3 2 Cournot Model Matilde Machado 1 3

3. 2. Cournot Model Matilde Machado 1

3. 2. Cournot Model Assumptions: n All firms produce an homogenous product n The market price is therefore the result of the total supply (same price for all firms) n Firms decide simultaneously how much to produce n Quantity is the strategic variable. If OPEC was not a cartel, then oil extraction would be a good example of Cournot competition. ? The equilibrium concept used is Nash Equilibrium (Cournot-Nash) Agricultural products? http: //www. iser. osaka-u. ac. jp/library/dp/2010/DP 0766. pdf n Industrial Economics- Matilde Machado 3. 2. Cournot Model 2

3. 2. Cournot Model Graphically: n Let’s assume the duopoly case (n=2) n MC=c n Residual demand of firm 1: RD 1(p, q 2)=D(p)-q 2. The problem of the firm with residual demand RD is similar to the monopolist’s. Industrial Economics- Matilde Machado 3. 2. Cournot Model 3

3. 2. Cournot Model Graphically (cont. ): P p* MC D(p) RD 1(q 2) = Residual demand q*1= q 2 R 1(q 2) MR Industrial Economics- Matilde Machado 3. 2. Cournot Model 4

3. 2. Cournot Model Graphically (cont. ): q*1(q 2)=R 1(q 2) is the optimal quantity as a function of q 2 Let’s take 2 extreme cases q 2: Case I: q 2=0 RD 1(p, 0)=D(p) whole demand Firm 1 should the q*1(0)=q. M produce Monopolist’ s quantity Industrial Economics- Matilde Machado 3. 2. Cournot Model 5

3. 2. Cournot Model Case 2: q 2=qc RD 1(p, qc)=D(p)-qc D(p) c Residual Demand qc c D( p) MR<MC q*1=0 qc MR Industrial Economics- Matilde Machado 3. 2. Cournot Model 6

3. 2. Cournot Model Note: If both demand cost functions are linear, reaction function will be linear as well. q 1 Reaction function of firm 1 q. M q* 1 (q 2) qc Industrial Economics- Matilde Machado 3. 2. Cournot Model q 2 7

3. 2. Cournot Model If firms are symmetric then the equilibrium is in the 45º line, the reaction curves are symmetric and q*1=q*2 q 1 qc ) (q 1 q* 2 q. M q*1 q 1=q 2 E q* 1 (q 2) 45º q*2 Industrial Economics- Matilde Machado q. M 3. 2. Cournot Model qc q 2 8

3. 2. Cournot Model Comparison between Cournot, Monopoly and Perfect Competition q. M<q. N<qc q 1 qc ) (q 1 q* 2 q 1+q 2=q. N q. M q 1+q 2=qc q* 1 (q 2) q 1+q 2=q. M Industrial Economics- Matilde Machado q. M q 1+q 2=q. N qc 3. 2. Cournot Model q 2 9

3. 2. Cournot Model Derivation of the Cournot Equilibrium for n=2 Takes the strategy of P=a-b. Q=a-b(q 1+q 2) firm 2 as given, i. e. takes q 2 as a constant. Note MC 1=MC 2=c the residual demand For firm 1: here Industrial Economics- Matilde Machado 3. 2. Cournot Model Reaction function of firm 1: optimal quantity firm 1 should produce given q 2. If q 2 changes, q 1 changes as well. 10

3. 2. Cournot Model We solve a similar problem for firm 2 and obtain a system of 2 equations and 2 variables. If firms are symmetric, then Industrial Economics- Matilde Machado 3. 2. Cournot Model Solution of the Symmetric equilibrium 11

3. 2. Cournot Model Solution of the Symmetric equilibrium Industrial Economics- Matilde Machado 3. 2. Cournot Model 12

3. 2. Cournot Model Comparing with Monopoly and Perfect Competition Where we obtain that: In perfect competition prices increase 1 -to-1 with costs. Industrial Economics- Matilde Machado 3. 2. Cournot Model 13

3. 2. Cournot Model In the Case of n 2 firms: If all firms are symmetric: Industrial Economics- Matilde Machado 3. 2. Cournot Model 14

3. 2. Cournot Model Total quantity and the equilibrium price are: If the number of firms in the oligopoly converges to ∞, the Nash-Cournot equilibrium converges to perfect competition. The model is, therefore, robust since with n→ ∞ the conditions of the model coincide with those of the perfect competition. Industrial Economics- Matilde Machado 3. 2. Cournot Model 15

3. 2. Cournot Model DWL in the Cournot model = area where the willingness to pay is higher than MC p. N DWL c QN Industrial Economics- Matilde Machado 3. 2. Cournot Model qc When the number of firms converges to infinity, the DWL converges to zero, which is the same as in Perfect Competition. The DWL decreases faster than either price or quantity (rate of n 2) 16

3. 2. Cournot Model In the Asymmetric duopoly case with constant marginal costs. The FOC (from where we derive the reaction functions): Replace q 2 in the reaction function of firm 1 and solve for q 1 Industrial Economics- Matilde Machado 3. 2. Cournot Model 17

3. 2. Cournot Model In the Asymmetric duopoly case with constant marginal costs. Which we replace back in q 2: Industrial Economics- Matilde Machado 3. 2. Cournot Model 18

3. 2. Cournot Model From the equilibrium quantities we may conclude that: If c 1<c 2 (i. e. firm 1 is more efficient): In Cournot, the firm with the largest market share is the most efficient Industrial Economics- Matilde Machado 3. 2. Cournot Model 19

3. 2. Cournot Model From the previous result, the more efficient firm is also the one with a larger price-Mcost margin: Industrial Economics- Matilde Machado 3. 2. Cournot Model 20

3. 2. Cournot Model Comparative Statics: The output of a firm ↓ when: ↑ own costs ↓ costs of rival q 2 Shifts the reaction curve of firm 1 to the left R 1 ↑c 1 E’ E ↑q*2 and ↓q*1 R 2 q 1 Industrial Economics- Matilde Machado 3. 2. Cournot Model 21

3. 2. Cournot Model Profits are: Increase with rival’s costs Decrease with own costs Symmetric to firm 2. Industrial Economics- Matilde Machado 3. 2. Cournot Model 22

3. 2. Cournot Model More generally… for any demand cost function. There is a negative externality between Cournot firms. Firms do not internalize the effect that an increase in the quantity they produce has on the other firms. That is when ↑qi the firm lowers the price to every firm in the market (note that the good is homogenous). From the point of view of the industry (i. e. of max the total profit) there will be excessive production. Externality: firms only take into account the effect of the price change in their own output. Then their output is higher than what would be optimal from the industry’s point of view. Industrial Economics- Matilde Machado 3. 2. Cournot Model 23

3. 2. Cournot Model If we define the Lerner index of the market as: Is the Herfindhal Concentration Index Industrial Economics- Matilde Machado 3. 2. Cournot Model 24

3. 2. Cournot Model The positive relationship between profitability and the Herfindhal Concentration Index under Cournot: Remember the FOC for each firm in that industry can be written as: The Industry-wide profits are then: The concentration index is up to a constant an exact measure of industry profitability. Industrial Economics- Matilde Machado 3. 2. Cournot Model 25

3. 2. Cournot Model Note: The Cournot model is often times criticized because in reality firms tend to choose prices not quantities. The answer to this criticism is that when the cournot model is modified to incorporate two periods, the first where firms choose capacity and the second where firms compete in prices. This two period model gives the same outcome as the simple Cournot model. Industrial Economics- Matilde Machado 3. 2. Cournot Model 26