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## Comparison of Cournot, Stackelberg and cartel duopoly

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In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

In the duopoly series of posts on here, I’ve used the same model to illustrate three different forms of duopoly competition when the firms are identical. We can now compare the different effects on market price, market quantity, profitability of each firm, amount of market power on each firm and welfare for consumers.

In the monopoly model:
Market price: 325
Market output: 175
Firm’s profits: 30625
Lerner index: 0.538
Consumer surplus: 15312.5
Producer surplus: 30625
Welfare: 45937.5

In the Cournot model:
Market price: 266.667
Market output: 233.333
Firm’s profits: 13611.11 each (combined: 27222.22)
Lerner index: 0.438
Consumer surplus: 27222.222
Producer surplus: 27222.222
Welfare: 54444.444

In the Stackelberg model:
Market price: 237.5
Market output: 262.5
Firm’s profits: leader: 15312.5, follower: 7656.25(combined: 22968.75)
Lerner index: 0.368
Consumer surplus: 34453.125
Producer surplus: 22986.75
Welfare: 57439.88

In the cartel model (where neither cheats):
Market price: 325
Market output: 175
Firm’s profits: 15312.5 each (combined: 30625)
Lerner index: 0.538
Consumer surplus: 15312.5
Producer surplus: 30625
Welfare: 45937.5

Consumer surplus:

Producer surplus:

## The duopoly cartel

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

The previous two models of competition, Cournot and Stackelberg, looked at different ways in which firms can compete with each other strategically by setting their quantity decisions.

Now we can look at the case where the firms decide to collude rather than compete, and form a cartel.

If the firms can trust each other, they can come together and behave like a joint monopoly firm, and just split the output.

When firm A had a monopoly, it produced output of 175 and sold at price of 325.

So if firms A and B come together, they can decide to each produce 87.5, so the market output is 175 and the price is 325. This means each firm gets profits of 325(87.5) – 150(87.5) = 15312.5.

However this arrangement relies on trust, because the cartel output is not on either firm’s best response curve. If firm A knows that firm B is going to produce 87.5, then given the best response function for firm A is $Q_A = 175-0.5Q_B$, the best response for firm A would be 131.25. As the firms are identical the same holds for firm B – if B knows A is producing 87.5, then the best response for B is to produce 131.25.

So what happens if one of them cheats and produces 131.25? Then the market output is 219 and the market price is 500-219 = 281. So the cheat will get profits of 281(131.25) – 150(131.25) = 17193.75, which is an improvement on the cartel equilibrium of 15312.5. The one that didn’t cheat will get profits of 281(87.5) – 150(87.5) = 11462.5, so they lose out.

But if they both cheat and produce 131.25, then the market output is 262.5 and the market price is 237.5, so they will both get profits of 237.5(131.25) – 150(131.25) = 11484.38. They would both have been better off by not cheating. So the cartel relies on trust!

## Stackelberg competition

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

We established that the best response curve for firm A is $Q_A = 175 - 0.5 Q_B$ and the best response curve for firm B is $Q_B = 175 - 0.5 Q_A$.

When the firms had no knowledge of each other’s output making decision, and made their decisions simultaneously, they reached a Cournot equilibrium where each firm produced 116.667.

The total market output was 233.333, and the market price was 266.667. Each firm made profits of 13611.11, and had a Lerner index of 0.438.

Now we can consider the situation where the firms did not make their decisions simultaneously, but one firm effectively ‘moved first’ – the other firm knew what output the first firm was producing, when it came to make its own decision. This gives the first firm an advantage, because now if the second firm is going to respond by choosing the appropriate point on its best response curve, it has no choice but to choose the output that corresponds to the output chosen by the first firm.

This form of competition is called Stackelberg competition and the firm moving first is called the Stackelberg leader. Again we assume that the firms are producing identical goods and are competing with each other based on their quantity decisions. We will continue to use the case where the firms are identical to allow us to compare this more easily with the Cournot example, so both firm A and firm B have marginal costs of 150. But now we will assume firm A is the first mover, so firm B is entering the market but has to respond to the output level firm A is producing.

If A knows B is entering then it can take advantage of being first mover by choosing to produced the Stackelberg leader output. To do this we find a profit function for firm A (the leader firm) in terms of the quantity it produces and the quantity firm B produces, and we optimise it.

