The King of Price

Forwards

This topic certainly demands a certain amount of mathematical computations. It is inevitable that we need to give a lot of definitions and math works. Just be patient! We're sure that you'll have fun once you get used to it!

We provided the least amount of definition that will help constructing the frame of discussion in the simplified version. A more formal and rigorous one is also provided to you. You may choose the one that you feel comfortable with!

Basic Notions and Global Assumptions

Let there be two countries, $A$ and $B$, where

Country $A$ has $2$ firms, we call them $1, 2$ respectively. Firm $i$ $(i=1, 2)$ can produce $q_i$ units of chips that are needed for manufacturing CPUs for computers. Note that these $2$ firms will together produce $$q=q_1 + q_2$$ units of chips.

Country $A$ has $l$ firms, which we denote by $1, 2, \cdots, l$ respectively. Firm $i$ $(i=1, 2, \cdots, l)$ can produce $q_i$ units of chips that are needed for manufacturing CPUs for computers. Note that the total output level is exactly $$q=\sum_{i=1}^l q_i.$$

Let us assume one unit of chip cost $c$ to produce. We further assume that $c = 5$ for convenience. This means producing $q$ units of chips need $cq$ (which is just $5q$) units of money, in which case, we'll write $c(q)$ to represent the cost to produce $q$ units of chips.

The cost of producing $q$ units of chips is provided by a cost function $c(q)$.

Country $B$ would like to pay $p$ for each unit of chip. It is generally known that the rarer a product is, the more valuable it is. So is the chips producued by Country $A$. Let us assume that Country B would like to pay $p(q) = 20 - q$ units of money for each unit of the chip Country A produced ($(q)$ behind the price $p$ is just a notation we use to remind you that the amount of chips Country A produced, i.e., $q$ is related to the price that Country B would like to pay, you can also see that the higher the $q$ is, the lower the price is).

Country $B$ has a certain demand $D(q)$ on the chips that Country A would like to produce, where
An inverse demand function $p(q)$ therefore exists, which was then regarded as the price function (Note that $D$ and its inverse resembles the relationship between supply and demand).

We shall globally assume $p(q)=20-q$. This means if the firms in Country $A$ produce $q$ units of good in total, then Country $B$ will need to pay $20-q$ for each unit of chips it purchases. The linear form is assumed merely for convenience of later discussion. Indeed, price function can be of quadratic form or takes on more abstract form. One can also solely manipulate on symbols. However, the result will not be that intuitive and interesting to see.
We shall globally assume $c(q)=5q$, that is, one unit of such chip requires 2 units of money to produce. Similarly, the linear form is assumed for convenience of later discussion. In reality, the cost function may contains a fixed cost term (e.g. consider salaries of staffs), different firms may produce products at various cost dependent on the firm's productivity, etc.

Cournot Monopoly

Let's start with the simplest case! Consider there is only one firm being capable of producing such chips in demand! Treat yourself as the head of the company and your staff all counts on you to give a number, telling them how many units of chips they should produce in next quarter of the year. Of course, you want to earn more profits.

So here's the first question! Feel free to see the solution by clicking on the question!

We shall denote the profit function of your firm by $\pi$. Then $$\pi = p(q)q - c(q)q = (20-q)q - 5q = -q^2+15q$$

We help you to plot the graph of the profit function against the output level. Explore the gragh, there is a obvious peak where profit is maximised.

Completing the square gives $$\pi=-q^2+15q=-(q-\dfrac{15}{2})^2+\dfrac{225}{4}, $$ which maximizes by $q=\dfrac{15}{2}$ and the corresponding profit is $\dfrac{225}{4}$ ($=56.25$).

As the head of your firm, based on your analysis about your profits, you will simply instruct your staffs to start producing $\dfrac{15}{2}$ such chips and wait for your fortune! The above graph also serves as an evidence that your profit will maximize by $\dfrac{15}{2}$.

