Silviase/jgov_v1.4_full_ann huggingface dataset:
Firm 1 would earn $ \gamma_{1}M $ if it were to impose a tie, which is less than it would earn by selling the component to Firm 2. In this equilibrium Firm 2 sets $ P_{2} = \gamma_{2} + z^{M} $ , sells systems to the entire market, and earns $ \pi_{2} = (P_{2} - w)M - r(z^{M}) $ . Firm 1 sets $ P_{1} = w $ and earns $ \pi_{1} = wM $ . This case corresponds closely to the standard Chicago School argument that a firm does not benefit from a technological tie if it can charge the monopoly price for an essential upstream input.

It is interesting that multiple equilibria exist when $ \gamma_{1} + z^{M} - r(z^{M})/M \leq w < \gamma_{1} + z^{M} $ . The equilibrium in which Firm 2 invests efficiently Pareto-dominates the equilibrium in which Firm 1 invests while threatening Firm 2 with a technological tie. In this case, the ability to impose a technological tie is a trap that the tying firm would prefer to avoid. The ability to tie can result in a “bad equilibrium” in which Firm 2 is discouraged from investing to improve its product, because it expects Firm 1 to improve its owns system and hence have an ex post incentive to foreclose any competition with a technological tie. When $ w \geq \gamma_{1} + z^{M} - r(z^{M})/M $ , Firm 1 has no profitable use for a technological tie, either threatened or actual, and would be better off relinquishing its ability to impose a tie.

The standard Chicago School argument is that the owner of an essential input has no incentive to foreclose access to the input if it can charge the monopoly price. Since it is possible that $ \gamma_{1} + z^{M} > \gamma_{2} + z^{M} - r(z^{M})/M = \bar{w} $ , this argument must be qualified in the presence of endogenous innovation by the possibility of multiple equilibria even when the wholesale price is set at the monopoly level $ (\bar{w}) $ . Although in equilibrium the integrated monopolist does not impose an actual technological tie when $ w \geq \gamma_{1} + z^{M} - r(z^{M})/M $ , its mere ability to tie can deter investment by the more efficient Firm 2 and therefore make it impossible to fully extract the monopoly profit. Moreover, the case in which $ \gamma_{1} + z^{M} > \bar{w} $ corresponds to $ (\gamma_{2} - \gamma_{1})M < r(z^{M}) $ , and from Proposition 1 this implies that efficient investment would be the unique equilibrium of the product improvement game if tying were not possible. Thus, in this case, there is a policy rationale for prohibiting technological tying, because a prohibition would be Pareto-improving. $ ^{20} $ There are however, other plausible modifications of the game in which technological tying can increase welfare in some instances.

## 5 Welfare

We now consider the welfare implications of the product improvement and technological tying games. The ability of Firm 1 to impose a technological tie obviously threatens social welfare whenever the product improvement game yields an equilibrium in which the more efficient Firm 2 invests. When $ r(z^{M}) < (\gamma_{2} - \gamma_{1})M $ , invest-
ment by Firm 2 is the unique equilibrium of the product improvement game and is the efficient market structure, but technological tying destroys the possibility of an efficient market structure when $ w < \gamma_{1} + z^{M} - r(z^{M})/M $ . If $ r(z^{M}) \geq (\gamma_{2} - \gamma_{1})M $ , there are multiple equilibria of the product improvement game. Technological tying does not improve total welfare in this case and can reduce welfare by preventing an efficient equilibrium. Thus Propositions 1 and 2 imply the following.

Corollary 1 If $ r(z^{M}) < (\gamma_{2} - \gamma_{1})M $ , then actual or threatened technological tying strictly reduces social welfare relative to the unique equilibrium of the product improvement game.

Corollary 2 If $ r(z^{M}) \geq (\gamma_{2} - \gamma_{1})M $ , then actual or threatened technological tying weakly reduces social welfare relative to equilibria of the product improvement game. If the efficient equilibrium is focal, then technological tying strictly reduces social welfare. Otherwise, the ability of Firm 1 to technologically tie is irrelevant for market structure and social welfare (but not for consumer welfare).

