Bertrand Price Competition

Firms will choose price as the strategic variable.

• Market Demand $x(p)=A-bp$ where $A,b$ are constants
• Simultaneous game
• Firm with lower price captures all of demand

Case: Constant and Same Marginal Cost §

• Marginal Cost is constant for both firms: $c(x)=cx$, $MC=c$
• Both firms will attempt to undercut each other.
  • $BR{C}_1:p_1(p_2)=max(p_2-ϵ,MC)$
  • $BR{C}_2:p_2(p_1)=max(p_1-ϵ,MC)$
  • $NE=\{p_1=c,p_2=c\}$ where both firms have zero profit
  • 

Case: Different but Constant Marginal Cost §

• Marginal Cost is constant, but different for both firms
  • $c_1(x)=c_{1}x$, $c_2(x)=c_{2}x$, $c_1<c_2$
• Then…
  • $BR{C}_1:p_1(p_2)=max(p_2-ϵ,c_1)$
  • $BR{C}_2:p_2(p_1)=max(p_1-ϵ,c_2)$
  • $NE=\{p_1=c_2-ϵ,p_2=c_2\}$
    • & firm 1 can set undercut the price because they still have $c_1<c_2-ϵ$
  • Firm 1 will make positive profit; firm 2 will make zero profit.

Case: Sequential Move §

• Change assumption: Dynamic game; firm 1 sets price first, then firm 2 sets price. ($c_1<c_2$)
  • Then, there cannot be a best response curve. Instead, consider the following subgame perfect Nash equilibirum:

$$\begin{array}{rl}1& :p_1=c_2-ϵ\\ 2& :p_2=\{\begin{array}{llc}min(p_2^M,p_2-ϵ)& \text{if}{c}_2<p_1& \text{(can undercut price)}\\ c_2& \text{if}{c}_2\ge p_1& \text{(can’t b/c MC)}\end{array}\end{array}$$

- The following is a non-subgame perfect Nash equilibirum. The non-credible threat is when firm 2 threatens "I'm just gonna lose money if you don't listen.". Given a constant $k$, 

$$\begin{array}{rl}1& :p_1=p_2^M+k\\ 2& :p_2=\{\begin{array}{ll}{p}_2^M& \text{if}{p}_1\text{ sets what I want}\\ 0& \text{if}{p}_1\text{isn’t what I want}\end{array}\end{array}$$

• Different order: Dynamic game; firm 2 sets price first, then firm 1 sets price. ($c_1<c_2$ still assumed)
  • Then, SPNE

$$\begin{array}{rl}2& :p_2=c_2\\ 1& :p_1(p_2)=\{\begin{array}{ll}min(p_1^M,p_2-ϵ)& \text{if}{p}_2>c_1\\ c_1& \text{else}{p}_2<c_1\end{array}\end{array}$$

If Product Can Be Differentiated §

When we assume that products can be differentiated (=diversified), then we can improve on the Bertrand model;.

1. Consumers’ preferences are uniformly distributed across differentiation space $\mathbb{R}$
2. Firms can produce a product with a certain value $P$ of differentiated characteristic $P\in \mathbb{R}$
3. A consumer with preference $C$ will pay a cost when consuming $P$ which is proportional to the distance $\text{Cost}=|P-C|$

• Assume there exists a possibility of differentiation space ($\exists \mathbb{R}\in [0,1]$)
• However firms are not allowed change product characteristics.
• The only strategic variable is price.

• First consider firm 1’s BRC:
  • If firm 1 gives away products for free $p_1=0$
    • …firm 2 can still charge a price since some consumers will prefer their products
    • …and $p_2>MC$ because firm 1 will exit otherwise.
  • If firm 2 charges a price $p_1>0$
    • …firm 2 can charge an even higher price since some consumers will prefer their products
  • Therefore the BRC is a positively sloping curve
• Firm 2’s BRC is symmetrical:
  • → Equilibrium is at the intersection of the two BRCs, and at this point $p_1^{\ast }=p_2^{\ast }$.
  • → Equilibirum price is above marginal cost, unlike pure Bertrand Price Competition

Info

Differentiation “softens” price competition. Therefore firms want more differentiation

Determining Firm 2’s Best Response Curve: §

1. when firm 1’s price < MC, firm 2’s best response is set price at MC
2. when firm 2’s price > MC, firm 2’s best response is the undercut firm 1’s price by a very small amount ($ϵ$)

• Firm 2’s BRC is symmetric → Firms’ prices will unravel to both pricing to p = MC (Bertrand price = MC) → At $p=MC$, market will purchase $2x^M$, and firms produce $x^M$ each.

p_{1}&=max[MC_{2}-\epsilon,MC_{1}] \\ p_{2}&=max[MC_{1}-\epsilon,MC_{2}] \end{align}

Hotelling Model §

• Products can be differentiated by one characteristic variable which ranges from 0 to 1.
• Differentiation is the only strategic variable; i.e. price, etc. cannot be changed.

Tip

The graph and its corresponding explaination in the textbook is wrong; the following is the correct explaination.

Thinking about Firm 1’s BRC:

1. when firm 2 produces a product with characteristic < 0.5, then firm 1 will produce a product that is just closer to 0.5 [=larger] to capture more consumers
2. when firm 2 produces a product with characteristic > 0.5, then firm 1 will produce a product that is just closer to 0.5 [=smaller] to capture more consumers
3. when firm 2 produces a product with characteristic = 0.5, then firm 1 will produce a product with characteristic = 0.5.

Info

In a single axis (1-dim. differentiation space)

→ Firm 2’s BRC is symmetrical; this unravels to the equilibrium where both firms produce product with characteristic = 0.5

Circle Model §

The circle model combines the Price Competition and Differentiation.

• This is a sequential game where firms will enter [first move] and then it will compete with price [second move]
  • Firms who enter incurs entry cost (an economic cost) of FC when they enter.
• Price and differentiation are the two strategic variables.
  • Product differentiation space is around a circle whose circumference is 1.

Then the following conclusions:

Info

Each firm is respresented as a point on the circle.

1. Firms will enter as long as $FC>\pi$ …①
2. Firms compete with their immediate neighbors → Firms will space themselves out around the circle

The number of firms entering $N$ and the equilibrium price given that $N$ firms have entered $p$ are determined in the following order:

1. $FC\mapsto N^{\ast }$ where $N\propto \frac{1}{FC}$ (∵ ①)
2. $N^{\ast }\mapsto p^{\ast }$ where $p\propto \frac{1}{N}$ but $p>MC$. Specifically: