We propose a scalable method for computing global solutions of nonlinear, high-dimensional dynamic stochastic economic models. First, within a time iteration framework, we approximate economic policy functions using an adaptive, high-dimensional model representation scheme, combined with adaptive sparse grids to address the ubiquitous challenge of the curse of dimensionality. Moreover, the adaptivity within the individual component functions increases sparsity since grid points are added only where they are most needed, that is, in regions with steep gradients or at nondifferentiabilities. Second, we introduce a performant vectorization scheme for the interpolation compute kernel. Third, the algorithm is hybrid parallelized, leveraging both distributed- and shared-memory architectures. We observe significant speedups over the state-of-the-art techniques, and almost ideal strong scaling up to at least $1,000$ compute nodes of a Cray XC$50$ system at the Swiss National Supercomputing Center. Finally, to demonstrate our method's broad applicability, we compute global solutions to two variates of a high-dimensional international real business cycle model up to $300$ continuous state variables. In addition, we highlight a complementary advantage of the framework, which allows for a priori analysis of the model complexity.