Monopoly vs Duopoly (Math and Economy) - 2

After the previous publication I have discussed the situation in which the Firm $A$ was on the market in a monopoly situation, in this publication I will discuss the case in which it appears a company $B$ willing to compete with the Firm $A$ by the sale of a same product.

As in the previous case, we will also assume that the Firm $B$, as the Firm $A$, sell what it produce and the cost of production of the two companies is the same, ie \$ 5. However, now the price $p$ will have to relate to the supply for a different way because the product supply now not only matches the number of products produced by the firm $A$, but matches the number of products produced by the Firms $A$ and $B$. Thus, if we consider $x$ the number of products produced by the Firm $A$ and $y$ the numbers of products produced by the company $B$, suppose that the price and supply are related as follows: $$ p = \frac{60}{x + y}, $$ where $x$ and $y$ are positive integers.

Thus, the Firm A's profit function , $\pi_{A}$ , is defined as follow $$\pi_{A}(x, y) =\frac{60}{x + y} \times x-5x, $$ and the Firm's $B$ profit function, $\pi_{B}$, is defined as follow $$\pi_{B} (x, y) = \frac{60}{x + y} \times y-5y.$$

Note that in this situation the profit function of each Firm depends not only on the choices that they make relatively to the amount of product that they supply, but also depends on the choices of your competitor. Not to complicate the situation very much we will think only in the case where firms $A$ and $B$ can only choose between supply 1 or 2 units. Recall that produce 1 unit was the best choice for the Firm A when it was in a monopoly situation.

Using the profit functions of each Firm we can through the following table summarize profits that firms will have considering all the possible choices that they can do.

Example: $\Pi_{A}(2,1) $ corresponds to the profit of the Firm A if it chooses to produce two units and the Firm $B$ a unit. Similarly, $\Pi_{B} (2,1)$ corresponds to the profit of the Firm $B$ if it chooses to produce a unit and the Firm $A$ chooses to produce two units.

And in this case, we fall in a case similar to the Prisoner's Dilemma. Firms would get more profit if they could coordinate strategies and produce a unit each, but as each of them does not know the choice that the other will take, the best choice for each considering possible options of the competitor is providing to the market 2 units each. This implies a reduction in price to \$15 and a profit of \$20 each.

Thus, in the end, the results illustrates how the competition is good for the consumer because with it is possible to reduce the price of the product from \$60 (price of monopoly situation) to \$15, and moreover, the supply ceases to be one product be extended to 4 products.

Also the transition to the duopoly situation requires the reduction of profits, including a reduction in the aggregate profit because both together only get a profit of €40, while in a monopoly situation the firm had a profit €55.

Note: The Firms also profited more if they sell a single product, act as a single monopolist, and divide the profit. But let's admit that this is not possible.

Monopoly vs Duopoly (Maths and economy) - 1

For those not familiar with the concepts of monopoly and duopoly, it is said that there is a monopoly situation when there is only one firm on the market to sell a product / service, similarly it is said that there is a duopoly when there are two firms in the market to compete for the sell of a product / service.

In the next two publications I propose to discuss, the effects of each of these types of markets for businesses and consumers. However, what I will do not want to be a practical application, but an analysis of a simplified theoretical formulation not to alienate the reader of what is essential. In reality all this is much more complex.

Let us assume that there is only one company in the market, Firm A, to produce and market a product. So we are in this first phase, in a monopoly situation.

For this firm, the production cost of each unit is \$5. Thus, if the number of units produced is equal to the number of units sold and this company wants to sell each unit at a price $p$, where $p$ is a positive number, your $ \ pi $ profit is given by

$$\pi = p \times x - 5x $$

where $x$ is the number of units produced / sold (demand) and a positive integer.

To maximize your profit the firm needs to analyze the market and estimate the number of quantities that will sell at a certain price $p$, that is, need to relate the price and demand. To this end, suppose that in this market price and demand are related as follows $x =\frac{60}{p}$. Note that, in this respect, a price increase implies a decrease in the number of units sold (demand) and a decrease in the price implies an increase in units sold (demand).

In this way porque,

$$ x =\frac{60}{p} = p\text{, that is } p = \frac{60}{x} $$

and $\pi=p\times x - 5x $ then

$$\pi =\frac{60}{x}\times x-5x = 60-5x $$ ,

i.e.

$$\pi = 60-5x. $$

This means that, in a monopoly situation, the option that maximizes profit of Firm A is to produce 1 unit at a price of \$60 to get a \$55 profit.

The next publication I will discuss the case in which it appears a competitor in the market to sell the same product, a Firm B, and discuss their implications both for companies and for consumers when compared to the monopoly situation.

