Lesson 9 – Marginal Analysis
Announcements
Quiz
Homework
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Given a model of a process – sometimes it is a good idea to get an idea of the costs
involved with producing just one more item at a given point. This process is quite
similar to blending the average rate of change with the derivative. The average is
taken over a denominator of 1…ie, where h = 1…and if the numbers involved are
large it is VERY close to taking the derivative at the original point. So we fudge
the matter and use the derivative and label the process “marginal”.
We have marginal profit, marginal revenue, marginal cost, marginal production…
Suppose the total cost in dollars per hour by ACME corporation for it’s bricks is
C ( x)  10,304  2597 x  .33x 2
Find C’(100) and interpret the results
C’(100) = 2531
At 100 bricks produced an hour, total costs are increasing at
$2531 per brick per hour
Now find the average rate of change of producing the 101st item:
AROC C’(101) – C(100)=2530.67
The cost of producing the 101st brick is $2530.67
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So we find that:
C’(x) is approximately C(x+1) – C(x)
The marginal cost is approximately the exact cost…and it is much less calculating
to do, too! And, note, we use 100 when we want the marginal cost of the 101st
item.
Popper 08
Question 1
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Now let’s look at AVERAGE COST and its derivative, marginal average cost.
We use a superscripted bar to denote average.
C ( x) 
C ( x)
x
If C '(a ) > 0, then the average cost of an item is increasing when x = a.
If C '(a ) < 0, then the average cost of an item is decreasing when x = a.
This is the main use of the marginal average cost – we want to find out if the
average cost is increasing or decreasing.
A company manufactures fuel tanks for automobiles. The total weekly cost in
dollars for producing x tanks is modeled by:
C ( x)  8, 000  75 x  0.06 x 2
Is the average cost of a tank increasing or decreasing when x = 1501?
What is the marginal cost? Where is its derivative at the given number – above or
below zero?
C ( x) 




8000
 75  0.06 x
x
grab GGB!
f(x) = 8000 / x + 75 - 0.06x
f'(x) = (-3x² - 400000) / (50x²)
Number
a = -0.06
negative! The average cost is decreasing.
So f (x) is the average cost, f’(x) is the marginal average cost and the number tells
me whether average cost per item is increasing or decreasing.
Now what happens to the average cost when x is VERY large
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C ( x)
is
x 
x
lim
Dependent on what the average cost function is! And you need to take a limit at
infinity to get the information!
Popper 08 Question 2
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Example
A company estimates that the demand for its product is modeled by
where p is the unit price and x is the quantity demanded.
p  .05 x  750
What is the revenue function?
What is the marginal revenue function?
Use the marginal revenue function to approximate the actual revenue realized on
the sale of the 5001st item:
Popper 08
Question 3
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Example
A company paid a research firm to gather pricing information on the best-selling
product in their line. Here’s the data, find the linear demand line!
x
Price
200
527
400
501
600
493
800
482
900
477
What is the demand function?
What is the revenue function?
Find the approximate revenue realized on the sale of the 1201st item.
Do you suppose there is marginal profit?
Given the cost and demand functions below find the marginal profit realized on the
sale of the 56th item.
C ( x)  .0003x3  500 x  25700
p  603  0.9 x
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Popper 08
Question 4
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Finally, pulling lots of things together, we have this problem:
Table 1 – the costs associated with producing x items.
x
cost
200
134, 350
400
145,480
600
157,750
800
174,300
1000
185,975
Find the exponential cost function
Table 2 – the price when x items are demanded
x
price
200
525
400
515
600
493
800
481
1000
468
Find the revenue function
Find the profit function and the marginal profit function.
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Use the marginal profit function to approximate the actual profit realized on the sale
of thee 751st item.
Popper 08
Question 5
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And now, a bit of economics:
Marginal Propensity to Consume or Save
The marginal propensity to consume is the increase in consumption than an
economy experiences when income increases. Now for this can be read with the
word decreases, too. The marginal propensity to consume is what consumers will
do when they have extra income. If the marginal propensity to consume is .75, then
for every dollar of increase in income, a consumer will spend 75 cents and save 25
cents.
Of course we start with a consumption function C(x). Now you will have to rely on
context to distinguish a consumption function from a cost function! If you take the
derivative of the consumption function you get the marginal propensity to consume:
dC
dx
Now there is only consumption and saving so that C + S = X, the income to be had.
So S(x), saving, = x – C(x).
If you take the derivative of both sides:
Marginal saving = 1 – marginal consumption
If you know one item you can get the other by subtracting from 1!
Suppose the consumption function for a country’s economy can be modeled by
C ( x)  0.783x1.389  34.47
where C(x) and x are in billions of dollars.
Find the marginal propensity to consume and the marginal propensity to save when
x = 7.
Note that we don’t reduce 7 billion by one dollar in the calculation this time
because the numbers are SO huge!
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Popper 08 – the rest!
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