Applied Industrial Economics Topic 2: Oligopoly (continuation) Juan A. Maez A. Sanchis 2.1.3 Cournot model: comparison (I) Comparison to monopoly and perfect competition Monopoly Let us assume q2 =0 Firm 1 is a monopolist Perfect competition p CMg c Max (a bq1)q1 cq1 p CMg c dp F.O.C. a 2bQ c 0 dQ a bQ c Q PC q1 Q M

M 1 q a c a c y pM 2b 2 1 (a c )2 4 b M PC 0 a c b 2.1.3 Cournot model: comparison (II) Duopoly versus monopoly:

Cournot duopoly production is higher than the monopoly production Cournot dupoloy price is lower than the monopoly price Duopoly versus perfect competition: Cournot duopoly production production is smaller than monopoly Cournot duopoly price is higher than perfect competition price 2.1.3 Cournot model: graphic analysis

q2 a c b a c 2b q1* q 2 q1* (q2) C q a c 3b * 2 q1 q2* (q1) a c 3b a c 2b

a c b q1 a c bq 2 2b a c bq1 2b 2.1.4 Cournot model: interpreteing the Cournot equilibrium (I) Cournots model is a static game in which firms simultaneously choose quantities. Just to understand how this equilibrium appears, we will consider (even if it si not correct), the Cournot model as dynamic model in which: Firm 1 chooses quantity odd periods Firm 2 chooses quantity even periods q2 q1* (q2)

C q20 q2* (q1) q1 2.1.5 Cournot model: N firms (I) Assumptions Each firm chooses each production level taken as given other firms production levels null conjectural variations in quantities Market price depend on the level of production of the N firms in the market. The profit of each firm depends not only on their own production but also on the production of the other firms strategic interdependence. Inverse demand function and cost function N p Q a bQ a b qi i 1 C qi cqi 2.1.5 Cournot model: N firms (II) Firm 1 maximization problem Max 1 p (Q )q1 cq1 q1

C.P.O. d dp dQ p q1 c 0 dq1 dQ dq1 a c 1 n q1 qi 2b 2 i 2 dp b dQ n d qi dQ dq1 i 2 1 dq1 dq1 dq1 2.1.5 Cournot model: N firms (III) Under symmetry qi = qN "i a c 1 n

q1 qi 2b 2 i 2 qN n 1 N a c q 2 2b qN Market production and price: QN n a c n 1 b p N a bQ N 1 n a c n 1 n 1 1 a c n 1 b

2.1.5 Cournot model: N firms (IV) Equilibrium properties: 1 n 1 n Lim p N Lim a c Lim a Lim c n n n 1 n n n 1 n 1 n 1 n Indetermination we apply LHopital 0 Lim c n n 1 theorem Lim p N c

n Limq N Lim n n 1 a c 0 n 1 b 2.1.5 Cournot model: Relationship between structure and results (I) Inverse demand function and cost function N p Q a bQ a b qi Firm i problem i 1 C i qi c i qi Max i p (Q )qi C i (qi ) qi

F.O.C. d dp dQ p qi MgCi 0 dqi dQ dqi p MgC i qi dp dQ 2.1.5 Cournot model: Relationship between structure and results (II) Multiplying and dividing the RHS of the equation by p and Q : dp q dp Q p MgC i qi p MgCi i p dQ Q dQ p qi s i Q p MgC si

i If we define p dQ p dp Q 1. Perfect competition 2. Monopoly Lerners index for one firm 2.1.5 Cournot model: Relationship between structure and results (III) Market Lerners index We use as starting point the one-firm Lerners index: p MgCi si p

Multiplying by s i 1 MgCi si p MgCi si si2 si p Summing for i n n si i 1 MgC s i i 1 p n

s i i 1 n 2 i MgC s i 1 i 1 p i H

2.1.5 Cournot model: Relationship between structure and results (IV) n MgCi si H 1 p Cowling-Waterson equation: the market share weighted price-cost margin is positively related to the concentration ratio and negatively related to the price elasticity of demand i 1 p n MgC s i i 1

p i H Market structure (H, h) Firms conduct (profit maximization under the assumption of null conjectural variations) Performance (MPC) 2.1.5 Cournot model: Relationship between structure and results (IV) Simplification of the C-W equation under symmetry : We assume that MgCi=MgC for all i p n

MgC s i i 1 p i H p MgC 1 p n 2.2 Stackelberg Model The Cournots model represent a symmetryc situation as both firms have identical conjectural variations both firms behave in the same way dqi 0"i j dq j

