Let's now consider the theory of natural monopoly

and the difference between what is often called weak and strong natural monopoly

Common examples of natural monopoly are,

electricity distribution systems, and water piping systems,

but often the concept of natural monopoly is used somewhat loosely or impressionistically.

Does it mean that in a market setting there would just be a single producer?

Or does it mean that there should just be a single producer.

Well, we're going to look at this a little more rigorously and break down the concept of natural monopoly into two different cases.

And as mentioned those cases will be strong natural monopoly and weak natural monopoly.

As we will see this difference will relate to whether average cost AC is decreasing as the strong natural monopoly or in weak natural monopoly AC is increasing.

We can define decreasing average costs by this equation here.

What this means is that as more units of the goods are produced,

the average cost of producing each unit falls

So you could also put this on a graph,

and here would be average cost, and as q is increasing, Ac is going down.

And of course this could be true across some range of the output, but not all ranges of the output

so this is defined across some particular range namely where the market will be.

Now let's consider the concept of subadditivity.

We can write subadditivity as follows, with a cost function here we have cost

and in parentheses we have a sum of q super script i, close parentheses,

being less than or equal to another sub here, c q super script i.

What the left hand side portrays here is simply the cost of producing all those units in a single firm.

What the right hand side portrays is the cost of taking those units and say producing them through using ten little different firms.

And of subadditivity holds, well it's cheaper to produce that given quantity of ouput,

in the single larger firm.

In this setting, subadditivity is simply an assumption.

With that which we don't have natural monopoly at all.

Decreasing average cost is something which as we will see, may or may not hold.

By definition we have strong natural monopoly

when both assumptions hold, and thus we are considering the case of decreasing average cost.

We can draw that case like this,

so here we have a demand curve, we have the marginal revenue curve,

we have an average cost curve

and we have a marginal cost curve,

and what's striking about this picture is that if we were too set price at marginal cost,

right here, well that would be normally the efficient price,

but at that price, you can see that average cost is above price

so if average cost is greater than price,

then the firm is earning profits which are less than zero,

or the firm is suffering loses.

In these cases, due to subadditvity in both strong and weak natural monopolies,

it is cheapest for a single firm to supply all output

but the general question, which arises,

is a single firm sustainable?

Well, in the case of strong natural monopoly,

if we set prices equal to marginal cost,

the tradition optimality condition, we can see that a subsidy is needed.

Again you can see in this picture that where demand meets marginal cost,

average cost is above that point,

and its the subsidy that would be needed to make up that difference.

And if you're wondering by the way, how is it that average costs are getting to be higher than marginal costs,

well basically, it has to do with the presence of fixed costs in the production function.

Namely costs that must be incurred no matter what,

just to get the firm up and running.

Those fixed costs will not contribute to marginal costs

but they will make average costs higher.

Sometimes, the regulators will allow a firm to set price equal to average costs,

and then that means there will not be any subsidy needed for the firm,

This is done for a few reasons ,

one is simply that subsidies to firms are not politically popular,

but another reason is that subsidies to firms are often distorting or cause corruption,

or there's a problem, how do you know how much subsidy to send to the firm?

What do you do, just call up the firm and ask, "Hey, what's your cost curve?"

Well, they're probably going to lie to you, to get a higher subsidy.

So in practice, with regulation, prices equals marginal costs under strong monopoly

is not always attainable and often the regulators settle for something more in the neighborhood of price equals average costs.

We'll now consider the case of weak natural monopoly

where subadditivity agains holds

but we do not have decreasing average cost.

Rather, we have increasing average cost.

The important mathematical point here is that for AC to be rising, marginal costs has to be greater than average cost

that is, you only get an average to be rising by tacking on some larger numbers to that average.

But if marginal costs is greater than average cost,

and we're going to be trying to set prices equal to marginal cost,

then indeed price itself will be greater than average cost

and the firm will earn profits rather than suffering loses.

We can see here in this diagram that, that price equal to marginal cost

that is indeed above average cost, and thus profits will be greater than zero.

The good news is, we don't need that subsidy anymore.

The bad news, that we have a new problem, and that is entry is possible

and that conceivably could raise total production costs.

So the new firms, they engage in what is sometimes called cream skimming

namely trying to serve the most profitable segments of the market,

and leaving the initial enterent to cover all the others

It's sometimes claimed for instance, if we opened the US Postal Service

to full private competition you might get say, three or four suppliers

mostly fighting over the lucrative urban segments of the market.

What you would have is three or four different firms, varying those fixed costs

we talked about.

because of subadditivity this may not be socially efficient

to be having three or four different firms paying the fixed costs of setting up a delivery network

yet because of the struggle,

over these profits based on monopoly or market power

maybe, that's what you would get, at least according to this theory.

So what does this look like?

Well here you would imagine a bunch of firms,

fighting over the top portion of this demand curve,

the consumer surplus, for people who demand service the most,

and the people who use this model to favor entry restrictions,

well they focus on the fixed cost.

They say you would have here, say three different firms varying those blocks of fixed costs

to set up a delivery network,

as opposed to the one, possibly state protected monopolist,

and that is an argument for creating a state protected monopolist.

Other economists will accept the logic of this view,

but they come to a different policy conclusion, they place a lot of stress on innovation, and also the dynamic benefits of competition.

So they understand that by having three competitors, as here you will pay more in terms of fixed costs,

but over the longer run you'll have more innovation,

saying how packages are delivered and tracked,

and the benefits of that dynamic competition will out weigh the initial payment of fire to fixed costs.

And these individuals would argue, then that we should allow a competitive market to operate,

and how you will make this judgment call depends on,

how important you think the fixed costs are,

compared to these factors of innovation and dynamic competition.

But in any case, sits the theory of weak natural monopoly, which helps you see this trade off.

There are plenty of places where you can pursue these topics,

one route to go is simply to study specific sectors and industries,

but more generally, you can google strong weak natural monopoly.

Also google natural monopoly regulation.

And here's a very useful source, it's not online, it's not free,

but it's a very good book, and it's called,

"Natural Monopoly Regulation".