11.5 Economic Applications of the Derivative - PBL …pblpathways.com/calc/C11_5.pdf · The total...
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11.5 Economic Applications of the Derivative
Question 1: What does the term marginal mean?
Question 2: How are derivatives used to compute elasticity?
Derivatives are perfect for examining change. By their definition, they tells us how one
variable changes when another variable changes. In business and economics, this
allows us to examine how revenue and cost change as the quantity produced and sold
changes. Marginal revenue and marginal cost help a business determine compute
these changes.
Elasticity is used to determine how changes in price affect the quantity demanded by
consumers. Understanding this relationship helps us to determine whether a price
should be increased or decreased.
In this section we’ll examine these terms and apply them to several examples drawn
from businesses operating in the real world.
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Question 1: What does the term marginal mean?
In economics, the term marginal is used to indicate the change in some benefit or cost
when an additional unit is produced. For instance, the marginal revenue is the change in
total revenue when an additional unit is produced. If we let the total revenue function be
represented by TR Q , where Q is the number of units produced and sold, then the
marginal revenue is calculated with the difference
Marginal Revenue 1TR Q TR Q
Since this is a difference, it corresponds to a change in revenue. The production levels
Q and 1Q differ by one units, so 1TR Q TR Q describes the change in total
revenue when production is changed by one unit.
The marginal revenue can also be interpreted as an average rate of change. Using the
definition of average rate of change, the average rate of change of R Q over , 1Q Q
is
Average rate of change 1
of over , 1
TR Q TR Q
T Q QR Q Q
1 Q 1TR Q TR Q
Let’s label these quantities on a graph of a revenue function ( )TR Q .
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Figure 1 – A revenue function TR(Q) (blue) with a secant line (red) passing through two points (Q, TR(Q)) and (Q+1, TR(Q+1)).
In addition to being equal to the average rate of change of ( )TR Q over , 1Q Q , we can
view the marginal revenue as a slope. If we calculate the slope of the secant line
between the points ,Q TR Q and 1, 1Q TR Q , we get a numerator equal to
1TR Q TR Q and denominator equal to 1. This yields the same expression,
1TR Q TR Q , as the marginal revenue.
Now let’s compare this slope to the slope of a tangent line to the revenue function
( )TR Q .
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Figure 2 - A revenue function R(Q) (blue) with a tangent line (green) at (Q, R(Q)).
In Figure 2, we have placed the point of tangency on the graph at ,Q TR Q . Another
point is placed on the tangent line at 1Q . Since these points are separated by 1 unit
and the slope of the tangent line is TR Q , the points must be separated vertically by
TR Q . This insures that the slope of the tangent line between these points is 1
TR Q
or TR Q .
These graphs may appear almost identical, so let’s compare them side by side.
Figure 3 – A revenue function R(Q) with a secant line (red) and a tangent line (green).
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The secant line (red) and the tangent line (green) both pass through ,Q TR Q .
However, the tangent line is slightly steeper and passes through a slightly higher point
than the secant line. This means the slope of the tangent line is approximately the same
as the slope of the secant line. In terms of the revenue,
Slope of the secant line Slope of the tangent line
1
1 1
TR Q TR Q TR Q
The marginal revenue at a production level Q is
approximately equal to the derivative of the total revenue
function at Q,
1TR Q TR Q TR Q
We can evaluate the derivative of the revenue function to estimate the marginal revenue
at any production level.
Example 1 Find and Interpret the Marginal Revenue
Based on sales data from 2000 to 2009, the relationship between the
price per barrel of beer P at the Boston Beer Company and the number
of barrels sold annually, Q, can be modeled by the power function
0.0209209.7204P Q
where Q is in thousands of barrels.
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a. Find the revenue function ( )TR Q .
Solution To find the revenue, we must multiply the quantity times the
price. In this example, the quantity of beer is represented by Q in
thousands of barrels and the price per barrel is reprensented by
0.0209209.7204P Q in dollars per barrel. The revenue function is
0.0209
quantity price
( ) 209.7204TR Q Q Q
Simplifying the function by combining the factors,
0.9791209.7204TR Q Q
where the exponents have been added on the Q factors.
The units of the revenue function are very important. By multiplying the
units on the price and the quantity, we can determine the units on the
revenue function:
units on units onthousands of
the quantity the prbarrels
ice
dollars
barrel
thousands of dollars
We complete the total revenue function by labeling the units on the
function and write
0.9791209.7204 thousand dollarsTR Q Q
b. Find the annual revenue when 1,500,000 barrels of beer are sold.
