Optimality of a Linear Decision Rule in Discrete Time AK Model

Surprisingly, formal proof on the optimality of a linear decision rule in the discrete time AK model with a CRRA utility function has not been established in the growth literature while that in the continuous time counterpart is well-established. This note fills such a gap: I provide a formal proof that consumption being linearly related to investment is a sufficient and necessary condition for Pareto optimality in the discrete time AK model.

Optimal Order Strategy in Uncertain Demands with Free Shipping Option

Free shipping with conditions has become one of the most effective marketing tools; more and more companies especially e-business companies prefer to offer free shipping to buyers whenever their orders exceed the minimum quantity specified by them. But in practice, the demands of buyers are uncertain, which are affected by weather, season, and many other factors. Firstly, we model the centralization ordering problem of retailers who face stochastic demands when suppliers offer free shipping, in which limited distributional information such as known mean, support, and some deviation measures of the random data is needed only. Then, based on the linear decision rule mainly for stochastic programming, we analyze the optimal order strategies of retailers and discuss the approximate solution. Further, we present the core allocation between all retailers via dual and cooperative game theory. The existence of core shows that each retailer is pleased to cooperate with others in the centralization problem. Finally, a numerical example is implemented to discuss how uncertain data and parameters affect the optimal solution.