Katz and Shapiro (1985)

Transcription

1 Katz and Shapiro (1985) 1 The paper studies the compatibility choice of competing firms in industries with network externalities. Also investigated are the social vs. private incentives of compatibility choice and its welfare implications. n firms, competing à la Cournot. The output of a firm can either be compatible or incompatible with others. A continuum of consumers distributed on (, A] with density 1. A > 0. Consumers buy either 1 or 0 unit of good from a firm.

2 xi e : number of consumers firm i is expected to have. x i : its actual value. y e i : expected network size of firm i. y i : its actual value. When all firms are compatible, y e i = n j=1 x e j i. If no two firms goods are compatible, then y e i = x e i Consumers are heterogeneous, with a utility of γ + v(y e ) if the expected network size of the good he buys is y e, u(0) = 0, v > 0, v < 0, v (y) = 0 as y. i.

3 3 γ (, A] uniformly. p i : Price of firm i s good. A consumer with utility γ + v(y e ) buys good i iff γ + v(y e i ) p i γ + v(y e j ) p j j and that γ + v(y e i ) p i 0. p i v(y e i ) can be called hodonic price of good i. Let φ be this common value. There are A φ consumers.

4 4 z n i=1 x i: total output. It must be the case that A φ = z for market to clear, i.e., A + v(y e i ) p 1 = z i with x i > 0. p i = A + v(y e i ) z. Matching cost is F i for firm i. Assume (1) (y e 1,..., y e n ) is given; (2) j i x j x i is given for firm i, i.

5 Profit for firm i is π i = x i p i = x i (A + v(y e i ) z). FOC: x i = A + v(y e i ) n j=1 x i. Solving for FOCs we have x i = A+nv(y e i ) P j i v(y e j ) n 1. This solution defines a mapping from (y e 1,..., y e n ) to (y 1,..., y n ). Let Γ(y e ) be the mapping. A RE equilibrium is the one with Γ(y ) = y.

6 6 Welfare calculation: The profit of firm i is π i = (x i ) 2. Consumer surplus is S(z) = A A z (w + z A)dw = z 2 /2. Total social welfare = n i=1 π i + S(z) = n i=1 x 2 i + z 2 /2.

7 Case of complete compatibility z e = n i=j x e j = y e i i. x i = (A + v(z e ))/(n + 1). In RE equilibrium, z e = x x n. We thus have z c = n n+1 (A + v(zc )). $ n+1 n z A + v(z) A z c z Total Output Figure 1. Equilibrium with Complete Compatibility The unique equilibrium is symmetric with output of each firm z c /n.

8 Case of Incompatibility In this case y e i = x e i. x i = A + v(x i ) z. There are three subcases: (a) Symmetric case with n active firms. (b) Symmetric case with k < n active firms. (c) Asymmetric case.

9 In case (a), there is a unique symmetric equilibrium in which x i = z I /n, where n+1 n zi = A + v( zi n ). $ n+1 n z A + v(z) A z I z Total Output Figure 2. Unique Symmetric Equilibrium with Complete Incompatibility

10 In case (b), equilibrium exists iff v( A k ) A k. $ k+1 k z A + v(z) A z k z Total Output Figure 3. Natural Oligopoly

11 11 A monopolist s profit might be lower than a duopolist in 2-active-firm equilibrium. x 2 Monopoly Equilibrium Firm 2 s Reaction Curve Symmetric Equilibrium Firm 1 s Reaction Curve Monopoly Equilibrium x1 Figure 4. Natural Monopoly Equilibria

12 For k n 1, if a k-active-firm symmetric equilibrium exists, then a (k + 1)-active-firm symmetric equilibrium exists. For any k n 1, if a k-active-firm symmetric equilibrium exists, then z k < z k+1.

13 Case (c): Asymmetric output. x 2 A Asymmetric Equilibria Firm 1 s Reaction Curve Symmetric Equilibrium Firm 2 s Reaction Curve x1 Figure 5. Asymmetric Duopoly

14 The case of partial compatibility Partition {1,..., n} into {G 1,..., G T }. If Firm i G j, y i = k G x j k y i. FOC: x i = A z + v(m j x j ); where x j and m j are output and number of firm in group j. Output is greater under complete compatibility than any equilibrium less than so.

15 15 $ na Complete Compatibility z I z c (n + 1)z Incomplete Compatibility na + nv(z) na + v(y i ) z Total Output Figure 6. Complete vs. Incomplete Compatibility

16 Private and social incentives for compatibility There are several ways to make product compatible. One is to set up product standard, the other is to produce adaptor. In the first case, all firms need to agree to have a standard. In the 2nd case, a firm can unilaterally make its product compatible to others by producing an adaptor. We also have to distinguish the case whether the firms can make side payments when they make their product compatible. In the case when side payments are feasible, since firms choose to be compatible iff it increases joint profit, W = π + S, and S(2) = z 2 /2 so that S(2) increases iff joint output increases, we know that any move to complete compatibility that increases industry profit is socially beneficial. 16

17 Since S > 0, W > π. As a result, if the compatibility cost F is such that π < F < W, then private incentive for compatibility is not strong enough from social viewpoint. Suppose side payment are feasible only between firms adopting the same technology, then a group of firms G adopt standard when i G F i < i G π i. If side payments are not feasible, then it must be that F i < π i i G. That is, it is more difficult to adopt standard when side payments are not feasible. There is possibility of excess standardization when standardization is less than complete. This occurs when π > W, but i G π i > i G F i for some G.

18 In the adaptor case, a firm can unilaterally make its product compatible with others. Assume the firm which makes adaptor pays its cost. Private incentive for i to use adaptor: π i F. Social incentive: π i + j i π j + S F. There is a gap between the two.

19 In an example of two firms, suppose the initial equilibrium (before there is adaptor) is symmetric. Then firm 1 has incentive to adapt if π 1 > 0. Symmetry implies π 2 > 0. Since total network size increases, S. As a result, W > π i : Incentive for adaptor is socially too low. When, on the other hand, the original equilibrium is asymmetric there might be excess compatibility. For example, suppose firm 2 is a small firm. When it adapts, the loss in firm 1 s profit might be so large that π 1 + S < 0. Consequently W < π 2 : There is excess incentive to adapt.