Stochastic optimization in linear economic models

Abstract

Linear economic models containing variablesx,y and parametersθ, that may be subject to a stochastic generating mechanism pose two basic problems of optimization. One is the problem of estimation of parametersθ from observations onx andy. This is dealt with in econometrics. The second is the problem of optimal decision-making under uncertainty under the stochastic environment. Two most basic questions in this theory are: how to select an optimal decision and how to update the solution if the information structure is changing? For instance, the model may be specified by a set of linear equations

$$f\left( {x,y,\theta } \right) = 0$$

((1.1))

in state variablesy control variablesx and the stochastic componentsθ entering linearly. Then one type of selection problem is: how to choose the control vectorx, if the objective is to reach a target levely ⁰? An alternative form of the objective is to minimize the expected valueEL(y−y ⁰) of the loss function L(y−y ⁰) defined as a scalar convex function of the deviations ofy fromy ⁰ Some conventional loss functions used in economics and other applied decision models are quadratic as a sum ∑∈, ²₁ of squared errors ∈_i =y _i−y ⁰_i or linear as a sum ∑|∈_i| of absolute deviations. These functions are symmetric and to that extent, asymmetry of information structure, if present in the stochastic generating mechanism may not be well captured here.