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I have seen the following demand function

$q=a-p+c\bar{p}$

where $p$ is the price, $\bar{p}$ is the so called "average price". The values $a=1-c$ are competition parameters.

I have basically two question:

1) in classical demand functions there is no $\bar{p}$. Where does is come from?

2) I want to construct, if possible, the demand function, based on the one above, which has the form $e^{a-p+c\bar{p}}$. Does it make any sense? Is there any application in "real life" for such a demand function?

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Probably economics stack exchange would be more correct for this.

It is quite common to have such demand functions in non perfectly competitive markets.

Take for example two firms (1) and (2), which strategically interact. Their demand functions should depend on prices of the one another.

$q_1 = a - p_1 + c p_2$

$q_2 = a - p_2 + c p_1$

The basic intuition is that if my competitor increases the price I sell more and if I increase the price I sell less.

Now assume there are $N$ firms and let's say parameter $c$ is a scaling constant then a given firm $i$ could have this demand:

$q_i = a - p_i + c/N \sum_{j \neq i}^N p_j$

which is your demand function. You can definitely microfound a demand such as the linear one you mentioned. Not sure about the exponential one and I see no reason to have that exponential. Just take logs on both sides and use the linear one.

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