Research Question Germany has high and persistent current account surpluses since the start of the millennium. To many, (projected) population ageing in Germany is one of the most important drivers for these developments. In an ageing society, people want to prepare for increased longevity by augmenting savings. As these savings cannot all be invested domestically, net foreign assets and the current account surplus rise. Contribution We analyze the effects of population ageing in a three-region New Keynesian life-cycle model. Two of the regions form a monetary union. The model is calibrated to Germany, the rest of the Eurozone and the remaining OECD countries. In an additional analysis, we add China to the latter region. Results We find that population ageing in Germany is indeed a significant driver of net foreign asset developments. It has the potential of generating current account surpluses of up to 15% of GDP. However, this only happens when the demographic trends of the ageing region (Germany) and its trading partners are unsynchronized. Put differently, this happens when Germany ages while the demographic structure in the rest of the world remains constant. Assuming Germany's most important trading partners to be the other Euro area member states and the remaining OECD countries (and potentially China), we see that these regions face a similar, but delayed ageing process. Feeding these developments into our model reduces the average current account-to-GDP ratio from 2000 to 2018 in Germany to 2.8% (1.2% when taking into account China). It turns negative around 2035.

Many quantitative economic models have to be solved with numerical methods. This is also true for many household models, e.g., standard models of consumption and savings. With an increasing number of variables that are of relevance for a household's decision (=state variables), this may become very costly (in terms of computational time). One reason for this complexity is that household decisions are characterized as functions of state variables which have to be approximated on grids. The solution of the economic decision problem has to be computed numerically for each gridpoint. State variables are, e.g., financial assets and nonfinancial assets such as housing, educational background or age of the household.The numerical solution of such a prototypical model is basically characterized by two computationally demanding numerical operations: the solution of a non-linear system of equations and interpolation of functions. Typically, the solution of the non-linear system of equations does not have a closed form. Recently, a method that has received considerable attention in the literature is the method of endogenous gridpoints (ENDGM). It is mainly applied to one-dimensional problems. In contrast to a standard method with exogenous grids (EXOGM), a smart redefinition of state variables in ENDGM may make it possible to solve the non-linear system of equations analytically. This greatly enhances speed of computations. However, we highlight that there exists a trade-off in higher dimensions: while the solution of the non-linear system of equations in ENDGM may still have a closed form, interpolation becomes much more complex. It is not clear how this trade-off is resolved vis-a-vis a standard EXOGM method.Against this background, we compare three numerical methods: The standard exogenous grid method (EXOGM), the method of endogenous gridpoints (ENDGM), and a hybrid method (HYBGM) as a combination of the former two. We do this comparison by solving a dynamic model with two continuous state variables (financial assets and health of the household) and occasionally binding borrowing constraints. Evaluation is based on speed and accuracy of the methods. Our conclusion is that HYBGM and ENDGM both dominate EXOGM. In dynastic models where representative households have long or infinite horizons, ENDGM also always dominates HYBGM. In a finite horizon model, the choice between HYBGM and ENDGM depends on the number of gridpoints in each dimension. For a standard choice of gridpoints, ENDGM is always faster. These insights will be very useful in applications of economic models with more than one continuous state variable. Such models include, amongst others, models with an explicit notion of portfolio choice decisions with respect to financial wealth and housing investments over a household's life-cycle, models in health economics such as our application in this paper or human capital models. Enhancing the speed of computations in such models will simplify their use for policy analysis. Endogenous Grids in Higher Dimensions

Aluminum gallium nitride (AlGaN) based ultraviolet light emitting diodes (UV LEDs) have a variety of applications in areas of biological sensing/detection, optical data communication, medical treatments, and sterilization. However, it is challenging to increase the efficiency of UV LEDs emitting in the UVB and UVC spectral range. [1] One of the approaches to achieve this goal is to develop AlGaN-based UV LEDs on low threading dislocation density (TDD) AlGaN layers.For UV LEDs, AlGaN layers with an Al mole fraction x Al above 0.5 are required to avoid absorption of the UV light generated in the active region of the LEDs. [2][3][4][5] However, it is challenging to reduce the TDD in AlGaN layers. Without improvements in the growth and fabrication process, relaxed or partially relaxed AlGaN layers with x Al > 0.5 usually exhibit a TDD of more than mid-10 9 cm À2 . [6] This is associated with the lattice mismatch between AlGaN and AlN that becomes larger with increasing Ga content. Moreover, relaxation of the AlGaN when growing the layers above the Matthews-Blakeslee critical thickness can lead to surface roughening [7] and generation of new dislocations. [8] Low dislocation density AlGaN layers with the absence of relaxation can be obtained by pseudomorphic growth of the layers on bulk AlN substrates. Nonetheless, pseudomorphic growth is limited by x Al and layer thickness as reported by. [8] For example, pseudomorphic growth of n-Al 0.6 Ga 0.4 N can be achieved with a thickness of up to 0.5 μm, while for n-Al 0.7 Ga 0.3 N, the thickness is up to 1.0 μm. Dalmau et al. found that a 0.4 μm thick Al 0.65 Ga 0.35 N layer already shows 8% of relaxation. [9] In a recent work, [10] pseudomorphic growth was demonstrated for thicker Al 0.6 Ga 0.4 N layers with thicknesses between 0.95 and 3.5 μm. However, these layers suffered from high compressive stress of 3-4 GPa, which could lead to detrimental relaxation in subsequently grown layers with higher Ga content (x Al < 0.6) and huge wafer bow. The latter results in non-uniform growth of the active region of devices and problems in the LED chip process. In the same report, [10] the authors attempted to relieve the stress in the layers by introducing two types of buffer layers, that is, with continuously graded composition and stepwise changed composition, neither of which was successful. From this result, a large barrier for the nucleation of misfit dislocation segments in Al-rich AlGaN layers was concluded. Hence, AlGaN strain relaxation on AlN is challenging especially when growing on low dislocation density AlN buffer layers.At present, UV-transparent bulk AlN substrates are expensive and their availability is still limited. Therefore, AlGaN layers are widely grown on AlN/sapphire templates. A thick fully strained Al 0.8 Ga 0.2 N layer with a TDD of %5 Â 10 8 cm À2 was obtained by growing the layer on maskless epitaxial lateral overgrowth (ELO) AlN/sapphire template. [6,11] In contrast, the TDD was higher

and the members of the ESCB Working Group on Public Finance for their useful comments and suggestions. We are also grateful to an anonymous referee for very useful comments and suggestions on the ECB working paper. Any remaining errors are ours. The views expressed in this paper are those of the authors and do not necessarily reflect those of the European Central Bank or the ESCB Working Group on Public Finance and its members. This paper has also been published as an ECB Working Paper with number 2450, July 2020.