Katz and Shapiro (1985)
The paper studies the compatibility choice of competing firms in industries with network externalities. Also investigated are the social vs. private incentives of compatibility and its welfare implications.
n firms, competing ála Cournot. The output of a firm can either be compatible or incompatible with others.
A continuum of consumers distributed on with density 1.
Consumers buy either 1 or 0 unit of good from a firm.
: number of consumers firm i is expected to have. : its actual value.
: expected network size of firm i. : its actual value.
When all firms are compatible, .
If no two firms’ goods are compatible, then .
Consumers are heterogeneous, with a utility of if the expected network size of the good he buys is as .
uniformly.
: Price of firm i’s good.
A consumer with utility buys good i iff
and that .
Firms i and j both have positive sales only if .
can be called hodonic price of good i.
Let be this common value.
There are consumers.
: total output.
It must be the case that for market to clear, i.e., with .
.
Matching cost is for firm i.
Assume (1) is given; (2) is given for firm i, .
Profit for firm i is .
FOC: .
Solving for FOCs we have .
This solution defines a mapping from to . Let be the mapping.
A RE equilibrium is the one with .
Welfare calculation:
The profit of firm i is . Consumer surplus is
Total social welfare =
=.
Case of complete compatibility:
.
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.
In RE equilibrium, . We thus have .
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The unique equilibrium is symmetric with output of each firm .
Case of Incompatibility:
In this case .
.
There are three subcases:
(a)Symmetric case with n active firms.
(b)Symmetric case with active firms.
(c)Asymmetric case.
In case one, there is a unique symmetric equilibrium in which , where .
In case (b), equilibrium exists iff .
A monopolist’s profit might be lower than a duapolist in 2-active-firm equilibrium.
For , if a k-active-firm symmetric equilibrium exists, then a - active-firm symmetric equilibrium exists.
For any , if a k-active-firm symmetric equilibrium exists, then .
Case (c): Asymmetric output.
The case of partial compatibility: Partition into . If firm .
FOC: ;where and are output and number of firms in group j.
Output effects of compatibility change:
- Output is greater under complete compatibility than any equilibrium less than so.
Private and social incentives for compatibility:
There are several ways to male product compatible. One is to set up product standard, the other is to produce adaptor. For the first case, all firms need to agree to have a standard. For the 2nd case, a firm can unilaterally make its product compatible to others by producing an adaptor.
We also have to distinguish to 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, , and so that increases iff joint output increases, we know that any move to complete compatibility that increases industry profit is socially beneficial.
Since , . As a result, if the compatibility cost F is such that , 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 . If side payments are not feasible, then it must be that . 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 , but for some G.
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: . Social incentive: . There is a gap between the two.
In an example of two firms, suppose the initial equilibrium (before there is adaptor) is symmetric. Then firm 1 has incentive to adapt if . Symmetry implies . Since total network size increases, . As a result, : 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 . Consequently : There is excess incentive to adapt.
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