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.