Prove that profit maximization implies cost minimization but not vice versa.

dime365-ga:

Thank you for your Question regarding profit maximization.	

The simplest answer to your question is this:

Profit is defined as the difference between the price that a product
or service is sold for, and the cost associated with manufacturing and
selling the product or service.

The profit for a single item sold can be increased in one of two ways.
You can either increase the price of the item, or you can lower the
cost of the item. To maximize the profit for this single item,
therefore, you need to do both: you need to maximize the price that
the item is sold for, and you need to minimize the cost of the item.

   Profit = Price - Cost

The maximum price that an item can be sold for is essentially
infinite. A prime example of this would be rare items sold at an
auction. The maximum price that a large quantity of like items can be
sold for, however, is dictated by market conditions. In a free (or
quasi-free) market, the price for an item will be influenced by, among
other things, supply versus demand, the level of competition in the
marketplace for the item and the availability of substitutes.

The cost of an item is comprised of the fixed costs associated with
the existence of the business, and the variable costs associated with
the manufacture of the product or service. Even if no items are
manufactured, the fixed costs will always be greater than zero, as
there are always some costs associated with establishing and
maintaining a legal business entity.

If the price for an item is held constant while the fixed or variable
costs associated with the manufacture of a product or service is
reduced, then the profit for the item will increase.

   Profit(+) : Price(=) - Cost(-)

However, even if the cost for an item is reduced, the profit will
still be dependent on the price of the item. If the price of the item
is also reduced by the same amount that the cost is reduced, then the
profit remains the same. If the price of the item is reduced more than
the cost is reduced, the the profit will actually decrease. Raising
the price, or holding it constant (per the previous paragraph) while
the cost is reduced, will increase the profit.

Therefore, while it is correct to state that profit maximization
requires cost minimization, it would not be correct to state that cost
minimization implies profit maximization. It only creates the
possibility of profit maximization through appropriate pricing
strategy.

If you do not want to take my word for this, then please refer to the
following article for a more indepth proof, complete with formulas:

http://www.anderson.ucla.edu/faculty/ely.dahan/content/unit_cost.pdf


I hope this helps!

aht-ga
Google Answers Researcher

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Request for Answer Clarification by dime365-ga on 02 Feb 2004 14:11 PST

is there any way to do this proof via contradiction??

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Clarification of Answer by aht-ga on 02 Feb 2004 15:01 PST

dime365-ga:

By contradiction, do you mean "to show that anything but the above
would result in a contradiction"?

For example: 

"Since profit is the difference between the selling price and the
total cost to manufacture and sell a good, the profit can always be
increased by either increasing the price or decreasing the cost, as
long as someone is willing to buy the good.

If it is possible to further decrease the cost without impacting the
price at which the good is sold, then it is impossible to say that
profit has been maximized until the cost cannot be reduced any further
without creating an equal or greater reduction in the price that the
good can sell for."

Similarly:

"While reducing the cost of a good can help increase the profit, it is
impossible to say that cost minimization will always result in profit
maximization, since the cost reductions may result in price decreases
that are equal to or greater than the reduction in cost."

A situation to illustrate the second concept, can be found in the auto
industry. Let's say that a basic model of a car sells for $16000, with
an actual cost of $12000. The profit is therefore $4000. I can then
sell a 'loaded' version of the same car, with lots of options, for
$20000. However, the options only cost me $2000, so my profit is now
$6000. In this case, even though I raised my cost by $2000, I was able
to raise my price by more (since the market is willing to pay the
difference), so I have increased my profit.

Does this help?

aht-ga
Google Answers Researcher

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Request for Answer Clarification by dime365-ga on 02 Feb 2004 16:43 PST

ah, i meant solving via contradiction using mathematical proofs.  I
think it would be something having to do with concavity (?? not sure)

the proofs i learned in class are

Direct proof.  Assume that A is true, deduce various consequences and
use them to show that B must also hold

Contrapositive proof.  Assume that B does not hold, then by deducing
various consequences show that A cannot hold.

