Three canonical models of Oligopoly
Antoine Augustin Cournot
1801-1877
"Cournot competition": two (or more) firms compete on quantity to sell the same good
Firms set their quantities simultaneously
Firms' joint output determines the market price faced by all firms
Suppose two firms (1 and 2), each have an identical constant cost $$MC(q)=AC(q)=c$$
Firm 1 and Firm 2 simultaneously set quantities, \(q_1\) and \(q_2\)
Total market demand is given by
$$\begin{align*} P&=a-bQ\\ Q&=q_1+q_2\\ \end{align*}$$
$$\begin{align*} \pi_1&=q_1(P-c)\\ \pi_1&=q_1(a-b(q_1+q_2)-c)\\ \end{align*}$$
And, symmetrically same for firm 2
Note each firm's profits depend (in part) on the outputs of the other firm!
Consider each the demand each firm faces to be a residual demand
e.g. for firm 1
$$\begin{align*} p&=a-b(q_1+q_2)\\ p&=\underbrace{(a-bq_2)}_{intercept}-\underbrace{b}_{slope}q_1\\ \end{align*}$$
Firm 2 will produce some amount, \(\color{purple}{q_2}\).
Firm 1 takes this as given, to find its own residual demand
Firm 2 will produce some amount \(q_2\)
Firm 1 will take this as a given, a constant
Firm 1's choice variable is \(q_1\), given \(q_2\)
Example: Assume Saudi Arabia \((sa)\) and Iran \((i)\) are the only two oil producers, each with a constant \(MC=AC=20\). The market (inverse) demand curve is given by: $$\begin{align*} P&=200-3Q\\ Q&=q_{sa}+q_i\\ \end{align*}$$
Example: Assume Saudi Arabia \((sa)\) and Iran \((i)\) are the only two oil producers, each with a constant \(MC=AC=20\). The market (inverse) demand curve is given by: $$\begin{align*} P&=200-3Q\\ Q&=q_{sa}+q_i\\ \end{align*}$$
$$P=200-3q_{sa}-3q_i$$
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
Firms maximize profit (as always), by setting \(q^*: MR(q)=MC(q)\)
Solve for Saudi Arabia
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
Firms maximize profit (as always), by setting \(q^*: MR(q)=MC(q)\)
Solve for Saudi Arabia
$$MR_{sa}=200-3q_i-6q_{sa}$$
Solve for \(q^*\) for each firm (where \(MR(q)=MC(q))\), we derive each firm's reaction function or best response function to the other firm's output
Symmetric marginal costs and marginal revenues
Solve for \(q^*\) for each firm (where \(MR(q)=MC(q))\), we derive each firm's reaction function or best response function to the other firm's output
Symmetric marginal costs and marginal revenues
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Iran's reaction curve to Saudi Arabia's output
We can graph Iran's reaction curve to Saudi Arabia's output
We can graph Iran's reaction curve to Saudi Arabia's output
Combine both curves on the same graph
Cournot-Nash Equilibrium: $$\big( \color{red}{20}, \color{blue}{20} \big)$$
Both are playing mutual best response to one another
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
$$P=200-3q_{sa}-3q_i$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=20(80-20)\\ \pi_{sa}&=1,200\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} P&=200-3(30)\\ P&=\$110\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} P&=200-3(30)\\ P&=\$110\\ \end{align*}$$
$$\Pi=30(110-20)=\$2,700$$
with $1,400 going to each firm
Cournot Competition: each firm produces 20 and earns $1,200
Cournot Collusion: each firm produces 15 and earns $1,400
Cournot Competition: each firm produces 20 and earns $1,200
Cournot Collusion: each firm produces 15 and earns $1,400
But is collusion a Nash equilibrium?