The inverse market demand function is $P= 500 - Q$ and as $Q = Q_A + Q_B$ we can write it $P = 500 - (Q_A + Q_B)$.

Because firm A, the Stackelberg leader, chooses its output first, then firm B has to respond by choosing its output based on the decision of A. The best response function for firm B is $Q_B = 175 - 0.5 Q_A$ so we know this is how B’s output will relate to A’s output. Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 175 - 0.5 Q_A) \Rightarrow P = 325 - 0.5Q_A$.

Now we think about the profit function for firm A. This will be $\pi_A = TR_A - TC_A = PQ_A - (MC_A)Q_A = (P-MC_A)Q_A$. As we have an expression for the market price, and we know the marginal cost is 150, the profit function for firm A becomes $\pi_A = (325 - 0.5Q_A - 150)Q_A \Rightarrow \pi_A = 175Q_A-0.5{Q_A}^2$.

Now to optimise the profits of firm A we differentiate the function with respect to the output of firm A and set that equal to 0, $\frac{d\pi_A}{dQ_A} = 175-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 175$. This means the Stackelberg leader output for firm A is 175.

If A produces 175, then B will produce $Q_B = 175 - 0.5 (175) = 87.5$.

So with A producing 175 and B producing 87.5, market output is 262.5 and market price is 500-262.5=237.5. Firm A will get profits of 237.5(175) – 150(175) = 15312.5, and firm B will have profits of 237.5(87.5) – 150(87.5) = 7656.25. Both firms have a Lerner index of 0.368.

Again we can look at the case of what would happen if the firms were not identical and had different marginal costs. First consider the case where the new entrant, firm B, was more efficient than the incumbent firm, firm A, and whereas firm A had a marginal cost of 150, firm B had a marginal cost of 120. On the Cournot section I found the best response curve for firm B here to be $Q_B = 190 - 0.5 Q_A$.

Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 190 - 0.5 Q_A) \Rightarrow P = 310 - 0.5Q_A$.

The profit function for firm A will be $\pi_A = (P-MC_A)Q_A = (310 - 0.5Q_A - 150)Q_A =160 Q_A- 0.5{Q_A}^2$. Differentiating this we get $\frac{d\pi_A}{dQ_A} = 160-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 160$. The Stackelberg leader output for firm A this time is 160. If A produces 160, then B will produce $Q_B = 190 - 0.5 (160) = 110$. The total market output will be 270 so the market price will be 230.

Alternatively we could consider the case where it was firm A that was more efficient than firm B as well as being the Stackelberg leader. What if A had the marginal cost of 120 and B had the marginal cost of 150. This time firm B’s best response curve would be $Q_B = 175 - 0.5 Q_A$ as before, because the best response curve depends on its own marginal cost. Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 175 - 0.5 Q_A) \Rightarrow P = 325 - 0.5Q_A$.

Again the profit function for firm A will be $\pi_A = (P-MC_A)Q_A$, but this time the marginal cost for A is 120, so the profit function for firm A becomes $\pi_A = (325 - 0.5Q_A - 120)Q_A \Rightarrow \pi_A = 205Q_A-0.5{Q_A}^2$.

Differentiating this we get $\frac{d\pi_A}{dQ_A} = 205-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 205$. The Stackelberg leader output for firm A this time is 205. If A produces 160, then B will produce $Q_B = 175 - 0.5 (205) = 72.5$. The total market output will be 277.5 so the market price will be 222.5.

The Stackelberg leader is able to be more aggressive (produce a greater share of the market) when it has a lower marginal cost.

## Cournot competition

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

We established that the best response curve for firm A is $Q_A = 175 - 0.5 Q_B$

What about firm B? Again the inverse residual demand curve for firm B is $P= 500 - Q_A - Q_B$, so the total revenue is $TR_B = PQ_B = 500Q_B - {Q_B}^2 - {Q_A}{Q_B}$.

The marginal revenue is $MR_B = \frac{d(TR_B)}{dQ_B} = 500 - 2Q_B - Q_A$.

The profit maximising quantity of output is where MR = MC, so $500 - 2Q_B - Q_A = 150 \Rightarrow 2 Q_B = 350 - Q_A \Rightarrow Q_B = 175 - 0.5 Q_A$.

This is the best response function for firm B, so we can add this to the best response curve graph.