Solution The annual revenue from 1,500,000 barrels of beer is found by
substituting this production level into ( )R Q . Since Q is in thousands of
barrels, we need to divide the production level by 1000 to scale it
properly,
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1,500,000 barrels 1 thousand barrels
1 1000 bar rels 1500 thousand barrels
The conversion factor, 1 thousand barrels
1000 barrels, is equal to 1 and converts
1,500,000 barrels to 1500 thousand barrels. The revenue at this product
level is
0.9791(1500) 209.7204 1500 269,992.4558 thousand dollarsTR
We can convert this amount to dollars by multiplying by 1000,
thou269, sand992.4558 dol lars
t
1000 dollars
1 housand do1 llars 269,992, 455.8 dollars
c. Approximate the marginal revenue when 1,500,000 barrels of beer
are sold.
Solution The marginal revenue at 1,500,000 barrels of beer is
approximated by (1500)TR . We can find the derivative using the power
rule for derivatives,
0.9791
0.9791
.0209
0.0209
( ) 209.7204
209.7204
209.7204 0.9791
205.3372
dTR Q Q
dQ
dQ
dQ
Q
Q
The marginal revenue is approximately
0.0209(1500) 205.3372 1500 176.2330TR
Use the Constant Times a Function Rule
Use the Power Rule
Multiply the constants
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d. How will revenue change if production is increased from 1,500,000
barrels?
Solution The marginal revenue is the same as the instantaneous rate of
change and has the same units. These units are found by dividing the
units on the dependent variable by the units on the independent
variable,
units on the dependent variable thousand
units on the independent variable
dollars
thousand
dollars
barrels barrels
The marginal revenue is about 176.2330 dollars per barrel meaning that
an increase in production of one barrel will result in an increase in
revenue of approximately 176.23 dollars.
The actual increase is found by subtract the revenue at each level,
0.9791 0.9791(1500.001) (1500) 209.7204 1500.001 209.7204 1500
0.17623309 thousand dollars
TR TR
or 176.23309 dollars.
The marginal cost is the change in cost when an additional unit is produced. If Q units
are produced at a total cost ( )TC Q , the marginal cost is defined as
Marginal Cost 1TC Q TC Q
This definition is identical to the definition of marginal revenue except that the total cost
function is used instead of the total revenue function. Like the marginal revenue, the
marginal cost at a production level Q is approximately the same as the derivative of the
total cost function.
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The marginal cost at a production level Q is approximately
equal to the derivative of the total cost function at Q,
1TC Q TC Q TC Q
Example 2 Find and Interpret the Marginal Cost
The total cost TC Q to produce Q thousand barrels of beer at the Craft
Brewers Alliance from 2000 to 2009 is given by the function
3 2( ) 0.0024 2.9978 961.4000 119249.2929TC Q Q Q Q
where the cost is in thousands of dollars.
a. Approximate the marginal cost for a production level of 300,000
barrels of beer.
Solution The marginal cost at a production level of 300,000 barrels of
beer is approximated by ( )TC Q at that production level. The derivative
of the cost function is
3 2
2
2
( ) 0.0024 2.9978 961.4000 119249.2929
0.0024 3 2.9978 2 961.400 1 0
0.0072 5.9956 961.4000
dTC Q Q Q Q
dQ
Q Q
Q Q
Since the quantity Q is in thousands of barrels, we must substitute 300
thousand into ( )TC Q to estimate the marginal revenue at 300,000
barrels. When we do this, we get
2(300) 0.0072 300 5.9956 300 961.4000
189.28
TC
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Since this number is a rate, the units on this value is
units on the dependent variable thousand
units on the independent variable
dollars
thousand
dollars
barrels barrels
a. What does the marginal cost at this production level tell you about
beer production at the Craft Brewers Alliance?
Solution Since costs are increasing at 189.28 dollars per barrel, an
increase in production of 1 barrel (from 300,000 to 300,001 barrels) will
result in an increase in cost of 189.28 dollars.
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Question 2: How are derivatives used to compute elasticity?
In economics, the term elasticity refers to the responsiveness of one economic variable
to changes in another economic variable. The elasticity is measured in terms of
percentage changes instead of absolute changes. This means we measure the change
in a variable as a percentage of the original amount of the variable. For instance, the
percent change in a variable X is defined as
Change in the variable Percent change in
Original value of
XX
X
This definition can be symbolized in a compact form by symbolizing the change as ∆ X.
If a variable X changes from one value X to another value X
+ ∆X, then
Percent change in X
XX
Suppose that the value of X changes from 20 to 30. The percentage change is
30 20Percent change in 0.5
20X
Written as a percentage, this is a percentage change in X of 50%.
