Modeling Demand with Competitor Prices

Summary

The document explains why a demand function may depend on competitors’ prices in markets where firms interact strategically. In a two-firm example, a firm’s demand falls as its own price rises and increases as its rival’s price rises. For multiple firms, the rival-price term can be represented by a scaled average of other firms’ prices, making the model a way to express cross-price effects under imperfect competition.

The answer says that a linear form can be grounded in a model of firms competing on price, while it offers no justification for exponentiating the linear expression. It suggests that the exponential version may lack a clear motivation and points out that taking logarithms can return to a linear relationship. The response is brief and qualitative: it gives intuition and functional forms, but no parameter estimation, empirical evidence, or conditions for choosing among demand specifications. The framework is an economic modeling concept, not a trading strategy.

Key ideas

• In imperfectly competitive markets, a firm’s demand can depend on its rivals’ prices.
• A higher own price reduces demand in the stated linear specification.
• A rival’s higher price can increase a firm’s demand by making its offer relatively attractive.
• With multiple firms, the cross-price effect can use the average price of competitors.
• The answer provides no empirical validation or rationale for exponentiating the linear form.

Tags

# Demand Function


# Demand Function












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?

## Answer by phdstudent (score 1)

https://quant.stackexchange.com/a/41726

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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