Social welfare is obviously at a maximum in the efficient equilibrium of the product improvement game. Nonetheless, consumers (weakly) prefer the inefficient equilibrium to the efficient one. With Bertrand competition, the equilibrium price is the quality level of the investing firm less the margin between quality and cost for the rival firm, provided this margin is positive. This margin determines consumer surplus. The margin is (weakly) larger for Firm 2 because Firm 2's quality level exceeds Firm 1's when neither firm invests. Firm 2 is a greater competitive threat to Firm 1 in the inefficient equilibrium than Firm 1 is to Firm 2 in the efficient equilibrium. $ ^{21} $ Consumers benefit directly from the greater competitive threat of Firm 2 in the inefficient equilibrium.

Corollary 3. Consumer surplus is weakly higher in the inefficient equilibrium of the product improvement game than in the efficient equilibrium, and strictly higher when $ w < \gamma_{2} $ .

Technological tying is not in the interests of consumers in this model. It is evident from Proposition 2 that consumer surplus is zero for all values of w in the technological tying game. Actual or threatened technological tying eliminates Firm 2 as a potential competitor in an equilibrium in which Firm 1 wins the market, and a high component price eliminates Firm 1 as a potential competitor in an equilibrium in which Firm 2 wins the market. Thus, consumer surplus is fully extracted in all equilibria of the technological tying game.
Corollary 4 Consumer welfare is weakly lower in the technological tying game than in the product improvement game. If $ w < \gamma_{2} $ , then consumer surplus is strictly lower in the technological tying game relative to the inefficient equilibrium of the product improvement game. If $ w < \gamma_{1} $ , then consumer surplus also is strictly lower relative to the efficient equilibrium of the product improvement game.

Some important caveats are in order. These conclusions depend on the assumption that Firm 2 is initially the more efficient system supplier. We consider the case in which Firm 1 is initially more efficient in the next section. Furthermore, the above welfare analysis is premised on the assumption that the wholesale price is the same in the product improvement game and the corresponding technological tying game. The assumption is most plausible when the wholesale price is determined by regulation. The assumption also makes sense for our specific model in which the wholesale price is determined by the monopoly price of the component in its stand-alone use. More generally, the non-discriminatory profit-maximizing price that the upstream monopolist would charge without technological tying is a compromise between the profit-maximizing prices for each possible use. Technological tying allows the firm effectively to set a higher price for the component when it is used as part of a system, and a lower price for other uses. When such a mixed bundling strategy is profitable, a prohibition against technological tying could cause the price for alternative uses to increase, depending on price elasticities. This possibility tempers the case against technological tying.

## 6 Some Extensions

In this section we consider several modifications to the product improvement and technological tying games. These include reversing the assumption that Firm 2 has the initial quality advantage, allowing the losing firm to exit the market, reversing the order of moves in the technological tying game, and allowing for systems with network effects.

### 6.1 Firm 1 has the initial quality advantage

The analysis of the equilibria of the product improvement game is little changed if Firm 1 has the initial quality advantage. Suppose $ \gamma_{1} > \gamma_{2} \geq 0 $ and define $ \bar{w}_{1} \equiv \gamma_{1} + z^{M} - r(z^{M}) / M $ . Payoffs are shown in Table 2 below for $ \gamma_{1} \leq w \leq \min(\gamma_{2} + z^{M}, \bar{w}_{1}) $ . There is an efficient equilibrium in which Firm 1 invests $ z^{M} $ and Firm 2 does not invest. This is the unique equilibrium if $ r(z^{M}) < (\gamma_{1} - \gamma_{2}) M $ . There is a second equilibrium in which Firm 2 invests $ z^{M} $ and Firm 1 does not invest if and only if $ r(z^{M}) \geq (\gamma_{1} - \gamma_{2}) M $ . Note that $ \bar{w}_{1} \leq \gamma_{2} + z^{M} $ if and only if $ r(z^{M}) \geq (\gamma_{1} - \gamma_{2}) M $ . Consequently, Table 2
describes the outcomes for all feasible $ w \geq \gamma_{1} $ when $ r(z^{M}) \geq (\gamma_{1} - \gamma_{2})M. $ ^{22}

Table 2. Firm payoffs when $ \gamma_{2} < \gamma_{1} \leq w \leq \min(\gamma_{2} + z^{M}, \bar{w}_{1}) $

