XIII. MAXIMIZING PROFITS (IN GENERAL)
We have seen that net benefits are maximized where marginal benefits
are equal to marginal costs as long as marginal costs "cut"
marginal benefits from below. Profit is one type of net
benefit, revenues net of costs, that is, revenues minus costs. Marginal benefits that we
use here are called marginal revenues. We need to see how we figure
out marginal revenues, which we will do after we look at the rules from profit maximization.
Rules for profit maximization
Profits are maximized where:
marginal revenues and marginal costs are equal (This is the PRIMARY RULE for profit maximization),
if:
1) marginal costs "cut" marginal revenue from below (Secondary Rule)
AND
2) revenues exceed the sum of all non-sunk costs. (Another Secondary Rule)
First, revenue is equal to price times quantity, assuming that
we sell each unit for the same price. We get these from
the demand for the firm's product (notice that the demand for
the firm's product is the same as the industry or market demand
for that product ONLY IF the firm is the only seller, that is,
if the firm has a monopoly on that product, otherwise, the firm's
demand curve is more elastic than the market demand). Revenue
can easily be seen on a demand curve of FIGURE XIII as the area
of the rectangle that has from zero to the quantity OQ sold as
its base and from zero to the price OP as its height, or OP, the
price, times OQ, the quantity, as seen below.
We can see that for us to sell one more unit, we must drop our price. If we sell one more unit we add the price that we sell one more unit for to our revenues, but we also have to sell the units before this at the same new price (perhaps you should recall now that quantity demanded is a FLOW variable, that is, we are dealing with units sold per week or per month or per year). Dropping the price on the units before this last one decreases our revenues on all of those units, so marginal revenue is less than price.
Perhaps a numerical example may help. In TABLE 4, we have
a demand relation between price and quantity. Multiplying
those two columns together gives us the revenue column.
Finding the change in revenues divided by the change in quantity
sold gives us marginal revenue. In this example we change
quantity by only one unit each time. In this example, we
make up a demand function and a cost function. We use "Prof" for profit.
We have seen the logic behind our primary rule for profit maximization, that marginal revenues and marginal costs must equal for profits to be maximized, in note 10, Marginal Analysis & Maximizing Net Benefits. As you recall, benefits, such as revenues, are additions to net benefits, or profits. Costs, on the other hand, reduce net benefits or profits and so, are subtracted from benefits or revenues. We also must recall that Marginal Revenues are the additions to total revenues, not the total of revenues. Also, Marginal Costs are the additions to total costs, not total costs.
If marginal revenues exceed marginal costs, then we are adding more to our revenues than to our costs. Profits must be rising. The profit function has a positive sloper here.
On the other hand, if marginal costs exceed marginal revenues, then we are adding more to our costs than to our revenues. Profits must be falling. The profit function has a negative slope here. The place where we go from profits rising to profits falling is where marginal revenues and marginal costs are equal. Here, total profits are neither rising nor falling. The slope of the profit function is defined as marginal revenues minus marginal costs and so it is zero where marginal revenues equal marginal costs. A zero slope is a necessary condition for a function to be maximized. To see this, draw a graph where the curve increases, comes to a peak or maximum, and then falls. What is the slope at the peak? So, for profits to be maximzed, marginal revenues must equal marginal costs.
Now let us examine our first subsidiary rule, that the equality of marginal revenues and marginal costs gives us maximum profits only if marginal costs cut marginal revenues from below. What this means is that for profits to be maxmized at the quantity where marginal revenues and marginal costs are equal, marginal costs must be below marginal revenues prior to their equality. If this were not true, if marginal costs intersected marginal revenues from above, then all the units produced before the point where marginal revenues and marginal costs are equal must have added more to costs than to revenues. This would continually push profits downward until that point where marginal costs = marginal revenue. After that profits would begin to rise, because after that point marginal revenues would exceed marginal costs. The point of equality between marginal costs and marginal revenues would then give us a profit minimum, not a profit maximum. A maximum of any function occurs where that function has a zero slope AND where the slope went from positive to negative, never where the slope goes from negative to positive.
Look back at the graph you were told to draw a moment ago. It rises, comes to a peak, and then falls. For profit to rise, come to a peak, and then fall, marginal costs must first be below marginal revenues, then cut, and then above marginal revenues.
VC=all non-sunk costs
TC=sunk and non-sunk costs
We see two things in the numerical example in TABLE 4. First, marginal revenues are below price for a downwardsloping demand curve. Second, we see that profits are maximized where marginal costs equal marginal revenues and marginal costs cut marginal revenue from below.
We need to examine the second part of the IF of the rule above:
2) revenues exceed all non-sunk costs.
One way to show that this must also be true is to first suppose
that it isn't. Suppose that VC > R. The firm is
then making a loss of FC (we will treat this as sunk costs) +
(VC - R). If the firm were to shut down completely,
it would still have to pay the sunk costs, yet would have NO revenue,
making a loss of FC. Where would losses be greater,
where the only loss is FC or where the loss is FC + (VC - R). If VC > R then losses would be minimized by shutting
down. This gives us our shutdown rule. Another
version of this rule is "sunk costs are sunk costs"
or even "don't cry over spilt milk". Again, we
have the concept of opportunity cost staring us in the face!
If the cost is changeable by your actions, it is a relevant cost,
but if you can't change that cost, it is irrelevant for decision
making purposes.
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