by ohanian » Mon 12 Sep 2005, 08:11:10
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')You have fallen into a trap. If a company produces a product and later find a way to reduce the cost of production by 10%, they will not make more product because because it may reduce their profits.
You have fallen into a trap of missing the big picture. It
may reduce their profit to produce more, therefore they have to find the
optimal amount of additional production that will increase their profits to the maximum.
With lower costs, that optimal amount is always higher than it was before.
Ha! I caught you red handed.
you say "
With lower costs, that optimal amount is always higher than it was before."
What a load of bullshit!
I have written a mathematical proof that
With lower costs, that optimal amount is
NOT ALWAYS higher than it was before.
Here is da Proof
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$this->bbcode_second_pass_code('', '
Q) If the cost of a product goes down by 10%, will the company producing it sell more of that product?
dQ |
A) Only if == | < Gmax
dS | S=S0
where S = selling price
Q = quantity sold
Gmax = Max gradient
S0 = initial selling price (old selling price)
Q0 + 1
Gmax = ==================================
Q0 * C0 * (R - 1) - (S0 - R * C0)
where Q0 = initial quantity sold (old quantity sold)
C0 = initial cost per item (old cost per item)
R = New Cost fraction (10% cost reduction -> R=0.9 )
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Proof as follows.
Step 1) Assume that 10% cost reduction
Thus R = 0.9
Step 2) Describe the situvation before cost reduction
S = S0 Selling price is initial selling price
C = C0 Cost per item is initial cost per item
Q = Q0 Quantity sold is initial quantity sold
Profit = Q * (S - C) # An equation so simple even a child can understand
Profit_before = Q0 * (S0 - C0)
Step 3) Describe the situvation after the company has a 10% cost reduction
S = S0 + deltaS New selling price. Please note that [ deltaS < 0 ]
C = R * C0 New cost per item. Please note that R = 0.9 for 10% cost reduction
Q = Q0 + deltaQ New quantity sold. Please note that [ deltaQ > 0 ]
Two constraints are
Constraint_1 : deltaS < 0 # To sell more the price point must drop
Constraint_2 : deltaQ > 0 # Company is selling more than before
The quantity sold and selling price equation
dQ |
deltaQ = == | * deltaS
dS | S=S0
= G * deltaS
where G is just
dQ |
G = == |
dS | S=S0
Any way, profit is
Profit = Q * (S - C) # An equation so simple even a child can understand
Profit_after = (Q0 + deltaQ)(S0 + deltaS - R * C0)
Step 4) Assume that Profit_after > Profit_before
(Q0 + deltaQ)(S0 + deltaS - R * C0) > Q0 * (S0 - C0)
Now substitute deltaS with G and deltaQ
deltaQ = G * deltaS # From above
Hence
deltaQ
deltaS = ======
G
deltaS = K * deltaQ where K = 1/G
Substitute with "deltaS = K * deltaQ"
(Q0 + deltaQ)(S0 + deltaS - R * C0) > Q0 * (S0 - C0)
becomes
(Q0 + deltaQ)(S0 + K * deltaQ - R * C0) > Q0 * (S0 - C0)
Expand the left hand side and right hand side
Q0*S0 + Q0*K*deltaQ - Q0*R*C0 + deltaQ*S0 + K*deltaQ*deltaQ - deltaQ*R*C0 > Q0*S0 - Q0*C0
Simplify
K (Q0*deltaQ + deltaQ*deltaQ) - Q0*R*C0 + deltaQ*S0 - deltaQ*R*C0 > - Q0*C0
Simplify
K (deltaQ(Q0 + deltaQ)) - Q0*R*C0 + deltaQ(S0 - R*C0) > - Q0*C0
Isolate K
K (deltaQ(Q0 + deltaQ)) > Q0*R*C0 - deltaQ(S0 - R*C0) - Q0*C0
Q0*R*C0 - deltaQ(S0 - R*C0) - Q0*C0
K > ===================================
deltaQ(Q0 + deltaQ)
Q0*R*C0 - Q0*C0 - deltaQ(S0 - R*C0)
K > ===================================
deltaQ(Q0 + deltaQ)
Q0*C0 (R - 1) - deltaQ(S0 - R*C0)
K > =================================
deltaQ(Q0 + deltaQ)
As K = 1/G , we can then write G as
deltaQ(Q0 + deltaQ)
G < =================================
Q0*C0 (R - 1) - deltaQ(S0 - R*C0)
Step 5) Assume the absolute minimum increase in quantity sold. ie deltaQ = 1
Given deltaQ = 1
deltaQ(Q0 + deltaQ)
G < =================================
Q0*C0 (R - 1) - deltaQ(S0 - R*C0)
Q0 + 1
G < ===========================
Q0*C0 (R - 1) - (S0 - R*C0)
G < Gmax
where
Q0 + 1
Gmax = ===========================
Q0*C0 (R - 1) - (S0 - R*C0)
Thus the proof is completed.
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Consider the following. A company is selling an MP3 player/recorder
Initially the company is selling
S0 = 100 dollars (selling price per item)
Q0 = 10000 units (quantity sold at S0 price point)
C0 = 14 dollars (cost per item)
Quantity sold goes down 0.5 unit per $1 increase in selling price
dQ |
G = == | = -0.5
dS | S=S0
Profit_before = 10000*(100-14) = $860,000 ***VERY IMPORTANT***
Now the company has slash the production cost by 10%
C1 = 0.9 * C0 = $12.60 (cost per item)
Profit_after = Q1 * (S1 - C1)
Where Q1 = Q0 + deltaQ and deltaQ = G * deltaS
deltaS Q1 S1 Profit_after
====== ====== ======= ============
2.0 9999 $102.00 $893,910
0.0 10000 $100.00 $874,000
-2.0 10001 $98.00 $854,085
So there you have it. The company should sell less MP3 players/recorder
after it has reduced its cost by 10% in order to generate more profits.
Why? Because
Q0 + 1
Gmax = ================================== = -0.71
Q0 * C0 * (R - 1) - (S0 - R * C0)
and G = -0.5
so
G is MORE THAN Gmax Thus this violates the constraint "G < Gmax" at the end of (Step 4)
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