$ z_{2}=z^{M} $	$ z_{2}=0 $
$ z_{1}=z^{M} $	$ (\gamma_{1}-\gamma_{2}+w)M-r(z^{M});-r(z^{M}) $	$ (\gamma_{1}+z^{M})M-r(z^{M});0 $
$ z_{1}=0 $	$ wM;(\gamma_{2}+z^{M}-w)M-r(z^{M}) $	$ \gamma_{1}M;0 $

In the technological tying game, if Firm 1 has the initial quality advantage, then for $ w < \gamma_{2} $ there is an equilibrium in which Firm 1 invests $ z^{M} $ , Firm 2 does not invest, and Firm 1 forecloses Firm 2 with a technological tie. Firm 1 sets $ P_{1} = \gamma_{1} + z^{M} $ , sells systems to the entire market, and earns $ \pi_{1} = P_{1}M - r(z^{M}) $ . Firm 2 sets $ P_{2} = w $ and earns $ \pi_{2} = 0 $ . For $ w \geq \gamma_{2} $ , there is a second equilibrium in which Firm 1 invests $ z^{M} $ , Firm 2 does not invest, and Firm 1 does not impose a technological tie. The component price eliminates Firm 2 as a potential competitor when it does not invest, and if Firm 2 were to invest, then Firm 1 would have a credible threat to impose a tie. Payoffs are the same as in the first equilibrium. These are the only equilibrium outcomes of the tying game when Firm 1 is the more efficient supplier of systems. For example, if Firm 2 invested $ z^{M} $ , Firm 1 could earn no more than $ (\gamma_{2} + z^{M})M - r(z^{M}) $ by selling the component to Firm 2; but by investing and imposing a technological tie in the second stage of the game it would earn $ (\gamma_{1} + z^{M})M - r(z^{M}) $ . When Firm 1 has the initial quality advantage, the technological tie has the beneficial result of selecting the efficient equilibrium of the product improvement game.

### 6.2 Losing firm exits the market

Our analysis assumed that a firm can be a competitive threat even if it does not win the market. By “waiting in the wings” a lower-quality firm can discipline the price charged by a higher-quality firm, provided that it is not otherwise foreclosed from competing. An alternative assumption is that a firm exits the market if it cannot make sales. $ ^{23} $ This implies that Firm 2 exits the market and is not a competitive constraint if Firm 1 invests and Firm 2 does not, or if it is foreclosed by a high component price. Similarly, Firm 1 does not constrain the prices charged by Firm 2 if Firm 2 invests and wins the market. Table 3 shows the payoff matrix for the product
improvement game for investments $ z^{M} $ or 0 under this alternative exit assumption when $ w > \gamma_{2} $ . In this case the efficient equilibrium in which Firm 2 invests $ z^{M} $ and Firm 1 invests 0 is the unique equilibrium of the product improvement game. Firm 2 sets a price equal to $ P_{2} = \gamma_{2} + z^{M} $ , sells systems to all M customers, and earns $ \pi_{2} = (P_{2} - w)M - r(z^{M}) \geq 0 $ . Firm 1 sets price $ P_{1} = w $ , sells M units of component A and no systems, and earns $ \pi_{1} = wM $ .

Table 3. Firm payoffs when $ w > \gamma_{2} $ and the losing firm exits

$ z_{2}=z^{M} $	$ z_{2}=0 $
$ z_{1}=z^{M} $	$ wM-r(z^{M});(\gamma_{2}+z^{M}-w)M-r(z^{M}) $	$ (\gamma_{1}+z^{M})M-r(z^{M});0 $
$ z_{1}=0 $	$ wM;(\gamma_{2}+z^{M}-w)M-r(z^{M}) $	$ \gamma_{1}M;0 $

If $ w \leq \gamma_{2} $ , Table 3 would be unchanged except for the cell corresponding to no investment by either firm. Payoffs in this case would be $ [wM; (\gamma_{2} - w)M] $ , because Firm 2 would not be foreclosed by a high component price when it does not invest. Thus, if the losing firm exits the systems market, the efficient equilibrium in which Firm 2 invests $ z^{M} $ and Firm 1 does not invest is the unique equilibrium of the product improvement game for all $ w < \bar{w} $ . $ ^{24} $

