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The linear demand function or curve is widely used in economic models and described by:

$$ Q = a - \lambda P $$

where $Q$ is quantity produced or sold and $P$ is the selling price (and $\lambda > 0$).

When this demand curve is used to calculate the profit = (price - cost) * quantity, we get:

$$ \pi = (P - C) * Q $$

Substituting the linear demand curve for $P$ we get an equation where $\pi$ is quadratic in $Q$ with a negative coefficient for the quadratic term. This means that beyond a certain point, the profits will start to decline as more quantities of the product are sold. How can this be a realistic model of what happens in the real-world? I do not have any background in economics so sorry if the question is naive but would like to understand.

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This means that beyond a certain point, the profits will start to decline as more quantities of the product are sold. How can this be a realistic model of what happens in the real-world?

This is completely realistic. Profits are, as your equation above shows, price minus cost times the quantity.

At some point when firm wants to sell large quantity of product, it will have to start selling it below cost of production ($P<C$). At that point profit would become negative since $P<C \implies \pi <0$.

This is why firms simply can’t always just maximize the quantity they produce. Producing as much as possible is not necessarily profitable.

Of course, firm will try its best to avoid entering the negative portion of its profit function, but it is still there.

Furthermore, note this is not a property that is exclusive to linear demand. Although exceptions can exist, almost any downward sloping demand function will at some point require price to be smaller than cost to sell ever increasing quantities of product.

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the price $P$ you sell all the products is unique, you seem not taking in consideration this thing when you say

This means that beyond a certain point, the profits will start to decline as more quantities of the product are sold.

even in the case in which the price is not unique but you extract all the surplus from your consumers with a strategy of price discrimination you need to stop at some point because you have that some consumers are willing to pay less than your production costs.

Anyway in the case of unique price (for example in a perfect competitive market) you have the extreme case that if you sell at $P=0$ the quantity demanded is $Q=a$ and so if you pursue this strategy you will have a negative profit $\pi = -a C$.

In this context you want to produce till your marginal revenue equal your marginal costs because every time you produce and want to sell a new unit of your product you have to lower the price to sell it (and sell all the others at the same price).

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