Cournot Duopoly

Now that you've figured out what to do when you're the only firm monopolizing the market. A natural question to ask is what happened when more competitors joined. Let us assume now that another firm cuts into your market and start selling chips of same qualities. How should you defend yourself in this competition?

Denote your firm as Firm 1, with profit $\pi_1$, your competitor's firm as Firm 2, with profit $\pi_2$. Assume the outputs from Firm 1 and Firm 2 are $q_1, q_2$ respectively.

Evidently, $\pi_1$ and $\pi_2$ are both functions in terms of $q_1$ and $q_2$. Hence we may write notations like $\pi_1(q_1, q_2)$ and $\pi_2(q_1, q_2)$. Then, it is easy to see that $$\pi_1(q_1, q_2) = (20-q_1-q_2)q_1-2q_1=-q_1^2+(15-q_2)q_1$$ and $$\pi_2(q_1, q_2) = (20-q_1-q_2)q_2-2q_2=-q_2^2+(15-q_1)q_2.$$

Based on your previous analysis, we again see that the profit functions for both firms are of quadratic form. Moreover, we notice that $\pi_1$ is maximized by $\hat{q_1}=\dfrac{15-q_2}{2}$ and $\pi_2$ is maximizd by $\hat{q_2}=\dfrac{15-q_1}{2}$. It is reasonable for us to call such $\hat{q_1}$ (red line) and $\hat{q_2}$ (blue line) the "best responses" for Firm 1 and Firm 2 respectively, since such output level is sort of a "response" that maximizes one's profit given one's opponent's move. Naturally, we come up with the following plot.

Recall that these two lines correspond to the "best response" of both firms in terms of output level that will maximize their profits. Hence there is a special meaning about the intersection point, where both firms have their profits maximized at it. It follows that no one can gain by unilateral change the output level. Therefore, we reach a Nash Equilibrium at the intersection.

This means, if the other firm also has a smart mind like you, you and your opponent will likely to choose the output quantity corresponds to the intersection point!!! We'll denote this special point from now on by $(q_1^e, q_2^e)$ ($e$ refers to equilibrium).

We leave the computation of the exact profits of both firms to you. See if you can find anything interesting.

Solving the system of equations $$ \begin{cases} q_1 = \dfrac{15-q_2}{2} \\ q_2 = \dfrac{15-q_1}{2} \end{cases} $$ gives $$ \begin{cases} q_1 = 5 \\ q_2 = 5 \end{cases} $$ Putting $(q_1, q_2) = (5, 5)$ into $\pi_1$ and $\pi_2$ gives $\pi_1 = \pi_2 = 25$.

Note that $\pi_1 + \pi_2 = 50 < \dfrac{225}{4} (=56.25)$. This means when the two firms cooperate and act like one firm, they behave worse than the scenario we've first discussed in the Cournot Monopoly section.

Cooperated Cournot Duopoly

Why is that? This is because of two firms are simultaneously striving for their own best interest. Hence they lost the grip of the bigger picture. What if these two firms cooperate with each other? Can they do a better job?

With cooperation, they will not only strive for their own best interest, but will work together (behave as a single firm) to achieve the best result and reasonably split their profits afterwards. Of course, by working together, here we mean that they together adjust their output level to maximize their total profits.

Note that $$\pi = \pi_1 + \pi_2 = -q_1^2 + (15-q_2)q_1 - q_2^2 + (15-q_1)q_2 = -(q_1+q_2)^2 + 15(q_1 + q_2)$$

Taking $Q = q_1 + q_2$, we arrives at the same expression as that in the Cournot Monopoly section, i.e. they maximize the profits when $Q = q_1 + q_2 = \dfrac{15}{2}$.

It is reasonable to split the workload equally, i.e. two firms produce half of the optimal quantity. We'll call these quantities $q_1^c, q_2^c$ respectively ($c$ stands for cooperation). and hence they should also split the profits equally. We leave it for you to do this. Comparing the profit of each firm when there is cooperation with the scenario where there is no such cooperation, what you can find now?