Read either firm's reaction curve at the collusive outcome
Suppose Saudi Arabia knows Iran is producing 15 (as per the cartel agreement)
Saudi Arabia's best response to Iran's 15 is to produce 22.5
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=22.5(87.50-20)\\ \pi_{sa}&=\$1,518.75\\ \end{align*}$$
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=22.5(87.50-20)\\ \pi_{sa}&=\$1,518.75\\ \end{align*}$$
$$\begin{align*} \pi_{i}&=q_{i}(P-c)\\ \pi_{i}&=15(87.50-20)\\ \pi_{i}&=\$712.50\\ \end{align*}$$
Imagine Bertrand competition between Saudi Arabia and Iran instead (price competition)
Nash equilibrium: Firms will set \(P=MC\), so:
$$\begin{align*} P&=MC\\ 200-3Q&=20\\ Q&=60\\ \end{align*}$$
Both countries split demand equally, each selling 30 units
Profit for both countries would be 0, since \(P=MC\)
Type | Output | Price | Profits |
---|---|---|---|
Collusion | 30 | $110 | $2,700 |
Cournot | 40 | $80 | $2,400 |
Bertrand | 60 | $20 | $0 |
Type | Output | Price | Profits |
---|---|---|---|
Collusion | 30 | $110 | $2,700 |
Cournot | 40 | $80 | $2,400 |
Bertrand | 60 | $20 | $0 |
Where subscript \(m\) is monopoly (collusion), \(c\) is Cournot, \(b\) is Bertrand
Example: Suppose Firm 1 and Firm 2 have a constant \(MC=AC=8\). The market (inverse) demand curve is given by:
$$\begin{align*} P&=200-2Q\\ Q&=q_1+q_2\\ \end{align*}$$
Find the Cournot-Nash equilibrium output and profit for each firm.
Find the output and profit for each firm if the two were to collude.
Find the price and output if the two were to compete on price instead of quantity.
Antoine Augustin Cournot
1801-1877
Cournot Theorem: as the number of firms \((N)\) in the market increases, market output \(Nq\) goes to the competitive level, and price converges to \(c\).
More (fewer) firms reduce (increase) market distortions from market power
Antoine Augustin Cournot
1801-1877
Major implications from Cournot:
As \(\uparrow\) number of firms: \(\uparrow Q\), \(\downarrow p\), \(\downarrow \pi\), \(\uparrow CS\), \(\downarrow DWL\) (closer to perfect competition)
If a firm has lower costs than others, it earns greater profit. Firms will want to lower their own costs OR raise rivals' costs (same effect)
Three canonical models of Oligopoly
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Three canonical models of Oligopoly
Antoine Augustin Cournot
1801-1877
"Cournot competition": two (or more) firms compete on quantity to sell the same good
Firms set their quantities simultaneously
Firms' joint output determines the market price faced by all firms
Suppose two firms (1 and 2), each have an identical constant cost $$MC(q)=AC(q)=c$$
Firm 1 and Firm 2 simultaneously set quantities, \(q_1\) and \(q_2\)
Total market demand is given by
$$\begin{align*} P&=a-bQ\\ Q&=q_1+q_2\\ \end{align*}$$
$$\begin{align*} \pi_1&=q_1(P-c)\\ \pi_1&=q_1(a-b(q_1+q_2)-c)\\ \end{align*}$$
And, symmetrically same for firm 2
Note each firm's profits depend (in part) on the outputs of the other firm!
Consider each the demand each firm faces to be a residual demand
e.g. for firm 1
$$\begin{align*} p&=a-b(q_1+q_2)\\ p&=\underbrace{(a-bq_2)}_{intercept}-\underbrace{b}_{slope}q_1\\ \end{align*}$$
Firm 2 will produce some amount, \(\color{purple}{q_2}\).
Firm 1 takes this as given, to find its own residual demand
Firm 2 will produce some amount \(q_2\)
Firm 1 will take this as a given, a constant
Firm 1's choice variable is \(q_1\), given \(q_2\)
Example: Assume Saudi Arabia \((sa)\) and Iran \((i)\) are the only two oil producers, each with a constant \(MC=AC=20\). The market (inverse) demand curve is given by: $$\begin{align*} P&=200-3Q\\ Q&=q_{sa}+q_i\\ \end{align*}$$
Example: Assume Saudi Arabia \((sa)\) and Iran \((i)\) are the only two oil producers, each with a constant \(MC=AC=20\). The market (inverse) demand curve is given by: $$\begin{align*} P&=200-3Q\\ Q&=q_{sa}+q_i\\ \end{align*}$$
$$P=200-3q_{sa}-3q_i$$
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
Firms maximize profit (as always), by setting \(q^*: MR(q)=MC(q)\)
Solve for Saudi Arabia
