Strategy for Information Markets/Background/Cournot competition

Cournot competition is an economic model used to describe industry structure. It so called after Antoine Augustin Cournot (1801-1877) after he observed competition in a spring water duopoly. It has the following features:

• There are two firms producing homogeneous products;
• Firms do not cooperate.
• Firms have market power;
• There are barriers to entry;
• Firms compete in quantities, and choose quantities simultaneously;
• There is strategic behaviour by the firms;

Price is a commonly known decreasing function of total output. All firms know N and take the output of the others as given. Each firm has a cost function ci(qi) (cost per unit multiply quantity). Normally the cost functions are treated as common knowledge. The cost functions are normally the same for all firms. The market price is set at a level such that demand equals the total quantity produced by both firms.

Graphically finding the Cournot duopoly equilibrium

p1 = firm 1 price, p2 = firm 2 price

q1 = firm 1 quantity, q2 = firm 2 quantity

c = marginal cost (assumed to be constant)

Equilibrium prices will be:

p1 = p2 = P(q1+q2)

This implies that firm i’s profit is given by $\Pi \ i=qi(P(q1+q2)-c)$

• Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity q2. What is firm 1s optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by P(0+q2)=P(q2). If firm 1 sets produces q1’ then price is given by P(q1’+q2). More generally, for each quantity that firm 1 might decide to set, price is given by the curve d1(q2). The curve d1(q2) is called firm 1’s residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of q2.

• Determine firm 1’s optimum output: To do this we must find where marginal revenue equals marginal cost. Marginal cost (c) is assumed to be constant. Marginal revenue is a curve - r1(q2) - with twice the slope of d1(q2) and with the same vertical intercept. The point at which the two curves intersect corresponds to quantity q1’’(q2). Firm 1’s optimum q1’’(q2), depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1’s optimum for other possible values of q2. Diagram 2 considers two possible values of q2. If q2=0, then firm firms residual demand is effectively the market demand, d1(0)=D. The optimal solution is for firm 1 to choose the monopoly quantity; q1’’(0)=q^m (q^m is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, q2=qc P(q^c)=c, then firm 1’s optimum would be to produce nil: q1’’(q^c)=0. This is the point at which marginal cost intercepts the marginal revenue corresponding to d1(q^c).

• It can be shown that, given the linear demand and constant marginal cost, the function q1’’(q2) is also linear. Because we have two points, we can draw the entire function q1’’(q2), see diagram 3. Note the axis of the graphs has changed, The function q1’’(q2) is firm 1’s reaction function, it gives firm 1’s optimal choice for each possible choice by firm 2. In other words, it gives firm 1’s choice given what it believes firm 2 is doing.

• The last stage in finding the Cournot equilibrium is to find firm 2’s reaction function. In this case it is symmetrical to firm 1’s as they have the same cost function. The equilibrium is the interception point of the reaction curves. See diagram 4.

• The prediction of the model is that the firms will choose Nash equilibrium output levels.

Calculating the equilibrium

In very general terms, let the price function for the (duopoly) industry be $P(q_{1}+q_{2})$ and firm i have the cost structure $C_{i}(q_{i})$. To calculate the Nash equilibrium, the best response functions of the firms must first be calculated.

The profit of firm i is revenue less cost. Revenue is the product of price and quantity and cost is given by the firm's cost structure, so profit is (as described above): $\Pi \ i=P(q_{1}+q_{2}).q_{i}-C_{i}(q_{i})$. The best response is to find the value of $q_{i}$ that maximises $\Pi \ i$ given $q_{j}$, with $i\neq \ j$, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of $\Pi \ i$ with respect to $q_{i}$ is to be found. First derive $\Pi \ i$ with respect to $q_{i}$:

${\frac {\partial \Pi \ i}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}.qi+P(q1+q2)-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}$

Setting this to zero for maximisation:

${\frac {\partial \Pi \ i}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}.qi+P(q1+q2)-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}=0$

The values of $q_{i}$ that satisfy this equation are the best responses. The Nash equilibria are where both $q_{1}$ and $q_{2}$ are best responses given those values of $q_{1}$ and $q_{2}$.

si===An example===

Suppose the industry has the following price structure: $P(q_{1}+q_{2})=a-b(q_{1}+q_{2})$ The profit of firm i (with cost structure $C_{i}(q_{i})$ such that ${\frac {\partial ^{2}C_{i}(q_{i})}{\partial q_{i}^{2}}}=0$ and ${\frac {\partial C_{i}(q_{i})}{\partial q_{j}}}=0,j\neq \ i$ for ease of computation) is:

$\Pi \ i={\bigg (}a-b(q_{1}+q_{2}){\bigg )}.q_{i}-\partial C_{i}(q_{i})$

The maximisation problem resolves to (from the general case):

${\frac {\partial {\bigg (}a-b(q_{1}+q_{2}){\bigg )}}{\partial q_{i}}}.qi+a-b(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}=0$

Without loss of generality, consider firm 1's problem:

${\frac {\partial {\bigg (}a-b(q_{1}+q_{2}){\bigg )}}{\partial q_{1}}}.q1+a-b(q_{1}+q_{2})-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0$

$\Rightarrow \ -bq_{1}+a-b(q_{1}+q_{2})-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0$

$\Rightarrow \ q_{1}={\frac {a-bq_{2}-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}$

By symmetry:

$\Rightarrow \ q_{2}={\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}}$

These are the firms' best response functions. For any value of $q_{2}$, firm 1 responds best with any value of $q_{1}$ that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for $q_{2}$ in firm 1's best response:

$\ q_{1}={\frac {a-b({\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}})-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}$

$\Rightarrow \ q_{1}*={\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{3b}}$

$\Rightarrow \ q_{2}*={\frac {a+{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{3b}}$

The Nash equilibria are all $(q_{1}*,q_{2}*)$. This yields a market price of 5a/3.

Implications

• Output is greater with Cournot duopoly than monopoly, but lower than perfect competition.
• Price is lower with Cournot duopoly than monopoly, but not as low as with perfect competition.

Bertrand versus Cournot

• Although both models have similar assumptions, but both have very different implications.
• Bertrand predicts a duopoly is enough to push prices down to marginal cost level, meaning that duopoly will result in perfect competition.
• Neither model is ‘better’, it depends on the industry as to which is more accurate.
• If capacity and output can be easily changed, Bertrand is a better model of duopoly competition. Or, if output and capacity are difficult to adjust, then Cournot is generally a better model.