If we wish to find the elasticity of Y with respect to X, we find the ratio of the percentage
change in Y to the percentage change in X.
The elasticity of Y with respect to X is
Percent change in Elasticity of with respect to
Percent change in
YY X
X
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For most elasticity calculations, the percent change in one variable corresponds to an
increase while the other percent change corresponds to a decrease. This means the
numerator and denominator will have opposite signs resulting in a negative value. Some
textbooks will define the elasticity with a negative sign or in absolute values, but we’ll
preserve the sign to emphasize the nature of the percent changes. An elasticity more
negative than -1 indicates that a percent increase in X corresponds to a greater percent
decrease in Y. In this case, we would say that the variable Y is elastic with respect to
variable X.
An elasticity between -1 and 0 indicates that a percent increase in X corresponds to a
smaller percent decrease in Y. In this case, the variable Y is inelastic with respect to the
variable X.
An elasticity of -1 means that a percent increase in X corresponds to an identical
percent decrease in the variable Y. In this case, the variable Y is said to be unit-elastic
with respect to the variable X.
The most common use of elasticity in economics is price elasticity of demand or
elasticity of demand with respect to price. This concept allows us to explore the
responsiveness of the consumer demand for some product or service to changes in the
price of that product or service. If the price of a cup of coffee were to increase, how
would this effect the quantity sold? In 2006, Starbuck’s increased prices on all coffee
drinks by 5 cents. At one franchise in Sacramento, California, the price on a tall coffee
increased from $1.65 to $1.70. As expected, the demand for tall coffees dropped from
440 units per day to 436 units per day. The percent change in price P is
1.70 1.65.03
1.65
P
P
or about 3%. The percent change in the demand Q is
436 440.009
440
Q
Q
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or about -0.9%. An 3% increase in price corresponds to a 0.9% decrease in the quantity
of tall coffees sold.
The elasticity of demand with respect to price or price elasticity of demand is
Percent change in 0.0090.3
Percent change in P .03
QE
Since this is a ratio, we can think of it as 0.3
1
and say that a 1% increase in price
corresponds to a 0.3% drop in demand for tall coffee. On the other hand, we could also
think of this ratio as 0.3
1 and say that a 1% drop in price corresponds to a 0.3% increase
in demand. Many economics textbooks work in terms of larger price changes and will
interpret this ratio as 3
10
. In this case, we would say that a 10% increase in the price of
a tall coffee is accompanied by a 3% decrease in demand.
If the relationship between demand and price is given by a function, we can utilize the
derivative of the demand function to calculate the price elasticity of demand. If we write
think of the change in price as being from P to P P and the corresponding change in
demand as being from Q to Q Q , the corresponding percent changes in price and
demand are P
P
and
Q
Q
. The definition of elasticity leads to
Percent change in
Percent change in P
QE
QQP
P
Q P
Q P
P Q
Q P
Definition of price elasticity of demand
Replace the percent changes with appropriate symbols
Dividing by P
P
is the same as
multiplying by the reciprocal PP
Rearrange the factors in the numerators of each fraction.
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The factor Q
P
represents the average rate of change of demand Q with respect to
price P. If we assume the change in price is small, we can replace the average rate of
change with the instantaneous rate of change, dQ
dP.
If the demand Q and the price P are related by a demand
function ( )Q f P , then the price elasticity of demand E is
EP dQ
Q dP
If 1E , then demand is elastic and a percent increase in
price yields a larger percent decrease in demand.
If 1 0E , then demand is inelastic and a percent
increase in price yields a smaller percent decrease in
demand.
Several other cases of elasticity should be mentioned. If 1E , then the demand is
unit-elastic and a percent increase in price yields the same percent decrease in
demand. Demand is perfectly elastic if any increase in price causes the quantity
demanded to decrease to 0. Demand is perfectly inelastic if and increase or decrease in
price makes no change in the quantity demanded.
Some economists prefer to define elasticity as a positive number by taking the absolute
value of P dP
Q dQ . This change loses the idea that an increase in price leads to a drop in
the quantity sold, For this reason, we’ll use the definition above for computing price
elasticity of demand.
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Example 3 Calculate the Price Elasticity of Demand
Using data from market years 2000 through 2010, the relationship
between the price per bushel of oats and the number of bushels of oats
sold is given by
0.543152.07Q P
where P is in dollars and Q is in millions of bushels.
a. Find the price elasticity of demand when the price per bushel is
$1.75.
Solution Since the relationship between the quantity and price is given
as a function of price, we may compute the price elasticity of demand
from
P dQE
Q dP
The derivative dQ
dP is found from the relationship between quantity and
price. Using the Product with a Constant Rule and the Power Rule, we
get
0.543
1.543
1.543
152.07
152.07 0.543
82.574
dQ dP
dP dP
P
P
Substitute the expression 1.54382.574dQ
PdP
for the derivative and the
quantity 0.543152.07Q P into the elasticity formula yields
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1.5430.543
82.574152.07
0.543
P dQE
Q dP
PP
P
Since the elasticity simplifies to a form that does not contain P, the
elasticity does not depend upon price.
b. Is the demand for oats elastic or inelastic when the price per bushel
is $1.75?
Solution Since the elasticity, 0.543E , is between -1 and 0, the
elasticity is inelastic. As a consequence, a change in price of 1% will
result in a 0.543% decrease in the quantity demanded.
The product of the price per unit and the quantity demanded is equal to the revenue
from the product. If the quantity demanded decreases as the price is increased, what
will be the overall effect on the total revenue? To answer this question, let’s examine a
table of prices, corresponding quantities, revenue and the elasticity for a product where
the demand function is 120Q P .
price P 20 40 60 80 100
quantity Q 100 80 60 40 20
total revenue TR 2000 3200 3600 3200 2000
price elasticity of demand E
-0.2 -0.5 -1 -2 -5
Even though the slope of the demand curve is -1, the price elasticity of demand
changes as the price increases. For lower prices (green), the price elasticity of demand
is between -1 and 0 and increasing the price increases revenue. For higher prices
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(blue), the elasticity is more negative than 1 and increasing the price leads to lower
revenue. The highest revenue occurs when the elasticity is equal to -1.
Should prices be increased?
1. If demand is inelastic 1 0E , then a price increase
yields an increase in total revenue.
2. If demand is elastic 1E , then a price increase yields
a decrease in total revenue.
3. Total revenue is highest at a price where demand is unit-
elastic.
For inelastic demand, price increases are countered by small decreases in the quantity
resulting in more revenue. However, when demand is elastic, price increases lead to
large drops in the quantity sold that lowers the overall revenue. If maximizing revenue is
the overall goal, setting the price where the price elasticity of demand is equal to -1 is
ideal. This goal ignores cost and should be approached cautiously.
Example 4 Calculate the Price Elasticity of Demand
A small technology company in Northern California is developing a
tablet PC to compete with Apple’s Ipad. Based on market surveys, the
company believes the quantity Q (in thousands of units) that will be
demanded by consumers is related to the price P (in thousands of
dollars) by the relationship
24000 250Q P
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a. Find the price elasticity of demand when the price of the tablet PC
is $3000.
Solution To compute the price elasticity of demand, we need to find the
derivative of the demand function 24000 250Q P :
2
2
4000 250
4000 250
0 250 2
500
dQ dP
dP dP
d dP
dP dP
P
P
Use the derivative and the expression for the quantity Q in the formula
for price elasticity of demand,
2500
4000 250
P dQE
Q dP
PP
P
We can simplify the right hand side of this equation to make it easier to
substitute the price in:
2
2
2
500
4000 250
500
4000 250
P PE
P
P
P
A tablet PC priced at 3000 dollars corresponds to 3P . If we put this
value into the expression for price elasticity of demand, we get
2
2
500 32.57
4000 250 3E
Use the Sum / Difference Rule and the Constant Times a Function Rule
Use the Power Rule
Combine factors
Simplify numerator
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Since the elasticity, 2.57E , is more negative than -1, the demand is
elastic. As a consequence, revenue will drop as the price increases.
b. At what price is the price elasticity of demand equal to -1?
Solution From part a, we know that a price of $3000 is elastic. At lower
prices the demand might be inelastic. To find the dividing line between
elastic and inelastic, we set the price elasticity of demand equal to -1,
2
2
5001
4000 250
P
P
Now solve for P:
2
2 22
2 2
2
2
5004000 250 1 4000 250
4000 250
500 4000 250
750 4000
16
3
16
3
PP P
P
P P
P
P
P
The negative price is not a possible value for the price of a tablet PC.
The other price, 163P , is approximately 2.309. Since P has units of
thousands of dollars, this value corresponds to a price of $2309.
Multiply both sides by the denominator to clear fraction
Add 250P2 to both sides
Divide both sides by 750 and reduce
Square root both sides of the equation
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Figure 4 – At the point of intersection, the elasticity is equal to -1. To the left or right of the point of intersection, the price elasticity of demand is either lower or higher than -1.
If the price is lower than $2309, the value of E is lower than -1 meaning
demand is inelastic. Prices higher than $2309 result in elastic demand.
This means revenue is maximized at a price of $2309.
(P,E ) (2.309, -1)
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