

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4 Average Value

Take this rainfall graph from climate-data.org

month Rainfall in (mm)1 2302 1103 1354 1655 3006 3507 2858 1709 9510 11011 18012 330

Find the average rainfall for the year:

To find average we sum up all of the data points and divide by thenumber of data points used:

But that assumes each data point is independent and our function isdiscrete. What if our function f(t) is a continuous, varying function?

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From our definite integral section we said that

∆t =b− a

n

So....

Thus, AVERAGE VALUE of a function is given by:

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b− a

∫ b

a

f(t)dt

5 Average Value Examples

1. Find the average value of the function f(x) = x2+3x−2 between x = 0and x = 3. Use your calculator

(A) 5.5

(B) 16.5

(C) 9.5

(D) 0.5

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2. Suppose C(t) = 5(1.05)t represents the daily cost of heating your house,in dollars per day, where t is time in days and t = 0 corresponds toJanuary 1st.

(a) Find the TOTAL cost to heat your house over the first 2 weeks.

(A) $10.50

(B) $5.26

(C) $7.17

(D) $100.42

(b) Find the AVERAGE cost to heat you house over the first 2 weeks.

(A) $10.50

(B) $5.26

(C) $7.17

(D) $100.42

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3. The rate at which the population of a town can be modeled by

P (t) = 430(1.026)t

Where is in thousands people/year and t is in years since 2000. Whatis the average change in population from 2020 to 2040?

(A) 9.25×1024 thousand people

(B) 469.5 people

(C) 469.5 thousand people

(D) 18779.2 thousand people

4. Suppose I have $5000 in an account that receives 4% interest com-pounded continuously. The money will remain in the account for 6years. What is the average balance?

(A) $5,637.31

(B) $5,001.13

(C) $16,171.60

(D) $33,823.80

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6 Supply and Demand Curves

• Supply curve:

• Demand curve:

• Equilibrium price p∗:

• Equilibrium quantity q∗:

• p0:

• p1:

• q1:

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7 Consumer and Producer Surplus

Most times in a free market the price goes to the equilibrium price.Note: Some consumers would have bought the item at a higher price!Note: Some suppliers would have been willing to produce the item for a

lower price

• Consumer Surplus: Measures the consumers gain from trade. It is thetotal amount gained by consumers by buying the item at the currentprice rather than the price they would have been willing to pay

• Producer Surplus: Measures the suppliers gain from trade. It is thetotal amount gained by producers by selling at the current price, ratherthan the price they would have been willing to accept

Why?

• Let’s think about units of the area

• Demand curve D(q) has units=

• The quantity line q has units =

• Units of area on this graph

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8 Why Teach this in Calculus?

Those shaded regions sure look like areas.

Can you come up with away to grab that area using a definite

integral?

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Example

Suppose supply and demand are given in the graph below.

At the equilibrium price, calculate the consumer and producer

surplus.

Consumer Surplus=

Producer Surplus=

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9 How To Do Consumer and Producer Sur-plus:

1. Find (p∗, q∗), intersection between curves, if not given

2. Consumer surplus ∫ q∗

0

D(q)dq − p∗q∗

3. Producer surplus

p∗q∗ −∫ q∗

0

S(q)dq∗

Example

Suppose demand is given by D(q) = 40− 2q2 and supply is given by

S(q) = 5q2 + 3, find the consumer and producer surplus when the

market is in equilibrium.

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10 Wage and Price Controls

Without outside forces the price of a product often moves to the

equilibrium point.

What happens if the price is set artificially?

The artificial price: Price set by outside source = p+

When the artificial price is higher:

• Consumer surplus

• Producer surplus

Total gain from trade =

At p+ the total gain is less than it would be at p∗ → so we say

the total financial effect of is negative.

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Example

Suppose demand is given by p = 20e−0.002q and supply is given by

p = 0.02q + 1. If an artificial price $10 per unit is set, find the

consumer and producer surplus.

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Example

Given the graph of the supply and demand functions below, fin the

consumer surplus when the market is in equilibrium:

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