The Stackelbergs model represents an asymmetric situation: Leader (firm 1): it choose its optimal level of profits (the one tha maximize its profits) taking into account its rivals (firm 2) reaction function (empresa 2) Follower (firm 2): null conjectural variation (as in the Cournot model) firm 2 decides its optimal production level taken as given firm 1 production level Inverse demand funciton and cost function are as follows: p Q a bQ a b q1 q2 C i cqi 2.2.1 Stackelberg model: equilibrium (I) Leaders problem Max 1(q1 ,q2 ) p q1 ,q2 q1 cq1 q1 s .a. q2* (q1)

a c q1 2b 2 Max 1 q1 ,q2* q1 p q1 ,q2* q1 q1 cq1 q1 a c q1 q1 a b q1 cq1 2 b 2 F.O.C.

a c bq1 0 2 q1L a c 2b 2.2.1 Stackelberg model: equilibrium (II) Followers problem Substituting into the followers (firm 2) reaction function we obtain its optimal level of production L a c q a c 1 a c q2S 1 2b 2

2b 2 2b q2F a c 4b q1L q2F With identical linear demand fucntion and identical and constant marginal costs, the follower produces just half the quantity produced by the leader 2.2.2 Stackelberg model: comparison Price and production comparison: various models p Q a bQ a b q1 q2 Leader Follower q1 q2 Q P Cournot

C i cqi Stackelberg Monopoly 1a c 3 b 1a c 2 b 1a c 3 b 2a c 3 b 1a c 4 b 3a c 4 b a 2c 3 a 3c 4 PC 1a c 2 b

0 1a c 2 b 1 (a c ) 2 a c b c 2.2.3 Stackelberg model: isoprofit curves analysis (I) Firm 1 isoprofit curves : Combinations of de q1 y q2 that report the same level of profits to firm1. How can we obtain firm 1 isoprofit curves? 1 a bq1 bq2 q1 cq1 We set 1 1 and then solve out for q2as a function of q1 1 aq1 bq12 bq2q1 cq1

Solving out for q2 1 1 cq1 q2 a q1 b q1 2.2.3 Stackelberg model: isoprofit curves analysis (II) 1 1q1 1 c a q1 b In order to know the shape of the function, we proceed to calculate the first and second derivative with respec to q1 dq2 1 1q1 2 1 dq1 b q2 d 2q2 2 21 3 q 0 1 1

2 3 b dq1 bq1 Therefore, firm 1 isoprofit curves are concave with respet to the q1 axis (in the same way, firm 2 isoprofit curves are concave with respect to the q2 axis) 2.2.3 Stackelberg model: isoprofit curves analysis (III) Properties of the isoprofits curves (firm 1) 2.2.3 Stackelberg model: isoprofit curves analysis (IV) q2 q2 q1* q2 q1* q2 13 q2 13 12 q2

1 1 q10 M 1 12 q1 11 q1No q10 1M q1 2.2.3 2.2.3 Stackelberg model: isoprofit curves analysis (V) q2 q1* (q2 ) S2 C

2 q2c q C1 C S S 2 1S q2* (q1 ) c 1 q q1S q1 2.2.3 Stackelberg model: isoprofir curves analysis (VI) Equilibrium in the Cournots and Stackelbergs models:

The equilibrium in the Cournots model occurs in the intersection (crossing) between the reaction functions of the two firms in this point, there is also a crossing between the isoprofit curves of the both firms. In the Stackelberg model, the leader (firm 1) knows the followers (firm 2) reaction function it will choose the point on the followers reaction function that makes its profits maximum lower isoprofit curve of firm 1 that is compatible with a tangency between this isoprofit curve and firm 2 reaction function. 1L F2 1L C1 C2 F2 2.2.3 Stackelberg model: Who does decide the leader? The leader can have cost (technological) advantages more efficient firms The leader can detent a prestige and have loyal custom national brands over private labels Asymmetric information on the demand side that favors the leader Mercadona en Valencia versus Caprabo 2.3 Collusion (I) Firms agree to maximize profits jointly. In fact, they act as if they were a monopoly : Assumptions

n firms in a market reach a collusive agreement (they form a cartel) for joint maximization of the industry profits. Costs C (q )=cq "i and industry total costs are: i i i n n n i 1 i 1 i 1 C (Q ) Ci (qi ) cqi c qi cQ 2.3 Collusion (II) The n firms of the industry maximize profits jointly: Max p (Q )Q cQ Q

C.P.O. d dp p Q c 0 dQ dQ Q dp p 1 c p dQ Let us define the price elasticity of demand as: =- dQ p dp Q p c 1 p

Lerner Index 2.3 Collusion (II):duopoly Using the duopoply case as in former models, and with inverse demand and costs function given by: p Q a bQ a b q1 q2 Collusive behavior implies: C i cqi Max 1 2 p (Q )q1 cq1 p (Q )q2 cq2 q1 ,q2 p (Q ) q1 q2 c q1 q2 F.O.C. d a 2bq1 2bq2 c 0 dq1 F.O.C. d a 2bq2 2bq1 c 0 dq2

2.3 Collusion (III): duopoly The symmetric equilibrium implies: q1 = q2.= q. Therefore from any one of the two FOCs a 4bq c 0 q q1 q2 a c 4b Total industry production and market price are: Q q1 q2 a c 2b p a b (q1 q2 ) a c 2 2.3 Collusion (IV): comparison Price and product comparison for the different models p Q a bQ a b q1 q2

Leader Follower q1 q2 Q P Cournot C i cqi Stackelberg Collusion 1a c 3 b 1a c 2 b 1a c 3 b 2a c 3 b 1a c 4 b 3a c

4 b a 2c 3 a 3c 4 1a c 4 b 1a c 4 b 1a c 2 b 1 (a c ) 2 2.3 Collusin (V): isoprofit curves analysis q2 q1* (q2 ) Col C 2 S2 2

C1 q2c q C S 2 q2Col S COL 1S Col 1 q2* (q1 ) Col 1 q c 1 q q1S

q1 2.4 Bertrand model Assumptions Two firms Homogeneous product there is no product differentiation each firm sells to the whole market undercutting its rival by an infinitesimal greater than 0 The two firms choose price simultaneously: MgC = MgC =c 1 2 Demand rules: All consumers buy from the firm setting the lowest price If both firms set the same price, firms share market demand evenly (every firm sells half of the market) If both firms sell an identical product and costs are equal for both firms equilibrium price must be equal for both firms. Crucial assumption in the Bertrand model: conjectural

variations in price is zero : 2.4.1. Modelo de Bertrand: firm 1 demand obtention Let us assume that firm 2 price is given and it is set to p2 P1 0 si p1 p2 D p1 q1 si p1 p2 2 D p1 si p1 p2 P2 q1 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (I) 1. If c

p1 c D p1 si p1 p2 Which is firm 1 optimal reaction? 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (II) 2. If p2 > pM 3. Si p2 < c Therefore, firm 1 reaction functions is given by: p1* p2 p M si p2 p M p2 - si c p2 p M c si p c 2 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (III) RF1 P2 PM p1* (p2 ) RF2

p1* (p2 ) p1* p2 c B c PM P1 p M si p2 p M p2 - si c p2 p M c si p c 2 2.4.2. 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (IV) If we assume (such as we did) that both firms can freely access the same technology (they have identical cost functions) then firm 1 reaction function is symmetrical to firm 2 reaction function Bertrand equilibrium is given by the crossing of both reaction functions p1=p2=c price competition between just two firms results in marginal cost price setting as in perfect competition.

Bertrand equilibrium is a Nash equilibrium in prices, no firm has an incentive to deviate from pi = c 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (V) How do we reach the equilibrium? Let us assume to start that both firms set a high price Firms have null cojectural variations dpi 0 Each firm assumes that the other firm does not dp j change its price when it changes its own price Thus, each firm expect that undercutting (infinitesimally) its rival price it will get all the market each firm will undercut its rival price up to the point in which p = c as in the perfect competition equilibrium 2.4.2. Modelo de Bertrand: reaction functions and Nash equilibrium (VI) Bertrand paradox:

2.4 Cournot vs. Bertrand (I) Even if we assume identical cost functions Cournot and Bertrand model predictions are different: Cournot: MgC = pCP < pC < pM Bertrand MgC = p CP = pB < pM Empirical evidence shows that depending on the characteristics of the industries we can find both results An interesting way to make compatible both models is to consider a two-stage game in which firms have to choose capacity and price: : Stage 1: Log run choice (LR) Stage 2: Short run choice (CP) 2.4 Cournot vs. Bertrand (II) The relative sequence of choices is crucial to determine which model to choose (which model is the more appropriate model) :

If capacity is more difficult to adjust than price Long run choice: capacity/output Short run choice: price Cournot model is the appropriate model. Example: cement, car, steel If price is more difficult to adjust than capacity (capacity and production can be easily adapted) Long run choice : price Short run choice: capacity/output Bertrand model is the appropriate model. Example: insurance, banking, software 2.4 Cournot vs. Bertrand (III) Although static model have some problems, they are useful as they allow us model in an easy form a complicated phenomenon (intra-industry firm competition). In fact, using these model we can answer questions such as: which is the effect of the number of firms on market price? However, these models are not able to explain questions as :

Why in some very concentrated oligopolistic markets (as the US cigarettes industry until the early 1990) were firms able to mantain high prices without reaching explicit collusive agreements? Why in other concentrated oligopolistic markets (as the regional cement markets price competition is very intense (tough)? 2.4 Cournot vs. Bertrand (IV) In order to analyze these question we have to use dynamic models in which firms interact repeatedly (in consecutive periods)