Transcript Lesson 10.7: Marginal Analysis

10.7 Marginal Analysis in
Business and Economics
• This is the non-linear cost
function.
• It is an increasing function
because it costs more if
you produce more and
more items
• The cost function
increases quickly at first
and then slowly because
producing larger quantities
is often more efficient then
producing smaller
quantities.
Cost C
Fixed cost
The sharp
increases may
occur when new
factories have to
be built and
when resources
become scarce
Quantity q
• This is the non-linear
revenue function.
• In this section, however,
there are problems that we
will use linear revenue
function
• It is an increasing function
because the company
receives more money if
they sell more and more
items
Revenue
R
In reality, for a
large values of q,
the price is
dropping
Quantity q
• For what production
quantities does the
company make a profit?
• When revenue function is
above cost function
$$$
R
• Production between 150
and 520 items will
generate a profit
C
150
520
q
• Estimate the maximum
profit
• Arrow goes up represent a
profit
• Arrow goes down
represent a loss
$$$
R
• Maximum Profit can occur
where R’=C’
C
150
520
q
Marginal Cost
Remember that marginal refers to an instantaneous rate of
change, that is, a derivative.
Definition:
If x is the number of units of a product produced in some time
interval, then
Total cost = C(x) for producing x units
Marginal cost = C’(x)
Marginal cost is the instantaneous rate of change of cost relative
to a given production level
Marginal Revenue and
Marginal Profit
Definition:
If x is the number of units of a product sold in some time
interval, then
Total revenue = R(x) for selling x units
Marginal revenue = R’(x)
If x is the number of units of a product produced and sold in
some time interval, then
Total profit = P(x) = R(x) – C(x)
Marginal profit = P’(x) = R’(x) – C’(x)
Marginal Cost and Exact Cost
Assume C(x) is the total cost of producing x items. Then the exact
cost of producing the (x + 1)st item is
C(x + 1) – C(x).
The marginal cost is an approximation of the exact cost.
C’(x) ≈ C(x + 1) – C(x).
Similar statements are true for revenue and profit.
Review
Application
The total cost of producing x electric guitars is
C(x) = 1,000 + 100x – 0.25x2.
1. Find the exact cost of producing the 51st guitar.
The exact cost is C(x + 1) – C(x).
C(51) – C(50) = 5,449.75 – 5375 = $74.75.
2. Use the marginal cost to approximate the cost
of producing the 51st guitar.
The marginal cost is C’(x) = 100 – 0.5x
C’(50) = $75.
Example 1:
A company manufactures automatic transmissions for automobiles. The
total weekly cost (in dollars) of producing x transmissions is given by
C(x) = 50000 + 600x -.75x2
(A) Find the marginal cost function
C’(x) = 600 – 1.5x
(B) Find the marginal cost at a production level of 200 transmissions per week
and interpret the results.
C’(200) = 600 -1.5(200) = 600 - 300 = 300
At a production level of 200 transmissions, total costs are increasing at
the rate of $300 per transmission.
(C) Find the exact cost of producing the 201st transmission
C(201) – C(200) = 140299.25 – 140000= 299.25
Example 2:
Price-demand equation: x = 10000 -1000 p or p = 10 - .001x
where x is the demand at price p (or x is the number of headphones
retailers are likely to buy at $p per set).
Cost function: C(x) = 7000 + 2x
where $7000 is the fixed costs and $2 is the estimate of variable costs per
headphone set (materials, labor, marketing, transportation, storage, etc.)
(A) Find the domain of the function defined by the price-demand equation.
p = 10 - .001x ≥ 0; -.001x ≥ -10; so 0 ≤ x ≤ 10,000
(B) Find the marginal cost function C’(x) and interpret.
C’(x) = 2 means it costs an addition $2 to produce one more headset
(C) Find the revenue function (R = xp) as a function of x, and find its domain.
R = xp = x(10 - .001x) = 10x - .001x2
Domain: x(10 - .001x) ≥ 0; so 0 ≤ x ≤ 10,000
(D) Find the marginal revenue at x = 2000, 5000, and 7000. Interpret the results
Use G.C or find R’(x), R’(2000) = 6; revenue is increasing at $6/headphone
R’(5000) = 0; revenue stays the same with an increase in production
R’(7000)= -4; revenue is decreasing with an increase in production
continue
(E) Graph the cost function and the revenue function in the same
coordinate system, find the intersection points of these two graphs,
and interpret the results.
R(x) = 10x - .001x2
C(x) = 7000 + 2x
C
G.C window: -10,10000,-10,30000
To find the intersections using GC:
you can trace the cursor or press
2nd trace, 5, enter, enter, then move
the cursor to the intersection, then
press enter again. You can also
Solve algebraically, set R=C
R
Break-even-points:
points where revenues and costs are the same. In this
problem, they are: (1000, 9000) and (7000, 21000)
continue
(F) Find the profit function, and sketch its graph.
R(x) = 10x - .001x2
C(x) = 7000 + 2x
P(x) = R(x) – C(x)
P(x) = -.001x2 + 8x – 7000
Maximum
point
Use GC to find the production level to maximize the profit:
2nd, Trace, 4, move cursor to the left, press enter, move cursor to the right,
press enter, move to the maximum point then press enter again. You should
get 4000
(G) Find the marginal profit at x = 1000, 4000, and 6000. Interpret these
results.
P’(x) = -.002x + 8
P’(1000) = 6 profit is increasing if produce more
P’(4000) = 0 profit stays the same if produce one more
P’(6000) = -4 profit is decreasing if produce more
Marginal Average Cost
Definition:
If x is the number of units of a product produced in some time
interval, then
C ( x)
C ( x) 
x
Marginal average cost = C ' ( x )  d C ( x)
dx
Average cost per unit =
Marginal Average Revenue
Marginal Average Profit
If x is the number of units of a product sold in some time
interval, then
R ( x)
Average revenue per unit = R ( x) 
x
Marginal average revenue = R ' ( x)  d R ( x)
dx
If x is the number of units of a product produced and sold in
some time interval, then
P ( x)
Average profit per unit = P ( x) 
x
d
P ( x)
Marginal average profit = P ' ( x) 
dx
Example 3:
The cost function for the production of headphone sets :
C(x) = 7000 + 2x
(A) Find C(x) and C’ (x)
C ( x) 
C ( x) 7000  2 x

 7000 x 1  2
x
x
C '( x)  7000 x
2
7000
 2
x
(B) Find C(100) and C’(100) and interpret
C(100) = 72; the average cost per headphone is $72
C’(100) = -0.70; the average cost is decreasing at a rate of 70 cents
per headphone
(C) Use the result in part B to estimate the average cost per headphone at
a production level of 101 headphone sets.
72 – 0.7 = 71.30; about $71.30 per headphone