Production bottlenecks are central to the performance of manufacturing and service systems, yet their economic analysis typically assumes exogenous capacity. This paper studies a single-period, multi-product production system with stochastic demand. In this system, products compete for a shared bottleneck whose capacity is determined endogenously through costly investment. Extending the classical bottleneck framework, we incorporate inventory holding costs and salvage values, allowing inventory risk to directly influence both production and capacity decisions. We characterize the optimal solution and show that bottleneck capacity behaves as an endogenous economic resource whose shadow price equals the marginal cost of capacity investment. The resulting optimality conditions yield a capacity-adjusted newsvendor rule that links inventory overage risk and bottleneck usage to production quantities. Comparative statics demonstrate that higher holding costs discourage capacity expansion, while greater salvage opportunities justify larger investments. A numerical example illustrates the analytical results and highlights the importance of coordinating production planning, inventory management, and capacity investment under demand uncertainty.