Previously we considered what the best response would be for firm A if firm B produced 200: firm A would be best to produce 75.

But if firm A was to produce 75, then it is no longer optimal for firm B to produce 200. If firm A produces 75, then the best response for firm B is to produce $Q_B = 175 - 0.5 (75) = 137.5$.

So if firm B thinks that firm A is basing its decision on B producing 200, then firm A is going to produce 75. Knowing this, firm B will be better to actually produce 137.5.

But if firm B was to produce 137.5, then it is no longer optimal for firm A to produce 75! If B produces 137.5, then A should produce $Q_A = 175 - 0.5 (137.5) = 106.25$.

This kind of decision making and guesswork can go on and on, this is strategic thinking. You are trying to guess what your rival will do, and choosing the optimal response to that, but you are taking into account the fact that your rival is basing its decision on what it thinks your firm will do, and choosing its optimal response to that.

Where will it end? The answer is when you reach a Nash equilibrium, which is a position where neither firm wants to change its decision based on the output decision of the other. So you need to find a point where both firms are happy to ‘stick’ based on the output they expect from the other.

In this duopoly model, where both firms are identical and are competing with each other based on the quantity of output they produce, the Nash equilibrium we are looking for is called a Cournot equilibrium. We are assuming here that neither firm knows the output decision of the other when it makes its own decision – it is just guessing, and both firms are making their decision simultaneously, one does not announce its output decision before the other.

Notice how on the graphs, each time the firms change their decision based on what they think the other will do, their output decisions approach the point where the two best response curves intersect. This is a clue as to how to find the answer – it is at the intersection point of the curves. So you can solve this by setting the best response curves equal to each other.

We had the best response curve for firm A: $Q_A = 175 - 0.5 Q_B$, and the best response curve for firm B: $Q_B = 175 - 0.5 Q_A$, so we can just substitute one into the other:

$Q_A = 175 - 0.5 (175 - 0.5 Q_A) \Rightarrow Q_A = 175 - 87.5 + 0.25Q_A \Rightarrow 0.75Q_A = 87.5 \Rightarrow Q_A = 116.667$. So firm A will produce 116.667.

Substituting this back into the best response function for firm B, $Q_B = 175 - 0.5 (116.667) \Rightarrow Q_B = 116.667$. So firm B will also produce 116.667.

Now we have reached a Cournot equilibrium. If firm A produces 116.667 then firm B’s best response is to also produce 116.667. And if firm B produces 116.667 then firm A’s best response is to also produce 116.667. No other combination will be on the best response curves of both firms, so this is the Cournot equilibrium.

In the monopoly form of the model, the firm was producing output of 175, selling at a price of 325. It was making profits of 30625 and had a Lerner Index of 0.538.

Now in the Cournot duopoly, each firm produces output of 116.667 which means the total market output is 233.333, and the market price is 500-233.333 = 266.667. So the price to consumers has fallen. Each firm is making profits of 266.667(116.667) – 150(116.667) = 13611.11, and has a Lerner Index of $\frac{266.667-150}{266.667}$ 0.438 showing that the amount of market power has dropped.

What would happen if the firms were not identical and had different marginal costs? Consider the case where the new entrant, firm B, was more efficient than the incumbent firm, firm A, and whereas firm A had a marginal cost of 150, firm B had a marginal cost of 120.

This time, firm B’s best response curve would be different. The marginal revenue is $MR_B = \frac{d(TR_B)}{dQ_B} 500 - 2Q_B - Q_A$, so for firm B to produce at the profit maximising point where MR = MC, then $500 - 2Q_B - Q_A = 120 \Rightarrow 2Q_B = 380 - Q_A \Rightarrow Q_B = 190 - 0.5 Q_A$.

Firm A’s best response curve is unchanged as the marginal cost is still 150 so setting MR = MC will give us the same result. But this time, to find the Cournot equilibrium we solve these simultaneous equations:
Best response curve for firm A: $Q_A = 175 - 0.5 Q_B$, and best response curve for firm B: $Q_B = 190 - 0.5 Q_A$, so we substitute one into the other:

$Q_A = 175 - 0.5 (190 - 0.5 Q_A) \Rightarrow Q_A = 175 - 95 + 0.25Q_A \Rightarrow 0.75Q_A = 80 \Rightarrow Q_A = 106.667$. This time it is best for firm A to produce a lower amount, 106.667.

