Wealth in the Quadratic Loss Function of the Ramsey Malinvaud Cass Koopmans Model of Optimal Savings

Using the second order Taylor expansion of the Lagrangian of the Ramsey model of optimal savings, wealth is included in the quadratic loss function, and not only consumption. Its weight is given by the degree of concavity of the decreasing returns to scale production function times the marginal utility of consumption, whereas the weight of consumption is given by the degree of concavity of the utility function. This quadratic loss function implies that the speed of convergence is explicitly driven by the trade-off between wealth smoothing (fostering convergence, related to technology) versus consumption smoothing (delaying convergence, related to preferences). By contrast, the second order Taylor expansion of the utility instead of the Lagrangian leads to a quadratic loss function with a weight of wealth equal to zero, which is false for a decreasing returns to scale production function.