The Differentiated Bertrand Model is a theoretical model used in economics that explores competition and pricing behavior among firms producing, as inferred, differentiated products. The Bertrand model alone, on the other hand, focuses on homogeneous products.
In the differentiated Bertrand model, firms compete for imperfect substitutes. So, consumers have preferences for certain product characteristics, and as a result firms have some degree of market power, letting them charge prices above marginal cost. The more differentiated the products are, the higher the markup over marginal costs.
The model is often used to explore the entry and exit of firms, as well as deriving profit-maximizing prices, quantity produced, and revenue. In this article, we’ll focus on those applications, namely how to derive price, quantity, and revenue given the marginal costs and demand functions of two firms. There’s more to the model than such, but this both provides the best groundwork and is the most common application of the differentiated Bertrand model among for students.
As an example, let’s look at a market with two firms called Firm 1 and Firm 2. Both firms have a marginal cost of 0 and respectively have demand functions of q1 = 72–3p1 + 2p2 and q2 = 72–3p2 + 2p1. This means that the pricing behavior of Firm 1 affects Firm 2, and vice versa.
To calculate the optimal price, let’s first find marginal revenue, since marginal revenue = marginal cost at the competitive equilibrium:
For firm 1:
- TR1 = p1*q1 = p1(72–3p1 + 2p2) = 72p1–3p1² + 2p1p2
*to calculate marginal revenue, we’ll take the partial derivative of TR (dTr1/dP1).
- MR1 = 72 – 6p1 + 2p2
- *now, solve for Firm 1’s price.
- MR1 = MC
- 72 – 6p1 + 2p2 = 0
- -6p1 = -2p2 – 72
- p1 = 12 + 1/3p2
*reaction function; given any P2 price, we’ll get P1’s best response, e.g., it’s profit maximizing price.
For firm 2:
- TR2 = p2*q2 = p1 (72–3p2 + 2p1) = 72p2–3p2² + 2p2p1
- MR1 (dtr2/dp2) = 72 – 6p2 + 2p1
- 72–6p2 + 2p1 = 0
- -6p2 = -2p1–72
- p2 = 12 + 1/3p1
*now, let’s substitute Firm 1’s reaction function into Firm 2’s reaction function:
- p1 = 12 + 1/3(12 + 1/3p1)
- p1 = 12 + 4 + 1/9p1 = 16 + 1/9p1
- 8/9p1 = 16
- 8p1 = 144
- p1 = 18
Therefore, the profit-maximizing price for Firm 1 is 18. If we do the math for Firm 2, we’ll get the same result given the symmetric demand curves and marginal costs. We can furthermore solve for quantity by plugging the profit maximizing price into the original demand functions. For Firm 1:
- q1 = 72–3p1 + 2p2
- q1 = 72–3(18) + 2(18)
- q1 = 72- 54 + 36
- q1 = 54
- TR = PQ
- TR = (18)(54) = 972
So, in sum, both firms will sell 54 goods for a price of $18, resulting gross revenue of $972.
That’s all for our introduction to the differentiated Bertrand model! Let me know in the comments if you would like to see any additions to this article, or further articles on similar subjects.
