3. Partial Equilibrium under Imperfect Competition Competitive Equilibrium

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1 3. Imperfect Competition 3. Partial Equilirium under Imperfect Competition Competitive Equilirium Partial equilirium studies the existence of equilirium in the market of a given commodity and analyzes its properties. Prices in other markets as well as any other variale (income, cost parameters, etc) are kept constant. We study the market of a homogeneous product simultaneously offered y a numer F of firms and with a demand function that represents the ehavior of the consumers. p d (p) s (p) Under the assumption of perfect competition firms have no market power and make decisions taking prices as given. p 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. 3. Imperfect Competition 3. Imperfect Competition Plan 3.1 Cournot Competition: Firms simultaneously choose the quantity to produce 3. Bertand Competition: Firms simultaneously choose the price 3.3 Stackelerg Competition: Firms sequentially choose the quantity to produce 3.4 Monopolistic Competition: Firms sell differentiated products that consumers view as close sustitutes In all cases: F = {1,,..., F } is the set of firms Market demand function Inverse market demand function where and = α βp p = a = f F q f a = α β = 1 β

2 3. Imperfect Competition 3. Imperfect Competition 3.1. Cournot Competition (After Auguste Cournot (1838)) There are F firms in the market, all producing the same homogeneous (indistinguishale) good. Market demand is therefore given y p = a q f (1) f F where a > c, > 0 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) Profit Maximization Each firm f F acts as a profit maximizer, were profits are given y From equation (3) we get q f = 1 c) ((a q k ) k f π f (q 1,..., q F ; p) = pq f cq f = (p c)q f According to the demand function (1), we have π f (q 1,..., q F ) = (a f F q f c)q f () 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 () yields q f = ( k f q f + q f ) (3) which is known as the Reaction function or Best-Reply function of firm f. It shows what is the est action for firm f for any comination of actions of the other firms. Definition. Cournot-Nash Equilirium In a market of a homogeneous product with demand function p = a and F identical firms with cost functions c(q f ) = cq f, a Cournot-Nash equilirium is a F tuple (q c1,..., q cf ) such that, for every f F, q cf = 1 c) ((a q ck ) k f Since in equilirium the right-hand side of equation (3) does not depend on f, we conclude that all firms produce the same quantity in equilirium, that is, q cf = q c f F

3 Hence, according to (3) we have Therefore, q c = q c = f F q c = F q c, f F (4) (F + 1) Accordingly, from (4) we can compute the full set of the Cournot equilirium values q c = (F +1), f F c = f F qc = p c = a c = a π c = (F +1), f F F F +1 F F +1 Comparison with the competitive equilirium Under the assumption of perfect competition, firms ehave 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 equilirium,, is = Therefore, the following relationships etween the two levels of competition hold: q c = F +1 c = F F +1 p c p = F +1 π c > 0 whereas π = 0 We oserve that the main difference etween the two sets of equilirium values comes from the numer of firms F. Since a typical interpretation of the perfect competition assumption is that p... the numers of firms is large enough so that no firms has enough market power and, hence, all ehave as price takers... we could investigate what happens to these equilirium values as the numer of firms grows large. In this sense, we find: p c p d (p) F s (p) lim F qc = 0 F lim F c = c 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 )

4 Example: Cournot Duopoly When there are only two firms (F = {1, }), we get Graphically, q 1 = 1 c) ((a q ) q = 1 c) ((a q 1 ) as the Best-reply functions. Accordingly, the equilirium values in this case are: q q 1 q c = 1 3 c = 3 p c = a 3 π c = 1 9 q c q c q q 1 3. Imperfect Competition 3. Imperfect Competition 3. Bertrand Competition 3. 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 (ecause, according to the demand function, price determines quantity and quantity determines price), the results contradict this intuition. For simplicity, F = {1, } The two firms have identical marginal cost c > 0 and no fixed cost Market demand is given y = α βp Firms set prices simultaneously and must supply all that is demanded at their prices Consumers uy from the cheapest firm. If prices are equal, consumers split equally. In this case profits take the form Profit Maximization (p 1 c)(α βp 1 ) c < p 1 < p π 1 (p 1, p 1 ) = (p1 c)(α βp 1 ) c < p 1 = p 0 otherwise (similarly for firm ) Definition. Bertrand-Nash Equilirium In a market of a homogeneous product with demand function = α βp and identical firms with cost functions c(q f ) = cq f, a Bertrand-Nash equilirium is a pair (p 1, p ) such that, for every f p f arg max π f (p 1, p ) Proposition The pair (p 1, p ) = (c, c) is the unique Bertrand-Nash equilirium