The assumption that the losing firm exits the market narrows the range of component prices for which Firm 1 would impose a technological tie. One reason for tying in the technological tying game is to eliminate Firm 2 as a competitive constraint when Firm 1 invests and $ w < \gamma_{2} $ . The tie is unnecessary if Firm 2 were to exit the market. Thus the alternative exit assumption eliminates the equilibrium described in case (i) of Proposition 2. This leaves two cases of equilibria of the technological tying game. If $ w \leq \gamma_{1} + z^{M} $ , there is an equilibrium in which Firm 1 invests $ z^{M} $ and Firm 2 does not invest. Firm 1 does not impose a technological tie, sets $ P_{1} = \gamma_{1} + z^{M} $ , sells systems to the entire market, and earns $ \pi_{1} = (\gamma_{1} + z^{M})M - r(z^{M}) $ . Firm 2 sets $ P_{2} = w $ and earns $ \pi_{2} = 0 $ . In this equilibrium, the mere threat of a technological tie is sufficient to deter investment by Firm 2. As in the previous model, no actual anticompetitive behavior is observed in this equilibrium, even though the outcome is inefficient and the product improvement game without tying has a unique equilibrium in which the more efficient Firm 2 invests. If $ w \geq \gamma_{1} + z^{M} - r(z^{M}) / M $ , there is a second equilibrium in which Firm 2 invests $ z^{M} $ , and Firm 1 does not invest and does not impose a technological tie. Firm 2 sets $ P_{2} = \gamma_{2} + z^{M} $ , sells systems to the entire market, and earns $ \pi_{2} = (P_{2} - w)M - r(z^{M}) $ . Firm 1 sets $ P_{1} = w $ and earns $ \pi_{1} = wM $ . Either equilibrium outcome could occur if $ \gamma_{1} + z^{M} - r(z^{M}) / M \leq w \leq \min(\gamma_{1} + z^{M}, \bar{w}) $ .
systems at price $ P_{1} = \gamma_{1} + z^{M} $ and earns $ \pi_{1} = (\gamma_{1} + z^{M})M - r(z^{M}) $ ; Firm 2 sets $ P_{2} = w $ and earns zero. Firm 1 also prefers the outcome in which it invests when $ \gamma_{2} \leq w < \gamma_{1} + z^{M} - r(z^{M})/M $ . $ ^{25} $ In this case there are, however, two possible equilibria, only one of which involves a tie. In both equilibria, Firm 1 invests $ z^{M} $ and Firm 2 does not invest. In one equilibrium, Firm 1 imposes a tie in the first stage. Given the tie, Firm 2 has no reason to invest. If Firm 1 were to deviate and not impose the tie, then Firm 2 would invest and win the market in the continuation game. In the other equilibrium, Firm 1 does not impose a tie and Firm 2 does not invest. Given that Firm 2 does not invest and is foreclosed by a high component price, Firm 1 has no incentive to tie in the first stage of the game. $ ^{26} $

Firm 1 would not impose a tie in the first stage if $ w \geq \gamma_{1} + z^{M} - r(z^{M}) / M $ . In this case the possible equilibrium outcomes are the same as in the product improvement game without tying. There are two equilibria, corresponding to investment by Firm 2 or by Firm 1. If Firm 2 invests, its equilibrium price is $ P_{2} = \gamma_{2} + z^{M} $ , it sells systems to the entire market, and it earns $ \pi_{2} = (P_{2} - w) M - r(z^{M}) $ ; Firm 1 earns $ \pi_{1} = w M $ . If Firm 1 invests, its equilibrium price is $ P_{1} = \gamma_{1} + z^{M} $ , it sells systems to the entire market, earns it $ \pi_{1} = (\gamma_{1} + z^{M}) M - r(z^{M}) $ ; Firm 2 sets $ P_{2} = w $ and earns zero.

If Firm 1 is the more efficient firm and could impose a tie in the first stage of the game, it would do so if $ w < \gamma_{2} $ or if $ r(z^{M}) \geq (\gamma_{1} - \gamma_{2})M $ . In the first case the tie eliminates costly potential competition from Firm 2 when Firm 1 invests. In the second case the tie eliminates the possibility of an inefficient equilibrium in which Firm 2 invests. A tie would be unnecessary if the more efficient Firm 1 could impose a tie in the first stage of the game and the less efficient firm would exit the market. In this case the efficient equilibrium in which Firm 1 invests and earns the maximum profit is the unique equilibrium of the product improvement game, so the tie is of no benefit.

### 6.4 Network effects

Another extension to the model is to allow for network effects. For example, suppose that the perceived quality of the $ j^{th} $ system is $ q_{j} = \gamma_{j} + z_{j} + v(y_{j}) $ , where $ y_{j} $ is the network of consumers that choose technology j and $ v(y_{j}) $ is a positive network externality with $ v(0) = 0 $ and $ v'(y) \geq 0 $ . If the networks are fully compatible, and if all consumers purchase a system, then $ v(y_{j}) = v(M) $ for all $ j = 1, \ldots, n $ . With full compatibility, the network externality is common to all systems and raises each system's quality by the same level. As a consequence, the relative payoffs from product improvement are unchanged, assuming that unsuccessful firms remain in
the market as potential competitors. Under the assumption of full compatibility, network effects do not change the predictions of the basic product improvement and technological tying games.

If networks are incompatible, multiple equilibria may arise because the values that consumers place on each system depend on the number of consumers who are expected to adopt the system. Thus, in the absence of investments in quality improvement, a less efficient Firm 1 may win the market simply because consumers expect it to win and $ \gamma_{1} + v(M) > \gamma_{2} $ .²⁷ Moreover, consumer expectations and product improvement incentives are mutually reinforcing. Firm 1 may become the system of choice because each consumer expects other consumers to purchase that system, which in turn creates an incentive for Firm 1 to improve its product, and which in turn deters product improvement by Firm 2.

## 7 Conclusions

We have examined the causes and consequences of technological tying in a winner-take-all market for systems. In our basic model a vertically integrated upstream monopolist supplies an essential component to a more efficient independent competitor in the downstream systems market. The two firms compete on price and quality for sales to consumers with homogeneous preferences over these vertically differentiated products. If the wholesale price of the essential component is insufficiently remunerative, then the upstream monopolist has an incentive to foreclose rival systems, either by selling only systems, contractually tying components, or designing an essential component so that it works better with its own systems (technological tying). A technological tying strategy has the advantage of facilitating price discrimination for alternative uses of the essential component. However, a technological tie, or even in some cases the mere threat of a tie, may reduce social welfare by distorting market structure.

The ambiguity regarding the welfare effects of technological tying concerns equilibrium selection when technological tying is infeasible for the vertically-integrated firm, e.g., due to regulation. When firm heterogeneity is not too great, there is an equilibrium of the product improvement game in which the less capable vertically integrated firm improves its quality and wins the market for systems. If this inefficient equilibrium is focal, then technological tying is irrelevant for social welfare. If instead an efficient equilibrium is focal, in which a more capable competitor wins the market, then prohibiting technological tying can increase social welfare by preserving the efficient market structure.
If the wholesale price of the essential component is sufficiently near the monopoly price, then the vertically-integrated upstream monopolist and the more efficient independent downstream firm both prefer the equilibrium in which the upstream monopolist supplies the component and the independent firm wins the downstream market. Nonetheless, if the initial quality difference is not too great, then an equilibrium exists in which the vertically-integrated monopolist forecloses its more efficient rival with a technological tie. This unfortunate coordination failure would be prevented by a ban on technological tying. If, however, the owner of the essential component were the more efficient supplier of systems, then technological tying could increase welfare by eliminating an equilibrium in which a less efficient independent system supplier invests and wins the market. Hence we cannot conclude that a prohibition against technological tying would generally enhance welfare.

The simple vertical differentiation model does not admit equilibria in which both firms invest in product improvement. In a richer model that allows for both vertically and horizontally differentiated systems, such that some consumers prefer the system sold by Firm 1 and others prefer Firm 2's system even when each has the same (vertical) quality and is sold at the same price, both firms may have an equilibrium incentive to improve their products, and the ability of Firm 1 to technologically tie might reduce Firm 2's market share short of complete foreclosure. The welfare effects of technological tying are more subtle in this case.
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