Defend MSE over MAE for car prices

You are building a regression model that predicts car prices in U.S. dollars, and the target variable is left in its original units. Describe the situations and reasons for preferring mean squared error over mean absolute error as the objective. Cover: (a) optimization behavior—compare gradients with subgradients at zero and what that means for SGD/Adam; (b) convexity—state whether each loss is convex and flag the mistaken claim that 'MAE is non-convex'; (c) outlier sensitivity and bias—when it is useful to impose a larger penalty on big errors; (d) probabilistic assumptions—derive the noise distribution for which MSE, rather than MAE, is the maximum likelihood estimate; (e) business alignment—provide one concrete case where squared dollar errors align better with actual cost (such as luxury models), and one where they do not; (f) impact of leaving the target unscaled—how the dollar scale affects learning rate and regularization.

Overview: This question tests a data scientist's understanding of choosing regression loss functions—MSE versus MAE in particular—including optimization behavior, convexity, sensitivity to outliers, probabilistic noise assumptions, and how well the loss matches business cost when the target is unscaled car prices in USD.