4. Partial Equilibrium under Imperfect Competition

Transcription

1 4. Partial Equilibrium under Imperfect Competition Partial equilibrium studies the existence of equilibrium in the market of a given commodity and analyzes its properties. Prices in other markets as well as any other variable (income, cost parameters, etc) are kept constant. We study the market of a homogeneous product simultaneously offered by a number F of firms and with a demand function that represents the behavior of the consumers. Under the assummption of perfect competition firms have no market power and make decisions taking prices as given. Under the assumption of imperfect competition firms might have some market power. Hence, they make decision in a strategic way as they know that their decisions determine (to some extent) the price in the market. IDEA - Microeconomics II

2 4. Imperfect Competition Competitive Equilibrium p Q d (p) Q s (p) p Q Q IDEA - Microeconomics II

3 4. Imperfect Competition Plan 4.1 Cournot Competition: Firms simultaneously choose the quantity to produce 4.2 Bertand Competition: Firms simultaneously choose the price 4.3 Stackelberg Competition: Firms sequentially choose the quantity to produce 4.4 Monopolistic Competition: Firms sell differentiated products that consumers view as close substitutes IDEA - Microeconomics II

4 4. Imperfect Competition In all cases: F = {1, 2,..., F } is the set of firms Market demand function Q = α βp Inverse market demand function p = a bq where Q = f F q f and a = α β b = 1 β IDEA - Microeconomics II

5 4. Imperfect Competition 4.1. Cournot Competition (After Auguste Cournot (1838)) There are F firms in the market, all producing the same homogeneous (indistinguishable) good. F = {1,..., F } Price is not taken as given There is no free entry Firms set quantities simultaneously All firms f F have identical costs C(q f ) = cq f (c 0) Market demand is therefore given by p = a b f F q f (16) where a > c IDEA - Microeconomics II

6 4. Imperfect Competition 4.1 Cournot Competition Profit Maximization Each firm f F acts as a profit maximizer, were profits are given by π f (q 1,..., q F ; p) = pq f cq f = (p c)q f According to the demand function (16), we have π f (q 1,..., q F ) = (a b f F q f c)q f (17) Profit maximization implies that, for every firm f F, the optimal quantity q f must satisfy the first order condition π f (q 1,..., q f,..., q F ) q f = 0 which, applied to (17) yields q f = b ( k f q f + q f ) (18) IDEA - Microeconomics II

7 4. Imperfect Competition 4.1 Cournot Competition From equation (18) we get q f = 1 c) ((a 2 b k f q k ) which is known as the Reaction function or Best-Reply function of firm f. It shows what is the best action for firm f for any combination of actions of the other firms. Definition. Cournot-Nash Equilibrium In a market of a homogeneous product with demand function p = a bq and F identical firms with cost functions c(q f ) = cq f, a Cournot-Nash equilibrium is a F tuple (q c1,..., q cf ) such that, for every f F, q cf = 1 c) ((a 2 b k f q ck ) Since in equilibrium the right-hand side of equation (18) does not depend on f, we conclude that all firms produce the same quantity in equilibrium, that is, q cf = q c f F IDEA - Microeconomics II

8 4. Imperfect Competition 4.1 Cournot Competition Hence, according to (18) we have q c = b f F q c = b F q c Therefore, q c =, f F (19) b(f + 1) Accordingly, from (19) we can compute the full set of the Cournot equilibrium values q c = (a c) b(f +1), f F Q c = f F qc = F (a c) F +1 b p c = a bq c = a π c = (a c)2 b(f +1) 2, f F F F +1 IDEA - Microeconomics II

9 4. Imperfect Competition 4.1 Cournot Competition Comparison with the competitive equilibrium Under the assumption of perfect competition, firms behave as price takers and, therefore, the optimal quantity to produce is the one that equals price with marginal cost. In our setting, p = c Hence, from the demand function we get that the aggregate quantity in a competitive equilibrium, Q, is Q = b Therefore, the following relationships between the two levels of competition hold: q c = Q F +1 Q c = F F +1 Q p p c = (a c) F +1 π c > 0 whereas π = 0 IDEA - Microeconomics II