5 3. Imperfect Competition 3. Bertrand Competition 3. Imperfect Competition 3. Bertrand Competition Proof. That (c, c) is a Bertrand-Nash equilirium is clear since π 1 (c, c) = π (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 ). To see that this is the unique equilirium, suppose that firm 1 chooses p 1 > c. Then, 0 p > p 1 π (p 1, p 1 ) = (p c)(α βp ) p = p 1 (p c)(α βp ) c < p < p 1 equilirium p = p = c, f {1, } = = (α βc) q = 1 (α βc), f {1, } π = π = 0, f F Clearly, p 1 > c p > c and p < p 1 (5) Switching the roles of firms 1 and we get p > c p 1 > c and p 1 < p (6) Since (5) and (6) are incompatile we conclude that thare is no other equilirium. Therefore, in this case the set of equilirium values almost coincide with those of the competitive 3. Imperfect Competition 3. Imperfect Competition 3.3 Stackelerg Competition Dynamic version of the cournot model Two firms, F = {1, } 3.3 Stackelerg Competition Solving at t = The profits of firm, once q 1 is known, are given y π (q 1, q ) = (a (q 1 + q ))q cq At t = 1 firm 1 (leader) chooses q 1 At t = firm (follower) knows q 1 and chooses q As efore, p = a = a (q 1 + q ) Taking π (q 1,q ) q and = 0 yields a q 1 q c = 0 q = 1 c) ((a q 1 ) (7) c(q f ) = cq f f {1, } which corresponds to the Best Reply Function found in the Cournot model The model is solved y ackwards induction (Sugame Perfect Equilirium) Solve firms s decision at t = Knowing what the result is at t =, solve firm 1 s decision at t = 1

6 3. Imperfect Competition 3.3 Stackelerg Competition Solving at t = 1 3. Imperfect Competition 3.3 Stackelerg Competition and solving for q 1 we get the Stackelerg solution for firm 1 (the leader) The profits of firm 1 ar given y π 1 (q 1, q ) = (a (q 1 + q ))q 1 cq 1 q s1 = Taking into account the expected ehavior of firm, profits can e written as π 1 (q 1 ) = (a (q 1 + q {}} { 1 c) ((a q 1 )))q 1 cq 1 From the est reply function (7) of firm we get the Stackelerg solution for the follower Taking δπ1 (q 1 ) δq 1 = 0 yields a q 1 + q 1 c = 0 q s = 4 3. Imperfect Competition 3.3 Stackelerg Competition 3. Imperfect Competition 3.3 Stackelerg Competition From here we can compute the rest of the Stackelerg equilirium values q s1 = 1 q s = 1 4 s = 3 4 p s = a 3 4 Graphically, p d (p) s (p) π 1 (q s1, q s ) = 1 8 π (q s1, q s ) = 1 16 p c p s p Notice that π 1 (q s1, q s ) > π }{{} 1 (q c1, q c ) = π (q c1, q }{{ c > π } (q s1, q s ) }{{} c s } 3 {{ } c < 3 } 4 {{ } s < }{{ }

7 3. Imperfect Competition 3.4 Monopolistic Competition First, we will study the ehavior of a single firm (monopoly) in the market Next, we will study the competition etween F monopolies. Monopolist Behavior Consider a monopolist facing an invers demand function p() Thus, the profit function takes the form π() = p() c() The F monopolies are involved in competition with each other ecause, although selling different products, consumers view them as close sustitiutes. Thus, the demand for the product of each of the monopolist depends on all the prices with the standard interpretation q f = q f (p 1,..., p F ) q f p f < 0 qf p k > 0 k f where c() is the cost function Profit maximization yields p( m ) + mδp(m ) δ }{{} mr() Marginal revenue mr() can e re-written as mr() = p()[1 + δp() δ = p()[1 1 ǫ() ] = δc(m ) δ }{{} mc() p() ] = (8) where ǫ() is the elasticity of the demand function. Using this in equation (8) we get which can e rought to the market at the price p( m ) mc( m ) p( m ) = 1 ǫ( m ) Example: Linear Demand - Linear Cost If the cost function is c() = c and rgw demand function takes the form p m = a = + c p = a then marginal reveneu is mr() = a Hence, the optimal monopolistic ehavior is The monopolist profits are mr( m ) = mc( m ) a m = c m = 1 π( m ) = (p m c) m = 1 = 4

