21.4 Market Structure and Framework
Basic model utilizing aggregate demand
Bertrand Equilibrium: Firms compete on price
Cournot Market structure: Firm compete on quantity
Stackelberg Market structure: Leader-Follower model
Because we start with the quantity demand function, it is important to know where it’s derived from Richard and Martin (1980)
K. S. Moorthy (1988)
- studied how two firms compete on product quality and price (both simultaneous and sequential)
21.4.1 Cournot - Simultaneous Games
\[ TC_i = c_i q_i \text{ where } i= 1,2 \\ P(Q) = a - bQ \\ Q = q_1 +q_2 \\ \pi_1 = \text{price} \times \text{quantity} - \text{cost} = [a - b(q_1 +q_2)]q_1 - c_1 q_1 \\ \pi_2 = \text{price} \times \text{quantity} - \text{cost} = [a - b(q_1 +q_2)]q_1 - c_2 q_2 \\ \]
\[\begin{equation} \frac{d \pi_1}{d q_1} = a - 2bq_1 - bq_2-c_1 = 0 (1) \tag{21.1} \end{equation}\] \[\begin{equation} \frac{d \pi_2}{d q_2} = a - 2bq_2 - bq_1-c_2 = 0 \tag{21.2} \end{equation}\]From (21.1)
\[\begin{equation} q_1 = \frac{a-c_1}{2b} - \frac{q_2}{2} = R_1 (q_2) \tag{21.3} \end{equation}\]is called reaction function, for best response function
From (21.2)
\[\begin{equation} q_2 = \frac{a-c_2}{2b} - \frac{q_1}{2} = \tag{21.4} \end{equation}\]\[ q_1 = \frac{a-c_1}{2b} - \frac{a-c_2}{4b} + \frac{q_1}{4} \]
Hence,
\[ q_1^* = \frac{a-2c_1+ c_2}{3b} \\ q_2^* = \frac{a-2c_2 + c_1}{3b} \]
Total quantity is
\[ Q = q_1 + q_2 = \frac{2a-c_1 -c_2}{3b} \]
Price
\[ a-bQ = \frac{a+c_1+c_2}{3b} \]
21.4.2 Stackelberg - Sequential games
also known as leader-follower games
Stage 1: Firm 1 chooses quantity
Stage 2: Firm 2 chooses quantity
\[ c_2 = c_1 = c \]
Stage 2: reaction function of firm 2 given quantity firm 1
\[ R_2(q_1) = \frac{a-c}{2b} - \frac{q_1}{2} \]
Stage 1:
\[ \pi_1 = [a-b(q_1 + \frac{a-c}{2b} - \frac{q_1}{2})]q_1 - cq_1 \\ = [a-b( \frac{a-c}{2b} + \frac{q_1}{2}]q_1 + cq_1 \]
\[ \frac{d \pi_1}{d q_1} = 0 \]
Hence,
\[ \frac{a+c}{2} - b q_1 -c =0 \]
The Stackelberg equilibrium is
\[ q_1^* = \frac{a-c}{2b} \\ q_2^* = \frac{a-c}{4b} \]
Under same price (c), Cournot =
\[ q_1 = q_2 = \frac{a-c}{3b} \]
Leader produces more whereas the follower produces less compared to Cournot
\[ \frac{d \pi_W^*}{d \beta} <0 \]
for the entire quantity range \(d < \bar{d}\)
As \(\beta\) increases in \(\pi_W^*\) Firm W wants to reduce \(\beta\).
Low \(\beta\) wants more independent
Firms W want more differentiated product
On the other hand,
\[ \frac{d \pi_S^*}{d \beta} <0 \]
for a range of \(d < \bar{d}\)
Firm S profit increases as \(\beta\) decreases when d is small
Firm S profit increases as \(\beta\) increases when d is large
Firm S profit increases as as product are more substitute when d is large
Firm S profit increases as products are less differentiated when d is large