Proof by contradiction.  Assume that A is true and B is not true, then
show that these assumptions imply a logical contradiction

I'm looking for a proof like one of the 3 to satisfy the question..
does the proof in the example from
http://www.anderson.ucla.edu/faculty/ely.dahan/content/unit_cost.pdf
show that fact? im a bit confused

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Clarification of Answer by aht-ga on 02 Feb 2004 23:47 PST

dime365-ga:

The UCLA paper I referenced above does illustrate the proof as a
direct proof, but with several degrees of complexity greater than is
required for the statement we are trying to prove here.

Please also review the following paper, which may help illustrate the
situation better for you:

http://econfloat.tamu.edu/deere/class/Econ202/Second%20Portion%20Archive/Lectures%20&%20Notes/Notes%20on%20firm%20behavior.pdf

(you may need to cut and paste this URL into your browser's address field)

Also see:

http://en.wikipedia.org/wiki/Profit_maximization

And, finally, here a couple of other links to define profit
maximization and cost minimization:

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http://www.amosweb.com/cgi-bin/gls.pl?fcd=dsp&key=profit+maximization

profit maximization: The process of obtaining the highest possible
level of profit through the production and sale of goods and services.
The profit-maximization assumption is the guiding principle underlying
short-run production by a firm. In particular, it is assumed that
firms undertake actions and make the decisions that increase profit.
The profit-maximization assumption is the production counterpart to
the utility-maximization assumption for consumer behavior.

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http://www.amosweb.com/cgi-bin/gls.pl?fcd=dsp&key=cost+minimization

cost minimization: The process or goal of incurring the least possible
opportunity cost in the pursuit of a given activity. Cost minimization
is comparable to other objectives, including utility maximization and
profit maximization. This goal, however, is generally used when
circumstances constrain a decision. For example, a government agency
has been assigned the task of building a bridge. It must now do so at
the lowest cost possible.

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The fundamental assumption here is that, in a perfectly-competitive
market, also known as a free market, every manufacturer will try to
maximize their profit. In a free market, there is also the assumption
that there are multiple manufacturers, each with a share of the market
for a given good. Finally, there is the assumption that the market
itself will determine the price at which a given good is sold.

In a free market, therefore, a manufacturer can increase their revenue
by selling more units at the market price. This market price is based
on the total number of units available on the market from all
manufacturers, so the assumption is that as the number of units
increases, the law of supply and demand will cause the market price to
decline. The revenue from each additional unit sold is known as
marginal revenue (defined in the links above). Each additional unit
that is sold, is manufactured at what is known as the marginal cost.
In order to maximize their profit, a manufacturer will continue to
increase the volume of units manufactured, until the marginal cost
equals the marginal revenue. Going beyond this point will result in a
reduction in profit, since the additional units will be sold at a
loss.

Now, within the limitations of the Google Answers text interface, I
will try to illustrate how my previous explanation can be used as
proofs.

The two statements you are interested in are:

a) Profit Maximization implies Cost Minimization

b) Cost Minimization does not imply Profit Maximization

We will prove the first using a direct proof, and prove the second
using a proof by contradiction. Before we can do this, there are some
additional assumptions that we must make.

Definitions
-----------

First, we need to define what the terms profit and cost mean.

For this exercise, cost means the cost to produce a single unit of a
good or service. Profit is the amount that the seller earns after
accounting for cost. In other words, for a single unit:

 Profit (pr) = Market Price (P) - Cost (C)

To simplify the model, we will look only at the cases where both P and
C are >= 0. What this means is that we are not going to pay someone to
take a unit off our hands, and that we cannot pay a negative amount to
make a unit.

We also assume that P >= C; otherwise, we are selling at a loss
(negative profit), and it would make better business sense (in this
simple model) to get out of this business.

Next, we use a linear model where we assume a fixed volume of
production in order to simplify the model. The simplified model can
then be run at different fixed volumes of production, which drives
different marginal costs and, through the law of supply and demand,
different market prices.

With these assumptions for the model, we can now proceed with the proofs.


Profit Maximization implies Cost Minimization
---------------------------------------------

Stating this another way in order to derive a direct proof, we wish to
prove that if profit is maximized, then cost is minimized.

