Cournot - YouTube

Hi.

Today we look at Antoine Augustin Cournot

whose great tragedy was that he was so far ahead of his time

that he was never recognized as great until long after his death.

Cournot is a French mathematician, a philosopher,

and only peripherally an economist.

He was born in 1801 and died in 1877,

and his major work in economics, a thin volume, was published in 1838.

Nevertheless, in this thin volume, Cournot has a number of firsts.

He was the first to draw a demand curve.

He was the first to solve for the profit-maximizing monopoly price

and output.

He pioneered comparative statics, including tax analysis.

He pioneered many topics in industrial organization,

most notably duopoly and oligopoly theory, with a little game theory thrown in.

He was the first to understand the double marginalization problem

in a way which would not be exceeded for over a hundred years.

He was the first to use second-order conditions

as stability conditions in a dynamic game.

Most importantly, he pioneered the use of calculus

and constrained optimization techniques.

Nevertheless,

in his lifetime as an economist, Cournot was almost completely ignored.

Simply put, other economists did not have the mathematical training

to be able to understand what Cournot was doing.

So it wasn't until long after his death that people were able to look back

and to see just how far Cournot had advanced.

Let's take a closer look at some of his insights.

The classical economists were not very clear about demand.

Is it a quantity? Is it an expenditure of money?

Is it a desire?

This led to confusions over quantity demanded

and the shifting of the demand curve;

that's how we would put it today; and other things.

Cournot was clear.

Demand was a function,

and he said the quantity demanded over a given time period

is a function of the price,

and he wrote it exactly as we would today.

This seems very simple,

and, yet, writing demand as a function allowed Cournot

to make big advances in monopoly theory and a whole bunch of other areas as well.

By the way, this allows us to resolve a little bit of confusion.

Sometimes physicists look at the demand and supply curve,

and they're a little bit confused because the rule is

you put the dependent variable on the vertical axis

and the independent variable, the variable which changes,

or which can be adjusted, like time,

or which can be manipulated, the independent variable,

on the horizontal axis.

Cournot did exactly this.

So Cournot's demand curve has quantity on the vertical axis

and price, which adjusts, on the horizontal axis.

Marshall would later reverse this into our contemporary framework,

putting price on the vertical axis and quantity on the horizontal.

But Marshall also kept the rule

because for Marshall it was quantity which was the independent variable,

quantity which adjusted.

Modern economists, however, have adopted the Cournot story about what changes,

namely price, what's independent,

but they adopted the Marshall diagram,

giving us the sort of mixed-up version, mixed up at least according to physicists.

Let's have a look at some of Cournot's other insights.

Classicals had a good understanding of competition.

They knew that competition would push prices down to cost.

Average cost, marginal cost, a little bit unclear,

but they understood the idea of pushing prices down to cost.

They had only a weak understanding of the monopoly price, however.

They knew the monopoly price would be higher than the competitive price.

But how much higher? Under what conditions and so forth?

They really didn't know.

Cournot solved the monopoly profit maximization problem

almost exactly as is done in a modern textbook.

Here, for example, is a clip from Cournot's book,

and it's exactly what we would write today

with some slight changes in notation

due to the fact that Cournot had --

quantity on the vertical axis and price on the horizontal axis.

But other than changes in notation, this is exactly what you would see.

Cournot did not talk about marginal revenue and marginal cost,

but mathematically that's exactly what he showed.

Using mathematics, Cournot was able to do sophisticated comparative statics.

We will illustrate one case with a diagram, the modern diagram.

This shows the optimal price and quantity for the monopolist.

Now, Cournot asks, "Suppose the monopolist's costs go up."

Let's say there's a tax by a dollar.

What happens then to the price the monopolist charges?

Now, many people when asked this question will say, "Well,

if the monopolist's costs go up by a dollar,

the monopoly is just going to pass all of those costs on to consumers,

so price will go up by a dollar as well."

Cournot showed that this was not the case.

Price, in fact, could go up by less than a dollar or by more than a dollar.

It did not necessarily have to go up by a dollar.

In a linear case, let's just take a look right here, see what happens.

In fact, you can see price goes up by less than a dollar,

and if you work out the mathematics,

it turns out that in the linear case

a one-dollar increase in marginal cost will lead to a 50-cent increase in price.

Cournot is able to solve problems like this

and much more sophisticated problems as well

decades before anyone else.

Beginning with the monopoly problem which Cournot had solved,

he then expanded to duopoly and oligopoly and towards competition,

so Cournot could handle all kinds of markets.

Let's take a look at his most famous example, the Cournot duopoly.

Let's use some graphics to illustrate Cournot's solution to the duopoly problem.

We know that the profit-maximizing output for a monopolist

is where marginal revenue is equal to marginal cost,

and that gives us a profit-maximizing output at Q1*.

Now, let's imagine that there's a second firm in the market,

and suppose that the second firm produces nothing.

Well, Q1* is clearly also the optimum amount for Firm 1 to produce

if the second firm produces nothing.

What, however, if the second firm produces Q2 units?

What's the profit-maximizing output for Firm #1 if Firm #2 produces Q2 units?

Well, to solve for this,

we can think about what's going on here

as Q2 units of output have simply been subtracted from the market,

have simply disappeared.