Previous question leads to $q_1 = q_2 = \dfrac{Q}{2} = \dfrac{15}{2}$. One computes that $\pi_1 = \pi_2 = \dfrac{225}{8}$. Note that $$\dfrac{225}{8} (=28.125) > 25.$$ This means that with cooperation, they together indeed achieve a better result.

Note further that this cooperation is only beneficial to the two firms in Country A, but not the customer (Country B), for it has to purchase chips at a higher price! Indeed, when the two firms doesn't collaborate, the price per unit of chip is $p(q_1+q_2)= p(5+5)=10$. However, with collaboration, the price per unit of chip rises to $p(q_1+q_2)=p(\dfrac{15}{4}+\dfrac{15}{4})=12.5$.

Incentive to Cheat

However, this cooperation is by no means stable.

Let us assume that your opponent is convinced by the invitation of cooperation that you proposed. This leaves you a choice to cheat on your opponent!!! Try the next question to see how you can take advantage of your opponent.

Note that $$\pi_1(q_1, q_2^c) = -q_1^2 + (15-\dfrac{15}{4})q_1 = -q_1^2 + \dfrac{45}{4}q_1.$$ This will maximize by $$q_1 = \dfrac{45}{8} (=5.625)$$ and $\pi_1 = \dfrac{2025}{64}$ ($=31.640625$), $\pi_2 = \dfrac{675}{32}$ ($=21.09375$).

We note that the profit of firm 1 is increased (From $\dfrac{225}{8} (=28.125)$ to $\dfrac{2025}{64} (=31.640625)$), while the profit of firm 2 dropped (From $\dfrac{225}{8} (=28.125)$ to $\dfrac{675}{32} (=21.09375)$). This means that given firm 2 cooperates, firm 1 can choose to betray them by outputing some $q_1^b$ ($b$ stands for betray) that maximizes his own profit at the cost of sacrificing the firm 2's profit. The same is to Firm 2.

The Tale of A Greedy World

Up until now, we've learnt a lot about Cournot Duopoly! Recall that

Two firms striving for their best interest without cooperation lead to a Nash Equilibrium ($(q_1^e, q_2^e) = (5, 5)$, with $(\pi_1, \pi_2) = (25, 25)$).
Two firms can cooperate, which grants them higher profits ($(q_1^c, q_2^c) = (\dfrac{15}{4}, \dfrac{15}{4})$, with $(\pi_1, \pi_2) = (\dfrac{225}{8}, \dfrac{225}{8})$).
One firm can cheat on the other, by providing them fake intention of cooperation. The cheater gets even higher payoff, and the one get cheated gets worse payoff than he's supposed to receive ($(q_1^b, q_2^c) = (\dfrac{45}{8}, \dfrac{15}{4})$, with $(\pi_1, \pi_2) = (\dfrac{2025}{64}, \dfrac{675}{32})$).

It seems that one of the case is missing, i.e., when both firms provide with each other a fake intention of cooperation and all decide to cheat the other. We'll show that in this case, both players "reap what they've sown" and eventually hurt themselves. But before that, we invite you to try to explore yourself.

Cournot Duopoly Game

> You're given the price function and the cost function: $p(q) = 20 - q$ and $c(q) = 5q$.
> You're free to adjust the output level of two firms, however, $q_1, q_2, q_1+q_2$ must be within the range $[0, 20]$ and of 4 decimal places.
> "Show Result" will help you to compute the profits earned by both firms providing that they abide the output level you gave.

Firm 1 produces:
Firm 2 produces:

The profit of Firm 1 is:
The profit of Firm 2 is:
The total profit of 2 firms is:

We hope you noticed that

This is by no means an accident. Indeed, it can be shown that, no matter how one alter the price function and the cost function, as long as they are of the linear form, this is always true.