8 p a d (p) When p = a and c(q) = cq Comparison p m Competitive Stackelerg Cournot Monopoly (Bertrand) c mr() m 1 a a p c Monopolistic competition Suppose that there are F monopolies selling products that are close sustitute of each other. Each firm f has a cost function c f (q f ) The demand for the product of each of the monopolist depends on all the prices q f = q f (p 1,..., p F ) The profits of firm f are given y Profit maximization yields π f (p 1,..., p F ) = p f q f (p 1,..., p F ) c f (q f (p 1,..., p F )) q f (p 1,..., p F ) + p f qf (p 1,..., p F ) p f δcf (q f (p 1,..., p F )) δq f q f (p 1,..., p F ) p f = 0 Equilirium exists when all firms choose p f that maximizes profits with the standard interpretation q f p f < 0 qf p k > 0 k f Example Linear Demand - Linear Cost - firms Let c f (q f )) = c f q f e the cost function of firm f {1, } that operates in a market with demand functions For each product f there is a price p f > 0 such that q f (p 1,..., p f,..., p F ) = 0 q 1 = a p 1 + cp q = α βp + γp 1

9 Profits, then, are Profit maximization produces π 1 (p 1, p ) = p 1 (a p 1 + cp ) c 1 (a p 1 + cp ) π (p 1, p ) = p (α βp + γp 1 ) c (α βp + γp 1 ) p 1 = a c 1 p = α c β + c 1 p + c β p 1 that are the Best reply functions of firms 1 and. Solving we get The linear city model of product differentiation This model is a generalization of the Hotelling model of spatial competition. It encompasses many different model of competition in product differentiation. Two firms selling the same product, with identical marginal cost c > 0, operate in a linear city of lenght 1. In this version of the model, firms live in the opposite ends of the city. A mass of M consumers is uniformly distriuted along the city. A consumer s location is denoted y z [0, 1]. All consumers have the same valuation (v > 0) of the product. The cost of uying form firm j {1, } at price p j for a consumer that lives at a distance of d form firm j is given y p j + td p mc 1 = ((a c 1) + βc 1 (α c )) 4 βc 1 c p mc = β((α c ) + c (a c 1 )) 4 βc 1 c The model has different difficulties depending on the parameter (v, c, t) configuration Proposition. When v > c + t and firms follow est responses, all consumers will uy from one of the two firms Proof. Suppose, wlog, that Firm 1 is following a est response to the choice of p y Firm. Suppose also that some consumers don t uy (as in Figure ). Then, the sales volume for Firm 1 are determined y consumer z 1, who is indifferent etween uying and not. Consumer z 1 is given y p 1 + tz 1 = v z 1 = v p 1 t Thus, as long as Firm 1 does not start to compete with Firm, Firm 1 s profits are given y Figure 1: All consumers uy Figure : Some comsumers don t uy Taking derivatives, π 1 (p 1 ) = (p 1 c) v p 1 t dπ 1 (p 1 ) = v + c p 1 p 1 t By assumption, some consumers prefer not to uy than uying form Firm 1. This must e true, (9)

10 in particular, for consumer living at the point z = 1. Thus, it must e true that p 1 + t > v p 1 > v t In the case that all consumers uy from one of the two firms, the consumer who is indifferent etween the two firms is the point z such that Susittuting this into the equation 9 we otain that dπ 1 (p 1 ) = v + c p 1 < c + t v p 1 t t By assumption, c + t v < 0 and thus we have that dπ 1 (p 1 ) p 1 < 0 Therefore, Firm 1 could increase its profits y lowering p 1, which contradicts the assumption that Firm 1 is playing a Best response to p Therefore, y assuming that v is large enough (v > c + t) we can ignore the possiliity of nonpurchase p 1 + tz = p + t(1 z ) Thus, z = t + p p 1 t Hence, the demand for FIrm 1 is given y 0 z < 0 x 1 (p 1, p ) = Mz z [0, 1] M z > 1 Using (10) it can e writen as a function of prices: 0 p 1 > p + t x 1 (p 1, p ) = M (t+p p 1 ) t p 1 [p t, p + t] M p 1 < p t (10) (11) Symmetry implies 0 p > p 1 + t x (p 1, p ) = M (t+p 1 p ) t p [p 1 t, p 1 + t] M p < p 1 t It is clear that, for each firm j {1, } and for any p j y its competitor, any price p j > p j + t yields the same profits as p j = p j + t (zero profits) any price p j < p j + t yields lower profits than p j = p j + t (all result in sales of M units) Thus, we can concentrate in the interval p j [p j t, p j + t]. Therefore, the prolem for Firm j is max (p j c)m (t+p j p j ) p t j s.t. p j [p j t, p j + t] (1) The Khun-Tucker conditions for this prolem are 0 p j = p j t t + p j + c p j = 0 p j (p t, p j + t) 0 p j = p j + t Solving these conditions we find the Best response for each firm j p j + t p j c t t+p p j = j +c p j (c t, c + 3t) p j t p j c + 3t By symmetry, we must have that p 1 = p = p. This can happen only in the middle case.

11 Thus and so is the Nash equilirium in this model p = t + p + c p = t + c Each firm sells M Profits are t M As t 0, firms ecome undifferentiated and then p c as in the Bertrand model