$$P=\underbrace{200-3q_i}_{intercept}-3q_{sa}$$
Firms maximize profit (as always), by setting \(q^*: MR(q)=MC(q)\)
Solve for Saudi Arabia
$$MR_{sa}=200-3q_i-6q_{sa}$$
Solve for \(q^*\) for each firm (where \(MR(q)=MC(q))\), we derive each firm's reaction function or best response function to the other firm's output
Symmetric marginal costs and marginal revenues
Solve for \(q^*\) for each firm (where \(MR(q)=MC(q))\), we derive each firm's reaction function or best response function to the other firm's output
Symmetric marginal costs and marginal revenues
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Saudi Arabias's reaction curve to Irans's output
We can graph Iran's reaction curve to Saudi Arabia's output
We can graph Iran's reaction curve to Saudi Arabia's output
We can graph Iran's reaction curve to Saudi Arabia's output
Combine both curves on the same graph
Cournot-Nash Equilibrium: $$\big( \color{red}{20}, \color{blue}{20} \big)$$
Both are playing mutual best response to one another
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
$$\begin{align*} q_{sa}^*&=30-0.5q_i\\ q_i^*&=30-0.5q_{sa}\\ \end{align*}$$
$$P=200-3q_{sa}-3q_i$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} P&=200-3(20)-3(20)\\ P&=\$80\\ \end{align*}$$
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=20(80-20)\\ \pi_{sa}&=1,200\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} P&=200-3(30)\\ P&=\$110\\ \end{align*}$$
$$\begin{align*} MR&=MC\\ 200-6Q&=20\\ 30&=Q^*\\ \end{align*}$$
$$\begin{align*} P&=200-3(30)\\ P&=\$110\\ \end{align*}$$
$$\Pi=30(110-20)=\$2,700$$
with $1,400 going to each firm
Cournot Competition: each firm produces 20 and earns $1,200
Cournot Collusion: each firm produces 15 and earns $1,400
Cournot Competition: each firm produces 20 and earns $1,200
Cournot Collusion: each firm produces 15 and earns $1,400
But is collusion a Nash equilibrium?
Read either firm's reaction curve at the collusive outcome
Suppose Saudi Arabia knows Iran is producing 15 (as per the cartel agreement)
Saudi Arabia's best response to Iran's 15 is to produce 22.5
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=22.5(87.50-20)\\ \pi_{sa}&=\$1,518.75\\ \end{align*}$$
$$\begin{align*} \pi_{sa}&=q_{sa}(P-c)\\ \pi_{sa}&=22.5(87.50-20)\\ \pi_{sa}&=\$1,518.75\\ \end{align*}$$
$$\begin{align*} \pi_{i}&=q_{i}(P-c)\\ \pi_{i}&=15(87.50-20)\\ \pi_{i}&=\$712.50\\ \end{align*}$$
Imagine Bertrand competition between Saudi Arabia and Iran instead (price competition)
Nash equilibrium: Firms will set \(P=MC\), so:
$$\begin{align*} P&=MC\\ 200-3Q&=20\\ Q&=60\\ \end{align*}$$
Both countries split demand equally, each selling 30 units
Profit for both countries would be 0, since \(P=MC\)
Type | Output | Price | Profits |
---|---|---|---|
Collusion | 30 | $110 | $2,700 |
Cournot | 40 | $80 | $2,400 |
Bertrand | 60 | $20 | $0 |
Type | Output | Price | Profits |
---|---|---|---|
Collusion | 30 | $110 | $2,700 |
Cournot | 40 | $80 | $2,400 |
Bertrand | 60 | $20 | $0 |
Where subscript \(m\) is monopoly (collusion), \(c\) is Cournot, \(b\) is Bertrand
Example: Suppose Firm 1 and Firm 2 have a constant \(MC=AC=8\). The market (inverse) demand curve is given by:
$$\begin{align*} P&=200-2Q\\ Q&=q_1+q_2\\ \end{align*}$$
Find the Cournot-Nash equilibrium output and profit for each firm.
Find the output and profit for each firm if the two were to collude.
Find the price and output if the two were to compete on price instead of quantity.
Antoine Augustin Cournot
1801-1877
Cournot Theorem: as the number of firms \((N)\) in the market increases, market output \(Nq\) goes to the competitive level, and price converges to \(c\).
More (fewer) firms reduce (increase) market distortions from market power
Antoine Augustin Cournot
1801-1877
Major implications from Cournot:
As \(\uparrow\) number of firms: \(\uparrow Q\), \(\downarrow p\), \(\downarrow \pi\), \(\uparrow CS\), \(\downarrow DWL\) (closer to perfect competition)
If a firm has lower costs than others, it earns greater profit. Firms will want to lower their own costs OR raise rivals' costs (same effect)