Cornout Quantity Competition

Firms will choose quantity as the strategic variable.

• Determining Firm 2’s Best Response Curve:
  1. When firm 1’s quantity = 0, firm 2’s best response is to produce $x^M$ (=monopoly quantity)
  2. When firm 1’s quantity > 0, firm 2’s best response derived by the reduction in demand in graph (c):
     1. $D$ (=full market demand) is shifted left by $x_1$ (=firm 1’s produced quantity) to $D^r$; Firm 2’s BR is to optimize at $\text{MC}=\text{MR}$
     2. → This shows that when Firm 1 produces $2x^M$, $D$ will shift left so much that $\text{MR = MC}$ at $x_1=0$.
• Firm 1’s BRC is symmetrical, as shown in (b)
  • → Firms produce $x^C$ (= cournot quantity)
  • → $\frac{1}{2}{x}^M<x^C<x^B$

Info

Cornout Price $p^C$ converges to competitive prices $p=MC$ as the number of firms increase. (Take my word for this; it’s proven mathematically but not shown above)

Sequential Move: Stackleberg Competition §

• Stackleberg Leader is Firm 1, Stackleberg Follower is Firm 2

Firm 1 considers Firm 2’s expected BRC before making a move:

Profit maximize knowing firm 2’s BRC:

1. Graph (a) is the expected Firm 2’s BRC
2. When Firm 1’s quantity $>x^{\ast }$ then firm 2 produces nothing → Firm 2 faces the full market demand ⇒$D=D^r$
3. When Firm 1’s quantity $\in [0,x^M]$ then firm 2 faces the residual demand $D^r$ which is calculated by:
   • at $x_1=x^M$,
     • …then BR: $x_2=\frac{1}{2}{x}^M$ and $\Sigma x=\frac{3}{2}{x}^M$
     • …thus $D^r=D-\frac{1}{2}{x}^M$
     • …equivalently $D^r(x^M)=D(\frac{3}{2}{x}^M)$
   • at $x_1=0$,
     • …then BR: $x_2=x^M$ and $\Sigma x=x^M$
     • …thus $D^r=D-x^M=0$
     • …equivalently $D^r(x_1[=0])=D(x^M)$

Sequential Move: Entry Deterrence §

Game Structure §

• First move is Firm 2’s entry decision
• Second move is either Cournot Quantity, Stackleberg Quantity competition (Bertrand isn’t considered)
• If firm 2 enters it must pay a Fixed Cost (FC) which is an economic cost to them (=matters in profit calculation).

1. Firm 2 chooses entry;
   • …if it enters it must pay FC, and two players cornout compete;
   • …if it doesn’t its profit is zero, and Firm 2 is a monopoly
2. Firm 2 chooses entry;
   • …if it enters it must pay FC, and two players stackleberg compete with firm 1 as stackleberg leader;
   • …if it doesn’t its profit is zero, and Firm 2 is a monopoly
3. Firm 1 chooses quantity first;
   • …Firm 2 enters, paying FC, and Firm 2 gets stackleberg follower profit
   • …Firm 2 doens’t enter, and Firm 1 gets profit from its original quantity

⇒ The game structure of (c) allows for strateigic entry deterrance by firm 1:

Entry Deterence §

• Firm 1 may deter entry to maximize profit by setting a certain quantity
• FC is an exogenous variable → firm 2’s entry decision will depend on how big FC is → thus, firm 1’s entry deterrence will depend on how big FC is

(a) is the profit against a choice of production quantity for a monopoly.

→ Then, see (b):

• If $FC>\overline{FC}$: Second firm will not enter no matter …when $\overline{FC}={\pi }^M$
• If $\underline{FC}<FC<\overline{FC}$: First firm will deter entry by increasing production …when $\underline{FC}=(x\text{ s.t. }\pi (x)={\pi }^{SL})$ …i.e. Firm will deter entry until $\pi$ drops so much that it’s better to compete (and get Stackelberg profit)
• If $FC<\underline{FC}$: normal stackleberg competition …where firm 1 is Stackelberg Leader (SL) and get ${\pi }^{SL}$ …and firm 2 is Stackelberg Follower (SF) and get ${\pi }^{SF}$