Prisoner's Dilemma

Invented by Merrill Flood and Melvin Dresher in 1950 and later formalized by Albert Tucker, one of the most famous dilemmas in Game Theory is the Prisoner's Dilemma.

Unless the names, the story is as follows:

In Chicago district a crime was committed and the district attorney, although it has no sufficient evidence, know that it was committed by two criminals who, for simplicity we will call Bob and Adam. However, the attorney may convict only if at least one of them confess to the crime. For this reason, he orders arrest the criminals, put them in separate rooms and offers each of them the following agreement:

In Chicago district a crime was committed and the district attorney, although it has no sufficient evidence, know that it was committed by two criminals who, for simplicity we will call Ivo and Bruno. However, the attorney may order only if at least one of them confess to the crime. For this reason, he orders arrest the criminals, put them in separate rooms and offers each of them the following agreement:

• If you confess and your accomplice does not confess, you go in freedom and to your accomplice will be sentenced to maximum penalty of 10 years in prison. But if your accomplice confess and you do not confess, he goes free and you will be sentenced to 10 years in prison.
• If both confess, both will be arrested but will not be sentenced to maximum sentence for collaborating with justice, that is, will be sentenced to seven years.
• If no confession, the evidence already gathered is sufficient to give a sentence of one year in prison each.

The question can be summarized as the following table, where in each cell the first number corresponds to the prison sentence to be awarded to Ivo, and the second corresponds to the prison sentence to be given to Bruno according to the possible options that each of the criminals can take.

Note: The use of negative numbers is only because the prison sentence is a negative consequence.

In this case, criminals are faced with a dilemma: confess or not to confess?

Let us analyze the situation in Bob's perspective having only in mind the effect of the attorneys's agreement:

If Adam does not confess, Bob will have a penalty of one year in prison if not confess to the crime and leave freely if he confess;

If Adam confess, Bob will have a 10-year prison sentence if not confess to the crime and a 7-year prison sentence if he confess.

The same analysis can be made for the case of Bruno. Which means that, overall, taking into account what the accomplice can do to either of them, confess to the crime is always the best option, even at the risk of being sentenced to a prison sentence of seven years.

And what this dilemma is interesting is precisely that. Although there is the possibility of having both a prison sentence of a year, which would be possible if the criminals were able to coordinate the best strategy to respond to the agreement of the prosecutor and not confess to the crime, this not being possible, the best strategy for both is to confess the crime and sentenced to a prison term of seven years.

As this, there are many situations in life where we could take advantage if it were possible to coordinate our action with the response of other actors, but as this is not always possible, we ended up taking the option that puts us in a position to minimize losses, whatever the choices of other players. Even if this option can bring us to very negative consequences.

But, it is clear that this is only a theoretical formulation, in real life, each person is an individual and there are more averse to the risk then others, more cooperative and less cooperative. In addition, in real life often an option of this nature can be taken considering that a person has knowledge of the other. But this is nonetheless a model who turns out to illustrate many aspects of life.

Theorem of Pythagoras demonstration and its geometric illustration (using the Thales theorem)

In this publication will be made a formal proof of the Pythagorean theorem, using the theorem of Thales, together with an geometric illustration of the results that are being achieved.

Consider the triangle $ \left [ABC \right] $ rectangle in $ C $ and $ \left [CD \right] $ its height relative to the side $ \left [AB \right] $. Let $ a = \overline{BC} $, $ b = \overline{AC} $, $ c = \overline {AB} $, $ x = \overline {AD} $ and $ y = \overline {DB} $ .

Step 1: Prove that the triangle $ \left [ABC \right] $ is similar to the triangle $ \left [ADC \right] $ and the triangle $ \left [ABC \right] $ is similar to the triangle $ \left [BDC \right] $.

• As the triangle $\left [ABC \right] $ and the triangle $ \left [ADC \right] $ are both rectangles and have the included angle of the vertex $A$ in common, by the AA criterion of similar triangles, the two triangles are similar.
• Similarly, as the triangle $\left [ABC \right] $ and the triangle $ \left [BDC \right] $ are both rectangles and have the included angle of the vertex $ B $ in common, by the AA criterion of similarity of triangles, two triangles are similar.

Step 2: Prove that $ \frac{\overline {AC}}{\overline {AB}} = \frac{\overline{AD}}{\overline {AC}}$ and $ \frac{\overline {BC} }{\overline {AB}} = \frac {\overline{DB}}{\overline{BC}} $. 

As by the first step we conclude that the triangle $\left [ABC \right] $ is similar to the triangle $\left [ADC \right] $ and 

• $\left [AC \right] $ corresponds to $\left[AB \right] $;
• $\left [AD \right] $ corresponds to $\left[AC \right] $. 

Then $$ \frac{\overline{AC}}{\overline{AB}} = \frac{\overline{AD}}{\overline{AC}} $$. 

Similarly, as at step 1 we conclude that the triangle $\left [ABC \right] $ is similar to the triangle $ \left [BDC \right]$ and 

• $\left[BC \right] $ corresponds to $ \left[AB \right] $;
• $\left[DB \right] $ corresponds to $ \left[BC \right] $.

Then $$\frac{\overline{BC}}{\overline{AB}} = \frac{\overline{DB}}{\overline{BC}}$$.

Step 3: Given the figure below: conclude that $b^{2} = x \times c$, that is, the area of the square $ \left[ACKL \right]$ is equal to the rectangle area $\left [ADOP \right]$; And conclude that $a^{2} = y \times c$, that is, the area of the square $\left [BCNM \right]$ is equal to the rectangle area $\left[DBOQ\right]$.

As in step 2 we conclude that $$\frac{\ overline{AC}}{\overline{AB}} = \frac{\overline{AD}}{\overline{AC}}$$ then, 

$$\frac{b}{c} = \frac{x}{b}. $$ Therefore, $ b ^ {2} = x \times c $.

Similarly as in step 2, we conclude that, $$\frac{\overline{BC}}{\overline{AB}} = \frac{\overline {DB}}{\overline {BC}}$$ then $$\frac{a}{c} = \frac{y}{a}. $$ 

So $a^{2} = y \times c $.

Step 4: Prove that $ a^{2} + b^{2} = c^{2} $. That is, the area of squares that are on the sides is equal to the square on the hypotenuse area.

After performing these four steps, we conclude intuitively that the sum of the area of the square $\left[ACKL\right] $ with the area of the square $\left[BCNM\right]$ is equal to the area of the square $\left[ACKL\right] $. 

However, being more rigorous, at step 3, because $b^{2} = x \times c $ and $ a^{2} = y \times c $, then $$ a^{2} + b^{2} = y \times c + x \times c = (y + x) \times c = c \times c = c^{2} $$ because $ x + y = c $. Therefore, the $ a^{2} + b^{2} = c^{2}$.

The following application Geogebra illustrates perfectly all this demonstration.

Euclide's algorithm (illustration)

Using the Euclide's algorithm computes the greatest common divisor between $24$ and $10$.

Resolution:

Using the integer division algorithm repeatedly we can easily conclude that:

Therefore, because the remainder of the division betwwen $\color{blue}{4}$ and $\color{red}{2}$ is 0, the greatest common divisor between $24$ and $10$ is $2$.

Game Theory

John Nash (1928-...)

In our life, situations such as setting the price for a particular product, the bargaining between two people, internet auctions, art auctions, are very common. In each of these cases, the goal is always to determine the best strategy to maximize gain and / or minimize losses.

For example, in setting the price of a certain product the seller always wants to maximize its profit. Thus, despite being a monopoly position in the market, he can not set the price it pleases because if he does, he may run the risk of asking such an high price that keep away all demand. And even fewer can do it if he has competitors in the market because he runs the risk of consumers prefer to buy the product at a competitor.

Something similar may apply to trade between two people. To illustrate this case we can take the example of wage bargaining between employers and trade union representatives of the employees of a company. In this case, although employers prefer not to raise wages, or at worst, increase them as little as possible, representatives of trade unions wished otherwise. Which means, in this case, nor may employers go for negotiating for a proposal of this type with a very disadvantageous wage for workers nor unions can demand a very high salary increase, under penalty of an agreement is impossible.

For the case of auctions, the goal is always to sell at the highest price and the goal of those who buy is always buy at the lowest price. In this case, the auction format and the rules to which it complies influence the final result. For example, in an auction, all participants can have access to proposals that are being made by the participants but on the other hand, it may also be the case with anyone having knowledge of the proposals submitted. Surely that is not equivalent to opt for an auction model or another.

It's about this kind of topics that focuses the Theory of Games. The Game Theory is a branch of mathematics really booming these days and we can find these and other examples of their application in situations involving defining the best strategy in such popular games like poker and chess, and in many areas of knowledge as economics, finance, biology, sociology, psychology, physics and anthropology.

This branch of mathematics, though not invented by John Nash (1928 -...) and the claims of its invention is attributed to John von Neumann (1903-1957), he was the expanded and endowed with sufficiently powerful tools to solve real problems in various areas. Your work has an importance such that mathematicians made of Newton and Einstein is to physics as well as for Nash are the biological and social sciences. In 1994 John Nash shared the Nobel Prize in Economics with John Harsanyi and Selten Reihnard.

Noteworthy that many have been awarded Nobel prizes in economics for work in the field of Game Theory. This year, Jean Tirole is an example. He has won for his work on power analysis and market regulation. A very timely and relevant topic.