As mentioned above, you can plot the market price and marginal cost on
a graph where the y-axis is $ (dollar amount), and the x-axis is
quantity produced. In a free market, both market price and marginal
cost will start high for low quantities produced; in fact, marginal
cost may actually be higher than market price for certain low
quantities, up to a minimum quantity required to break even. To the
'right' of this on the graph, the market price curve will be 'higher'
(ie. greater $ value) than the marginal cost curve. The area starting
from the break-even point, and bounded by the two curves to the right
of the break-even point, is the profit. Eventually, for the reasons
mentioned above, the cost to produce an additional unit will exceed
the marginal revenue for that unit. This defines another break-even
point, and closes off the area of the graph that represents the
profit.

To maximize profit in a free-market economy, a manufacturer will want
to maximize the price they can sell their product at, and maximize the
number of units that can be sold at that price without going beyond
the point where marginal cost exceeds marginal revenue. Again, as we
are assuming a free-market economy, we can assume that the perfectly
competitive environment will mean that the market price curve remains
static regardless of whether any particular manufacturer is in the
market or not. So, in order for any manufacturer to maximize their
profit, they need to shift the position or shape of their marginal
cost curve. By lowering the various costs that go into defining the
marginal cost curve, the manufacturer can cause the marginal cost
curve to be 'lower' on the graph, or can 'widen' the curve on the
graph, meaning that a greater area of the graph is bounded by the
market price and marginal cost curves, resulting in greater profit.

What this means, then, is that in order to maximize profits in a
free-market economy, a profit-seeking manufacturer must attempt to
increase the number of units sold before the marginal cost of
producing one more unit exceeds the marginal revenue commanded by that
unit. Increasing the number of units sold is accomplished by lowering
the marginal cost curve, also referred to as cost minimization. As
long as the curve can be lowered without affecting the market price
curve, then profit maximization requires that the cost curve be
lowered as far as possible. So, profit maximization must imply cost
minimization.

Since this is not the easiest thing to describe using just words, here
is one more link to a research paper on this topic:

http://business.baylor.edu/Beck_Taylor/PAPERS/Factor%20Demands.pdf

In this paper, you will see the math involved, along with some illustrative charts.


Cost Minimization does not imply Profit Maximization
----------------------------------------------------

In a micro-economic sense, cost minimization means seeking the lowest
possible cost to accomplish a given activity.

If we apply a proof of contradiction to this, we can say that the
pursuit of the lowest overall cost must lead to the maximization of
profit.

The pursuit of lowest overall cost necessarily is modeled at a given
volume of production. If the volume of production is not held as a
constant in the model, then the lowest overall cost would always be at
a volume of zero, as then only fixed costs remain.

For a given volume of production, then, cost minimization will result
in the maximum profit possible for that volume.

However, as we have already illustrated above, profit maximization
entails adjusting the volume of production until marginal cost equals
marginal revenue. While cost minimization will allow the volume of
production to be optimized to maximize profit, a goal of cost
minimization for a given volume of production will not necessarily
maximize the profit that is possible, due to this constraint on
volume. Therefore, the pursuit of the lowest overall cost contradicts
with the pursuit of maximum profit in a free-market economy. Only if
cost minimization is pursued along with optimization of the volume of
production will profit maximization occur.

----------------------------------------------------

Does this work better for you? The nature of the mathematical model
makes it difficult to illustrate within the text interface of Google
Answers. The most important part of the proofs is really a proper
understanding of the meaning of the two terms, 'profit maximization'
and 'cost minimization' as drivers of business decisions. A business
seeking to maximize profit will always need to drive out unnecessary
costs in order to increase the spread between cost and selling price.
A business seeking to minimize costs, must choose a volume of
production to minimize costs for. This condition (fixed volume)
contradicts with the requirement of volume optimization in order to
maximize profit.

This has been a long Answer, so please let me know how else I can
clarify this for you, within the capabilities of the text interface.

Regards,

aht-ga
Google Answers Researcher

Thank you so much!