Let's imagine that these Q2 units were never there to begin with,

just disappeared from the market.

Well, then we would have actually a new demand curve.

This is the residual demand curve.

It's a demand curve left over when we subtract the Q2 units

which are being produced by Firm #2.

With this residual demand curve, there's a residual marginal revenue curve,

and once we have that, we can calculate the new optimal price and quantity.

Now, notice if this Q1* is the optimal quantity

given that Firm 2 is producing Q2 units.

So, the intuition is that by repeating this procedure,

we can find the optimal response of Firm 1 to whatever Firm 2 decides to produce.

So, for every assumption of Q2,

there is a Q1*

where we can write this now as Q1* as a function of,

now a variable, how much Firm 2 produces.

Of course,

Firm 2 is also going to want to respond optimally to Firm 1.

So Q1* is the profit-maximizing output of Firm 1

given what Firm 2 decides to do.

There's a similar story for Firm 2.

For every assumption of what Firm 1 does,

there's a profit-maximizing output, Q2*,

which tells you what's the best response of Firm 2.

Now, a lesser mind would have given up at this stage

because what Cournot has done so far is show that

for every Q2 there's a best response, Q1*,

and for every Q1, there's the best response, Q2*.

But it's not at all obvious that these are consistent.

It's not at all obvious that one can find a solution

to these two equations.

Nevertheless, Cournot was able to solve the equations and produce a solution.

Let's take a look.

Cournot was able to derive mathematically the best response functions,

that is, for any Q2,

what was the profit-maximizing output for Firm 1?

Similarly, for any Q1, what was the profit-maximizing output for Firm 2?

We can show what these functions look like in this linear case on this diagram.

Let's put the quantity that Firm 1 produces on the vertical axis,

and the quantity that Firm 2 produces on the horizontal axis.

Then Q*1 gives the best response of Firm 1 to any Q2.

For example, we already showed that if Q2 is equal to 0,

if Firm 2 produces nothing,

then the best response of Firm 1 is to produce the monopoly output.

Similarly, if Firm 2 were to produce the competitive output,

that is, if Firm 2 were to drive prices down to costs,

then the best that Firm 1 can do is to produce nothing.

There's no profit in doing anything else.

For any other output, this red line, this red curve,

tells us the optimal amount of production for Firm 1

given any output produced by Firm2.

Similarly, there's a similar curve for Firm 2.

This tells us the best response of Firm 2 given any output produced by Firm 1.

Now, notice that there is only one set of outputs

which is mutually consistent,

that is, if Firm 2 produces Q*2,

then the best response to that is Q*1,

and if Firm 1 produces Q*1,

then the best response to that is Q*2.

Q*<i>1</i> and Q*<i>2</i> are the Cournot-Nash equilibrium,

Nash because Nash was later able to generalize this

and show that in games of this kind there's always an equilibrium like this.

Cournot gave the first example of such an equilibrium.

Cournot also showed the conditions under which this equilibrium is stable,

and he gave a dynamic argument for the stability.

Let's take a look.

For example, suppose that Firm 2 were to produce this amount of output.

What's the best response of Firm 1?

Well, we read the best response of Firm 1 on the red line is this amount right here.

Now, let's consider if Firm 1 produces this amount.

What's the best response of Firm 2?

Well, we read that off the green line.

That tells us that Firm 2's best response is right here.

Now, if Firm 2 produces this much, how much will Firm 1 want to produce?

They will want to produce this much right here.

We keep making this argument so forth and so forth,

and we show, or Cournot showed,

that wherever we begin on this diagram,

the dynamics push us towards the Cournot-Nash equilibrium,

so it's a stable equilibrium.

This was really a tour de force on the part of Cournot.

Cournot also had a tremendous analysis of the double marginalization problem.

Briefly, this problem is:

suppose we have a monopolist involved in a vertical relationship,

so there's a monopoly producer of jet engines

which sells its engines to an aircraft manufacturer,

which is also a monopolist, and then which sells to consumers.

Now, what Cournot showed here is that the monopolist can work

in a way against their own interests.

The jet engine producer in trying to maximize its profit

sells at a high price of the aircraft manufacturer,

which in turn in trying to maximize its profit

sells at a high price to consumers.

But because the aircraft manufacturer doesn't take into account

the interests of the jet engine producer

and the jet engine producer doesn't take into account

the interests of the aircraft manufacturer,

the monopoly prices that they set can be too high,

too high not simply from the viewpoint of consumers

but from their own viewpoint,

from the viewpoint of maximizing their joint profits.

Cournot showed that vertical integration would make sense in this case.

I give a much more extensive analysis of the double marginalization problem

in its own video,

so take a look at that for more insight into that issue.

For more information on the Cournot model and Cournot-Nash equilibrium,

just take a look at any graduate textbook in microeconomics.

On double marginalization, have a look at my video on this topic

which goes into more detail and gives some applications.

On the legacy of Augustin Cournot, James Friedman has a nice paper.

Let me finish on this note.

Cournot in many ways was the Samuelson of his time.

He brought mathematics to economics,

but when Samuelson did this in the post-World War II era,

the world was ready.

The world was not ready for Cournot.

As a result, Samuelson was incredibly influential.

Cournot was not.

It raises an interesting question of how much genius is in the man

and how much genius is in the audience.

Thanks.