*In representation learning, we massively use the Kullback-Leibler (KL) divergence in our model's loss, and while the KL divergence between two Gaussian distributions admits a closed-form expression, this is not the case for mixtures of Gaussians, for which no analytical solution is available. Current state-of-the-art solutions rely on Monte Carlo estimations, which is computationally expensive, specially in machine learning when it must be recomputed at each loss evaluation during training. In this work, we propose a more efficient method to estimate the KL divergence between Gaussian mixtures, aiming to reduce computational cost while maintaining accuracy. We expose analytically and evaluate on a set of toy experiments with controlled distributions.

**Credit supply shocks can have heterogeneous effects across sectors. In this paper, I focus on commercial real estate (CRE) investors. I first establish empirically that these investors are particularly reliant on credit, as their assets are more easily pledegeable compared to that of other non-financial firms. I then show in a structural general equilibrium model that this feature creates sectoral misallocation in face of credit supply shocks. CRE investors benefit more from credit expansion than other non-financial firms, feeding growth in the expansion phase of a credit cycle. However, their overaccumulation of capital limits the effect of credit expansion on aggregate productivity. In addition, CRE investors are more likely to face a binding capital irreversibility constraint in the bursting phase of the cycle, amplifying the recession. Macroprudential policy, through redistributive taxes on credit, can correct this sectoral misallocation.