Proposition 2 In the technological tying game:

(i) If and only if $ w < \gamma_{2} $ , there exists an equilibrium in which Firm 1 invests $ z^{M} $ , Firm 2 invests 0, and Firm 1 forecloses Firm 2 with a technological tie. In this equilibrium, 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 $ .

(ii) If $ \gamma_{2} \leq w < \gamma_{1} + z^{M} $ , there exists an equilibrium in which Firm 1 invests $ z^{M} $ and Firm 2 invests 0. 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} = P_{1}M - r(z^{M}) $ . Firm 2 sets $ P_{2} = w $ and earns $ \pi_{2} = 0 $ .

(iii) If $ w \geq \gamma_{1} + z^{M} - r(z^{M}) / M $ , there exists an equilibrium in which Firm 2 invests $ z^{M} $ and Firm 1 invests 0. Firm 1 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} = w M $ .

(iv) There are no other equilibrium outcomes.

We sketch the proof under the assumption that firms are restricted to invest $ z^{M} $ or 0. The Appendix shows that there is no loss of generality in this restriction because some other investment never is more profitable than the best response from the restricted choice set. When $ w < \gamma_{2} $ , Firm 1 prefers to invest $ z^{M} $ and imposes a tie. The tie eliminates Firm 2 as a potential competitor and allows Firm 1 to set $ P_{1} = \gamma_{1} + z^{M} $ and earn its stand-alone value $ \pi_{1} = (\gamma_{1} + z^{M})M - r(z^{M}) $ . Firm 2 cannot leapfrog Firm 1's quality because it is foreclosed by the tie, and given that Firm 2 does not invest, Firm 1 is better off with the tie. If Firm 1 chose instead to sell the component to Firm 2, it would earn wM, which by Assumption A3 is less than $ (\gamma_{1} + z^{M})M - r(z^{M}) $ when $ w < \gamma_{2} $ . Investing and foreclosing Firm 2 with a technological tie is therefore a dominant strategy for Firm 1 if $ w < \gamma_{2} $ .

Assumption A3 implies that $ \gamma_{2} < \gamma_{1} + z^{M} $ . For $ \gamma_{2} \leq w < \gamma_{1} + z^{M} $ , there is an equilibrium in which Firm 1 invests $ z^{M} $ , Firm 2 does not invest, and Firm 1 does not impose a technological tie. An actual tie is unnecessary for these parameter values because the cost of component A forecloses competition from Firm 2 when it does not invest, and Firm 1 credibly threatens to impose a tie if Firm 2 were to invest sufficiently to become a competitive constraint absent the tie. Firm 1's credible threat to impose a tie in the second stage when $ w < \gamma_{1} + z^{M} $ is sufficient to deter investment by Firm 2. Expecting Firm 2 to invest 0, Firm 1 invests $ z^{M} $ and provides a superior system to Firm 2's unimproved system. In this case, no actual anticompetitive behavior is ever observed. Yet market structure is distorted, even when efficient investment by Firm 2 is the unique equilibrium of the product improvement game, i.e., when $ (\gamma_{2} - \gamma_{1})M > r(z^{M}) $ . The more efficient supplier of systems is effectively foreclosed by the ability of the vertically integrated firm to impose a technological tie and its incentive to do so off the equilibrium path.

For $ w \geq \gamma_{1} + z^{M} - r(z^{M}) / M $ , an equilibrium exists in which Firm 2 invests $ z^{M} $ and Firm 1 neither invests nor imposes a technological tie. In the second stage,