10 4. Imperfect Competition 4.1 Cournot Competition We observe that the main difference between the two sets of equilibrium values comes from the number of firms F. Since a typical interpretation of the perfect competition assumption is that... the numbers of firms is large enough so that no firms has enough market power and, hence, all behave as price takers... we could investigate what happens to these equilibrium values as the number of firms grows large. In this sense, we find: lim F qc = 0 lim F Qc = Q lim F (p p c ) = 0 lim F πc = 0 = π According to this results, Cournot model of imperfect competition is consistent with the neoclassical model of perfect competition as a limit case (F ) IDEA - Microeconomics II

11 4. Imperfect Competition 4.1 Cournot Competition p Q d (p) Q s (p) p c F p F Q c Q Q IDEA - Microeconomics II

12 4. Imperfect Competition 4.1 Cournot Competition Example: Cournot Duopoly When there are only two firms (F = {1, 2}), we get q 1 = 1 c) ((a 2 b q 2 = 1 c) ((a 2 b q 2 ) q 1 ) as the Best-reply functions. Accordingly, the equilibrium values in this case are: q c = 1 (a c) 3 b Q c = 2 (a c) 3 b p c = a 2 3 π c = 1 (a c) 2 9 b IDEA - Microeconomics II

13 4. Imperfect Competition 4.1 Cournot Competition Graphically, q 2 b q 1 2b q c q 2 q c 2b b q 1 IDEA - Microeconomics II

14 4. Imperfect Competition 4.2 Bertrand Competition (After Joseph Bertrand (1883)) Fifty years after Cournot s work, Bertrand argued that competition in prices, rather than competition in quantities, seemed closer to reality. Although the difference in the two approaches might appear as meaningless (because, according to the demand function, price determines quantity and quantity determines price), the results contradict this intuition. For simplicity, F = {1, 2} The two firms have identical marginal cost c > 0 and no fixed cost Market demand is given by Q = α βp Firms set prices simultaneously and must supply all that is demanded at their prices Consumers buy from the cheapest firm. If prices are equal, consumers split equally. IDEA - Microeconomics II

15 4. Imperfect Competition 4.2 Bertrand Competition Profit Maximization In this case profits take the form π 1 (p 1, p 2 ) = (p 1 c)(α βp 1 ) c < p 1 < p (p1 c)(α βp 1 ) c < p 1 = p 2 0 otherwise (similarly for firm 2) Definition. Bertrand-Nash Equilibrium In a market of a homogeneous product with demand function Q = α βp and 2 identical firms with cost functions c(q f ) = cq f, a Bertrand-Nash equilibrium is a pair (p b1, p b2 ) such that, for every f p bf arg max π f (p b1, p b2 ) Proposition The pair (p b1, p b2 ) = (c, c) is the unique Bertrand- Nash equilibrium IDEA - Microeconomics II

16 4. Imperfect Competition 4.2 Bertrand Competition Proof. That (c, c) is a Bertrand-Nash equilibrium is clear since π 1 (c, c) = π 2 (c, c) = 0 and π 1 (p 1, c) 0 for p 1 c. Hence, firm 1 has no incentives to deviate from p 1 = c (and so does firm 2). To see that this is the unique equilibrium, suppose that firm 1 chooses p 1 > c. Then, π 2 (p 1, p 2 ) = 0 p 2 > p (p2 c)(α βp 2 ) p 2 = p 1 (p 2 c)(α βp 2 ) c < p 2 < p 1 Clearly, p 1 > c p 2 > c and p 2 < p 1 (20) Switching the roles of firms 1 and 2 we get p 2 > c p 1 > c and p 1 < p 2 (21) Since (20) and (21) are incompatible we conclude that thare is no other equilibrium. IDEA - Microeconomics II

17 4. Imperfect Competition 4.2 Bertrand Competition Therefore, in this case the set of equilibrium values almost coincide with those of the competitive equilibrium p b = p = c, f {1, 2} Q b = Q = (α βc) q b = 1 2 (α βc), f {1, 2} π b = π = 0, f F IDEA - Microeconomics II