# "Cournot competition" with different prices for different firms

This operations research paper considers a "Cournot" competition modeled as follows.

[E]ach [firm] $$i$$ incurs a cost $$w$$ in obtaining one unit of the product, sells $$q_i$$ units of the product at a unit price $$p_i$$ , and generates a proﬁt $$π_i (q_1 , q_2)$$ when the two retailers sell $$q_1$$ and $$q_2$$ units of the product, respectively. We have $$p_i = z_i − (q_1 + q_2)$$ and $$π_i (q_1 , q_2) = (p_i − w)q_i = (z_i − (q_1 + q_2) − w)q_i$$, for $$i = 1, 2$$, where for the same quantity sold, a retailer with a larger $$z_i$$ charges a higher price for the product.

(Presumably, the prices are normalized so that they have the same unit as quantities.) Based on what little I have read about Cournot competitions, it seems odd that there are two different prices for the same goods (unless $$z_1 = z_2$$, but interesting things happen only if $$z_1 \neq z_2$$ in this article). Is this model reasonably be called a Cournot competition?

I concede that this model gives the same equilibrium as the Cournot duopoly in which the marginal cost incurred to each firm is $$w_i$$ and the common price is $$p_0 - (q_1 + q_2)$$, where $$p_0$$, $$w_1$$, $$w_2$$ are chosen so that $$p_0 - w_i = z_i - w$$ and that the relevant quantities are nonnegative. Still I find that presentation in the article weird.

(I guess it's possible, albeit unfortunate, that this is what's called a Cournot competition in operations research even if it's not in economics.)

• What is $z_i$? Is it $i$'s capacity? Commented Aug 21 at 2:12
• @HerrK. It's a mystery parameter not explained in the paper. Cho and Tang, who cite the paper, say that it "represents retailer i’s “market power” that captures her comparative advantage over her competitors due to certain factors including location, customer loyalty, and brand reputation" and put a reference to a seemingly unrelated paper. Commented Aug 21 at 2:56
• I only took a quick peak at the paper. At the beginning of Section 2, the author does say: "We consider Cournot competition, that is, the two retailers compete on the quantity of the product they sell." This is consistent with the common understanding of Cournot competition as one on quantities. It is conceivable that (a variant of) such competition occurs between rivals with differentiated characteristics (here the differentiation is measured by $z_i$), which could lead to differentiated prices in equilibrium. Commented Aug 21 at 3:36
• @HerrK My reservation is that the goods from each firm (or a retailer in their example) are sourced from the same person and thus complete substitutes, in which case I don't see there being two different prices. Commented Aug 21 at 9:52
• The bottom line is, as long as the competition is about simultaneous decisions on quantity, then it can be called a Cournot competition, or at least a variant thereof. Whether the competition leads to the same result as the classic Cournot model that appears in most textbooks is a different matter. Commented Aug 21 at 15:07