Defend MSE over MAE for car prices

You’re training a regression model to predict car prices in USD; the target is not scaled. Explain when and why you would choose to minimize MSE instead of MAE. Address: (a) optimization properties—contrast gradients vs subgradients at zero and implications for SGD/Adam; (b) convexity—state whether each loss is convex and identify any incorrect claim that 'MAE is non-convex'; (c) sensitivity to outliers and bias—when a greater penalty on large errors is desirable; (d) probabilistic assumptions—derive the noise model under which MSE (vs MAE) is the MLE; (e) business fit—give a concrete example where squaring dollar errors better matches cost (e.g., luxury models), and one where it does not; (f) effect of not scaling the target—how dollar magnitude interacts with learning rate and regularization.