The first order profit Maximising Condition reformatted Profit function of firm i H1=R()-C(Q) where, R(Qi) is the revenue function and C(Qi) is the cost function of firm For profit maximisation the derivative of prof it is zero, which amounts to saying that the derivative of revenue is equal to the derivative of cost MR= MC where MRi is the marginal revenue and MCi the marginal cost. Revenue is R=pQ where p=f(@), where Q is the sum of all individual outputs of the n firms in the industry(g=∑Q) Marginal revenue, the derivative with respect to the firms output level, Q i, is then MR=p Q=p 00C cO MR=p+Q(1+,) where 1= the conjectural variation term, ie the firms conjecture as to its rivals'response to a change in its own output. If we both multiply and divide equation 3 by then this becomes MR=P I
The First order Profit Maximising Condition reformatted Profit function of firm i: ( ) ( ) i = R Qi −C Qi 1 where, R(Qi) is the revenue function and C(Qi) is the cost function of firm i. For profit maximisation the derivative of profit is zero, which amounts to saying that the derivative of revenue is equal to the derivative of cost: MRi = MCi 2 where MRi is the marginal revenue and MCi the marginal cost. Revenue is R=pQi, where p = f (Q) , where Q is the sum of all individual outputs of the n firms in the industry ( = = n i Q Qi 1 ). Marginal revenue, the derivative with respect to the firm’s output level, Qi , is then + = + = + = + i i n j i j i i i i i i i Q Q Q Q Q Q p Q p Q Q Q p Q p Q p MR p MRi = (1 ) Qi i Q p p + + 3 where i n j i j i Q Q = is the conjectural variation term, i.e. the firm’s conjecture as to its rivals’ response to a change in its own output. If we both multiply and divide equation 3 by Q p then this becomes: + = − i i MRi p S 1 1 4
where, n=_ P, and, -g. is the market share of firm i. Substituting 4 into 2 ap o reformatted to with referenced to the structure-conduct-performance framewor theory gives the profit maximising condition taken orthodox microeconomic R,=Pl Alternative market structures Perfect competition 1+λ=0分A Hence equation 5 becomes MR monopoly Hence equation 5 becomes MR= p 1 MC Joint Profit maximisation Firms seek to maintain market share following therule Q,∑QQ-g 0 O
where, , Q p p Q = − and Q Q S i i = is the market share of firm i. Substituting 4 into 2 gives the profit maximising condition taken orthodox microeconomic theory reformatted to with referenced to the structure-conduct-performance framework: i i i MRi p S = MC + = − 1 1 5 Alternative market structures ➢ Perfect competition = 0 1+ = 0 = −1 i i Qi Q Hence equation 5 becomes MRi = p = MCi 6 ➢ Monopoly = 1 = = i i i i Q Q Q Q Q Q Hence equation 5 becomes MR p = MC = − 1 1 7 ➢ Joint Profit maximisation Firms seek to maintain market share following the rule: i n Q Q Q Q i j i j = = 1,2,..., = − − = = = 1 i i i i n j i j i n j i j i Q Q Q Q Q Q Q Q Q
Q Hence equation 5 becomes MR MC Cournot Each firm is assuming that its rivals output will not change in response to a change in his own output, hence 00 Hence equation 5 becomes MR The Cournot Model: An Example Set n=2, MCi=c and p=a-b0, b>0 and 1i =0. Profits are equal to I,=(a-bgke -@ nce n the profit maximising cond ition 9 be (a-b@ O =c今 0a-bo a-2b0-b2Q ∑,Q O b Writing the two equations explicitly
i i i Q S Q 1 1+ = = Hence equation 5 becomes MRi p = MCi = − 1 1 8 ➢ Cournot Each firm is assuming that its rivals output will not change in response to a change in his own output, hence: = 0 = i n j i j i Q Q Hence equation 5 becomes i i i MC S MR p = = − 1 9 The Cournot Model: An Example Set n=2, MCi=c and p=a-bQ, b> 0 and i=0. Profits are equal to ( ) i i i = a − bQ Q − cQ and since Q a bQ b − = 1 , the profit maximising condition 9 becomes: ( ) ( ) = − − − c a bQ Q b Q Q a bQ i 1 a − bQ − bQ = c a − bQ − b Q = c n j i 2 i 2 i j 2 2 − − = n j i j i Q b a c Q for i=1,2. Writing the two equations explicitly
2 a-c O Solving the system of the two equations we get Q=Q2=-,Q (a a+2 36 In the case of monopoly, since n=/ 1 b0 Qi=Q, equation 7 becomes a-bo a-c M a+c a-2b0=c0= In the case of perfect competition, since p=coa-bo=c, and therefore a-bg=c→OCa-c Stackelberg’ s Mode Instead of simultaneity in the firms' action as in the Cournot model, here the follower (say, firm 2)moves second playing Cournot Q eL while the leader, (say, firm 1)will maximise its profits by taking into account the follower's reaction function as he knows that the latter will act as a Cournot firm, (correctly) treating the leader's output as fixed Hence equation 1 becomes ela-b(@ +02 ) a-c a-c Consequently
b a c Q Q b Q a c Q 2 2 2 2 2 1 2 1 − + = − + = Solving the system of the two equations we get: 3 2 , 3 2( ) , 3 * 2 * 1 a c p b a c Q b a c Q Q C C + = − = − = = In the case of monopoly, since n=1, a bQ bQ − = 1 , Qi=Q, equation 7 becomes: 2 , 2 2 a c p b a c a bQ c Q M M + = − − = = In the case of perfect competition, since p=c a-bQ=c, and therefore: p c b a c a bQ c Q PC PC = − − = = , Stackelberg’s Model Instead of simultaneity in the firms’ action as in the Cournot model, here the follower (say, firm 2) moves second playing Cournot: 2 2 1 2 Q b a c Q − − = while the leader, (say, firm 1) will maximise its profits by taking into account the follower’s reaction function as he knows that the latter will act as a Cournot firm, (correctly) treating the leader’s output as fixed. Hence equation 1 becomes: ( ( )) ( ) + − − = − = − + − = − − 1 1 1 1 1 1 1 2 1 1 2 2 c Q Q b b a c pQ cQ Q a b Q Q cQ a bQ b 1 1 1 2 2 Q Q b a c − − = Consequently
di, a 0今O a 2b2 Q+o 46 46 In other words, there is a first mover advantage as the leader produces more relative to the Cournot model, at the expense of the follower who know produces less, while the total output(price)is greater (lower) than the total output(price)in the Cournot model
4 3 4 3( ) , 4 3( ) , 2 2 4 , 2 0 2 2 2 1 2 1 1 2 1 1 1 a c b a c p a b b a c Q Q Q b Q a c b a c Q b a c Q Q b a c dQ d S S L L F + = − = − − = + = − − = − = − − = = − = In other words, there is a first mover advantage as the leader produces more relative to the Cournot model, at the expense of the follower who know produces less, while the total output (price) is greater (lower) than the total output (price) in the Cournot model