Proof. We first generalize the price function and the cost function by writing $p(q) = D - q$ and $c(q) = Cq$ for some constants $C, D > 0 $ such that $D > C$. Then $$\pi_1(q_1, q_2) = (D-q_1-q_2)q_1 - Cq_1 = -q_1^2 + (D-C-q_2)q_1.$$ Similarly, $$\pi_2(q_1, q_2) = (D-q_1-q_2)q_2 - Cq_2 = -q_2^2 + (D-C-q_1)q_2.$$ Hence $$\pi = \pi_1 + \pi_2 = -q_1^2 + (D-C-q_2)q_1 - q_2^2 + (D-C-q_1)q_2 = -(q_1+q_2)^2+(D-C)(q_1+q_2),$$ which maximizes by $q_1+q_2 = \dfrac{D-C}{2}$.
One checks that $\{(\pi_1(q_1, q_2), \pi_2(q_1, q_2)): q_1 + q_2 = \dfrac{D-C}{2}, q_1, q_2 \ge 0\}$ forms the bargaining set of this game and the maximin point is $0, 0$. Applying the Nash Arbitration Method gives $(\pi_1^c, \pi_2^c) = (\dfrac{11}{16}(D-C)^2, \dfrac{11}{16}(D-C)^2)$, obtained by $(q_1^c, q_2^c) = (\dfrac{D-C}{4}, \dfrac{D-C}{4})$.

One checks that $\pi_1(q_1, q_2^c) = -q_1^2 + \dfrac{3}{4}(D-C)q_1$, which maximizes by $q_1^b = \dfrac{3}{8}(D-C)$. This then gives $\pi_2(q_1^b, q_2^c) = \dfrac{3}{32}(D-C)^2$. One also checks that $\pi_1(q_1^b, q_2^b) = \pi_2(q_1^b, q_2^b) = \dfrac{3}{32}(D-C)^2$. The equalities then arises naturally.

So what this theorem tells us is that, if both of the players are trying to cheat on the other, they themselves eventually reap what they've sown. As far as the author known, there is no documentation of such result. We shall name the result as

Cournot Oligopoly

All About the Cournot Duopoly

We shall use this last section to help you recap everything we've traversed up until now.

Two firms striving for their best interest without cooperation lead to a Nash Equilibrium ($(q_1^e, q_2^e) = (5, 5)$, with $(\pi_1, \pi_2) = (25, 25)$).

Two firms can have cooperation which can help them to achieve better result, yet there is also room for betrayal. We can formalize a $2 \times 2$ payoff matrix that documents all possibilities (Cells corrected to 3 decimal places).

Firm 1 \ Firm 2	Cooperate (Produce $\frac{15}{4}$)	Betray (Produce $\frac{45}{8})$
Cooperate (Produce $\frac{15}{4}$)	$(28.125, 28.125)$	$(31.641, 21.094)$
Betray (Produce $\frac{45}{8})$	$(21.094, 31.641)$	$(21.094, 21.094)$

Among which, both firms choose to cooperate is the only Nash Equilibrium (No one can gain by unilateral change).

By cooperating, two firms earn more profits, however, the customers suffer from increased prices ($10$ to $12.5$).
The Tale of A Greedy World: If both players choose to betray the other, then they eventually reap what they've sown.

Recall that back in the introduction part, we mentioned the story of the automobiles. Now that you've gone through this maze of mathematics! You should know that why the Japanese automobiles companies decreased their productivity but still earned their fortune! But! ... This automobiles competition is never a one time movie. In reality, the competition will be repeated every year, every season, or even every day! Therefore, we need to further investigate what happened when we repeatedly compete with other firms!. This directs us to the next section.

This wraps up the whole section on Cournot Competition. In the next section, we'll proceed to explore the beauty of the repeated Cournot Competition.

Firm 1 \ Firm 2	Cooperate (Produce \(\frac{15}{4}\))	Betray (Produce \(\frac{45}{8})\)
Cooperate (Produce \(\frac{15}{4}\))	\((28.125, 28.125)\)	\((31.641, 21.094)\)
Betray (Produce \(\frac{45}{8})\)	\((21.094, 31.641)\)	\((21.094, 21.094)\)