And if firm A produces 106.667 then firm B should produce $Q_B = 190 - 0.5 (106.667) = 136.667$. This time it is best for firm A to produce a higher amount, 136.667.

Firm B can produce more than firm A in the new Cournot equilibrium because it has a lower marginal cost – it is a more competitive producer.

With firm A producing 106.667 and firm B producing 136.667 the total market output is 243.333 and so the market price is 500-243.333 = 256.667. Firm A will produce profits of 11377.778 and have a Lerner index of 0.416 and firm B will produce profits of 18677.778 and have a Lerner index of 0.532.

## Best response curves

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

Given that we know each firm will want to produce at the profit maximising point, where MR = MC, we can use these equations to find out an expression for the profit maximising output for each term in terms of the quantity produced by the other – a best response function.

For firm A, the inverse demand function is $P= 500 - Q_A - Q_B$.

So the total revenue is $TR_A = PQ_A = 500Q_A - {Q_A}^2 - {Q_B}{Q_A}$.

The marginal revenue is $MR_A = \frac{d(TR_A)}{dQ} = 500 - 2Q_A - Q_B$.

The profit maximising quantity of output is where MR = MC, so $500 - 2Q_A - Q_B = 150 \Rightarrow 2 Q_A = 350 - Q_B \Rightarrow Q_A = 175 - 0.5 Q_B$.

This is the best response function for firm A.

This graph has the quantity produced by firm A on the horizontal axis and the quantity produced by firm B on the vertical axis. From this curve you can see what the optimum amount for firm A to produce is, when the amount produced by firm B is at a particular level.

For instance if firm B produces 200, what would be the optimum for firm A to produce?

The best response function is $Q_A = 175 - 0.5 Q_B$ so that means $Q_A = 175 - 0.5 (200) = 75$.

If firm B produces output of 200, then firm A’s best response is to produce output of 75.

## Duopoly and residual demand

The model of a monopoly firm I made had a demand function of Q = 500 – P , no fixed cost and a constant marginal cost of 150.

The firm was producing output of 175, selling at a price of 325. It was making profits of 30625 and had a Lerner Index of 0.538.

Now what would happen if another firm entered the market? First consider this being an identical firm, again with no fixed cost and a constant marginal cost of 150, and assume that the two firms could not differentiate their product in any way, they were producing identical products. The market has now gone from monopoly to duopoly.

Call the first firm (the model firm) firm A and the new rival firm B.

Now that there is another firm in the market, firm A does not have a monopoly on the whole market demand curve of Q = 500 – P. Instead it faces a residual demand curve. The residual demand curve is the market demand curve minus the quantity supplied by other firms, we can write this $D^r (P) = D (P) - S^o (P)$. The P in brackets indicates that the quantities are functions of price, like the original demand curve.

So in this model, firm A now faces a demand curve of$Q_A = 500 - P - Q_B$. This is fairly logical. When there was no other firm supplying the market, it faced a demand curve of Q = 500 – P, but if another firm comes in and supplies 100 to the market, then firm A faces a demand curve of 400 – P.

Each firm’s residual demand curve depends on the output decision of the other – or if they do not know the output decision of the other, the amount they expect the other to produce. Each firm will need to have a good estimate of what its residual demand curve will be in order to choose the correct quantity of output to produce, as the firm will want to produce the quantity where MR = MC.

Consider what would happen if firm B entered the market unknown to firm A, and produced 175 just like firm A did. Now with both firms producing 175, the total amount of output supplied to the market is 350, so the market price falls to 500-350 = 150.

On this graph the duopoly market price, Pd, is 150, which is just the same as each firm’s marginal cost. The quantity each firm is producing (QA and QB) is 175, and the duopoly market quantity, Qd, is 350. As each firm is selling at a price equal to its marginal cost this is like a competitive market, each firm is breaking even, neither firm is making a profit. This is clearly not an optimal choice for either firm.

The key concept to duopoly strategic behaviour (which can be extended to strategic behaviour with any number of firms) is that the output decision of each firm depends on the output decision it expects from its rival firm.

If both firms know that there is another firm in the market, then they need a framework for thinking about what is the optimal output to produce bearing in mind the decision of the other. This is where best response curves come in.