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The links of international reserves, exchange rates, and monetary policy can be understood through the lens of a modern incarnation of the “impossible trinity” (aka the “trilemma”), based on Mundell and Fleming’s hypothesis that a country may simultaneously choose any two, but not all, of the following three policy goals: monetary independence, exchange rate stability, and financial integration. The original economic trilemma was framed in the 1960s, during the Bretton Woods regime, as a binary choice of two out of the possible three policy goals. However, in the 1990s and 2000s, emerging markets and developing countries found that deeper financial integration comes with growing exposure to financial instability and the increased risk of “sudden stop” of capital inflows and capital flight crises. These crises have been characterized by exchange rate instability triggered by countries’ balance sheet exposure to external hard currency debt—exposures that have propagated banking instabilities and crises. Such events have frequently morphed into deep internal and external debt crises, ending with bailouts of systemic banks and powerful macro players. The resultant domestic debt overhang led to fiscal dominance and a reduction of the scope of monetary policy. With varying lags, these crises induced economic and political changes, in which a growing share of emerging markets and developing countries converged to “in-between” regimes in the trilemma middle range—that is, managed exchange rate flexibility, controlled financial integration, and limited but viable monetary autonomy. Emerging research has validated a modern version of the trilemma: that is, countries face a continuous trilemma trade-off in which a higher trilemma policy goal is “traded off” with a drop in the weighted average of the other two trilemma policy goals. The concerns associated with exposure to financial instability have been addressed by varying configurations of managing public buffers (international reserves, sovereign wealth funds), as well as growing application of macro-prudential measures aimed at inducing systemic players to internalize the impact of their balance sheet exposure on a country’s financial stability. Consequently, the original trilemma has morphed into a quadrilemma, wherein financial stability has been added to the trilemma’s original policy goals. Size does matter, and there is no way for smaller countries to insulate themselves fully from exposure to global cycles and shocks. Yet successful navigation of the open-economy quadrilemma helps in reducing the transmission of external shock to the domestic economy, as well as the costs of domestic shocks. These observations explain the relative resilience of emerging markets—especially in countries with more mature institutions—as they have been buffered by deeper precautionary management of reserves, and greater fiscal and monetary space. We close the discussion noting that the global financial crisis, and the subsequent Eurozone crisis, have shown that no country is immune from exposure to financial instability and from the modern quadrilemma. However, countries with mature institutions, deeper fiscal capabilities, and more fiscal space may substitute the reliance on costly precautionary buffers with bilateral swap lines coordinated among their central banks. While the benefits of such arrangements are clear, they may hinge on the presence and credibility of their fiscal backstop mechanisms, and on curbing the resultant moral hazard. Time will test this credibility, and the degree to which risk-pooling arrangements can be extended to cover the growing share of emerging markets and developing countries.

Solving dynamic economic models that capture salient real-world heterogeneity and nonlinearity requires the approximation of high-dimensional functions. As their dimensionality increases, compute time and storage requirements grow exponentially. Sparse grids alleviate this curse of dimensionality by substantially reducing the number of interpolation nodes, that is, grid points needed to achieve a desired level of accuracy. The construction principle of sparse grids is to extend univariate interpolation formulae to the multivariate case by choosing linear combinations of tensor products in a way that reduces the number of grid points by orders of magnitude relative to a full tensor-product grid and doing so without substantially increasing interpolation errors. The most popular versions of sparse grids used in economics are (dimension-adaptive) Smolyak sparse grids that use global polynomial basis functions, and (spatially adaptive) sparse grids with local basis functions. The former can economize on the number of interpolation nodes for sufficiently smooth functions, while the latter can also handle non-smooth functions with locally distinct behavior such as kinks. In economics, sparse grids are particularly useful for interpolating the policy and value functions of dynamic models with state spaces between two and several dozen dimensions, depending on the application. In discrete-time models, sparse grid interpolation can be embedded in standard time iteration or value function iteration algorithms. In continuous-time models, sparse grids can be embedded in finite-difference methods for solving partial differential equations like Hamilton-Jacobi-Bellman equations. In both cases, local adaptivity, as well as spatial adaptivity, can add a second layer of sparsity to the fundamental sparse-grid construction. Beyond these salient use-cases in economics, sparse grids can also accelerate other computational tasks that arise in high-dimensional settings, including regression, classification, density estimation, quadrature, and uncertainty quantification.