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.